Connecting Ocean Circulation withOcean Heat Storage: Experiments withan Idealised Aquaplanet Modela thesis submitted in partial fulfilmentof the requirements for the degree ofDoctor of Philosophyat Imperial College LondonbyPeter ShatwellSupervised by:Dr Arnaud Czaja (Imperial College London, Physics)Dr David Ferreira (University of Reading, Meteorology)Prof. Greg Pavliotis (Imperial College London, Mathematics)Space and Atmospheric PhysicsDepartment of PhysicsImperial College London180 Queen’s Gate, London, SW7 2AZAugust 2022I certify that this thesis, and the research to which it refers, are the product ofmy own work, and that any ideas or quotations from the work of other people,published or otherwise, are fully acknowledged in accordance with the standardreferencing practices of the discipline.Signed:iPeter ShahidiiCopyrightThe copyright of this thesis rests with the author and is made available under a Cre-ative Commons Attribution Non-Commercial No Derivatives licence. Researchersare free to copy, distribute or transmit the thesis on the condition that they at-tribute it, that they do not use it for commercial purposes and that they do notalter, transform or build upon it. For any reuse or redistribution, researchers mustmake clear to others the licence terms of this work.iiiivAbstractGlobal warming is really ‘ocean warming’ as ⇠93% of the excess heat energy enteringthe climate system is taken up by the oceans. The heat storage distribution ishighly nonuniform in space, affected by a multitude of ocean heat uptake (OHU)processes across many spatial scales. These processes contribute to a substantialspread in climate model projections of future heat storage, sea-level rise, and surfacewarming. A better understanding of OHU processes is essential in order to reduceour uncertainty in climate projections.I probe OHU processes using a coupled climate model with idealised geometry(‘Double-Drake’) under an abrupt CO2-doubling. By projecting the Double-Drakecirculation onto depth-temperature coordinates, I find that the ocean circulation re-sponse leads to an increase in the downward vertical heat transport of ⇠0.2 W m2,enabling more interior heat storage. This supports the idea that changes in oceancirculation may further reinforce the ocean’s role as a buffer of surface warming.I examine in more detail the circulation response of an analogous Atlantic merid-ional overturning circulation (AMOC) in Double-Drake. Many models project aweakening of the AMOC in the future, where the implications for ocean warmingare highly uncertain. I find that the link between the AMOC and heat storagerate is counterintuitive in Double-Drake, and that AMOC weakening may not be asimportant as traditionally thought.Finally, I explore the representation of OHU in two-layer energy-balance models(EBMs), and find that increasing/decreasing the sensitivity of OHU to the strati-fication in the EBM improves/worsens the representation for Double-Drake and asuite of other coupled climate models. This suggests that improving OHU represen-tation might be achieved via a simple change to the two-layer EBM.The results reaffirm that an understanding of OHU is incomplete without a graspof the underlying ocean circulation processes.vvi“Our mission is to teach the world what man ne’er knew before; all truth, by scienceis unfurl’d, which nature has in store.” — Crew song from the HMS Challengerexpedition, 1872–1876“When a problem comes along, you must whip it.” — Devo, 1980viiviiiList of AbbreviationsAbbreviationAABW Antarctic Bottom WaterACC Antarctic Circumpolar CurrentAMOC Atlantic Meridional Overturning CirculationAOGCM Atmosphere-Ocean General Circulation ModelCMIP5 Coupled Model Intercomparison Project phase 5DWBC Deep Western Boundary CurrentDWF Deep Water FormationEBM Energy Balance ModelECCO Estimating the Circulation and Climate of the OceanECS Equilibrium Climate SensitivityEEI Earth’s Energy ImbalanceGCM General Circulation ModelIPCC Intergovernmental Panel on Climate ChangeLB Large Basin (DDrake region)MHT Meridional Heat TransportMOC Meridional Overturning CirculationNADW North Atlantic Deep WaterNOAA National Oceanic and Atmospheric AdministrationOHC Ocean Heat ContentOHS Ocean Heat StorageOHT Ocean Heat TransportOHU Ocean Heat UptakeOHUE Ocean Heat Uptake EfficiencyOSNAP Overturning in the Subpolar North Atlantic ProgramRAPID The ‘Rapid’ climate change monitoring array at 26.5NRMSE Root Mean-Square ErrorSAMBA South Atlantic MOC Basin-wide ArraySAT Surface Air TemperatureSB Small Basin (DDrake region)SO Southern Ocean (DDrake region)SST Sea Surface TemperatureTCR Transient Climate ResponseTOA Top Of the AtmosphereVHT Vertical Heat TransportWBC Western Boundary CurrentWMT Water Mass TransformationixModelACCESS-OM2 Australian Community Climate and Earth System Sim-ulator Ocean Model version 2CANESM2 Canadian Earth System Model version 2CCSM4 Community Climate System Model version 4CNRM-CM5 Centre National de Recherches Météorologiques Cou-pled Model version 5GFDL-ESM2M Geophysical Fluid Dynamics Laboratory Earth SystemModel version ‘2M’HadCM3 Hadley Centre Coupled Model version 3MIROC-ESM Model for Interdisciplinary Research On Climate EarthSystem ModelMITgcm Massachusetts Institute of Technology general circula-tion modelMPI-ESM-LR Max Planck Institute for meteorology Earth SystemModel (Low Resolution)MRI-CGCM3 Meteorological Research Institute Coupled Global Cli-mate Model version 3NASA-GISS-E2-H National Aeronautics and Space Administration God-dard Institute for Space Studies (United States) modelversion ‘E2-H’NEMO Nucleus for European Modelling of the OceanNorESM-1M Norwegian Earth System Model version ‘1M’UVic ESM University of Victoria Earth System ModelxList of Figures1.1 The Spilhaus projection or world ocean map in a square, renderedin ArcGIS Pro 2.5 (reproduced from https://arcg.is/1vTXHn, byBojan Šavrič, David Burrows, and Melita Kennedy) . . . . . . . . . . 21.2 Estimates of zonal-mean meridional ocean heat transports over 1984-2006 based on the COREv2 climatological surface air-sea flux dataset. Green shading indicates the interannual variation. The trian-gles and squares are observational heat transport estimates derivedfrom the basin-wide hydrographic sections compiled by [Bryden andImawaki, 2001]. Figure reproduced from [Large and Yeager, 2009]. . . 31.3 Temperature reconstructions from the Greenland ice core GISP2 (blue,[Grootes et al., 1993]), and ocean sediment proxy data from the sub-tropical Atlantic (green, [Sachs and Lehman, 1999]). D/O warmevents are numbered, and Heinrich events (corresponding to large icesheet discharges into the North Atlantic) are marked as red squares.The grey vertical lines are spaced 1470 years apart, illustrating thetendency for D/O events to occur with this spacing. Figure repro-duced from [Rahmstorf, 2002]. . . . . . . . . . . . . . . . . . . . . . . 51.4 Schematic of the main surface ocean currents, highlighting the dom-inance of the subtropical gyres. The zonally-averaged zonal sur-face winds are shown in the left-hand panel. Figure taken from[Vallis, 2012], kindly made available on his personal website: http://empslocal.ex.ac.uk/people/staff/gv219/. . . . . . . . . . . . . 71.5 Schematic showing the ‘Upper’ and ‘Lower’ cells of the MOC, repro-duced from [Marshall and Speer, 2012]. Arrows indicate the directionof the circulation, with colours representing the average density ofwater masses from lightest (red-yellow) to densest (green-blue). Sur-face buoyancy fluxes and wind stress directions are also indicated.In the background is the zonally-averaged distribution of dissolvedoxygen, and the thin black line roughly indicates the depth of theMid-Atlantic Ridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Schematic of the global overturning circulation, reproduced from [Tal-ley, 2013]. Arrows indicate the direction of the circulation and coloursrepresent the average density of water masses from lightest (purple-red) to densest (green-blue). A small leakage of less than 1 Sv makesits way from the Pacific to the Atlantic through the Bering Strait. . . 9xi1.7 Two-dimensional schematic of the MOC, reproduced from [Ferrariet al., 2017]. Green is NADW, blue is AABW, red is Indian andPacific deep waters (IDW/PDW), and yellow is intermediate wa-ters. The vertical dashed lines indicate diapycnal mixing transform-ing AABW into NADW, or AABW into IDW/PDW. The jagged linerepresents the main bathymetric features in the Indo-Pacific, belowwhich diapycnal mixing is enhanced. . . . . . . . . . . . . . . . . . . 101.8 Schematic of the global annual-mean energy flows through the climatesystem (in W m2). Earth’s energy imbalance, or the net absorbed,is estimated at 0.9± 0.2 W m2. Figure reproduced from [Trenberth,2020], based on [Trenberth et al., 2009] and [Wild et al., 2015]. . . . . 141.9 Annual global-mean surface temperature anomalies relative to 1961-1990 from three different data sets (HadCRUT4, NASA-GISS, andNCDC-MLOST). Reproduced from [Hartmann et al., 2013]. . . . . . 151.10 Observational estimates of changes in ocean heat content since 1950from the IPCC’s fifth assessment report [Rhein et al., 2013]. a)Annual global-mean upper-ocean (0-700 m) heat content adaptedfrom [Levitus et al., 2012, Ishii and Kimoto, 2009, Domingues et al.,2008, Palmer et al., 2007, Smith and Murphy, 2007]. b) Five-yearrunning means of global-mean mid-depth (700-2000 m, data from[Levitus et al., 2012]) and deep (2000-6000 m, data from [Purkey andJohnson, 2010]) ocean heat content changes. . . . . . . . . . . . . . . 161.11 Spatial maps of the annual mean net heat flux divided into com-ponents of (a) Latent Heat, (b) Sensible Heat, (c) Longwave Flux,and (d) Shortwave Flux in W m2. Positive values are directeddownwards. Panels (a) and (b) use the Objectively Analyzed Air-sea Fluxes (OAFlux) dataset [Yu and Weller, 2007], while panels (c)and (d) use the satellite-only International Satellite Cloud Climatol-ogy Project (ISCCP) fields [Zhang et al., 2004]. Figure reproducedfrom [Josey et al., 2013]. . . . . . . . . . . . . . . . . . . . . . . . . . 181.12 Climatological annual-mean net air-sea heat flux from OAFlux [Yuand Weller, 2007] and ISCCP [Zhang et al., 2004], reproduced from[Josey et al., 2013]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.13 Linear trends in 0-700 m OHC anomalies (OHCA) over the period1993-2019 (W m2) from (a) NOAA’s Pacific Marine EnvironmentalLaboratory (PMEL) [Lumpkin et al., 2020], (b) Japan Meteorolog-ical Agency (JMA) [Ishii et al., 2017], (c) Institute of AtmosphericPhysics (IAP) [Cheng et al., 2017], and (d) NOAA’s National Centerfor Environmental Information (NCEI) [Levitus et al., 2012]. Figurereproduced from [Johnson and Lyman, 2020]. . . . . . . . . . . . . . 201.14 Schematic of the simple model developed by [Gnanadesikan, 1999].The effects of Southern Ocean winds and compensating eddies arerepresented by the volume flux Ts, NADW formation is representedby the flux Tn, and diapycnal mixing is represented by a low-latitudeupwelling flux Tu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23xii2.1 Time-averaged sea surface temperature (in C) in the control inte-gration, showing the model DDrake geometry from above (left) andbelow (right). The small basin (SB), large basin (LB), and southernocean (SO) regions are labelled. . . . . . . . . . . . . . . . . . . . . . 352.2 Meridional ocean heat transports (OHT) and freshwater transports(FWT) for (left) the small and large basins of DDrake and (right) theAtlantic and Indo-Pacific basins. Observations for OHT are takenfrom [Trenberth and Caron, 2001], while observations of FWT aretaken from [Wijffels et al., 1992]. Figure reproduced from [Ferreiraet al., 2010]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Zonally-averaged atmospheric potential temperature (a) and zonalwinds (b) for the DDrake control climate. . . . . . . . . . . . . . . . . 422.4 Control climate zonally and sector-averaged potential temperature(top), salinity (middle), and potential density 0 (bottom) distribu-tions for the small basin plus southern ocean (SBSO, left column)and large basin plus southern ocean (LBSO, right column) DDrakeregions. Potential temperature and potential density fields are refer-enced to the surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.5 Spatial maps of the DDrake control climate (a) sea surface temper-ature (SST, in C), (b) sea surface salinity (SSS, in psu), and (c)surface density (0, in kg m3). . . . . . . . . . . . . . . . . . . . . . 432.6 Hydrographic sections of potential temperature (top, in C) and salin-ity (bottom, in psu) for the Atlantic (left) and Pacific (right) basinstaken during the World Ocean Circulation Experiment (WOCE),from [Schlitzer, 2000]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.7 Decadally averaged surface climatological fields over the period 1955-2017 of (a) sea surface temperature (in C), (b) sea surface salinity (inpsu), and (c) surface density (0, in kg m3), from the World OceanAtlas 2018 [Boyer et al., 2018]. . . . . . . . . . . . . . . . . . . . . . . 452.8 Control climate residual overturning stream functions for the (a) smallbasin, (b) large basin, and (c) global regions of DDrake (in Sv, 1 Sv⌘ 106 m3 s1). The southern ocean residual overturning is shownsouth of 35S (thick black line) in panels (a) and (b). . . . . . . . . . 482.9 Zonally-averaged potential density referenced to 2100 dbar (top) andmeridional overturning circulations (bottom) for the Atlantic (left)and Indo-Pacific (right), calculated from the ECCOv4 ocean reanal-ysis [Forget et al., 2015]. Panels a and b are reproduced from [Cessi,2019] while panels c and d are reproduced from [Ferreira et al., 2018a]. 493.1 DDrake control climate vertical heat balance for advective (ADV) anddiffusive (DIFF) processes, and shortwave (SW) radiation penetration(in W m2). A downward heat flux is positive. The zero line is blackdotted. Only the top 1.5 km are shown for clarity. . . . . . . . . . . . 59xiii3.2 Comparison of model-outputted vertical advective heat fluxes (solidlines) with those calculated from the thermal overturning ✓z (dash-dotted lines), for the DDrake control climate. The zero line is blackdotted. Only the top 1.5 km are shown for clarity. . . . . . . . . . . . 613.3 DDrake control climate zonally-averaged ctrlres (colour, in Sv) and ✓distributions (black dashed lines, in C) for the (a) global, (b) south-ern ocean, (c) small basin, and (d) large basin regions. Contour in-terval is 5 Sv for panels a, c, and d. For panel b, the interval is 2 Sv;also, note the different colour-bar scale. Red cells circulate clockwisewhile blue cells circulate anti-clockwise. . . . . . . . . . . . . . . . . . 623.4 Control climate DDrake global thermal overturning circulations (inSv). (a) Resolved circulation ✓z, (b) eddy-induced circulation 0✓z,and (c) net circulation ✓z = ✓z + 0✓z. Red cells circulate clockwisewhile blue cells circulate anti-clockwise. . . . . . . . . . . . . . . . . . 643.5 DDrake global net control thermal overturning circulation ( ✓z, inSv), decomposed into a warm cell (red solid, for ✓ > 15C), a coolcell (blue dashed, for ✓ < 20C), and a tropical cell (green dashed,for ✓ > 20C). Dashed lines denote negative values and circulate anti-clockwise, while solid lines denote positive values and circulate clock-wise. Only the top 2 km are shown. . . . . . . . . . . . . . . . . . . . 653.6 DDrake net control thermal overturning circulations ( ✓z, in Sv) cal-culated for the (a) global, (b) southern ocean, (c) small basin, and(d) large basin regions. The large basin dominates the global ✓zstructure. Red cells circulate clockwise while blue cells circulate anti-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.7 Vertical heat transport profiles for the three thermal overturning cellsof DDrake in the control climate (in W m2). Here, a positive heattransport is directed downwards. The solid black line is the sum ofthe red, blue, and green lines. The dash-dotted black line is the totalglobal advective VHT, accounting for cells not included in the ✓zthree cell decomposition. Only the top 2 km are shown. . . . . . . . . 723.8 DDrake control global net overturning circulation in depth-densitycoordinates, as a function of potential density 0. (a) The net 0zoverturning (in Sv) and (b) its decomposition into a warm cell (reddashed, for 0 < 25.2 kg m3), a cool cell (blue solid, for 0 > 24 kgm3), and a tropical cell (green solid, for 0 < 24 kg m3). In panela, red cells circulate clockwise and blue cells circulate anti-clockwise.In panel b, solid lines denote positive values and circulate clockwise,while dashed lines denote negative values and circulate anti-clockwise. 76xiv3.9 (a) DDrake control climate (solid lines) and perturbed climate (dashedlines) vertical heat balance for advective (ADV) processes split into re-solved and eddy-induced advection (EIA), diffusive (DIFF) processesand shortwave (SW) radiation penetration (in W m−2). (b) Changein vertical heat flux due to the CO2-doubling perturbation (differencebetween the dashed and solid lines in panel a). A downward heat fluxis positive. Only the top 2 km are shown. . . . . . . . . . . . . . . . . 773.10 Response of the DDrake global thermal overturning circulation to anabrupt CO2-doubling (in Sv). Top row: (a) control climate ther-mal overturning Ψctrlθz , (b) perturbed climate thermal overturningΨperturbedθz , (c) their difference ΔΨθz = Ψperturbedθz − Ψctrlθz . Red cellscirculate clockwise while blue cells circulate anticlockwise. Bottomrow: same as the top row but decomposed into the tropical, warm,and cool thermal overturning cells as defined in the text. Here, solidlines circulate clockwise and dashed lines circulate anticlockwise. . . . 793.11 Small basin thermal overturning circulation response to an abruptCO2-doubling. (a) control climate thermal overturning Ψctrlθz , (b)perturbed climate thermal overturning Ψperturbedθz , (c) their differenceΔΨθz = Ψperturbedθz − Ψctrlθz . Red cells circulate clockwise while bluecells circulate anticlockwise. . . . . . . . . . . . . . . . . . . . . . . . 793.12 Large basin thermal overturning circulation response to an abruptCO2-doubling. (a) control climate thermal overturning Ψctrlθz , (b)perturbed climate thermal overturning Ψperturbedθz , (c) their differenceΔΨθz = Ψperturbedθz − Ψctrlθz . Red cells circulate clockwise while bluecells circulate anticlockwise. . . . . . . . . . . . . . . . . . . . . . . . 803.13 Southern ocean thermal overturning circulation response to an abruptCO2-doubling. (a) control climate thermal overturning Ψctrlθz , (b)perturbed climate thermal overturning Ψperturbedθz , (c) their differenceΔΨθz = Ψperturbedθz − Ψctrlθz . Red cells circulate clockwise while bluecells circulate anticlockwise. . . . . . . . . . . . . . . . . . . . . . . . 803.14 Zonally-averaged zonal surface wind stress τx (solid lines) and itsresponse to an abrupt CO2-doubling (dashed lines) (in N m−2). Solidcoloured lines correspond to 50-year time-means throughout the 200year experiment. Dashed lines are equivalent to the solid colouredlines with the control zonal-average subtracted. . . . . . . . . . . . . 813.15 Zonally-averaged Ekman pumping velocity WEk (solid lines) and itsresponse to an abrupt CO2-doubling (dashed lines) (in m yr−1). Solidcoloured lines correspond to 50-year time-means throughout the 200year experiment. Dashed lines are equivalent to the solid colouredlines with the control zonal-average subtracted. . . . . . . . . . . . . 83xvGo to page 142 3.16 Changes in vertical heat transports (VHTs) carried by the tropical,warm, and cool cells of the DDrake thermal overturning circulation.Positive = downward. (a) Control climate (solid lines) and perturbedclimate (dashed lines) VHT profiles. (b) VHT responses due to theCO2-doubling. The black dash-dotted line is the net global advectiveVHT response. The net global response takes into account circu-lations not included in the three-cell decomposition of the thermaloverturning. (c) Comparison of the three-cell sum and net globalVHT responses. Their difference (black dashed) is attributed to thedeep warm cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.17 Changes in the peak (positive or negative) VHT carried by the trop-ical, warm, and cool cells of the DDrake thermal overturning, inresponse to the CO2-doubling. (a) Control and perturbed climatevalues, and their difference (in W m2), (b) percentage changes. . . . 873.18 Changes in the work performed by the tropical, warm, and cool cellsof the DDrake thermal overturning, in response to the CO2-doubling(using equation 3.8 with 0z). (a) Control and perturbed climatevalues, and their difference (in GW), (b) percentage changes. . . . . . 883.19 Changes in the work performed by the global resolved and eddy-induced circulations, and their sum (net circulation), in response toa CO2-doubling (using equation 3.8 with 0z). (a) Control and per-turbed climate values, and their difference (in GW), (b) percentagechanges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.1 Control residual overturning stream functions ctrlres for the (a) smallbasin, and (b) large basin (in Sv, 1 Sv ⌘ 106 m3 s1). The southernocean residual overturning is shown south of 35S (thick black line). . 1004.2 Time series of annually-averaged (a) SB and LB heat storage rates(@t(OHC), in W m2); (b) OHC anomalies (OHC, in J m2);(c) the difference in heat storage rates i.e. the heat storage contrast(@t(OHCSB OHCLB), in W m2); and (d) SBMOC strength(max( pertres ), in Sv) following an abrupt doubling of CO2. Thicklines are decadal running means and horizontal dash-dot lines indicatetime-mean values. The time-mean heat storage contrast is 0.6 ± 0.2W m2 (standard error, assuming 20 degrees of freedom accountingfor decadal oscillations). . . . . . . . . . . . . . . . . . . . . . . . . . 1024.3 (a) Time-series of heat storage responses (in J) for all DDrake basins.(b-d) Comparison of actual versus expected heat storage responses(in J) considering the fractional coverage of the total surface area for(b) the small basin (SB), (c) the large basin (LB), and (d) the south-ern ocean (SO) of DDrake. For each basin B, the ‘expected’ curvesare (B’s surface-area/global surface-area)*(global response) i.e. thesurface-area-weighted fraction of the black curve in (a). From thisperspective, we see that the SB overperforms (while the LB under-performs) with respect to its size. . . . . . . . . . . . . . . . . . . . . 103xvi4.4 Depth-time plots of the anomalous temperature response (in K) inthe (a) small and (b) large basins of DDrake following an abruptCO2-doubling. The contour interval is 0.1 K, and the 0.1, 0.5, 1.0,and 1.5 K contours are highlighted. The upper ocean responses aresimilar, but the differences at depth are substantial. . . . . . . . . . . 1054.5 Vertically-averaged ✓ (in K) after 200 years following an abruptdoubling of atmospheric CO2 in DDrake for the depth intervals (a) 0-1km and (b) 1-3 km. The temperature anomaly at depth follows a deepwestern boundary current in the small basin. (c) Zonally-averaged✓ (colour, in K) in the small basin after 200 years’ warming, andstreamlines (white dashed contours, in Sv) for the control residualoverturning res. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.6 Time series of time-integrated net surface heat flux (OHU) and changesin OHC (OHS), and their difference (OHSOHU) (in J m2) for the(a) small basin, (b) large basin, and (c) southern ocean regions ofDDrake. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.7 (a) Map of time-mean SST warming trends during the first 30 years(i.e. SBMOC weakening regime) of the CO2-doubling experiment(in K/decade), (b) Zonally-averaged time-mean SST warming trendsover the same period. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.8 Time-mean vertically-averaged convective index for (a) the controlclimate, (b) the first 50 years following the CO2-doubling, and (c)their difference. The dark blue lines indicate the meridional barriersenclosing the small basin (they don’t contribute to the convection). . 1094.9 Time series of warming responses for the small basin MOC down-welling (✓#, red) and upwelling (✓", blue) regions, and their differ-ence ((✓), black) (in K). The downwelling region warms at a fasterrate than the upwelling region. . . . . . . . . . . . . . . . . . . . . . . 1104.10 Zonal-mean structures of the SBMOC response. (a) 0-30 years time-mean pertres (colour), and res (white dashed contours) overturning inthe small basin (in Sv) during the weakening regime; (b) 0-30 yearstime-mean res = pertres res overturning response (colour andpink dashed contours) in the small basin (in Sv) during the weakeningregime; (c-d) same as panels (a) and (b) except time-means are takenover the period 30-200 years during the non-weakening regime. Inpanels (a) and (c), the contour interval is 5 Sv and the zero contouris highlighted in black. . . . . . . . . . . . . . . . . . . . . . . . . . . 111xvii4.11 (a) Decomposition of the change in SB downward (positive) MOCheat flux (( res✓), magenta) into differential warming ( res(✓),blue), SBMOC weakening ( res✓, orange), and nonlinear ( res(✓),yellow) terms (annual averages, in W m2). A vertical dotted line isplotted at 30 years to separate the weakening and non-weakeningSBMOC regimes. (b) Decadal running means of the SB heat uptakerate (FSBs , blue), heat storage rate (@t(OHCSB), red), SBMOCheat flux (HMOC , magenta), and leakage rate to the southern ocean(Hleak = @t(OHCSB) FSBs , black) (in W m2). Note thatthe sum of the heat storage rate and SBMOC heat flux (light blue,dashed) almost matches the heat uptake rate. Horizontal dashed linesare centennial time-means. . . . . . . . . . . . . . . . . . . . . . . . . 1124.12 (a) Atlantic vs. Pacific top-3 km column-averaged annual OHC anoma-lies (in J m2) immediately following an abrupt quadrupling of at-mospheric CO2 in CMIP5 models (each dot represents one year).Atlantic and Pacific basins defined from 30S to 65N. The devia-tion from the identity line (black dashed) highlights the Atlantic’senhanced warming rate relative to the Pacific in these experiments.(b) Decadal running means of individual model Atlantic–Pacific heatstorage contrasts @t(OHCAtl OHCPac) (in W m2). . . . . . . . 1145.1 Diagram of the simple two-layer energy balance model. Arrows rep-resent energy fluxes. F is the effective forcing at the ocean surface,↵ is the globally-averaged climate feedback parameter, is the heatexchange parameter, T and T0 are temperature perturbations, and Cand C0 are effective heat capacities. . . . . . . . . . . . . . . . . . . . 1225.2 Time-series of annually-averaged surface air temperature (SAT, or-ange line) and sea surface temperature (SST, blue line); and vertically-averaged upper-layer temperature (T , yellow line) and lower-layertemperature (T0, purple line) corresponding to the derived layer thick-nesses presented in table 5.1 for the (a) global, (b) small basin, (c)large basin, and (d) southern ocean regions of DDrake, relative to thecontrol climate state. The fitted Geoffroy EBM responses for T andT0 are shown as black dashed lines. . . . . . . . . . . . . . . . . . . . 1285.3 Comparison of fitted EBM responses (black dashed) for the GeoffroyEBM (left column) and quadratic EBM (right column). Presentedare time-series of annually-averaged SST responses (magenta), andvertically-averaged upper-layer (red, T ) and lower-layer (blue, T0)temperature responses corresponding to the layer thicknesses h and Himplied by the EBM (see table E.1), for the DDrake CO2-doubling ex-periment. Panels (a-d) correspond to the different basins of DDrake:(a) Global responses, (b) Small basin responses (blue shading), (c)Large basin responses (red shading), and (d) southern ocean responses(yellow shading). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133xviii5.4 Percentage improvement in the RMSE of the fitted DDrake tempera-ture responses after changing from the Geoffroy EBM to the quadraticEBM. The change improves fits for all basins but the southern ocean.A substantial increase (> 50%) is seen for the global lower-layer tem-perature response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.5 Comparison between the annual-average global-mean SST responsefrom CMIP5 model abrupt 4⇥CO2 experiments, and the upper-layerT responses fitted from the two-layer (a) Geoffroy and (b) quadraticEBMs. Points for the abrupt CO2-doubling experiment in DDrakeare also plotted for comparison. . . . . . . . . . . . . . . . . . . . . . 1365.6 Comparison between the inferred upper-layer T responses (top row)and lower-layer T0 responses (bottom row) from CMIP5 models withthe corresponding fitted responses from both the Geoffroy (left col-umn) and quadratic (right column) two-layer EBMs. Points for theabrupt CO2-doubling experiment in DDrake are also plotted for com-parison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.7 Comparison between the RMSEs of the fits found from the Geof-froy and quadratic two-layer EBMs for CMIP5 model temperatureresponses in abrupt 4⇥CO2 experiments. The (a) total, (b) upper-layer, and (c) lower-layer RMSEs are shown. Points correspondingto the abrupt CO2-doubling experiment in DDrake are also shown asblue diamonds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.8 Percentage improvement in the RMSE of the fitted CMIP5 model(and DDrake) temperature responses after changing from the Geof-froy EBM to the quadratic EBM. The change improves fits for thelower-layer response for 4 out of the 5 CMIP5 models analysed. . . . 1395.9 Comparison of fitted EBM responses (black dashed) for the GeoffroyEBM (left column) and RY EBM (right column). Presented are time-series of annually-averaged SST responses (magenta), and vertically-averaged upper-layer (red, T ) and lower-layer (blue, T0) temperatureresponses corresponding to the layer thicknesses h and H implied bythe EBM (see table E.3), for the DDrake CO2-doubling experiment.Panels (a-d) correspond to the different basins of DDrake: (a) Globalresponses, (b) Small basin responses (blue shading), (c) Large basinresponses (green shading), and (d) southern ocean responses (yellowshading). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.10 Percentage improvement in the RMSE of the fitted DDrake temper-ature responses after changing from the Geoffroy EBM to the RYEBM. The change worsens fits overall, mainly due to the especiallypoor fit for the SB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.11 Percentage improvement in the RMSE of the fitted CMIP5 model(and DDrake) temperature responses after changing from the Geof-froy EBM to the RY EBM. The change leads to poorer fits across theboard apart from MRI-CGCM3. . . . . . . . . . . . . . . . . . . . . . 145xix6.1 Census of water masses for the DDrake control climate in temperature-salinity coordinates for (a) global, (b) southern ocean, (c) small basin,and (d) large basin regions (colour, in log10m3/C/psu). In white arecontours of potential density 0 (kg m3). . . . . . . . . . . . . . . . 1566.2 (a) Global water mass census for the DDrake control climate in temperature-salinity coordinates in log10m3/C/psu (equivalent to panel a from fig-ure 6.1). (b) Average water mass transformation rate between the con-trol climate and year 200 of the CO2-doubling run in 0.1 Sv/C/psu.Black lines are contours of potential density 0 (kg m3). . . . . . . . 158D.1 Two-layer EBM reduces to two separate one-layer EBMs in the limit << ↵. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222D.2 Two-layer EBM reduces to a one-layer system with effective heat ca-pacity C + C0 when >> ↵. . . . . . . . . . . . . . . . . . . . . . . . 224xxList of Tables5.1 Upper-layer heat capacity C, lower-layer heat capacity C0, the corre-sponding upper- and lower-layer thicknesses h and H, heat exchangecoefficient , and fitted slow and fast timescales ⌧s and ⌧f for each ofthe DDrake basins, following the fitting procedure in [Geoffroy et al.,2013b]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.2 Calculated radiative parameters for each of the DDrake basins usingthe SAT, following the fitting procedure in [Geoffroy et al., 2013b]:the radiative forcing F , the climate feedback ↵, and the correspondingequilibrium temperature Teq. . . . . . . . . . . . . . . . . . . . . . . . 1265.3 Root mean-square errors (RMSEs) for the EBM fits for the upper- andlower-layer temperature responses T and T0 for each basin of DDrake.The RMSE for the upper-layer EBM fit is also evaluated against theSAT response, shown in the left-most column for comparison. . . . . 1295.4 Root mean-square errors (RMSEs) for the (Geoffroy; quadratic) EBMfits for the upper- and lower-layer temperature responses T and T0for each basin of DDrake. The RMSE for the upper-layer EBM fitevaluated against the SST response is also shown in the left-mostcolumn for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.5 Root mean-square errors (RMSEs) for the (Geoffroy; quadratic) EBMfits for the upper- and lower-layer temperature responses T and T0from CMIP5 abrupt 4⇥CO2 experiments. The RMSE for the upper-layer EBM fit evaluated against the SST response is also shown inthe left-most column for comparison. Equivalent RMSEs for DDrake(same as the ‘Global’ row in table 5.4) are shown in the bottom rowfor reference. Note that the ensemble means and standard deviationsdon’t include the DDrake values. . . . . . . . . . . . . . . . . . . . . 1385.6 Root mean-square errors (RMSEs) for the (Geoffroy; RY) EBM fitsfor the upper- and lower-layer temperature responses T and T0 foreach basin of DDrake. The RMSE for the upper-layer EBM fit evalu-ated against the SST response is also shown in the left-most columnfor comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.7 Root mean-square errors (RMSEs) for the (Geoffroy; RY) EBM fitsfor the upper- and lower-layer temperature responses T and T0 fromCMIP5 abrupt 4⇥CO2 experiments. The RMSE for the upper-layerEBM fit evaluated against the SST response is also shown in the left-most column for comparison. Equivalent RMSEs for DDrake (sameas the ‘Global’ row in table 5.4) are shown in the bottom row forreference. Note that the ensemble means and standard deviationsdon’t include the DDrake values. . . . . . . . . . . . . . . . . . . . . 145xxiE.1 Fitted physical parameters for both the (Geoffroy; quadratic) EBMsusing an interior point algorithm: the radiative forcing F , radiativefeedback ↵, upper-layer heat capacity C, lower-layer heat capacityC0, the corresponding upper-layer and lower-layer thicknesses h andH, and the heat exchange coefficient (or ⇤) for each of the DDrakebasins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226E.2 Fitted physical parameters for both the (Geoffroy; quadratic) EBMsusing an interior point algorithm. The effective radiative forcing Fand climate feedback ↵ values for CMIP5 models are taken from [Kos-tov et al., 2014]. Upper-layer heat capacity C, lower-layer heat ca-pacity C0, the corresponding upper-layer and lower-layer thicknessesh and H, and the heat exchange coefficient (or ⇤) estimates forthe five CMIP5 models used. The equivalent parameters found forDDrake (same as the ‘Global’ row in table E.1) are shown in the bot-tom row for comparison. Ensemble means and standard deviationsdon’t include the values from DDrake. . . . . . . . . . . . . . . . . . 227E.3 Fitted physical parameters for both the (Geoffroy; RY) EBMs usingan interior point algorithm: the radiative forcing F , radiative feed-back ↵, upper-layer heat capacity C, lower-layer heat capacity C0,the corresponding upper-layer and lower-layer thicknesses h and H,and the heat exchange coefficient (or ⇤) for each of the DDrakebasins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228E.4 Fitted physical parameters for both the (Geoffroy; RY) EBMs usingan interior point algorithm. The effective radiative forcing F andclimate feedback ↵ values for CMIP5 models are taken from [Kostovet al., 2014]. Upper-layer heat capacity C, lower-layer heat capacityC0, the corresponding upper-layer and lower-layer thicknesses h andH, and the heat exchange coefficient (or ⇤) estimates for the fiveCMIP5 models used. The equivalent parameters found for DDrake(same as the ‘Global’ row in table E.3) are shown in the bottomrow for comparison. Ensemble means and standard deviations don’tinclude the values from DDrake. . . . . . . . . . . . . . . . . . . . . . 229xxiiContents1 Introduction 11.1 Setting the stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The ocean’s role in climate . . . . . . . . . . . . . . . . . . . . . . . . 21.3 A brief overview of large-scale ocean circulation . . . . . . . . . . . . 61.4 The ocean’s response to warming . . . . . . . . . . . . . . . . . . . . 131.4.1 Ocean Heat Storage Patterns . . . . . . . . . . . . . . . . . . 171.4.2 Ocean Heat Uptake Processes . . . . . . . . . . . . . . . . . . 211.5 A roadmap for this thesis . . . . . . . . . . . . . . . . . . . . . . . . 281.5.1 First results chapter . . . . . . . . . . . . . . . . . . . . . . . 281.5.2 Second results chapter . . . . . . . . . . . . . . . . . . . . . . 281.5.3 Third results chapter . . . . . . . . . . . . . . . . . . . . . . . 292 Experiment Design 312.1 Modelling philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 The MITgcm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 The DDrake configuration . . . . . . . . . . . . . . . . . . . . . . . . 352.3.1 Technical description . . . . . . . . . . . . . . . . . . . . . . . 372.3.2 Control Climatology . . . . . . . . . . . . . . . . . . . . . . . 412.4 Forcing experiment: an abrupt CO2-doubling . . . . . . . . . . . . . . 503 Changes in vertical heat transports by ocean circulation un-der global warming 533.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Stream functions in other coordinates . . . . . . . . . . . . . . . . . . 563.3 Control climate vertical heat balance . . . . . . . . . . . . . . . . . . 583.3.1 The DDrake thermal overturning circulation . . . . . . . . . . 613.3.2 Vertical heat transports . . . . . . . . . . . . . . . . . . . . . 723.3.3 Thermally or mechanically driven? . . . . . . . . . . . . . . . 733.4 Perturbed climate vertical heat balance . . . . . . . . . . . . . . . . . 773.4.1 Changes in ocean circulations and the associated vertical heattransports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.4.2 Changes in energetics . . . . . . . . . . . . . . . . . . . . . . . 873.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.5.2 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . 913.5.3 Key points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944 Ocean heat storage response to an AMOC slowdown 954.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96xxiii4.2 Control climate overturning circulations . . . . . . . . . . . . . . . . 994.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.3.1 Small and Large Basin Heat Storage Rates . . . . . . . . . . . 1014.3.2 Role of the Small Basin MOC . . . . . . . . . . . . . . . . . . 1044.3.3 Role of the southern ocean region . . . . . . . . . . . . . . . . 1074.3.4 Heat storage response to SBMOC Weakening . . . . . . . . . 1084.3.5 Parallels with CMIP5 Models . . . . . . . . . . . . . . . . . . 1134.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 1144.4.1 Key points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175 Representation of OHU in Two-Layer EBMs 1195.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.2 The traditional EBM framework . . . . . . . . . . . . . . . . . . . . . 1235.2.1 Fitting procedure . . . . . . . . . . . . . . . . . . . . . . . . . 1235.2.2 Results with traditional EBM: DDrake fitted responses . . . . 1255.3 A new EBM framework . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.3.1 New fitting procedure . . . . . . . . . . . . . . . . . . . . . . . 1305.3.2 Increasing the sensitivity (⇣ = 2) . . . . . . . . . . . . . . . . 1315.3.3 Decreasing the sensitivity (⇣ = 2/3) . . . . . . . . . . . . . . . 1405.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.4.1 Key points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486 Conclusion 1496.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.2 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 153Bibliography 163Appendix A Eddy parameterisation in MITgcm 207Appendix B Derivation of the change in SBMOC heat flux 211Appendix C Analytical solution of the Geoffroy EBM under astep-forcing 213C.1 Particular solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214C.2 Solution to the homogeneous equation . . . . . . . . . . . . . . . . . 214C.2.1 Eigenvalues of A . . . . . . . . . . . . . . . . . . . . . . . . . 215C.2.2 Eigenvectors of A . . . . . . . . . . . . . . . . . . . . . . . . . 215C.2.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217C.3 General solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Appendix D Non-dimensionalisation of the Geoffroy EBM andLimiting Cases 219D.1 Limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221D.1.1 Radiative feedbacks dominate, /↵ << 1 . . . . . . . . . . . . 221D.1.2 Heat exchange dominates, /↵ >> 1 . . . . . . . . . . . . . . 222xxivAppendix E Fitted parameter values for two-layer EBMs 225Appendix F Vertical buoyancy transport within a wind-drivengyre 231xxvxxviOcean, n. : A body of water occupying abouttwo-thirds of a world made for man — whohas no gills.Ambrose Bierce, 19111Introduction1.1 Setting the stageThe global ocean covers about 71% of the Earth’s surface, and contains more than99% of Earth’s habitable space. There is the oceanographer’s quip that if extrater-restrials came to visit our planet, they would have good reason to call it ‘Water’rather than ‘Earth’. Carl Sagan’s coinage of Voyager 1’s famous photograph wasthe pale blue dot, after all, and not the pale green dot.According to the United Nations (UN) Convention on the Law of the Sea, any watersbeyond 370 km from a nation’s coastline are the common heritage of humankind.The oceans thus act as a bridge connecting all nations, and they should be protectedfor the benefit of future generations. This call has come to great prominence recently,with 2021-2030 being declared the UN Decade of Ocean Science for SustainableDevelopment (https://www.oceandecade.org). The protection of the oceans isalso enshrined in the UN sustainable development goals ‘Climate Action’ and ‘Lifebelow Water’ for their 2030 Agenda (https://sdgs.un.org/goals).Perhaps a visual will help further cement the fundamental importance of the world’soceans. In 1942, the oceanographer Athelstan Spilhaus presented two world ocean1Connecting ocean circulation with ocean heat storagemaps [Spilhaus, 1942], and discussed a cut on the surface of the globe that wouldintersect the oceans as little as possible. He later collaborated with geodesists RobertHanson and Erwin Schmid to create a new conformal map projection in 1983 thatplaced the oceans front and centre, now often dubbed the ‘Spilhaus projection’[Spilhaus, 1983]. This is shown in figure 1.1.Figure 1.1: The Spilhaus projection or world ocean map in a square, rendered in ArcGIS Pro 2.5(reproduced from https://arcg.is/1vTXHn, by Bojan Šavrič, David Burrows, and Melita Kennedy)It is quite a remarkable picture, showing the seas as an almost uninterrupted whole.As a change in perspective often does, it belies our previous biases; in this case, ourbias towards the land. One could imagine that on a different evolutionary timeline,intelligent aquatic life would come up with this projection as the default picture ofthe world. We do not live on that timeline, but this change in perspective is useful.It sets the stage for the rest of this thesis, the main focus of which is the oceans.1.2 The ocean’s role in climateThe ocean is a major component of the Earth’s climate system. Two of the mainways it affects global climate include: 1. moderating/damping the atmospherictemperature fluctuations above it, and 2. transporting/redistributing heat aroundthe globe via its circulation.The first effect is due to the ocean’s heat capacity being, overall, about 2000 times2 Chapter 1Connecting ocean circulation with ocean heat storagethat of the atmosphere. In fact, the top ⇠3 m of an ocean column can store the sameamount of heat as the corresponding atmosphere above it. The ocean also has atypically lower albedo than that of land. These facts help explain why coastal regionshave considerably attenuated seasonal cycles compared to land-locked regions. Theocean’s thermal inertia is also the principal reason why the coldest and warmest daysof the year do not fall on the winter or summer solstices, but are in fact delayed toa few weeks later. And in the context of long-term global warming, it is because ofthe ocean that our warmest days are yet to come.The second effect is a result of large-scale ocean circulation, which we discuss insection 1.3. The ocean (like the atmosphere) transports heat away from tropical re-gions and towards higher latitudes. Estimates of the climatological-mean meridionalheat transports are shown in figure 1.2. One noteworthy aspect is that the polewardtransport is significantly larger in the Northern Hemisphere than in the SouthernHemisphere; the heat transport in the Atlantic ocean is directed northward at alllatitudes. This is a result of the pole-to-pole overturning circulation that is unique tothe Atlantic basin, and leads to a global, positive cross-equatorial heat transport of⇠0.5 PW. This likely contributes to the Northern Hemisphere being ⇠2C warmerthan the Southern Hemisphere [Marshall et al., 2014b].Figure 1.2: Estimates of zonal-mean meridional ocean heat transports over 1984-2006 based on theCOREv2 climatological surface air-sea flux data set. Green shading indicates the interannual varia-tion. The triangles and squares are observational heat transport estimates derived from the basin-wide hydrographic sections compiled by [Bryden and Imawaki, 2001]. Figure reproduced from [Largeand Yeager, 2009].Chapter 1 3Connecting ocean circulation with ocean heat storageWithout this ocean heat transport (OHT), Earth would probably develop large icesheets in the mid and high latitudes, making the climate much colder than it isat present. In model experiments by [Winton, 2003], the ocean was replaced by amixed-layer, removing the effect of ocean currents and thus ocean heat transports.Despite the continuing presence of atmospheric meridional heat transports, the lackof ocean heat transports allowed sea ice to grow, which was then further enhancedby a positive sea ice-albedo feedback. In two coupled climate models, each with twodifferent sea ice components, all simulations without ocean heat transport developedlarge sea ice extents. Furthermore, in fixed-current runs where the circulation wassystematically increased and decreased by 50%, sea ice extent retreated and grewcommensurately, with the slow current experiments having the most sea ice and thefast current experiments the least [Winton, 2003].Earth has existed through many different climate states in the past, ranging froma possible completely ice-covered state (‘snowball Earth’) during the Cryogenianperiod (⇠700 million years ago, e.g. [Hoffman et al., 1998]), to an ice-free statein the Cretaceous (⇠100 million years ago, e.g. [Hay, 2008]). In the more recentpast, throughout the Late Quaternary (from ⇠1 million years ago to the present),Earth has cycled through glacial and interglacial periods on a cycle of approxi-mately 100,000 years, characterised by the growth and decay of continental icesheets (e.g. [Imbrie et al., 1993]). Within these glacial/interglacial periods areshorter stadials (cold states, lasting up to ⇠1000 years) and interstadials (warmstates, lasting 1-10 thousand years). Measurements from Greenland ice cores reveallarge and abrupt climate changes (at least in the North Atlantic) during the lastglacial period (last 120,000 years) [Dansgaard et al., 1993]. These abrupt changes(‘Dansgaard-Oeschger’ or D/O events) have a characteristic ‘sawtooth’ pattern withrapid warming (a few decades) into an interstadial, followed by a slow cooling intoa stadial over several centuries (figure 1.3). They tend to occur around every 1500years [Alley et al., 2001].It is believed that changes in insolation due to the Milankovitch cycles cause theglacial/interglacial cycles [Imbrie et al., 1993]. However, changes in ocean circula-tion, particularly the overturning in the Atlantic, have been invoked to explain theD/O events (e.g. [Rahmstorf, 2002]). This is inspired by Stommel’s original work4 Chapter 1Connecting ocean circulation with ocean heat storageFigure 1.3: Temperature reconstructions from the Greenland ice core GISP2 (blue, [Grootes et al.,1993]), and ocean sediment proxy data from the subtropical Atlantic (green, [Sachs and Lehman,1999]). D/O warm events are numbered, and Heinrich events (corresponding to large ice sheet dis-charges into the North Atlantic) are marked as red squares. The grey vertical lines are spaced 1470years apart, illustrating the tendency for D/O events to occur with this spacing. Figure reproducedfrom [Rahmstorf, 2002].using an idealised two-box model of the overturning circulation, which under certainconditions exhibited two stable states: a strong mode and a weak mode [Stommel,1961]. Carbon isotopic ratios in sediments suggest that the Atlantic has occupiedthree distinct circulation modes in its past: a strong ‘warm’ mode (current condi-tions), a weak ‘cool’ mode, and an ‘off’ state [Sarnthein et al., 1994]. The idea isthat some outside trigger causes transitions between these stable circulation states,leading to abrupt changes in climate. What this outside trigger is remains unknown.Man-made CO2 emissions have been put forward as a novel trigger candidate [Broecker,1997], causing significant worry about the Atlantic overturning being forced into an‘off’ state. This could bring with it stark Northern Hemisphere cooling and extensiveArctic sea ice (e.g. [Vellinga and Wood, 2008]), similar to the results from [Winton,2003] (imagine subtracting the Atlantic OHT contribution from figure 1.2). Suchsevere consequences have prompted international observational monitoring of the At-lantic circulation (https://www.rapid.ac.uk), as well as scientifically-inaccurateHollywood disaster movies (The Day After Tomorrow).Significant attention has thus been directed to studying changes in ocean circulationaffecting OHT. The moniker ‘ocean heat transport’ has come to be synonymous withmeridional ocean heat transport (e.g. [Herweijer et al., 2005, Ferrari and Ferreira,2011, Macdonald and Baringer, 2013, Forget and Ferreira, 2019]). But the ocean alsotransports and redistributes heat in the vertical, which has received less attention.This is perhaps surprising as, in the context of transient climate change, the surfaceChapter 1 5Connecting ocean circulation with ocean heat storagewarming response is greatly dependent on the rate of heat sequestration in theocean interior, which is communicated via changes in vertical heat transports (e.g.[Gregory, 2000, Exarchou et al., 2015]).This thesis concerns the role of ocean circulation in transient climate change, andhow vertical heat transports are effected by circulation. We will examine the out-puts of a coarse-resolution, coupled general circulation model with idealised geom-etry under an abrupt doubling of atmospheric CO2. This CO2-doubling simulationis run for 200 years, so we are primarily concerned with the top 2 km of the modelocean responding on multi-decadal to centennial timescales. The model (describedin Chapter 2) has a flat-bottomed ocean with a constant depth of 3 km. This pre-cludes a realistic abyssal branch of the ocean circulation, so our neglect of millennialtimescale responses is appropriate.Before considering how the ocean is responding to climate change, and how it mayrespond in the future, we first take a look at our current understanding of the presentEarth’s large-scale ocean circulation.1.3 A brief overview of large-scale ocean circulationAt the most basic level, Earth’s ocean circulation can be split into two distinctcomponents: a quasi-horizontal circulation that is primarily wind-driven, giving riseto great circulating gyres, and a vertical meridional overturning circulation (MOC)that is driven by winds, tidal forces, and surface buoyancy fluxes.There exist subtropical gyres in the North and South Atlantic and Pacific oceans,and in the Indian ocean, from about 15 to 45 (figure 1.4). The flow is westwardon their equatorial branches and eastward in the midlatitudes. These directionscorrespond to the surface easterly winds in equatorial regions, and westerly windsin midlatitudes in the atmosphere. In the Northern Hemisphere, there also existsubpolar gyres that circulate in an opposite sense. For each subtropical gyre, thereis an associated intense western boundary current, formed via recirculation of equa-torward Sverdrup interior flow due to the variation of the Coriolis parameter withlatitude (or ‘beta effect’) [Sverdrup, 1947, Stommel, 1948].6 Chapter 1Connecting ocean circulation with ocean heat storageAlthough the gyres appear isolated to their respective basins in figure 1.4, the sub-tropical gyres in the Southern Hemisphere are also connected via the Tasman andAgulhas leakages in a ‘supergyre’ circulation, with the main gyre circulations nestedwithin (e.g. [Roemmich, 2007, Ridgway and Dunn, 2007, Qu et al., 2019]).-80-400408030E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0WINDS (m/s)–5 0 5 10 CurrentsFigure 1.4: Schematic of the main surface ocean currents, highlighting the dominance of the sub-tropical gyres. The zonally-averaged zonal surface winds are shown in the left-hand panel. Figuretaken from [Vallis, 2012], kindly made available on his personal website: http://empslocal.ex.ac.uk/people/staff/gv219/.Gyres are traditionally considered wind-driven, but it has been shown that gyre-likecirculations can be driven by surface buoyancy fluxes alone in an eddy-permittingocean model [Hogg and Gayen, 2020]. So, the plain language of “wind-driven gyres”is potentially misleading. In reality, both surface winds and surface buoyancy fluxeslikely play a role.The MOC is characterised by cold, dense water sinking at high-latitudes and subse-quently upwelling at lower latitudes. The classic schematic of the MOC is a zonally-averaged view of two global-scale ‘Upper’ and ‘Lower’ overturning cells stacked ontop of each other (e.g. [Lumpkin and Speer, 2007, Marshall and Speer, 2012]). See,for example, figure 1.5. The Upper and Lower cells are also known as the ‘mid-depth’and ‘abyssal’ cells in other literature (e.g. [Nikurashin and Vallis, 2012, Cessi, 2019]).The Upper Cell of the MOC is associated with deep water forming at convectivesites in the Nordic and Labrador seas in the North Atlantic, while the Lower Cellinvolves sites in the Weddell and Ross seas in the Southern Ocean. At these sites,Chapter 1 7Connecting ocean circulation with ocean heat storagethe stratification is sufficiently weak, and both heat fluxes to the atmosphere andbrine rejection from sea ice formation cause large buoyancy losses. Once the surfacewater becomes denser than water at depth, it becomes convectively unstable andsinks in localised plumes on the order of 1 km in horizontal scale [Marshall andSchott, 1999].As sinking within the Upper Cell is confined to the Atlantic basin, the basic structureof the Upper Cell is dominated by the Atlantic overturning (e.g. [Lumpkin andSpeer, 2007]). Because of this, the Upper Cell is often used synonymously with theAtlantic MOC, or AMOC (e.g. [Buckley and Marshall, 2016, Cessi, 2019, Johnsonet al., 2019]), and it is the AMOC that has received extensive study due to itsclimatic importance (e.g. [Kuhlbrodt et al., 2007, Weaver et al., 2012, Wunsch andHeimbach, 2013, Kostov et al., 2014, Chen and Tung, 2018]). Although the 2d-schematic in figure 1.5 is a useful mnemonic, with sinking in the North Atlanticand upwelling in the Southern Ocean, it necessarily ignores zonal transports andinterbasin exchanges. The idea of two distinct, separated cells is an artefact oftaking a zonal average.Figure 1.5: Schematic showing the ‘Upper’ and ‘Lower’ cells of the MOC, reproduced from [Mar-shall and Speer, 2012]. Arrows indicate the direction of the circulation, with colours representing theaverage density of water masses from lightest (red-yellow) to densest (green-blue). Surface buoyancyfluxes and wind stress directions are also indicated. In the background is the zonally-averaged distri-bution of dissolved oxygen, and the thin black line roughly indicates the depth of the Mid-AtlanticRidge.According to [Talley, 2013], once water sourced from the North Atlantic (North At-lantic Deep Water/NADW) upwells in the Southern Ocean, only a small fractionimmediately returns northwards to complete the circuit. The majority of NADWthat surfaces in the Southern Ocean is first transformed into denser Antarctic Bot-8 Chapter 1Connecting ocean circulation with ocean heat storagetom Water (AABW). Once converted into AABW, it then flows northwards into theIndian and Pacific basins where it is transformed into lighter waters via turbulentdiapycnal mixing. These lighter waters then resurface around Antarctica, and onlythen are they transformed into intermediate waters to return to the North Atlantic.The true structure of the MOC is thus a complex, three-dimensional network ofintertwining pathways, with interbasin transports mediated by the Southern Ocean.A summary schematic is shown in figure 1.6.Figure 1.6: Schematic of the global overturning circulation, reproduced from [Talley, 2013]. Arrowsindicate the direction of the circulation and colours represent the average density of water massesfrom lightest (purple-red) to densest (green-blue). A small leakage of less than 1 Sv makes its wayfrom the Pacific to the Atlantic through the Bering Strait.If we wish to retain the two-dimensional view, a more accurate picture of the MOCis given by [Ferrari et al., 2017], reproduced in figure 1.7. Here, the Upper andLower cells are connected by vertical dashed lines, indicating diapycnal mixing-driven upwelling between the different water masses. In this simpler picture, theMOC is best described as a ‘figure eight’ circulation that spans the Atlantic andIndo-Pacific basins, consistent with the more complicated picture put forward byChapter 1 9Connecting ocean circulation with ocean heat storageFigure 1.7: Two-dimensional schematic of the MOC, reproduced from [Ferrari et al., 2017]. Green isNADW, blue is AABW, red is Indian and Pacific deep waters (IDW/PDW), and yellow is intermedi-ate waters. The vertical dashed lines indicate diapycnal mixing transforming AABW into NADW, orAABW into IDW/PDW. The jagged line represents the main bathymetric features in the Indo-Pacific,below which diapycnal mixing is enhanced.[Talley, 2013] in figure 1.6.Now, the upwelling of deep waters involves a vast range of processes across a rangeof spatial scales (e.g. [Huang, 2010]), but figure 1.7 provides a useful overview. Thesurfacing of the green and red ribbons from depth indicates one type of upwelling,and the vertical dashed lines between the blue and green/red ribbons indicate an-other type of upwelling.The first type of upwelling is a wind-driven pathway, where waters are drawn up adia-batically along density surfaces by Ekman suction, caused by strong surface westerlywinds over the Southern Ocean (e.g. [Toggweiler and Samuels, 1998, Gnanadesikan,1999, Webb and Suginohara, 2001, Wolfe and Cessi, 2011]). The second type ofupwelling involves diapycnal mixing — that is, the mixing of water masses acrossdensity surfaces. Both the surface winds and diapycnal mixing thus exert primarycontrols on the strength of the MOC (e.g. [Nikurashin and Vallis, 2011, Nikurashinand Vallis, 2012]).Since isopycnals in the ocean are approximately horizontal across large areas of theocean, diapycnal mixing is also often referred to as vertical mixing. It is typicallymodelled as a diffusive process, with a canonical diapycnal (vertical) diffusivity of104 m2 s1 over the entire abyssal ocean, first put forward by [Munk, 1966] (notethat molecular diffusivities of heat and salt are on the order of 107 and 109 m2s1, respectively [Gill, 1982]).10 Chapter 1Connecting ocean circulation with ocean heat storageNow, diapycnal mixing implies a mass flux across density surfaces and, in a stablystratified fluid, pushes light water down and heavy water up. This raises the po-tential energy of the system, so requires an external source of mechanical energy tomaintain it. Munk originally suggested that this diapycnal mixing could be facili-tated by breaking internal waves in the ocean interior [Munk, 1966]. These internalwaves are generated via a variety of processes such as internal tides, wind-drivennear-inertial waves, and lee waves from abyssal flows impinging on topography (e.g.[Whalen et al., 2020]). In a follow-up paper, [Munk and Wunsch, 1998] estimatedthat about 2 TW of power would be needed to mix bottom waters up to the surfacevia breaking internal waves. About 1 TW is supplied by tidal forces [Ferrari andWunsch, 2009], and perhaps ⇠0.2-0.8 TW is supplied by surface winds [Zhai et al.,2012]. This is not enough to bring abyssal waters to the surface.However, in-situ observations show that internal waves tend to break close to bottomtopography, thus confining mixing to close to the ocean bottom (e.g. [Waterhouseet al., 2014]). The emerging view is that diapycnal mixing need only bring watersfrom the ocean bottom to a depth of ⇠2 km, the characteristic height of the mostprominent oceanic topographic features (indicated by the jagged line in figure 1.7).Once waters reach this depth, a ‘hand-off’ occurs whereby they are then brought upadiabatically to the surface via the westerly winds over the Southern Ocean [Ferrari,2014]. This is what is suggested in figure 1.7. As mixing only brings dense waters to⇠2 km depth, the power needed is about half that estimated by [Munk and Wunsch,1998].In addition to breaking internal waves, diapycnal mixing can be facilitated by otherprocesses such as geothermal heating (e.g. [Mashayek et al., 2013, De Lavergne et al.,2016b]), dissipation from mesoscale eddies (e.g. [Eden and Greatbatch, 2008, Yanget al., 2019]) and shear-induced turbulence from entrainment in localised overflows(e.g. [Legg et al., 2009, Salehipour et al., 2015, De Lavergne et al., 2016a]).Until quite recently, the consensus was that abyssal diapycnal mixing would driveupwelling of waters in the ocean interior. But, as pointed out by [Ferrari et al.,2016], bottom-enhanced mixing should drive sinking and not rising of waters throughdensity surfaces. They conclude that water parcels only rise along the sloping oceanicChapter 1 11Connecting ocean circulation with ocean heat storageboundaries, highlighting the crucial role of bottom topography. So, the net upwellingof abyssal waters is really a residual between downwelling of waters in the oceaninterior and upwelling in bottom boundary layers along topography.This is at odds with numerical models, where mixing is often represented with aconstant vertical diffusivity i.e. with no bottom-enhancement. Because of this,waters are expected to rise rather than sink in a model’s abyssal ocean interior.This is true for the idealised model used in this thesis, and the implications of thiswill be discussed further in Chapter 2. Suffice it to say here, our idealised modelhas no bottom topography, and precludes a realistic abyssal branch of the oceancirculation, so we are primarily concerned with the Upper Cell and not the LowerCell of the MOC.Finally, the structure of the MOC highlights a key difference between the Atlanticand Pacific basins. Deep water formation is confined to the northern high lati-tudes of the Atlantic basin, resulting in a pole-to-pole overturning circulation. Nosuch deep water formation exists in the northern high latitudes of the Pacific (e.g.[Warren, 1983]), nor a pole-to-pole overturning (figure 1.6). This is a fundamentalglobal asymmetry in overturning and has persisted for perhaps 2-3 million years (e.g.[Raymo et al., 2004]). There is still no clear, settled explanation for this overturningasymmetry.The asymmetry is fundamentally linked to the salinity contrast between the twobasins [Ferreira et al., 2018a]. Salty water is heavier than fresh water at the sametemperature, such that adding salt can have a similar effect to that of cooling thesurface. The North Atlantic is more saline than the North Pacific, so it can moreeasily maintain an overturning circulation. This salinity contrast could be set bydifferences in net precipitation over the basins (e.g. [Craig et al., 2017, Wills andSchneider, 2015]), or by differences in salt transport resulting from ocean circulation(e.g. [Nilsson et al., 2013, Cessi and Jones, 2017]). However, the relative contribu-tions of these atmospheric and oceanic processes in setting the salinity contrastbetween the two basins remain to be found [Ferreira et al., 2018a].12 Chapter 1Connecting ocean circulation with ocean heat storage1.4 The ocean’s response to warmingEarth’s surface is warmed by solar radiation and emits infrared (longwave) radiationupward in response. Greenhouse gases in the atmosphere (mainly CO2 and watervapour) absorb this longwave radiation, thus reducing the proportion that escapesto space. This absorption by greenhouse gases warms the atmosphere, which inturn re-emits infrared radiation. Some of this radiation is then absorbed at Earth’ssurface (‘Downwelling Radiation’ in figure 1.8). This is the greenhouse effect, themain mechanism for global warming.There is now greater absorbed solar radiation than outgoing long-wave radiationover the surface of the Earth, leading to an imbalance, designated Earth’s energyimbalance (EEI) (e.g. [Hansen et al., 2011, Trenberth et al., 2014, Von Schuckmannet al., 2016, Von Schuckmann et al., 2020]). More precisely, this imbalance is theresult of the difference between the incoming solar radiation and the reflected solarradiation plus outgoing longwave radiation (OLR) (figure 1.8). The difference islargely due to humans changing the composition of the atmosphere by increasingthe concentration of greenhouse gases by burning fossil fuels (e.g. [Trenberth, 2018,Masson-Delmotte et al., 2021]).It is now undeniable that global-mean surface temperatures are rising, and that weare the cause [Masson-Delmotte et al., 2021]. Figure 1.9 shows a time series of theannual global-mean surface temperature since 1850, relative to a 1961-1990 mean.The upward warming trend is unmistakable.Instead of greenhouse gas emissions, another possible candidate to explain the ob-served warming is that of natural variability; specifically, there being a variationin the ocean such that it releases heat to the atmosphere. If the ocean were tohave given up heat to the atmosphere, it would have become cooler over the lastfew decades. In fact, the heat content of the ocean has increased over the last fewdecades (figure 1.10). This implies that the ocean is warming because of the surfacetemperature increase, and is not the cause.The EEI estimated from [Trenberth, 2020] is 0.9 ± 0.2 W m2 (figure 1.8). Thevast majority (⇠93%) of the excess energy resulting from this imbalance goes intoChapter 1 13Connecting ocean circulation with ocean heat storageFigure 1.8: Schematic of the global annual-mean energy flows through the climate system (in Wm−2). Earth’s energy imbalance, or the net absorbed, is estimated at 0.9± 0.2 W m−2. Figure repro-duced from [Trenberth, 2020], based on [Trenberth et al., 2009] and [Wild et al., 2015].the ocean, leading to an increase in ocean heat content (OHC) (figure 1.10). Only∼4% has contributed to warming the atmosphere and the continents [Rhein et al.,2013]. This increase in OHC is also tightly linked to rising sea-levels (e.g. [Churchet al., 2013]), with about 40% of the global-mean increase in sea-level since 1957accounted for by thermal expansion [Frederikse et al., 2020].Over an ocean column, a change in OHC is due to an anomalous net air-sea heatflux Fs (W m−2) as well as the heat transport into or out of the column:∂t∫ 0−Hρ0cpθ dz = Fs − ρ0cp∫ 0−H∇ · (vθ) dz (1.1)where θ is the oceanic potential temperature, ρ0 the seawater density, and cp thespecific heat capacity of seawater. Ocean heat uptake (OHU) is synonymous withFs (i.e. energy transfer from the atmosphere to the ocean surface), while oceanheat storage (OHS) refers to an increase in OHC (l.h.s. of equation 1.1). Thus,OHU and OHS are equivalent globally, but can be very different regionally due tothe ocean heat transport divergence ∇ · (vθ). (Note that the terms OHU and OHSare sometimes used interchangeably in the literature, and ‘OHU’ is often used as a14 Chapter 1You’re a remolesee page 190Connecting ocean circulation with ocean heat storageFigure 1.9: Annual global-mean surface temperature anomalies relative to 1961-1990 from threedifferent data sets (HadCRUT4, NASA-GISS, and NCDC-MLOST). Reproduced from [Hartmannet al., 2013].shorthand for ‘OHU processes’, which are processes that lead to a change in OHC,as discussed in section 1.4.2.)Ocean heat uptake and subsequent storage redirects heat that would otherwise havewarmed the surface. OHU thus plays a critical role in slowing the pace of globalwarming (e.g. [Gregory, 2000, Held et al., 2010, Kostov et al., 2014]). For example,the so-called ‘surface warming hiatus’ during the beginning of the 21st century(figure 1.9) has been linked to enhanced OHU (e.g. [Meehl et al., 2011, Watanabeet al., 2013, Drijfhout et al., 2014]).Globally, over the period 2005-2017, CMIP5 models estimate an OHS rate of 0.68± 0.02 W m2 over the top 2 km [Cheng et al., 2019], while observations give ratesof 0.64 ± 0.02 W m2 [Domingues et al., 2008] and 0.54 ± 0.02 W m2 [Chenget al., 2017]. Over the period 2000-2010, a full depth estimate of the OHS ratefrom CMIP5 models is 0.73 ± 0.21 W m2 [Smith et al., 2015]. And a full-depthestimate from observations over the period 1993-2017 is 0.66 ± 0.22 W m2 (from[Meyssignac et al., 2019], combining contributions from [Blunden et al., 2018] and[Purkey and Johnson, 2010]). Other observational full-depth estimates vary from⇠0.5-0.8 W m2 depending on the time period and dataset used (see [Meyssignacet al., 2019]).About two thirds of the global OHS is accounted for by the upper 0-700 m of theocean (figure 1.10). This change in upper-ocean heat content only corresponds toChapter 1 15Connecting ocean circulation with ocean heat storageFigure 1.10: Observational estimates of changes in ocean heat content since 1950 from the IPCC’sfifth assessment report [Rhein et al., 2013]. a) Annual global-mean upper-ocean (0-700 m) heatcontent adapted from [Levitus et al., 2012, Ishii and Kimoto, 2009, Domingues et al., 2008, Palmeret al., 2007, Smith and Murphy, 2007]. b) Five-year running means of global-mean mid-depth (700-2000 m, data from [Levitus et al., 2012]) and deep (2000-6000 m, data from [Purkey and Johnson,2010]) ocean heat content changes.a mean temperature increase of about 0.15C, whereas the corresponding meansurface temperature increase is about 0.6C (figure 1.9). This discrepancy meansthat the ocean still has a long way to go in terms of warming, as we assume thatat equilibrium the deep ocean temperature increase will be similar to that at thesurface.The ocean is the pacemaker of global climate change due to its slow circulation andlarge thermal inertia. In response to greenhouse gas forcing, the climate systemadjusts towards a new equilibrium on multiple timescales. The stratosphere andtroposphere respond on timescales of days to weeks [Gregory and Webb, 2008, An-drews et al., 2012], and the land-surface and oceanic mixed-layer on timescales ofmonths to years (e.g. [Hansen et al., 1985, Wallace et al., 1990, Vallis, 2012]). Butthe deeper ocean responds on timescales of centuries to millennia (e.g. [Stouffer,2004, Yang and Zhu, 2011, Li et al., 2013]). So, even if we stopped emitting green-16 Chapter 1Connecting ocean circulation with ocean heat storagehouse gases, it would take the full-depth ocean several millennia to fully equilibrate,during which time the Earth’s surface temperature would nevertheless continue toincrease [Held et al., 2010].To properly quantify the response timescale of the full-depth ocean, we rely on multi-millennial integrations of atmosphere-ocean general circulation models (AOGCMs),run to equilibrium. But this is rarely done due to its enormous computationalexpense [Li et al., 2013]. Instead, the focus of climate prediction has shifted towardsthe time-development of climate change, or ‘transient’ climate change, rather thanthe eventual equilibrated state. This is more practical, as our immediate interest liesin what the climate might be in the next several decades. To that end, this thesis isprimarily concerned with the oceanic response above 2 km depth on multidecadal tocentennial timescales, so we will not be concerned with timescales longer than this.1.4.1 Ocean Heat Storage PatternsFigure 1.8 shows the annual-mean energy fluxes absorbed by Earth’s surface inthe global average, but doesn’t show how these fluxes vary spatially. Air-sea fluxdatasets that combine in-situ observations, satellite observations and atmosphericreanalyses can provide us with this spatial information. However, these datasetsusually have a global mean net surface heat flux from the atmosphere to the oceanin the range of 10-30 W m2 [Josey et al., 2013]. This is much higher than the globalOHS rate estimates of 0.5-0.8 W m2 [Meyssignac et al., 2019], indicating that thereare still major unresolved biases in these air-sea flux datasets, with the discrepancyhaving been termed the ‘ocean heat budget closure problem’. Model biases in air-seafluxes are also one of the key sources of uncertainty in climate projections [Huberand Zanna, 2017]. The qualitative spatial pattern these air-sea flux datasets affordis still valuable, but the quantitative values are nevertheless secondary.Climatological annual-means of surface heat flux components are shown in figure1.11. As expected from the global-mean values in figure 1.8, latent, sensible, andlongwave fluxes are all directed upward, while the shortwave flux is directed down-ward. The shortwave flux decreases towards the poles as expected, with the ex-ception of a band of reduced solar radiation near the equator that marks cloudChapter 1 17Connecting ocean circulation with ocean heat storageFigure 1.11: Spatial maps of the annual mean net heat flux divided into components of (a) La-tent Heat, (b) Sensible Heat, (c) Longwave Flux, and (d) Shortwave Flux in W m−2. Positive valuesare directed downwards. Panels (a) and (b) use the Objectively Analyzed Air-sea Fluxes (OAFlux)dataset [Yu and Weller, 2007], while panels (c) and (d) use the satellite-only International SatelliteCloud Climatology Project (ISCCP) fields [Zhang et al., 2004]. Figure reproduced from [Josey et al.,2013].18 Chapter 1Connecting ocean circulation with ocean heat storagecover over the intertropical convergence zone (ITCZ) (figure 1.11d). In contrast, thelongwave flux is a much more homogeneous field with a range of about 40 W m−2(figure 1.11c). Major latent and sensible heat losses are associated with the westernboundary currents (figure 1.11a, b), although globally the sensible heat flux tendsto be an order of magnitude smaller than the latent heat flux.Figure 1.12: Climatological annual-mean net air-sea heat flux from OAFlux [Yu and Weller, 2007]and ISCCP [Zhang et al., 2004], reproduced from [Josey et al., 2013].By summing panels a-d from figure 1.11, we can get an estimate for the net air-seaheat flux, shown in figure 1.12. The spatial pattern for the net heat flux primarilycomes from the balance between the latent heat and shortwave flux components.Heat is mainly taken up in the tropics (particularly the tropical Eastern Pacific),and lost over the western boundary currents and at high northern latitudes. Theglobal-mean net heat flux from this OAFlux/ISCCP combination is 29 W m−2,however, when it should be close to zero. So, it cannot be used to infer OHS.To do that, we’d need to be able to detect a change in the global-mean net heatflux of ∼0.5-0.8 W m−2 on multidecadal timescales in accord with the OHS rate[Meyssignac et al., 2019], which is not possible with the heat flux datasets we havecurrently.Estimates of OHS can be made using in-situ observations of oceanic temperatureand satellite altimeter observations of sea-level anomalies associated with thermalexpansion. A study from [Johnson and Lyman, 2020] examined gridded ocean heatcontent maps from four research groups, three based on ocean temperature aloneand one combining ocean temperature with satellite altimeter data. Their analysisrevealed that, over the period 1993-2019, about 60% of the global ocean area hada significant positive trend in 0-700 m OHC anomalies, while only about 7% had aChapter 1 19Connecting ocean circulation with ocean heat storageFigure 1.13: Linear trends in 0-700 m OHC anomalies (OHCA) over the period 1993-2019 (W m2)from (a) NOAA’s Pacific Marine Environmental Laboratory (PMEL) [Lumpkin et al., 2020], (b)Japan Meteorological Agency (JMA) [Ishii et al., 2017], (c) Institute of Atmospheric Physics (IAP)[Cheng et al., 2017], and (d) NOAA’s National Center for Environmental Information (NCEI) [Levituset al., 2012]. Figure reproduced from [Johnson and Lyman, 2020].significant negative trend (their Table 1).The large-scale patterns of OHS from the four maps are all very consistent (figure1.13). There is a clear asymmetry in OHS between the Atlantic and Pacific basins,with the Atlantic taking up more heat per unit area than the Pacific. This is afact we’ll explore further in Chapter 4, but it is possibly due to the presence of theAMOC [Shatwell et al., 2020].The patterns of OHS reveal pronounced warming over the western boundary currents(WBCs), which is consistent with enhanced sea surface temperature (SST) trendsover these regions [Wu et al., 2012]. The authors in [Wu et al., 2012] suggested thata poleward shift and/or intensification of the ocean gyres were associated with theseSST trends. Further work has provided more evidence from observations and modelsfor both these effects (e.g. [Yang et al., 2016, Yang et al., 2020]), perhaps drivenby changes in surface westerly winds (e.g. [Yin, 2005, Chen et al., 2008, Swart and20 Chapter 1Connecting ocean circulation with ocean heat storageFyfe, 2012]).Another consistent pattern in OHS from figure 1.13 is the dominance of the SouthernHemisphere (particularly the Southern Ocean), which has been seen in previousstudies (e.g. [Roemmich et al., 2015, Wijffels et al., 2016, Durack et al., 2018]). Amechanism put forward to explain this is the wind-driven upwelling of cold deepwaters around Antarctica, and their subsequent warming, northward transport, andsubduction into the ocean interior (e.g. [Marshall et al., 2014a, Marshall et al.,2015, Armour et al., 2016]).The other conspicuous feature across the four panels in figure 1.13 is the blue regionof cooling in the subpolar North Atlantic. This is highly reminiscent of the so-called‘warming hole’, a region of surface cooling in the same area, which has been linkedto a weakening AMOC (e.g. [Drijfhout et al., 2012, Rahmstorf et al., 2015, Caesaret al., 2018]). The given explanation is that a weakening of the AMOC leads toa weakening meridional heat transport into the warming hole region. However, anincrease in northward heat transport out of the warming hole region and a shortwavecloud feedback have also been implicated in driving the warming hole [Keil et al.,2020].Patterns of OHS in models are similar, with CMIP5 model simulations over the 1861-2005 historical period also revealing an Atlantic/Pacific asymmetry, an enhancementover western boundary currents, a dominance of the Southern Ocean, and a regionof cooling in the subpolar North Atlantic [Frölicher et al., 2015]. But how does heatget stored in the ocean in the first place?1.4.2 Ocean Heat Uptake ProcessesWe noted previously that around 93% of the excess energy from EEI ends up in theocean [Rhein et al., 2013]. This value (93%) is significantly larger than the surfacearea covered by the oceans (71%). This is due to a number of reasons, such asthe ocean typically having a lower albedo than land, and the ability of sunlight topenetrate the upper ocean to ⇠200 m depth. But most importantly, it is becausethe ocean is in constant motion, carrying heat downwards from the surface to theinterior via small-scale mixing processes and large-scale circulation.Chapter 1 21Connecting ocean circulation with ocean heat storageThe mechanisms by which heat is taken from the ocean surface into the interior arenormally denoted ‘OHU processes’ (e.g. [Exarchou et al., 2015, Kuhlbrodt et al.,2015]). This is confusing language, as OHU commonly refers to the net air-sea heatflux (e.g. [Raper et al., 2002, Gregory et al., 2004, Gregory and Forster, 2008]),which suggests that OHU processes only pertain to the atmosphere-ocean boundarylayer. This is not the case, as OHU processes can involve circulations that spanthe full-depth ocean column (e.g. [Newsom et al., 2016]). So, in a broader sense,‘OHU’ can refer to the energy transfer from the atmosphere to the ocean surface, aswell as its subsequent transfer into the ocean interior via a suite of mechanisms i.e.processes that lead to a change in OHC. A reduction in energy lost from the oceanto the atmosphere is also classed as OHU.Stratification and the PycnoclineOn timescales longer than a day, the ocean is in most places stably stratified, withlighter water sitting atop heavier water (e.g. [Talley, 2011]). A decrease in OHCnear the surface destabilises the stratification, resulting in overturning and mixingof the water column to remove the instability. Whereas, an increase in OHC nearthe ocean surface increases the stability of the stratification, further isolating surfacewaters from deep waters. This difference means that warm anomalies penetrate theocean interior significantly slower than cool anomalies (e.g. [Stouffer, 2004]). Theocean’s stable stratification thus acts as a barrier to heat uptake.This is supported by evidence from climate models, where it’s been found that thevertical temperature gradient (a proxy for the stratification) negatively correlateswith the ocean heat uptake efficiency (OHUE, defined as the net heat flux into theclimate system divided by the global surface air temperature change [Gregory andMitchell, 1997, Raper et al., 2002]) in CMIP3 and CMIP5 models [Kuhlbrodt andGregory, 2012]. This is because a weaker temperature gradient implies a weakerstratification, thus enabling more heat to be taken up. In addition, [Kuhlbrodt andGregory, 2012] suggest that models may overestimate OHU and thus underestimatesurface warming as most models’ stratification is too weak compared to observations.Related to the stratification is the pycnocline, the region in the upper ocean where22 Chapter 1Connecting ocean circulation with ocean heat storagedensity increases rapidly with depth, typically several hundred metres in extent.A useful framing of OHU processes is to consider the processes that maintain thepycnocline, which in turn effectively control the rate of OHU. A simple conceptualmodel was put forward by [Gnanadesikan, 1999], consisting of two homogeneouslayers with the top layer representing the pycnocline and bottom layer representingthe deeper ocean below. The pycnocline layer thickness (or ‘pycnocline depth’) iscontrolled by volume transports between the two layers, caused by four key processes:Southern Ocean Ekman transport, Southern Ocean eddies, NADW formation, anddiapycnal mixing (see figure 1.14).Figure 1.14: Schematic of the simple model developed by [Gnanadesikan, 1999]. The effects ofSouthern Ocean winds and compensating eddies are represented by the volume flux Ts, NADW for-mation is represented by the flux Tn, and diapycnal mixing is represented by a low-latitude upwellingflux Tu.Over the Southern Ocean, strong surface westerlies drive an equatorward Ekmantransport that acts to steepen isopycnals (weakening stratification). At the sametime, mesoscale eddies within the Antarctic Circumpolar Current (ACC), formed viabaroclinic instability, extract available potential energy from the flow which acts toflatten isopycnals (strengthening stratification). The collective action of the eddiesresults in a poleward return flow that compensates the wind-driven equatorwardEkman transport above the depth of the Drake Passage (e.g. [Marshall and Radko,2003, Hallberg and Gnanadesikan, 2006]). In Gnanadesikan’s model [Gnanadesikan,1999], the pycnocline is deepened by the Ekman transport and shallowed by theeddies (Ts). The other processes are diapycnal mixing (leading to a net upwellingof abyssal waters as discussed in section 1.3), which serves to deepen the modelpycnocline (Tu), and NADW formation, which shallows it (Tn).Chapter 1 23Connecting ocean circulation with ocean heat storageThis simple model was extended by [Marshall and Zanna, 2014], where they devel-oped a multilayer generalisation in order to probe OHU processes. In their model, adeepening/shallowing of the pycnocline is related to an increase/decrease in OHU.In a pure surface warming scenario, they found that the dominant process governingOHU was due to the Southern Ocean Ekman transport, corresponding to Ekmanpumping of the surface heat anomaly into the ocean interior. Southern Ocean eddiesoppose this Ekman heat uptake, controlling the long-term adjustment over manycenturies, with a lower eddy diffusivity leading to a longer adjustment period. Theimportance of Southern Ocean processes in OHU has been highlighted in many otherstudies (e.g. [Gregory, 2000, Xie and Vallis, 2012, Marshall et al., 2014a, Frölicheret al., 2015, Armour et al., 2016]).Winds over the Southern Ocean have shifted poleward and increased in strength byabout 20% over the last few decades (e.g. [Thompson and Solomon, 2002, Swartand Fyfe, 2012]). From modelling studies, it’s expected that this trend will lead toa strengthening of the overturning in the Southern Ocean (e.g. [Abernathey et al.,2011, Gent, 2016]), although the effects of eddy compensation may mean that thenet increase in overturning strength is greatly reduced (e.g. [Downes and Hogg,2013, Munday et al., 2013]). Nevertheless, the response of the Southern Oceanoverturning will have knock-on effects on Southern Ocean heat uptake and surfacewarming (e.g. [Fyfe et al., 2007, Morrison et al., 2013, Gao et al., 2018, Doddridgeet al., 2019]). The effect of Southern Ocean winds on OHU is captured by the simplemodel of [Marshall and Zanna, 2014], where an increase in the wind stress of 10%(in conjunction with surface warming) leads to an increase in the peak OHU rateby about 50%. The impact of the winds on OHU more generally will be exploredfurther in Chapter 3.Finally, coupled climate models project a decline in the AMOC strength over the21st century (e.g. [Gregory et al., 2005, Cheng et al., 2013, Weijer et al., 2020]),with CMIP6 models suggesting a decline of 34-45% by 2100 [Weijer et al., 2020],although the strength of the future decline is highly uncertain (e.g. [Reintges et al.,2017]). In-situ observations don’t preclude the possibility of a weakening AMOC,but large interannual and decadal variability make the detection of long-term trendsdifficult [Frajka-Williams et al., 2019, Jackson et al., 2022], especially as continuous24 Chapter 1Connecting ocean circulation with ocean heat storagedirect observations from RAPID only go as far back as 2004 [McCarthy et al., 2015].Indirect evidence from proxy observations, however, supports the hypothesis thatthe AMOC is weakening (e.g. [Rahmstorf et al., 2015, Caesar et al., 2018, Zhu andLiu, 2020]).A weakening of the AMOC is expected as a result of positive buoyancy forcingin the northern high latitudes from heat fluxes due to Arctic amplification (e.g.[Serreze and Barry, 2011, Dai, 2022]) and freshwater fluxes from increased high-latitude precipitation and river runoff (e.g. [Stouffer et al., 2006]), melting of theGreenland Ice Sheet (e.g. [Lenaerts et al., 2015, Van den Broeke et al., 2016]) andArctic sea ice (e.g. [Jahn and Holland, 2013, Sévellec et al., 2017]). This positivebuoyancy forcing both suppresses deep convection at the sites of NADW formation,leading to a commensurate reduction in the AMOC strength (e.g. [Thornalley et al.,2018]), and reduces the meridional density gradient, which is thought to scale withthe AMOC strength (e.g. [Stommel, 1961, Griesel and Maqueda, 2006, Gregory andTailleux, 2011, Ragen et al., 2022]).It is thought that a reduction in AMOC strength will lead to an increase in OHU,owing to a reduction in heat lost from the ocean to the atmosphere over the sitesof NADW formation (e.g. [Winton et al., 2013, Drijfhout et al., 2014, Garuba andKlinger, 2016]), which we will discuss further in Chapter 4. An increase in OHU witha reduction in NADW formation is still captured by the simple model of [Marshalland Zanna, 2014], as this deepens the pycnocline, weakening the stratification, thusleading to more heat uptake. Their model thus highlights the importance of high-latitude dynamical processes in the North Atlantic and Southern Ocean on OHU,while the role of diapycnal mixing is secondary.Partitioning in GCMsAnother framing of OHU processes is through the treatment and partitioning ofdifferent processes in ocean models. Early ocean models represented OHU as adiffusive (e.g. [Hansen et al., 1985]) or upwelling-diffusive process (e.g. [Hoffert et al.,1980, Wigley and Raper, 1987]) based on the classic upwelling-diffusion balance putforward by [Munk, 1966]. Modern general circulation models (GCMs) or coupledChapter 1 25Connecting ocean circulation with ocean heat storageatmosphere-ocean GCMs (AOGCMs) reveal a variety of other OHU processes suchas advection by the mean circulation or isopycnal diffusion (e.g. [Gregory, 2000]).Recent model intercomparisons (e.g. [Exarchou et al., 2015, Kuhlbrodt et al., 2015])divide OHU processes according to how they are treated in a model’s temperature(T ) tracer equation. The convergence of heat is determined by advective processes(of the form r · (vT )) and diffusive processes (of the form r · (KrT )), as well assurface fluxes and parameterisations of subgrid-scale processes such as convectionand near-surface mixing.According to the study of [Exarchou et al., 2015], the most important OHU pro-cesses found across three global climate models in CO2-quadrupling experimentswere: convection, near-surface mixing, and resolved advection. In their study, con-vection and near-surface mixing are lumped together as ‘vertical mixing’ (not tobe confused with diapycnal mixing in the ocean interior). They find that ‘verticalmixing’ and resolved advection contribute more than 80% to the OHUE (definedwith respect to SST rather than surface air temperature), with each process con-tributing approximately 40%. The remaining 20% is due to the remaining processesof diapycnal mixing, isopycnal diffusion, and eddy-induced advection.In another CO2-quadrupling experiment, with an eddy-permitting model (1/3 res-olution), [Kuhlbrodt et al., 2015] find that warming below 1 km is mainly causedby advection, while warming in the top 1 km is dominated by vertical mixing (asdefined in the previous paragraph). Both [Exarchou et al., 2015] and [Kuhlbrodtet al., 2015] (as well as previous work from [Gregory, 2000]) also reveal that OHU isdominated by advection at low latitudes, and by vertical mixing at high latitudes,where there is weak stratification.Recent work from [Saenko et al., 2021] analysed a suite of 11 AOGCMs under1% yr1 CO2-increase experiments (all with an ocean resolution ⇠1), and broadlyconfirmed this latitudinal dependence of OHU processes. In the control simulations,they find a leading-order global heat balance between resolved advection warmingthe subsurface ocean, and eddy-induced advection/diffusion cooling it, a findingechoed by several previous studies (e.g. [Gregory, 2000, Hieronymus and Nycander,2013, Griffies et al., 2015]). And in the warming simulations, they find that below26 Chapter 1Connecting ocean circulation with ocean heat storage400 m the OHU is dominated by the sum of these processes (dubbed the ‘super-residual transport’ (SRT) [Kuhlbrodt et al., 2015, Dias et al., 2020a]).However, [Saenko et al., 2021] also perform a heat budget analysis in a densityframework, revealing that most of the interior OHU processes are isopycnal, contraryto older upwelling-diffusion models where it is assumed that diapycnal diffusion isthe main OHU process (e.g. [Raper et al., 2001, Meinshausen et al., 2011]). Thisresult suggests that it is the advective component of the SRT that accounts for mostof the OHU.Connection with circulationA common thread throughout all of section 1.4.2 above is that an understanding ofOHU is incomplete without an understanding of ocean circulation processes, partic-ularly advective transports (e.g. [Gregory, 2000, Xie and Vallis, 2012, Marshall andZanna, 2014, Exarchou et al., 2015, Kuhlbrodt et al., 2015, Zika et al., 2015, Diaset al., 2020b, Saenko et al., 2021]). Far from being an inert diffusive heat reservoir,a myriad of intricate and intertwined OHU processes affect the vertical and lat-eral distributions of OHS making it highly nonuniform in space (e.g. [Zanna et al.,2019, Johnson and Lyman, 2020], and see figure 1.13).Such intricacies means that there is a substantial spread in the contributions to OHSfrom different OHU processes across models (e.g. [Exarchou et al., 2015, Saenkoet al., 2021]), which could, in part, explain the spread in OHUE across CMIP3 andCMIP5 models [Kuhlbrodt and Gregory, 2012]. This spread in OHUE also accountsfor about half the intermodel spread in thermal expansion, which is the largestcontributor to sea-level rise (e.g. [Church et al., 2013, Frederikse et al., 2020]). Asa result, spread in OHU processes significantly contributes to the uncertainty inclimate projections of OHS and sea-level rise, which in turn leads to uncertainty inprojections of transient surface warming.The lack of process-based observations with a global coverage (e.g. [Gould et al.,2013]) means that models are valuable for analysing these OHU processes. Yet,only a handful of model-based OHU process studies have been performed in thepast (e.g. [Gregory, 2000, Huang et al., 2003, Exarchou et al., 2015, KuhlbrodtChapter 1 27Connecting ocean circulation with ocean heat storageet al., 2015, Saenko et al., 2021]). In order to reduce our uncertainty in futureclimate projections, a better understanding of OHU processes and their connectionto circulation processes is essential. That is the aim of this thesis.1.5 A roadmap for this thesisThis thesis concerns the role of ocean circulation in transient climate change, whichwe will probe using an idealised aquaplanet model, the ‘Double-Drake’ configuration(described in Chapter 2), and its response to an abrupt CO2-doubling. The mainbody of this thesis consists of the results Chapters 3, 4, and 5, with a concludingChapter 6.1.5.1 First results chapterHow do particular ocean circulation components transport heat vertically? Andhow will these components respond to global warming? In Chapter 3, we take ideasfrom [Zika et al., 2013] and project the Double-Drake ocean circulation onto depth-temperature coordinates, revealing ‘thermal overturning cells’. In response to anabrupt CO2-doubling, we find that changes in these thermal cells lead to an increasein the downward vertical heat transport of ⇠0.2 W m2 in the top 2 km. Rather than‘resisting’ the warming, the circulation response enables more warming of the oceaninterior. Our results suggest that changes in ocean circulation will further reinforcethe ocean’s role as a buffer of global surface warming, complementing findings from[Winton et al., 2013].1.5.2 Second results chapterWhat is the link between the AMOC and ocean heat storage? And what will happenif this circulation weakens in the future? In Chapter 4, we focus on the weakening ofthe deep meridional overturning circulation in the small basin of Double-Drake. Welook in detail at the transient warming response in Double-Drake following a weak-ening (slowdown) of the overturning circulation, and explain why this weakening issurprisingly unimportant for changes in the small basin’s heat storage rate. This28 Chapter 1Connecting ocean circulation with ocean heat storagechallenges previously established links between the AMOC and ocean heat storage(e.g. [Kostov et al., 2014, Chen and Tung, 2018]).1.5.3 Third results chapterCan we emulate OHU in the Double-Drake model using a simple two-layer energy-balance model (EBM) [Geoffroy et al., 2013b]? And can we improve the emulationby increasing/decreasing the sensitivity of OHU to the stratification in the EBM?We find that increasing the sensitivity improves the emulation for Double-Drake andfour other CMIP5 models, while decreasing the sensitivity worsens the emulationacross all models analysed. The improvement following an increase in the sensitivitymight highlight the importance of the stratification for OHU, and it suggests thata simple change to the two-layer EBM could lead to better representation of OHU.We conclude in Chapter 6, where we summarise the main findings of the threeresults chapters and assess their relevance for the real world and the field of climatemodelling. We address the limitations of the work presented and establish directionsfor possible future work.Chapter 1 29Connecting ocean circulation with ocean heat storage30 Chapter 1“What is the Earth but a lump of clay sur-rounded by water?”From VairāgyaśatakaBhartr.hari, c. 625 CE2Experiment DesignThis chapter details the model used for the thesis results presented in Chapters 3,4, and 5. We use a coupled atmosphere-ocean model of intermediate complexity(Double-Drake, or ‘DDrake’), underpinned by the MITgcm. A strong radiativeforcing mimicking an abrupt CO2-doubling has not been implemented before in thismodel. We study the model’s response to such a radiative forcing on multidecadalto centennial timescales (primarily the top 2 km of the model ocean) to aid ourunderstanding of OHU processes, particularly those related to ocean circulation.We give an overview of the MITgcm in section 2.2, and in section 2.3, we introducethe DDrake configuration and provide a description of the model set-up. A few keydiagnostics from the control climate of DDrake are presented in section 2.3.2. Weconclude with a brief history of the CO2-doubling thought-experiment in section2.4, and how this is implemented in DDrake. Our results and discussion throughoutthe rest of the thesis will centre around differences between the control and CO2-perturbed climates of DDrake.31Connecting ocean circulation with ocean heat storage2.1 Modelling philosophyAccording to [Katzav and Parker, 2015], there are currently three approaches to thedevelopment of climate modelling: the unified approach, the pluralist approach, andthe hierarchy approach. The unified approach seeks to pool international resourcesto create a small number of multinational climate modelling centres, aiming todevelop very high resolution models with an eye towards more accurate predictions.The pluralist approach calls for substantially increasing the diversity of models usedin climate research, aiming to more widely sample the range of model structuraluncertainties. This will allow a better quantification of uncertainties in predictions.The hierarchy approach is the one exhorted by [Held, 2005], which aims to establisha hierarchy of models of increasing complexity, ultimately leading to improvementsin understanding the climate system.There has arguably been a lot more focus on the unified and pluralist approachesin the modern development of climate science. This has led to large collaborationsof scientists, software engineers, and other staff developing sophisticated generalcirculation models (GCMs) and Earth system models (ESMs). These models havegrown enormously in complexity over the last few decades, and are so resourceintensive that only a few dozen of them can be developed and maintained at anyone time. The results from these models are compared periodically in global ‘coupledmodel intercomparison projects’ or CMIPs (e.g. [Meehl et al., 2007, Taylor et al.,2012, Eyring et al., 2016]).Rather than follow the unified or pluralist approaches, the work in this thesispresents a contribution to the hierarchy approach. Due to the overwhelming collab-orative nature of the current climate modelling enterprise, and the resulting com-plexity of the models developed, no single person can wholly understand what one ofthese models is doing. According to [Held, 2005], there is a gap between comprehen-sive climate models (GCMs, ESMs) and simple idealised models (e.g. energy-balancemodels, layered models) that has not been successfully closed. This thesis analysesthe outputs of a coarse-resolution, coupled atmosphere-ocean GCM with highly-idealised geometrical constraints on ocean circulation. This model (‘Double-Drake’,[Ferreira et al., 2010]) is put forward to fill this gap and thus occupy an intermediate32 Chapter 2Connecting ocean circulation with ocean heat storageposition in the model hierarchy.We will analyse the output of an abrupt CO2-doubling experiment using the Double-Drake model. Such a scenario is an established test case in aiding understanding ofthe climate response to CO2 increase in the atmosphere (e.g. [Manabe and Wether-ald, 1975, Washington and Meehl, 1984, Schlesinger and Mitchell, 1987]). Under-standing gained from this idealised forcing experiment is then extrapolated to otherforcing scenarios. The Double-Drake model is not the real world, but, in a similarway, it is hoped that improvements in understanding this model will extrapolate tofeatures of the real world. Such an understanding then feeds back onto the develop-ment of the state-of-the-art GCMs and ESMs that inform the IPCC in assessmentreports (e.g. [Solomon et al., 2007, Stocker et al., 2013, Masson-Delmotte et al.,2021]), aiding improvements in the accuracy and uncertainty quantification of cli-mate projections.2.2 The MITgcmThe MITgcm (or Massachusetts Institute of Technology General Circulation Model)is a numerical computer model that solves the equations of motion governing theEarth’s atmosphere or ocean [Marshall et al., 1997a, Marshall et al., 1997b]. It isable to model processes on a wide range of scales from tens of metres to thousandsof kilometres. The MITgcm is used widely by the atmosphere, ocean, and climatemodelling communities, with over 100 research articles using the MITgcm publishedin 2021 alone (https://mitgcm.org/2022/01/06/2021-research-roundup/).The model code exploits an isomorphism between equation sets for the atmosphereand ocean so that the same dynamical kernel can be used to drive either fluid[Marshall et al., 2004]. The equations are written in a generic vertical coordinater, and this is interpreted as pressure if modelling the atmosphere, and as height ifmodelling the ocean. The standard equations of motion used for the ocean are the‘semi-compressible’ Boussinesq equations:DuDt+ fk⇥ u+ 1⇢0rzp0 = F (2.1)Chapter 2 33Connecting ocean circulation with ocean heat storage✏nhDwDt+ g⇢0⇢0+1⇢0@p0@z= ✏nhFw (2.2)rz · u+@w@z= 0 (2.3)Here, D/Dt is the total derivative, u is the horizontal velocity and w the verticalvelocity, f is the Coriolis parameter, and rz is the gradient operator at constant z.The dashed quantities p0 and ⇢0 are deviations away from some reference state. Forthe pressure, p0 is a deviation from a reference profile p0(z) such that p(x, y, z, t) =p0(z) + p0(x, y, z, t). For the density, ⇢0 is a deviation from a constant referencedensity: ⇢0 = ⇢(✓, S, p0(z)) ⇢0. The F terms are forcings provided by differentforcing packages, and the ✏nh parameter encapsulates the non-hydrostatic terms.The ‘hydrostatic primitive equations’ are recovered if ✏nh = 0. The hydrostaticprimitive equations are the model equations solved for the DDrake configurationused for the results in this thesis (the DDrake configuration is described in thefollowing section).Equations 2.1 and 2.2 are the horizontal and vertical momentum equations, respec-tively, while equation 2.3 ensures continuity. Together with an appropriate equationof state for ⇢, and tracer equations for the potential temperature ✓ and salinity S:D✓Dt= Q✓ (2.4)DSDt= QS (2.5)where the Q terms are forcing and dissipation of each tracer, equations 2.1 to 2.5form a complete set describing the hydrodynamics and thermodynamics of the modelocean.These equations are then discretised in space onto a model grid and solved usingfinite volume methods. The grid can be initialised using Cartesian, spherical-polar,or general orthogonal curvilinear coordinate systems. The grid used for the DDrakeconfiguration is the conformally expanded spherical cube [Rančić et al., 1996, Ad-34 Chapter 2Connecting ocean circulation with ocean heat storagecroft et al., 2004a]. This grid avoids the problem of converging meridians in tradi-tional latitude-longitude model grids where the smallest grid-element length scalesas xmin ⇠ N2, where N is the number of points around the equator. For thelatitude-longitude grids, doubling the number of points thus halves the resolutionat the equator but quarters the resolution near the poles. The cube-sphere gridprovides a more uniform coverage of the sphere and has a much better scaling ofxmin ⇠ N4/3. This means it can be taken to a much higher resolution beforebecoming computationally unmanageable [Adcroft et al., 2004b].2.3 The DDrake configurationOur model set-up is the ‘Double-Drake’ or ‘DDrake’ configuration, first introducedin [Ferreira et al., 2010]. In this geometry, two sea-floor to sea-surface barriers are set90 apart and joined at the North Pole. This separates the ocean into a large basin(LB) and a small basin (SB), with both of them connected by a ‘southern ocean’region (SO) south of 35S. The small and large basins in the DDrake configurationexhibit distinctive Atlantic-like and Pacific-like characteristics, respectively, withthe small basin being warmer and saltier, and exhibiting a deep interhemisphericoverturning circulation (see section 2.3.2). The configuration captures two importantasymmetries relevant to the Earth’s climate: a zonal asymmetry splitting the oceaninto small and large basins, and a meridional asymmetry allowing for circumpolarflow in the Southern Hemisphere, but not in the Northern Hemisphere.Figure 2.1: Time-averaged sea surface temperature (in C) in the control integration, showing themodel DDrake geometry from above (left) and below (right). The small basin (SB), large basin (LB),and southern ocean (SO) regions are labelled.Chapter 2 35Connecting ocean circulation with ocean heat storageIn [Ferreira et al., 2010], the DDrake configuration was used to help explain thelocalisation of deep water formation to the Atlantic basin in the real-world presentclimate. A deep overturning circulation is seen in the Atlantic basin (AMOC) butnot in the Pacific. Similarly, a deep overturning circulation is seen in the smallbasin of DDrake, but not in the large basin. This is because of the difference inthe zonal widths of the two basins, which results in a net atmospheric freshwatertransport out of the small basin and into the large basin. The freshwater transportsets up a salinity contrast between the two basins, which localises the sinking to thesmall basin. Note this is only one suggested mechanism to explain the real-worldAtlantic/Pacific salinity contrast, and it is not yet known what the true mechanismis (e.g. [Craig et al., 2017, Cessi and Jones, 2017, Ferreira et al., 2018a]).Figure 2.2: Meridional ocean heat transports (OHT) and freshwater transports (FWT) for (left)the small and large basins of DDrake and (right) the Atlantic and Indo-Pacific basins. Observationsfor OHT are taken from [Trenberth and Caron, 2001], while observations of FWT are taken from[Wijffels et al., 1992]. Figure reproduced from [Ferreira et al., 2010].The meridional barriers in DDrake act as geometrical constraints on the large-scaleocean circulation, having a profound effect on the climate of the coupled system.The constraints are sufficient in reproducing fundamental properties of the present36 Chapter 2Connecting ocean circulation with ocean heat storageEarth’s climate. For example, figure 2.2 shows the meridional ocean heat transportsand freshwater transports for the small and large basins of DDrake in comparisonwith estimates from observations. The transports are strikingly similar to those ofthe Atlantic and Indo-Pacific basins, despite DDrake’s highly idealised geometry.2.3.1 Technical descriptionThe DDrake configuration uses the MITgcm code to drive the dynamics [Marshallet al., 1997a, Marshall et al., 1997b]. An isomorphism between the atmosphericand oceanic dynamics is exploited so that either fluid can be driven by the samedynamical core [Marshall et al., 2004]. The isomorphic vertical coordinates used are:a rescaled pressure coordinate p⇤ for the compressible atmosphere, and a rescaledheight coordinate z⇤ for the Boussinesq ocean (as described in [Adcroft and Campin,2004]). The use of the z⇤ coordinate allows for perfect (machine accuracy) conser-vation of freshwater, heat, and salt during extended integration, as discussed in[Campin et al., 2008].Both the atmosphere and ocean component models use the same horizontal cubed-sphere grid [Adcroft et al., 2004a] at a coarse C24 resolution (24x24 points perface, giving a resolution of 3.75, or approximately 400 km, at the equator). Fluxesof momentum, heat, and freshwater are exchanged between the atmosphere andocean every hour (constrained by the ocean model time step). The seasonal cycleof insolation is represented, but there is no diurnal cycle.The atmosphere has a low vertical resolution of five levels, and its physics is based onthe ‘simplified parameterisations primitive-equation dynamics’ (SPEEDY) scheme[Molteni, 2003]. The lowest vertical level represents the planetary boundary layer,the middle three levels are placed in the troposphere, and the top vertical levelcrudely represents the stratosphere. The physics package comprises a four-bandradiation scheme, a boundary layer scheme, resolved baroclinic eddies, and param-eterisations of moist convection, large-scale condensation, and cloud cover.The ocean is flat-bottomed with a constant depth of 3 km, and is split into 15 levelswith increasing vertical resolution from 30 m at the surface to 400 m at depth. Adepth of 3 km is chosen so that the resulting heat capacity of the global ocean isChapter 2 37Connecting ocean circulation with ocean heat storagecomparable to that of Earth’s. The absence of land in the configuration means wemust compensate by having this shallower depth (the average depth of the presentEarth’s oceans is about 3.7 km). A big caveat is that the flat-bottom topographymeans that topographic form drag plays no role in balancing momentum inputfrom the surface winds. Hence, an enhanced linear bottom friction is employedand tuned to damp abyssal currents towards zero at the ocean floor. The seawaterequation of state is the nonlinear international equation of state for seawater [Milleroand Poisson, 1981], modified in terms of potential rather than in-situ temperature[Jackett and Mcdougall, 1995].Oceanic mesoscale eddies are parameterised as an advective process (‘GM’) [Gentand Mcwilliams, 1990] and an isopycnal diffusion [Redi, 1982], both with a transfercoefficient of 1200 m2 s1 (both transfer coefficients can be interpreted as ‘eddydiffusivities’; see Appendix A for a full description). Although it is numericallyconvenient to set both coefficients to the same value, this is a simplification, as theisopycnal diffusivity is generally larger than the GM diffusivity (e.g. [Smith andMarshall, 2009, Vollmer and Eden, 2013]).Values of O(103) m2 s1 are standard in representing mesoscale eddy processes incoarse-resolution ocean models (e.g. [Griffies, 2018]). This is consistent with esti-mates of lateral eddy diffusivities that use direct observations from tracer releaseexperiments (e.g. [Ledwell et al., 1998]), dispersion of subsurface floats (e.g. [La-Casce et al., 2014]), and satellite altimetry (e.g. [Abernathey and Marshall, 2013]).However, these studies and others (e.g. [Groeskamp et al., 2017, Groeskamp et al.,2020]) reveal that eddy diffusivities vary widely with location (from ⇠0 to O(104)m2 s1), with enhanced mixing in western boundary currents and a general decaywith depth. So, a constant background value for the eddy diffusivity in DDrakeis a large simplification, but one that has been used in more sophisticated coupledclimate models with realistic geometries (e.g. [Salas-Mélia et al., 2005, Watanabeet al., 2010, Kuhlbrodt and Gregory, 2012]).Additionally, it’s been shown that models are quite sensitive to the value of the isopy-cnal diffusion (‘⇢’) and the GM diffusivity (‘GM ’). The isopycnal diffusion affectsthe uptake of anthropogenic carbon [Gnanadesikan et al., 2015], deep ventilation38 Chapter 2Connecting ocean circulation with ocean heat storage[Jones and Abernathey, 2019], and global surface climate [Pradal and Gnanade-sikan, 2014]. As argued in [Pradal and Gnanadesikan, 2014], an increase in isopyc-nal diffusion results in increasing high-latitude surface salinity, which decreases thestratification and enhances convection. This mechanism could explain why isopyc-nal diffusion also affects deep ventilation and carbon uptake, and possibly OHU aswell (e.g. [Gregory, 2000]).The GM diffusivity negatively correlates with the ACC transport in coupled climatemodels [Kuhlbrodt et al., 2012], and negatively correlates with the ocean heat uptakeefficiency (OHUE) in CMIP3 models [Kuhlbrodt and Gregory, 2012]. A possibleexplanation linking these two results is that an increased ACC transport leads tosteepened isopycnals in the Southern Ocean, resulting in a weakened stratificationand hence greater OHU (e.g. [Gnanadesikan, 1999, Marshall and Zanna, 2014]).Another mechanism involves the AMOC, where it’s been found that GM negativelycorrelates with the AMOC strength [Marshall et al., 2017], and that the AMOCstrength positively correlates with OHUE [Kostov et al., 2014]. Taking both theserelationships together suggests that GM controls OHU indirectly by controlling thestrength of the AMOC [Saenko et al., 2018].In addition to mesoscale eddies, the other key ocean processes that are parameterisedin DDrake are convection, which is represented by an enhanced vertical mixingof temperature and salinity [Klinger et al., 1996], and diapycnal mixing, which isrepresented with a uniform background vertical diffusivity set to 3 ⇥ 105 m2 s1.This is broadly in accord with observational estimates of diapycnal mixing in theocean interior from tracer release experiments (e.g. [Ledwell et al., 1993, Ledwellet al., 2011, Watson et al., 2013]) and microstructure profiles (e.g. [Gregg et al.,2003, Klymak et al., 2006]).Like eddy diffusivities, diapycnal mixing also varies widely with depth and location(e.g. [Whalen et al., 2012, Waterhouse et al., 2014]), and it is particularly enhancedover rough bottom topography (e.g. [Polzin et al., 1997, Ledwell et al., 2000]).The global survey of diapycnal mixing measurements from [Waterhouse et al., 2014]reveals that diapycnal diffusivities above 1 km depth are O(105) m2 s1 while thosebelow 1 km depth are O(104) m2 s1 (although values can reach O(103) m2 s1 inChapter 2 39Connecting ocean circulation with ocean heat storagespecific locations). A choice of v = 3⇥ 105 m2 s1 for DDrake lies between thesetwo regimes.As discussed in Chapter 1, the magnitude of diapycnal mixing plays a primary rolein maintaining the MOC, particularly the Lower Cell. In idealised model studies,it’s been found that the strength and structure of the Lower Cell is influenced bythe magnitude of diapycnal mixing [Ito and Marshall, 2008, Nikurashin and Vallis,2011] as well as the vertical profile of diapycnal mixing [Saenko and Merryfield,2005, Mashayek et al., 2015].A constant background value for diapycnal mixing in DDrake is an oversimplifi-cation, and this partly explains why there is no realistic Lower Cell circulation inDDrake (seen later in figure 2.8). Additionally, the lack of bottom topography meansthat diapycnal upwelling in the DDrake ocean interior is unrealistic, as, in the realworld, bottom-enhanced diapycnal mixing drives downwelling in the ocean interior,while upwelling takes place in bottom boundary layers along sloping topography[Ferrari et al., 2016].The lack of a realistic Lower Cell circulation in DDrake (and the relatively shallowocean depth of 3 km) means that we should limit our discussion of results to thetop ⇠2 km of the ocean, so we are primarily concerned with the Upper Cell of theMOC (AMOC). Although v does affect the strength of the AMOC (classical scalingarguments suggest that AMOC strength / 2/3v (e.g. [Bryan and Cox, 1967, Parkand Bryan, 2000])), the study of [Saenko and Merryfield, 2005] found that bottom-enhancement of v had very little effect on the AMOC and the corresponding OHT,so, a constant v profile for DDrake is not inappropriate for studying this circulation.However, the magnitude of v does raise questions surrounding spurious diapycnalmixing in DDrake. The model vertical coordinate is height (rescaled z⇤), and itwas pointed out in [Griffies et al., 2000] that, in z-coordinate ocean models, trunca-tion errors in numerical advection schemes can cause isopycnal processes to becomeerroneously projected onto the cross-isopycnal direction, leading to spurious diapyc-nal mixing. This spurious mixing could become non-negligible in particular regions,reaching values of O(104 m2 s1), thus swamping the prescribed vertical diffusivity.It is likely that there is some spurious diapycnal mixing in DDrake, but its magnitude40 Chapter 2Connecting ocean circulation with ocean heat storageis not known. However, experiments with a similar aquaplanet configuration∗ haveshown that the model is sensitive to the magnitude of v, which suggests that thespurious mixing does not dominate over the prescribed vertical mixing.Unlike the set-up in [Ferreira et al., 2010], we now include a frictional heating termin the atmospheric temperature equation that allows a conversion from mechanicaldissipation to thermal energy. This captures the effects of friction, mainly in theplanetary boundary layer, which generates turbulent eddies in the atmospheric flowfield, after which internal viscous forces eventually dissipate kinetic energy intoheat. Without such a term, the net energy imbalance at the top of the atmosphere(TOA) in the control climate is on the order of 1 W m2 downward, a feature ofmany contemporary climate models [Lucarini and Ragone, 2011]. This is becauseenergy dissipated through friction is simply lost from the system, and needs to becontinually replaced to close the energy cycle.Including frictional heating reduces the energy bias at the TOA to ⇠0.1 W m2,which greatly aids interpretation of the net energy imbalance in a CO2-doublingexperiment, which we describe in section 2.4. The model is spun up for 6000 yearsuntil a statistically steady state is reached. The time-mean of the last 50-yearintegration is used as the equilibrated control climate state.The simplifications and caveats detailed above mean we must not confuse DDrakewith the real-world ocean, and we should be careful about what conclusions we drawfrom the model. Throughout this thesis, we thus value qualitative over quantitativeresults. However, as mentioned in section 2.1, even if some conclusions only applyto DDrake, this is still useful as a contribution to the hierarchy approach of climatemodel development [Held, 2005].2.3.2 Control ClimatologyA discussion of the control climate of DDrake, particularly in comparison to theDrake configuration, is given in [Ferreira et al., 2010]. We recapitulate some of thatdiscussion here, but with a greater emphasis on the model ocean. (Note that theclimate state is slightly different owing to the addition of frictional heating in the∗Rheinlænder et al. 2022 (in prep.)Chapter 2 41Connecting ocean circulation with ocean heat storageatmosphere.) Recall that the distribution of the continents is shown in figure 2.1and the meridional ocean heat transports and freshwater transports in figure 2.2.All remaining diagnostics shown in this section are time-averages of the last 50 yearsof the 6000-year-long model spin-up.300300300320320320320340340340340360360360380 380 380400400400420420440 440460 460480-50 0 50100200300400500600700800900-500000000000000000555555555510101010101015151515-50 0 50100200300400500600700800900 -505101520baFigure 2.3: Zonally-averaged atmospheric potential temperature (a) and zonal winds (b) for theDDrake control climate.Figure 2.3 shows the zonal mean distributions of the atmospheric potential tem-perature (panel a) and zonal winds (panel b). There’s a good level of structuraldetail despite there being only 5 vertical levels in the atmospheric model. Westerlyjet streams are seen in both hemispheres around 40 (collocated with the strongmeridional temperature gradients), reaching near 20 m s1 at 250 mb. These jetsare surrounded by much weaker trade winds and polar easterlies.The southern hemisphere is slightly cooler than the northern hemisphere, due totwo principal reasons: 1. an absence of meridional barriers south of 35S permitscircumpolar flow in the ocean, acting as a barrier to southward heat transport;and 2. the deep overturning circulation of the small basin permits a northwardcross-equatorial heat transport, warming the northern hemisphere while cooling thesouthern hemisphere. Both these oceanic effects then imprint on the overlying at-mospheric temperature distribution.In figure 2.4 we plot the zonally-averaged temperature, salinity, and density dis-tributions for the small and large basins of DDrake. The surface distributions forthese fields are also plotted in figure 2.5. For comparison, north-south hydrographicsections of temperature and salinity across the Atlantic and Pacific basins from the42 Chapter 2Connecting ocean circulation with ocean heat storage4 6 810121414 1616 1820222426283032-50 0 50Latitude-2500-2000-1500-1000-500Depth (m)51015202530303131.53232.53333.53434.535 35 35.536-50 0 50Latitude-2500-2000-1500-1000-500Depth (m)2830323436212222.52323 23.5242424.5 24.5252525.5 25.52626-50 0 50Latitude-2500-2000-1500-1000-500Depth (m)2122232425264 6 81010 1212121414 1616 1820222426283032-50 0 50Latitude-2500-2000-1500-1000-500Depth (m)51015202530303131.53232.53333.533.53434 34.534.534.53535-50 0 50Latitude-2500-2000-1500-1000-500Depth (m)28303234362121.5222 .52323 23.5 242424.524.5252525.525.52626-50 0 50Latitude-2500-2000-1500-1000-500Depth (m)212223242526a bdecfFigure 2.4: Control climate zonally and sector-averaged potential temperature (top), salinity (mid-dle), and potential density 0 (bottom) distributions for the small basin plus southern ocean (SBSO,left column) and large basin plus southern ocean (LBSO, right column) DDrake regions. Potentialtemperature and potential density fields are referenced to the surface.Figure 2.5: Spatial maps of the DDrake control climate (a) sea surface temperature (SST, in C),(b) sea surface salinity (SSS, in psu), and (c) surface density (0, in kg m3).Chapter 2 43Connecting ocean circulation with ocean heat storageFigure2.6:Hydrographicsectionsofpotentialtemperature(top,inC)andsalinity(bottom,inpsu)fortheAtlantic(left)andPacific(right)basinstakenduringtheWorldOceanCirculationExperiment(WOCE),from[Schlitzer,2000].44 Chapter 2Connecting ocean circulation with ocean heat storageFigure2.7:Decadallyaveragedsurfaceclimatologicalfieldsovertheperiod1955-2017of(a)seasurfacetemperature(in C),(b)seasurfacesalinity(inpsu),and(c)surfacedensity(0,inkgm3),fromtheWorldOceanAtlas2018[Boyeretal.,2018].Chapter 2 45Connecting ocean circulation with ocean heat storageWorld Ocean Circulation Experiment (WOCE) are shown in figure 2.6, and clima-tologies of surface temperature, salinity, and density from the World Ocean Atlas(WOA) are shown in figure 2.7.The zonal-average structures of temperature and salinity for the small and largebasins are broadly in accord with those seen in observations (figures 2.4 and 2.6).The comparison is more favourable if we restrict our view to the top 3 km of thehydrographic sections from WOCE. The small basin (SB) is both warmer and saltierthan the large basin (LB), just as the Atlantic is warmer and saltier than the Pacific.There is a tongue of fresher water protruding under the subtropical salinity maximain the SB (figure 2.4c) reminiscent of Antarctic Intermediate Water (AAIW) inthe Atlantic basin (figure 2.6c). Similarly, the tongue of cooler, fresher water in thenorthern high latitudes of the LB is reminiscent of North Pacific Intermediate Water(NPIW) (figures 2.4d and 2.6d).However, there are clearly major differences. The biggest one perhaps is the lackof any analogous Antarctic Bottom Water (AABW) mass (purple in figure 2.6a,b) in DDrake. This means that the global water mass properties of DDrake arebiased towards ‘SB deep water’, which results in the DDrake ocean being almost10C warmer on average than the Earth’s. The absence of an Antarctic land massin DDrake also means there’s a deep bowl of cool, fresh water over the South Polethat’s not seen in the present Earth’s Southern Ocean (figure 2.4). Here, there isa temperature inversion, and the stable stratification is controlled by salinity (e.g.compare panels a and c with panel e in figure 2.4).Turning to the surface distributions, we see that DDrake reproduces reasonablemeridional gradients for the surface temperature, salinity, and density fields (figures2.5 and 2.7). Surface salinity maxima are created near 30 in both hemispheresof DDrake due to large evaporation under the descending branches of analogous‘Hadley cells’, in a similar mechanism to the present climate (figures 2.5b and 2.7b).The contrast between the small and large basins of DDrake is similar to the contrastbetween the Atlantic and Pacific basins. Again, the SB is warmer and saltier thanthe LB, particularly in the northern high latitudes (figure 2.5a, b). Although theSST contrast between the small and large basins is large at these latitudes, the46 Chapter 2Connecting ocean circulation with ocean heat storagesurface waters in the SB are still denser than those in the LB (figure 2.5c), just asthe North Atlantic is denser than the North Pacific (figure 2.7c). This is due to thenonlinear equation of state for seawater [Millero and Poisson, 1981], as the salinityexerts a greater control on the density at lower temperatures (e.g. [Nycander et al.,2015]).Unlike in the LB, the absence of relatively fresher water in the SB northern highlatitudes leads to a very weak stratification with many outcropping density surfaces(figure 2.4e, f). This difference in stratification is also readily apparent between theAtlantic and Indo-Pacific basins (panels a and b in figure 2.9).Figures 2.4 and 2.5 reveal a controlling influence of the northern high latitudes onthe key differences between the two basins of DDrake. Indeed, these differenceslead to profoundly different ocean circulations in both basins, with deep convectionbeing localised solely to the small basin. As noted in section 2.3, this is due to thesalinity contrast between the two basins (figures 2.4c, d and 2.5b). This contrastexists due to the narrowness of the SB relative to the LB, and is brought about bya net atmospheric freshwater transport out of the SB into the LB, as explained in[Ferreira et al., 2010].As could be expected from figure 2.4e, the weak stratification in the SB northernhigh latitudes leads to deep convection, which feeds the downwelling branch of a deepoverturning circulation. The control climate overturning circulations are shown infigure 2.8, revealing the radically different circulations of the SB and LB. The SBcirculation is dominated by a deep clockwise overturning cell (SBMOC) spanningthe entire length of the basin (35S to 90N). The two shallow, subtropical Ekmancells about the equator are dwarfed by the convectively-driven overturning cell. Inthe LB, however, the shallow Ekman cells are of primary importance, and there is anorthward intrusion of bottom water from the southern ocean, about half of whichupwells at the equator. There is no circulation associated with deep convection inthe LB.If we compare with the overturning circulations in the Atlantic and Indo-Pacificbasins, we find that the circulation structures are remarkably similar (figures 2.8a,b and 2.9c, d). Both the SB and the Atlantic are dominated by a deep clockwiseChapter 2 47Connecting ocean circulation with ocean heat storage-10-5 00555551010101015151515202020002224444666881012-80 -50 -20 0 20 50 80-3000-2000-10000 -4035-2- 0-15-10-10-10-5-5-5-50 0000000 0555555510101015 152053002224444666881012-80 -50 -20 0 20 50 80-3000-2000-10000-55- 04-35- 02- 0-15- 0-50 00005 5 555551010 101010101515151520 20202022525303030 3545-80 -40 0 40 80-3000-2000-10000-40-2002040ca bFigure 2.8: Control climate residual overturning stream functions for the (a) small basin, (b) largebasin, and (c) global regions of DDrake (in Sv, 1 Sv ⌘ 106 m3 s1). The southern ocean residualoverturning is shown south of 35S (thick black line) in panels (a) and (b).overturning (Upper Cell), while the LB and Indo-Pacific have shallow Ekman cellsand an anticlockwise overturning (Lower Cell). However, there is an unrealisticcancellation of the ‘Upper’ and ‘Lower’ Cells in the global-average in DDrake (figure2.8c), so the global MOC is dominated by the SBMOC (compare with figure 1.5from the Introduction).This lack of an abyssal Lower Cell is one of the major shortcomings of the DDrakemodel in its applicability to study OHU below 2 km. The warming of the oceanbelow 2 km is primarily driven by the warming of AABW [Purkey and Johnson,2010, Purkey and Johnson, 2013], and this is not captured in DDrake. However, thecontribution below 2 km to global OHU is likely less than 10% (e.g. [Desbruyèreset al., 2016, Meyssignac et al., 2019]).Another unrealistic aspect of the DDrake circulation is that the SBMOC strength(stream function maximum) is 26 Sv, which is approximately 40% stronger than thereal-world AMOC (e.g. measurements from RAPID are ⇠18 Sv [Frajka-Williamset al., 2019, Danabasoglu et al., 2021]), although this is still within the CMIP6 model48 Chapter 2Connecting ocean circulation with ocean heat storageFigure 2.9: Zonally-averaged potential density referenced to 2100 dbar (top) and meridional over-turning circulations (bottom) for the Atlantic (left) and Indo-Pacific (right), calculated from the EC-COv4 ocean reanalysis [Forget et al., 2015]. Panels a and b are reproduced from [Cessi, 2019] whilepanels c and d are reproduced from [Ferreira et al., 2018a].range [Weijer et al., 2020]. Additionally, there is an unusual clockwise overturningcell in the LB northern high latitudes between ⇠50 and 80N (figure 2.8b) that’snot seen in the Indo-Pacific (figure 2.9d) due to interruption from topography. InChapter 3, we associate this cell with polar easterlies encircling the DDrake NorthPole.The clockwise overturning cell in the DDrake southern ocean (south of 35S) is alsounrealistically deep, spanning the full ocean depth (compared with the Deacon Cellin the Southern Ocean, which reaches approximately mid-depth (e.g. [Döös et al.,2008])). The lack of topography in DDrake means that waters must plunge to theocean bottom, as only the linear bottom friction can provide the zonal pressuregradient needed to balance the Coriolis force for the southward geostrophic flow.Finally, the real-world MOC is inherently interbasin in nature (e.g. [Talley, 2013])Chapter 2 49Connecting ocean circulation with ocean heat storageand the DDrake circulation likely is as well (e.g. [Nilsson et al., 2013]), so the SBand LB overturning circulations cannot be cleanly separated. The three-dimensionalpathways of the DDrake circulation may thus be more similar to the real-world thanfigure 2.8c would suggest, but this is obscured in the zonal average. A properassessment of the three-dimensional pathways would require Lagrangian trajectorystudies (e.g. [Van Sebille et al., 2018, Rousselet et al., 2021]) but this isn’t exploredin this thesis.2.4 Forcing experiment: an abrupt CO2-doublingWhether the global ocean circulation will even strengthen or weaken in the futureis not settled (e.g. [Toggweiler and Russell, 2008, Hu et al., 2020, Wunsch, 2020]).What this means for future ocean heat storage is even less clear, and this is tiedto uncertainties regarding the future trajectory of global surface warming. A betterunderstanding of ocean circulation and heat storage is a worthy goal in itself (evenone specific to DDrake), but it also leads to concomitant advances in climate modelformulation, projection accuracy, and eventual policy recommendations.As a starting point, we go back to the canonical climate thought-experiment: adoubling of atmospheric CO2. This idea arguably came to prominence with the‘Charney Report’ in 1979, the first comprehensive assessment of CO2-induced cli-mate change [Charney et al., 1979]. The warming effects of increasing CO2 in theatmosphere, however, were first articulated by Eunice Newton Foote over a hun-dred years prior in 1856 [Foote, 1856]! And the effects of a doubling of CO2 onglobal climate have been thought about as early as 1896 by Arrhenius [Arrhenius,1896], and later revived by Callendar in 1938 [Callendar, 1938]. The notion of aCO2-doubling in climate science has endured to this day (to this thesis!), providingan easy thought-experiment to hone our understanding of the essentials of globalclimate.A CO2-doubling now underpins our concept of equilibrium climate sensitivity (ECS),defined as the expected increase in global mean surface temperature to occur oncea steady state has been reached after a CO2-doubling. Arrhenius estimated an ECSof 5-6C [Arrhenius, 1896], while Callendar’s estimate was around 2C [Callendar,50 Chapter 2Connecting ocean circulation with ocean heat storage1938]. In the Charney Report, the authors state: “We estimate the most probableglobal warming for a doubling of CO2 to be near 3C with a probable error of ±1.5C”[Charney et al., 1979]. This estimate has been lauded as remarkably prescient as, 34years later, the fifth IPCC assessment report (AR5) states: “there is high confidencethat ECS is extremely unlikely to be less than 1C and medium confidence that theECS is likely between 1.5C and 4.5C and very unlikely greater than 6C” [Stockeret al., 2013]. The ECS for DDrake is likely close to 3C, so well within the range forEarth (see Chapter 5).A CO2-doubling has framed our way of thinking about climate change for a very longtime, with many norms and concepts growing out of this initial thought-experiment.It is thus a natural experimental choice for the DDrake configuration, and this thesispresents the first results arising from that scenario for DDrake. Such an idealisedstep-forcing experiment is achieved in DDrake by abruptly changing the longwaveabsorption in the CO2-band in the model atmosphere. This causes an initial top-of-atmosphere forcing of approximately 4.4 W m2 (see Chapter 5), thus mimicking adoubling of atmospheric CO2 (e.g. [Myhre et al., 1998, Byrne and Goldblatt, 2014]).We then run the model from this state for 200 years. The imposed radiative forcingresults in a warming of the climate system as it attempts to reach a new equilibrium.We diagnose the ensuing climate responses relative to the control climate.The rest of this thesis explores how the DDrake model ocean responds to such anabrupt radiative forcing, and what this might say about other climate models, bothsimpler and more complex, as well as the future state of Earth’s oceans.Chapter 2 51Connecting ocean circulation with ocean heat storage52 Chapter 2“In the heat of the sun, the ocean is theboiler and condenser of a gigantic steam en-gine, a weather engine that governs crops,floods, droughts, frosts, hurricanes, at timesas serene as motherhood or as frightening asGod’s wrath.”Jacques-Yves Cousteau, 1981 3Changes in vertical heattransports by ocean circulationunder global warmingThe rate of surface warming depends on the rate of OHU, which, in turn, depends onvertical advective heat transports by ocean circulation. In this chapter, we projectthe DDrake global ocean circulation onto depth-temperature coordinates and de-compose it into three circulation cells: a thermally indirect ‘warm cell’, a thermallydirect ‘cool cell’, and a thermally direct, shallow, ‘tropical cell’. We identify each ofthese cells with distinct locations and physical processes, and diagnose their verticalheat transport (VHT) profiles. We then analyse the circulation responses of eachof these cells to an abrupt CO2-doubling, and the resultant changes in their advec-tive VHTs. We find that the warm cell strengthens and the cool cell weakens (thetropical cell response is negligible). Both these responses lead to an increase in thedownward VHT of ⇠0.2 W m2 in the top 2 km, enabling greater warming of theocean interior. This suggests that changes in ocean circulation will further reinforcethe ocean’s role as a buffer of global surface warming.53Connecting ocean circulation with ocean heat storage3.1 IntroductionVertical advective heat transport by ocean circulation is a key OHU process (e.g.[Gregory, 2000, Exarchou et al., 2015, Kuhlbrodt et al., 2015, Saenko et al., 2021]).In this chapter, we seek a description of OHU due to changes in ocean circulation di-rectly. One approach to this problem is the use of novel overturning stream functionsin thermodynamic coordinates.Overturning stream functions have previously been calculated in density-latitudecoordinates. Doing so reduces potential overestimates of spuriously strong diabaticcirculations such as the Deacon cell in the Southern Ocean (e.g. [Gregory, 2000, Döösand Webb, 1994, Nurser and Lee, 2004]). For example, a horizontal circulation alongan isopycnal surface can show up as a meridional cell when projected into depth-latitude coordinates, if there happens to be a zonal variation in the depth of theisopycnal surface for the northward and southward flow. Such a meridional cellwould vanish in density-latitude coordinates.Another choice is to calculate the overturning in depth-density coordinates. Theprojection into depth-density coordinates allows one to determine whether an over-turning cell is driven by mechanical or thermal means, as work needs to be done ona fluid to lift dense water upwards (e.g. [Nycander et al., 2007]). Stream functionshave also been calculated in depth-temperature and depth-salinity coordinates en-abling analysis of vertical transports [Zika et al., 2013], and even purely thermohalinecoordinates to analyse global water mass transformations [Zika et al., 2012, Dööset al., 2012].As our focus is on vertical heat transports by ocean circulation, the most pertinenttool for our needs is the overturning stream function in depth-temperature coordi-nates and its changes under global warming. Such a study was performed by [Zikaet al., 2015] using the Community Climate System Model 4 (CCSM4) [Gent et al.,2011]. In the depth-temperature diagram, they identify a ‘warm’ cell that fluxesheat downwards, coincident with the AMOC, and a compensating ‘cold’ cell thatfluxes heat upwards, coincident with Antarctic Bottom Water (AABW) formation(as was found previously in [Zika et al., 2013] for the University of Victoria Earth54 Chapter 3Connecting ocean circulation with ocean heat storageSystem Climate Model (UVic ESM)). Under anthropogenic forcing, they find thatthe cool cell collapses while the warm cell continues fluxing heat downwards at anincreasing rate, leading to net ocean heat uptake.In this chapter, we perform a similar study using the MITgcm DDrake configura-tion under an abrupt CO2-doubling, focussing on the top 2 km of the model ocean.Although the deep ocean below 2 km contributes ⇠10% to global OHU (e.g. [Des-bruyères et al., 2016]), we neglect the ocean below 2 km for three principal reasons:1. Warming below 2 km is primarily driven by warming of AABW (e.g. [Purkeyand Johnson, 2013]) — DDrake lacks a realistic Lower Cell of the MOC, sothis process is not captured.2. The ocean below 2 km responds on a multi-centennial to millennial timescale(e.g. [Stouffer, 2004, Yang and Zhu, 2011, Li et al., 2013]) — we are interestedin the transient response of the ocean circulation after 200 years followingthe CO2-doubling, so the slow response below 2 km is not relevant for ourpurposes.3. Bottom topography is increasingly being shown to have a controlling influenceon the abyssal overturning (e.g. [Ferrari et al., 2016, Holmes et al., 2018]) —DDrake has no bottom topography, and so cannot accurately capture circula-tion processes below 2 km.We find similar results to [Zika et al., 2015] whereby a warm cell strengthens anda cool cell weakens, both contributing to an increase in the downward vertical heattransport, enabling greater ocean heat uptake. In section 3.2 we introduce the con-cept of stream functions in thermodynamical coordinates and formally define thethermal overturning circulation. In section 3.3, we examine the DDrake verticalheat balance in the control climate, and decompose the global thermal overturningcirculation into specific circulation features, identifying them with physical circula-tions in geographical coordinates. In section 3.4, we analyse the thermal overturningresponse to an abrupt CO2-doubling and the resultant changes in vertical heat trans-port. We conclude in section 3.5.Chapter 3 55Connecting ocean circulation with ocean heat storage3.2 Stream functions in other coordinatesAn overturning stream function is normally defined as the zonally-integrated merid-ional volume transport (e.g. [Buckley and Marshall, 2016]): (y, z) =1tZ t+ttZ ⌘zZ xExWv dx dz dt (3.1)where v is the meridional velocity averaged over some period of timet, ⌘ is the free-surface height, and xW and xE define the longitudinal positions of the meridionalboundaries enclosing the basin. Here, a positive stream function circulates clockwise,with northward flow at the surface and southward flow at depth. The key conceptualleap to find in other coordinates is that the averaging in equation 3.1 need notsimply take place at constant position (latitude), it could take place over any scalarvariable identifiable with a water parcel.We choose to take a statistical approach to describing the circulation, following[Pauluis et al., 2011] and [Zika et al., 2013]. Generalising the presentation in [Zikaet al., 2013], for some tracer C(x, y, z, t), one can calculate a distribution functionM rC for the transport at concentration C⇤ in a particular direction r using:M rC(C⇤, r) =1tZ t+ttZA(C⇤ C) v · r|r| dA dt (3.2)where is the Dirac delta function, v is the velocity field, t is time, and A is a two-dimensional surface with normal parallel to r through which we wish to calculatethe tracer transport. In equation 3.2, the averaging takes place at constant C, incontrast to equation 3.1 where it takes place at constant y. A volumetric streamfunction can then be calculated by cumulatively integrating the transport up to aparticular value of C: Cr(C, r) =Z C1M rC(C⇤, r) dC⇤ (3.3)M rC is the volume transport per unit tracer, so the actual tracer transport (in C-unitsm3) is then related to M rC and Cr via:56 Chapter 3Connecting ocean circulation with ocean heat storageT rC(r) =Z +11M rC(C⇤, r)C dC =Z +11 Cr(C, r) dC (3.4)For vertical heat transport, the scalar tracer is the potential temperature ✓ anddirection the local vertical k, so that v · k = w, the vertical velocity. In the modelconfiguration used, there is both a resolved velocity w and an eddy-induced velocitywb due to the Gent-McWilliams parameterisation [Gent and Mcwilliams, 1990]. Bothof these are thus incorporated in the distribution function using a residual verticalvelocity wres = w + wb. Equation 3.2 then becomes:M z✓ (✓⇤, z) =1tZ t+ttZ yNySZ xExW(✓⇤ ✓)wres dx dy dt (3.5)where xW , xE, yS, and yN define the boundaries of the horizontal region we wish toconsider. At some depth z, M z✓ is the area integral of the vertical velocity where✓ = ✓⇤. Using equation 3.3, a thermal overturning stream function can then bedetermined from M z✓ by: ✓z(✓, z) =Z ✓1M z✓ (✓⇤, z) d✓⇤ (3.6)This framework can be extended to two (or even n) scalar variables to calculatejoint distributions. For example, this was done for dry entropy and moist entropyin the atmosphere by [Pauluis et al., 2008], who used it to help determine a newpicture of the global atmospheric circulation.In practice, we don’t use wres in equation 3.5 but rather wres hwresi, where hwresiis the time and spatially averaged residual vertical velocity over a model verticallayer (non-zero due to small numerical truncation errors in the calculation). Thisenforces mass conservation at each model vertical level, which ensures that thethermal overturning stream function contains closed streamlines. The global residualvolume flux resulting from hwresi is ⇠0.01 Sv, so is very small compared to thestrength of the circulations considered here.In order to build up distribution functions M zC for the control and perturbed cli-mates of DDrake, we perform two five-year integrations with data outputted everyChapter 3 57Connecting ocean circulation with ocean heat storagefive days: one starting from the end of the control integration, and one starting fromyear 200 following the abrupt CO2-doubling. The overturning stream functions wecalculate for the control and perturbed climates are thus five-year time-means. Wechoose a sampling frequency of five days to ensure a high degree of temporal cor-relation between the sampled fields. We note that [Zika et al., 2013] use monthly-averaged fields, finding a negligible difference between monthly- and five-day-averagederived vertical heat transport. Thus, a five day sampling frequency may be unnec-essarily high, but we have chosen to err on the side of caution here. We have notchecked whether using monthly-averaged fields instead modifies our results.We calculate M z✓ globally and regionally for both the DDrake control and perturbedclimates and use it to diagnose thermal overturning cells. Taking ideas from [Ferrariand Ferreira, 2011], we decompose the global thermal overturning into specific oceancirculation features and examine their responses to the CO2 perturbation, linkingthem to physical processes. By integrating over the thermal overturning cells, weexamine the vertical heat transport responses due to these circulation features anddetermine which features dominate the global response.3.3 Control climate vertical heat balanceBefore looking at the thermal overturning and diving into specific circulation fea-tures, we first consider how the advective transport in DDrake fits into the globalvertical heat balance. We plot the total vertical advective (ADV) and diffusive(DIFF) heat fluxes outputted from the model in figure 3.1. The advective heat fluxis further decomposed into contributions from resolved and eddy-induced circula-tions (due to the GM eddy parameterisation [Gent and Mcwilliams, 1990]).The diffusive flux outputs include contributions from the background uniform verti-cal diffusion (kv = 3⇥ 105 m2 s1), the Redi isopycnal mixing scheme [Redi, 1982],and the strong vertical mixing due to convective adjustment [Klinger et al., 1996].We take the convention that a downward heat flux is positive.As noted in Chapter 2, there will be a small amount of spurious vertical diffusiondue to truncation errors in the advection scheme (e.g. [Griffies et al., 2000]), and58 Chapter 3Connecting ocean circulation with ocean heat storageits magnitude is not known. However, this doesn’t overwhelm the prescribed back-ground vertical diffusion, as we do not need to factor it into the vertical heat balancein order to balance the other processes in the DDrake control climate (black line infigure 3.1).-15 -10 -5 0 5 10 15-1500-1000-5000ADVDIFFSW penetrationADV+DIFF+SWResolved ADVEddy-induced ADVFigure 3.1: DDrake control climate vertical heat balance for advective (ADV) and diffusive (DIFF)processes, and shortwave (SW) radiation penetration (in W m2). A downward heat flux is positive.The zero line is black dotted. Only the top 1.5 km are shown for clarity.In figure 3.1, we see that below ⇠150 m, the advective and diffusive fluxes balanceeach other closely (red and blue curves). This supports the classic vertical ‘upwelling-diffusion’ balance first popularised by [Munk, 1966], where a cooling effect from theupwelling of abyssal waters is balanced by a downward diffusion of heat (although,here, the upward advective flux is due to the sum of the resolved and eddy-inducedcirculations i.e. the residual circulation).In the top 150 m, the advective and diffusive processes sum to produce an upwardheat flux. The upward diffusive flux is most likely due to the convective mixing (e.g.[Gnanadesikan et al., 2005]). These upward fluxes are balanced by shortwave radia-tion penetrating below the surface (cyan curve). This penetration is an exponentialdecay, based on a two-band solar absorption model from [Paulson and Simpson,1977].Notably, advection by the resolved circulation is associated with a downward heatflux (magenta curve), confirming the findings of more modern modelling studiesChapter 3 59Connecting ocean circulation with ocean heat storagesuch as [Gregory, 2000, Zika et al., 2013, Exarchou et al., 2015, Saenko et al.,2018]. However, the (parameterised) eddy-induced circulation (green curve) morethan compensates this downward heat flux, leading to a net upward advective heatflux (red curve). This is similar to the vertical heat balance seen in other models,where the main process transporting heat downwards is the resolved advection, whilethe bulk of the upward heat transport is due to parameterised eddy advection (e.g.[Gnanadesikan et al., 2005, Hieronymus and Nycander, 2013, Griffies et al., 2015]).As pointed out by [Hieronymus and Nycander, 2013], isopycnal mixing by eddiescould also be interpreted as an advective flux, like the eddy-induced circulation.Thus, it is possible to add its contribution to the residual circulation, resultingin a ‘super-residual’ transport (SRT), first suggested by [Kuhlbrodt et al., 2015].The SRT framework was used by [Dias et al., 2020a], who found that the SRT heatbalance has two depth-dependent regimes. In the upper-ocean (0-700 m), downwardSRT heat flux is balanced by convection and near-surface mixing, while below 700m, upward SRT heat flux is balanced by diapycnal diffusion. We do not considerthe effects of the SRT here.It is the net upward advective heat flux we wish to zoom in on (i.e. that due tothe residual circulation, the sum of the resolved and eddy-induced circulations),and subsequently decompose its structure into particular ocean circulation compo-nents using ✓z. To check the soundness of this premise, we compare these model-outputted vertical advective fluxes (direct calculation) with those inferred from ✓z(an indirect diagnostic). Using equation 3.4, the vertical heat transport Hz✓ is givenby:Hz✓(z) = ⇢0 cpZ +11 ✓z(✓, z) d✓ (3.7)where ⇢0 is a reference seawater density, and cp is the specific heat capacity ofseawater at constant pressure. The resolved, eddy-induced, and net Hz✓ profiles areplotted in figure 3.2 as dash-dotted lines, alongside the same advective terms fromfigure 3.1 as solid lines for comparison.The correspondence between the model-outputted terms and the Hz✓ calculations isfairly good. The overall shapes of the profiles are well-captured, with discrepancies60 Chapter 3Connecting ocean circulation with ocean heat storage-8 -6 -4 -2 0 2 4 6-1500-1000-5000ADVResolved ADVEddy-induced ADVNet H zResolved H zEddy-induced H zFigure 3.2: Comparison of model-outputted vertical advective heat fluxes (solid lines) with thosecalculated from the thermal overturning ✓z (dash-dotted lines), for the DDrake control climate. Thezero line is black dotted. Only the top 1.5 km are shown for clarity.within ⇠0.5 W m2. The Hz✓ calculation systematically underestimates the netupward advective heat flux, due to an overestimation of the downward resolved heatflux in the top ⇠400 m, and an underestimation of the upward eddy-induced heatflux below ⇠400 m depth. This could be due to the shortness of the M z✓ averagingperiod (five years), or a sensitivity to the ✓⇤ bin size. Nevertheless, the mismatchbetween the solid and dash-dotted lines in figure 3.2 is small enough for our purposes.3.3.1 The DDrake thermal overturning circulationWe first look at the control DDrake overturning circulation ctrlres in geographical(y, z) coordinates. This is the zonally-integrated meridional volume transport asdefined in equation 3.1, where for v we use the model vres = v+vb. This is shown infigure 3.3, calculated for the different basins of DDrake in the control climate. Thezonally-averaged ✓ contours are superimposed as black dashes.Due to the DDrake ocean being only 3 km deep, the overturning in the small basin(SB) reaches almost the full depth of the basin (figure 3.3c). Although the real-worldAMOC also reaches a depth of ⇠3 km, this is approximately mid-depth in the At-lantic (e.g. [Buckley and Marshall, 2016, Cessi, 2019]). Thus, rather than the Upperand Lower MOC cells stacking on top of each other like in other zonally-averagedviews of the overturning circulation (e.g. [Lumpkin and Speer, 2007, Marshall andSpeer, 2012]), the SB and LB MOC cells in DDrake overlap and unrealistically can-Chapter 3 61Connecting ocean circulation with ocean heat storage10101212121215152020 2530-50 0 50-3000-2500-2000-1500-1000-5000-40-30-20-10010203040 4 68810101212131314161820-80 -70 -60 -50 -40-3000-2500-2000-1500-1000-5000-20-15-10-505101520131313141414151520202530-20 0 20 40 60 80-3000-2500-2000-1500-1000-5000-40-30-20-1001020304011121213131414151520202530-20 0 20 40 60 80-3000-2500-2000-1500-1000-5000-40-30-20-10010203040ca bdFigure 3.3: DDrake control climate zonally-averaged ctrlres(colour, in Sv) and ✓ distributions (blackdashed lines, in C) for the (a) global, (b) southern ocean, (c) small basin, and (d) large basin re-gions. Contour interval is 5 Sv for panels a, c, and d. For panel b, the interval is 2 Sv; also, note thedifferent colour-bar scale. Red cells circulate clockwise while blue cells circulate anti-clockwise.62 Chapter 3Connecting ocean circulation with ocean heat storagecel at depth in the global-average (figure 3.3a). This means that the global pictureis dominated by the clockwise overturning in the SB, and there is no realistic LowerCell of the MOC, as noted in Chapter 2.In both the small and large basins there are shallow tropical overturning cells withdownwelling in the subtropical gyres at ⇠30S and 30N and upwelling at the equa-tor. In the large basin, these shallow cells are deeper and stronger, and there is alsoa fairly deep (⇠1 km) clockwise cell at ⇠60N associated with the subpolar gyre(figure 3.3d). Such a deep subpolar gyre overturning does not exist in the present-day Indo-Pacific due to interruption from topography (e.g. [Talley, 2013, Newsomand Thompson, 2018, Ferreira et al., 2018a]), so the relevance of this LB cell to thereal world is questionable.In the DDrake southern ocean, the overturning is dominated by a single clockwisecell, mimicking the Deacon cell, with wind-driven upwelling ⇠55S and subsequentdownwelling in the convergence to the north (figure 3.3b). As discussed in Chapter2, this overturning cell is unrealistically deep, as waters must plunge to the oceanbottom where only the linear bottom friction can provide the zonal pressure gradientneeded to support the southward geostrophic flow. The lack of any Lower MOC cellis also readily apparent from this panel (figure 3.3b). In the DDrake southern oceanhigh latitudes, there is a weak anticlockwise circulation due to convergence anddownwelling at the South Pole. Of course, such a circulation does not exist in thereal world due to the presence of Antarctica.We will refer back to these circulation features when discussing the thermal over-turning in DDrake. But we should keep in mind that, throughout, we are discussingthe circulation of the DDrake model, which should not be confused with the realworld. The caveats above surrounding ctrlres will also carry over to the DDrake ther-mal overturning. There is no bottom topography, no bottom-enhanced mixing, andno realistic Lower MOC Cell, so we restrict our discussion to the top 2 km of themodel ocean.The DDrake control climate global ✓z, calculated from M z✓ , is shown in figure 3.4.By substituting wres for w or wb in equation 3.5, we decompose the circulation intoresolved (panel a) and eddy-induced (panel b) components, respectively. Here, redChapter 3 63Connecting ocean circulation with ocean heat storagecells circulate clockwise and transport heat downwards, while blue cells circulateanti-clockwise and transport heat upwards. The resolved circulation is dominatedby a clockwise cell for water masses between ⇠12C and 30C (figure 3.4a). Theeddy-induced circulation is dominated by an anti-clockwise cell occupying the sametemperature range (figure 3.4b). These cells cancel each other out, leaving a weaker,residual net circulation with a different structure (figure 3.4c).Figure 3.4: Control climate DDrake global thermal overturning circulations (in Sv). (a) Resolvedcirculation ✓z, (b) eddy-induced circulation 0✓z, and (c) net circulation ✓z = ✓z + 0✓z. Red cellscirculate clockwise while blue cells circulate anti-clockwise.In this net circulation, we identify a ‘warm’ (clockwise) cell at ⇠500 m depth, occu-pying the temperature range 20-30C fluxing heat downwards (‘warm cell’ in figure3.4c). This is compensated by a ‘cool’ (anti-clockwise) cell in the range 10-20C64 Chapter 3Connecting ocean circulation with ocean heat storagethat extends down to ⇠1.5 km depth, which fluxes heat upwards (‘cool cell’ in fig-ure 3.4c). The warmest water masses from ⇠25-30C are associated with a shallow,cool cell confined to the top ⇠200 m (‘tropical cell’ in figure 3.4c). We will referto this cell as the ‘tropical’ cell, as only at low ‘tropical’ latitudes do water massesexist with such high temperatures near the surface (see figure 3.3). There are alsotwo other weak warm cells: one at the surface ⇠5-10C, and one at a depth of ⇠2km near 13C.By using a temperature threshold, we define the ‘cool cell’ as all negative streamfunction values with ✓ < 20C, and the ‘tropical cell’ as all negative values with✓ > 20C. The ‘warm cell’ is defined as all positive stream function values where✓ > 15C. This decomposition is shown in figure 3.5. Choosing this stream functiondecomposition simplifies the analysis, but neglects the other two weak warm cellsseen in figure 3.4c. We associate the weak warm cell at the surface with the southernocean, and the deep warm cell ⇠2 km depth with a SB/SO exchange associated withthe SBMOC (seen later).-504-3 -20-20-1-100055510-20-20-15-15-15-10-10-10-10-10-5-5-5-5-50000005 10 15 20 25 30 35-2000-1500-1000-5000Tropical cellWarm cellCool cellFigure 3.5: DDrake global net control thermal overturning circulation ( ✓z, in Sv), decomposed intoa warm cell (red solid, for ✓ > 15C), a cool cell (blue dashed, for ✓ < 20C), and a tropical cell(green dashed, for ✓ > 20C). Dashed lines denote negative values and circulate anti-clockwise, whilesolid lines denote positive values and circulate clockwise. Only the top 2 km are shown.We now proceed to identify the cells in figure 3.5 with particular ocean circulationfeatures seen in figure 3.3. In order to geographically locate the warm, cool, andtropical cells, we repeat the M z✓ calculation basin-wise. A similar procedure wasdone by [Saenko et al., 2018], who calculated ✓z for the Northern and SouthernChapter 3 65Connecting ocean circulation with ocean heat storageHemispheres separately, using the Nucleus for European Modelling of the Ocean(NEMO) version 3.4 ocean model [Madec et al., 2017]. Such a partitioning does notensure closed circulation cells when considering the different ocean basins separately,and so it cannot be used to rigorously attribute heat transport to a particularcirculation. However, it does provide insight into the individual basin contributionsto the global ✓z picture (figures 3.4c and 3.5). These net thermal overturningcirculations for each of the DDrake basins are shown in figure 3.6.From figure 3.6, we see that the global thermal overturning structure is dominated bythe large basin. This is in contrast to figure 3.3 where, in geographical coordinates,the global overturning structure is dominated by the small basin. Comparing panelsa and d in figure 3.6 makes clear that the warm cell in figure 3.5 is almost exclusivelymade up of circulations in the large basin. Referring back to figure 3.3d, we thusattribute the warm cell to a wind-driven inter-gyre communication between thesubtropical and subpolar gyres in the large basin. Warm water (⇠25C) downwellsat the poleward side of the subtropical gyre near 40N, crossing isotherms to reach⇠15-20C water at a depth of approximately 750 m. This water subsequently upwellsin the subpolar gyre ⇠50N, before warming back up to 25C again as it approachesthe convergence north of the centre of the subtropical gyre once more.For the tropical cell, we compare panels a, c, and d in figure 3.6 and thus attributeit to shallow circulations existing in both the small and large basins. In figures 3.3cand d, these shallow circulations are the tropical overturning cells already identified,associated with downwelling at the centres of the subtropical gyres at approximately30S and 30N and subsequent upwelling at the equator.The cool cell in figure 3.5 from 10-20C is trickier to identify as, from figures 3.6cand d, we see that it is a combination of different circulations in the small and largebasins. The small basin thermal circulation (figure 3.6c) penetrates almost the fullocean depth, downwelling at ⇠13C. This is coincident with the SBMOC (figure3.3c) where water downwells at approximately 14C and must cool to ⇠13C andcolder, poleward of 60N. This water keeps its temperature throughout much of theocean interior below 1 km depth. It then warms along its upwelling branch as itreturns northwards, before finally cooling again as it approaches the small basin66 Chapter 3Connecting ocean circulation with ocean heat storageFigure 3.6: DDrake net control thermal overturning circulations ( ✓z, in Sv) calculated for the (a)global, (b) southern ocean, (c) small basin, and (d) large basin regions. The large basin dominatesthe global ✓z structure. Red cells circulate clockwise while blue cells circulate anti-clockwise.Chapter 3 67Connecting ocean circulation with ocean heat storagenorthern high latitudes.The large basin thermal circulation (figure 3.6d) penetrates to ⇠2 km and not thefull depth, so we rule out the ‘abyssal upwelling’ anti-clockwise cell seen in figure3.3d as part of the cool cell. The fact that, in the LB, ctrlres reaches the full depth(figure 3.3d) but ✓z only reaches ⇠2 km depth (figure 3.6d) indicates that waterparcels circulating below 2 km depth in the LB hold their temperature (i.e. do notwarm up or cool down). So, these deep water parcels do not contribute to the LBthermal overturning.The only other candidate for the cool cell in the LB is the deep clockwise cell associ-ated with the subpolar gyre at the large basin northern high latitudes (figure 3.3d).Here, cool ⇠10C water downwells at the highest northern latitudes (⇠70N) due topolar easterly winds encircling the North Pole. The weak stratification enables thewater to plunge to depths of 1-2 km before subsequently upwelling and warming to⇠15C in the subpolar gyre near 50N.In the DDrake southern ocean, there is a single warm cell at ⇠15-20C (figure3.6b), coincident with the clockwise ‘Deacon cell’-like circulation seen in figure 3.3b.Relatively warm water near 20C downwells in the circumpolar convergence ⇠30Sand cools to around 15C, before upwelling due to Ekman suction near 55S. Thewater then warms during the northward Ekman transport before downwelling again.This clockwise Ekman overturning is counteracted by the parameterised eddies,whose collective action induces an anti-clockwise circulation that acts to slump theisopycnal surfaces. The resultant clockwise overturning in the DDrake southernocean (figure 3.3b) is thus approximately half as strong as it would be without theeddies.The southern ocean warm cell does not affect the global ✓z structure much, apartfrom the weak extension into 5-10C water, which explains the weak surface warmcell (figure 3.6a). The weak deep warm cell ⇠2 km depth remains to be explainedfully.We have now identified the tropical, warm, and cool cells in figure 3.5 with particularocean circulation features in the small and large basins. In summary:68 Chapter 3Connecting ocean circulation with ocean heat storage• Tropical cell: Shallow overturning circulations in both the small and largebasins confined to the top ⇠200 m, associated with downwelling in the sub-tropical gyres at approximately 30S and 30N and subsequent upwelling atthe equator.• Warm cell: A wind-driven inter-gyre circulation in the large basin associatedwith downwelling in the subtropical gyre ⇠40N and upwelling in the subpolargyre ⇠50N.• Cool cell: A combination of the deep interhemispheric MOC in the smallbasin and an overturning gyre circulation in the large basin where polar watersdownwell ⇠70N and upwell in the subpolar gyre ⇠50N.This type of thermal overturning decomposition has only been done for a handful ofother ocean circulation models. In the intermediate complexity climate model UVicESM, [Zika et al., 2013] did a similar three-cell decomposition for the global ✓zinto ‘surface’, ‘deep’, and ‘bottom’ cells. They identify the surface cell with shallowoverturning circulations associated with the subtropical gyres, just as we have donewith the tropical cell of DDrake. Their deep cell is found to be a combination ofthe AMOC, the Southern Ocean overturning, and an Antarctic Intermediate Watercirculation in the Pacific. The bottom cell is identified with the AABW circulation,but no such circulation exists in DDrake.In a follow-up study, [Zika et al., 2015] calculated ✓z for the coupled climate modelCCSM4 [Gent et al., 2011] and identified ‘cold’ and ‘warm’ cells, similar to the‘bottom’ and ‘deep’ cells found in UVic ESM. Like in [Zika et al., 2013], they identifythe cold cell with AABW formation and the warm cell with the AMOC and SouthernOcean overturning. In addition to these cold and warm cells, they again find a coolsurface cell (similar to the surface cell in UVic ESM and the tropical cell in DDrake),but also a secondary warm cell below this at ⇠500 m depth. This secondary warmcell is similar to the warm cell identified in DDrake in figure 3.5 and so is possiblylinked to wind-driven gyre circulations in CCSM4, however this is not commentedon in their paper.The deep cell in [Zika et al., 2013] and the warm cell in [Zika et al., 2015] are bothChapter 3 69Connecting ocean circulation with ocean heat storagelinked to the AMOC, and both flux heat downwards. An AMOC that contributesto the net downward transport of heat was also reiterated by [Saenko et al., 2018]for the NEMOv3.4 ocean model [Madec et al., 2017]. This is in contrast to DDrake,whose AMOC analogue (the SBMOC) fluxes heat upwards (figure 3.6c), contributingto the cool cell in figure 3.5. The only cell in DDrake that fluxes heat downwards isthe warm cell, which we conclude is almost entirely due to wind-driven circulationsin the large basin (see section 3.3.3 and figure 3.6d).A consensus may be emerging that model AMOC cells in fact flux heat downwards,however, this kind of analysis will need to be done for more coupled climate modelsbefore any firm conclusions can be made. Whether a control model AMOC cellfluxes heat downwards or upwards may directly impact transient ocean heat storageresponses (we touch on this in the next chapter).Since most of the ocean’s density variations are controlled by temperature, fluxingheat downwards is synonymous to pushing light fluid downwards or lifting dense fluidupwards. This requires mechanical forcing from the winds or tides, for example (seesection 3.3.3). The theory of [Nikurashin and Vallis, 2012] suggests that the AMOCis primarily driven by winds over the Southern Ocean, given typical parameter rangesfor the present-day ocean. This depends on the adiabaticity of the AMOC, andrequires a set of isopycnals to outcrop in both the Southern Ocean and NorthernHemisphere (i.e. in the upwelling and downwelling regions) [Wolfe and Cessi, 2011,Johnson et al., 2019].However, dynamical theories such as [Nikurashin and Vallis, 2012] and those re-viewed in [Johnson et al., 2019] are based on density (buoyancy), so we cannotseparate the roles of temperature and salinity. This makes it difficult to infer thevertical heat transport associated with the AMOC. In particular, the key down-welling regions of the AMOC may occur in ‘beta oceans’ where the upper-oceanstratification is controlled by salinity, not temperature (e.g. [Carmack, 2007, Pe-tit et al., 2020, Caneill et al., 2022]). So, it is not especially clear that even awind-driven, adiabatic AMOC should transport heat downwards.Whether the real-world AMOC is essentially adiabatic, and whether it advectivelyfluxes heat downwards (warm cell) or upwards (cool cell) in the net is still an open70 Chapter 3Connecting ocean circulation with ocean heat storagequestion [Johnson et al., 2019]. Vertical heat fluxes estimated from observationsshow that the dominant contribution is a downward heat flux in the Southern Oceanfrom the upwelling of Circumpolar Deep Water [Cummins et al., 2016]. This esti-mate neglects the convective downwelling from NADW formation, however. So, evenif we assume that the dominant downward heat flux in the Southern Ocean is solelyassociated with the AMOC, the estimate from [Cummins et al., 2016] doesn’t tell uswhether the upward heat flux associated with NADW formation would cancel this.Furthermore, the connection between the downwelling and upwelling regions of theAMOC is unclear due to complex interbasin pathways (e.g. [Talley, 2013]).So, it is unclear whether the real-world AMOC is thermally direct or indirect. Es-tablishing this fact is important for projections of transient climate change, as theAMOC is projected to weaken with global surface warming (e.g. [Gregory et al.,2005, Weaver et al., 2012, Weijer et al., 2020]). Recent proxy observations are alsoconsistent with a weakening response (e.g. [Rahmstorf et al., 2015, Caesar et al.,2018, Zhu and Liu, 2020]). How this weakening affects the corresponding warm orcool cell in temperature space is particularly uncertain; it is possible that the weak-ening may not even affect the vertical heat transport, as the heat transport dependson the temperature structure as well as the strength of the circulation.In the next chapter, we will show that the SBMOC heat flux is well described bythe control overturning, as the SBMOC weakening terms contributing to the heatflux cancel each other out [Shatwell et al., 2020]. A similar result was found by [Zikaet al., 2015] for CCSM4 under a global warming scenario. They well approximatedthe heat transport carried by the warm cell considering only the changing temper-ature difference between the downwelling and upwelling branches and assuming noweakening. We will explore the CO2-doubling vertical heat transport (VHT) re-sponses for DDrake in section 3.4. We will look not just at the SBMOC (which isonly a part of the cool cell), but at all the DDrake thermal overturning cells. Butfirst, we examine how the ✓z cells flux heat vertically in the control climate.Chapter 3 71Connecting ocean circulation with ocean heat storage3.3.2 Vertical heat transportsHaving identified the cells in ✓z with particular ocean circulations, we now look atthe vertical heat transports Hz✓ carried by each of them (equation 3.7). These areshown in figure 3.7.-4 -3 -2 -1 0 1-2000-1500-1000-5000Tropical cellWarm cellCool cellCell sumNet globalFigure 3.7: Vertical heat transport profiles for the three thermal overturning cells of DDrake in thecontrol climate (in W m2). Here, a positive heat transport is directed downwards. The solid blackline is the sum of the red, blue, and green lines. The dash-dotted black line is the total global advec-tive VHT, accounting for cells not included in the ✓z three cell decomposition. Only the top 2 kmare shown.As expected, the warm cell fluxes heat downwards, while the cool and tropical cellsflux heat upwards. The warm cell VHT peaks at ⇠0.6 W m2 at 200 m depth. Thecool cell VHT is almost twice as strong, peaking at ⇠ 1 W m2 a little deeper, near300 m depth. The deeper vertical extent of the cool cell means that below around600 m depth, the net advective vertical heat transport is supplied almost entirely byit. The tropical cell completely dominates the upward vertical heat transport nearthe surface, peaking near 3 W m2.The solid black line in figure 3.7 is the total VHT supplied by the tropical, warm,and cool cells. Also plotted is the net global advective VHT (black dash-dotted line,equivalent to the red dash-dotted line in figure 3.2). Below around 1.5 km depth,there is actually a net downward VHT, not captured by the cell sum. Nevertheless,the differences between the black solid and dash-dotted lines are negligible, lendingsupport to our chosen ✓z three-cell decomposition.72 Chapter 3Connecting ocean circulation with ocean heat storage3.3.3 Thermally or mechanically driven?Before assessing how the three DDrake thermal overturning cells respond to anabrupt radiative forcing in the CO2-doubling experiment, we first take a brief forayinto their possible driving mechanisms. Knowing whether a cell is primarily ther-mally or mechanically driven greatly aids a physical description of its response.It’s been over a hundred years since Sandström first postulated his famous theo-rem [Sandström, 1908]. He concluded that a thermally-forced, steady, overturningcirculation could only be maintained if the heating occurred at a greater depth (pres-sure) than the cooling. In this way, the circulation is able to generate the mechanicalenergy necessary to overcome frictional dissipation. The fact that a vigorous over-turning circulation is observed in the ocean, which is primarily heated and cooledat the surface, stands in flat contradiction to this theorem.For a long time oceanographers assumed that a significant part of the MOC wasforced by buoyancy fluxes at the surface, namely, localised surface cooling feedingthe downwelling branch, and surface heating and subsequent diffusion feeding theupwelling branch (e.g. [Wyrtki, 1961, Munk, 1966]). The problem with cold, denserwater going down and warm, lighter water going up is that this results in a lowering ofthe centre-of-mass and hence a release of gravitational potential energy (GPE). Thissink of GPE requires an external energy source to sustain it, or else the circulationwould slow to a halt with cold, dense water filling the abyss (see, for example,Chapter 5 in [Huang, 2010]).As discussed in Chapter 1, there are two main upwelling pathways by which densewater can be brought back to the surface: one facilitated by surface winds, andone facilitated by deep diapycnal mixing. For the first pathway, it is the winds thatprovide the mechanical (kinetic) energy source, drawing deep waters up adiabaticallyalong density surfaces via Ekman suction (e.g. [Toggweiler and Samuels, 1998, Webband Suginohara, 2001, Wolfe and Cessi, 2011, Marshall and Speer, 2012]). Butthere is a wider array of mechanical energy sources involved that drive diapycnalmixing, including the winds, tides, and geothermal heating (e.g. [Munk and Wunsch,1998, Wunsch and Ferrari, 2004]), and possibly marine life (e.g. [Dewar et al., 2006]).Chapter 3 73Connecting ocean circulation with ocean heat storageThis array of mechanical energy sources then generates internal waves through avariety of mechanisms (see, for example, the review by [Whalen et al., 2020]), in-cluding wind-driven near-inertial waves (e.g. [Plueddemann and Farrar, 2006]),internal tides (e.g. [Nycander, 2005]), and lee waves from abyssal flows imping-ing on topography (e.g. [Wright et al., 2014]). Although the breaking of internalwaves provides a significant portion of the energy needed to lift dense waters up-wards (e.g [Nikurashin and Ferrari, 2013, Kunze, 2017]), observations reveal thatmixing is strongly enhanced above rough bottom topography (e.g. [Polzin et al.,1997, Garabato et al., 2004, Waterhouse et al., 2014]). This last fact is crucial, asbottom-enhanced mixing drives sinking and not upwelling in the stratified oceaninterior [Ferrari et al., 2016].The new paradigmatic view is that diapycnal upwelling of waters does not occur inthe interior, but instead is confined to thin, turbulent, boundary layers along slopingbottom topography [Ferrari et al., 2016]. This has led to new studies investigatingthe influence of topography on the abyssal overturning circulation (e.g. [McDougalland Ferrari, 2017, Mashayek et al., 2017, Holmes et al., 2018, Drake et al., 2022]).These developments are not especially relevant for DDrake, as the configuration hasno bottom topography. As such, we limit the discussion to the top 2 km, so therelevant source of diapycnal mixing represented in DDrake is the breaking of internalwaves in the ocean interior.Nevertheless, this new paradigm (that bottom-enhanced diapycnal mixing drivessinking not rising in the interior) does not say anything about the fundamentaldriver/energy source of the diapycnal mixing. The mechanical energy source isstill likely partitioned between the winds and tides, for the most part (e.g. [Munkand Wunsch, 1998, Wunsch and Ferrari, 2004, Kuhlbrodt et al., 2007, Ferrari andWunsch, 2009]).However, it is still not settled whether the ocean circulation is essentially a mechan-ically driven system, and the buoyancy-driven view may be making a comeback.For example, [Saenz et al., 2012] show in an idealised MITgcm configuration thatbuoyancy forcing and wind forcing each have a positive feedback on the other, soneither forcing can be isolated completely. In relation to GPE, [Butler et al., 2013]74 Chapter 3Connecting ocean circulation with ocean heat storagefind that surface buoyancy fluxes are a comparable GPE source to diapycnal mixingin HadCM3, while [Zemskova et al., 2015] find that diapycnal mixing contributesnegligibly to GPE in ECCO2.Idealised simulations have revealed that important circulation features such as theACC [Hogg, 2010] and ocean gyres [Hogg and Gayen, 2020] can be driven by surfacebuoyancy forcing. It’s possible that buoyancy forcing is a comparable driver of theocean circulation as the winds and tides (see, for example, the recent review by[Gayen and Griffiths, 2022]), a suggestion that has been supported by theoreticalenergetic arguments (e.g. [Hughes et al., 2009, Tailleux, 2009, Tailleux, 2010]).So, the current picture surrounding drivers of ocean circulation is nuanced andcomplex, and we are certainly not going to solve it here. But a useful diagnosticfor ocean models was put forward by [Nycander et al., 2007]. They suggest using astream function in depth-density coordinates as an analysis tool to help identify thedriving mechanisms of overturning cells. They show that the integral of this streamfunction gives the work W performed by the circulation:W = gZ Z ⇢0z dz d⇢0 (3.8)where g is the acceleration due to gravity and ⇢0 is the density anomaly away froma reference density ⇢0. For stationary circulations, they show that W is essentiallyequivalent to the rate of conversion from potential (GPE) to kinetic energy (KE).This means that if the sign of an overturning cell is positive, potential energy isbeing converted into kinetic energy. The cell is termed ‘thermally direct’, and couldbe partly thermally driven and partly mechanically driven. However, if the sign ofa cell is negative, kinetic energy is being converted into potential energy, and thuswork is being performed on the fluid. Here, the cell is termed ‘thermally indirect’and it is purely mechanically driven.We thus calculated M z0 (using the potential density 0) to find the global 0zoverturning for the DDrake control climate. This is shown in figure 3.8a. Comparingwith figure 3.5, we again identify the same ‘tropical’, ‘warm’, and ‘cool’ cells as foundin the ✓z overturning (figure 3.8b). We define the tropical cell as all positive valueswith 0 < 24 kg m3, the cool cell as all positive values with 0 > 24 kg m3,Chapter 3 75Connecting ocean circulation with ocean heat storage-10-555555 5552525 25250000 000000020 21 22 23 24 25 26-3000-2500-2000-1500-1000-5000-50-40-30-20-1001020304050 202020 4040 6060 80-10-50 0001010101010202020303020 21 22 23 24 25 26-3000-2500-2000-1500-1000-5000Tropical cellWarm cellCool cella bFigure 3.8: DDrake control global net overturning circulation in depth-density coordinates, as afunction of potential density 0. (a) The net 0z overturning (in Sv) and (b) its decomposition intoa warm cell (red dashed, for 0 < 25.2 kg m3), a cool cell (blue solid, for 0 > 24 kg m3), and atropical cell (green solid, for 0 < 24 kg m3). In panel a, red cells circulate clockwise and blue cellscirculate anti-clockwise. In panel b, solid lines denote positive values and circulate clockwise, whiledashed lines denote negative values and circulate anti-clockwise.and the warm cell as all negative values where 0 < 25.2 kg m3 (this last choiceeliminates the aforementioned ‘deep warm cell’ near 2 km depth).Being negative, the warm cell in figure 3.8b is unambiguously mechanically driven,which supports our previous conclusion that this is a wind-driven inter-gyre circula-tion in the large basin. Here, wind-driven Ekman pumping pushes light fluid downin the subtropics, and pulls dense fluid up in subpolar latitudes. Such a circulationis put forward as the mean contribution to energy transfer from KE to GPE in theupper ocean in [Ferrari and Wunsch, 2009].By integrating over each of the cells in figure 3.8b, we find estimates for the workperformed (or rate of energy conversion) by each corresponding circulation (equation3.8). The warm cell performs 38 GW of work i.e. it requires 38 GW of externalmechanical forcing in order to sustain it. The tropical cell converts GPE to KEat a rate of 129 GW, and the cool cell at a rate of 250 GW. The total GPE toKE conversion rate (including the ‘deep warm cell’) is 337 GW. This is significantlylarger than what [Nycander et al., 2007] found (94 GW) for the eddy-resolving OceanCirculation and Climate Advanced Model (OCCAM), but there are too many modeldifferences with DDrake to find a clear reason for this.We note that the expression for the work performed in equation 3.8 is strictly onlycorrect for the in-situ density and not the potential density, although the choice76 Chapter 3Connecting ocean circulation with ocean heat storageof potential density gives a much clearer visualisation of the circulation [Nycanderet al., 2007]. Hence, our calculated energy conversion rates for the warm and coolcells may be underestimates and so shouldn’t be taken at face value. Nevertheless,their sign tell you whether they are purely or partly mechanically-driven, and theirvalue give a qualitative indication of their relative importance within the globalcirculation.3.4 Perturbed climate vertical heat balanceWe now turn our attention to the change in the vertical heat balance after 200years due to an abrupt CO2-doubling in the DDrake perturbed climate. The strongincrease in net surface heat flux must lead to ocean heat uptake (OHU), but howthis heat uptake is partitioned between the various processes is far from obvious.The profiles for the advective and diffusive OHU processes (along with shortwavepenetration) for both the DDrake control and perturbed climates are shown in figure3.9a. The difference between the perturbed and control climates, i.e. the anomalousvertical heat flux, is shown in figure 3.9b.-15 -10 -5 0 5 10 15-2000-1800-1600-1400-1200-1000-800-600-400-2000ADVDIFFSW penetrationADV+DIFF+SWResolved ADVEIA-1 -0.5 0 0.5 1 1.5-2000-1800-1600-1400-1200-1000-800-600-400-2000ADVDIFFSW penetrationADV+DIFF+SWResolved ADVEIAa bFigure 3.9: (a) DDrake control climate (solid lines) and perturbed climate (dashed lines) verticalheat balance for advective (ADV) processes split into resolved and eddy-induced advection (EIA),diffusive (DIFF) processes and shortwave (SW) radiation penetration (in W m2). (b) Change invertical heat flux due to the CO2-doubling perturbation (difference between the dashed and solid linesin panel a). A downward heat flux is positive. Only the top 2 km are shown.The anomalous vertical heat fluxes are approximately an order of magnitude smallerthan the control climate values, with the total peaking near +0.7 W m2 in the top200 m (figure 3.9b). They are overwhelmingly positive i.e. downward, as expected,Chapter 3 77Connecting ocean circulation with ocean heat storagewith the sharpest responses close to the surface. Changes in diffusion (includingconvective mixing) dominate over most of the water column; only in a slim bandnear 1 km depth do the advective processes outcompete the diffusive processes.Interestingly, the resolved advection (magenta line) shows a cooling response in theupper ocean. It changes sign at around 500 m, peaking near 1 km depth. Thiscontribution from the resolved circulation means that the shape of the advectiveresponse (red line) is markedly different from the diffusive response (blue line),having an almost flat profile near +0.25 W m2 between 200 m and 1 km depth.We now seek to explain the shape of this advective response using the thermaloverturning circulations.3.4.1 Changes in ocean circulations and the associated vertical heattransportsThermal overturning responsesFigure 3.10 shows the control climate, perturbed climate, and difference plots for theDDrake global thermal overturning circulation. The -notation in these differenceplots refers to a difference between the perturbed and control climate 5-year time-means (200 years apart from each other). The figure shows a clear strengthening ofthe warm cell, and a weakening of the cool cell. There is also a marked strengtheningof the deep warm cell, now near 1.5 km depth (figure 3.10b).The weakening of the cool cell is not limited to either the small basin or the largebasin. The weakening is due to a combination of the small basin MOC weakening(explored further in Chapter 4), and a weakening of the large basin subpolar gyreoverturning. The small basin MOC weakening (figure 3.11) is caused by a weakeningof the deep convection at high latitudes. This less vigorous convection is due to anincreasingly stable stratification due to the amplified warming over polar regions.The weakening of the subpolar gyre overturning in the large basin (figure 3.12) islikely related to a weakening of the polar easterly winds near 60N.A plot of the zonally-averaged zonal surface wind stress, and its response throughoutthe 200 years, is shown in figure 3.14. Each of the curves corresponds to a 50-year78 Chapter 3Connecting ocean circulation with ocean heat storage-15-15-10 0-10-10-5-5-5-5 -55 55 55100000000000005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-30 -20 -10 0 10 20 30-10 -10-10-10-5-5-5 -55 5 5555101010150000000000 005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-30 -20 -10 0 10 20 30-10-5-5-5-55555555510 10101010000000000000 00 005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-30 -20 -10 0 10 20 30-50-4-30-2-100055 510-20-15-10-10-10-10-5-5-5-5000005 10 15 20 25 30 35-2000-1500-1000-5000Tropical cellWarm cellCool cell-50-4-302-10 055510 15-15-10-10-10-5-5-5000005 10 15 20 25 30 35-2000-1500-1000-5000Tropical cellWarm cellCool cell-100 01055555555101010-5000000-5-500 00005 10 15 20 25 30 35-2000-1500-1000-5000Tropical cellWarm cellCool cellb cad e fFigure 3.10: Response of the DDrake global thermal overturning circulation to an abrupt CO2-doubling (in Sv). Top row: (a) control climate thermal overturning ctrl✓z, (b) perturbed climatethermal overturning perturbed✓z, (c) their difference ✓z = perturbed✓z ctrl✓z . Red cells circulateclockwise while blue cells circulate anticlockwise. Bottom row: same as the top row but decomposedinto the tropical, warm, and cool thermal overturning cells as defined in the text. Here, solid linescirculate clockwise and dashed lines circulate anticlockwise.-15-15 -15-10-10-10-10-5-5-5-5-50000000000000005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-20 -10 0 10 20-10-10-10-5-5-5-52200 000000000000005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-20 -10 0 10 20-5-5-5-5-5555510100 0000000 000000000005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-20 -10 0 10 20a b cFigure 3.11: Small basin thermal overturning circulation response to an abrupt CO2-doubling. (a)control climate thermal overturning ctrl✓z, (b) perturbed climate thermal overturning perturbed✓z, (c)their difference ✓z = perturbed✓z ctrl✓z . Red cells circulate clockwise while blue cells circulateanticlockwise.Chapter 3 79Connecting ocean circulation with ocean heat storage-15-15-10-100-5-5-5-5-5555510000000000000000005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-20 -10 0 10 20-10-10-10-5-5-5551000000 000000000005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-20 -10 0 10 20-10-5 -5-5 -5-5-55555101010 00000000000000005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-20 -10 0 10 20a b cFigure 3.12: Large basin thermal overturning circulation response to an abrupt CO2-doubling. (a)control climate thermal overturning ctrl✓z, (b) perturbed climate thermal overturning perturbed✓z, (c)their difference ✓z = perturbed✓z ctrl✓z . Red cells circulate clockwise while blue cells circulateanticlockwise.222555100 0000000000 0000005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-20 -10 0 10 202 22222555 10101500 000000000 0000 00000005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-20 -10 0 10 20-2 -2-22 2222225000000000 00005 10 15 20 25 30 35-3000-2500-2000-1500-1000-5000-20 -10 0 10 20a b cFigure 3.13: Southern ocean thermal overturning circulation response to an abrupt CO2-doubling.(a) control climate thermal overturning ctrl✓z, (b) perturbed climate thermal overturning perturbed✓z,(c) their difference ✓z = perturbed✓z ctrl✓z . Red cells circulate clockwise while blue cells circulateanticlockwise.80 Chapter 3Connecting ocean circulation with ocean heat storagetime-mean. The dashed response curves are anomalies away from the control zonal-average, i.e. the black solid curve subtracted from the coloured solid curves. Thefigure shows that in response to a CO2-doubling, there is a poleward shift andweakening of both the midlatitude westerlies and polar easterlies. The positivebulge in the response near 60N highlights the weaker northern polar easterly. It isour understanding that this weakening signal is responsible for the weakening coolcell in the large basin (figure 3.12).-0.1 -0.05 0 0.05 0.1 0.15-80-60-40-20020406080 Control0-50 yrs50-100 yrs100-150 yrs150-200 yrsFigure 3.14: Zonally-averaged zonal surface wind stress ⌧x (solid lines) and its response to anabrupt CO2-doubling (dashed lines) (in N m2). Solid coloured lines correspond to 50-year time-means throughout the 200 year experiment. Dashed lines are equivalent to the solid coloured lineswith the control zonal-average subtracted.Other coupled climate models show a poleward shift and strengthening of the South-ern Hemisphere (SH) midlatitude westerlies in response to anthropogenic forcingscenarios (e.g. [Bouttes et al., 2012, Swart and Fyfe, 2012, Goyal et al., 2021]). TheSH surface westerly wind-stress response (after 70 years) to a 1% yr1 CO2-increasescenario in CMIP3 and CMIP5 models is O(102) N m2 [Swart and Fyfe, 2012](their figure 4d), which is comparable to the magnitude of the response in DDrake(dashed lines in 3.14). We note, however, that this CMIP3/CMIP5 model responseis a strengthening, not a weakening.Chapter 3 81Connecting ocean circulation with ocean heat storageThe lack of a strengthening surface wind-stress response in DDrake is in contrast tothe observed strengthening of the Southern Hemisphere westerlies by ⇠20% over thelast few decades (e.g. [Thompson and Solomon, 2002, Swart and Fyfe, 2012, Perrenet al., 2020]). This strengthening has been attributed to both greenhouse gas (GHG)forcing (e.g. [Kushner et al., 2001]) and ozone depletion (e.g. [Gillett and Thompson,2003]) in model simulations. Our experiment does not include the effects of ozonedepletion, however, and this may be the primary cause of the observed wind changes[Polvani et al., 2011].The strengthening of the warm cell in figure 3.10e is mostly confined to the largebasin (figure 3.12), however there is an appearance of a shallow extension near 200 mdepth caused by a strengthening of the southern ocean thermal overturning (figure3.13). The Ekman pumping velocity WEk is given by:WEk =1⇢0curlz⇣⌧sf⌘(3.9)where ⌧s is the wind stress on the ocean surface and f is the Coriolis parameter. Thecurlz operator is defined on a vector A by curlzA ⌘ @xAy@yAx. The poleward shiftand weakening of the winds in response to the CO2-doubling (figure 3.14) leads toa similar poleward shift and weakening of the Ekman pumping velocities. A plot ofthe zonally-averaged WEk and its response is shown in figure 3.15 (The calculationhas been omitted for values between 5S and 5N where f is close to zero).Previously, we identified the warm cell circulation with a wind-driven inter-gyrecirculation in the large basin, where warmer subtropical water downwells near 40N,cools, and subsequently upwells near 50N (see figures 3.3d and 3.5d). From figure3.15, we find that the Ekman pumping response serves to reinforce this circulation.There is a strengthening of approximately 8 m yr1 in the downwelling WEk near40N, and there is a modest strengthening of the upwelling WEk near 50N of ⇠3m yr1. Both these signals support a strengthening of the warm cell circulation asseen in figure 3.10e.The poleward shift of the Southern Hemisphere winds also leads to a modest strength-ening of the southern ocean overturning circulation in the top 1 km. This results ina strengthening of the southern ocean’s thermal overturning, seen in figure 3.13b. It82 Chapter 3Connecting ocean circulation with ocean heat storage-50 -40 -30 -20 -10 0 10 20 30 40 50-80-60-40-20020406080Control0-50 yrs50-100 yrs100-150 yrs150-200 yrsFigure 3.15: Zonally-averaged Ekman pumping velocity WEk (solid lines) and its response to anabrupt CO2-doubling (dashed lines) (in m yr1). Solid coloured lines correspond to 50-year time-means throughout the 200 year experiment. Dashed lines are equivalent to the solid coloured lineswith the control zonal-average subtracted.is this strengthening in the southern ocean that is responsible for the shallow warmcell extension near 200 m depth, as seen in figure 3.10e.The relationship between the winds and the Southern Ocean MOC has been stud-ied extensively in models (e.g. [Saenko et al., 2005, Abernathey et al., 2011, Gent,2016]). The general current understanding is that the resolved (or mean) MOCstrength increases linearly with the maximum of the Southern Hemisphere zonalsurface wind stress (e.g. [Nikurashin and Vallis, 2012]). There might also be adependence on the bottom drag, as it was found by [Marshall et al., 2017] thatan increase in the bottom friction leads to an increase in the ACC transport. Al-though, this assumes that the Southern Ocean MOC and ACC transport are closelydynamically linked, which may not be the case (e.g. [Morrison and Hogg, 2013]).More importantly, however, is that an increase in the Southern Hemisphere windsalso leads to an increase in the eddy-induced MOC (e.g. [Viebahn and Eden, 2010,Abernathey et al., 2011]). The eddy-induced MOC compensates the increase in theresolved MOC, and the degree of compensation varies among models (e.g. [DownesChapter 3 83Connecting ocean circulation with ocean heat storageand Hogg, 2013]). This ‘eddy compensation’ effect means that the overall increase inoverturning strength is reduced. Nevertheless, it is expected that the net SouthernOcean MOC will intensify in response to an intensification of the winds.The relationship between this strengthening of the Southern Ocean MOC and ver-tical heat transport is less clear, however. In the study by [Nycander et al., 2007],they found that only with non-zero wind forcing was there significant downwardvertical heat transport in an idealised model using the MITgcm. A similar resultwas found by [Zika et al., 2013], where they varied the wind stress south of 30Sin UVic ESM. The downward vertical heat transport increased almost linearly withthe strength of the wind forcing. The implication here is that an increase in thewind forcing leads to a more vigorous Southern Ocean MOC, which in turn leads tostronger downward vertical heat transport.However, DDrake lacks a representation of an abyssal cell associated with AABWformation. Such a cell might weaken with increased wind forcing (e.g. [Nikurashinand Vallis, 2012]), and this could additionally contribute to a downward verticalheat transport, as the abyssal cell has been found to be thermally direct in othermodels (e.g. [Zika et al., 2013, Zika et al., 2015, Saenko et al., 2018]).We observe intensified warm cells in both the southern ocean and large basin ofDDrake (figures 3.12b and 3.13b), however, this is not due to a significant increasein the wind forcing (figure 3.14). The key driver is really a change in the curl ofthe surface wind stress (equation 3.9), leading to changes in the Ekman pumpingvelocities that reinforce the downwelling and upwelling branches of the cell circula-tions (figure 3.15). We should note that as our simulation keeps GM a constant, wecannot capture the effects of eddy compensation (e.g. [Gent, 2016]), so the responseof the DDrake warm cells to the surface winds is likely exaggerated.The southern ocean strengthening in figure 3.13b also provides us with one clue forthe deep warm cell near 1.5 km depth, seen most prominently in figure 3.10b. Asimilar deep warm cell strengthening is seen in the small basin (figure 3.11b), so weare led to believe that the deep warm cell must be associated with a deep SB/SOexchange, associated with the deep SBMOC. But it is unclear what the physicalstructure of this circulation might be.84 Chapter 3Connecting ocean circulation with ocean heat storageThe Ekman pumping response (figure 3.15) also supports our previous conclusionthat part of the weakening cool cell response is wind-driven. Before, we pointed tothe weakening polar easterly near 60N in figure 3.14. In figure 3.15, the changingpattern of the high-latitude surface winds contributes to an upwelling WEk responsepoleward of 60N. This increase in high-latitude upwelling damps the downwellingof polar waters in the large basin, leading to a weakening of the cool cell.Notably, the overall ✓z structure remains remarkably resilient to the CO2-doublingperturbation (figure 3.10). This is especially clear when comparing the three-cell de-compositions in panels d and e. However, there is a translational shift of ⇠1-2C tothe right in panel e, owing to the wholesale warming of the ocean. This shift makesthe ✓z structure more difficult to interpret. In the difference plot in panel f, solidlines denote a clockwise response and dashed lines denote an anticlockwise response.The cool and warm cells are ‘cut in half’ by opposing clockwise/anticlockwise re-sponses, owing to this shift. Rather than attempting to explain this difference plotin panel f (and in panel c), it may be easier to explain the vertical heat transport(VHT) responses resulting from this change in the thermal overturning. This shouldserve as an explanation of the total advective VHT response as plotted in figure 3.9b.Vertical heat transport responsesWe now investigate the advective VHT responses resulting from the changes inthermal overturning for the tropical, warm, and cool cells (equation 3.7). The controlclimate VHT profiles were explored in section 3.3.2. These profiles are plotted againin figure 3.16a (solid lines) alongside the perturbed climate profiles (dashed lines).Figure 3.16b shows the difference between the perturbed and control climate VHTprofiles, i.e. the VHT responses.The total VHT response is downward, as expected, peaking at +0.45 W m2 at 200m depth (the average forcing at the ocean surface is 4.44 W m2 (Chapter 5)). Thissharp peak is absent in the model-outputted advective response in figure 3.9b. Butwe should not expect these two profiles to match, owing to discrepancies betweenthe two calculations, as shown in figure 3.2. Nevertheless, the values in figure 3.16bare of the correct size, and the net global shape below 500 m depth is well-captured.Chapter 3 85Connecting ocean circulation with ocean heat storage-4 -3 -2 -1 0 1-2000-1500-1000-5000Tropical cellWarm cellCool cellCell sum-0.2 0 0.2 0.4 0.6-2000-1500-1000-5000Tropical cellWarm cellCool cellCell sumNet global-0.2 0 0.2 0.4 0.6-2000-1500-1000-5000Cell sumNet globalDeep warm cella cbFigure 3.16: Changes in vertical heat transports (VHTs) carried by the tropical, warm, and cool cellsof the DDrake thermal overturning circulation. Positive = downward. (a) Control climate (solid lines)and perturbed climate (dashed lines) VHT profiles. (b) VHT responses due to the CO2-doubling.The black dash-dotted line is the net global advective VHT response. The net global response takesinto account circulations not included in the three-cell decomposition of the thermal overturning. (c)Comparison of the three-cell sum and net global VHT responses. Their difference (black dashed) isattributed to the deep warm cell.We find that the cool cell dominates the VHT response in the top 1 km, peakingnear +0.25 W m2 through 250-750 m depth. The warm cell response supplies anadditional ⇠ +0.1 W m2 in the top 500 m. These increases in downward VHTare due to a weakening of the cool cell (transporting less heat upwards) and astrengthening of the warm cell (transporting more heat downwards), as explored inthe previous section. Changes in the tropical cell overturning are unimportant forthe VHT response. In [Zika et al., 2015], using ✓z for CCSM4, they found thattheir ‘cold cell’ collapsed while their ‘warm cell’ strengthened in a global warmingscenario. This is a similar response to the cool and warm cells in DDrake; both cellresponses contribute to an increase in downward VHT. There is no compensationhere. This is bad news for fish, as it suggests that changes in ocean circulation willenable even more warming of the ocean.We have plotted both the cell sum and the net global VHT responses in figure3.16b. The net global response (black dash-dotted) takes into account circulationsnot captured by our chosen three-cell decomposition of the DDrake net global ✓z.This was not a problem for the control climate VHT profiles (see figure 3.1), however,a strengthening of the deep warm cell now leads to a discrepancy between the twoblack curves. This discrepancy is plotted in figure 3.16c as the black dashed curve,which we attribute to the deep warm cell. As noted in the previous section, wesuspect that the deep warm cell is related to the deep interhemispheric overturning86 Chapter 3Connecting ocean circulation with ocean heat storagelinking the small basin and southern ocean (see figures 3.11b and 3.13b). Thus,our chosen three-cell decomposition neglects this deep circulation, and we miss anadditional ⇠0.1 W m2 of downward VHT below 1 km depth.A plot of changes in only the peak (positive or negative) VHT for each of the thermaloverturning cells is shown in figure 3.17. Panel a reiterates that the response ispositive (downwards) for all three cells. The percentage changes relative to thecontrol climate values in panel b are more revealing. The plot shows that thelargest proportional response comes from the warm cell, whose peak (downward)VHT increases by 32%. This is followed not too far behind by the cool cell, whosepeak (upward) VHT decreases by 25%. Again, the change in VHT due to thetropical cell is negligible.Cool cell Tropical cell Warm cell-3.5-3-2.5-2-1.5-1-0.500.51ControlPerturbedChangeCool cell Tropical cell Warm cell-40-30-20-10010203040a bFigure 3.17: Changes in the peak (positive or negative) VHT carried by the tropical, warm, and coolcells of the DDrake thermal overturning, in response to the CO2-doubling. (a) Control and perturbedclimate values, and their difference (in W m2), (b) percentage changes.3.4.2 Changes in energeticsBefore concluding, we take a brief look at the changes in work performed by theDDrake ocean circulation, using the overturning in potential density space, as in-troduced in section 3.3.3. There, we linked the tropical, warm, and cool cells of ✓z to similar features in 0z (figure 3.8). Here, we integrate over each of thesecells in both the control and perturbed climates to find the work performed by thesecirculation features (equation 3.8). These are shown in figure 3.18.Chapter 3 87Connecting ocean circulation with ocean heat storageCool cell Tropical cell Warm cell-150-100-50050100150200250300ControlPerturbedChangeCool cell Tropical cell Warm cell-50-40-30-20-1001020304050a bFigure 3.18: Changes in the work performed by the tropical, warm, and cool cells of the DDrakethermal overturning, in response to the CO2-doubling (using equation 3.8 with 0z). (a) Controland perturbed climate values, and their difference (in GW), (b) percentage changes.Eddy-induced Net Resolved-600-400-2000200400600800ControlPerturbedChangeEddy-induced Net Resolved-50-40-30-20-1001020a bFigure 3.19: Changes in the work performed by the global resolved and eddy-induced circulations,and their sum (net circulation), in response to a CO2-doubling (using equation 3.8 with 0z). (a)Control and perturbed climate values, and their difference (in GW), (b) percentage changes.88 Chapter 3Connecting ocean circulation with ocean heat storageThere is a 48% decrease in the work performed by the cool cell as a result of theCO2-doubling. This means much less gravitational potential energy (GPE) is beingconverted into kinetic energy (KE), which reflects the weakening of the (mostlybuoyancy-driven) small basin MOC. However, there is a commensurate 49% increasein the (negative) work performed by the warm cell. This means more kinetic energyis being converted into potential energy, which is supplied by the winds. This isconsistent with our use of the surface winds in explaining the strengthening of thewarm cell. There is a modest increase of almost 15% in the work performed by thetropical cell, which may reflect the pattern of surface buoyancy forcing, with greaterwarming at the equator than in the subtropics.Again, the analysis in [Nycander et al., 2007] and equation 3.8 are only formallycorrect for the in-situ density, not the potential density. So, the precise values forthe work performed in the control and perturbed climates may be underestimates.Nevertheless, this underestimation cancels out when viewing the changes in the workperformed, so the most meaningful values in figures 3.18 and 3.19 are these changes.We performed the same calculation for the global 0z overturning, decomposedinto resolved and eddy-induced components. This is shown in figure 3.19. Thisplot reveals an important difference between the resolved and eddy-induced circula-tions. Resolved circulations perform negative work, eddy-induced circulations per-form positive work. While resolved circulations convert KE into GPE, eddy-inducedcirculations counteract this by converting GPE into KE (this is expected, as theGent-McWilliams parameterisation explicitly serves to flatten isopycnal surfaces).In response to the CO2-doubling, there is a weakening of the eddy-induced circula-tion (8%), and a strengthening of the resolved circulation (+17%). Perhaps mostinteresting is that both these changes result in a much more substantial weakeningof the net circulation (40%). Figure 3.19 provides a definitive answer to the ques-tion: “Does the DDrake ocean circulation weaken in response to a CO2-doubling?”The answer is yes. So, despite weakening by 40%, the circulation response still leadsto an increase in downward VHT by ⇠0.2 W m2 in the top 1 km (figure 3.16).Note, however, that it is still unclear whether the real-world ocean circulation willweaken or strengthen with global warming. Many modelling studies (e.g. [GregoryChapter 3 89Connecting ocean circulation with ocean heat storageet al., 2005, Weaver et al., 2012, Weijer et al., 2020]) and observational studies(e.g. [Caesar et al., 2018, Rahmstorf et al., 2015, Smeed et al., 2018]) point toa weakening of the AMOC (with an implied concomitant weakening of the globalocean circulation). Recent observations, however, reveal a wholesale strengtheningof the global ocean circulation since 1990, accompanied by an increase in the global-mean sea-surface wind speed [Hu et al., 2020]. The authors in [Hu et al., 2020] alsonote that the CMIP5 ensemble mean sea-surface wind speed displays a significantincreasing trend in the RCP8.5 scenario, indicating that ocean circulation may alsoincrease commensurately with CO2 emissions. A follow-up study, however, findsthat surface warming, not the winds, is the main driver of intensified currents in theupper 200 m in an abrupt 4⇥CO2 experiment [Peng et al., 2022].3.5 Conclusion3.5.1 SummaryIn this chapter, we have examined the advective vertical heat transport response ofthe top 2 km of the global DDrake ocean circulation using the thermal overturningstream function ✓z. We decomposed the global thermal overturning into particularcirculation features — a warm cell, a cool cell, and a tropical cell — and connectedthese with physical processes and circulations in geographical coordinates. We foundthat, in response to a CO2-doubling, the warm cell strengthens and the cool cellweakens (the tropical cell response is negligible) (figure 3.10). Both these responsesincrease the downward vertical heat transport, enabling greater warming of theocean interior (figure 3.16).The warm cell circulation is identified with a wind-driven, subtropical-subpolar inter-gyre circulation in the large basin (figure 3.3d), and its strengthening is linked tochanging patterns in the surface winds. The surface wind response reinforces thedownwelling and upwelling regions of this overturning cell via changes in Ekmanpumping (figure 3.15). The cool cell is identified partly with a wind-driven circula-tion in the large basin, and partly with a buoyancy-driven circulation in the smallbasin. The wind-driven circulation is a subpolar gyre overturning (figure 3.3d),90 Chapter 3Connecting ocean circulation with ocean heat storagewhose weakening is associated with damped Ekman pumping at high latitudes (fig-ure 3.15). The buoyancy-driven circulation is the deep small basin MOC (figure3.3c), whose weakening is due to reduced deep convection at northern high latitudesresulting from amplified polar warming (see Chapter 4).3.5.2 Discussion and ConclusionsThe DDrake warm and cool cells have similar-sized VHT responses (figure 3.17),which suggests that both wind- and buoyancy-driven circulation changes are ofcomparable importance in transient climate change. However, there is a robustweakening of the net global circulation in response to the CO2-doubling (figure3.19), which is largely due to the weakening of the (buoyancy-driven) small basinMOC (figure 3.11).This weakening of the small basin MOC is similar to the weakening AMOC responseseen in models (e.g. [Gregory et al., 2005, Weaver et al., 2012, Cheng et al., 2013,Weijer et al., 2020]), proxy observations (e.g. [Rahmstorf et al., 2015, Caesar et al.,2018, Zhu and Liu, 2020, Caesar et al., 2021]), and possibly direct observations(e.g. [Smeed et al., 2018]). As the small basin MOC in DDrake is thermally direct,the weakening contributes to an increase in downward VHT. This is in contrast toother models, where a thermal overturning cell coincident with the AMOC has beenfound to be thermally indirect (e.g. [Zika et al., 2013, Zika et al., 2015, Saenko et al.,2018, Savita et al., 2021]).The VHT response also depends on the temperature structure, however, and notjust the strength of the circulation in geographical coordinates [Shatwell et al., 2020],so it is not obvious that a weakening circulation would result in a weakening VHT(upwards or downwards). It would be valuable to link the AMOC weakening andthermal overturning responses in other models. Following the spirit of the AMOCobserving arrays RAPID [McCarthy et al., 2015] and SAMBA [Meinen et al., 2018]in both hemispheres, a greater observational focus on the downwelling and upwellingbranches of the AMOC would also help answer the question of whether the circula-tion is thermally direct or indirect.In CMIP5 historical simulations, the Southern Ocean takes up as much as 75% ofChapter 3 91Connecting ocean circulation with ocean heat storagethe global ocean heat gain [Frölicher et al., 2015]. This is partly associated witha wind-driven upwelling of cold deep water, which delays the surface warming re-sponse [Armour et al., 2016]. The relatively cool sea-surface temperature (SST)enables a large amount of excess heat to be taken up from the atmosphere, whichis then advected northward and downward by the MOC. A strengthening of theSouthern Ocean overturning and its associated thermal overturning (as seen in fig-ure 3.13b) could possibly further secure the Southern Ocean’s dominance in globalOHU. However, recent considerations of the projected relative changes in GHG and,particularly, aerosol forcings reveal that the Southern Ocean’s importance may de-crease substantially in the future, with the North Atlantic becoming a comparableplayer in global OHU [Shi et al., 2018].However, we should be careful about making any strong inferences from DDrakeabout the North Atlantic or Southern Ocean circulation (and hence VHT) responses.DDrake lacks a realistic representation of diapycnal mixing, especially its bottom-enhancement, as seen in many observations (e.g. [Polzin et al., 1997, Garabato et al.,2004, Waterhouse et al., 2014]). Although we have focussed on advective transports,the magnitude and profile of diapycnal mixing can affect the strength and structureof the MOC (e.g. [Saenko and Merryfield, 2005, Ito and Marshall, 2008, Nikurashinand Vallis, 2011, Mashayek et al., 2015]), which has implications for the thermaloverturning.Admittedly, diapycnal mixing is more important for the abyssal circulation or LowerCell of the MOC, which DDrake does not realistically represent. So, we have limitedthe discussion to the top 2 km, relevant to the Upper Cell of the MOC. But thisdoes not completely solve the issue, as the Upper and Lower Cells are connected,and the strength of their coupling through interbasin transports is a topic of ongoingresearch (e.g. [Talley, 2013, Thompson et al., 2016, Ferrari et al., 2017, Newsom andThompson, 2018, Johnson et al., 2019]).In addition, the depth of the DDrake ocean is only 3 km, so we cannot addressthe deep ocean’s role in the heat budget. This is not a major issue, as warmingbelow 2 km is mainly due to warming of AABW (e.g. [Purkey and Johnson, 2013]),which DDrake does not represent, and the contribution to global OHU below 2 km is92 Chapter 3Connecting ocean circulation with ocean heat storageprobably only ⇠10% (e.g. [Desbruyères et al., 2016, Meyssignac et al., 2019]). Theocean below 2 km also responds on a multi-centennial to millennial timescale (e.g.[Yang and Zhu, 2011, Li et al., 2013]), so our 200 year CO2-doubling simulation isnot long enough to capture the deep response. Nevertheless, the ocean below 2 kmwould become more relevant at longer timescales, and so the relevance of DDrakewould become increasingly tenuous.These caveats aside, our results demonstrate that the thermal overturning circu-lation ✓z is a powerful model diagnostic for analysing both circulation and VHTresponses in transient climate change. Particularly, when addressed regionally (fig-ure 3.6), and in comparison with the geographical overturning (figure 3.3), thermaloverturning cells can be linked to physical circulations. The overturning in depth-density coordinates (figure 3.8) can then also indicate the driving mechanisms forthese physical circulations [Nycander et al., 2007].Our calculation of ✓z incorporates the resolved and parameterised eddy advectionterms, capturing the residual circulation. A new diagnostic, called the ‘super residualcirculation’ (SRC), was recently put forward by [Savita et al., 2021] incorporatingsubgridscale isopycnal mixing due to mesoscale eddies i.e. ‘residual circulation +isopycnal mixing’. Previous studies find that the residual circulation transports heatdownward overall (e.g. [Kuhlbrodt et al., 2015, Griffies et al., 2015]). Adding theeffect of isopycnal mixing leads to the SRC transporting heat upward overall [Savitaet al., 2021] (although the residual circulation in DDrake already transports heatupward overall, (figure 3.1)).The study of [Savita et al., 2021] combines efforts from [Dias et al., 2020a] by ex-tending the concept of ‘super residual transport’ (originally suggested by [Kuhlbrodtet al., 2015]), and from [Holmes et al., 2019] by combining the residual circulationwith the effects of local temperature tendencies to construct a ‘diathermal’ circula-tion (crossing temperature surfaces).The SRC has the advantage that it allows models with different grid resolutions andeddy process parameterisations to be compared in a consistent framework. A high-resolution model (say, 1/12) might not need parameterisations for eddy advectionor isopycnal mixing. For such a model, the residual and resolved circulations areChapter 3 93Connecting ocean circulation with ocean heat storageequivalent, and the residual circulation implicitly contains effects due to isopycnalmixing. As such, comparing this model’s residual circulation with a lower-resolutionmodel’s that does parameterise isopycnal mixing is problematic, as pointed out in[Kuhlbrodt et al., 2015]. Whereas, comparing these two models’ SRCs would bemore consistent.We have not considered such intricacies for the DDrake (residual) thermal over-turning, so a direct comparison with the thermal overturning circulations in othermodels (e.g. [Zika et al., 2013, Zika et al., 2015, Saenko et al., 2018]) may not beespecially meaningful. We encourage efforts like [Savita et al., 2021] to enable moremeaningful model intercomparison, and for researchers to compute thermal over-turning diagnostics for a greater range of coupled climate models in order to gain abetter understanding of the structure and response of the global ocean circulationunder continued global warming.3.5.3 Key points1. Overturning stream functions in thermodynamic coordinates can be used todecompose the global ocean overturning into particular circulation features,defined by whether they pump heat downward (warm cell) or upward (coolcell).2. In response to an abrupt CO2-doubling in DDrake, the warm cell strengthensand the cool cell weakens, resulting in an increase in the downward advectivevertical heat transport of ⇠0.2 W m2 in the top 2 km. The circulationresponse does not ‘resist’ the warming.3. The warm cell strengthening is linked to changes in the surface winds, whilethe cool cell weakening is linked to both changes in surface winds and surfacebuoyancy forcing.94 Chapter 3“I perceive I have not really understood anything, not a single object, and that no manever can,Nature here in sight of the sea taking advan-tage of me to dart upon me and sting me,Because I have dared to open my mouth tosing at all.”From As I Ebb’d with the Ocean of LifeWalt Whitman, 1860 4Ocean heat storage response toan AMOC slowdownIn this chapter∗, we use the DDrake model to explore the role of the Atlantic merid-ional overturning circulation (AMOC) in transient climate change, likening the smallbasin MOC (SBMOC) to the AMOC. We introduce the concept of a heat storagecontrast: the difference between the small and large basins’ heat storage rates. Overthe simulated 200 years following the CO2-doubling, we find that the small basinstores heat at a faster rate than the large basin by 0.6 ± 0.2 W m2. We argue thatthis is due to the SBMOC. However, we also find that as the SBMOC weakens sig-nificantly, it has little impact on the heat storage contrast. We suggest this is due tothe effects of compensating warming patterns and interbasin heat transports. Thus,although the presence of the SBMOC is important for the small basin’s enhancedheat storage, the SBMOC weakening is surprisingly unimportant.∗Note this chapter is an adapted and extended version of an article originally published inGeophysical Research Letters [Shatwell et al., 2020]. This work has been reviewed by MarthaBuckley, Yavor Kostov, an anonymous reviewer, and GRL editor Kathleen Donohue.95Connecting ocean circulation with ocean heat storage4.1 IntroductionThe AMOC plays a central role in global climate. It is an important conduit insequestering anthropogenic heat into the ocean interior (e.g. [Chen and Tung, 2014,Kostov et al., 2014]), and its interhemispheric reach is responsible for a northwardmeridional heat transport in both hemispheres (e.g. [Trenberth and Caron, 2001]).This latter feature results in a global non-zero cross-equatorial heat transport (⇠0.5PW), contributing to the Northern Hemisphere being slightly warmer (by ⇠2C)than the Southern Hemisphere [Marshall et al., 2014b].The AMOC’s role in global climate is especially interesting due to the possibility ofit weakening in the future. A weakening response of the AMOC with global surfacewarming is seen across coupled climate models (e.g. [Weaver et al., 2012, Chenget al., 2013, Weijer et al., 2020]), and indirect proxy observations point to the AMOChaving weakened since the mid-twentieth century (e.g. [Caesar et al., 2018, Zhu andLiu, 2020]), with one study suggesting that the AMOC is in its weakest state in overa thousand years [Caesar et al., 2021]. Model biases may favour a stable AMOC,and it is still a concern that the AMOC could collapse in the future, leading toabrupt changes in global climate (e.g. [Liu et al., 2017, Caesar et al., 2018]).A shutdown of the AMOC could potentially lead to a colder Northern Hemispherewith more extensive Arctic sea-ice [Vellinga and Wood, 2008], an equatorward shiftof the ITCZ [Stouffer et al., 2006], and significantly colder winters in Europe [Liuet al., 2017]. Large-scale reorganisations and transitions between different AMOCstates have previously been invoked to explain past abrupt paleoclimatic shifts (e.g.[Broecker, 1997, Alley and Clark, 1999]).The basis for the concern of an AMOC collapse is the potential existence of these dif-ferent stable AMOC states, in particular, the possibility that the AMOC is bistablewith ‘on’ and ‘off’ states. Such bistable behaviour originates from Stommel’s earlywork with an idealised two-box model [Stommel, 1961]. Bistable behaviour has sub-sequently been found in a range of other idealised climate models (e.g. [Manabe andStouffer, 1988, Rahmstorf, 1995, Johnson et al., 2007]). The presence of multipleequilibria depends critically on the existence of a destabilising salt advection feed-96 Chapter 4Connecting ocean circulation with ocean heat storageback, whereby, if the AMOC were to weaken, less salt would be transported to theNorth Atlantic, encouraging further weakening and so on (e.g. [Rahmstorf, 1996]).Although multiple equilibrium climate states have been found in very similar mod-elling set-ups to the DDrake configuration [Ferreira et al., 2011, Ferreira et al.,2018b], the states are not caused by bistability of the overturning circulation, butinstead by an ice-albedo feedback modified by meridional ocean heat transport [Fer-reira et al., 2018b]. As such, we are not concerned with a complete collapse ofthe overturning circulation in DDrake, but we are still interested in a substantialweakening of the circulation. A weakening of the AMOC is still projected over the21st century (e.g. [Kirtman et al., 2013]). For example, CMIP6 models project anAMOC decline of 34-45% by 2100 [Weijer et al., 2020], although the true strengthof the future decline is highly uncertain (e.g. [Reintges et al., 2017]). It is alsouncertain what effect this will have on ocean heat storage and surface climate in thefuture (e.g. [Bellomo et al., 2021]).As discussed in Chapter 1, a weakening of the AMOC is expected as a result of pos-itive buoyancy forcing in the northern high latitudes from a combination of positiveheat and freshwater fluxes. Anomalous heat fluxes are largely the result of Arcticamplification — enhanced surface warming in the Arctic relative to the rest of theworld (e.g. [Cohen et al., 2014]). Multiple closely related processes are responsiblefor Arctic amplification, such as changes in atmospheric and oceanic heat transports,sea-ice extent, cloud cover, water vapour, and black carbon aerosols (e.g. [Serrezeand Barry, 2011, Previdi et al., 2021]).However, it’s been argued that Arctic sea-ice loss is the dominant contributor toArctic amplification [Dai et al., 2019]. The key mechanism put forward is an increasein absorbed solar radiation over larger ice-free ocean regions from May to September,and a subsequent release to the atmosphere from October to April (through outgoinglongwave radiation, latent, and sensible heat fluxes) [Serreze and Barry, 2011]. Thewarmer atmosphere will then radiate more strongly back to the surface.Arctic sea-ice loss also increases the freshwater flux, further contributing to a weak-ening of the AMOC (e.g. [Jahn and Holland, 2013, Sévellec et al., 2017]). Freshwaterfluxes from melting of the Greenland Ice Sheet (e.g. [Lenaerts et al., 2015, Van denChapter 4 97Connecting ocean circulation with ocean heat storageBroeke et al., 2016]) and increased high-latitude precipitation and river runoff (e.g.[Stouffer et al., 2006]) also play a role.Both the anomalous heat and freshwater fluxes act to reduce the surface density inthe North Atlantic, although the relative importance of each type of flux still remainsto be determined (e.g. [Kostov et al., 2019, Haskins et al., 2020]). The reduction insurface density suppresses deep convection at the sites of NADW formation, leadingto a commensurate reduction in the AMOC strength (e.g. [Thornalley et al., 2018]),and also reduces the meridional density gradient, which is thought to scale with theAMOC strength (e.g. [Stommel, 1961, Griesel and Maqueda, 2006, Gregory andTailleux, 2011, Ragen et al., 2022]).The depth and strength of the AMOC positively correlates with the depth of globalocean heat storage (OHS) across CMIP5 models [Kostov et al., 2014]. Multidecadalvariability of the AMOC has been linked to periods of enhanced global surfacewarming and cooling [Chen and Tung, 2018]. It is possible that the presence of theAMOC gives the Atlantic its enhanced heat storage rate compared to the Pacific,as seen in observations (e.g. [Chen and Tung, 2014, Desbruyeres et al., 2017, Zannaet al., 2019, Johnson and Lyman, 2020]).A weakening AMOC results in a weakening of the northward oceanic meridional heattransport (MHT), which may explain the conspicuous region of cooling (‘warminghole’) in the subpolar North Atlantic found in observational maps of temperaturetrends (e.g. [Rahmstorf et al., 2015]) and model projections (e.g. [Gervais et al.,2018]). It has been suggested that this North Atlantic surface cooling reduces theannual-mean sea-air temperature difference, and so reduces the sensible heat fluxfrom the ocean to the atmosphere, leading to an increase in OHU (e.g. [Wintonet al., 2013, Drijfhout et al., 2014]). It should be emphasised that the OHU andOHS responses are distinct in this context. Here, a local increase in OHU couldpossibly be accompanied by a global decrease in OHS, even though both responsesare associated with a weaker AMOC.Recent work [Saenko et al., 2018] has cast doubt on the observed model correlationbetween AMOC strength and heat storage [Kostov et al., 2014], suggesting thatthe eddy parameterisation affects both AMOC strength and global OHU efficiency,98 Chapter 4Connecting ocean circulation with ocean heat storagethus causing a spurious correlation between the two quantities. But without abetter conceptual grasp on how the AMOC affects OHS, it is unclear whether thiscorrelation is spurious or not. Furthermore, these studies [Kostov et al., 2014, Saenkoet al., 2018] establish a link between the AMOC and global OHS, while it may beeasier to first consider the AMOC’s influence on heat storage within the Atlanticbasin itself. Given the importance of constraining and monitoring Earth’s energyimbalance through OHC observations (e.g. [Von Schuckmann et al., 2016]), and thepossibility that the AMOC may continue to weaken into the future, it is imperativeto better understand the AMOC’s role in ocean heat uptake and storage as theworld continues to warm.To this end, we examine the ocean response of DDrake over 200 years, after anabrupt doubling of atmospheric CO2. We first define the overturning circulation insection 4.2. The small basin exhibits an overturning circulation akin to the AMOC,while the large basin does not. We look at the basins’ individual responses, andsuggest isolating the effect of the small basin MOC by focusing on small–large basindifferences. In section 4.3, we present results where we find and define a heat storagecontrast between the small and large basins. We explore the role of the small basin’sMOC (SBMOC) in establishing this contrast, particularly its time-evolution as theSBMOC weakens. We also present preliminary results from four CMIP5 modelsdemonstrating the existence of a heat storage contrast between the Atlantic andPacific basins in abrupt CO2-quadrupling experiments, which parallels the contrastbetween our model’s small and large basins. Conclusions are presented in section4.4.4.2 Control climate overturning circulationsIn this chapter, an overturning stream function is defined as the zonally-integratedmeridional volume transport: (y, z) =1tZ t+ttZ 0zZ xExWv dx dz dt (4.1)Chapter 4 99Connecting ocean circulation with ocean heat storagewhere v is the meridional velocity averaged over some period of time t, and xWand xE define the longitudinal positions of the meridional boundaries enclosing thebasin. A positive stream function circulates clockwise, with northward flow at thesurface and southward flow at depth. The strength of the circulation can be definedas the maximum of this stream function over particular depth and latitude rangesdepending on one’s focus. For our purposes, we define the small basin MOC strengthas the maximum over all depths and latitudes within the basin i.e. 3000 m < z <0 m, 35S < y < 90N. This picks up only the deep overturning cell as the shallowsubtropical overturning cells do not dominate in the small basin (figure 4.1a).-10-500555510101015151520202000222444666881012-50 0 50-3000-2500-2000-1500-1000-5000-40-2002040 -4035-25-20-15-10-10-5-5-5-500000000055555551010 105 152053000222444666881012-50 0 50-3000-2500-2000-1500-1000-5000-40-2002040a bFigure 4.1: Control residual overturning stream functions ctrlresfor the (a) small basin, and (b) largebasin (in Sv, 1 Sv ⌘ 106 m3 s1). The southern ocean residual overturning is shown south of 35S(thick black line).In the DDrake model, the effects due to mesoscale eddies are parameterised by meansof a ‘bolus velocity’ ub, which depends on the slope of the isopycnal surfaces [Gentand Mcwilliams, 1990] (see Appendix A). As we did in Chapter 3 for the thermaloverturning, in order to capture eddy effects in the overturning circulation definedin equation 4.1, we use a ‘residual velocity’ vres, which is a sum of the resolvedvelocity v and the eddy bolus velocity vb. This defines a residual overturning streamfunction given by: res(y, z) =1tZ 0zZ xExWvres dx dz dt =1tZ 0zZ xExW(v + vb) dx dz dt (4.2)Calculations for the small basin (SB) and large basin (LB) res in the DDrakecontrol climate (where t = 50 years) are shown in figure 4.1. The maximum of the100 Chapter 4Connecting ocean circulation with ocean heat storageSB stream function is defined as the SBMOC strength, and it is the weakening ofmax( res) with time that characterises the SBMOC weakening in response to theCO2-doubling.4.3 Results4.3.1 Small and Large Basin Heat Storage RatesIn response to the CO2-doubling, an arbitrary quantity C = C(t, x, y, z) in themodel changes from its control value C(x, y, z) to a perturbed value Cpert(t, x, y, z)by an amount C(t, x, y, z) = Cpert C, where C is a 50-year time-average. TheSB stores heat at a faster rate than the LB, and both heat storage rates decreasewith time, reflecting the adjustment to a new equilibrium (figure 4.2a). The largebasin’s surface area is three times larger than that of the small basin, yet it onlytakes up 2.21 times more heat energy in joules over the course of the simulation. Byconsidering areal proportions of the total global OHC increase, we find that this isdue to a combination of the SB taking up more heat than expected for its size, andthe LB taking up less than expected (figure 4.3). We also note that the southernocean starts out overperforming for its size, but this discrepancy decreases to almostzero over the rest of the 200 years.To compare the two basins’ efficiencies in storing heat, we look at basin OHC changesdivided by the respective basin’s surface area (in J m2). Here, a change in OHCis calculated as OHC =RV ⇢0cp✓ dV , where the integral is taken over the entirebasin volume, ⇢0 is the seawater density, cp is the specific heat capacity of seawater,and ✓ is the potential temperature.After 200 years’ warming, the final anomalous OHC difference,OHCSBOHCLB,is 3.45 ⇥ 109 J m2 (figure 4.2b), and in terms of heat storage rates in W m2, thistranslates to a time-mean heat storage contrast of 0.6 ± 0.2 W m2 (figure 4.2c,the difference between the red and blue curves in figure 4.2a). There is large inter-annual variability in the heat storage rates, and the heat storage contrast shows nodiscernible trend over the 200 years. At the same time, we see that the SBMOCstrength (stream function maximum) weakens rapidly by ⇠25% during the first 30Chapter 4 101Connecting ocean circulation with ocean heat storage0 50 100 150 200-2-1012345SBLB0 50 100 150 200051015109SBLB0 50 100 150 200-3-2-101230 50 100 150 20016182022242628a bdcFigure 4.2: Time series of annually-averaged (a) SB and LB heat storage rates (@t(OHC), in Wm2); (b) OHC anomalies (OHC, in J m2); (c) the difference in heat storage rates i.e. the heatstorage contrast (@t(OHCSB OHCLB), in W m2); and (d) SBMOC strength (max( pertres ),in Sv) following an abrupt doubling of CO2. Thick lines are decadal running means and horizontaldash-dot lines indicate time-mean values. The time-mean heat storage contrast is 0.6 ± 0.2 W m2(standard error, assuming 20 degrees of freedom accounting for decadal oscillations).102 Chapter 4Connecting ocean circulation with ocean heat storage0 50 100 150 200012341024GlobalSBLBSO0 50 100 150 200-202468101023SB - actualSB - expectedDifference0 50 100 150 200-0.500.511.522.51024LB - actualLB - expectedDifference0 50 100 150 20002468101023SO - actualSO - expectedDifferencebadcFigure 4.3: (a) Time-series of heat storage responses (in J) for all DDrake basins. (b-d) Compar-ison of actual versus expected heat storage responses (in J) considering the fractional coverage ofthe total surface area for (b) the small basin (SB), (c) the large basin (LB), and (d) the southernocean (SO) of DDrake. For each basin B, the ‘expected’ curves are (B’s surface-area/global surface-area)*(global response) i.e. the surface-area-weighted fraction of the black curve in (a). From thisperspective, we see that the SB overperforms (while the LB underperforms) with respect to its size.Chapter 4 103Connecting ocean circulation with ocean heat storageyears (figure 4.2d), after which it remains quite stable at ⇠19 Sv (1 Sv ⌘ 106 m3s1), with small oscillations of ⇠1 Sv which likely reflect internal oscillations of thesystem (see [Buckley et al., 2012]).Figure 4.4 shows the anomalous horizontal-mean temperature response ✓ as afunction of depth and time for the two basins. The SB temperature signal penetratesinto the ocean interior more quickly than the LB signal. The 0.1 K contour reachesthe sea floor in the SB within 150 years, whereas in the LB it only extends downto ⇠2 km depth after the full 200 years. The vertical structures of the two basins’temperature responses are significantly different. While the upper-ocean responses(⇠top 500 m) are similar, the differences at depth are substantial. This is expectedas the depth to which the wind-driven circulation penetrates is similar in both basins.At a depth of ⇠800 m in figure 4.4, there is an abrupt change in the SB responsewhere heat is taken up more rapidly. The gradient of ✓ with depth reveals that,in this region, deeper waters’ warming is preceding that of the waters above it.This suggests that temperature anomalies are being advected downwards to depthsbelow 800 m faster than that allowed by diffusion, Ekman pumping, and other pro-cesses associated with the waters above. Rapid downward advection of temperatureanomalies points to deep convection [Marshall and Schott, 1999], which we take tobe closely associated with the downwelling branch of the small basin’s MOC.4.3.2 Role of the Small Basin MOCTo make the connection between ocean heat storage and the SBMOC more explicit,we look at the spatial pattern of the vertically-averaged potential temperature re-sponse ✓ for different depth intervals (figure 4.5a, b). The upper 1 km reveals pro-nounced warming at high latitudes in every basin, a conspicuous pool of warming at40-60N in the SB, and enhanced warming at 40S where there is zonal circumpolarflow, a feature reminiscent of the warming behaviour along the Antarctic Circum-polar Current (ACC) [Armour et al., 2016]. Below 1 km depth, the temperatureanomaly in the SB flows along the deep western boundary current, coincident withthe lower limb of its MOC. Note there are no large temperature anomalies at depthin the LB or southern ocean regions.104 Chapter 4Connecting ocean circulation with ocean heat storage0.10.10.50.50.51111.550 100 150 200-3000-2500-2000-1500-1000-500000.511.50.10.10.10.50.50.51111.51.550 100 150 200-3000-2500-2000-1500-1000-500000.511.5baFigure 4.4: Depth-time plots of the anomalous temperature response (in K) in the (a) small and(b) large basins of DDrake following an abrupt CO2-doubling. The contour interval is 0.1 K, andthe 0.1, 0.5, 1.0, and 1.5 K contours are highlighted. The upper ocean responses are similar, but thedifferences at depth are substantial.The SBMOC’s role is made even clearer when we plot the control residual overturn-ing ( res) on top of the final zonally-averaged ✓ in the SB (figure 4.5c). Thereis distinctive deep warming at 60-80N, collocated with the downwelling branch of res. The ✓ structure also approximates the pattern of the streamlines, and we seean isolated pool of warm water between 1 and 1.5 km depth near the equator, sug-gesting the equatorward advection of temperature anomalies into this region awayfrom the high latitudes of deep water formation.Figures 4.2 and 4.5 together highlight that the key process governing the enhancedheat storage rate in the SB is the rapid advection of surface temperature anomaliesinto the interior by the SBMOC. This relationship between the small basin’s MOCand heat storage rate highlights the importance of the present-climate AMOC, whichcould help explain the Atlantic’s observed enhanced warming rate compared to thePacific (e.g. [Chen and Tung, 2014, Desbruyeres et al., 2017, Zanna et al., 2019]).Chapter 4 105Connecting ocean circulation with ocean heat storageFigure 4.5: Vertically-averaged ✓ (in K) after 200 years following an abrupt doubling of atmo-spheric CO2 in DDrake for the depth intervals (a) 0-1 km and (b) 1-3 km. The temperature anomalyat depth follows a deep western boundary current in the small basin. (c) Zonally-averaged ✓(colour, in K) in the small basin after 200 years’ warming, and streamlines (white dashed contours,in Sv) for the control residual overturning res.106 Chapter 4Connecting ocean circulation with ocean heat storage4.3.3 Role of the southern ocean regionIn DDrake, the large basin dominates global ocean heat uptake, while the southernocean and small basin are comparable (figure 4.3a). In the real-world, the SouthernOcean dominates global ocean heat uptake in CMIP5 models [Frölicher et al., 2015]and observations (e.g. [Desbruyeres et al., 2017]). From the historical simulationsin [Frölicher et al., 2015], approximately half of the heat taken up by the South-ern Ocean is exported northwards. This northward transport is also supported byobservations, which results in a delayed warming of the Southern Ocean [Armouret al., 2016]. Thus, another possibility is that the SB–LB heat storage contrast isdue to a difference in northward heat transports from the southern ocean region ofDDrake.However, if we plot the integrated net air-sea heat flux (OHU) compared with theheat storage (OHS) for the DDrake basins, we find that the southern ocean gainsheat rather than exporting heat northwards. The heat storage in the SB is consis-tently less than the heat uptake (figure 4.6a). This implies that the SB actually leaksheat across 35S to the southern ocean region in DDrake. At the same time, thereis little difference between heat uptake and heat storage in the LB, and although itstarts out losing heat to the southern ocean, it ends up gaining heat from it afterabout 100 years. Thus, the SB–LB contrast cannot be explained by heat transportswith the southern ocean. For this to be the case, we would need the LB to leak heatto the SB. Currently, the opposite is observed.0 50 100 150 200-5051015 109OHUOHS0 50 100 150 200-5051015 109OHUOHS0 50 100 150 200-5051015 109OHUOHSa b cFigure 4.6: Time series of time-integrated net surface heat flux (OHU) and changes in OHC (OHS),and their difference (OHSOHU) (in J m2) for the (a) small basin, (b) large basin, and (c) south-ern ocean regions of DDrake.Chapter 4 107Connecting ocean circulation with ocean heat storage4.3.4 Heat storage response to SBMOC WeakeningIt is now clear that the SBMOC plays an important role in setting the heat storagecontrast between the two basins of DDrake. However, the SBMOC strength weakensrapidly by ⇠25% during the first 30 years, after which it remains stable between18 and 20 Sv (figure 4.2d). As has been suggested in other modelling studies (e.g.[Gregory, 2000, Gregory et al., 2005, Liu et al., 2017]), this weakening is primarilya result of a thermally-forced buoyancy increase in the northern high latitudes (wehave not investigated salinity-forced changes in buoyancy due to freshwater fluxes).Figure 4.7 shows the time-mean sea surface temperature (SST) trend during theperiod of SBMOC weakening (first 30 years). The warming pattern shows a clearpolar amplification in both hemispheres.Figure 4.7: (a) Map of time-mean SST warming trends during the first 30 years (i.e. SBMOC weak-ening regime) of the CO2-doubling experiment (in K/decade), (b) Zonally-averaged time-mean SSTwarming trends over the same period.There is also a distinctive ‘dipole-like’ pattern in the SB northern high latitudes,reminiscent of the AMOC weakening ‘fingerprint’ described in [Caesar et al., 2018](although we should emphasise that this is not a dipole per se as all the trendsare positive; the blue spot near 70N merely indicates less warming, not cooling).The SB blue spot resembles the ‘warming hole’, a region of surface cooling observedin the subpolar North Atlantic that is linked to a weakening of the AMOC (e.g.[Drijfhout et al., 2012, Rahmstorf et al., 2015]). The superficial resemblance of thispattern in DDrake is consistent with the SBMOC decline (figure 4.2d).The amplified warming at the northern high latitudes leads to a reduction in con-108 Chapter 4Connecting ocean circulation with ocean heat storagevection at the sites of deep water formation. The reduction in deep water formationrates feeding the downwelling branch of the SBMOC leads to a commensurate re-duction in the SBMOC strength. This is demonstrated in figure 4.8 where we plotthe vertically-averaged convective index for the control climate (panel a) and overthe first 50 years following the CO2-doubling (panel b). The index measures thepercentage of time over which convective adjustment is active. There is a markedreduction in the average convective index of ⇠30% over the first 50 years, consistentwith the reduction in the SBMOC strength.-1-11-0.5-0.50.5YX000Z0.5-0.50.51-11-0.2 -0.1 0 0.1 0.2-1-11-0.5-0.50.5YX000Z0.5-0.50.51-11-0.2 -0.1 0 0.1 0.2-1-11-0.5-0.50.5YX000Z0.5-0.50.51-11-0.2 -0.1 0 0.1 0.2a b cFigure 4.8: Time-mean vertically-averaged convective index for (a) the control climate, (b) the first50 years following the CO2-doubling, and (c) their difference. The dark blue lines indicate the merid-ional barriers enclosing the small basin (they don’t contribute to the convection).Despite the clear reduction in the SBMOC strength, the heat storage contrast,however, remains approximately constant over the 200 years (figure 4.2c). Thus,we find that the SBMOC weakening has little, if any, impact on the heat storagecontrast. To explain this counter-intuitive result, consider the vertical heat fluxassociated with the SBMOC, approximated by ⇢0cp res✓ (in W), where ⇢0 = 1030kg m3 is a reference seawater density, cp = 3994 J K1 kg1 is the specific heatcapacity of seawater (at constant pressure), and ✓ is the temperature difference (inK) across the downwelling and upwelling branches of the circulation i.e. ✓# ✓".This choice is motivated by similar scaling arguments used to represent the MHTassociated with the AMOC (e.g. [Buckley and Marshall, 2016]). We take vertically-averaged ✓ values in the latitude bands 60-80N and 30-50S in the SB sector for✓# and ✓", respectively. Following the CO2-doubling, we consider the change in theSBMOC heat flux, HMOC = ⇢0cp( res✓). This is given by (see Appendix B):Chapter 4 109Connecting ocean circulation with ocean heat storageHMOC⇢0cp⇡ res(✓)| {z }>0+ res✓| {z }>0+ res(✓)| {z }<0(4.3)where we take the convention that a downward heat flux is positive. There aretwo processes to consider: one due to res (SBMOC weakening) and one due to(✓) (differential warming). We find that ✓# warms at a faster rate than ✓", sothat (✓) > 0 (figure 4.9). The first term in equation 4.3 is then positive, leadingto an increase in the anomalous downward heat flux.0 50 100 150 200-0.200.20.40.60.811.21.4Figure 4.9: Time series of warming responses for the small basin MOC downwelling (✓#, red) andupwelling (✓", blue) regions, and their difference ((✓), black) (in K). The downwelling regionwarms at a faster rate than the upwelling region.Now, we know that the SBMOC weakens ( res < 0), so one might think that thisprocess compensates the differential warming (as (✓) > 0). Importantly, however,in the control integration, ✓ = 0.70 K (< 0), indicating that the SBMOC isthermally direct and transports heat upwards. So, the second term in equation 4.3is in fact positive. We also concluded this more directly in Chapter 3, where weprojected the circulation into depth-temperature coordinates, demonstrating thatthe downwelling occurs at a cooler temperature than the upwelling.Both the differential warming and the SBMOC weakening processes thus contributeto an increase in the anomalous downward heat flux. Only their interaction res(✓)is negative, leading to an upward heat flux. This is because the zonally-averaged res circulates counterclockwise, with upwelling at the northern end of the SB anddownwelling at the southern end (figure 4.10). Thus, combined with an increase in110 Chapter 4Connecting ocean circulation with ocean heat storageδθ, this leads to warmer water being upwelled than downwelled.Overall, this means that ΔHMOC > 0. These terms are plotted (in W m−2) in figure4.11a. Notably, the two terms involving ΔΨres (i.e. SBMOC weakening) compensateeach other, while the dominant term is ΨresΔ(δθ), so it is the control overturningthat plays the prominent role. We acknowledge that equation 4.3 assumes thatthe latitudinal structures of ΔΨres and Ψres are similar, and this assumption isnot entirely appropriate during the period of SBMOC weakening (figure 4.10b).However, it is appropriate during the non-weakening regime (figure 4.10d), so ourresults — the ΔΨresδθ and ΔΨresΔ(δθ) terms compensate each other, and ΔHMOCincreases with time — are robust.55555 101010101010 1515151520202020-20 0 20 40 60 80Latitude-3000-2000-10000Depth (m)-20-1001020-5-5-2-2-2-2-2-20000000000-20 0 20 40 60 80Latitude-3000-2000-10000Depth (m)-10-8-6-4-2055555 101010101010 1515151520202020-20 0 20 40 60 80Latitude-3000-2000-10000Depth (m)-20-1001020-10-10-10-10-8-8-8-8-8-5-5-5-5-2-2 -2-2-2-20 00 000000-20 0 20 40 60 80Latitude-3000-2000-10000Depth (m)-10-8-6-4-20bcadFigure 4.10: Zonal-mean structures of the SBMOC response. (a) 0-30 years time-mean Ψpertres(colour), and Ψres (white dashed contours) overturning in the small basin (in Sv) during the weaken-ing regime; (b) 0-30 years time-mean ΔΨres = Ψpertres − Ψres overturning response (colour and pinkdashed contours) in the small basin (in Sv) during the weakening regime; (c-d) same as panels (a)and (b) except time-means are taken over the period 30-200 years during the non-weakening regime.In panels (a) and (c), the contour interval is 5 Sv and the zero contour is highlighted in black.The anomalous downward SBMOC heat flux increases with time, and most rapidlyduring the SBMOC weakening. Why this increase in ΔHMOC does not lead to anincrease in the heat storage contrast remains to be explained. As noted previously insection 4.3.3, if we consider the increase in air-sea heat flux (heat uptake) comparedto the increase in OHC (heat storage) in the small basin, we find that the smallbasin leaks heat across 35◦S to the southern ocean region at a rate of ∼0.5 W m−2,Chapter 4 111Find equation C.17Connecting ocean circulation with ocean heat storageand this leakage rate increases with time (figure 4.11b). For a change in OHC inthe small basin, we can write:@t(OHCSB) = FSBs +⇢0cpZat 35SZ 0HSB(v✓SB) dz dx| {z }Hleak(4.4)where FSBs is the change in net surface heat flux over the small basin, HSB isthe small basin depth, and we define Hleak as the SB heat leakage rate across35S. From figure 4.11b, we see that Hleak and HMOC compensate each other,especially on long timescales (see dashed lines). This is made even clearer in anenergy budget sense, where we find that @t(OHCSB) + HMOC ⇡ FSBs (lightblue, dashed), which implies that HMOC ⇡ Hleak. So, as the SBMOC heatflux increases, this permits more heat to penetrate the ocean surface, causing anincrease in the net surface heat flux FSBs . However, this additional heat inputis then lost to the southern ocean, which ensures that the heat storage contrastremains stationary.0 50 100 150 200-0.4-0.200.20.40.60.80 50 100 150 200-1.5-1-0.500.511.522.533.5a bFigure 4.11: (a) Decomposition of the change in SB downward (positive) MOC heat flux(( res✓), magenta) into differential warming ( res(✓), blue), SBMOC weakening ( res✓,orange), and nonlinear ( res(✓), yellow) terms (annual averages, in W m2). A vertical dot-ted line is plotted at 30 years to separate the weakening and non-weakening SBMOC regimes. (b)Decadal running means of the SB heat uptake rate (FSBs, blue), heat storage rate (@t(OHCSB),red), SBMOC heat flux (HMOC , magenta), and leakage rate to the southern ocean (Hleak =@t(OHCSB) FSBs , black) (in W m2). Note that the sum of the heat storage rate and SB-MOC heat flux (light blue, dashed) almost matches the heat uptake rate. Horizontal dashed lines arecentennial time-means.112 Chapter 4Connecting ocean circulation with ocean heat storageThis heat loss to the southern ocean is similar to the ‘redistribution temperature’response seen in [Xie and Vallis, 2012] where, in an idealised model of the Atlanticocean, the AMOC weakening serves to transport heat from the Northern Hemispherehigh latitudes towards the Southern Hemisphere. We noted previously that, acrossCMIP5 models, under historical simulations, the Southern Ocean dominates oceanheat uptake and exports approximately half of the energy it takes up northwards[Frölicher et al., 2015]. However, a weakening of this northward heat transport(perhaps as a result of ozone hole recovery in the near future) could manifest asan anomalous southward transport from the Atlantic basin to the Southern Ocean,which is consistent with our results (figures 4.6 and 4.11b).4.3.5 Parallels with CMIP5 ModelsOur analysis suggests new ways to look at the AMOC’s role in ocean heat storagein observations and more complex climate models. We performed a preliminaryanalysis of four CMIP5 models, and show that there is a heat storage contrastbetween the Atlantic and Pacific basins (defined from 30S to 65N) in abrupt CO2-quadrupling experiments (figure 4.12). Under this more intense forcing scenario, themulti-model time-mean heat storage contrast is 2.2 W m2. Looking at individualmodels, the contrast persists in the models CANESM2 and NASA-GISS-E2-H, butweakens in MIROC-ESM and GFDL-ESM2M (figure 4.12b).This could be due to different model AMOC responses and, particularly, whetherthe control model AMOC cells are thermally direct (flux heat upward, like the SB-MOC) or indirect (flux heat downward). For example, [Zika et al., 2013] diagnosedoverturning cells in UVic ESM and found that the cell coincident with the AMOCwas thermally indirect. Additionally, model AMOC cells have been found to trans-port heat downwards in CCSM4 [Zika et al., 2015], NEMOv3.4 [Saenko et al., 2018],and ACCESS-OM2 [Savita et al., 2021]. So, our results might not apply to thesemodels. However, it is still an open question whether the real-world AMOC is ther-mally direct or indirect (see Chapter 3). Nevertheless, we suggest that the AMOCis at least responsible for the existence of a contrast in each of these CMIP5 models,just as the SBMOC is responsible for the contrast in DDrake.Chapter 4 113Connecting ocean circulation with ocean heat storageFigure 4.12: (a) Atlantic vs. Pacific top-3 km column-averaged annual OHC anomalies (in J m2)immediately following an abrupt quadrupling of atmospheric CO2 in CMIP5 models (each dot rep-resents one year). Atlantic and Pacific basins defined from 30S to 65N. The deviation from theidentity line (black dashed) highlights the Atlantic’s enhanced warming rate relative to the Pacificin these experiments. (b) Decadal running means of individual model Atlantic–Pacific heat storagecontrasts @t(OHCAtl OHCPac) (in W m2).4.4 Discussion and ConclusionsIn this study, we have used the DDrake model, where the small basin is endowedwith an overturning circulation akin to that of the present-day Atlantic Ocean, tostudy the transient response of the ocean to an abrupt doubling of atmospheric CO2.We find that the small basin experiences an enhanced heat storage rate due to arapid advection of surface temperature anomalies into its interior. Similar to [Kostovet al., 2014], who found no significant correlations between the AMOC weakeningand the depth of heat storage in CMIP5 models, we find no significant relationshipbetween the small basin’s MOC weakening and its weakening heat storage rate inour set-up. Moreover, we find that the SB–LB heat storage contrast remains almostconstant during the period of SBMOC weakening, and throughout the rest of thesimulated 200 years on multidecadal timescales (figure 4.2c).Weakening of the AMOC has been seen in proxy observations (e.g. [Rahmstorf et al.,2015, Caesar et al., 2018, Zhu and Liu, 2020, Caesar et al., 2021]) and climate models(e.g. [Gregory et al., 2005, Weaver et al., 2012, Cheng et al., 2013, Kirtman et al.,2013, Weijer et al., 2020]), and recent observations from RAPID are consistent witha declining AMOC response predicted from coupled climate models [Smeed et al.,114 Chapter 4Connecting ocean circulation with ocean heat storage2018]. However, large interannual and decadal variability in the AMOC make thedetection of long-term trends difficult (e.g. [Frajka-Williams et al., 2019, Jacksonet al., 2022, Latif et al., 2022]), and it’s still unclear whether the suggested weakeningfrom proxies is reliable (e.g. [Kilbourne et al., 2022]).It has been suggested that a weakening of the AMOC in the future could lead to areduction in deep ocean heat storage, resulting in an accelerated warming of surfacetemperatures [Chen and Tung, 2018]. However, this view perhaps focuses too nar-rowly on res and, as we have found in our experiments, considering ( res✓)paints a more complicated picture, by introducing differential warming as an addi-tional process to AMOC weakening (equation 4.3). Additionally, positive correla-tions between the global-mean surface temperature and AMOC strength have beenfound in observations [Caesar et al., 2020] and simulations [Bonnet et al., 2021], im-plying that a weakening of the AMOC should lead to a reduction in surface warming,not an acceleration.This is consistent with our results as, contrary to expectations, we find that theanomalous downward SBMOC heat flux HMOC increases as the SBMOC weakens.Furthermore, we find that the dominant term in its decomposition is from differentialwarming, with the control overturning playing the prominent role (figure 4.11a).Finally, although HMOC increases, this does not lead to an increase in the heatstorage contrast, as this additional heat input is subsequently lost to the southernocean region (figure 4.11b). Thus, although the presence of the SBMOC is importantfor the small basin’s enhanced heat storage rate, the change in SBMOC strengthis surprisingly unimportant during the simulated 200 years in our experiment. Itis not known whether this SBMOC weakening affects the difference in OHC atequilibrium between the small and large basins; further simulations would be neededto address this.We emphasise that, like all modelling studies, our results are model-dependent. Thecompensation we found between the res terms in figure 4.11a very likely dependson model-specific AMOC states and responses. It should also be underlined thatthis is an idealised model: there is no bottom topography, no representation ofAntarctic Bottom Water formation, and the control SBMOC strength is 26 Sv,Chapter 4 115Connecting ocean circulation with ocean heat storagewhich is ⇠40% stronger than the real-world AMOC (e.g. [Buckley and Marshall,2016, Danabasoglu et al., 2021]), although this is not outside the CMIP5 and CMIP6model ranges [Weaver et al., 2012, Weijer et al., 2020].Observational estimates of the AMOC strength vary. Transport estimates fromRAPID at 26.5N [McCarthy et al., 2015] give values ⇠18 Sv (e.g. [Smeed et al.,2018]), while estimates from the Meridional Overturning Variability Experiment(MOVE) at 16N [Send et al., 2011] give values ⇠24 Sv (e.g. [Frajka-Williamset al., 2018, Danabasoglu et al., 2021]). Although, the MOVE array is designed tomeasure variability rather than the absolute value.The control SBMOC strength is quite high. Nevertheless, our results rest on theSBMOC weakening and not its initial value. Many studies have shown a relation-ship between the control AMOC strength and the subsequent AMOC weakeningin climate change simulations, with a stronger AMOC generally leading to greaterweakening (e.g. [Gregory et al., 2005, Gregory and Tailleux, 2011, Weaver et al.,2012, Winton et al., 2014, Weijer et al., 2020]). Thus, an overprediction of theSBMOC strength likely means that the SBMOC weakening is also an overpredic-tion. This lends further support to our results, as the weakening doesn’t impact theSB–LB heat storage contrast despite being significantly large.Finally, the real-world MOC is three-dimensional, spanning multiple basins withinterbasin transports (e.g. [Talley, 2013, Ferrari et al., 2017, Newsom and Thompson,2018]). This may also be true for the DDrake MOC, but, in our analysis, we’veimplicitly assumed that the overturning circulations in the small and large basinsare not strongly connected. In section 4.3.3, we found that the SB leaks heat tothe LB. This is similar to a recent study from [Sun et al., 2022] who found that theAtlantic leaks heat to the Indo-Pacific as a result of AMOC weakening in a CO2-quadrupling experiment in the CCSM4 model. They attribute this result specificallyto an interbasin overturning circulation, and the same explanation could apply toour result.If so, this points to an interbasin overturning circulation connecting the small andlarge basins in DDrake, which makes it difficult to attribute the existence of theSB–LB heat storage contrast solely to the SBMOC. However, as we saw in section116 Chapter 4Connecting ocean circulation with ocean heat storage4.3.2, we associated the SB’s enhanced heat storage rate with deep convection atthe northern high latitudes i.e. the downwelling branch of the SBMOC. If theDDrake MOC is interbasin, it might take thousands of years for a heat anomaly fromthe SB northern high latitudes to traverse the SB and enter the LB (e.g. transittimes through overturning circulation components reach ⇠3600 years in ECCOv4[Rousselet et al., 2021]). So, our simulation length of 200 years is likely too shortfor this interbasin effect to have an impact on the heat storage contrast.To conclude, in this simplified set-up, we were able to provide a conceptual graspon how a deep overturning circulation affects a basin’s heat storage rate, and thecomparison with CMIP5 models in figure 4.12 is encouraging. A key conclusionthat extends beyond the specifics of DDrake is that the impact of the AMOC onOHS is driven not only by the AMOC’s strength. The temperature structure andits changes must also be considered for a complete evaluation of the AMOC’s role.Our results underline the importance of the AMOC in ocean heat storage, and for itsaccurate representation in other, predictive climate models. Continued observationalmonitoring efforts such as RAPID [McCarthy et al., 2015], MOVE [Send et al., 2011],the South Atlantic MOC Basin-wide Array (SAMBA) [Meinen et al., 2018], and theOverturning in the Subpolar North Atlantic Program (OSNAP) [Lozier et al., 2019],in conjunction with more advanced high-resolution climate models, should drive adeeper understanding of the AMOC. But we are also encouraged by the use ofsimpler, more conceptual models such as DDrake in order to make sense of thisincreasing complexity.4.4.1 Key points1. A deep meridional overturning circulation (MOC) connects the ocean surfaceto its interior, enhancing heat storage rate under global warming.2. This circulation in the Atlantic (AMOC) may enable its enhanced heat storagerate relative to the Pacific in recent decades.3. AMOC weakening may not necessarily impact ocean heat storage rate due tocompensating physical processes.Chapter 4 117Connecting ocean circulation with ocean heat storage118 Chapter 4“Essentially all models are wrong, but someare useful.”George Box, 19795Representation of OHU inTwo-Layer EBMsCurrent two-layer energy-balance models (EBMs) successfully emulate the global-mean surface air temperature responses of CMIP5 models in abrupt 4⇥CO2 exper-iments [Geoffroy et al., 2013b], yet little attention has been paid to how well theyemulate the subsurface ocean temperature response. Using the fitting procedureoutlined in [Geoffroy et al., 2013b], we fit a two-layer EBM to the temperature re-sponses of the DDrake configuration under an abrupt CO2-doubling. We find that wecan interpret the fitted EBM parameters physically, in light of what we know aboutOHU processes in DDrake from Chapters 3 and 4. Unlike in previous approaches,which focus primarily on surface warming, we assess both the upper- and lower-layerEBM-fitted responses, and find that the fits for the lower-layer (subsurface ocean)are poor. This indicates that the current EBM formulation is unsatisfactory, andany inferred subsurface ocean warming from the EBM is misleading.We attempt to improve the representation of subsurface warming in the traditionaltwo-layer EBM by adjusting the power-law dependence of the heat exchange betweenthe two layers. Increasing or decreasing this power-law dependence conceptually in-119Connecting ocean circulation with ocean heat storagecreases or decreases the sensitivity of OHU to the stratification in the EBM. We findthat increasing the sensitivity (quadratic power-law) improves the representation ofsubsurface warming in DDrake and in 4 out of 5 CMIP5 models analysed underabrupt CO2-increase experiments. Decreasing the sensitivity (2/3 power-law) leadsto a poorer representation of subsurface warming across all models analysed. Thisdifference in behaviour between the quadratic and 2/3 power-laws might highlightthe importance of the stratification for OHU, but why a quadratic power-law shouldfare better than a linear one remains to be explained.5.1 IntroductionCoupled atmosphere-ocean general circulation models (AOGCMs) have grown greatlyin complexity, commensurate with advances in computing technologies over thelast few decades. They are tremendously useful and comprehensive tools to studythe climate and perform projections, yet they are computationally very expensive.Furthermore, the resulting overwhelming complexity of state-of-the-art AOGCMsmeans that no single person can wholly understand what one of these models isdoing. As we’ve seen in Chapters 3 and 4, even a greatly simplified, highly idealisedaquaplanet model like DDrake still exhibits rich, complex behaviour, with multipledistinct circulation processes playing an important role in OHU. An alternative isto use simple climate models to emulate the AOGCM responses. These are far sim-pler to understand, and can cover a wider range of potential climate scenarios withnegligible computational cost.One branch of simple climate models encompasses physically-based energy balancemodels (EBMs). Energy balance models have been used extensively in climatescience to aid the understanding of heat transfers within the climate system (e.g.[Budyko, 1969, Sellers, 1969, Hoffert et al., 1980, Murphy, 1995]). Alongside theadvancement of AOGCMs, EBMs have been used to emulate and make sense ofAOGCM results (e.g. [Schlesinger and Jiang, 1990, Raper and Cubasch, 1996, Raperet al., 2001]) and, more recently, to help identify and quantify the sources of spread inclimate change projections (e.g. [Padilla et al., 2011, Geoffroy et al., 2012, Bodmanand Jones, 2016, Colman and Soldatenko, 2020]).120 Chapter 5Connecting ocean circulation with ocean heat storageEBMs provide a useful framework for summarising and comparing bulk physicalproperties of different AOGCMs (e.g. [Raper et al., 2002, Gregory and Forster,2008, Geoffroy et al., 2013b]), and the parameters describing these properties canbe interpreted in terms of physical processes (e.g. [Kostov et al., 2014]) so mayprovide insight into the underlying physics.A recent popular EBM is a form of two-layer EBM, which is capable of capturingthe ‘two-timescale’ surface temperature response seen in abruptly-forced AOGCMexperiments [Held et al., 2010, Geoffroy et al., 2013b]. In this two-layer EBM,the climate system is aggregated and split into two vertical layers: the first layerrepresents the atmosphere, land-surface, and oceanic mixed-layer; and the secondlayer represents the deeper ocean below.Due to the ocean’s overwhelming heat capacity compared to that of the atmosphereand land-surface, one can approximate the upper-layer as simply representing theoceanic mixed-layer, making the EBM an ocean-only model (as was done previouslyin [Kostov et al., 2014]). We will do the same in this chapter. This allows us toconcentrate on physical processes within the ocean, transferring energy from themixed-layer to the thermocline and deeper waters below.The state of each layer is described by the temperature perturbations T and T0, andthese change according to a net surface heat flux at the ocean surface and a heatexchange between the two layers. Following [Geoffroy et al., 2013b], the temperatureperturbations obey the equations:CdTdt= F ↵T (T T0)C0dT0dt= (T T0)(5.1)Here, F is an effective forcing at the ocean surface (in W m2); for an abruptCO2-doubling, the resulting forcing F is, to a good approximation, a constant.The parameter ↵ encompasses different radiative feedbacks of the system, and parameterises the heat exchange (or thermal coupling) between the two layers. Theconstants C and C0 are effective heat capacities for each layer (in J m2 K1). AChapter 5 121Connecting ocean circulation with ocean heat storageFigure 5.1: Diagram of the simple two-layer energy balance model. Arrows represent energy fluxes.F is the effective forcing at the ocean surface, ↵ is the globally-averaged climate feedback parame-ter, is the heat exchange parameter, T and T0 are temperature perturbations, and C and C0 areeffective heat capacities.diagram of the energy fluxes in the EBM is shown in figure 5.1.Equations 5.1 can be solved to give (see Appendix C for derivation):T (t) = Teq(1 afet/⌧f aset/⌧s)T0(t) = Teq(1 fafet/⌧f saset/⌧s)(5.2)where Teq = F/↵ is the temperature at equilibrium, and the parameters af , as, ⌧f , ⌧s,f , and s are as defined in [Geoffroy et al., 2013b]. The parameters ⌧f and ⌧s aredistinct fast and slow timescales, respectively. The fast timescale corresponds tothe timescale of adjustment of the upper-layer to the equilibrium temperature (afew years to a decade), while the slow timescale corresponds to a joint adjustmentof the upper and lower layers (order of centuries). These roughly correspond totimescales of adjustment for the oceanic mixed-layer and the top 2-3 km of theocean, respectively.The parameters af , as,f , and s are ‘mode parameters’ that partition the adjust-ment towards equilibrium into the fast and slow components such that af + as = 1and faf + sas = 1. As t ! 1, both T and T0 tend towards the equilibriumtemperature Teq, under the two timescales.Can we accurately emulate OHU in DDrake using this EBM (equation 5.1)? TheEBM has found success in emulating the forced surface temperature response inother AOGCMs (e.g. [Held et al., 2010, Geoffroy et al., 2013b, Rohrschneider et al.,122 Chapter 5Connecting ocean circulation with ocean heat storage2019]), but the emulated subsurface temperature response (i.e. the OHU) has re-ceived less attention. The problem of subsurface warming emulation was recentlyhighlighted by [Nadiga and Urban, 2019], who found that this EBM could not simul-taneously reproduce both the surface and subsurface ocean warming of an AOGCMunder an abrupt 4⇥CO2 forcing. If the two-layer EBM is to be used as a physically-based emulator, we would like it to be able to reproduce both the surface andsubsurface warming.We first use the two-layer EBM in equations 5.1 (the ‘Geoffroy EBM’) to fit thesurface and subsurface temperature responses of DDrake forced by an abrupt CO2-doubling. The fitting procedure is briefly described in section 5.2.1, and section5.2.2 presents results using this traditional EBM framework. We find that we caninterpret the fitted EBM parameters physically, in light of what we know aboutDDrake from Chapters 3 and 4. However, we also find that the Geoffroy EBM isunable to represent the subsurface warming in DDrake, confirming the shortcomingof this EBM highlighted by [Nadiga and Urban, 2019].We attempt to improve the representation of subsurface warming by modifying thepower-law for the heat exchange between the two layers, ⇤(T T0)⇣ , in section5.3. In the Geoffroy EBM, the heat exchange is linear i.e. ⇣ = 1. By increasingthe power-law dependence to ⇣ = 2, we improve the representation of subsurfacewarming for DDrake and 4 out of 5 other CMIP5 models under abrupt CO2-increaseexperiments. Decreasing the power-law dependence to ⇣ = 2/3 (suggested fromquasigeostrophic theory [Rhines and Young, 1982b]) worsens the representation ofsubsurface warming across all models. Some conclusions are offered in section 5.4.5.2 The traditional EBM framework5.2.1 Fitting procedureTo begin our investigation, we follow the same procedure as in [Geoffroy et al.,2013b], where they successfully fit this two-layer EBM (equations 5.1) to CMIP5model surface air temperature (SAT) responses to an abrupt CO2-quadrupling.Here, we fit to annually-averaged SAT anomalies relative to the DDrake controlChapter 5 123Connecting ocean circulation with ocean heat storageclimate rather than monthly snapshots. This does not impact our results.The radiative parameters F and ↵ are found by linearly regressing the first 50years of the net radiative imbalance against the global mean surface air temperatureresponse, following the method of [Gregory et al., 2004]. The mode parameters arefound making use of the solutions (equations 5.2), and C,C0, and are then foundthrough analytical relationships between the fitted parameters (see [Geoffroy et al.,2013b] for the full procedure):C =↵af/⌧f + as/⌧s(5.3)C0 = ↵(⌧faf + ⌧sas) C (5.4) =C0⌧fas + ⌧saf(5.5)Throughout this procedure, only the global mean surface air temperature responseis used. The EBM radiative and mode parameters are fitted to each CMIP5 modeloutput in order to reproduce the SAT response. Arguably, this is a practical choice ashumans live in the atmospheric surface boundary layer, so the SAT is most relevantto us when making climate projections. However, if we wish to be consistent withthe meaning of the symbols in the two-layer EBM, T is not the SAT response; it isthe upper-layer temperature response, which in this case is the oceanic mixed-layertemperature.The logic is thus flawed here, as the authors in [Geoffroy et al., 2013b] conflate theEBM’s predicted mixed-layer temperature with a surface air temperature, and theseare quite different things (if they were the same, there would be no atmosphere-oceansensible heat fluxes). Moreover, the EBM-predicted T0 and its correspondence withthe deep temperature response in the CMIP5 models is not examined in [Geoffroyet al., 2013b].124 Chapter 5Connecting ocean circulation with ocean heat storage5.2.2 Results with traditional EBM: DDrake fitted responsesHere, we present results for the two-layer ‘Geoffroy EBM’ fits to the DDrake tem-perature responses to an abrupt CO2-doubling. The fitting procedure is performedfor the globally-averaged response, and also for the basin-average responses for thesmall and large basins, and the model southern ocean. The effective layer heat ca-pacities C and C0 (in W yr m2 K1) are found using equations 5.3 and 5.4, andthe corresponding upper- and lower-layer thicknesses (h and H, respectively) canthus be found using:h =86400⇥ 360⇥ C⇢0cpH =86400⇥ 360⇥ C0⇢0cp(5.6)where ⇢0 is a reference seawater density (kg m3), and cp the specific heat capacity ofseawater at constant pressure (J kg1 K1). In this way, the fitted EBM parametersgive ‘predictions’ for the mixed-layer depth (h) and the depth to which the deep-ocean is thermally involved over the 200 years (h+H).Note that it is difficult to pin down what the lower-layer in the EBM specificallyrepresents, as it is simply treated as a large heat sink. The lower-layer thicknessH merely arises from the fitting procedure, and the fitted parameters are optimisedto the SAT. In [Geoffroy et al., 2013b] (and in many other studies e.g. [Gregory,2000, Held et al., 2010, Kostov et al., 2014, Rohrschneider et al., 2019]), the lower-layer is simply called the ‘deep ocean’ or ‘lower ocean’. We will continue using thelanguage ‘deep ocean’ to refer to the lower-layer, which we can take to mean ‘allocean layers below the mixed-layer’.A key aim of this chapter is to critically evaluate the upper- and lower-layer temper-ature responses given by the EBM, and hence the representation of OHU. We do thisby plotting the vertically-averaged DDrake temperature responses that correspondto the predicted layer thicknesses (equation 5.6). Using these plots, we can assesshow well the EBM actually captures the subsurface ocean warming i.e. the transferof heat from the mixed-layer to the thermocline and deeper waters below.Chapter 5 125Connecting ocean circulation with ocean heat storageThis transfer is characterised by the heat transfer coefficient (in W m2 K1),calculated using equation 5.5. The thermal physical parameters (along with thefitted slow and fast timescales) are displayed in table 5.1. Radiative parameters andcorresponding equilibrium temperatures are shown in table 5.2.C C0 h H ⌧s ⌧f(W yr m2 K1) (W yr m2 K1) (m) (m) (W m2 K1) (yr) (yr)Global 20.3 108.1 153 817 1.02 185 7.9SB 21.5 113.6 163 859 1.48 141 6.1LB 21.3 109.0 161 824 1.39 141 6.3SO 14.7 98.8 111 747 0.55 313 10.7Table 5.1: Upper-layer heat capacity C, lower-layer heat capacity C0, the corresponding upper- andlower-layer thicknesses h and H, heat exchange coefficient , and fitted slow and fast timescales ⌧sand ⌧f for each of the DDrake basins, following the fitting procedure in [Geoffroy et al., 2013b].F ↵ Teq(W m2) (W m2 K1) (K)Global 4.44 1.48 3.00SB 4.64 1.92 2.42LB 4.73 1.90 2.49SO 3.93 0.79 4.97Table 5.2: Calculated radiative parameters for each of the DDrake basins using the SAT, followingthe fitting procedure in [Geoffroy et al., 2013b]: the radiative forcing F , the climate feedback ↵, andthe corresponding equilibrium temperature Teq.The global value for h+H gives a total ocean depth of 970 m, close to the CMIP5ensemble mean found in [Geoffroy et al., 2013b] of 1182 m. The fitted physical pa-rameters in table 5.1 are largely consistent with our findings from previous chapters.The stratification in the SB is weaker than in the LB (Chapter 2). We know thatdeep convection is localised to the SB northern high latitudes, which enables anoverturning circulation in the SB akin to the AMOC (SBMOC, Chapter 4). Theseresults suggest that the fitted mixed-layer and deep-ocean layer thicknesses (h andH) should be largest in the SB. This is what we see in table 5.1 (columns 3 and 4).In Chapter 4, we determined that the SBMOC is responsible for the small basin’senhanced heat storage rate, as it connects the SB surface to its interior. We alsoattributed the existence of the SBMOC to the heat storage contrast between thesmall and large basins. The heat exchange coefficient characterises the couplingbetween the two layers in the EBM. As the surface and interior are more strongly126 Chapter 5Connecting ocean circulation with ocean heat storageconnected in the SB due to the SBMOC, this suggests that the fitted should belargest for the SB, which is what we see in table 5.1 (column 5).This is also consistent with our findings from Chapter 3. There, we associated theSBMOC with a deep thermal overturning circulation ( ✓z) that spans the full depthof the basin. Thermal overturning circulations in the LB reach approximately mid-depth, while the thermal circulation in the SO reaches ⇠500 m depth (figure 3.6). Ifwe view the ✓z depths as a measure of H, then we expect that the order of H fromlargest to smallest should go: SB, LB, SO. If we view the ✓z depths as a measureof the thermal coupling between the mixed-layer and deeper ocean, then we expectthat from largest to smallest should also go: SB, LB, SO. This is precisely whatwe see in table 5.1 (columns 4 and 5).Notably, the longest adjustment timescales are seen for the SO — almost twiceas long as those for the small and large basins (table 5.1 columns 6 and 7). Thisis consistent with the relatively slow response of the real-world Southern Ocean(e.g. [Li et al., 2013, Marshall et al., 2014a]), due to wind-driven upwelling of old,unmodified deep water dampening the surface warming around Antarctica [Armouret al., 2016]. It’s possible that a similar mechanism exists in DDrake, but the lack ofan AABW mass or a realistic Lower Cell of the MOC makes this unlikely. Furtheranalysis would be needed to address this.Nevertheless, we can interpret the fitted EBM parameters in terms of the DDrakeocean circulation. The fitted parameters thus have physical meaning, and this lendsfurther support to the EBM’s use as a physically-based emulator. In spite of this,however, the resulting fitted temperature responses are poor.Figure 5.2 shows the fitted EBM temperature responses. The EBM (black dashedlines) reproduces the DDrake SAT response not only globally, but also regionally foreach basin. This is unsurprising as the SAT response is what the EBM radiative andmode parameters were fitted to. However, if we compare the EBM fits to the SSTresponses, a good fit is only seen for the southern ocean (figure 5.2, panel d), wherethe SST and SAT are comparable. Globally, and in the small and large basins, theSAT is ⇡ 0.2 K warmer than the SST.Even worse, the vertically-averaged temperature responses corresponding to theChapter 5 127Connecting ocean circulation with ocean heat storage0 50 100 150 20000.511.522.53SSTSATTT00 50 100 150 20000.511.522.50 50 100 150 20000.511.522.50 50 100 150 20000.511.522.533.54dbacFigure 5.2: Time-series of annually-averaged surface air temperature (SAT, orange line) and seasurface temperature (SST, blue line); and vertically-averaged upper-layer temperature (T , yellow line)and lower-layer temperature (T0, purple line) corresponding to the derived layer thicknesses presentedin table 5.1 for the (a) global, (b) small basin, (c) large basin, and (d) southern ocean regions ofDDrake, relative to the control climate state. The fitted Geoffroy EBM responses for T and T0 areshown as black dashed lines.EBM-predicted upper-layer thicknesses h as presented in table 5.1 (yellow lines)show very little correspondence with the upper-layer black dashed fits. The same istrue for the lower-layer responses, corresponding to the EBM-predicted lower-layerthicknesses H (purple lines). As mentioned in section 5.2.1, if we wish to be consis-tent with the original formulation of the two-layer EBM, the fitted black dashed linesfor T and T0 should correspond to the yellow and purple lines respectively in figure5.2. Evidently, they do not. So, although the SAT response is well-reproduced, theother curves suggest this two-layer EBM poorly represents OHU in DDrake.Root mean-square errors for each of these fits are shown in table 5.3. The upper-layer fit is evaluated against the upper-layer T response (yellow lines), as well asthe SAT response (orange lines) for comparison. The lower-layer fit is evaluatedagainst the lower-layer T0 response (purple lines). The small basin and southern128 Chapter 5Connecting ocean circulation with ocean heat storageocean span the extremes; the small basin has the poorest upper-layer fit, but thebest lower-layer fit, while the southern ocean is the opposite. Taking both layerstogether, the EBM captures the small and large basin responses almost equally well,with the worst overall fit found for the southern ocean.SAT Upper-layer T Lower-layer T0 Total (Upper + Lower)RMSE (K) RMSE (K) RMSE (K) RMSE (K)Global 0.07 0.45 0.23 0.68SB 0.07 0.47 0.21 0.68LB 0.06 0.39 0.28 0.67SO 0.12 0.37 0.40 0.77Table 5.3: Root mean-square errors (RMSEs) for the EBM fits for the upper- and lower-layer tem-perature responses T and T0 for each basin of DDrake. The RMSE for the upper-layer EBM fit isalso evaluated against the SAT response, shown in the left-most column for comparison.5.3 A new EBM frameworkWe’ve now seen that the traditional two-layer EBM (equations 5.1) cannot representthe subsurface warming in DDrake (e.g. figure 5.2). To remedy this, we suggestmodifying the heat exchange between the two layers by introducing a power, ⇣:CdTdt= F ↵T ⇤(T T0)⇣C0dT0dt= ⇤(T T0)⇣(5.7)Here, we’ve recast as ⇤ to reflect the change in dimensions this coefficient willhave (dependent on ⇣). In the traditional Geoffroy EBM (equations 5.1), the heatexchange is linear, so ⇣ = 1.The temperature difference between the two layers (T T0) can be interpreted asa crude representation of the change in stratification. A stronger/weaker temper-ature difference might indicate a stronger/weaker stratification. In the real ocean,a stronger stratification further isolates the upper ocean from waters below the py-cnocline, thus hindering OHU (e.g. [Kuhlbrodt and Gregory, 2012, Marshall andZanna, 2014, Bourgeois et al., 2022]).Chapter 5 129Connecting ocean circulation with ocean heat storageIf the upper-layer in the EBM represented the pycnocline and the lower-layer rep-resented the deeper ocean below (like in the model of [Gnanadesikan, 1999]), thenan increase in the temperature difference would entail an increase in the stratifica-tion. But this is not the case. In the two-layer EBM, the upper-layer represents themixed-layer, and the lower-layer represents the ocean below the mixed-layer, whichincludes the pycnocline. So, we cannot simply link a stronger/weaker temperaturedifference with a stronger/weaker stratification in the EBM.In the EBM, a stronger temperature difference results in a larger heat exchangebetween the two layers (for constant ⇤), thus enhancing OHU. For the GeoffroyEBM solution to a step-function forcing (equations 5.2), this temperature differencegrows to a peak on the fast timescale ⌧f and decays to zero over the slow timescale⌧s. It is thus the decrease in the temperature difference over the slow timescale thathinders OHU in the EBM, mimicking the effect of an increasing stratification.Going back to equations 5.7, increasing or decreasing ⇣ conceptually increases ordecreases the sensitivity of OHU to the stratification. Given that a decrease in thelayer temperature difference mimics the effect of an increase in the stratification, it isnot clear whether increasing ⇣ or decreasing ⇣ will improve the EBM’s performancein emulating the DDrake subsurface warming. We will try both, first increasing thesensitivity with ⇣ = 2, and then decreasing the sensitivity with ⇣ = 2/3.5.3.1 New fitting procedureIn section 5.2.1, the fitting of the mode parameters for the Geoffroy EBM relieson prior knowledge of the analytical solution to equations 5.1, but we do not knowwhat the analytical solution is for equations 5.7 with an arbitrary ⇣. Also, in orderto compare the two different EBM formulations, we should use the same fittingprocedure for both; hence we cannot follow the same procedure as in section 5.2.1.Instead, we take a more pragmatic approach.The radiative parameters F and ↵ are still found in the same way using the methodoutlined in [Gregory et al., 2004], but fitting to the mean SST rather than the SATresponse. For a given value of ⇣, we co-vary the other model parameters ⇤, h, andH aiming to minimise the RMSE of the fit to the SST response, using an interior130 Chapter 5Connecting ocean circulation with ocean heat storagepoint algorithm [Byrd et al., 2000]. These three parameters are subject to lower andupper bounds to eliminate unphysical solutions: 0 < ⇤ (W m2 K⇣) < 10, 10 < h(m) < 500, and 100 < H (m) < 3000. The optimal triple (⇤, h,H) that minimisesthe RMSE is then taken along with the radiative parameters F and ↵ to define thefit.This fitting method is in keeping with the spirit of the initial fitting procedure asoutlined in section 5.2.1, as the calibration data used comprises only the surfacewarming (SST) response and net radiative imbalance. The effective heat capacitiesC and C0 (and hence the layer thicknesses) are determined a posteriori, as before.This approach allows for a direct assessment of the EBM’s ability in inferring thedeep ocean response, as C0 arises from the fitting procedure; no information aboutthe true deep ocean response is given.5.3.2 Increasing the sensitivity (⇣ = 2)We first try increasing the sensitivity to (T T0) by setting ⇣ = 2. This makes theheat exchange in the EBM quadratic, rather than linear, giving the new ‘quadraticEBM’:CdTdt= F ↵T ⇤(T T0)2C0dT0dt= ⇤(T T0)2(5.8)Fitted parametersUsing the fitting procedure outlined in section 5.3.1, we fit both the Geoffroy andquadratic EBMs to the DDrake temperature responses. The five fitted model pa-rameters (and corresponding layer thicknesses) for each DDrake basin, for both theGeoffroy and quadratic EBMs, can be found in table E.1 in Appendix E. The fittedparameters are qualitatively very similar to what was found before. The upper-layer heat capacities are approximately an order of magnitude smaller than thelower-layer, however the C and C0 values are systematically larger for the quadraticChapter 5 131Connecting ocean circulation with ocean heat storageEBM compared to the Geoffroy EBM, with the ratio C/C0 being slightly smaller(except in the southern ocean). Again, the largest values of (⇤) and H are foundfor the small basin where there is a deep overturning circulation, consistent with thesmall basin’s enhanced heat storage rate (Chapter 4).DDrake fitted responsesA comparison of the newly fitted Geoffroy and quadratic EBM responses for DDrakefrom the interior point algorithm method is shown in figure 5.3. Clearly, thequadratic EBM does a better job than the Geoffroy EBM at capturing the deep-ocean temperature response globally, and for the small and large basins (blue linesin figure 5.3, panels a-c). It is less clear whether a cost is incurred with regard tothe upper-layer responses.Root mean-square errors for each of these fits are shown in table 5.4. The upper-layerfits are evaluated against the implied upper-layer responses (red lines) as well as theSST responses (magenta lines) for comparison. The lower-layer fits are evaluatedagainst the implied lower-layer responses (blue lines). Almost no cost is incurredupon moving from the Geoffroy EBM to the quadratic EBM as the RMSEs for theupper-layer T responses are identical for each EBM (to two significant figures). Atthe same time, large gains are made for the lower-layer T0 responses (except in thesouthern ocean).SST Upper-layer T Lower-layer T0 Total (Upper + Lower)RMSE (K) RMSE (K) RMSE (K) RMSE (K)Global 0.07; 0.07 0.23; 0.23 0.19; 0.10 0.42; 0.33SB 0.06; 0.06 0.20; 0.20 0.12; 0.05 0.32; 0.25LB 0.05; 0.05 0.14; 0.14 0.23; 0.09 0.37; 0.23SO 0.12; 0.13 0.41; 0.41 0.57; 0.59 0.98; 1.00Table 5.4: Root mean-square errors (RMSEs) for the (Geoffroy; quadratic) EBM fits for the upper-and lower-layer temperature responses T and T0 for each basin of DDrake. The RMSE for the upper-layer EBM fit evaluated against the SST response is also shown in the left-most column for compari-son.132 Chapter 5Connecting ocean circulation with ocean heat storageFigure 5.3: Comparison of fitted EBM responses (black dashed) for the Geoffroy EBM (left column)and quadratic EBM (right column). Presented are time-series of annually-averaged SST responses(magenta), and vertically-averaged upper-layer (red, T ) and lower-layer (blue, T0) temperature re-sponses corresponding to the layer thicknesses h and H implied by the EBM (see table E.1), for theDDrake CO2-doubling experiment. Panels (a-d) correspond to the different basins of DDrake: (a)Global responses, (b) Small basin responses (blue shading), (c) Large basin responses (red shading),and (d) southern ocean responses (yellow shading).Chapter 5 133Connecting ocean circulation with ocean heat storageFigure 5.4: Percentage improvement in the RMSE of the fitted DDrake temperature responses afterchanging from the Geoffroy EBM to the quadratic EBM. The change improves fits for all basins butthe southern ocean. A substantial increase (> 50%) is seen for the global lower-layer temperatureresponse.It is more instructive to see the percentage change improvement in the RMSE of thefits upon moving from the Geoffroy EBM to the quadratic EBM (using the samefitting procedure):Improvement =RMSEGeoffroy RMSEquadraticRMSEGeoffroy=RMSERMSE(5.9)This performance improvement is shown in figure 5.4. As deduced from table 5.4,almost no change is seen for the upper-layer fits. However, large and substantialimprovements are seen for the lower-layer fits. Notably, the switch to the quadraticEBM improves the global-average fits, as well as fits for the small and large basins.Only in the southern ocean is no improvement seen (actually, there is a small de-cline). Switching to the quadratic EBM leads to a global improvement of +23%,with a striking improvement of +51% for the lower-layer response. The largest in-creases (⇠+60%) are seen for the lower-layer responses of the small and large basins,with the large basin enjoying the largest overall improvement of +38%.Note that we have also introduced an improvement in the lower-layer fits simply bychanging the fitting procedure. For example, comparing tables 5.3 and 5.4, we see134 Chapter 5Connecting ocean circulation with ocean heat storagethat the global lower-layer RMSE has changed from 0.23 to 0.19 — same GeoffroyEBM, different fitting procedure. This is only an improvement of +17%, so palesin comparison to the (further) +51% improvement we find upon moving from theGeoffroy EBM to the quadratic EBM (0.19 to 0.10 in table 5.4). Nevertheless, thisconcealed improvement due to the change in fitting procedure is worth mentioning.Results with CMIP5 modelsWe see large improvements for the lower-layer fits upon moving to the quadraticEBM (increasing ⇣ from 1 to 2) for DDrake (figure 5.4). However, this is only for anidealised configuration of the MITgcm. We now test the quadratic EBM on fittingthe temperature responses from a suite of five CMIP5 models under abrupt 4⇥CO2experiments. We take the radiative parameters F and ↵ from [Kostov et al., 2014],who tabulated their fitted EBM parameters in their supporting information. Likeus, they conceptualised the two-layer EBM as an ocean-only model, and so theirradiative forcing values F include a contribution from the heat exchange betweenthe land and ocean domains. This is not the case for the fitted values for CMIP5models found in [Geoffroy et al., 2013b], so we do not use their fitted parameters.We find the other model parameters (⇤), h, and, H in the same way as describedin section 5.3.1, by co-varying each of them using an interior point algorithm, withthe aim to minimise the RMSE of the fit to the CMIP5 model SST response. Theparameters (⇤) and h are subject to the same upper and lower bounds as before;H is now allowed to vary between 100 and 5000 m, on account of 3000 m no longerbeing the maximum depth, as is the case in DDrake. Our fitted parameter values canbe found in table E.2 in Appendix E. As found before when fitting to the differentbasins of DDrake (table E.1), the quadratic EBM produces C and C0 values thatare systematically larger than those found using the Geoffroy EBM.We note that the C and C0 values (for both EBMs) are unusually large, with verythick corresponding layer thicknesses. The ensemble mean C and C0 (as well as )are ⇠5 times larger than those found in [Geoffroy et al., 2013b]. We suggest this isdue to methodological differences. However, there is also large intermodel spread inthe fitted thermal parameters ((⇤), h,H). We note that the model MRI-CGCM3Chapter 5 135Connecting ocean circulation with ocean heat storageis an outlier with regard to our fitting method, as the fitted h from the quadraticEBM is the maximum allowed by the procedure (500 m), and the fitted H fromboth EBMs is the maximum of 5000 m. This gives this particular model especiallyhigh values for C and C0.Using these fitted parameters, we first check whether the quadratic EBM can skilfullyrepresent the surface warming (SST) response. As mentioned in section 5.2.1, wenote this is logically inconsistent as we should be comparing the upper-layer Tresponses, but this is how two-layer EBMs have typically been used (e.g. [Heldet al., 2010, Geoffroy et al., 2013b, Kostov et al., 2014]).Figure 5.5: Comparison between the annual-average global-mean SST response from CMIP5 modelabrupt 4⇥CO2 experiments, and the upper-layer T responses fitted from the two-layer (a) Geoffroyand (b) quadratic EBMs. Points for the abrupt CO2-doubling experiment in DDrake are also plottedfor comparison.Figure 5.5 plots the fitted T responses from both the Geoffroy and quadratic EBMsagainst the global mean SST responses from abrupt 4⇥CO2 CMIP5 model exper-iments. Panel (a) confirms the findings of [Geoffroy et al., 2013b], demonstratingthe Geoffroy EBM’s skill in reproducing an AOGCM’s global surface temperatureresponse (SST rather than SAT, in this case). Panel (b) shows a similar perfor-mance for the quadratic EBM, apart from a few points in GFDL-ESM2M withoverestimated warming (red crosses).Comparing the actual AOGCM vertically-averaged layer temperature responses, cor-responding to the inferred layer thicknesses h and H in table E.2, paints a differentpicture. This is shown in figure 5.6, and we see that both EBMs systematically over-136 Chapter 5Connecting ocean circulation with ocean heat storageFigure 5.6: Comparison between the inferred upper-layer T responses (top row) and lower-layer T0responses (bottom row) from CMIP5 models with the corresponding fitted responses from both theGeoffroy (left column) and quadratic (right column) two-layer EBMs. Points for the abrupt CO2-doubling experiment in DDrake are also plotted for comparison.estimate the AOGCM warming response in both the upper and lower layers. It isunclear from this plot whether the quadratic EBM better represents the upper-layerresponse (top row of figure 5.6), but there is a subtle shift of the curves towardsthe black dashed line in panel (d) (comparing with panel (c)), indicating that thequadratic EBM better represents the lower-layer temperature response.To see this more directly, we tabulate the RMSEs for each of these fits in table 5.5,just as we did for the DDrake basin fits in table 5.4. The upper-layer EBM fits areagain evaluated against the inferred upper-layer AOGCM responses, as well as theAOGCM SST responses for comparison. The lower-layer EBM fits are evaluatedagainst the inferred lower-layer AOGCM responses.The RMSEs for each EBM are plotted against each other in figure 5.7, and theChapter 5 137Connecting ocean circulation with ocean heat storageSST Upper-layer T Lower-layer T0 Total (Upper + Lower)RMSE (K) RMSE (K) RMSE (K) RMSE (K)CNRM-CM5 0.08; 0.08 0.59; 0.62 0.17; 0.13 0.76; 0.75GFDL-ESM2M 0.09; 0.15 0.21; 0.15 0.24; 0.31 0.45; 0.46MPI-ESM-LR 0.09; 0.10 0.15; 0.18 0.34; 0.22 0.49; 0.40MRI-CGCM3 0.09; 0.11 0.30; 0.34 0.48; 0.46 0.79; 0.80NorESM-1M 0.06; 0.07 0.15; 0.21 0.22; 0.20 0.37; 0.41MITgcm DDrake 0.07; 0.07 0.23; 0.23 0.19; 0.10 0.42; 0.33Ensemble mean 0.08; 0.10 0.28; 0.30 0.29; 0.26 0.57; 0.56Std. dev. 0.01; 0.03 0.18; 0.19 0.12; 0.13 0.19; 0.19Table 5.5: Root mean-square errors (RMSEs) for the (Geoffroy; quadratic) EBM fits for the upper-and lower-layer temperature responses T and T0 from CMIP5 abrupt 4⇥CO2 experiments. TheRMSE for the upper-layer EBM fit evaluated against the SST response is also shown in the left-mostcolumn for comparison. Equivalent RMSEs for DDrake (same as the ‘Global’ row in table 5.4) areshown in the bottom row for reference. Note that the ensemble means and standard deviations don’tinclude the DDrake values.Figure 5.7: Comparison between the RMSEs of the fits found from the Geoffroy and quadratic two-layer EBMs for CMIP5 model temperature responses in abrupt 4⇥CO2 experiments. The (a) total,(b) upper-layer, and (c) lower-layer RMSEs are shown. Points corresponding to the abrupt CO2-doubling experiment in DDrake are also shown as blue diamonds.percentage improvements (equation 5.9) are shown in figure 5.8 (as was done for thebasins of DDrake in figure 5.4). From figure 5.7c, we see that DDrake plus 4 outof the 5 CMIP5 models lie below the black dashed line, showing that the quadraticEBM provides a better fit for the lower-layer temperature response (this can also beseen from table 5.5).In section 5.3.2, we found that no cost was incurred for the DDrake upper-layer fitin moving to the quadratic EBM. We can see this from figure 5.7b, as DDrake (bluediamond) lies almost exactly on the black dashed line. This is not the case for theCMIP5 models; those models that lay below the line for the lower-layer fit (panelc) now lie above the line for the upper-layer fit (panel b). This demonstrates that138 Chapter 5Connecting ocean circulation with ocean heat storageFigure 5.8: Percentage improvement in the RMSE of the fitted CMIP5 model (and DDrake) temper-ature responses after changing from the Geoffroy EBM to the quadratic EBM. The change improvesfits for the lower-layer response for 4 out of the 5 CMIP5 models analysed.a trade-off in the goodness of fits from the upper to the lower layers takes placewhen moving from the Geoffroy EBM to the quadratic EBM (GFDL-ESM2M beingthe exception, where the opposite occurs). As the quadratic EBM produces C andC0 values that are systematically larger (table E.2), the inferred warming for bothlayers is smaller. So, a previous overestimate for the lower-layer warming becomesa better fit, but the upper-layer warming may now be an underestimate, becominga worse fit.This is more easily seen in figure 5.8 — where the lower-layer improvements are posi-tive, the upper-layer improvements are negative (and vice versa for GFDL-ESM2M).The results in figure 5.4 for the individual DDrake basins were quite striking. Uponlooking at other AOGCMs, we see that the results are mixed. Apart from DDrake,only CNRM-CM5 and MPI-ESM-LR show >10% improvement in the lower-layerfits, and it is only MPI-ESM-LR that shows an overall improvement >10%. Never-theless, the quadratic EBM improves the lower-layer fits for 4 out of 5 of the CMIP5models chosen (by an average of 21%), demonstrating that this effect is not limitedto the idealised DDrake configuration.Chapter 5 139Connecting ocean circulation with ocean heat storage5.3.3 Decreasing the sensitivity (⇣ = 2/3)We’ve now seen that increasing the sensitivity of OHU to the stratification in thetwo-layer EBM (i.e. increasing ⇣ from 1 to 2) improves the representation of sub-surface warming in AOGCMs (e.g. figure 5.8). This suggests that decreasing thesensitivity will lead to the opposite effect, but we still need to test this.First, we need to choose a value for ⇣ < 1. When we increased the sensitivity, therewas a natural progression from linear (⇣ = 1) to quadratic (⇣ = 2). Here, however,there is no natural progression from linear to sublinear guiding our decision. Butwe can look to a physical justification for why we might decrease the sensitivity tothe stratification.Overlapping with the mixed-layer in the top ⇠50 m is the surface Ekman layer.In the subtropical gyres, convergence of Ekman transport leads to downwelling,or ‘Ekman pumping’. Surface heat anomalies can thus be pumped down into theocean interior by the surface winds. Such a mechanism has been found in manymodelling studies (e.g. [Zhang and Vallis, 2013, Exarchou et al., 2015, Kuhlbrodtet al., 2015, Saenko et al., 2021]). This is also consistent with our findings fromChapter 3, where we showed that heat is transported downwards by the large-scalewind-driven circulation in DDrake.We can describe the effect of this Ekman pumping OHU process using the quasi-geostrophic (QG) theory of [Rhines and Young, 1982b] for the wind-driven circula-tion. Starting with the assumption that potential vorticity is homogenised withinthe gyre [Rhines and Young, 1982a] (and ignoring relative vorticity), we can derivean expression for the vertical buoyancy transport (wb) as a function of the Ekmanpumping velocity wEk and the stratification @b/@z:wb|z=0 ="6f 20(x xE)#1/3w4/3Ek @b@z!2/3(5.10)where f0 is a reference value for the Coriolis parameter, is the variation of theCoriolis parameter with latitude, and xE is the eastern boundary of the gyre. Thefull derivation of this equation can be found in Appendix F.140 Chapter 5Connecting ocean circulation with ocean heat storageEquation 5.10 shows that the vertical buoyancy transport is proportional to w4/3Ek ,which is consistent with the idea that a stronger wind stress curl should lead togreater OHU. But the key result here is that wb is also proportional to the 2/3-power of the stratification i.e. the relationship is sublinear.We can liken the vertical buoyancy transport to the vertical heat exchange in theEBM, where ⇤ captures the effects of wEk, and (T T0) is analogous to @b/@z.This motivates a choice of ⇣ = 2/3, giving us the new ‘Rhines and Young EBM’ or‘RY EBM’:CdTdt= F ↵T ⇤(T T0)2/3C0dT0dt= ⇤(T T0)2/3(5.11)Fitted parametersLike before, we fit the RY EBM to the DDrake temperature responses, using thefitting procedure outlined in section 5.3.1. The fitted parameters for each DDrakebasin can be found in table E.3 in Appendix E. Similar to the Geoffroy and quadraticEBMs, the RY EBM gives the small basin the largest value for ⇤, consistent withthe SB’s enhanced heat storage rate (Chapter 4).Unlike the other EBMs, however, the RY EBM gives the large basin the largestvalue for H (1000 m). This is inconsistent with the relative ✓z depths we foundin Chapter 3, as we expect H to be largest for the small basin. Processes scalingwith (T T0)2/3 might be more pronounced in the large basin, which is consistentwith a dominance of wind-driven circulations and Ekman pumping there (Chapter3). This could explain why the fitted H is larger for the LB than for the SB, butthis is very speculative.Chapter 5 141Connecting ocean circulation with ocean heat storageDDrake fitted responsesA comparison of the fitted Geoffroy and RY EBM responses for DDrake are shown infigure 5.9. Changing to the RY EBM leads to consistently poorer fits. Initially, theGeoffroy EBM overestimates the DDrake subsurface warming (T0 curves). Changingto the quadratic EBM reduces this overestimation, leading to an improvement in thefits (figure 5.3). Changing to the RY EBM, however, increases this overestimation,leading to even poorer fits than before.This can be seen more clearly from the RMSEs for each of the fits in table 5.6,and from the percentage change improvement in figure 5.10. There is actually aslight improvement in the fits for the LB and SO, but this is overwhelmed by theespecially poor fit for the SB leading to poorer fits overall. The difference in the RYEBM performance between the SB and LB is consistent with their respective meancirculations. Wind-driven circulations dominate in the LB (most relevant to the RYEBM scaling), while the convectively-driven SBMOC dominates the SB (Chapter3). So, we expect that the RY EBM would work better for the LB than for the SB,and this is what we see in figure 5.10.SST Upper-layer T Lower-layer T0 Total (Upper + Lower)RMSE (K) RMSE (K) RMSE (K) RMSE (K)Global 0.06; 0.07 0.23; 0.23 0.19; 0.28 0.42; 0.51SB 0.06; 0.06 0.20; 0.20 0.12; 0.20 0.32; 0.40LB 0.05; 0.08 0.14; 0.13 0.23; 0.23 0.37; 0.36SO 0.12; 0.12 0.41; 0.33 0.57; 0.63 0.97; 0.96Table 5.6: Root mean-square errors (RMSEs) for the (Geoffroy; RY) EBM fits for the upper- andlower-layer temperature responses T and T0 for each basin of DDrake. The RMSE for the upper-layerEBM fit evaluated against the SST response is also shown in the left-most column for comparison.Results with CMIP5 modelsJust as we did for the quadratic EBM in section 5.3.2, we now test the RY EBMon fitting temperature responses from the same suite of five CMIP5 models underabrupt 4×CO2 experiments. We take the radiative parameters F and α from [Kostovet al., 2014] like before, and the thermal parameters γ(∗), h, and H are again foundby co-varying each of them using an interior point algorithm, minimising the RMSE142 Chapter 5Find figure 4.10Connecting ocean circulation with ocean heat storageFigure 5.9: Comparison of fitted EBM responses (black dashed) for the Geoffroy EBM (left column)and RY EBM (right column). Presented are time-series of annually-averaged SST responses (ma-genta), and vertically-averaged upper-layer (red, T ) and lower-layer (blue, T0) temperature responsescorresponding to the layer thicknesses h and H implied by the EBM (see table E.3), for the DDrakeCO2-doubling experiment. Panels (a-d) correspond to the different basins of DDrake: (a) Global re-sponses, (b) Small basin responses (blue shading), (c) Large basin responses (green shading), and (d)southern ocean responses (yellow shading).Chapter 5 143Connecting ocean circulation with ocean heat storagePercentage improvement for changing to RY EBMGlobal LB SB SO-80-70-60-50-40-30-20-1001020TotalUpper layerLower layerFigure 5.10: Percentage improvement in the RMSE of the fitted DDrake temperature responsesafter changing from the Geoffroy EBM to the RY EBM. The change worsens fits overall, mainly dueto the especially poor fit for the SB.of the fit to the CMIP5 model SST response (subject to the same upper and lowerbounds as before).The fitted parameter values can be found in table E.4 in Appendix E. Whereasthe quadratic EBM produced C and C0 values that were systematically larger thanthose found using the Geoffroy EBM, the RY EBM produces C and C0 values thatare systematically smaller. This leads to an overestimate of the subsurface warming,resulting in poorer fits overall, as we found for DDrake in the previous section. Thiscan be seen from the RMSEs for each of the fits in table 5.7, and from the percentagechange improvement in figure 5.11.From the SST column in table 5.7, we see that the RY EBM has a similar skillto the Geoffroy EBM in reproducing an AOGCM’s surface temperature response.However, the RY EBM gives poorer fits than the Geoffroy EBM across the board (byan average of 10%), except for the model MRI-CGCM3 (blue bars in figure 5.11).Furthermore, the RY EBM worsens the lower-layer fits for all the models analysedby an average of 29% (yellow bars in figure 5.11). So, changing to the RY EBM(i.e. decreasing the sensitivity of OHU to the stratification) does not improve theemulation of subsurface warming.144 Chapter 5Connecting ocean circulation with ocean heat storageSST Upper-layer T Lower-layer T0 Total (Upper + Lower)RMSE (K) RMSE (K) RMSE (K) RMSE (K)CNRM-CM5 0.08; 0.08 0.59; 0.59 0.17; 0.23 0.76; 0.82GFDL-ESM2M 0.09; 0.08 0.21; 0.19 0.24; 0.33 0.45; 0.52MPI-ESM-LR 0.09; 0.09 0.15; 0.14 0.34; 0.42 0.49; 0.56MRI-CGCM3 0.09; 0.07 0.30; 0.25 0.48; 0.50 0.79; 0.75NorESM-1M 0.06; 0.06 0.15; 0.11 0.22; 0.29 0.37; 0.40MITgcm DDrake 0.06; 0.07 0.23; 0.23 0.19; 0.28 0.42; 0.51Ensemble mean 0.08; 0.08 0.28; 0.26 0.29; 0.35 0.57; 0.61Std. dev. 0.01; 0.01 0.18; 0.19 0.12; 0.11 0.19; 0.17Table 5.7: Root mean-square errors (RMSEs) for the (Geoffroy; RY) EBM fits for the upper- andlower-layer temperature responses T and T0 from CMIP5 abrupt 4⇥CO2 experiments. The RMSE forthe upper-layer EBM fit evaluated against the SST response is also shown in the left-most column forcomparison. Equivalent RMSEs for DDrake (same as the ‘Global’ row in table 5.4) are shown in thebottom row for reference. Note that the ensemble means and standard deviations don’t include theDDrake values.Percentage improvement for changing to RY EBMCNRM-CM5 GFDL-ESM2M MITgcm DDrake MPI-ESM-LR MRI-CGCM3 NorESM1-M-50-40-30-20-100102030TotalUpper layerLower layerFigure 5.11: Percentage improvement in the RMSE of the fitted CMIP5 model (and DDrake) tem-perature responses after changing from the Geoffroy EBM to the RY EBM. The change leads topoorer fits across the board apart from MRI-CGCM3.Chapter 5 145Connecting ocean circulation with ocean heat storage5.4 ConclusionsUsing the two-layer EBM framework of [Geoffroy et al., 2013b] (equations 5.1), wesuccessfully fitted surface air temperature (SAT) responses for each of the DDrakebasins under an abrupt CO2-doubling (figure 5.2). This skill in reproducing surfacewarming has typically been taken as good evidence for the EBM accurately repre-senting the climate system (e.g. [Held et al., 2010, Geoffroy et al., 2012, Geoffroyet al., 2013b, Kostov et al., 2014]). However, these studies neglect the subsurfacewarming (i.e. the OHU).We found that we could interpret the Geoffroy EBM fitted parameters physically, interms of the DDrake ocean circulation (section 5.2.2). But despite this, the GeoffroyEBM poorly represents the subsurface warming in DDrake (figure 5.2). Work from[Nadiga and Urban, 2019] found that this EBM poorly represents the subsurfacewarming in the AOGCM CCSM4. From the work in this chapter, we find poorrepresentation of subsurface warming in DDrake and in a suite of five other CMIP5models (figure 5.6c). We believe that this poor representation should extend toother AOGCMs if probed by other researchers.We attempted to improve the representation of subsurface warming by proposinga new EBM framework (equations 5.7), which modifies the power law for the heatexchange between the two layers. Increasing or decreasing the power law dependence(⇣) conceptually increases or decreases the sensitivity of OHU to the stratificationin the two-layer EBM.We found that increasing the sensitivity (⇣ = 1 ! ⇣ = 2) improves the representa-tion of subsurface warming in DDrake by 51% (figure 5.4), and in 4 other CMIP5models by an average of 21% (figure 5.8). Decreasing the sensitivity (⇣ = 1 !⇣ = 2/3) leads to a poorer representation of subsurface warming across all modelsanalysed, however, by an average of 29% (figure 5.11).We suggested ⇣ = 2/3 by likening the vertical heat exchange in the EBM to thevertical buoyancy transport within a wind-driven gyre, using the theory of [Rhinesand Young, 1982b] (equation 5.10, derived in Appendix F). The derivation of equa-tion 5.10 rests on the assumption that potential vorticity is homogenised within the146 Chapter 5Connecting ocean circulation with ocean heat storagegyres [Rhines and Young, 1982a]. Although large regions of homogenisation havebeen observed in the Atlantic and Pacific Oceans (see e.g. Chapter 20 in [Vallis,2017]), this may not be the case for the AOGCMs we analysed. Additionally, it isthe Southern Ocean, and not the Atlantic and Pacific Oceans, that dominates OHUin models and observations (e.g. [Frölicher et al., 2015, Wijffels et al., 2016, Duracket al., 2018]). These reasons help explain why the RY EBM may not be applicable,and so leads to poorer fits across the board.The improvement in the fits using ⇣ = 2 is more puzzling, however. The increasedsensitivity to (T T0) in the quadratic EBM might highlight the importance ofthe stratification for OHU. But it is unclear why processes scaling with (T T0)2should be dominant, leading to improvements in the lower-layer fits (although theimprovement is quite small). We have no physical justification for the choice of⇣ = 2; it was a simply a natural progression from ⇣ = 1. Clearly, why the quadraticEBM better represents subsurface warming (especially so in DDrake) demands aphysical explanation. I hope that future researchers might explore this.Although we do see improvements in the fits using the quadratic EBM, these im-provements are marginal. Previous EBMs have treated as a constant (e.g. [Gre-gory, 2000, Held et al., 2010, Geoffroy et al., 2013b, Kostov et al., 2014, Rohrschnei-der et al., 2019]), and so cannot represent ocean circulation change. The quadraticand RY EBMs also suffer from this limitation, despite changes in ocean circulationgreatly affecting ocean heat uptake (e.g. [Banks and Gregory, 2006, Xie and Vallis,2012, Winton et al., 2013, Garuba and Klinger, 2016, Dias et al., 2020b]). (See alsoChapters 3 and 4.) Introducing a time-dependent may be a more fruitful approachto better represent subsurface warming.The use of only two layers for the EBM should also be questioned. To properlyrepresent subsurface ocean warming, a third layer might be added to account forabyssal warming. (Although, the contribution below 2 km to global OHU is likelyless than 10% (e.g. [Desbruyères et al., 2016, Meyssignac et al., 2019])). Thiswould necessarily introduce a millennial timescale in addition to the fast and slowtimescales found previously. The model runs used in this study only last 200 years(for DDrake) and 150 years (for the CMIP5 models), so a millennial timescale wouldChapter 5 147Connecting ocean circulation with ocean heat storageprobably not impact our results significantly. But a three-layer model would becomemore relevant if investigating longer-term warming.5.4.1 Key points1. Current two-layer energy-balance models (EBMs) poorly represent subsurfacewarming.2. Increasing the sensitivity of OHU to the stratification in the EBM improvesthe representation of subsurface warming in 5 out of 6 AOGCMs analysed.3. Decreasing the sensitivity worsens the representation of subsurface warmingacross all AOGCMs analysed.148 Chapter 5“The search for the truth is in one way hardand in another easy — for it is evident thatno one of us can master it fully, nor miss itwholly. Each one of us adds a little to ourknowledge of nature, and from all the factsassembled arises a certain grandeur.”Aristotle, c. 350 BCE 6Conclusion6.1 SummaryIn this thesis, I have presented the first study of the response of the DDrake modelto an abrupt atmospheric CO2-doubling. Noting that a better understanding ofOHU processes is crucial to reduce our uncertainty in climate projections, I haveattempted to address this by focussing on OHU processes in DDrake. The model’sidealised geometry favours a clear separation of ocean basins with distinct circulationprocesses. This allows for an evaluation of the role of different circulations for oceanheat storage under transient climate change.In Chapter 3, we projected the DDrake control-climate ocean circulation onto depth-temperature coordinates to reveal thermal overturning cells. We identified threedistinct cells: a warm cell fluxing heat downward, a cool cell fluxing heat upward,and a shallow tropical cell fluxing heat upward (figure 3.5).Unlike in [Zika et al., 2013], who only looked at the global structure of the thermaloverturning, we performed the same calculation regionally for each of the DDrakebasins, exploiting the model’s simple geometry (figure 3.6). By carefully comparingthe SB, LB, and SO thermal overturning structures with the global picture, and with149Connecting ocean circulation with ocean heat storagethe overturning in geographical coordinates, we identified the circulation featuresresponsible for the warm, cool, and tropical cells. The warm cell is a wind-drivensubtropical-subpolar inter-gyre circulation in the large basin. The cool cell is acombination of the deep small basin MOC and a polar gyre circulation in the largebasin. And the tropical cell consists of the shallow Ekman overturning cells in thesmall and large basins, with upwelling at the equator. We were able to conclude thatthe warm cell is primarily wind-driven by identifying it in depth-density coordinatesas thermally indirect.We examined how these circulations and their associated vertical heat transports(VHTs) responded to an abrupt CO2-doubling. We demonstrated that a shift in thezonal surface wind stress leads to an Ekman pumping response that serves to rein-force the large basin warm cell circulation (figure 3.15), supporting our conclusionthat this cell is wind-driven. This leads to an increase in the downward advectiveVHT (figure 3.16). At the same time, amplified warming at northern high latitudesresults in less vigorous convection feeding the downwelling branch of the SBMOC.The resultant SBMOC weakening is the dominant contributor to a weaker cool cellcirculation (figure 3.11). A weaker cool cell also results in an increase in the down-ward VHT, and both the warm and cool cell responses sum to give ⇠ +0.2 W m2in the top 2 km (figure 3.16).This is similar to results from [Zika et al., 2015] for CCSM4, who also found astronger warm cell and weaker cool cell in a global warming scenario. The keytakeaway is that the DDrake circulation response does not ‘resist’ the warming,but rather permits even greater warming of the ocean interior. This interior oceanwarming most likely delays the surface temperature response (e.g. [Gregory, 2000]),making the circulation changes a negative climate feedback. Notably, the interiorwarming response occurs in spite of the fact that the DDrake ocean circulation alsoweakens substantially by 40% (figure 3.19). This chapter provides a link betweenocean circulation and heat storage, drawing a direct correspondence between circu-lation features and vertical heat transports using thermal overturning cells.In Chapter 4, we focussed on the SBMOC weakening response, drawing a comparisonwith the possible weakening of the AMOC in the future (e.g. [Kirtman et al.,150 Chapter 6Connecting ocean circulation with ocean heat storage2013, Weijer et al., 2020]). Noting the absence of such a deep overturning circulationin the large basin, we introduced the concept of a ‘heat storage contrast’ (defined asthe difference between the SB and LB heat storage rates) to quantify the differencebetween the two basins, and perhaps isolate the role of the SBMOC. We found thatthe small basin warms at a faster rate than the large basin by 0.6 ± 0.2 W m2(figure 4.2). This resembles the Atlantic warming at a faster than the Pacific inobservations (by ⇠0.3 W m2 [Desbruyeres et al., 2017]). We also demonstrated theexistence of this Atlantic-Pacific contrast in four CMIP5 models in CO2-quadruplingruns (figure 4.12).We showed how a rapid advection of surface temperature anomalies into the inte-rior associated with the SBMOC enhances the heat storage rate. The temperatureresponse below 1 km depth in the small basin follows a deep western boundary cur-rent, and a plume of deep warming is collocated with the downwelling branch of theSBMOC (figure 4.5). We also ruled out the possibility of heat exchanges with thesouthern ocean setting the heat storage contrast, as the small basin actually leaksheat to the large basin via the southern ocean (figure 4.6). We thus concluded thatthe reason why the small basin warms at a faster rate than the large basin is due tothe presence of the SBMOC. This suggests that the reason why the Atlantic warmsat a faster rate than the Pacific is due to the presence of the AMOC.The more interesting finding, however, is that the SB-LB heat storage contrastremains approximately constant during the simulated 200 years, despite the fact thatthe SBMOC weakens substantially by ⇠25%. The weakening thus has little or noimpact on the heat storage contrast. We derived a simple expression for the verticalheat flux associated with the SBMOC (Appendix B), enabling us to decomposethe response into SBMOC weakening and differential warming components. Thisdecomposition highlighted the dominant role of the control climate overturning, asthe terms involving SBMOC weakening compensate each other (figure 4.11). TheSBMOC heat flux does actually increase with time however, but this additional heatinput is lost to the southern ocean, ensuring that the heat storage contrast does notincrease in concert. The surprising takeaway is that the presence of the SBMOC isimportant for setting the heat storage contrast, but its subsequent weakening doesnot lead to a weakening of the contrast. This chapter gives a second link betweenChapter 6 151Connecting ocean circulation with ocean heat storageocean circulation and heat storage, finding a counterintuitive relationship for theSBMOC.Finally, in Chapter 5, we asked whether we could emulate OHU in DDrake using atwo-layer energy-balance model (EBM) [Geoffroy et al., 2013b]. We found that wecould interpret the fitted EBM parameters physically, and that we could reproducethe DDrake surface warming following a CO2-doubling both globally and basin-wise.However, we were unable to accurately represent the subsurface warming in DDrake,confirming a shortcoming of this EBM pointed out by [Nadiga and Urban, 2019].We attempted to improve the subsurface warming representation by modifying thepower-law dependence ⇣ of the heat exchange between the two layers in the EBM,⇤(T T0)⇣ . We noted that increasing or decreasing ⇣ was analogous to increasingor decreasing the sensitivity of OHU to the stratification in the EBM. We foundthat increasing the sensitivity from ⇣ = 1 ! ⇣ = 2 improved the representation ofsubsurface warming in DDrake by 51% (figure 5.4) and in 4 out of 5 other CMIP5models by an average of 21% (figure 5.8). Whereas decreasing the sensitivity from⇣ = 1 ! ⇣ = 2/3 worsened the subsurface warming representation across all modelsby an average of 29% (figure 5.11).The suggestion of ⇣ = 2/3 came from likening the vertical heat exchange in the EBMto the vertical buoyancy transport within a wind-driven gyre (see Appendix F). Thefact that this choice leads to poorer subsurface warming representation suggests thatwind-driven gyres do not dominate OHU in the models we analysed. The suggestionof ⇣ = 2, however, had no physical justification. Why this choice improves on theoriginal EBM remains to be explained.This final results chapter presents a third link between ocean circulation and heatstorage, as we found we could interpret the fitted EBM parameters in terms of theDDrake ocean circulation. But a stronger link may be within grasp, if a robust phys-ical explanation concerning the circulation were offered to explain the improvementupon changing ⇣ = 1 ! ⇣ = 2.152 Chapter 6Connecting ocean circulation with ocean heat storage6.2 Conclusions and OutlookAn important finding from Chapter 5 is that current evaluation of two-layer EBMs isphysically inconsistent, where the EBM’s success is judged by how well the upper-layer (mixed-layer) T reproduces an AOGCM’s surface air temperature response.If we are to use this EBM as a physics-based emulator, future studies need toacknowledge this inconsistency and compare like for like quantities. In addition, aspointed out by [Nadiga and Urban, 2019], we should ask of a physics-based emulatorto reproduce both the surface and subsurface warming. They found that the two-layer EBM could not adequately represent the subsurface warming in CCSM4.This subsurface warming representation has been neglected by other researchers,who have instead chosen to focus on the two-timescale nature of the surface warmingresponse (e.g. [Held et al., 2010, Geoffroy et al., 2013b, Geoffroy et al., 2013a]). Thisis surprising, as it neglects consideration of OHU processes. Given an AOGCM’ssurface warming response (from an abrupt CO2-increase, for example), the two-layerEBM tells you what the corresponding mixed-layer T and deep ocean T0 shouldbe for that AOGCM. If we reduce the AOGCM’s ocean into the two layers thatcorrespond to the EBM’s predictions (by vertically averaging), we can verify whetherthe AOGCM is actually behaving as the EBM tells us. The key idea is that if theEBM predictions for T and T0 are poor, then the energy transfer parameterisationsin the EBM are likely inaccurate.We did this procedure for DDrake and five other CMIP5 models in Chapter 5.Rather than focussing on bettering the fit for the surface warming response, wewanted to interrogate the EBM formulation itself. Does heat really transfer fromthe mixed-layer to the ocean below as (T T0)? Other researchers have probablyneglected this question for the obvious answer being “no”. But it is a question worthasking. Why choose (T T0) otherwise? We hope our suggestion of ⇤(T T0)⇣has at least opened up this discussion. And I hope that future researchers mightexplain why a choice of ⇣ = 2 leads to broad improvements in subsurface warmingemulation.I’m not sure if it is possible to solve the RY or quadratic EBM equations analytically.Chapter 6 153Connecting ocean circulation with ocean heat storageA nice feature of the Geoffroy EBM equations is that exact expressions for the fastand slow response timescales can be derived in terms of the radiative and thermalproperties of the system (see Appendix C). It would be helpful if a similar two-timescale decomposition existed for the quadratic EBM equations’ solution, butthis is not known. If this were found, this would lend further support for adoptingthe quadratic EBM equations in emulating AOGCM results, and would bring theseequations into the discussion surrounding two-timescale approaches.The key original finding from Chapter 3 is that the DDrake ocean circulation re-sponse permits more heat to enter the ocean interior, not less. This happens at thesame time as the circulation weakening by 40%. The response is due to a strength-ening of the warm cell and a weakening of the cool cell, both contributing to anincrease in the downward advective heat flux. This complements findings from [Zikaet al., 2015], who found a strengthening warm cell and weakening cool cell in a globalwarming scenario for the model CCSM4. Notably, the circulations responsible forthe warm and cool cells in CCSM4 are most likely very different to those in DDrake.For the CCSM4 warm and cool cells, [Zika et al., 2015] draw a comparison with thethermal overturning found for UVic ESM in [Zika et al., 2013]. They do not analysethe circulations or driving mechanisms responsible for specific cells in CCSM4, butinstead assume them to be the same as those diagnosed in UVic ESM becauseof the close correspondence between their thermal overturning structures. Hence,the warm cell is linked to the Southern Ocean and Atlantic MOC maintained bySouthern Hemisphere winds, and the cool cell is linked to AABW formation (as inUVic ESM [Zika et al., 2013]). In DDrake, the warm cell is a wind-driven inter-gyrecirculation in the large basin. No such AABW mass exists in DDrake, and the coolcell is a combination of the small basin MOC and a gyre circulation in the largebasin.Due to the very different circulations responsible for the warm and cool cells inDDrake compared to those for UVic ESM, it is perhaps inaccurate to assume thatthe thermal overturning circulations in UVic ESM and CCSM4 should correspond(as do the authors in [Zika et al., 2015]). Arguably, these two more realistic coupledclimate models should bear more resemblance with each other than they do with the154 Chapter 6Connecting ocean circulation with ocean heat storageidealised DDrake model, but they do have distinct ocean component models. (UVicESM uses GFDL’s Modular Ocean Model (MOM) [Pacanowski and Griffies, 1996]while CCSM4 uses the Los Alamos Parallel Ocean Program (POP) [Smith et al.,2010]).Nevertheless, perhaps there are some unifying physical processes shared betweenthe DDrake, UVic ESM, and CCSM4 thermal overturning circulations. In eachof these models, the ‘warm cell’ is identified as wind-driven, and the ‘cool cell’ isat least partly buoyancy-driven. In response to global warming, increasing surfacestratification (e.g. [Li et al., 2020]) should hinder deep water formation (see e.g.[Heuzé et al., 2020] pre-print) and thus weaken cool cell circulations. Additionally,an intensification and poleward shift of the winds (e.g. [Yin, 2005, Swart and Fyfe,2012, Hu et al., 2020, Goyal et al., 2021]) may lead to a strengthening of (wind-driven) warm cell circulations.So, irrespective of the specific geographical circulations responsible for the warmand cool cells in each of these models, their driving mechanisms may still provide aclue to their response under transient climate change. The notion that downwardadvective VHT in the ocean should increase in the future is plausible, as we expectwarm cells to strengthen and cool cells to weaken. Of course, future process-basedanalyses of the thermal overturning circulations for other climate models shouldprovide more support for this hypothesis.Perhaps the key difference between DDrake and these other models is the lack ofa realistic Lower MOC Cell, associated with AABW formation. From the depth-density overturning (figure 3.8 in Chapter 3), we deduce that water masses with0 ⇡ 26.5 kg m3 sink in the SB northern high-latitudes and fuel the downwellingbranch of the SBMOC. In figure 6.1, I plot an inventory of the DDrake controlclimate ocean volume in temperature-salinity coordinates, both globally and basin-wise. Overlaid are contours of potential density 0.From the figure, it is clear that no denser water masses exist in the SO (panel b) that‘SB deep water’ (panel c) could mix with during its upwelling. In the real world,mixing takes place between North Atlantic Deep Water (NADW) and AABW alongthe upwelling branch of the AMOC (e.g. [Talley, 2013]). There is thus a densificationChapter 6 155Connecting ocean circulation with ocean heat storage202021212222232324242525262626.526 28 30 32 34 36 3805101520253035121314151617202021212222232324242525262626.526 28 30 32 34 36 3805101520253035121314151617202021212222232324242525262626.526 28 30 32 34 36 3805101520253035121314151617202021212222232324242525262626.526 28 30 32 34 36 3805101520253035121314151617ac dbFigure 6.1: Census of water masses for the DDrake control climate in temperature-salinity coor-dinates for (a) global, (b) southern ocean, (c) small basin, and (d) large basin regions (colour, inlog10m3/C/psu). In white are contours of potential density 0 (kg m3).156 Chapter 6Connecting ocean circulation with ocean heat storageof the upwelling waters sourced from the North Atlantic high-latitudes, making theupwelling AMOC branch denser than the downwelling branch. This leads to theAMOC cell being thermally indirect in depth-temperature coordinates, as found in[Zika et al., 2013, Saenko et al., 2018, Savita et al., 2021], which suggests it mustbe driven by the Southern Hemisphere winds (ignoring haline effects). The lackof an analogous AABW mass in DDrake means that no such mixing takes placeduring the SBMOC upwelling, and so the water cannot become any denser. Hence,the SBMOC cell ends up being thermally direct rather than indirect (as found inChapter 3).Staying within the simple aquaplanet modelling framework, a natural next stepwould be to introduce a circular Antarctic continent at the South Pole of DDrake,with a suitable choice of continental slope. An increase in the model sea ice thick-ness diffusion would promote formation and equatorward transport of sea ice nearthe South Pole. This should hopefully encourage brine rejection, bottom waterformation, and the development of an AABW-like overturning cell. (The introduc-tion of the Antarctic land mass would also eliminate the easterly current runningcounter-clockwise around the South Pole of DDrake.) Projecting again the resultantcirculation state onto depth-temperature (or depth-sigma) coordinates would helpassess the role of the AABW circulation.It would be most interesting to see if this changed the character of the SBMOC fromthermally direct to thermally indirect. The introduction of this AABW circulationmight then bring the DDrake thermal overturning structure into line with the moresophisticated climate models, and provide avenues to explore the influence of LowerCell MOC weakening on transient climate change (e.g. [Suzuki et al., 2022]), ratherthan Upper Cell (AMOC) weakening (as we did in Chapter 4).A major theme of this thesis has been to try and find a mechanistic link betweenocean circulation and heat storage. Following on from the data presented in fig-ure 6.1, another future direction is to use the ‘water mass transformation’ (WMT)framework first introduced by [Walin, 1982]. In this framework, the ocean is par-titioned into conceptually distinct water masses characterised by particular scalarproperties (such as temperature and salinity). Changes in water masses are causedChapter 6 157Connecting ocean circulation with ocean heat storageby property exchanges at boundary layers (e.g. air-sea fluxes, geothermal heating)or interior mixing processes. Defining water masses by buoyancy (approximated bypotential density) gives a direct connection to circulation. The approach thus offersmeasures of the ocean circulation within the WMT phase space without needing anydirect measures of the circulation itself (see review by [Groeskamp et al., 2019]).202020212121222222232323242424252525262626.526.526 28 30 32 34 36 38051015202530351212.51313.51414.51515.51616.517202020212121222222232323242424252525262626.526.526 28 30 32 34 36 3805101520253035-5-4-3-2-1012345a bFigure 6.2: (a) Global water mass census for the DDrake control climate in temperature-salinity co-ordinates in log10m3/C/psu (equivalent to panel a from figure 6.1). (b) Average water mass trans-formation rate between the control climate and year 200 of the CO2-doubling run in 0.1 Sv/C/psu.Black lines are contours of potential density 0 (kg m3).In figure 6.2, I plot the water mass change between the control climate and year 200of the CO2-doubling experiment, divided by the 200 years (to give a WMT rate).Water masses are eroded from the blue regions and created in the red regions inpanel b. The challenge is to estimate the ‘map’ from the blue regions to the redregions, and infer the mechanisms associated with those water mass transformations.For example, we can already see that for water masses with temperatures below⇠12C, there appears to be a shift ‘upward’ in temperature space from the blue armof water masses to the red arm (possibly along isopycnals). This indicates strongradiative/sensible heating without much role for evaporation/precipitation, and thismust be occurring in the southern ocean region (compare with figure 6.1b).A recent paper from [Zika et al., 2020] attempted to infer mechanisms of OHC changein this way, only considering changes in water masses from 11 years of observationalestimates from the EN4 data set [Good et al., 2013]. They estimate such a map from‘early’ water masses (2006-2011) to ‘late’ water masses (2012-2017) in temperature-158 Chapter 6Connecting ocean circulation with ocean heat storagesalinity (T-S) space using an Earth Mover Distance (EMD) algorithm [Pele andWerman, 2009]. This minimises the ‘distance’ in T-S space a water mass has tomove to go from the ‘early’ to the ‘late’ states. As advection plays no role in T-Sspace (assuming it does no mixing), the method allows an estimation of the materialwarming added to the ocean (distinct from warming caused by the redistribution ofbackground OHC due to changes in ocean circulation).The role of circulation in the redistribution warming can be found as a residualbetween the total warming and the inferred material warming due to water masschanges (i.e. Tredistribution = Ttotal Tmaterial). This is similar to the ‘added’and ‘redistributed’ heat components seen in previous studies (e.g. [Banks and Gre-gory, 2006, Xie and Vallis, 2012, Garuba and Klinger, 2016]). Plotting maps of thematerial and redistributed warming components, they find that the redistributedsignal is very heterogenous, while the material warming shows a coherent increaseover the entire globe [Zika et al., 2020]. Notably, the material OHC change peaksat 35S and 35N, collocated with peaks of Ekman downwelling. This is consistentwith the theme of wind-driven OHU for the warm cell in Chapter 3.Another striking feature from their results is the clear Atlantic-Pacific contrast inthe material OHC change. They find that the North Atlantic (< 44N) stores 24%of the global material OHC change despite representing less than 12% of the globalsurface area. A similar calculation for DDrake finds that, 200 years after the CO2-doubling, the SB is responsible for 24% of the global OHC increase despite covering18% of the global surface area.In Chapter 4 we defined a heat storage contrast between the small and large basinsof DDrake, finding the SB to warm at a faster rate than the LB by 0.6 ± 0.2 Wm2. Using the total OHC change data from Table 1 in [Zika et al., 2020], there isan Atlantic-Pacific contrast of ⇠1 W m2. (The Atlantic OHC increase is 135 TW,and the Pacific increase is 49 TW. I take the Atlantic and Pacific surface areas to be81.8 million km2 and 161.8 million km2, respectively∗.) Notably, if we perform thesame calculation using only the material OHC change data, we still find an Atlantic-Pacific contrast of ⇠0.5 W m2. This crude calculation highlights that the contrast∗Using estimates from NOAA at https://www.ngdc.noaa.gov/mgg/global/etopo1_ocean_volumes.html.Chapter 6 159Connecting ocean circulation with ocean heat storageis not simply due to circulation changes (e.g. AMOC weakening). It suggests theimportance of background ocean circulations in setting the heat storage contrast,which we argued for in Chapter 4 with the case of the SBMOC setting the contrastbetween the small and large basins of DDrake.Performing a similar WMT analysis for DDrake would help isolate the role of circu-lation changes through the material/redistributed decomposition. Although thereexist other methods to form such a decomposition, either through the use of pas-sive tracers (e.g. [Banks and Gregory, 2006]) or by ‘fixing’ currents to their controlclimate state [Winton et al., 2013], the added information from the water masschanges could also help shed light on the causes of those circulation changes. Themethod used in [Zika et al., 2020] is also novel and understudied, so applying it toa simple model such as DDrake brings with it the benefits of easier evaluation andinterpretation.The key message from Chapter 4 is that the Atlantic punches above its weight interms of ocean heat storage. This is seen in models in CO2-increase experiments(e.g. [Kuhlbrodt and Gregory, 2012, Shatwell et al., 2020]), and in observations (e.g.[Desbruyeres et al., 2017, Zika et al., 2020]). We argued that the analogous enhancedSB heat storage in DDrake was due to the presence of the SBMOC. However, it’sbeen suggested that enhanced ocean heat storage in the Atlantic is in fact due toan enhanced inter-ocean heat transport from the Indian to the Atlantic basin viaAgulhas leakage [Lee et al., 2011]. This contradicts our findings from Chapter 4,where we showed that the SB-LB heat storage contrast persists despite the fact thatheat actually leaks from the SB to the LB via the SO, and not the other way around(figure 4.6).We did not show the pathways of this SB-LB heat exchange. Different Agulhas-likeleakages were found by [Nilsson et al., 2013] in a very similar modelling set-up toDDrake. Here, however, the leakage depended on the meridional boundary extentsbeing equatorward or poleward of the zero wind line. Only if the ‘African’ continent(i.e. the Eastern boundary of the basin with deep water formation) were equatorwardof the zero wind line would this permit an Agulhas-like leakage just south of thisboundary. In DDrake, both meridional boundaries extend to 35S, poleward of the160 Chapter 6Connecting ocean circulation with ocean heat storagezero wind line near 30S. We therefore do not expect such an Agulhas-like leakagefrom the LB to the SB in DDrake. There could instead be an ‘anti-Agulhas-like’leakage slipping under the SB’s eastern boundary into the LB (explaining the SB-to-LB heat leakage), but this would mean the analogous Atlantic/Indian basin roleswould be reversed.So, although we showed the SB leaking heat to the LB (via the SO) in Chapter 4, itis not clear exactly what causes this. It is worth noting that the redistribution com-ponent in [Zika et al., 2020] shows an anomalous southward transport of heat fromthe Atlantic into the Southern Ocean (in contrast to the northward heat transportfrom Agulhas leakage assumed in [Lee et al., 2011]). The role of Agulhas leakagein transient climate change is thus another open question that could be exploredusing the DDrake configuration (perhaps with higher resolution, due to the key roleof mesoscale eddies).Agulhas leakage is expected to increase with a poleward shift of the Southern Hemi-sphere winds (e.g. [Biastoch et al., 2009]). This also directly impinges on AMOCweakening, as an increase in warm, salty water from the Indian Ocean enteringthe Atlantic could help strengthen the AMOC [Beal et al., 2011], thus counteract-ing weakening due to increased surface buoyancy forcing from Arctic amplificationand sea ice melt (e.g. [Sévellec et al., 2017, Dai et al., 2019, Dai, 2022]). FutureLagrangian studies with simulated virtual particles may help better reveal the cir-culation pathways in DDrake (see review by [Van Sebille et al., 2018]), and perhapsshed light on the interplay between Agulhas-like leakage, the heat storage contrast,and MOC weakening.The work in this thesis has provided the first results revealing the DDrake model’sresponse to an abrupt CO2-doubling. It has also hopefully put this model on themap for its utility in answering fundamental climate dynamics questions relatedto transient climate change. As mentioned in Chapter 1, this thesis provides acontribution to the hierarchical approach to climate modelling (as exhorted by [Held,2005]), where DDrake occupies an intermediate position in the model hierarchy,between simple climate models (e.g. EBMs) on one end, and high-resolution Earthsystem models (ESMs) on the other. I have attempted to connect DDrake to theseChapter 6 161Connecting ocean circulation with ocean heat storageother models spanning the hierarchy through Chapters 3 to 5. I hope the uniqueinsights garnered from using an intermediate model, aided by its simple geometryand clean circulation decomposition, clearly reveal its utility.I hope this thesis builds towards a stronger, process-based understanding of OHU,and helps to underpin a link between ocean circulation and heat storage. 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The ‘Redi’ scheme mixes tracer properties along isopycnal surfaces,introducing a tendency term on the right-hand side of the tracer equations 2.4 and2.5 of the form r · (⇢KRedir⌧). Here, ⇢ is the isopycnal diffusivity and KRedi is arank 2 tensor that projects the gradient of the tracer ⌧ onto the isopycnal surface.The KReditensor has the form:KRedi=11 + |S|20BB@1 + S2y SxSy SxSxSy 1 + S2x SySx Sy |S|21CCA ⇡0BB@1 0 Sx0 1 SySx Sy |S|21CCA (A.1)where Sx = @x/@z and Sy = @y/@z define the components of the slope ofthe isopycnal surfaces . To get the final expression, we have made the small slopeapproximation (|S| ⌧ 1). This is appropriate, as typical slopes in the ocean interior207are very small (⇠ 104–102).The GM advective or ‘transport’ effect of the eddies is parameterised by means ofa ‘bolus velocity’ ub which depends on Sx and Sy. This adds another term on theright-hand side of the model tracer equations of the form r · (⌧ub). The bolusvelocity is defined as the curl of a stream function F⇤ (and hence is automaticallynon-divergent):ub = r⇥ F⇤ = r⇥0BB@F ⇤xF ⇤y01CCA = r⇥0BB@GMSyGMSx01CCA =0BB@@z(GMSx)@z(GMSy)@x(GMSx) + @y(GMSy)1CCA(A.2)where GM is the GM eddy transfer coefficient. It was pointed out by [Griffies, 1998]that the eddy bolus flux ⌧ub could be written as the sum of a non-divergent fluxand a skew flux:⌧ub =0BB@@z(GMSx)⌧@z(GMSy)⌧(@x(GMSx) + @y(GMSy))⌧1CCA=0BB@@z(GMSx⌧)@z(GMSy⌧)@x(GMSx⌧) + @y(GMSy⌧)1CCA+0BB@GMSx@z⌧GMSy@z⌧GMSx@x⌧ GMSy@y⌧1CCA(A.3)As the first vector in this decomposition is non-divergent, it has no effect on thetracer field, so it can be dropped. This means that we can redefine ⌧ub ⌘ GMKGMr⌧ ,where KGMis an antisymmetric tensor with the form:KGM=0BB@0 0 Sx0 0 SySx Sy 01CCA (A.4)It is this formulation of the GM scheme that is used in the DDrake configuration.208The horizontal eddy buoyancy transfer is equivalent to conventional downgradientdiffusion, while the vertical transfer is upgradient. The combination of these fluxesacts to rotate and flatten the isopycnal surfaces, extracting kinetic energy from theavailable potential energy of the flow.The GM scheme is used in conjunction with the Redi isopycnal mixing scheme sothat: ⇢KRedir⌧ ⌧ub = (⇢KRedi + GMKGM)r⌧ . By choosing the same transfercoefficient (⇢ = GM = ) for both KRedi and KGM , substantial cancellations takeplace leaving us with:⇢KRedi + GMKGM = (KRedi +KGM) = 0BB@1 0 00 1 02Sx 2Sy |S|21CCA (A.5)This is the essence of the ‘GMREDI’ scheme used in the MITgcm for the parame-terisation of mesoscale oceanic eddies.209210BDerivation of the change inSBMOC heat fluxAs introduced in Chapter 4, a model quantity C = C(t, x, y, z) changes from itscontrol value C(x, y, z) to a perturbed value Cpert(t, x, y, z) by an amount C dueto the CO2 perturbation:C(t, x, y, z) = Cpert(t, x, y, z) C(x, y, z) (B.1)where C is a 50-year time average taken from the last 50 years of the model spin-up, after it has reached an equilibrated state. We note that we do not analyseinstantaneous values in the perturbed climate, but rather annual averages. Thisdetail is unimportant when considering changes in one quantity, but has an impactwhen considering changes in a combination of quantities. Hence, we first rewriteequation B.1 as a difference between an annually averaged quantity in the perturbedclimate, and a 50-year averaged quantity in the control climate:C(t, x, y, z) = C1yrpert(t, x, y, z) C50yrctrl (x, y, z) (B.2)211Then, an instantaneous value of res in the perturbed simulation is: res = res1yr+ 0res = res + res50yr+ 0res (B.3)where 0res is the sub-annual variability in res, and the ‘ctrl’ and ‘pert’ subscriptshave been omitted for clarity. A similar expression can be obtained for ✓. Then, achange in the SBMOC heat flux HMOC is given by:HMOC⇢0cp= res✓1yrpert res✓50yrctrl= ( res + res50yr+ 0res)((✓) + ✓50yr+ ✓0)1yr res✓50yr= res(✓)1yr+ res✓50yr1yr+ res✓01yr+ res50yr(✓)1yr+ res50yr✓50yr1yr+ res50yr✓01yr+ 0res(✓)1yr+ 0res✓50yr1yr+ 0res✓01yr res✓50yr= res(✓) + res✓50yr+ res50yr(✓)+ res50yr✓50yr+ 0res✓01yr res✓50yr| {z }T(B.4)Note that C1yr = C and C 01yr = 0 for some quantity C. The term T representsthe change in variability between the control and perturbed climates:T = 0res✓01yr| {z }Tpert ( res✓50yr res50yr✓50yr)| {z }Tctrl(B.5)In our analysis, we assume that this term T is small and hence neglect it in ourexpression for HMOC . This leaves us with:HMOC⇢0cp⇡ res(✓) + res✓50yr+ res50yr(✓) (B.6)which is our equation 4.3 as stated in Chapter 4.212CAnalytical solution of theGeoffroy EBM under astep-forcingThe two-layer EBM utilised in [Geoffroy et al., 2013b] was introduced in Chapter 5in equation 5.1, which we reprint here for reference:CdTdt= F ↵T (T T0)C0dT0dt= (T T0)(C.1)For a step-forcing, F is a constant. We can rewrite these equations as a matrixdifferential equation in the form dX/dt = AX+B, giving:ddt0@TT01A =0@(↵ + )/C /C/C0 /C01A0@TT01A+0@F/C01A (C.2)213The general solution to this system of ODEs is the solution to the homogenousequation, plus a particular solution (or particular integral).C.1 Particular solutionFor a steady-state solution X⇤, dX/dt = 0, which implies that AX⇤ +B = 0. Fromthis, we see that X⇤ = A1B and that B = AX⇤. From this last expression, wecan rewrite equation C.2 as:dXdt= A(X(t)X⇤) (C.3)which reveals that the steady-state solution X⇤ is in fact a particular solution. Hence,noting that detA = ↵/CC0 6= 0, we can invert the matrix A and evaluate thesolution as:X⇤ = A1B = CC0↵0@/C0 /C/C0 (↵ + )/C1A0@F/C01A =0@F/↵F/↵1A =0@TeqTeq1A (C.4)So, the steady-state solution is the equilibrium temperature Teq, which is to beexpected.C.2 Solution to the homogeneous equationWe now seek a solution to the homogeneous equation dXh/dt = AXh. A generalform of solution to this equation is:Xh(t) = c1e1tu1 + c2e2tu2 (C.5)where i and ui are eigenvalues and eigenvectors of A, respectively, and the ci areconstants. We now proceed to find these eigenvalues and eigenvectors.214C.2.1 Eigenvalues of AThe eigenvalues must obey the characteristic equation det(A I) = 0:det(A I) =(↵ + )/C /C/C0 /C0 =⇣ (↵ + )C ⌘⇣ C0 ⌘ 2CC0= 2 +⇣ ↵ + C+C0| {z }b⌘+↵CC0= 0(C.6)Following the notation in [Geoffroy et al., 2013b], we define the discriminant =b2 4 ↵CC0 , which allows us to write the solution:± =b±p2(C.7)To preserve the ‘two-timescales’ approach particular to this EBM, we define ⌧s =1/ and ⌧f = 1/+, giving:⌧s = 1=CC02↵(b+p)⌧f = 1+=CC02↵(bp)(C.8)which are the ‘slow’ and ‘fast’ timescales that characterise the temperature responseto a step-forcing.C.2.2 Eigenvectors of AHaving found the eigenvalues ± (equation C.7), we can proceed in finding theeigenvectors of A, which obey Au± = ±u±. Firstly, for u = (µ, ⌫), we haveAu = u:2150@(↵ + )/C /C/C0 /C01A0@µ⌫1A = 0@µ⌫1A (C.9)which can be readily solved to give:µ =⇣1 +C0⌘⌫ (C.10)We can rewrite the prefactor for ⌫ as:1 +C0 = C02⇣ ↵ + C C0| {z }b⇤p⌘= C02⇣b⇤ p⌘=1s(C.11)where we’ve introduced the quantities b⇤ and s, following the presentation in [Ge-offroy et al., 2013b]. We now choose µ = 1, which forces ⌫ = s. Hence, our‘slow’ eigenvector is:u =0@ 1s1A (C.12)Following a similar procedure for u+, we find:µ+ =⇣1 +C0+⌘⌫+ (C.13)and we rewrite the ⌫+ prefactor:1 +C0+ = C02⇣b⇤ +p⌘=1f(C.14)where, again, we’ve followed the presentation in [Geoffroy et al., 2013b], introducingthe parameter f . As before, if we choose µ+ = 1, we arrive at our ‘fast’ eigenvector:u+ =0@ 1f1A (C.15)216C.2.3 SolutionHaving found the eigenvalues (equation C.7) and eigenvectors (equations C.12 andC.15) of A, we can now simply substitute these into equation C.5 to find the solutionto the homogeneous equation:Xh = c−eλ−tu− + c+eλ+tu+ (C.16)which can be written:⎛⎝ Th(t)T0h(t)⎞⎠ = c−e−t/τs⎛⎝ 1φs⎞⎠+ c+e−t/τf⎛⎝ 1φf⎞⎠=⎛⎝ 1 1φs φf⎞⎠⎛⎝c−e−t/τsc+e−t/τf⎞⎠(C.17)C.3 General solutionThe general solution is simply the sum of the solution to the homogeneous equation(equation C.17) and the particular solution (equation C.4). Hence:T (t) = Teq + c−e−t/τs + c+e−t/τfT0(t) = Teq + φsc−e−t/τs + φfc+e−t/τf(C.18)In order to find the constants c±, we need the initial conditions. If T and T0 arelayer temperature perturbations, we have that at t = 0, T = T0 = 0. Substitutingin, we eventually find that:c+ = Teq(φs − 1)(φf − φs) = −Teqaf (C.19)where the last equality follows to stay in keeping with the notation in [Geoffroyet al., 2013b]. This allows us to write:217Go to figure 1.8af =(s 1)(s f )(C.20)Similarly, we find that:c = Teq(1 f )(f s)= Teqas (C.21)where, again, the last equality follows in order to be consistent with notation, whichimplies that:as =(1 f )(s f )(C.22)We note that as + af = 1, and sas + faf = 1. The ai and i are the modeparameters that partition the adjustment towards equilibrium into the fast andslow components. Finally, we can write the full general solution:T (t) = Teq(1 afet/⌧f aset/⌧s)T0(t) = Teq(1 fafet/⌧f saset/⌧s)(C.23)which is the same as equation 5.2 as presented in Chapter 5.218DNon-dimensionalisation of theGeoffroy EBM and LimitingCasesThe two-layer EBM equations utilised in [Geoffroy et al., 2013b] are:CdTdt= F ↵T (T T0)C0dT0dt= (T T0)(D.1)where C and C0 are effective heat capacities, and T and T0 are temperature pertur-bations for the mixed-layer ocean and deep ocean layers, respectively. The symbolF is an effective radiative forcing, ↵ is the climate feedback parameter, and is theheat exchange parameter.We can rewrite the effective heat capacities as C = ⇢0cph and C0 = ⇢0cpH, where ⇢0is a reference seawater density, cp the specific heat capacity of seawater (at constantpressure), and h and H are layer thicknesses. We can then define the scalings hscale219and Hscale for the upper and lower layers, giving h = hscaleĥ and H = HscaleĤ,where hatted variables are non-dimensional.To isolate parameters affecting the dynamics, we need to non-dimensionalise the timevariable by a timescale tscale, such that t = tscalet̂. There are many natural timescalesto choose, but here we choose the ‘fast’ timescale ⌧F = (⇢0cphscale)/(↵+) as definedin [Held et al., 2010], as this timescale is most immediately relevant to transientclimate change. Rewriting the EBM equations in terms of these non-dimensionalvariables gives us:⇢0cphscale ĥ1tscaledTdt̂= F ↵T (T T0)⇢0cpHscale Ĥ1tscaledT0dt̂= (T T0)(D.2)Substituting in tscale = ⌧F and dividing by ↵ results in the equations:↵ + ↵ĥdTdt̂=F↵ T ↵(T T0)↵ + ↵HscalehscaleĤdT0dt̂=↵(T T0)(D.3)or, equivalently:ĥdTdt̂=⇣1 +↵⌘1⇣F↵ T ↵(T T0)⌘ĤdT0dt̂=hscaleHscale⇣1 +↵⌘1⇣↵(T T0)⌘ (D.4)where it is apparent that the ratio /↵ is crucial in controlling the dynamics. Thissimple procedure highlights that it is not just climate feedbacks that are importantin transient climate change. The interplay of climate feedbacks with the efficiencyof ocean heat uptake, or ocean vertical heat transports, is key.220D.1 Limiting casesD.1.1 Radiative feedbacks dominate, /↵ << 1In this case, we presume that there is very little heat exchange between the twolayers, and so the dominant heat flux in the model is that radiated upwards to spacefrom the upper layer. We make use of the notation used in Appendix C, where wederived the analytical solution for the Geoffroy EBM under a step forcing. We firstlook at the parameters b and :b =↵ + C+C0⇡ ↵C = b2 4↵CC0= b2 4↵2(/↵)CC0⇡ b2 ⇡⇣↵C⌘2 (D.5)These then feed back into the slow and fast timescales, giving us:⌧s =CC02↵(b+p) ⇡ CC02↵⇣↵C+↵C⌘=C0=) ⌧f ⇡C↵(D.6)where we’ve used the fact that detA = ↵/CC0 is the product of the eigenvalues,i.e. 1/(⌧f⌧s), in order to find the fast timescale ⌧f . Now, like b, b⇤ ⇡ ↵/C, whichmeans we can find the parameters s and f by:s =C2(b⇤ +p) ⇡ C2⇣↵C+↵C⌘=↵f =C2(b⇤ p) ⇡ 0(D.7)We can then find the parameters af and as:af =s 1s f⇡ 1 ↵as =1 fs f⇡ ↵(D.8)221Note that the relationships af +as = 1 and faf +sas = 1 still hold, which ensuresthe satisfaction of the initial conditions T (0) = T0(0) = 0. Finally, we can writedown the limiting solutions:T (t) ⇡ Teq ↵Teqet/C0 ⇣1 ↵⌘Teqe↵t/C ⇡ Teq1 e↵t/CT0(t) ⇡ Teq1 et/C0 (D.9)Note that if = 0, then T0(t) = 0 as expected. What’s interesting here is thatthe two-layer EBM has reduced to two separate one-layer EBMs in its behaviour.If << ↵, the coupling between the two layers disappears. Each layer simply ap-proaches the equilibrium temperature Teq with their respective fast or slow timescale.This decoupling of the two-layer EBM into two one-layer EBMs is shown schemati-cally in figure D.1. The lower layer heat fluxes are of the same form as those of theupper layer, but are scaled by /↵ i.e. a very small number.Figure D.1: Two-layer EBM reduces to two separate one-layer EBMs in the limit << ↵.D.1.2 Heat exchange dominates, /↵ >> 1In this limit case, we presume that heat is much more rapidly transported downwardsfrom the upper layer to the lower layer than is radiated upwards to space, i.e. >> ↵. Following the procedure in the previous section, we approximate b and 222first:b =↵ + C+C0⇡ C+C0=(C + C0)CC0 = b2 4↵CC0=(↵ + )2C2+2( ↵)CC0+2C20⇡ 2C2+22CC0+2C20=⇣ C+C0⌘2(D.10)We find that ⇡ b2, as we found for the previous limit case. We now look at thetimescales ⌧s and ⌧f , giving us:⌧s =CC02↵(b+p) ⇡ CC02↵2(C + C0)CC0=C + C0↵=) ⌧f ⇡CC0(C + C0)(D.11)where, again, we’ve made use of the fact that detA = 1/(⌧f⌧s). Note that in thiscase ⌧f ⇡ 1/b. Similarly, as for b, we find the b⇤ approximation:b⇤ =↵ + C C0⇡ C C0=(C0 C)CC0(D.12)This, in turn, lets us find the parameters:f =C2(b⇤ p) ⇡ C2 (C0 C)CC0 (C0 + C)CC0!= CC0s =C2(b⇤ +p) ⇡ 1(D.13)Now, we find approximations for the mode parameters af and as:af =s 1s f⇡ 0as =1 fs f⇡ 1(D.14)As before, af + as = 1 and faf + sas = 1 still hold. Finally, we can write downthe limiting solutions for this case as:223T (t) ⇡ Teq1 e↵t/(C+C0)T0(t) ⇡ Teq1 e↵t/(C+C0) (D.15)So, T (t) = T0(t), and both layers in the system behave in exactly the same way.This effectively reduces the two-layer EBM to a well-mixed, one-layer system withan effective heat capacity of C + C0, as shown in figure D.2.Figure D.2: Two-layer EBM reduces to a one-layer system with effective heat capacity C + C0 when >> ↵.In the real-world system, and ↵ are probably comparable, so the oceanic mixed-layer is neither completely decoupled from, nor completely coupled to the deepocean. The behaviour lies somewhere in between these two limiting cases.224EFitted parameter values fortwo-layer EBMs225F↵CC0hH(⇤)(Wm2)(Wm2K1)(Wyrm2K1)(Wyrm2K1)(m)(m)(Wm2K1(2))Global4.261.5617.8;18.8113.6;135.0134;143859;10211.36;1.05Smallbasin4.502.1413.6;15.3135.5;159.4103;1161025;12052.12;2.20Largebasin4.562.0612.5;15.0121.5;141.495;114919;10702.10;2.05southernocean3.830.7617.0;19.2104.0;128.4128;145786;9710.52;0.18TableE.1:Fittedphysicalparametersforboththe(Geoffroy;quadratic)EBMsusinganinteriorpointalgorithm:theradiativeforcingF,radiativefeedback↵,upper-layerheatcapacityC,lower-layerheatcapacityC0 ,thecorrespondingupper-layerandlower-layerthicknesseshandH,andtheheatexchangecoefficient(or⇤)foreachoftheDDrakebasins.226F↵CC0hH(⇤)(Wm2)(Wm2K1)(Wyrm2K1)(Wyrm2K1)(m)(m)(Wm2K1(2) )CNRM-CM59.882.3328.7;32.4226.3;292.6217;2451711;22121.40;0.55GFDL-ESM2M9.692.7624.7;1.5463.6;491.2187;113505;37143.48;3.42MPI-ESM-LR11.802.0923.4;29.0391.6;454.2177;2192961;34343.69;1.84MRI-CGCM39.782.5455.8;66.1661.3;661.3422;5005000;50006.24;6.60NorESM-1M10.852.4835.3;43.0538.7;611.6267;3254073;46244.98;3.52MITgcmDDrake4.261.5617.8;18.8113.6;135.0134;143859;10211.36;1.05Ensemblemean10.402.4433.6;34.4456.3;502.2254;2603450;37973.96;3.19Std.dev.0.910.2513.3;23.4162.8;144.6100;1771231;10931.81;2.27TableE.2:Fittedphysicalparametersforboththe(Geoffroy;quadratic)EBMsusinganinteriorpointalgorithm.TheeffectiveradiativeforcingFandcli-matefeedback↵valuesforCMIP5modelsaretakenfrom[Kostovetal.,2014].Upper-layerheatcapacityC,lower-layerheatcapacityC0,thecorrespond-ingupper-layerandlower-layerthicknesseshandH,andtheheatexchangecoefficient(or⇤ )estimatesforthefiveCMIP5modelsused.TheequivalentparametersfoundforDDrake(sameasthe‘Global’rowintableE.1)areshowninthebottomrowforcomparison.Ensemblemeansandstandarddeviationsdon’tincludethevaluesfromDDrake.227F↵CC0hH(⇤)(Wm2)(Wm2K1)(Wyrm2K1)(Wyrm2K1)(m)(m)(Wm2K1(2/3))Global4.261.5617.7;17.6113.6;103.7134;133859;7841.36;1.43Smallbasin4.502.1413.6;13.2135.6;125.3103;1001025;9472.12;2.02Largebasin4.562.0612.6;13.2121.5;132.395;100919;10002.10;1.83southernocean3.830.7616.9;15.7104.0;90.3128;119786;6830.52;0.74TableE.3:Fittedphysicalparametersforboththe(Geoffroy;RY)EBMsusinganinteriorpointalgorithm:theradiativeforcingF,radiativefeedback↵,upper-layerheatcapacityC,lower-layerheatcapacityC0 ,thecorrespondingupper-layerandlower-layerthicknesseshandH,andtheheatexchangecoeffi-cient(or⇤)foreachoftheDDrakebasins.228F↵CC0hH(⇤)(Wm2)(Wm2K1)(Wyrm2K1)(Wyrm2K1)(m)(m)(Wm2K1(2/3) )CNRM-CM59.882.3328.7;26.6226.3;191.6217;2011711;14491.40;1.92GFDL-ESM2M9.692.7624.7;22.1463.6;405.2187;1673505;30643.48;4.00MPI-ESM-LR11.802.0923.4;20.2391.6;352.1177;1532961;26623.69;4.65MRI-CGCM39.782.5455.8;48.0661.3;661.3422;3635000;50006.24;6.16NorESM-1M10.852.4835.3;30.8538.7;489.2267;2334073;36994.98;5.58MITgcmDDrake4.261.5617.7;17.6113.6;103.7134;133859;7841.36;1.43Ensemblemean10.402.4433.6;29.5456.3;419.9254;2233450;31753.96;4.46Std.dev.0.910.2513.3;11.1162.8;173.2100;841231;13101.81;1.65TableE.4:Fittedphysicalparametersforboththe(Geoffroy;RY)EBMsusinganinteriorpointalgorithm.TheeffectiveradiativeforcingFandclimatefeedback↵valuesforCMIP5modelsaretakenfrom[Kostovetal.,2014].Upper-layerheatcapacityC,lower-layerheatcapacityC0,thecorrespondingupper-layerandlower-layerthicknesseshandH,andtheheatexchangecoefficient(or⇤ )estimatesforthefiveCMIP5modelsused.Theequivalentpa-rametersfoundforDDrake(sameasthe‘Global’rowintableE.3)areshowninthebottomrowforcomparison.Ensemblemeansandstandarddeviationsdon’tincludethevaluesfromDDrake.229230FVertical buoyancy transportwithin a wind-driven gyreWe start with the quasigeostrophic (QG) theory of [Rhines and Young, 1982b] todescribe the circulation within a wind-driven gyre. Ignoring relative vorticity (ap-propriate for basin-scale circulations), the QG potential vorticity q is given byq = y +f 20N2@2 @z2(F.1)where f0 is a reference value for the Coriolis parameter, is the variation of theCoriolis parameter with latitude, N is the buoyancy frequency, and is the streamfunction.We assume that potential vorticity is homogenised within closed contours of [Rhines and Young, 1982a], so that within the gyre, q = q0 is a constant. Andwe define a depth D(x, y) that marks the lower boundary of the gyre. This alsomarks the depth of the wind’s influence, so, below this depth, = 0. The flow isthen given by solving the following equations:231@2 @z2=N2f 20(q0 y) ; D(x, y) < z < 0 = 0 ; z  D(x, y)(F.2)At the lower boundary of the gyre, we require = z = 0 to ensure that and zremain continuous across the boundary. This leads to the solution: =N22f 20(z +D)2(q0 y) ; D(x, y) < z < 0 (F.3)Next, we can obtain an expression for D using the QG buoyancy equation andrearranging for w giving:w = 1N2J( , b) = f0N2J( , z) (F.4)where the perturbation buoyancy b = f0 z, J denotes the Jacobian, and we haveassumed that bt = 0 as we are only interested in statistically-steady solutions.We assume that at z = 0, w = wEk is the Ekman pumping velocity. Evaluating ourexpression for w (equation F.4) at z = 0 gives:wEk = w|z=0 =N22f 30D2@D@x(q0 y) (F.5)We define a barotropic stream function as: B =Z xExwEk dx0 (F.6)where xE is the eastern boundary of the gyre. Then, integrating the expression forwEk and rearranging for D gives:D ="6f 30N2 B(q0 y)#1/3(F.7)If we further assume that wEk is independent of x, then B = wEk(xE x) and wecan rewrite equation F.7 as:232D ="6f 30N2wEk(x xE)(q0 y)#1/3(F.8)Substituting either expression for D into equation F.3 then gives the full streamfunction solution.Next, we wish to find an expression for the vertical buoyancy transport wb. This is:wb = f0N2J( , z)| {z }w· f0 z|{z}b=f 20N2 z( y zx x zy) (F.9)After expanding and simplifying, this leads to an expression that reads:wb =2 N2f 20!2Dx(z +D)3(q0 y)2 (F.10)Noting that bz = f0 zz = N2f0(q0 y), we can rewrite equation F.10 as:wb =2f 20Dx(z +D)3 @b@z!2(F.11)And finally, substituting in equation F.8 for D and evaluating at z = 0 gives:wb|z=0 ="6f 20(x xE)#1/3w4/3Ek @b@z!2/3(F.12)which is the same as equation 5.10 as presented in Chapter 5. We thus find thatthe vertical buoyancy transport at the surface is proportional to the 2/3-power of@b/@z. Further, we find that wb is proportional to w4/3Ek , indicating that a strongerwind-stress curl may lead to greater OHU.233

Warm water mass formation

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Warm water mass formation

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