Balance, gravity waves and jets in turbulent shallow water flows

This thesis contains a thorough investigation of the properties of freely decaying turbulence in a rotating shallow water layer on a sphere. A large number of simulations, covering an extensive range of Froude and Rossby numbers, have been carried out using a novel numerical algorithm that exploits the underly- ing properties of the flow. In general these flows develop coherent structures; vortices interact, merge and migrate polewards or equatorwards depending or their sign, leaving behind regions of homogenized potential vorticity separated by sharp zonal jets. In the first half of the thesis we investigate new ways of looking at these structures. In the second half of the thesis we examine the properties of the potential vorticity (PV) induced, balanced component and the residual, unbalanced component of the flows. Cyclone-anticyclone asymmetry has long been observed in atmospheric and oceanic data, laboratory experiments and numerical simulations. This asymmetry is usually seen to favour anticyclonic vorticity with the asymmetry becoming more pronounced at higher Froude numbers (e.g. Polvani et al. [1994a]). We find a similar result but note that the cyclones, although fewer, are significantly more intense and coherent. We present several ways of quantifying this across the parameter space. Potential vorticity homogenization is an important geophysical mechanism responsible for sharpening jets through the expulsion of PV gradients to the edge of flow structures or domains. Sharp gradients of PV are obvious in contour plots of this field as areas where the contours are bunched together. This suggests that we can estimate the number of zonal jets by performing a cluster analysis on the mean latitude of PV contours (this diagnostic is also examined by Dritschel and McIntyre [2007]). This provides an estimate rather than an exact count of the number of jets because the jets meander signficantly. We investigate the accuracy of the estimates provided by different clustering techniques. We find that the properties of the jets defy such simple classification and instead demand a more local examination. We achieve this by examining the palinstrophy field. This field, calculated by taking the gradient of the PV, highlights the regions where PV contours come closer together, exactly what we would expect in regions of strong jets. Plots of the palinstrophy field reveal the complex structure of these features. The potential vorticity field is even more central to the flow evolution than the strong link with jets suggests. From a knowledge of the spatial distribution of PV, it is possible to diagnose the balanced components of all other fields. These components will not contain inertia-gravity waves but will contain the dominant, large scale features of the flow. This inversion, or decomposition into balanced (vortical) and unbalanced (wave) components, is not unique and can be defined to varying orders of accuracy. We examine the results of four dfferent definitions of this decomposition, two based on truncations of the full equations and two based on an iterative procedure applied to the full equations. We find the iterative procedure to be more accurate in that it attributes more of the flow to the PV controlled, balanced motion. However, the truncated equations perform surprisingly well and do not appear to suffer in accuracy at the equator, despite the fact that the scaling on which they are based has been thought to break down there. We round off this study by considering the impact of the unbalanced motion on the flow. This is accomplished by splitting the integration time of the model into intervals τ < t < τ+dτ and comparing, at the end of each interval, the balanced components of the flow obtained by a) integrating the model from t = 0 and b) integrating the full equations, initialised at t = τ with the balanced components from a) at t = τ. We find that any impact of the unbalanced component of the flow is less than the numerical noise of the model