A new gauge-invariant method for diagnosing eddy diffusivities

Coarse resolution numerical ocean models must typically include a parameterisation for mesoscale turbulence. A common recipe for such parameterisations is to invoke mixing of some tracer quantity, such as potential vorticity or buoyancy. However, it is well known that eddy fluxes include large rotational components which necessarily do not lead to any mixing; eddy diffusivities diagnosed from unfiltered fluxes are thus contaminated by the presence of these rotational components. Here a new methodology is applied where by eddy diffusivities are diagnosed directly from the eddy force function. The eddy force function depends only upon flux divergences, is independent of any rotational flux components, and is inherently non-local and smooth. A one-shot inversion procedure is applied, minimising the mis-match between parameterised force functions and force functions derived from eddy resolving calculations. This enables diffusivities associated with the eddy potential vorticity and Gent–McWilliams coefficients associated with eddy buoyancy fluxes to be diagnosed. This methodology is applied to multi-layer quasigeostrophic ocean gyre simulations. It is found that:(i) a strictly down-gradient scheme for mixing potential vorticity and quasi-geostrophic buoyancy has limited success in reducing the mis-match compared to one with no sign constraint on the eddy diffusivity or Gent–McWilliams coefficient, with prevalent negative signals around the time-mean jet; (ii) the diagnostic is successful away from the jet region and wind-forced top layer; (iii) the locations of closed mean stream lines correlate with signals of positive eddy potential vorticity diffusivity; (iv) there is indication that the magnitude of the eddy potential vorticity diffusivity correlates well with the eddy energy. Implications for parameterisation are discussed in light of these diagnostic results