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 whereby 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. The methodology is applied to multi-layer quasi-geostrophic
ocean gyre simulations. It is found that: (i) a strictly down-gradient scheme for mixing potential vorticity and quasigeostrophic buoyancy has limited success in reducing the mis-match compared to one with no sign constraint on the
eddy diffusivity or coefficient, with prevalent negative signals of eddy diffusivity and coefficient around the timemean jet; (ii) the parameterisations are able to reproduce eddy flux patterns in regions away from strong forcing and
dissipation; (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.