The Vortex Gas Closure of Baroclinic Turbulence

The eddy heat flux generated by the statistically equilibrated baroclinic instability of a uniform, horizontal temperature gradient is studied using a two-level f-plane quasigeostrophic model. In the limit of small friction, relevant for the extratropical atmosphere and possibly the Southern Ocean, the eddy field consists in a gas of well-defined vortices. Building on previous work by Thompson, Young and Held, we derive scaling laws for the dependence of the eddy diffusivity D on the deformation radius, the vertical variations of the large-scale flow, and the bottom friction. We consider both linear and quadratic bottom friction. The scaling laws are very well supported by a suite of simulations where we vary all the relevant non dimensional parameters. Finally we present an example of a laterally varying large-scale temperature gradient and show that the scaling laws apply locally for that problem as well. The predictions from this local closure agree quantitatively with the simulations, as long as the temperature gradient varies on scales larger than the mixing length scale associated with the vortices.