Parameterization of ocean eddies: Potential vorticity mixing, energetics and Arnold's first stability theorem

A family of eddy closures is studied that flux potential vorticity down-gradient and solve an explicit budget for the eddy energy, following the approach developed by Eden and Greatbatch (2008, Ocean Modelling). The aim of this manuscript is to demonstrate that when energy conservation is satisfied in this manner, the growth or decay of the parameterized eddy energy relates naturally to the instability or stability of the flow as described by Arnold's first stability theorem. The resultant family of eddy closures therefore possesses some of the ingredients necessary to parameterize the gross effects of eddies in both forced-dissipative and freely-decaying turbulence. These ideas are illustrated through their application to idealized, barotropic wind-driven gyres in which the maximum eddy energy occurs within the viscous boundary layers and separated western boundary currents, and to freely-decaying turbulence in a closed barotropic basin in which inertial Fofonoff gyres emerge as the long-time solution. The result that these eddy closures preserve the relation between the growth or decay of eddy energy and the instability or stability of the flow provides further support for their use in ocean general circulation models.