A geometric interpretation of zonostrophic instability

The zonostrophic instability that leads to the emergence of zonal jets in barotropic beta-plane turbulence was analyzed through a geometric decomposition of the eddy stress tensor. The stress tensor is visualized by an eddy variance ellipse whose characteristics are related to eddy properties. The tilt of the ellipse principal axis is the tilt of the eddies with respect to the shear, the eccentricity of the ellipse is related to the eddy anisotropy, while its size is related to the eddy kinetic energy. Changes of these characteristics are directly related to the vorticity fluxes forcing the mean flow. The statistical state dynamics of the turbulent flow closed at second order was employed as it provides an analytic expression for both the zonostrophic instability and the stress tensor. For the linear phase of the instability, the stress tensor was analytically calculated at the stability boundary. For the non--linear equilibration of the instability the tensor was calculated in the limit of small supercriticality in which the amplitude of the jet velocity follows Ginzburg--Landau dynamics. It is found that dependent on the characteristics of the forcing, the jet is accelerated either because it primarily anisotropizes the eddies so as to produce upgradient fluxes or because it changes the eddy tilt. The instability equilibrates as these changes are partially reversed by the non--linear terms. Parameterizations of the ellipse characteristics are also discussed.