On calculating the potential vorticity flux

We discuss and compare different approaches to calculating the dynamics of anisotropic flow structure formation in quasi two-dimensional turbulence based on potential vorticity (PV) transport in real space. The general structure of the PV flux in the relaxation processes is deduced non-perturbatively. The transport coefficients of the PV flux are then systematically calculated using perturbation theory. We develop two non-perturbative relaxation models: the first is a mean field theory for the dynamics of minimum enstrophy relaxation based on the requirement that the mean flux of PV dissipates total potential enstrophy but conserves total fluid kinetic energy. The results show that the structure of PV flux has the form of a sum of a positive definite hyper-viscous and a negative or positive viscous flux of PV. Turbulence spreading is shown to be related to PV mixing via the link of turbulence energy flux to PV flux. In the relaxed state, the ratio of the PV gradient to zonal flow velocity is homogenized. ...