Potential Vorticity Dynamics and Models of Zonal Flow Formation

We describe the general theory of anisotropic flow formation in quasi two- dimensional turbulence from the perspective on the potential vorticity (PV) transport in real space. The aim is to calculate the vorticity or PV flux. In Chapter 2, the general structure of PV flux is deduced non-perturbatively using two 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 analyses 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. This structure of the relaxed state is consistent with PV staircases. The homogenized quantity sets a constraint on the amplitudes of PV and zonal flow in the relaxed state. The second relaxation model is derived from a joint reflection symmetry principle, which constrains the PV flux driven by the deviation from the self- organized state. The form of PV flux contains a nonlinear convective term in addition to viscous and hyper-viscous terms. The nonlinear convective term, how- ever, can be viewed as a generalized diffusion, on account of the gradient- dependent ballistic transport in avalanche-like systems. For both cases, the detailed transport coefficients can be calculated using perturbation theory in Chapter 3. For a broad turbulence spectrum, a modulational calculation of the PV flux gives both a negative viscosity and a positive hyper-viscosity. For a narrow turbulence spectrum, the result of a parametric instability analysis shows that PV transport is also convective. In both relaxation and perturbative analyses, it is shown that turbulent PV transport is sensitive to flow structure, and the transport coefficients are nonlinear functions of flow shear. In Chapter 4, the effect of mean shear flows on zonal flow formation is considered in the contexts of plasma drift wave turbulence and quasi-geostrophic turbulence models. The generation of zonal flows by modulational instability in the presence of large-scale mean shear flows is studied using the method of characteristics as applied to the wave kinetic equation. It is shown that mean shear flows reduce the modulational instability growth rate by shortening the coherency time of the wave spectrum with the zonal shear. The scalings of zonal flow growth rate and turbulent vorticity flux with mean shear are determined in the strong shear limit