On Turbulent Behavior of the Generalized Surface Quasigeostrophic Equations

Turbulent behavior of the two-parameter family of generalized surface quasigeostrophic equations is examined both rigorously and numerically. We adapt a cascade mechanism argument to derive an energy spectrum that scales as $\kappa^{2\beta/3-3}$ where $\beta$ controls the regularity of the velocity ($\beta=1$ in the special case of the SQG). Direct numerical simulations indicate that this fits better than $\kappa^{\beta/3-3}$ which was derived in earlier work. Guided by earlier work on the 2D Navier-Stokes equations, we prove a certain condition implies a direct cascade of enstrophy, as well as an upper bound on the enstrophy dissipation rate, and sharp bounds on a dissipation wavenumber. The dependence of these rigorous results on the two parameters is demonstrated numerically.