On turbulent behavior of the generalized surface quasi-geostrophic 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 compares favorably with $ \kappa^{\beta/3-3} $ which was derived in Chaos, Sol. & Frac., 4, 1994. Guided by earlier work on the two-dimensional 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.