Oscillatory damping in long-time evolution of the surface quasi-geostrophic equations with generalized viscosity: a numerical study

We study numerically the long-time evolution of the surface quasi-geostrophic equation with generalized viscosity of the form (−)α, where global regularity has been proved mathematically for the subcritical parameter range α ≥ 1/2. Even in the supercritical range, we have found numerically that smooth evolution persists, but with a very slow and oscillatory damping in the long run. A subtle balance between nonlinear and dissipative terms is observed therein. Notably, qualitative behaviours of the analytic properties of the solution do not change in the super and subcritical ranges, suggesting the current theoretical boundary α = 1/2 is of technical nature