## Abstract

We study numerically the long-time evolution of the surface quasi-geostrophic
equation with generalised viscosity of the form $(-¥Delta)^¥alpha$, where global regularity has
been proved mathematically for the subcritical parameter range $¥alpha ¥geq ¥frac{1}{2}$. 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 $¥alpha =¥frac{1}{2}$ is of technical nature.