Metastability for the Dissipative Quasi-Geostrophic Equation and the Non-local Enhancement

Abstract

In this paper, we study the linear metastability for the linearized 2D dissipative surface quasi-geostrophic equation with small viscosity $$\nu $$ around the quasi-steady state $$\Theta _{\sin }=-e^{-\nu t}\sin y$$ . We proved the linear enhanced dissipation and obtained the dissipation rate. Moreover, the new non-local enhancement phenomenon was discovered and discussed. Precisely we showed that the non-local term $$\cos y\partial _x(-\Delta )^{-\frac{1}{2}}\theta $$ re-enhances the enhanced diffusion effect by the shear-diffusion mechanism.