The quasi-geostrophic approximation for the rotating stratified Boussinesq equations

Abstract

We prove that the solution of the 3D inviscid Boussinesq equations converges to the solution of the quasi-geostrophic (QG) equations in an asymptotic regime where the intensities of rotation and stratification increase to infinity while the rotation-stratification ratio tends to any positive number other than one. Despite the non-uniformity of the generic convergence rates near the region where the ratio is one, we further show that such quasi-geostrophic approximation continues to be valid even when the ratio goes to one as long as both intensities increase to infinity fast enough. In contrast, we prove non-convergence when such intensities grow sufficiently slow to infinity with the ratio tending to one. Our proof of the non-convergence result contains the first mathematical proof of the Devil's staircase paradox that was originally cast in Theoret. Comput. Fluid Dynamics 9, 223-251 (1997). Combining both results for convergence and non-convergence, we give a lower bound for the growth of the convergence rates when the ratio tends to one. Independently of the aforementioned results proved with respect to the mixed norm for $L^q(0,T;W^{1,\infty})$ which is spatially a type of $L^\infty$, we investigate as well whether spatially $L^2$ type spaces are appropriate for analyzing the quasi-geostrophic approximation or not. We verify that non-convergence actually takes place for any fixed positive rotation-stratification ratio with respect to the $H^s$ norm with any integer $s\geq 3$. Nevertheless, in view of the projections onto the eigenspaces of the linear propagator, we specify a particular class of initial data which ensures convergence even in such $H^s$ regardless of the ratio