The hydrostatic approximation of the Boussinesq equations with rotation in a thin domain

TL;DR

This paper establishes the global existence of strong solutions to primitive equations with horizontal viscosity and diffusivity, and rigorously justifies the hydrostatic approximation of scaled Boussinesq equations with rotation in thin domains.

Contribution

It improves existing results on global solutions and provides a rigorous convergence proof of the scaled Boussinesq equations to primitive equations as the aspect ratio tends to zero.

Findings

Global existence of strong solutions under weaker initial data assumptions.

Strong convergence of scaled Boussinesq equations to primitive equations as aspect ratio approaches zero.

Convergence rate of order O(λ^{η/2}) depending on initial data regularity.

Abstract

In this paper, we improve the global existence result in [9] slightly. More precisely, the global existence of strong solutions to the primitive equations with only horizontal viscosity and diffusivity is obtained under the assumption of initial data $(v_0,T_0) \in H^1$ with $\partial_z v_0 \in L^4$. Moreover, we prove that the scaled Boussinesq equations with rotation strongly converge to the primitive equations with only horizontal viscosity and diffusivity, in the cases of $H^1$ initial data, $H^1$ initial data with additional regularity $\partial_z v_0 \in L^4$ and $H^2$ initial data, respectively, as the aspect ration parameter $\lambda$ goes to zero, and the rate of convergence is of the order $O(\lambda^{{\eta}/2})$ with $\eta=\min\{2,\beta-2,\gamma-2\}(2<\beta,\gamma<\infty)$. The convergence result implies a rigorous justification of the hydrostatic approximation.