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

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 (v0, T0) ∈ H 1 with ∂zv0 ∈ 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 ∂zv0 ∈ L 4 and H 2 initial data, respectively, as the aspect ration parameter λ goes to zero, and the rate of convergence is of the order O(λ η/2 ) with η = min{2, β − 2, γ − 2}(2 < β, γ < ∞). The convergence result implies a rigorous justification of the hydrostatic approximation.