Rigorous derivation of the full primitive equations by the scaled Boussinesq equations with rotation

TL;DR

This paper rigorously derives the full primitive equations of geophysical fluid dynamics from the scaled Boussinesq equations with rotation, establishing strong, global-in-time convergence with a rate of O(ε).

Contribution

It extends previous results by including temperature effects and proves convergence of the scaled Boussinesq equations to the primitive equations with rotation.

Findings

Proves strong convergence of scaled Boussinesq to primitive equations

Establishes convergence rate of O(ε)

Results hold globally in time

Abstract

The primitive equations of large-scale oceanic dynamics form a fundamental model in geophysical flows. It is well-known that the primitive equations can be formally derived by the hydrostatic approximation. On the other hand, the mathematically rigorous derivation of the primitive equations without coupling with the temperature is also known. In this paper, we generalize the above result from the mathematical point of view. More precisely, we prove that the scaled Boussinesq equations with rotation converge to the full primitive equations in a strong sense, globally in time, with the convergence rate $O(\varepsilon)$, as the aspect ratio $\varepsilon$ goes to zero.