Interpretation of stochastic primitive equations with relaxed hydrostatic assumption as a higher order approximation of 3D stochastic Navier-Stokes

TL;DR

This paper studies the convergence of stochastic primitive equations to 3D Navier-Stokes equations, exploring relaxed hydrostatic assumptions to include non-hydrostatic effects as a higher-order approximation.

Contribution

It introduces a stochastic generalized hydrostatic model that relaxes the hydrostatic assumption, capturing non-hydrostatic effects within the primitive equations framework.

Findings

The stochastic model converges to Navier-Stokes solutions under certain conditions.

The relaxed hydrostatic model remains well-posed with regularized noise.

The model provides a higher-order approximation of 3D Navier-Stokes equations.

Abstract

In this paper, we investigate the convergence of solutions of a stochastic representation of the three-dimensional Navier-Stokes equations to those of their primitive equations counterpart. Our analysis covers both weak and strong convergence regimes, corresponding respectively to rigid-lid and "fully periodic" boundary conditions. Furthermore, we explore the impact of relaxing the hydrostatic assumption in the stochastic primitive equations by retaining martingale terms as deviations from hydrostatic equilibrium. This modified model, obtained through a specific asymptotic scaling accessible only within the stochastic framework, captures non-hydrostatic effects while remaining within the primitive equations formalism. The resulting generalized hydrostatic model has been shown to be well-posed when the additional terms are regularized using a suitable filter for divergence-free noises under suitable assumptions. Within this setting, we demonstrate that the model provides a higher-order approximation of the 3D Navier-Stokes equations for appropriately scaled noises.