Some properties of a non-hydrostatic stochastic oceanic primitive equations model

TL;DR

This paper investigates the mathematical properties of a non-hydrostatic stochastic oceanic primitive equations model, focusing on well-posedness and the relation to deterministic solutions, with implications for numerical modeling of non-hydrostatic phenomena.

Contribution

It establishes the well-posedness of a stochastic non-hydrostatic primitive equations model under specific boundary and noise conditions, linking stochastic and deterministic solutions.

Findings

Proved well-posedness of the stochastic model.

Showed convergence of stochastic solutions to deterministic ones as noise vanishes.

Identified conditions under which the model accurately captures non-hydrostatic effects.

Abstract

In this paper, we study how relaxing the classical hydrostatic balance hypothesis affects theoretical aspects of the LU primitive equations well-posedness. We focus on models that sit between incompressible 3D LU Navier-Stokes equations and standard LU primitive equations, aiming for numerical manageability while capturing non-hydrostatic phenomena. Our main result concerns the well-posedness of a specific stochastic interpretation of the LU primitive equations. This holds with rigid-lid type boundary conditions, and when the horizontal component of noise is independent of z. In fact these conditions can be related to the dynamical regime in which the primitive equations remain valid. Moreover, under these conditions, we show that the LU primitive equations solution tends toward the one of the deterministic primitive equations for a vanishing noise, thus providing a physical coherence to the LU stochastic model.