Stochastic interpretations of the oceanic primitive equations with relaxed hydrostatic assumptions
Arnaud Debussche, Étienne Mémin, Antoine Moneyron
TL;DR
This work analyzes stochastic LU primitive equations with relaxed hydrostatic balance, situating models between the LU-Navier–Stokes and hydrostatic primitive equations. By formulating abstract low-pass filtered problems and eddy-viscosity variants, the authors prove global martingale solutions and local pathwise well-posedness, with global-pathwise solvability under stronger assumptions and regularization. They develop a rigorous functional-analytic framework, leveraging Galerkin approximations, energy estimates, tightness, and Skorohod representations to handle stochastic transport noise and pressure terms. The results illuminate energy balance mechanisms and the impact of transport-noise regularization (via low-pass filtering and eddy-viscosity terms) on the well-posedness of nonhydrostatic ocean models, with implications for grey-zone ocean dynamics and numerical approximations.
Abstract
In this paper, we investigate how weakening the classical hydrostatic balance hypothesis impacts the well-posedness of the stochastic LU primitive equations. The models we consider are intermediate between the incompressible 3D LU Navier-Stokes equations and the LU primitive equations with standard hydrostatic balance. As such, they are expected to be numerically tractable, while accounting well for phenomena within the grey zone between hydrostatic balance and non-hydrostatic processes. Our main result is the well-posedness of a low-pass filtering-based stochastic interpretation of the LU primitive equations, with rigid-lid type boundary conditions, in the limit of ``quasi-barotropic'' flow. This assumption is linked to the structure assumption proposed in the work of Agresti et al., which can be related to the dynamical regime where the primitive equations remain valid. Furthermore, we present and study two eddy-(hyper)viscosity-based models.