On the hydrostatic limit of stably stratified fluids with isopycnal diffusivity

TL;DR

This paper rigorously justifies the hydrostatic limit for stably stratified fluids with isopycnal diffusivity, without viscosity, using energy estimates in isopycnal coordinates, relevant for oceanographic modeling.

Contribution

It provides the first rigorous hydrostatic limit justification considering only diffusivity effects, without viscosity, for stratified fluids in isopycnal coordinates.

Findings

Established well-posedness of hydrostatic and non-hydrostatic equations.

Proved convergence in the hydrostatic limit with vanishing shallow-water parameter.

Developed energy estimates using isopycnal coordinates.

Abstract

This article is concerned with rigorously justifying the hydrostatic limit for continuously stratified incompressible fluids under the influence of gravity. The main distinction of this work compared to previous studies is the absence of any (regularizing) viscosity contribution added to the fluid-dynamics equations; only thickness diffusivity effects are considered. Motivated by applications to oceanography, the diffusivity effects in this work arise from an additional advection term, the specific form of which was proposed by Gent and McWilliams in the 1990s to model the effective contributions of geostrophic eddy correlations in non-eddy-resolving systems. The results of this paper heavily rely on the assumption of stable stratification. We establish the well-posedness of the hydrostatic equations and the original (non-hydrostatic) equations for stably stratified fluids, along with their convergence in the limit of vanishing shallow-water parameter. These results are obtained in high but finite Sobolev regularity and carefully account for the various parameters involved. A key element of our analysis is the reformulation of the systems using isopycnal coordinates, enabling us to provide meticulous energy estimates that are not readily apparent in the original Eulerian coordinate system.