Geometric Lagrangian averaged Euler-Boussinesq and primitive equations

TL;DR

This paper derives new geometric Lagrangian averaged equations for rotating stratified fluids, incorporating turbulence effects and conservation laws, resulting in models similar to Euler-Boussinesq-$\alpha$ but with a distinct regularization operator.

Contribution

It introduces a geometric Lagrangian averaged framework for Euler-Boussinesq and primitive equations, combining turbulence closure hypotheses with geometric methods, leading to novel regularized models.

Findings

Derived Euler-Poincaré equations for stratified fluids

Models conserve energy and potential vorticity

Feature a new regularizing operator similar to $\alpha$ models

Abstract

In this article we derive the equations for a rotating stratified fluid governed by inviscid Euler-Boussinesq and primitive equations that account for the effects of the perturbations upon the mean. Our method is based on the concept of geometric generalized Lagrangian mean recently introduced by Gilbert and Vanneste, combined with generalized Taylor and horizontal isotropy of fluctuations as turbulent closure hypotheses. The models we obtain arise as Euler-Poincar\'{e} equations and inherit from their parent systems conservation laws for energy and potential vorticity. They are structurally and geometrically similar to Euler-Boussinesq-$\alpha$ and primitive equations-$\alpha$ models, however feature a different regularizing second order operator.