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

Arnaud Debussche, Étienne Mémin, Antoine Moneyron

TL;DR

This work advances stochastic geophysical fluid dynamics by analyzing a location-uncertainty (LU) formulation of the 3D Navier–Stokes equations and their convergence to stochastic primitive equations in a thin-domain limit. It introduces regularised weak hydrostatic primitive equations (PE$_{\alpha_\sigma}^{\varepsilon}$) and a corresponding regularised NS (rSNS$_{\alpha_\sigma}^{\varepsilon}$) within a robust abstract framework built on modified Leray projectors and carefully chosen function spaces. The main results establish weak convergence under rigid-lid BC with bidimensional noise and improved estimates under fully periodic BC, identifying a grey zone where weak hydrostatic PE provides a higher-order approximation than strong hydrostatics; strong hydrostatic convergence occurs under stricter scaling $\alpha_\sigma = o(\varepsilon^{-1/2})$, with the weak regime achievable for $\alpha_\sigma = o(\varepsilon^{-1})$. Collectively, the results extend stochastic well-posedness for LU models and quantify asymptotic errors in stochastic thin-domain limits, underpinning efficient reduced models for geophysical flows.

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.

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

TL;DR

This work advances stochastic geophysical fluid dynamics by analyzing a location-uncertainty (LU) formulation of the 3D Navier–Stokes equations and their convergence to stochastic primitive equations in a thin-domain limit. It introduces regularised weak hydrostatic primitive equations (PE) and a corresponding regularised NS (rSNS) within a robust abstract framework built on modified Leray projectors and carefully chosen function spaces. The main results establish weak convergence under rigid-lid BC with bidimensional noise and improved estimates under fully periodic BC, identifying a grey zone where weak hydrostatic PE provides a higher-order approximation than strong hydrostatics; strong hydrostatic convergence occurs under stricter scaling , with the weak regime achievable for . Collectively, the results extend stochastic well-posedness for LU models and quantify asymptotic errors in stochastic thin-domain limits, underpinning efficient reduced models for geophysical flows.

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.
Paper Structure (27 sections, 10 theorems, 146 equations)

This paper contains 27 sections, 10 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. The LU formalism
  3. Scaled Navier-Stokes equations and primitive equations in the LU formalism
  4. Navier-Stokes equations with linear law of state (NS)
  5. Discussion about the noise structure
  6. The bidimensional case
  7. The tridimensional case
  8. Scaled Navier-Stokes equations and primitive equations
  9. Scaled Navier-Stokes equation (SNS$_{\alpha_\sigma}^\epsilon$)
  10. Primitive equations with strong hydrostatic hypothesis (PE)
  11. Non-regularised primitive equations with weak hydrostatic assumption
  12. (Regularised) primitive equations with weak hydrostatic assumption (PE$_{\alpha_\sigma}^\epsilon$)
  13. Scaled LU Navier-Stokes equations with regularised noise term (rSNS$_{\alpha_\sigma}^\epsilon$)
  14. Boundary conditions
  15. Abstract formulation of the problem
  16. ...and 12 more sections

Key Result

Proposition 1

The problem $(\mathcal{P}_\epsilon^{rSNS})$, equipped with the initial condition $(u_0, \theta_0) \in H$, admits a least one global-in-time martingale solution.

Theorems & Definitions (15)

  • Remark 3.1
  • Definition 1: Global martingale Leray-Hopf weak solution
  • Proposition 1
  • Theorem 1
  • Remark 4.1
  • Proposition 2
  • Theorem 2
  • Remark 4.2
  • Proposition 3
  • Theorem 3
  • ...and 5 more