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

Debussche, Arnaud, Étienne Mémin+1 more|arXiv (Cornell University)|Feb 4, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper studies convergence of a stochastic 3D Navier-Stokes model to stochastic primitive equations under various boundary conditions and introduces a relaxed hydrostatic model that captures non-hydrostatic effects as a higher-order approximation.

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.

Motivation & Objective

• Motivate the use of stochastic modeling to represent unresolved variability in geophysical flows.
• Show convergence of LU (location uncertainty) 3D Navier-Stokes to LU primitive equations under rigid-lid and fully periodic boundaries.
• Introduce a relaxed hydrostatic primitive equations model with martingale pressure deviations to capture non-hydrostatic effects.
• Establish well-posedness of the regularized relaxed hydrostatic model under appropriate noise regularization.
• Demonstrate that the relaxed hydrostatic LU primitive equations can provide a higher-order approximation of LU 3D Navier-Stokes for suitable noise scaling.

Proposed method

• Define a scaled LU Navier-Stokes framework on thin domains with aspect ratio epsilon.
• Introduce a scaled gradient and a modified Leray projector to handle scaled pressure terms in Itô calculus.
• Derive scaled LU Navier-Stokes equations and the LU primitive equations with bidimensional and tridimensional noise structures.
• Prove convergence results for weak solutions under rigid-lid boundaries and strong solutions under fully periodic boundaries.
• Analyze the role of vertical noise scaling alpha_sigma and epsilon to identify regimes where weak hydrostatic, strong hydrostatic, or higher-order approximations are preferred.
• Discuss the energy balance and stochastic transport terms within the LU framework.

Experimental results

Research questions

• RQ1 Under what conditions do solutions of the LU 3D Navier-Stokes equations converge to the LU primitive equations with rigid-lid boundary conditions?
• RQ2 How does relaxing hydrostatic balance via stochastic martingale pressure deviations affect convergence and well-posedness?
• RQ3 What noise scaling regimes (in terms of alpha_sigma and epsilon) yield weak hydrostatic or strong hydrostatic primitive equations as higher-order approximations?
• RQ4 Is the relaxed hydrostatic LU primitive equations well-posed after regularization of the additional terms?
• RQ5 How do bidimensional versus tridimensional noise structures influence convergence and approximation quality?

Key findings

• Weak convergence of LU 3D Navier-Stokes to LU primitive equations under rigid-lid boundaries is established.
• Strong convergence is shown for fully periodic domains in the presence of appropriate noise scaling.
• A relaxed hydrostatic primitive equations model with martingale pressure deviations captures non-hydrostatic effects while remaining within the primitive equations framework.
• There exist regimes where regularised LU Navier-Stokes can be efficiently approximated by the LU primitive equations with weak hydrostatic balance, but not by strong hydrostatic ones.
• Convergence analyses rely on a scaled gradient and a modified Leray projector to manage stochastic pressure terms in the Itô calculus setting.