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Some properties of a non-hydrostatic stochastic oceanic primitive equations model

Arnaud Debussche, Étienne Mémin, Antoine Moneyron

TL;DR

This work introduces a non-hydrostatic stochastic oceanic primitive equations model within the LU framework, relaxing the strong hydrostatic balance via a weak low-pass filtered hypothesis implemented with a smoothing kernel $K$ and a covariance operator $a^K$. It formulates an abstract SPDE for the state $U=(v,T,S)^T$ with a martingale pressure term $dp_t^\sigma$ and a pressure relation expressed through vertical integrals, and proves well-posedness results: global martingale solutions for the filtered problem $(\mathcal{P}_K)$, local pathwise solutions with stopping times, and uniqueness; these results are strengthened under the Barotropic Horizontal Noise (BHN) assumption, yielding globally defined pathwise solutions and continuous dependence on data. The key contribution is establishing global-in-time well-posedness and a vanishing-noise limit $U^{\sqrt{\Upsilon}\sigma} \to U^0$, connecting the stochastic LU model with the deterministic primitive equations and providing a coherent physical interpretation of the stochastic terms. The work thus offers a mathematically rigorous pathway to incorporate non-hydrostatic stochastic effects into LU-based ocean models, with potential impact on climate-relevant simulations and uncertainty quantification under rigid-lid boundary conditions.

Abstract

In this paper, we study how relaxing the classical hydrostatic balance hypothesis affects theoretical aspects of the LU primitive equations well-posedness. We focus on models that sit between incompressible 3D LU Navier-Stokes equations and standard LU primitive equations, aiming for numerical manageability while capturing non-hydrostatic phenomena. Our main result concerns the well-posedness of a specific stochastic interpretation of the LU primitive equations. This holds with rigid-lid type boundary conditions, and when the horizontal component of noise is independent of z. In fact these conditions can be related to the dynamical regime in which the primitive equations remain valid. Moreover, under these conditions, we show that the LU primitive equations solution tends toward the one of the deterministic primitive equations for a vanishing noise, thus providing a physical coherence to the LU stochastic model.

Some properties of a non-hydrostatic stochastic oceanic primitive equations model

TL;DR

This work introduces a non-hydrostatic stochastic oceanic primitive equations model within the LU framework, relaxing the strong hydrostatic balance via a weak low-pass filtered hypothesis implemented with a smoothing kernel and a covariance operator . It formulates an abstract SPDE for the state with a martingale pressure term and a pressure relation expressed through vertical integrals, and proves well-posedness results: global martingale solutions for the filtered problem , local pathwise solutions with stopping times, and uniqueness; these results are strengthened under the Barotropic Horizontal Noise (BHN) assumption, yielding globally defined pathwise solutions and continuous dependence on data. The key contribution is establishing global-in-time well-posedness and a vanishing-noise limit , connecting the stochastic LU model with the deterministic primitive equations and providing a coherent physical interpretation of the stochastic terms. The work thus offers a mathematically rigorous pathway to incorporate non-hydrostatic stochastic effects into LU-based ocean models, with potential impact on climate-relevant simulations and uncertainty quantification under rigid-lid boundary conditions.

Abstract

In this paper, we study how relaxing the classical hydrostatic balance hypothesis affects theoretical aspects of the LU primitive equations well-posedness. We focus on models that sit between incompressible 3D LU Navier-Stokes equations and standard LU primitive equations, aiming for numerical manageability while capturing non-hydrostatic phenomena. Our main result concerns the well-posedness of a specific stochastic interpretation of the LU primitive equations. This holds with rigid-lid type boundary conditions, and when the horizontal component of noise is independent of z. In fact these conditions can be related to the dynamical regime in which the primitive equations remain valid. Moreover, under these conditions, we show that the LU primitive equations solution tends toward the one of the deterministic primitive equations for a vanishing noise, thus providing a physical coherence to the LU stochastic model.
Paper Structure (9 sections, 2 theorems, 48 equations)

This paper contains 9 sections, 2 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Oceans dynamics models in the LU framework
  3. The LU primitive equations
  4. Weak low-pass filtered hydrostatic hypothesis:
  5. Definition of the spaces
  6. Abstract formulation of the problems
  7. Low-pass filtered problem $(\mathcal{P}_K)$:
  8. Main results
  9. Remark:

Key Result

Theorem 1

Suppose $K \in H^{3}(\mathcal{S}, \mathbbm{R})$. Then, the following propositions hold,

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2