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

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.

💡 Research Summary

This paper investigates the relationship between three‑dimensional stochastic Navier‑Stokes equations (NS) and their primitive‑equation (PE) counterpart within the stochastic Location‑Uncertainty (LU) framework. The authors first recall that large‑scale oceanic and atmospheric flows involve a huge range of interacting scales, making deterministic direct numerical simulation infeasible. Stochastic parameterisations, especially those grounded in the LU theory, provide a mathematically rigorous way to represent unresolved variability while preserving physical conservation laws.

In the LU setting the Lagrangian particle trajectory Xₜ satisfies the Itô‑type stochastic differential equation
 dXₜ = u(Xₜ,t) dt + σ(Xₜ,t) dWₜ,
where u is the resolved velocity field and σ dW represents highly oscillatory unresolved motions. The noise operator σ(t) is Hilbert‑Schmidt, inducing a covariance tensor a(x,t) that appears in the Itô‑Stokes drift uₛ = ½∇·a. Applying a stochastic Reynolds transport theorem to mass and momentum balances yields a stochastic Navier‑Stokes system with a semimartingale pressure p = pᵈᵗ dt + p^σ dWₜ.

The authors then consider a thin domain S_ε = S_H × (‑ε, ε) with aspect ratio ε = H/L (vertical over horizontal length scale). Spatial derivatives are rescaled as ∇_ε = (∇_H, ε⁻¹∂_z) and the pressure gradient appears as ∇_εp. To handle the scaled pressure term in the stochastic setting, they introduce a “modified Leray projector” P_ε that removes both the scaled pressure gradient and the Itô‑Stokes drift contributions. This projector is essential because Itô’s lemma generates additional covariance terms that would otherwise prevent the use of classical Leray theory.

Two convergence regimes are studied.

1. Weak (L²–H¹) convergence under rigid‑lid boundary conditions. Here the horizontal component of the noise is assumed two‑dimensional (independent of the vertical coordinate). By letting ε → 0, the weak solutions of the stochastic NS converge to weak solutions of a weak‑hydrostatic primitive equation, i.e., the hydrostatic balance is relaxed and the vertical acceleration appears as a martingale term.
2. Strong (H¹–H²) convergence under fully periodic boundary conditions. In this case the vertical noise component is retained and scaled by a parameter α_σ. The authors identify two scaling windows:
   • If α_σ = o(ε⁻¹) but α_σ ≠ o(ε⁻¹⁄²), the weak‑hydrostatic PE provides a higher‑order approximation of the stochastic NS.
   • If α_σ = o(ε⁻¹⁄²), the convergence improves further and the strong‑hydrostatic PE (the classical primitive equations) is recovered.

Thus the aspect ratio ε and the vertical noise amplitude α_σ control whether the non‑hydrostatic effects are significant. The analysis shows that when the vertical noise is sufficiently small relative to ε, the traditional hydrostatic assumption is justified; otherwise, the relaxed model captures essential non‑hydrostatic dynamics.

To guarantee well‑posedness of the relaxed PE, the authors regularise the divergence‑free noise by applying a low‑pass filter (e.g., a spectral cutoff). The filtered noise remains L²‑integrable, and the resulting stochastic PE admits global existence and uniqueness of solutions. Energy estimates are obtained at the level of second moments (mean‑square), which is typical for SPDEs where almost‑sure bounds are hard to achieve. Consequently, the blow‑up time of the scaled stochastic NS tends to infinity in probability as ε → 0, extending deterministic thin‑domain well‑posedness results to the stochastic context.

The main contributions of the paper are:

• Introduction of a stochastic primitive‑equation model that retains vertical martingale terms, thereby relaxing the hydrostatic balance while staying within the primitive‑equation formalism.
• Development of a modified Leray projector that cancels scaled pressure and Itô‑Stokes drift terms, enabling the use of classical Leray techniques in a stochastic setting.
• Rigorous weak and strong convergence theorems linking stochastic NS to both weak‑hydrostatic and strong‑hydrostatic primitive equations, with explicit scaling conditions on ε and α_σ.
• Demonstration that the relaxed primitive‑equation model constitutes a higher‑order approximation of the full 3‑D stochastic Navier‑Stokes system when the vertical noise scaling satisfies α_σ = o(ε⁻¹) but not o(ε⁻¹⁄²).

In conclusion, the work extends deterministic thin‑domain convergence results to a stochastic framework, provides a mathematically sound way to incorporate non‑hydrostatic effects via stochastic vertical acceleration, and offers a pathway for more accurate probabilistic climate and ocean modelling where vertical mixing and deep convection are important. Future directions include exploring more general boundary conditions, non‑linear noise structures, and data‑driven calibration of the covariance operator σ.