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arxiv: 2309.12077 · v3 · pith:ZLAI7TCCnew · submitted 2023-09-21 · ⚛️ physics.flu-dyn

Stochastic compressible Navier-Stokes equations under location uncertainty and their approximations for ocean modelling

Gilles Tissot , \'Etienne M\'emin , Quentin Jamet This is my paper

Pith reviewed 2026-05-24 07:10 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords stochastic Navier-Stokeslocation uncertaintyocean modelingBoussinesq approximationvertical mixinglarge-eddy simulationpenetrative convectioncompressible fluids
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The pith

Stochastic compressible Navier-Stokes equations under location uncertainty model upper ocean vertical mixing via Boussinesq approximations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a stochastic version of the compressible Navier-Stokes equations by extending the Reynolds transport theorem to include location uncertainty and stochastic source terms. Conservation statements for mass, species mass, momentum, and total energy yield transport equations for density, mass fraction, velocity, and temperature. Boussinesq approximations applied to this system recover prior stochastic incompressible and Boussinesq formulations. Large-eddy simulations of temperature-driven free convection show that stochastic transport reproduces penetrative convection at the mixed-layer base. Compression effects prove negligible in the internal-energy temperature equation but significant in the potential-energy form, exposing properties of stochastic pressure terms.

Core claim

The stochastic compressible Navier-Stokes equations under location uncertainty are obtained by applying an extended stochastic Reynolds transport theorem with source terms to the conservation of mass, species, momentum, and energy; after Boussinesq approximations these equations recover known stochastic models and, in large-eddy simulation of free convection, enable stochastic transport to capture penetrative convection while compression effects remain negligible for internal energy but significant for potential energy.

What carries the argument

The extended stochastic Reynolds transport theorem incorporating location uncertainty and stochastic source terms.

If this is right

  • Stochastic transport reproduces penetrative convection effects at the base of the mixed layer under the Boussinesq approximation.
  • Compression effects identified in the stochastic Boussinesq hydrostatic model are negligible in the temperature equation expressed with internal energy.
  • When the temperature equation uses potential energy, compression effects become significant and reveal properties of the stochastic pressure terms within the mixed layer.
  • The framework provides a basis for improving the energetic consistency of subgrid-scale vertical models used in ocean general circulation models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same derivation path could be tested on stratified or wind-driven cases to check whether stochastic pressure terms remain dominant only in potential-energy budgets.
  • If the energetic consistency gain holds, the approach supplies a route to reduce artificial dissipation in existing ocean subgrid closures without adding new tunable parameters.
  • Direct comparison of the derived stochastic pressure work against high-resolution deterministic reference runs would quantify how much of the observed mixed-layer compression is captured by location uncertainty alone.

Load-bearing premise

The stochastic Reynolds transport theorem with location uncertainty can be applied to the compressible continuity, momentum, and energy equations while preserving the validity of the subsequent Boussinesq approximations.

What would settle it

A large-eddy simulation or field observation in which the stochastic terms fail to produce penetrative convection at the mixed-layer base or produce inconsistent energy balances between internal and potential forms would falsify the modeling utility.

read the original abstract

This paper presents a joint theoretical and numerical study of a stochastic version of the compressible Navier-Stokes equations within the location uncertainty (LU) framework, applied to problems related to upper ocean vertical mixing. This approach builds on an extended stochastic form of the Reynolds transport theorem, incorporating stochastic source terms. As in the deterministic case, this conservation theorem is applied to mass, mass of species (such as salinity), momentum, and total energy, leading to transport equations for the primitive variables: density, mass fraction of species, velocity, and temperature. We subsequently apply the Boussinesq approximations to this general system, and recover existing formulations of the incompressible stochastic Navier-Stokes and stochastic Boussinesq equations. We employ this new framework in a Boussinesq large-eddy simulation of temperature-driven free convection event, and highlight the potential of stochastic transport to reproduce penetrative convection effects at the base of the mixed layer under the Boussinesq approximation. Compression effects identified in our stochastic Boussinesq hydrostatic model are found to be negligible in the temperature equation when expressed in terms of internal energy, in agreement with Boussinesq approximation. However, when expressed in terms of potential energy, compression effects become significant, and reveal interesting properties of the stochastic pressure terms within the mixed layer. We believe this later results open new physical modelling perspective enabling to represent oceanic dynamics within a stochastic framework that more fully accounts for physical uncertainties and approximations, while also providing a basis for improving the energetic consistency of subgrid-scale vertical models used in ocean general circulation models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper derives stochastic compressible Navier-Stokes equations via an extended stochastic Reynolds transport theorem with location uncertainty and stochastic source terms, applies this to mass, species, momentum and energy conservation, then invokes Boussinesq approximations to recover known stochastic incompressible and Boussinesq systems. It further presents a Boussinesq LES of temperature-driven free convection, claiming that stochastic transport reproduces penetrative convection at the mixed-layer base while compression effects remain negligible in the internal-energy temperature equation but become significant in the potential-energy form, opening perspectives for stochastic ocean modeling and improved subgrid energetics.

Significance. If the Boussinesq step remains valid, the derivation supplies a systematic route from compressible stochastic conservation laws to reduced ocean models that incorporate location uncertainty, with the numerical experiment suggesting that stochastic pressure work can affect mixed-layer energetics in ways relevant to vertical mixing parameterizations.

major comments (2)
  1. [Boussinesq application section] Boussinesq application section: the reduction to the stochastic Boussinesq system assumes that stochastic advection and source terms preserve the small relative density fluctuation scaling (typically O(10^{-3})) required to filter acoustics and treat density as constant except in buoyancy; however, no a-priori bound is supplied on the amplitude of the location-uncertainty field that would guarantee this scaling once the stochastic pressure-work terms identified in the numerical experiment are active.
  2. [Numerical experiment] Numerical experiment: the penetrative-convection claim and the distinction between internal-energy and potential-energy forms rest on a single LES configuration whose quantitative metrics, resolution sensitivity, and uncertainty quantification are not reported, leaving open whether post-hoc choices affect the reported compression-effect contrast.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We provide point-by-point responses to the major comments below.

read point-by-point responses

  1. Referee: [Boussinesq application section] Boussinesq application section: the reduction to the stochastic Boussinesq system assumes that stochastic advection and source terms preserve the small relative density fluctuation scaling (typically O(10^{-3})) required to filter acoustics and treat density as constant except in buoyancy; however, no a-priori bound is supplied on the amplitude of the location-uncertainty field that would guarantee this scaling once the stochastic pressure-work terms identified in the numerical experiment are active.

    Authors: We agree that an explicit a-priori bound on the location-uncertainty amplitude would strengthen the consistency argument. In the revised manuscript we will add a short scaling analysis in the Boussinesq section showing that, provided the uncertainty velocity remains O(10^{-2}) or smaller relative to the resolved velocity (consistent with its subgrid interpretation), the stochastic contributions to density fluctuations stay higher-order and the O(10^{-3}) relative-density scaling is preserved. revision: yes

  2. Referee: [Numerical experiment] Numerical experiment: the penetrative-convection claim and the distinction between internal-energy and potential-energy forms rest on a single LES configuration whose quantitative metrics, resolution sensitivity, and uncertainty quantification are not reported, leaving open whether post-hoc choices affect the reported compression-effect contrast.

    Authors: The experiment is presented as an illustrative demonstration of the framework. We accept that additional quantitative support is desirable. In revision we will report basic quantitative metrics (mixed-layer depth evolution, domain-integrated energy budgets) and a brief resolution-sensitivity check performed with the existing code. A full ensemble-based uncertainty quantification lies beyond the computational scope of the present study and is identified as future work. revision: partial

Circularity Check

1 steps flagged

Location-uncertainty framework presupposed from prior self-citation; compressible derivation and Boussinesq reduction remain independent

specific steps
  1. self citation load bearing [Abstract]
    "This approach builds on an extended stochastic form of the Reynolds transport theorem, incorporating stochastic source terms. As in the deterministic case, this conservation theorem is applied to mass, mass of species (such as salinity), momentum, and total energy, leading to transport equations for the primitive variables: density, mass fraction of species, velocity, and temperature. We subsequently apply the Boussinesq approximations to this general system, and recover existing formulations of the incompressible stochastic Navier-Stokes and stochastic Boussinesq equations."

    The load-bearing modeling premise (stochastic RTT under location uncertainty with source terms) is imported from prior work by overlapping authors rather than re-derived or externally validated here; the paper's new content is the application to compressible equations and Boussinesq recovery, which does not circularly reduce to the cited theorem but inherits its foundational assumptions without independent bounds on stochastic density fluctuations.

full rationale

The paper's core modeling step invokes the stochastic Reynolds transport theorem with location uncertainty from prior work (abstract: 'This approach builds on an extended stochastic form of the Reynolds transport theorem, incorporating stochastic source terms'). This is a self-citation load-bearing element for the overall framework, but the subsequent application to compressible continuity/momentum/energy equations, the primitive-variable transport equations, and the Boussinesq reduction are logically new steps that do not reduce to the cited input by construction. The numerical LES experiment is presented as illustrating potential rather than as an independent falsification test of the framework assumptions. No self-definitional, fitted-prediction, or ansatz-smuggling reductions are exhibited in the provided derivation chain. This yields a moderate circularity score of 4 with one non-load-bearing self-citation step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the location-uncertainty stochastic transport theorem (prior framework) and the Boussinesq approximation applied to the stochastic system; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Boussinesq approximation remains valid after addition of stochastic transport terms
    Invoked to recover incompressible stochastic NS and Boussinesq equations
  • domain assumption Stochastic Reynolds transport theorem with location uncertainty applies to compressible continuity, momentum and energy
    Starting point for the entire derivation

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discussion (0)

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Reference graph

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