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

This paper presents a joint theoretical and numerical study of stochastic compressible Navier-Stokes equations under location uncertainty, demonstrating their application to ocean modeling through Boussinesq approximations that reveal significant compression effects in potential energy and offer improved energetic consistency for subgrid-scale vertical mixing models.

The Gist

Imagine the ocean as a giant, churning pot of soup. For a long time, scientists have tried to write the "recipe" for how this soup moves using perfect, deterministic rules. They assumed that if you knew the exact position and speed of every water molecule, you could predict exactly where the soup would go next.

However, the ocean is too complex for that. It's full of tiny, chaotic swirls (turbulence) that are too small to see or measure directly. Trying to simulate every single swirl is like trying to count every grain of sand on a beach to predict the tide—it's impossible and computationally too expensive.

This paper proposes a new way to write the recipe. Instead of trying to track every grain of sand, the authors suggest adding a "fudge factor" based on uncertainty. They call this the Location Uncertainty (LU) framework.

Here is the core idea broken down into simple concepts:

1. The "Drunkard's Walk" Analogy

Imagine you are walking through a crowded market. You have a clear destination (the "large-scale" flow), but people bumping into you push you slightly off course in random directions.

• The Old Way: You try to calculate the exact path of every person bumping into you.
• The New Way (LU): You accept that you will be bumped. You model your movement as a smooth walk plus a random, jittery "Brownian motion" (like a drunkard's walk). You don't know exactly where the bumps will push you, but you know the statistics of the bumps (how strong they are and how they correlate).

2. The "Compressible" Soup

Most ocean models assume water is "incompressible"—meaning it's like a solid block of jelly that can't be squished. But in reality, water can be slightly squished, especially when pressure changes or temperature shifts.

• The authors start with the full, complex physics of compressible water (water that can be squished).
• They then apply their "random bump" math to this complex system.
• The Result: They derive a new set of equations that look like the old ones but include extra terms. These extra terms represent the "work" done by the random bumps. Think of it as the energy transferred when the crowd pushes you; it's not just a random nudge, it actually changes your speed and the heat of your body.

3. The "Hidden Heat" in the Mix

The paper focuses heavily on temperature and convection (hot water rising, cold water sinking).

• The Problem: In standard models, when cold water sinks, it often stops abruptly at the bottom of the mixed layer (the top part of the ocean). In reality, these "plumes" of cold water often punch through, like a spear, into the deeper, warmer water. This is called penetrative convection.
• The Discovery: When the authors ran their new stochastic model, they found that the "random bump" terms naturally recreated this punching-through effect.
• The Metaphor: Imagine a crowd of people (the ocean) trying to move a heavy box (a cold water plume). Standard models act like a rigid wall that stops the box. The new model acts like a chaotic crowd; the random jostling gives the box enough extra momentum to slip through the wall and go deeper than expected.

4. Two Ways to Measure Energy

The authors found something interesting about how they measured the energy of the system:

• Internal Energy (The "Hotness"): When they looked at just the heat, the "squishing" (compression) effects were tiny and didn't matter much. This matched the old, simpler models.
• Potential Energy (The "Height"): But when they looked at the energy related to height (how high the water is in the gravity field), the "squishing" effects became very important.
• The Takeaway: It's like measuring a bouncing ball. If you only measure how hot the ball gets when it hits the floor, the bounce doesn't seem to matter. But if you measure how high it bounces, the impact is huge. The authors found that the random pressure terms in their model act like a hidden spring, affecting how high the water "bounces" in the energy budget.

5. The "Drift" and the "Diffusion"

The math produces two specific new terms that act as the "fudge factors":

• The Drift (Itô-Stokes Drift): This is a systematic push caused by the fact that the random bumps aren't perfectly uniform. It's like a river current that flows slightly differently because the rocks (turbulence) are arranged in a specific pattern.
• The Diffusion: This is the spreading out effect caused by the random bumps.

Summary of the Achievement

The authors successfully built a bridge between the messy, chaotic reality of the ocean and the clean, mathematical models we use to predict it.

• They started with the most complex physics possible (compressible, random, thermodynamic).
• They showed that when you simplify it down to the "standard" ocean view (Boussinesq approximation), your new equations still work and actually improve the prediction of how deep cold water sinks.
• They proved that you don't need to simulate every tiny swirl to get the right answer; you just need to mathematically account for the uncertainty of where the water is going.

In short, they replaced the impossible task of "counting every grain of sand" with a smarter strategy: "account for the sand's tendency to scatter," and found that this approach captures the deep, penetrating currents of the ocean much better than previous methods.

Technical Summary

Technical Summary: Stochastic Compressible Navier–Stokes Equations and Their Approximations for Ocean Modelling

Problem Statement
Ocean modelling faces significant challenges in representing subgrid-scale processes, particularly vertical mixing and penetrative convection, which are critical for climate simulations (e.g., the Atlantic Meridional Overturning Circulation). Traditional approaches often rely on the Boussinesq approximation, which assumes incompressibility and neglects compressibility effects. While reasonable in many contexts, this approximation can obscure energetic consistency between temperature and momentum fluxes and fails to capture acoustic wave propagation or specific thermodynamic work terms associated with compression and dilatation. Furthermore, existing stochastic models for ocean dynamics have largely been derived for incompressible flows, leaving a gap in the theoretical framework for compressible, thermodynamically active fluids. The authors aim to develop a rigorous stochastic formulation that accounts for location uncertainty (LU) in compressible flows, explicitly deriving the necessary thermodynamic terms to improve the energetic consistency of subgrid-scale vertical models.

Methodology
The paper employs the Location Uncertainty (LU) framework, which decomposes fluid particle displacement into a time-differentiable resolved velocity and a highly fluctuating, time-decorrelated component modeled as a Brownian motion.

1. Theoretical Derivation:

   • The authors start from the fundamental conservation laws of mass, species mass, momentum, and total energy.
   • They apply the Stochastic Reynolds Transport Theorem (SRTT) extended to include stochastic source terms. This leads to a general system of stochastic compressible Navier–Stokes equations in both conservative and non-conservative (primitive variable) forms.
   • Crucially, the derivation explicitly accounts for the work of forces (pressure and viscous stresses) and heat fluxes acting on the stochastic displacement. This results in the emergence of specific stochastic terms, including noise-induced diffusion, Itô-Stokes drift, and quadratic covariation terms (arising from the product of stochastic differentials).
   • The system is then subjected to the Boussinesq and hydrostatic approximations. The authors carefully track the order of magnitude of terms to derive a stochastic Boussinesq system that retains thermodynamic effects, specifically the work of compression expressed through potential energy.

2. Numerical Application:

   • To validate the theoretical framework, the authors perform a Large Eddy Simulation (LES) of a temperature-driven free convection event in a stratified ocean using the CROCO model (non-hydrostatic, non-Boussinesq capabilities).
   • They apply a time-filtering technique to the LES data (using a 3-hour decorrelation timescale) to separate resolved scales from subgrid-scale fluctuations.
   • The subgrid-scale residuals are used to construct the stochastic noise (Brownian motion) and its variance tensor.
   • An offline 1D stochastic model is then forced by these diagnosed statistics to simulate the evolution of the horizontally averaged temperature profile. This allows for a direct comparison between the stochastic model's output and the reference LES results.

Key Contributions

• General Stochastic Compressible Formulation: The paper provides a complete derivation of the stochastic compressible Navier–Stokes equations, including the transport of species and the full energy budget. This extends previous LU work which was primarily limited to incompressible flows.
• Thermodynamic Consistency: The derivation explicitly includes the work of compression and dilatation. The authors demonstrate that while these terms are negligible when expressed in terms of internal energy (consistent with the Boussinesq approximation), they become significant when expressed in terms of potential energy. This reveals new stochastic pressure terms within the mixed layer that couple kinetic and potential energy.
• Stochastic Boussinesq Model with Thermodynamics: The authors derive a stochastic Boussinesq model that incorporates these thermodynamic effects, offering a more physically consistent framework than previous stochastic isochoric models.
• Diagnostic of Penetrative Convection: The study identifies that the stochastic terms, particularly the Itô-Stokes drift and stochastic diffusion, are capable of reproducing the vertical temperature fluxes at the base of the mixed layer. This is a region where standard parameterizations often fail to capture penetrative convection effects (warming at the base of the mixed layer).

Results

• Energy Budget: In the numerical experiment, the leading-order balance for temperature changes is governed by internal energy transport, consistent with the Boussinesq approximation. However, when analyzing potential energy, the stochastic terms (specifically the quadratic covariation $A_u$ and drift work $D_\sigma$ associated with stochastic pressure) show significant contributions.
• Mixed Layer Dynamics: The stochastic model successfully reproduces the warming at the base of the mixed layer associated with penetrative convection. This is attributed to the model's ability to capture non-local vertical dynamics driven by horizontal variability and inertial processes without relying on explicit diffusive closure assumptions.
• Limitations: Within the mixed layer itself, the stochastic model shows limited skill compared to the reference LES. The authors note that the relatively low variability of the turbulent field in this region leads to negligible contributions from the stochastic terms in their current diagnostic setup. The model also struggles near the surface, likely due to the inability of the stochastic representation to capture the specific small-scale fluctuations resolved by the LES.

Significance and Claims
The authors claim that this work provides a structured modelling framework that opens new perspectives for oceanic dynamics within a stochastic setting.

• Physical Justification: Unlike phenomenological approaches, the additional terms in the stochastic equations are derived rigorously from conservation laws and stochastic calculus, providing a clear physical justification for their form and role (e.g., analogous to baropycnal work in compressible LES).
• Energetic Consistency: The framework offers a path to improving the energetic consistency of subgrid-scale vertical models used in Ocean General Circulation Models (OGCMs) by explicitly accounting for physical uncertainties and approximations, particularly regarding the coupling between dynamics and thermodynamics.
• Penetrative Convection: The ability of the stochastic terms to qualitatively reproduce missing Reynolds stresses at the base of the mixed layer suggests a potential for representing penetrative convection effects without the computational cost of full non-hydrostatic solvers.
• Future Directions: The paper modestly suggests that while the current diagnostic approach is promising, future work should focus on prognostic modelling of the noise characteristics (e.g., via TKE-based models) and the inclusion of random surface pressure forcing to better capture dynamics within the mixed layer and near the surface.