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arxiv: 2512.04452 · v2 · pith:FLD4AS7Znew · submitted 2025-12-04 · ⚛️ physics.ao-ph · cs.AI· cs.LG· physics.comp-ph· physics.flu-dyn

NORi: An ML-Augmented Ocean Boundary Layer Parameterization

Xin Kai Lee , Ali Ramadhan , Andre Souza , Gregory LeClaire Wagner , Simone Silvestri , John Marshall , Raffaele Ferrari This is my paper

Pith reviewed 2026-05-21 18:52 UTC · model grok-4.3

classification ⚛️ physics.ao-ph cs.AIcs.LGphysics.comp-phphysics.flu-dyn
keywords ocean boundary layermachine learning parameterizationneural ODERichardson number closureentrainmentturbulence parameterizationlarge-eddy simulationnumerical stability
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The pith

A physics-based Richardson number closure augmented by neural ODEs captures ocean boundary layer entrainment and remains stable over century-scale integrations.

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

The paper develops NORi, a parameterization that starts from a local diffusive closure whose diffusivity and viscosity depend on the Richardson number and then adds neural ordinary differential equations to represent the non-local entrainment of fluid across the base of the mixed layer. These networks are trained in an a posteriori fashion directly on the time-integrated profiles produced by large-eddy simulations rather than on instantaneous fluxes. When tested under varied convection, stratification, rotation, and wind forcing, the resulting scheme reproduces seasonal mixed-layer evolution at Ocean Weather Station Papa at a level comparable to a state-of-the-art two-equation turbulence closure. Implemented inside a double-gyre ocean model, the parameterization integrates stably for at least one hundred years despite having seen only two-day training trajectories and permits time steps as long as one hour while conserving tracers by construction.

Core claim

NORi combines a Richardson-number-dependent local diffusivity and viscosity with neural ODEs that learn the additional entrainment flux needed at the base of the boundary layer. Trained a posteriori on short large-eddy simulation trajectories, the scheme reproduces observed entrainment under changing convective intensity, background stratification, rotation, and wind stress; it matches the seasonal cycle at Ocean Weather Station Papa as closely as a conventional k-ε closure; and it remains numerically stable for at least one hundred years inside a double-gyre configuration while allowing hourly time steps.

What carries the argument

The NORi closure, a hybrid model whose physical base is a Richardson-number-dependent diffusivity and viscosity and whose neural ODE component supplies the non-local entrainment flux across the mixed-layer base.

If this is right

  • Climate models can adopt larger time steps without loss of stability.
  • Training data volume and computational cost for developing new closures are sharply reduced because only short, high-resolution segments are required.
  • Inference performance on quantities of direct interest (mixed-layer depth, entrainment rate) can be optimized as the primary training objective.
  • Tracer conservation and realistic nonlinear thermodynamics are preserved by design rather than enforced after the fact.

Where Pith is reading between the lines

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

  • The same hybrid architecture could be applied to other sub-grid processes such as cloud microphysics or gravity-wave drag where local closures miss non-local transport.
  • Because stability emerges from the training process rather than from added constraints, similar methods may shorten the development cycle for parameterizations in other Earth-system components.
  • Long integrations without retraining suggest the learned entrainment correction may remain valid across a wider range of ocean states than the original training set explicitly covered.

Load-bearing premise

Neural networks trained only on two-day large-eddy simulation segments will continue to produce physically consistent, non-drifting solutions when integrated for decades or centuries inside a different ocean model.

What would settle it

A multi-decade integration of a global ocean model using NORi that develops a systematic drift in temperature, salinity, or tracer inventories larger than the drift seen in a comparable k-ε simulation would falsify the long-term stability claim.

Figures

Figures reproduced from arXiv: 2512.04452 by Ali Ramadhan, Andre Souza, Gregory LeClaire Wagner, John Marshall, Raffaele Ferrari, Simone Silvestri, Xin Kai Lee.

Figure 1
Figure 1. Figure 1: Schematic of the vertical profile of buoyancy as a function of depth below the ocean surface—buoyancy is negatively proportional to the density anomaly generated by a change in temperature and salinity. At time t = 0, the buoyancy decreases linearly with depth. After some time, in response to buoyancy loss at the surface, a well-mixed buoyancy layer develops in the upper ocean. In the absence of entrainmen… view at source ↗
Figure 2
Figure 2. Figure 2: Large-eddy simulations (LES) for free convection and pure wind stress scenarios in a horizontally doubly-periodic domain of size (Lx, Ly, Lz) = (128, 128, 128) m with a grid reso￾lution of 0.5 m and a Coriolis parameter of f = 8 × 10−5 s −1 . The top row shows a convective turbulence LES driven by surface cooling, while the bottom row shows a shear turbulence LES driven by surface wind stress. The first an… view at source ↗
Figure 3
Figure 3. Figure 3: The top panel shows the calibrated diffusivity and viscosity values as a function of the local gradient Richardson number Ri in the base closure model. In the convective range, the viscosity and diffusivity are constant, while they decrease linearly towards a background value in the shear range. The lower panels show the vertical profiles of momentum, temperature, salinity, and buoyancy from LES simulation… view at source ↗
Figure 4
Figure 4. Figure 4: Training and validation losses of NORi. Top row: mean and individual losses against epoch over the final integration horizon of 43.3 hours for the training suite (Tables B2, B3, and B4) in the left panel and the validation suite (Table B5) in the right panel. All losses are normalized only once at epoch 0—an epoch is defined as one iteration over the entire training suite. The gray lines are the individual… view at source ↗
Figure 5
Figure 5. Figure 5: Temperature, salinity, and buoyancy profiles for selected training (columns 1 through 3) and validation (columns 4 through 6) examples generated with the calibrated NORi model (black lines), the base closure (orange lines) as well as area-averaged large-eddy simulation (LES) solutions (green lines). The profiles are computed 1.75 days after the initial conditions (dashed lines). –19– [PITH_FULL_IMAGE:figu… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of NORi versus k-ϵ performance in a column model setting. The model profiles of temperature, salinity, and buoyancy are plotted together with the area-averaged LES profiles. The examples shown are samples of validation cases not seen by NORi during training (see Table B5 for a complete list). The profiles are computed 1.75 days after the initial conditions (dashed lines). So far, we have focused… view at source ↗
Figure 7
Figure 7. Figure 7: Velocity, temperature, and salinity profiles 60 days after initialization (dashed lines) for simulations forced with a constant wind stress, J top u = −1 × 10−4 m2 s −2 and time-dependent heat and freshwater fluxes given by Equations (37) and (38). The Coriolis parameter is set to f = 1 × 10−4 s −1 . Solutions are reported from the NORi and k-ϵ models together with the area￾averaged LES. 6.4 Time step depe… view at source ↗
Figure 8
Figure 8. Figure 8: Velocity, temperature, and salinity profiles 4 days after initialization generated with NORi using a range of time steps. The surface forcing parameters are given in the text. There is no v velocity component, because the simulation is not rotating and the surface momentum stress acts in the x-direction. The LES solution is also shown for reference. u (m s⁻¹) 0.0 0.1 0.2 0.3 z (m) −200 −100 0 T (°C) 16.5 1… view at source ↗
Figure 9
Figure 9. Figure 9: Same as [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The configuration of the double-gyre simulation. The results shown in this figure are produced using the NORi closure implemented in Oceananigans.jl, run on GPUs, taken at t = 100 years. The subfigure on the left shows a 3D snapshot of the buoyancy field as well as the zonally-averaged buoyancy of the simulation. The subfigures on the right show the surface temperature, salinity surface restoring profiles… view at source ↗
Figure 11
Figure 11. Figure 11: Zonally-averaged temperature slice from simulations using NORi, k-ϵ, and the base closure in the double-gyre simulation. The snapshots above are taken at t = 2.5 years. The rightmost column shows the temperature differences between NORi and the other two models. The results of the double-gyre solutions highlight the importance of recalibrating boundary layer parameterizations in global simulations with cl… view at source ↗
read the original abstract

NORi is a machine learning (ML) parameterization of ocean boundary layer turbulence that is physics-based and augmented with neural networks. NORi stands for neural ordinary differential equations (NODEs) Richardson number (Ri) closure. The physical parameterization is controlled by a Richardson number-dependent diffusivity and viscosity. The neural ODEs are trained to capture the entrainment through the base of the boundary layer, which cannot be represented with a local diffusive closure. The parameterization is trained using large-eddy simulations in an "a posteriori" fashion, where parameters are calibrated with a loss function that explicitly depends on the actual time-integrated variables of interest rather than the instantaneous subgrid fluxes, which are inherently noisy. NORi conserves tracers by design, uses realistic nonlinear thermodynamics, and demonstrates excellent prediction and generalization capabilities in capturing entrainment dynamics under different convective strengths, background stratifications, rotation, and wind forcings. NORi is shown to simulate the seasonal evolution of the boundary layer at Ocean Weather Station Papa with similar performance to the state-of-the-art two-equation $k$-$\epsilon$ closure. When implemented in a double-gyre simulation, it is numerically stable for at least 100 years, despite only being trained on two-day horizons, and can be run with time steps as long as one hour. The highly expressive neural networks, combined with a physically rigorous base closure, prove to be a robust paradigm for designing parameterizations for climate models: data required and training cost are drastically reduced, inference performance can be directly optimized as a primary objective, and numerical stability is implicitly promoted through training.

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 / 2 minor

Summary. The manuscript presents NORi, a hybrid physics-ML parameterization for ocean boundary layer turbulence. It augments a local Richardson-number-dependent diffusivity and viscosity closure with neural ODEs that are trained a posteriori on short (two-day) LES trajectories to reproduce integrated entrainment at the base of the mixed layer. The approach conserves tracers by design, incorporates nonlinear thermodynamics, and is evaluated for generalization across convective forcing, stratification, rotation, and wind stress. Results include comparable performance to a k-ε closure in seasonal simulations at Ocean Weather Station Papa and numerical stability over 100 years in a double-gyre configuration, despite training only on short horizons and allowing time steps up to one hour.

Significance. If the long-term stability and lack of bias accumulation hold under broader testing, the work offers a concrete example of how a physically grounded base closure combined with targeted neural corrections can achieve data-efficient training, direct optimization of integrated quantities, and implicit numerical stability. This paradigm could meaningfully reduce the data and compute burden for developing hybrid parameterizations suitable for climate models while preserving conservation properties.

major comments (2)
  1. The central claim that short-horizon a posteriori training on integrated entrainment produces unbiased long-term evolution rests on the 100-year double-gyre stability result and the OWS Papa seasonal match. However, these tests do not directly address possible slow accumulation of entrainment or stratification biases over many turnover times, as the neural correction is non-local and learned rather than derived from an asymptotic limit. A quantitative assessment of mixed-layer depth drift or buoyancy flux bias relative to observations or a reference closure over multi-year periods would be required to support the extrapolation.
  2. The abstract and results sections report promising generalization across convective strengths, stratifications, rotation, and wind forcings, yet no quantitative error metrics (e.g., RMSE on entrainment rate or mixed-layer depth), baseline comparisons against purely local Ri closures or other ML parameterizations, or ablation studies on the neural ODE component are provided. This makes it difficult to judge the magnitude of improvement attributable to the NODE augmentation versus the underlying physical closure.
minor comments (2)
  1. Clarify the precise architecture and input features of the neural ODEs (e.g., which variables are passed to the network and how the entrainment flux is injected into the prognostic equations) to allow reproducibility.
  2. The manuscript would benefit from an explicit statement of the loss function weights and any regularization terms used during a posteriori training, as these choices directly affect the balance between short-term fidelity and long-term stability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the strengths and limitations of our work. We address each major comment below and indicate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: The central claim that short-horizon a posteriori training on integrated entrainment produces unbiased long-term evolution rests on the 100-year double-gyre stability result and the OWS Papa seasonal match. However, these tests do not directly address possible slow accumulation of entrainment or stratification biases over many turnover times, as the neural correction is non-local and learned rather than derived from an asymptotic limit. A quantitative assessment of mixed-layer depth drift or buoyancy flux bias relative to observations or a reference closure over multi-year periods would be required to support the extrapolation.

    Authors: We agree that the existing tests, while supportive, do not explicitly quantify potential slow bias accumulation. The 100-year double-gyre run demonstrates stability with no visible drift, and the OWS Papa case matches the reference closure over a seasonal cycle, but a dedicated multi-year bias analysis would provide stronger evidence. We will add quantitative comparisons of mixed-layer depth and buoyancy flux drift relative to the k-ε closure over extended periods in the revised manuscript. revision: yes

  2. Referee: The abstract and results sections report promising generalization across convective strengths, stratifications, rotation, and wind forcings, yet no quantitative error metrics (e.g., RMSE on entrainment rate or mixed-layer depth), baseline comparisons against purely local Ri closures or other ML parameterizations, or ablation studies on the neural ODE component are provided. This makes it difficult to judge the magnitude of improvement attributable to the NODE augmentation versus the underlying physical closure.

    Authors: We acknowledge that the manuscript emphasizes qualitative generalization and visual agreement rather than explicit error metrics or controlled ablations. To address this, we will incorporate RMSE values for entrainment rates and mixed-layer depth in the generalization experiments, add direct baseline comparisons against the purely local Ri closure, and include ablation studies isolating the neural ODE contribution in the revised results section. revision: yes

Circularity Check

1 steps flagged

NN weights fitted a posteriori to LES-integrated entrainment; 'prediction' of entrainment dynamics reduces to the training fit for tested regimes

specific steps
  1. fitted input called prediction [Abstract]
    "The neural ODEs are trained to capture the entrainment through the base of the boundary layer, which cannot be represented with a local diffusive closure. The parameterization is trained using large-eddy simulations in an 'a posteriori' fashion, where parameters are calibrated with a loss function that explicitly depends on the actual time-integrated variables of interest rather than the instantaneous subgrid fluxes"

    The NN correction is calibrated directly against the time-integrated entrainment signal from the same LES data it is later asked to reproduce; for regimes inside the training distribution the reported 'prediction' of entrainment is therefore the fitted mapping rather than an independent derivation from the Ri base closure.

full rationale

The paper explicitly trains neural ODEs on short LES trajectories using a loss on time-integrated quantities to reproduce entrainment that a local Ri closure cannot capture. This is standard supervised ML augmentation rather than a first-principles derivation, so the entrainment 'prediction' in the coupled model is the learned correction by construction on the training distribution. However, the base Ri-dependent diffusivity/viscosity remains an independent physical closure, conservation is enforced by design, and generalization claims are supported by tests on varied forcings and a 100-year run. No self-citation load-bearing steps or self-definitional reductions appear in the provided text; the central result does not collapse to its inputs by definition.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the assumption that the Richardson-number local closure plus learned non-local entrainment term together reproduce the essential dynamics of boundary layer evolution across regimes, with the neural component calibrated to short-term LES trajectories.

free parameters (1)
  • Neural network weights and biases
    Fitted via optimization to minimize a loss on time-integrated variables from large-eddy simulations.
axioms (1)
  • domain assumption A local Richardson-number-dependent diffusivity and viscosity provides an adequate base representation of turbulent mixing.
    Invoked as the physical foundation that the neural ODEs augment.
invented entities (1)
  • Neural ODE component for entrainment no independent evidence
    purpose: To represent non-local mixing processes at the base of the boundary layer that cannot be captured by local diffusion.
    Postulated to close the parameterization; no independent falsifiable prediction outside the training data is provided.

pith-pipeline@v0.9.0 · 5845 in / 1608 out tokens · 57133 ms · 2026-05-21T18:52:14.066066+00:00 · methodology

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Works this paper leans on

2 extracted references · 2 canonical work pages

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    Retrieved 2025-04-25, fromhttps://www.mdpi.com/2311-5521/6/10/360 doi: 10.3390/fluids6100360 –47– manuscript submitted toJournal of Advances in Modeling Earth Systems (JAMES) Wagner, G. L., Hillier, A., Constantinou, N. C., Silvestri, S., Souza, A., Burns, K. J., . . . Ferrari, R. (2025, April). Formulation and Calibration of CATKE, a One-Equation Paramet...

    work page doi:10.3390/fluids6100360 2025

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    P., Zavala-Romero, O., Wan, X., & Cronin, M

    Retrieved 2025-01-21, fromhttp://journals.ametsoc.org/doi/ 10.1175/2007JCLI1714.1doi: 10.1175/2007JCLI1714.1 Yuan, J., Liang, J., Chassignet, E. P., Zavala-Romero, O., Wan, X., & Cronin, M. F. (2024, September). The K-Profile Parameterization Augmented by Deep Neu- ral Networks (KPP dnn) in the General Ocean Turbulence Model (GOTM). Journal of Advances in...

    work page doi:10.1175/2007jcli1714.1doi: 2025