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A Computational Model for Global Ocean Dynamics at all Scales

Peter Korn

arXiv:2608.25679v1math.NA

TL;DR

Ocean models omit small scales because computational limits have favored hydrostatic and Boussinesq approximations, leaving effects such as convection to parametrizations. The paper presents AC/DC, which combines artificial compressibility with a vertical column solve to recover non-hydrostatic dynamics while retaining computable mixing diagnostics, though feasibility is constrained by ocean-grid structure and solver costs.

  • Problem

    Computational limits force ocean models to exclude smaller scales through approximations, while their effects must be represented by parametrizations.

  • Method

    AC/DC replaces an instantaneous global constraint with finite-speed relaxation, removes the vertical constraint component through a column solve, and transports tracers consistently with the resulting flux.

  • Results

    The paper establishes that AC/DC recovers internal-gravity-wave dispersion with an error smaller than pure artificial compressibility by the square of the grid aspect ratio, while mixing remains a computed output.

  • Takeaways & Limitations

    The hydrostatic approximation's computational trade-off need not preclude computing smaller-scale ocean dynamics and mixing outputs.

  • Takeaways & Limitations

    AC/DC feasibility is established by scale analysis rather than timing measurements, and alternative approaches face substantial costs or unresolved obstacles on stretched, topographic, or unstructured ocean meshes.

Abstract

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The dynamics of the world's ocean spans a breathtaking range of scales. The circulation carries processes from thousands of kilometres down to the millimetre, and from seconds to millennia. Our understanding of global ocean dynamics is to a large extent based on the analysis of ocean model simulation data; computational models provide spatial and temporal coverage that observations can not deliver. However, the scales a computational ocean model can simulate are limited by the available computational capacity. Smaller scales, anticipated as out of reach, have been excluded a priori by shortcuts in the dynamical equations; examples are the hydrostatic and Boussinesq approximations. Hydrostatic Boussinesq models have been used since \citet{Bryan1969} up to now to simulate the planetary scale down to the mesoscale eddy resolving scale of the order of kilometers, and the hydrostatic approximation holds for most oceanic phenomena at the $\sim$1--100\,km resolutions that have been computationally affordable. The effect of removed scales on represented scales--the convective scale in the case of the hydrostatic approximation-had to be substituted by ad hoc parametrizations.

1 Introduction

Global ocean models have relied on hydrostatic approximations because non-hydrostatic dynamics make each time step computationally difficult. The paper presents AC/DC as a route to represent smaller scales while controlling approximation errors and preserving key physical properties.

  • 1 Introduction: Computational capacity limits the scales resolved by ocean models, motivating shortcuts such as the hydrostatic and Boussinesq approximations.Hydrostatic Boussinesq models commonly reach roughly 1–100 km resolution, while smaller-scale effects are represented through parametrizations.
  • 1 Introduction: At subkilometer scales, non-hydrostatic pressure carries vertical internal-wave dispersion, vortex tilting, frontal dynamics, and convective overturning.The hydrostatic approximation removes vertical acceleration and cannot grow resolved plumes from unstable stratification.
  • 1 Introduction: AC/DC computes the vertical non-hydrostatic pressure locally by tridiagonal column solves and relaxes only the horizontal residual with artificial compressibility.This avoids placing a global three-dimensional elliptic pressure solve at the center of every time step.
  • 1 Introduction: The method is intended to let hydrostatic and non-hydrostatic regimes coexist on telescoping meshes without an interface, with cost following the mesh.The paper frames this as opening new scales while retaining a controlled approximation and assessing feasibility through scale analysis rather than a direct timing comparison.
  • 1 Introduction: The paper does not perform a 1:1 performance comparison between projection and AC/DC because no suitable projection-side technology exists for that comparison.Instead, it prices algorithms in tracer passes and compares their global-stop requirements on memory-bound hardware.

2 Artificial Compressibility for Thin Fluids

AC/DC replaces the global incompressibility solve with a weakly compressible formulation whose pressure is split into an exact vertical column component and a horizontally relaxed residual. Its flux-form design restores exact conservation properties while retaining controlled approximation errors.

  • 2 Artificial Compressibility for Thin Fluids: AC/DC introduces pseudo-pressure ψ and pseudo-density, replacing the non-hydrostatic pressure and incompressibility constraint with a local compressible system.As α →∞, ψ approaches pNH and ∇·v approaches zero, formally recovering the incompressible non-hydrostatic equations.
  • 2 Artificial Compressibility for Thin Fluids: The horizontal residual is advanced through a local hyperbolic equation rather than a global Poisson solve, iteration, or preconditioner.The computational trade-off is carrying one additional prognostic variable per cell.
  • 2 Artificial Compressibility for Thin Fluids: The artificial sound speed is controlled by α, and the compressibility error is measured by the barotropic Froude number.For U ∼1 m s−1 and Hfull = 4000 m, the supplied estimate gives Frbaro ≈5×10−3, with AC errors proportional to Frbaro^2.
  • 2 Artificial Compressibility for Thin Fluids: Flux-form pseudo-density transport makes the continuity and tracer equations consistent and supports exact conservation of energy-related quantities.The paper states that the flux form is what makes the conservation statements exact, whereas the relaxation form leaves an O(1/α) remainder.
  • 2 Artificial Compressibility for Thin Fluids: The method decomposes the three-dimensional pressure problem into an exact vertical column solve and a horizontally relaxed residual.On prismatic meshes, the Laplacian separates into vertical and horizontal operators; the aggregated pressure is pNH := pV + ψ.
  • 2 Artificial Compressibility for Thin Fluids: The density-free kinetic-energy budget is not closed because its cubic obstruction Sdf is not sign-definite, whereas density-weighted kinetic energy removes that obstruction.The correction works through the pseudo-density weight and flux-form transport, causing the relevant cubic contributions to telescope or cancel.
  • 2 Artificial Compressibility for Thin Fluids: Anisotropic viscosity and diffusivity split dissipation into separately diagnosable reservoirs associated with horizontal and vertical shear.The assignment uses νh for vertical-axis vorticity generated by horizontal shear and νv for horizontal-axis vorticity generated by vertical shear.

3 AC/DC for Boussinesq Equations

The AC/DC formulation supports non-hydrostatic Boussinesq dynamics while controlling artificial-compressibility errors through an exact vertical column solve and a relaxed horizontal residual. Its analysis establishes energy, tracer, dispersion, and mixing-budget properties, with accuracy advantages strongest on anisotropic ocean meshes but limited on genuinely non-hydrostatic branches.

  • AC/DC assesses physical plausibility through energy conservation, volume-conservation error, tracer conservation, and internal-gravity-wave dispersion estimates.
  • At grid scale, AC/DC reduces divergence error by (∆z/∆x)^2; with ∆x/∆z = 100, its error is 10^-4 times the pure-AC error.This is a scaling estimate rather than an exact identity, and projection methods likewise retain a finite divergence tolerance.
  • The column solve removes the dominant vertical pressure component, reducing both artificial-compressibility error and the stiffest CFL constraint.Vertical modes are stiffest because their eigenvalues scale as 1/∆z^2.
  • The column solve retains its advantage for shallow convecting layers but loses it when narrow plumes span deep, weakly stratified layers.Increasing α can restore accuracy because divergence error falls as 1/α, at the cost of proportionally shorter sub-steps.
  • The column pressure cancels the vertical hydrostatic–buoyancy balance, while a Coriolis remainder survives in the rotating case.The cancellation leaves only the residual pseudo-pressure contribution in the vertical momentum balance.
  • The column solve improves accuracy in the near-hydrostatic sector, where kz ≫ kx, but pure AC is more accurate on the genuinely non-hydrostatic branch kx > kz.The advantage exploits anisotropy that is absent when kx ∼ kz.
  • AC/DC preserves total mass, the pseudo-density-weighted content, and the weighted tracer mean exactly, while bounding representation error uniformly in time.Under additional boundedness, the error bound improves to O(α^-1/2).
  • The tracer-variance budget closes algebraically: residual divergence is absorbed by pseudo-density weighting, and physical and numerical variance sinks are separately computable.For non-centred monotone transport, the numerical sink is an explicit face sum; for centred reconstruction, the variance budget is purely physical.

4 Implementation of AC/DC

AC/DC implements non-hydrostatic dynamics by removing the global velocity solve: vertical stiffness is handled by column-local operations, while horizontal residuals use acoustic relaxation. Its design preserves nearest-neighbour and column-local work, but explicit acoustic stepping has refinement-dependent accuracy and cost trade-offs.

  • Classical artificial compressibility: The classical artificial-compressibility step is limited by one global velocity solve, whose conditioning and cost stop it at ocean resolutions.This solve is the only global and expensive operation in Template A.
  • AC/DC step: AC/DC replaces the global three-dimensional solve with directional treatment: vertical operators are solved within columns, while horizontal residuals are handled acoustically.The vertical split uses tridiagonal column solves, leaving the horizontal component for the acoustic stage.
  • AC/DC step: The method separates pseudo-pressure into a two-dimensional depth average advanced quickly and a stiff vertical remainder updated once per step by column-local solves.This separation keeps the fast response two-dimensional while retaining vertical structure in the slower update.
  • AC/DC step: AC/DC’s acoustic sub-loop uses horizontal CFL control, nearest-neighbour stencils, and block-diagonal column solves rather than global reductions.The sub-step count scales as nsub = O(cac∆t/∆x).
  • Option S3-b: Under refinement, explicit S3-b cannot hold sub-step count and accuracy simultaneously; fixed accuracy makes it more expensive than S3-a.Reducing the artificial-compressibility parameter to hold nsub fixed increases pseudo-density departure, whereas S3-a’s treatment is unconditionally stable.
  • Initialization: Unprepared initialization errors enter at O(Frbaro), whereas the paper’s divergence, dispersion, and energy-residual errors enter at O(Frbaro^2).The paper characterizes the initialization error as one order lower in the small parameter.

5 Integration into General Ocean Circulation Model

The non-hydrostatic formulation is integrated into a primitive-equation ocean model by treating non-hydrostatic effects as corrections to the hydrostatic core. The pressure decomposition supports local closure, free-surface consistency, and error levels that are small relative to accepted ocean-model approximations.

  • Hydrostatic–Non-Hydrostatic Decomposition: The pressure split p = phyd + psfc + pNH determines hydrostatic pressure from buoyancy, carries surface pressure through the host free-surface solve, and leaves pNH as the residual.AC/DC closes this residual by solving its vertical part locally and treating its horizontal part separately.
  • Free Surface: Under a free surface, the barotropic solve carries surface-pressure effects, while non-hydrostatic pressure enforces volume conservation through ∇·v = 0.The free-surface elevation is advanced prognostically rather than obtained from a rigid-lid constraint inversion.
  • Hydrostatic–Non-Hydrostatic Decomposition: Non-hydrostatic dynamics make vertical velocity prognostic, whereas the hydrostatic system diagnoses it from incompressibility and vertical integration.This change enables dynamics absent from the hydrostatic approximation, including resolved vertical motion.
  • Hydrostatic–Non-Hydrostatic Decomposition: The non-hydrostatic equations are expressed as the hydrostatic primitive-equation core plus correction terms for velocity, nonlinearity, Bernoulli energy, and pressure.The correction formulation is intended to add non-hydrostatic capability to an existing operational ocean-model core.
  • Errors in Context of other Errors: AC/DC’s constraint-violation error is two orders of magnitude below the Boussinesq error and more than four orders below the hydrostatic velocity-field error.The comparison distinguishes failure to satisfy incompressibility from satisfying it on the wrong velocity field.
  • Time Stepping Structure: Because ocean meshes typically have ∆x/∆z ∼10–1000, AC/DC can use larger time steps without changing the host algorithms’ limiting conditions.The method adds fields but does not change what limits those algorithms.

6 Computational Feasibility

AC/DC is presented as computationally feasible because it replaces a costly global three-dimensional elliptic solve with local column solves and nearest-neighbour horizontal updates. Scale analysis indicates bounded overhead and no AC/DC-specific global synchronisation, while several alternatives remain impractical on realistic ocean meshes.

  • Computational obstruction: The computational cost of non-hydrostatic pressure is governed by the iteration count of the three-dimensional elliptic solve, which is close to one mesh pass per iteration.The paper measures cost in passes because memory movement and synchronisation dominate on memory-bound hardware.
  • Projection methods: Without preconditioning, conjugate-gradient iteration counts reach thousands or more, making the method impractical relative to a twelve-pass hydrostatic step.The count follows order √κ, where κ is the discrete Laplacian condition number.
  • Projection methods: A vertical-line preconditioner reduces the iteration count to 10^2–10^3, but the resulting step still costs 10^2–10^3 passes versus twelve hydrostatic passes and includes a global stop per iteration.For the stated 4000 m depth and 50 m spacing, the idealized count is 80 before degradation from realistic ocean geometry.
  • Alternative approaches: Exact vertical-mode splitting is unsuitable for stretched grids and real bathymetry because dense transforms are expensive and depth-dependent modes recouple the horizontal problem.The transform alone can require about forty passes per step and substantial storage under the stated stretched-grid conditions.
  • Alternative approaches: No price is assigned to multigrid because the paper reports no three-dimensional multigrid solver for major unstructured-mesh ocean models, citing topographic complexity among the obstructions.The relevant domain includes irregular coastlines, narrow passages, sills, and near-disconnected basins.
  • AC/DC cost: AC/DC adds about two passes, or 1.15–1.22 times the hydrostatic step, and performs no AC/DC-specific global stops because its column solve is local and horizontal sub-steps are nearest-neighbour.Its overhead is described as bounded, known before execution, and flat under refinement, although the figures are scale estimates rather than measurements or theorems.

7 Numerical Experiments

The experiments test AC/DC against projection-method references for physical correctness, dispersion, conservation, mixing, and convection. Across these tests, AC/DC reproduces reference behavior while its compressibility effects follow the predicted parameter dependence.

  • The two methods agree on front Froude number to 2.2×10^-4 and 1.8×10^-4, while mesh refinement changes it by 2.1 × 10^-2.Method differences are therefore two orders of magnitude below mesh differences in this diagnostic.
  • AC/DC converges to projection at fixed mesh, reaching Fr = 0.695636 versus 0.695590 and a relative kinetic-energy difference of 6.4 × 10^-6 as ∆t → 0.
  • Both arms close energy and tracer-variance budgets at the time-integrator rate, while Dnum remains non-negative and converges to a non-zero limit.The reported energy residuals decrease under refinement, and the variance residual is indistinguishable between methods.
  • AC/DC and projection agree on Kexpl within 0.41% and on Knum within 0.65% on the coarser mesh, with closer agreement after refinement.The convention-free Knum/Kexpl ratio agrees within 0.25% on the coarser mesh and 0.12% on the finer mesh.
  • The AC/DC closure error is O(10^-5) at ˜α = 1 and falls as 1/˜α, reaching O(10^-10) at the production value, below spatial error.
  • At α ∈ {10^5, 10^6, 10^7}, AC/DC develops runaway acoustic instability and NaNs, whereas α = 10^8 completes and tracks projection closely.

8 Telescoping Meshes: Where the Advantages Converge

On telescoping meshes, projection retains a globally conditioned elliptic solve, whereas AC/DC uses one equation set whose cost and error follow the locally refined mesh. This avoids an explicit hydrostatic–non-hydrostatic interface.

  • A telescoping mesh spans approximately 50 km in the open ocean and 500 m or finer in a target region, combining hydrostatic and non-hydrostatic regimes.
  • Projection solves a global elliptic problem whose conditioning follows the finest spacing and whose iterations sweep coarse regions where pNH is negligible.
  • AC/DC uses one equation set everywhere, so ψ remains near zero in the coarse region and the system reduces to hydrostatic primitive equations without switching or an interface condition.
  • The AC/DC coupling covers only the target region, with completion time L/cac set by that region’s size.
  • In the coarse region, AC/DC’s divergence error is ∼10^-13 versus pure AC’s ∼10^-9; in the refined patch it remains 10^-11–10^-9.
  • Neither cost nor error is switched or tuned at the transition; both follow the mesh.

9 Conclusions

The conclusion argues that AC/DC can remove the computational obstruction that has kept global ocean models hydrostatic. It presents this as a controlled, feasible route whose broader simulation consequences remain to be established.

  • The hydrostatic approximation trades computational speed against computability of small scales, but the paper argues that this price need not be paid.
  • The hydrostatic primitive equations are a subsystem of the non-hydrostatic equations, with the missing contribution supplied as a correction.
  • Artificial compressibility introduces a finite-speed relaxation error controlled by one parameter, while the column solve removes the vertical constraint and leaves only the horizontal residual.
  • The feasibility argument finds AC/DC overhead bounded, known before execution, and flat under refinement, using scale analysis rather than timing comparison.
  • The numerical experiments are neither performance measurements nor global simulations, and a convection-resolving world-ocean simulation is outside the paper’s scope.
  • The paper claims that the computational obstruction is removable at a small fixed multiple of the existing model cost, leaving consequences for simulation.

A.2 Proofs

The proofs establish conservation, budget identities, dispersion relations, and bounds for the AC/DC formulation. They rely on discrete cancellations, boundary conditions, monotone transport, and asymptotic expansion.

  • Summing the discrete transport equations cancels interior face contributions and gives exact conservation of mass and tracer sum on closed domains.
  • Under the stated transport hypotheses, constants remain stationary for any α because the tracer and density fluxes use the same advancing flux.
  • Monotone transport bounds tracer deviations through positivity, exact conservation, and the pseudo-pressure contribution, yielding α-dependent estimates.
  • The Proposition 3.6 proof derives semi-discrete energy and tracer-variance balances from the stated flux-form identities.
  • Summation by parts converts incidence-based discrete operators into cancellation identities used in the variance-budget proof.
  • The moving-boundary proofs use materiality and boundary conditions to cancel surface terms and recover fixed-domain identities.
  • The dispersion proof derives the biquadratic relation by eliminating Fourier amplitudes, then expands the slow root to obtain the asymptotic error expression.

A.3 Boundedness of Acoustic Energy

The semi-discrete AC/DC acoustic reservoir reduces to decoupled damped wave modes whose behavior can be bounded under stated assumptions. Well-prepared or low-frequency forcing yields progressively stronger bounds on the pseudo-pressure.

  • Energy caveat: The analysis does not establish finite energy solely from the section’s budgets; boundedness instead relies on curl–divergence dissipation controlling the divergent velocity component.The argument is semi-discrete because the relevant bound is discrete.
  • Modal reduction: The acoustic reservoir is represented exactly, without truncation error from the reduction, by a second-order equation involving the discrete Laplacian.The equation is q¨ + νLq˙ + ˜αLq = −DF + ϱ˙ + νLϱ.
  • Modal reduction: In the eigenbasis of the discrete Laplacian, the system decouples into independent forced damped oscillators.Each mode satisfies q¨_k + νλ_kq˙_k + ˜αλ_kq_k = f_k.
  • Damping and conservation: Every nonconstant mode decays unconditionally in α, ν, and mesh resolution, while the constant mode remains stationary because its forcing vanishes.The decay rate is νλ_k/2 when ν²λ_k < 4˜αλ_k and ˜α/ν beyond that regime.
  • Bounds: With bounded modal forcing, the pseudo-pressure norm remains uniformly bounded in time; well-prepared initial data gives ∥ψ∥h = O(α^1/2).Consequently, ∥ψ∥h/α = O(α^-1/2).
  • Bounds: When forcing lies well below the modal acoustic frequencies, the response is static and the bounds sharpen to ∥ψ∥h = O(1) and ∥ψ∥h/α = O(α^-1).This low-frequency condition is expressed through ω_k := √˜αλ_k.

A.4 Cost Accounting

The cost accounting estimates AC/DC overhead relative to a hydrostatic baseline using tracer-equivalents and source-based operation counts. Across counting conventions, the overhead remains near 1.2, while refinement leaves the ratio approximately flat.

  • Overall overhead: AC/DC adds about 1.2× the cost of a hydrostatic step, with estimates ranging from 1.15 to 1.22 across counting conventions.The appendix presents this as an overhead estimate.
  • Units and configuration: The accounting uses tracer-equivalents F1 = Ncol nz Cstencil ≈ 10^11 FLOP for the stated configuration.Table 3 uses Ncol = 4×10^6, nz = 500, nsub = 200, and ntracer = 5.
  • Operation counts: The contraction increment is estimated as 0.90 F1, obtained from 3.86 F1 for the non-hydrostatic implementation minus 2.96 F1 for the hydrostatic production code.The increment corresponds to one additional pass over edge triples for the Lamb-vector correction.
  • Sensitivity: Allowing 0.8–1.3 F1 for contraction and 0.1–0.4 F1 for subtracting the host advective term changes the overhead range only from 1.9–2.2 to 1.8–2.6 F1 against a baseline of about 12.The resulting factor stays near 1.2.
  • Barotropic solve: The barotropic pseudo-pressure solve uses the same stiffness as the free-surface solve, but its looser tolerance makes it cheaper.The stated tolerances are εη ∼ 10^-13 for surface height and εψ ∼ 10^-2–10^-3 for pseudo-pressure.
  • Refinement: Under refinement, AC/DC’s column solve remains O(nz) per column and its sub-step count scales as Δx^-1, so the cost ratio stays flat.The projection cost grows with H/Δx through κprec = 1 + H²/Δx².

A.5 The Alternatives in the Same Unit

The comparison prices alternative routes to non-hydrostatic pressure in the same pass-based unit. A vertical eigenbasis split is computationally and geometrically impractical, whereas a three-dimensional projection iteration costs 0.7–0.9 F1.

  • Comparison basis: The alternatives are evaluated through scale analysis in tracer-equivalent passes rather than measurements, theorem statements, or absolute iteration counts.The accounting is tied to the configuration in Table 3.
  • Projection method: One preconditioned conjugate-gradient iteration costs 0.7–0.9 F1, combining operator, preconditioner, inner-product, and vector-update work.The estimate counts one iteration as approximately one pass.
  • Vertical split: An exact vertical split on the stretched grid requires dense nz × nz transforms rather than fast trigonometric transforms.The transform is required in both directions for each column.
  • Vertical split: Reducing the average column depth to nz = 300 lowers the transform estimate to 14 F1, changing the objection’s size but not its direction.The reduction is quadratic in the average column depth.
  • Topographic limitation: Real bathymetry prevents the split because column-dependent vertical operators produce different eigenbases and a horizontally coupled dense mode-index operator.The decoupled two-dimensional problems therefore do not exist with realistic topography.

A.6 The Density Weighting of the Kinetic Energy

Artificial compressibility makes density-free kinetic energy unsuitable as a Lyapunov functional because divergent flow leaves a cubic term. The paper instead uses an exact density-weighted kinetic energy repair, preserving the relevant structure.

  • Cubic obstruction: The vector-invariant formulation carries kinetic energy inside the Bernoulli gradient, which vanishes silently under incompressibility but not under artificial compressibility.The remaining term is cubic in velocity.
  • Cubic obstruction: The density-free kinetic-energy budget contains a non-sign-definite cubic term proportional to |v|²∇·v, so Kdf is not a Lyapunov functional for the inviscid system.The term vanishes for all velocity fields only when ∇·v ≡ 0.
  • Structural constraint: Modifying the Lamb term cannot cancel the cubic contribution without violating Lamb antisymmetry and the Cartan identity underlying circulation conservation.This is the stated energy–circulation dichotomy.
  • Density-weighted repair: The repair is the density-weighted kinetic energy Kdw, for which the cubic sector telescopes and the remaining budget contains pressure, viscous, and buoyancy pairings.The weighting is defined through the density factor r.
  • Density-weighted repair: The density-weighted repair is exact at any compressibility, whereas advancing artificial compressibility in relaxation form would leave an O(1/α) remainder.The remaining non-pseudo-pressure work carries an explicit α^-1 and vanishes in the incompressible limit.

A.7 Porting an Existing Hydrostatic Model

Porting AC/DC requires extending a hydrostatic host with prognostic vertical dynamics, pseudo-density transport, column and barotropic coupling, diagnostics, and staged verification. Existing transport, solvers, time steps, and several physical components can otherwise be retained.

  • Host requirements: AC/DC requires flux-form transport, a column-wise tridiagonal solver, a barotropic solver, and a z- or z*-coordinate prismatic mesh from the host.Isopycnal and terrain-following coordinates are outside the stated host requirements.
  • The changes, in dependency order: AC/DC porting begins with prognostic vertical velocity and a pseudo-pressure field, adding vertical advection, three-dimensional vorticity, and non-hydrostatic pressure-gradient terms.The host also advances ψ prognostically, with α = ρ0gH_full and ψ = 0 at the free surface.
  • The changes, in dependency order: Pseudo-density must replace cell mass, and tracer transport must use the pseudo-mass flux while retaining the existing reconstruction, limiter, and tracer time step.Using volumetric flux instead forfeits the stated conservation propositions and is detected by verification test V1.
  • The changes, in dependency order: The existing vertical tridiagonal solver handles the column pressure solve, while barotropic coupling carries the depth-averaged pseudo-pressure and updates its residual baroclinically.The column solve uses Dirichlet surface and Neumann bottom conditions; surface weighting makes the surface condition exact across the coupling loop.
  • The changes, in dependency order: Diagnostics should report density-weighted kinetic energy and potential energy; the equation of state, hydrostatic pressure, time steps, free-surface solver, tracer advection, and vertical mixing remain unchanged.The unchanged components preserve much of the host model infrastructure.
  • Verification ladder: Verification proceeds proposition by proposition, beginning with constant and non-uniform tracer tests before energy, internal-wave, and artificial-compressibility checks.Arbitrary-Lagrangian–Eulerian hosts additionally require remapping ρα_AC consistently with tracers; AC/DC has not been implemented in such a host.
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