A Computational Model for Global Ocean Dynamics at all Scales

Motivation

• Hydrostatic models exclude smaller scales through approximations and replace their effects, including convective processes, with ad hoc parametrizations.They have traditionally simulated planetary to kilometre-scale dynamics, where the hydrostatic approximation is generally adequate.Source: §1 · p. 3

Necessity of Non-Hydrostatic Ocean Models

• At subkilometre resolution, non-hydrostatic pressure enables internal-wave dispersion, vortex tilting, frontal dynamics, instabilities, and explicitly simulated convective plumes.Hydrostatic models cannot grow resolved plumes from unstable stratification, so convection must otherwise be parametrised.Source: §1 · p. 3

The Computational Obstruction

• The elliptic pressure solve, rather than variable resolution itself, is the computational obstruction because its global cost and communication are determined by the finest spacing anywhere in the mesh.Other model components are local and scale with added cells, whereas confining the pressure solve to a patch introduces artificial boundaries that reflect internal waves.Source: §1 · p. 4

AC/DC

• AC/DC replaces the global elliptic solve with artificial compressibility and direct column solves, adding only ∼1.2× hydrostatic-model overhead independent of resolution and without global communication.Its artificial-compressibility error decreases with the ratio of flow speed to chosen acoustic speed, while its formulation supports telescoping meshes and existing hydrostatic models.Source: §1 · p. 5

Claims and Outline

• The paper claims that AC/DC combines hydrostatic and non-hydrostatic regimes under one mesh-dependent model while retaining the relevant energy, divergence, tracer, free-surface, and wave physics.It establishes tracer-variance-budget closure despite residual divergence, allowing dissipation and mixing quantities to be diagnosed from simulated fields rather than prescribed parametrizations.Source: §1 · pp. 3–5
• The paper does not perform a 1:1 projection-versus-AC/DC performance comparison, instead comparing costs in tracer-pass units and arguing that AC/DC has bounded order-unity overhead.Projection cost varies with an iteration count that grows with H/∆x and requires sequential global reductions, whereas AC/DC’s overhead is fixed a priori and resolution-independent.Source: §1 · p. 5

2.1 Prelude: Artificial Compressibility

• Artificial compressibility replaces the non-hydrostatic pressure and Poisson solve with a local hyperbolic pseudo-pressure equation, recovering incompressible dynamics formally as α →∞.The method adds one prognostic variable per cell and suppresses divergence through ∇ψ, with convergence established in the incompressible limit.Source: §2.1 · p. 6
• Choosing the artificial sound speed as the barotropic gravity-wave speed yields Frbaro ≈5×10−3 for U ∼1 m s−1 and Hfull = 4000 m, with errors proportional to Fr2baro.The artificial mode is therefore carried by machinery that already resolves the barotropic mode.Source: §2.1 · p. 7

2.2 AC/DC: Thin Fluids and Anisotropic Grids

• AC/DC decomposes thin-fluid pressure into a cheap local vertical column solve and a horizontal artificial-compressibility correction, then aggregates pNH = pV + ψ.The column solve removes vertical stiffness from pV, while the horizontal AC step also relaxes the stiff vertical operator acting on ψ.Source: §2.2 · pp. 7–8

2.3 Consequences of Artificial Compressibility

• AC/DC introduces pseudo-density and modified kinetic energy so flux-form continuity, tracer transport, and total energy retain exact conservation rather than only O(1/α) consistency.The flux form is essential: it makes the energy and tracer statements exact.Source: §2.2 · p. 7, §2.3 · p. 9, Tracer equation · p. 9

Kinetic Energy.

• Density-weighted kinetic energy removes the cubic obstruction Sdf because the pseudo-density weight contributes −Sdf and the remaining cubic correction telescopes under flux-form transport.Using the relaxation form instead leaves an O(1/α) remainder, so the conservation result is only asymptotic.Source: Kinetic Energy · pp. 10–11
• Anisotropic viscosity and diffusivity are embedded in the curl–divergence operator, producing separately diagnosable dissipation reservoirs with νh assigned to horizontal-shear vorticity and νv to vertical-shear vorticity.This assignment matches the physical decomposition of vorticity in thin anisotropic flows.Source: Kinetic Energy · p. 11

3 AC/DC for Boussinesq Equations

• Together, these estimates assess AC/DC through energy conservation, volume-conservation and tracer errors, and internal-gravity-wave dispersion fidelity for Boussinesq equations.The section frames these as quantitative tests of AC/DC’s physical plausibility.Source: §3 · p. 12

3.1 Total Energy

• The density-weighted total-energy budget closes without an α−1⟨ψb, w⟩ mismatch, while remaining AC/DC terms are pressure work against artificial compressibility.The discrete budget contains no cubic term or Lamb-term contribution, and all terms are diagnosable.Source: §3.1 · pp. 13–14

3.2.1 Divergence, volume, and tracer errors

• AC/DC’s column solve reduces grid-scale divergence error by (∆z/∆x)^2, yielding 10^-4 of pure AC on a mesh with ∆x/∆z = 100.The advantage disappears on isotropic meshes and can fail for narrow plumes spanning deep, weakly stratified layers; increasing α restores accuracy at shorter sub-step cost.Source: §3.2.1 · pp. 14–15

3.2.2 Dispersion relations

• AC/DC improves dispersion accuracy over pure AC by the wavenumber-dependent factor k2_z in the near-hydrostatic sector, but pure AC is more accurate when kx > kz.The column solve removes buoyancy coupling from the vertical momentum equation; the absolute worst-case frequency error is about 2 × 10^-3 in the supplied example.Source: §3.2.2 · p. 17

3.3 Consistent Tracer Transport

• Consistent pseudo-mass-flux tracer transport exactly conserves pseudo-density-weighted content and constant fields, preventing secular salt drift under the stated positivity and consistency conditions.The representation error is uniformly bounded in time and improves to O(α^-1/2) when ∥ψ∥h is bounded.Source: §3.3 · pp. 18–19

3.4 Dissipation Rate of Tracer Variance and its computability

• The Osborn–Cox tracer-variance dissipation budget closes exactly because pseudo-density weighting absorbs residual divergence and separates physical from numerical variance sinks.This makes the resulting mixing diagnostic computable despite AC velocity’s residual divergence.Source: §3.4 · p. 19
• AC/DC gives an exact tracer-variance identity for any face reconstruction, with residual divergence absorbed by pseudo-density weighting.For centred reconstruction, the advective contribution vanishes and only the explicit variance sink remains; upwind transport adds a separately computable numerical sink.Source: §3.4 · p. 20
• The Osborn–Cox diffusivity KT and irreversible buoyancy-mixing rate are computed outputs from discrete tracer or buoyancy fields, not prescribed inputs.The buoyancy specialization also supplies the weighted potential-energy budget, while kinetic-energy dissipation ε comes from Section 3.1.Source: §3.4 · p. 20
• For centred transport, the variance budget is purely physical and the energy budget closes; for monotone non-centred transport, physical and numerical sinks remain separately reportable as exact face sums.This contrasts with prior finite-volume diagnostics, which estimated local numerical sinks inside operators that dissipate everywhere; AC/DC preserves exact spatial conservation.Source: §3.4 · p. 20
• The paper distinguishes mixing-efficiency coefficient Γ from flux Richardson number Rf, related by Γ = Rf/(1−Rf), with Γ = 0.2 corresponding to Rf ≈0.167.Throughout the paper, Γ denotes εb/ε and becomes computable by combining Corollary 3.7 with the energy budget.Source: §3.4 · pp. 20–21

4 Implementation of AC/DC

• AC/DC replaces the global solve with nearest-neighbour stencil work and at most single-column solves, targeting throughput-oriented hardware.The stiff vertical direction is handled within columns, while the acoustic stage avoids reintroducing a global system.Source: §4 · p. 21, §4.2 · p. 23, §4.4 · p. 27

4.1 Classical Artificial Compressibility

• Classical AC separates velocity and pressure updates, but its only global operation is the expensive velocity solve that limits ocean-resolution simulations.The resulting grad–divergence strength is controlled by Δt/ε and becomes the dominant operator in the fixed-parameter ocean setting.Source: §4.1 · pp. 22–23

4.2 The AC/DC Step

• The AC/DC step removes the vertical divergence contribution through a column pressure solve, leaving an acoustic residual advanced over sub-steps without global operations.Tracer transport retains explicit horizontal diffusion and column-local implicit vertical diffusion.Source: §4.2 · pp. 23–24

4.3 Option S3-a: Pseudo-Pressure Splitting

• Option S3-a splits pseudo-pressure into a column-implicit remainder and an explicit depth-averaged component, so sub-stepping is governed by the horizontal condition nsub = O(cacΔt/Δx).Its sub-loop uses nearest-neighbour stencils and column-local blocks, matching existing barotropic sub-stepping.Source: §4.3 · pp. 24–25

4.4 Option S3-b: An Explicit Acoustic Step

• Option S3-b advances the full three-dimensional velocity–pressure pair explicitly with nearest-neighbour stencils, but refinement at fixed accuracy makes it more expensive than S3-a.Keeping the vertical acoustic channel forces the sub-step count to rise with refinement unless compressibility is reduced, which increases pseudo-density departure.Source: §4.4 · p. 26

4.5 Well-Prepared Initial Conditions

• Well-prepared initialization requires both discretely divergence-free velocity and compatible pseudo-pressure; otherwise the initial error is O(Frbaro), one order larger than the paper’s other controlled errors.An inconsistent pseudo-pressure error decays as α^-1/2, whereas initial divergence produces an α-independent velocity error; damping or an initial projection can enforce compatibility.Source: §4.5 · pp. 27–28

5.1 Hydrostatic–Non-Hydrostatic Decomposition

• AC/DC expresses the non-hydrostatic model as a hydrostatic core plus correction, solving the vertical residual exactly by tridiagonal columns and relaxing only the horizontal remainder locally.The pressure split removes the buoyancy-determined hydrostatic component and leaves a smaller residual; the AC closure is the section’s sole approximation.Source: §5.1 · pp. 28–30

5.2 Free Surface

• AC/DC supports a moving free surface by enforcing the kinematic boundary condition, requiring non-hydrostatic pressure to vanish at the surface, and retaining exact barotropic-reservoir energy accounting.The full three-dimensional pseudo-pressure formulation is a verification reference, while the split scheme reproduces it to round-off; full-mesh sub-stepping is too costly for production.Source: §5.2 · pp. 31–32

5.3 Errors in Context of other Errors

• 10^-10 grid-scale relative divergence is reached with AC/DC, two orders below the Boussinesq constraint error and more than four orders below the hydrostatic velocity-field error.The column solve reduces pure-AC divergence by (kx/kz)^2, making the residual comparable to real precipitation-minus-evaporation forcing rather than the hydrostatic approximation’s removed velocity-field error.Source: §5.3 · p. 33, Table 1 · p. 34

5.4 Time Stepping Structure

• AC/DC preserves the host model’s baroclinic and barotropic time stepping, introduces no new CFL restriction, and confines α-dependent acoustic sub-stepping to two-dimensional fields.The depth-averaged pseudo-pressure follows the barotropic machinery, the column-mean-free remainder updates once per outer step, and nsub grows as √α only for the acoustic sub-step.Source: §5.4 · pp. 34–35
• The four-step template combines hydrostatic baroclinic tendencies, barotropic updating, an AC/DC pseudo-pressure loop, vertical implicit solves, and tracer transport.The added solves include a two-dimensional depth-averaged pseudo-pressure update, a direct column solve for non-hydrostatic pressure, and a column-implicit baroclinic pseudo-pressure update.Source: §5.4 · pp. 35–36
• Distributed ψ′ updates during the fast loop close both the velocity-consistency and volume budgets, unlike an end-of-step velocity correction.The end-of-step correction removes divergence without carrying volume flux, causing the time-integrated divergence of the actual velocity path to drift linearly.Source: §5.4 · p. 36
• In split-explicit models, the depth-averaged pseudo-pressure shares the barotropic loop when α = ρ0gHfull, adding one two-dimensional field per sub-step without a new CFL constraint.The artificial acoustic speed then equals the barotropic gravity-wave speed.Source: §5.4 · p. 37
• In semi-implicit models, the free-surface and depth-averaged pseudo-pressure form a weakly coupled 2 × 2 system that can be advanced in one implicit block.The coupling is formally O(δ2), and the alternative is a separate second call to the free-surface solver.Source: §5.4 · p. 37

• Passes and global stops are the relevant cost units because memory-bound iterations reread the mesh and sequential reductions do not shrink with added nodes.A hydrostatic step costs about twelve passes, while an iterative pressure solve costs approximately one pass per iteration.Source: §6 · pp. 37, 40
• Without preconditioning, conjugate-gradient counts reach thousands or more, making non-hydrostatic pressure prohibitively expensive relative to a twelve-pass hydrostatic step.The count scales with the square root of the discrete Laplacian condition number, whose vertical contributions span many orders of magnitude.Source: §6 · p. 37
• A vertical-line preconditioner reduces iterations to 102–103 passes, but irregular ocean geometry still makes the solve seven to seventy times a hydrostatic step.The idealized count is H/∆x, or 80 for the stated 4000 m depth and 50 m spacing, before geometric degradation.Source: §6 · p. 39
• Exact vertical splitting fails on stretched grids and realistic bathymetry, while multigrid remains unpriced because unstructured ocean meshes face topographic and communication barriers.The vertical transform alone costs about forty passes and requires terabytes for basis storage; multigrid coarse transfers also become latency-dominated.Source: §6 · p. 39
• Projection methods add 102–103 sequential global reductions per step, whereas AC/DC uses local column solves and nearest-neighbour substeps beyond the host model’s existing barotropic solve.The reductions arise from two global inner-product operations per conjugate-gradient iteration and remain latency costs independent of node count.Source: §6 · p. 40
• Two passes and no global stops of its own make AC/DC about 1.15–1.22 times the hydrostatic step, versus 102–103 passes and stops for projection methods.These are scale estimates under the stated configuration, not measurements or theorems.Source: §6 · p. 40

7 Numerical Experiments

• The experiments use exact Leray projection as ground truth, while AC/DC replaces it with a column solve plus horizontal artificial compressibility and reports relative differences on identical meshes and timesteps.The expected discrepancy is the O(Fr2_barо) error analyzed earlier, except Experiment 5, which uses analytical references.Source: §7 · p. 40
• All experiments use z-coordinates, enabling orthogonal horizontal–vertical decomposition and a clean tridiagonal vertical Laplacian per column.Free-surface and moving-mesh extensions are treated separately, with conservation statements reported to hold on moving meshes.Source: §7 · p. 41

7.1 Experiment 1: Lock-exchange gravity current

• Lock-exchange AC/DC matches projection in front speed and gravity-current structure, while conserving tracer content and preserving measured mixing diagnostics.Front Froude differences are 2.2×10−4 and 1.8×10−4, versus 2.1 × 10−2 between meshes; mixing diagnostics agree more closely with refinement.Source: §7.1 · pp. 43–45

7.2 Experiment 2: Internal gravity wave dispersion

• Internal-wave measurements reproduce the analytical AC/DC dispersion formula exactly to all reported digits, with closure error O(10−5) at α̃ = 1 and O(10−10) at production α̃.Projection error is spatial truncation, whereas AC/DC closure error decreases as 1/α̃ and becomes far smaller than spatial error at production α̃.Source: §7.2 · p. 46

7.3 Experiment 3: Open-ocean deep convection

• Deep-convection AC/DC requires α = 108 for stable completion; at that value, plume depth is 0.8, max |w| is 2105, and min T ′ is −1.00.At α ∈{105, 106, 107}, AC/DC develops acoustic instability and NaNs by t ≈0.012–0.015, whereas α = 108 tracks projection closely.Source: §7.3 · p. 47
• The paired convection runs confirm 1/α scaling: with α2/α1 = 9, artificial-compressibility quantities have observed ratios between 8.94 and 9.01.The compared runs use α1 = 3.131 × 107 and α2 = 2.818 × 108 with identical mesh, timestep, initial condition, and integrator.Source: §7.3 · pp. 48–49
• Lower α worsens numerical diffusivity, variance residual, and energy-budget residual because its velocity field carries greater divergence, although the absolute errors remain small.The reported differences are 0.27%, 0.90%, and 30%, respectively; the α1 run advects with nine times the divergence, and the numerical diffusivity remains within one part in four hundred of 0.0894 against physical diffusivity unity.Source: §7.3 · pp. 49–50
• The full-integration comparison gives projection a 1.41× pressure-treatment speed advantage, but short early samples reverse the ranking because AC/DC cost grows with pseudo-density departure.The early measurements are 228 ms versus 333 ms, while later AC/DC cost rises from 228 to 429 ms as pseudo-density departure grows from 2.5 × 10−5 to 6.0 × 10−3.Source: §7.3 · p. 50
• The two runs agree on the flow while differing substantially in error measures, isolating artificial compressibility as the source of the discrepancy.Settled means differ by 0.12% on max |w|, 0.020% on the patch anomaly, and 0.22% on plume depth, versus a 66% max |w| spread; downstream quantities agree to better than one part in a thousand, while first-table errors differ by a factor nine.Source: §7.3 · pp. 49–50
• Across a factor-sixteen α range, every error follows 1/α within 7%, with no threshold, onset, or instability, while domain-scale surface signals move twenty times less than plume peak velocity.The results indicate that the practical lower bound is not failure of a domain-scale pressure response, since the domain scale survives best.Source: §7.3 · pp. 50–51
• After mesh refinement by eight, the 1/α law persists: measured errors are 4.63%, 0.931%, and 0.046%, while the lower-α pressure treatment costs 238.6 ms versus 247.0 ms for projection.At that lower α, AC/DC reproduces plume peak velocity to under 1% and the domain-scale signature to 0.05%; unconstrained advance remains within 0.6% across all three runs.Source: §7.3 · p. 51
• The wall-clock results come from one laptop and implementation, so they isolate pressure-treatment ratios rather than replacing transferable operation-count comparisons.The same code is used up to the pressure treatment, but the measurements are explicitly not a substitute for Appendix A.4 operation counts.Source: §7.3 · p. 51

Telescoping-mesh setup

• A single mesh can narrow from about 50 km in the open ocean to 500 m or finer in a target region, requiring hydrostatic and non-hydrostatic dynamics to coexist.Projection methods face global costs and interface-wave-reflection issues because the physics is localized but the solve is global.Source: §8 · p. 51

Telescoping-mesh advantages

• AC/DC uses one equation set globally, eliminates hydrostatic–non-hydrostatic interface conditions, and confines its local column-solve work to the refined patch.Its column solve costs O(nz) independently per column, remaining cheap where pV ≈ 0 while capturing dominant non-hydrostatic pressure.Source: §8 · pp. 51–52

Telescoping-mesh methods

• Pure AC avoids an interface but sub-steps the entire mesh at a rate set by its finest cell, so coarse regions are advanced more often than their dynamics require.AC/DC instead places the expensive work where the non-hydrostatic physics occurs.Source: §8 · p. 52

Telescoping-mesh advantages

• 10−13 versus 10−9 divergence error in the coarse region, while the refined patch reaches 10−11–10−9; both cost and error follow the mesh without transition tuning.The coarse-region estimate uses δ ≈ 7 × 10−3 for a 50 km-resolved flow, whereas the refined patch has δ = 0.2.Source: §8 · p. 52

Conclusions

• The hydrostatic primitive equations are an exact subsystem of the three-dimensional non-hydrostatic momentum equation, requiring only a correction rather than an alternative model.This decomposition introduces no approximation.Source: §9 · p. 53
• Artificial compressibility introduces a controlled finite-speed relaxation error, while the column solve removes vertical constraint components and improves internal-wave dispersion accuracy by the square of grid aspect ratio.The compressibility error equilibrates instead of growing with integration length.Source: §9 · p. 53
• The tracer-variance budget closes exactly despite residual divergence, making the numerical variance sink an explicit face sum and Osborn–Cox diffusivity computable from discrete fields.This preserves mixing as a computed output rather than a prescribed input.Source: §9 · p. 53
• Section 7 experiments show that AC/DC reproduces processes removed by hydrostatic modeling and agrees with hydrostatic results where agreement is expected.These experiments are neither performance measurements nor global simulations; a convection-resolving world-ocean simulation is out of scope.Source: §9 · p. 53
• AC/DC removes the computational obstruction to non-hydrostatic ocean modeling at a small, fixed multiple of existing model cost on emerging machines.The overhead is bounded, known before execution, and flat under refinement, but the feasibility argument is not a timing comparison.Source: §9 · pp. 53–54

A.2 Proofs

• Incidence-based cancellation proves conservation of M and Σ independently of mesh metric, reconstruction, or moving cell volumes.Interior-face contributions cancel pairwise, while boundary fluxes vanish.Source: §A.2 · p. 54
• Using the same face flux F_f for r and F_fC_f makes constant C stationary exactly for any α.The proof reduces the C equation to the r equation under H2.Source: §A.2 · p. 54
• The relaxation bounds follow from positivity and exact conservation, with O(α^-1) or O(α^-1/2) forms depending on whether C′ is controlled in L∞ or ψ in the discrete norm.A discrete maximum principle controls C′, while Cauchy–Schwarz yields the alternative scaling.Source: §A.2 · p. 54
• The Proposition 3.6 identities follow from semi-discrete balances, summation by parts, and cancellation of the jump-form contribution from DΦ.The argument applies the incidence convention and the upwind reconstruction to derive the stated identities.Source: §A.2 · pp. 54–55
• For moving domains, materiality makes boundary terms cancel or become total surface time derivatives, recovering the fixed-domain identities.The free-surface conditions remove hydrostatic and pseudo-pressure contributions, fix r=1 at the surface, and use the kinematic condition.Source: §A.2 · p. 55
• The dispersion proof derives the biquadratic relation by substitution, retains the necessary slow-root correction, and verifies limiting cases and rotation-dependent error reduction.The slow-root expansion includes the C2/B3 term because it matches O(1/α̃) coefficient shifts.Source: §A.2 · pp. 55–56

• The boundedness argument depends on curl–divergence dissipation damping both divergent and rotational velocity components, with the divergent component exchanging energy with pseudo-pressure.The proof is semi-discrete because the controlled bound is discrete and the reduction is exact on the mesh via d◦d = 0.Source: §A.3 · p. 56
• The semi-discrete acoustic reservoir is exactly a damped wave, reducing without truncation error to independent modal equations.The reduction uses discrete divergence, gradient, curl, and Hodge-dissipation identities, with the curl-of-gradient term vanishing identically.Source: §A.3 · p. 56
• Every nonconstant mode decays unconditionally in α, ν, and mesh size, while the constant mode remains stationary through exact pseudo-mass conservation.The decay rate is 2νλk when ν^2λk < 4α̃ and α̃/ν beyond that threshold.Source: §A.3 · p. 56
• Under bounded modal forcing, ∥ψ∥h stays uniformly bounded in time, while well-prepared initial data gives ∥ψ∥h = O(α^1/2) and ∥ψ∥h/α = O(α^-1/2).The bound follows by treating each eigenmode as a damped oscillator and summing the modal estimates; the stationary zero mode contributes nothing.Source: §A.3 · pp. 56–57
• When forcing lies well below each modal frequency ωk = √α̃λk, the response is static, with |qk| ≤ Fk/(α̃λk).In this low-frequency regime, the reservoir bounds sharpen to ∥ψ∥h = O(1) and ∥ψ∥h/α = O(α^-1).Source: §A.3 · p. 56

A.4 Cost Accounting

• About 1.2×: AC/DC overhead ranges from 1.15 to 1.22 across counting conventions, staying near 1.2 against a hydrostatic baseline of about 12 F1.For ntracer = 5, the ratio falls to a few per cent at twenty to fifty tracers because the baseline grows while contraction overhead remains fixed.Source: §A.4 · pp. 57–58
• The accounting uses tracer-equivalents F1 = Ncol nz Cstencil ≈1011 FLOP, with AC/DC adding contraction, column-solve, barotropic, and ψ′-update work.At reference intensity, AC/DC increments use 10–20 FLOP per cell rather than 50, so byte-limited costs are biased against the quoted values.Source: §A.4 · p. 57, Table 3 · p. 57
• The non-hydrostatic contraction is 3.86 F1 versus 2.96 F1 hydrostatic, a 0.90 F1 Lamb-vector correction ωH × v.The contraction range is 0.8–1.3 F1, while subtracting the host advective term contributes 0.1–0.4 F1.Source: §A.4 · p. 58
• The barotropic ¯ψ and free-surface η operators have identical stiffness, but ¯ψ needs εψ ∼10−2–10−3 versus εη ∼10−13, making tolerance the cost difference.On a split-explicit host, ¯ψ adds 0.16 →0.24 F1 as one extra two-dimensional field in the existing sub-step loop.Source: §A.4 · p. 58
• Under refinement, AC/DC’s cost ratio stays flat because its column solve is O(nz) per column and its sub-step count scales as ∆x−1 like the hydrostatic model.The projection cost instead grows with H/∆x through κprec = 1 + H2/∆x2.Source: §A.4 · p. 58

Scope

• All costs are scale-analysis estimates under Table 3’s configuration, not measurements or theorems.The subsection prices each route to a non-hydrostatic pressure in passes.Source: §A.5 · p. 58

Pressure iteration cost

• A preconditioned conjugate-gradient pressure iteration costs 0.7–0.9 F1 arithmetically or 1–1.5 F1 by data movement, supporting one iteration as one pass.At the 102–103 iteration band, pressure costs 102–103 passes versus about twelve for a hydrostatic step and requires two global reductions per iteration.Source: §A.5 · p. 59

Exact vertical split

• Exact vertical splitting costs 40 F1 for nz = 500 and Ncol = 4 × 106, exceeding three times the entire hydrostatic step before solving any two-dimensional problem.The stretched-grid transform is dense; storing bases costs about 8 TB, while rebuilding them costs O(n3) per-column decomposition of order 103 F1.Source: §A.5 · p. 59
• Reducing the average column depth to nz = 300 lowers the transform cost to 14 F1 but does not change the conclusion that exact splitting is unavailable.The transform scales quadratically with column depth, while basis storage and rebuilding remain unavailable at the larger configuration.Source: §A.5 · p. 59

Topographic limitation

• Real bathymetry destroys exact vertical splitting because column-dependent eigenbases make the horizontally coupled operator dense in mode space.Below sill depth, disconnected basins also create depth-varying singular operators with one undetermined constant per basin; whether an outer coupling iteration must fail remains untested.Source: §A.5 · p. 59

• The density-free kinetic energy Kdf is not a Lyapunov functional under artificial compressibility because its Bernoulli contribution produces a non-sign-definite cubic term.The term vanishes for incompressible flow but survives when ∇·v ≠ 0.Source: §A.6 · p. 60
• The cubic obstruction cannot be cancelled by modifying the Lamb term without violating the Lamb-antisymmetry axiom and the Cartan identity underlying circulation conservation.Under (A3), the Lamb contribution is identically zero in the energy budget.Source: §A.6 · p. 60
• The density-weighted kinetic energy Kdw removes all cubic terms exactly, leaving only pressure, viscous, and buoyancy pairings in the energy budget.The cancellation follows from the mass-flux form ∂tr = −∇·(rv) on a closed domain.Source: §A.6 · p. 60
• Within state-dependent corrections that vanish at ψ ≡ 0, the density weight is unique; flux-form advancement gives exact cancellation, whereas relaxation-form advancement leaves an O(1/α) remainder.The same weighting also telescopes a tracer-variance budget into a face sum.Source: §A.6 · pp. 60–61

Core changes

• The largest porting change is prognostic vertical velocity, requiring vertical advection, full three-dimensional Lamb-term vorticity, and the non-hydrostatic vertical pressure gradient.Hosts using the traditional approximation should revisit Coriolis terms coupling u and w.Source: §A.7 · p. 61
• AC/DC also requires prognostic pseudo-pressure, pseudo-density transport using pseudo-mass fluxes, a vertical column solve, barotropic pseudo-pressure coupling, and new energy diagnostics.Existing face reconstruction, limiter, tracer time step, equation of state, hydrostatic pressure, time steps, free-surface solver, tracer advection, and vertical mixing remain unchanged.Source: §A.7 · p. 61

Verification ladder

• Transporting tracers with pseudo-mass fluxes rather than volumetric fluxes is necessary to preserve Propositions 3.4 and 3.6, and V1 detects this implementation error.On sub-stepped hosts, V1 commonly fails when outer-step fluxes replace time-integrated sub-step fluxes.Source: §A.7 · pp. 61–62
• The verification ladder checks constant and passive-tracer conservation, positive pseudo-density, energy behavior, internal-wave dispersion, O(1/α) convergence, and absence of an α-dependent time-step restriction.Each step tests one proposition and should pass before the next; the first three require no dynamics.Source: §A.7 · p. 62

Host limitation

• For arbitrary-Lagrangian–Eulerian vertical grids, remapping must conserve pseudo-density-weighted tracer mass; AC/DC was not implemented in such a host, so this is a requirement rather than a result.Remapping that conserves |c| Cc but not |c| ρAC,cCc breaks Proposition 3.4 at every remap.Source: §A.7 · p. 62