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arxiv: 2605.15879 · v1 · pith:DN36FPBUnew · submitted 2026-05-15 · ⚛️ physics.flu-dyn · physics.ao-ph

Coarse-grained local available potential energy

Jacob O. Wenegrat , Tomas Chor , Roy Barkan This is my paper

Pith reviewed 2026-05-19 19:38 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.ao-ph
keywords available potential energycoarse-grainingspatial filteringstratified flowsenergy cyclemulti-scale energeticsKelvin-Helmholtz instability
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The pith

Spatial filtering applied to local available potential energy produces separate evolution equations for large and small scales plus a cross-scale flux term.

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

This paper develops a coarse-graining method to examine the available potential energy of stratified fluids across different spatial scales. Applying a spatial filter to the local APE definition separates the energy into components larger and smaller than the filter scale. The resulting equations include a term that tracks the transfer of APE between those scales. The framework is constructed so it can be combined with existing coarse-grained kinetic energy equations to track a complete energy cycle that includes both scale transfers and conversions between kinetic and potential energy.

Core claim

Applying a spatial filter to the local definition of available potential energy yields evolution equations for the filtered large-scale APE and the sub-filter small-scale APE, together with an explicit cross-scale APE flux term that accounts for transfers between the two ranges.

What carries the argument

The spatial filtering (coarse-graining) operator applied to the local APE, which isolates the filtered APE, the sub-filter APE, and the cross-scale flux term within the derived evolution equations.

If this is right

  • The new equations can be paired with existing coarse-grained kinetic energy frameworks to study a full energy cycle that includes both spatial-scale conversions and reservoir conversions.
  • The method applies directly to numerical simulations of stratified flows such as Kelvin-Helmholtz instability.
  • It supplies a tool for analyzing energetics and dynamics in flows ranging from three-dimensional turbulence to planetary-scale circulations.

Where Pith is reading between the lines

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

  • The scale-separated budgets could support development of sub-grid closures that respect both energy reservoirs and scale transfers in large-scale models.
  • Extending the same filtering procedure to three-dimensional turbulence cases would test whether the cross-scale flux behaves similarly in fully turbulent stratified flows.
  • Applying the framework to other canonical instabilities would show how APE cascades differ across flow types.

Load-bearing premise

The local definition of available potential energy keeps its original physical meaning and usefulness once a spatial filter has been applied.

What would settle it

In the two-dimensional Kelvin-Helmholtz simulation, if the sum of the large-scale and small-scale APE evolution equations does not recover the unfiltered APE equation, the derivation would be falsified.

Figures

Figures reproduced from arXiv: 2605.15879 by Jacob O. Wenegrat, Roy Barkan, Tomas Chor.

Figure 1
Figure 1. Figure 1: Schematic energy cycle in the coarse-graining framework. Single arrows [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Cross-scale fluxes of KE (Π𝐾 ) and APE (Π𝐴) averaged over 𝑡 = 0 − 140 as a function of filter scale 𝑙. Dashed lines indicate scales shown in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Domain-integrated subfilter-scale KE (top row) and subfilter-scale APE (bottom [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Snapshots of select energy budget terms for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

The available potential energy (APE) of a fluid can be defined locally in space, providing useful insights into both the energetics and dynamics of stratified flows ranging from three-dimensional turbulence to planetary scale circulations. Here we develop a framework for considering the multi-scale evolution of the local APE using a spatial filtering, or coarse-graining, approach. Evolution equations for the APE at scales larger, and smaller, than the filtering scale are derived -- including the cross-scale APE flux term. These results can be paired with existing frameworks for coarse-grained kinetic energy, offering the potential for examining a complete energy cycle that accounts for conversions between both spatial scales and energy reservoirs. An illustrative example of the application of this approach to a simulation of two-dimensional Kelvin-Helmholtz instability is provided.

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 paper develops a spatial filtering framework for the local available potential energy (APE) in stratified fluids. It derives evolution equations for the APE components at scales larger and smaller than the filter scale, including an explicit cross-scale APE flux term. These are intended to complement existing coarse-grained kinetic energy equations to enable analysis of complete multi-scale energy cycles involving conversions between KE and APE reservoirs. An application to a 2D Kelvin-Helmholtz instability simulation is presented as illustration.

Significance. If the filtered local APE quantities retain their physical interpretation as energy available for conversion to kinetic energy, the framework would allow systematic examination of inter-scale energy transfers and the full KE-APE cycle in stratified turbulence and geophysical flows. The work builds directly on prior local APE definitions and filtering methods for kinetic energy, providing a natural extension that could support falsifiable predictions about energy cascades once verified against known limits or exact solutions.

major comments (2)
  1. [Derivation of filtered APE equations] The central derivation (abstract and the evolution-equation section) obtains the filtered APE equations by direct application of the filter operator to the local APE expression. This does not address the non-commutativity between spatial filtering and the global reference-state construction (adiabatic rearrangement to minimum potential energy). Because the reference state is determined by a global sorting operation on the buoyancy field, the filtered APE is not in general equal to the APE of the filtered field; the resulting cross-scale flux term may therefore lack the claimed physical meaning as genuine inter-scale transfer of available energy.
  2. [Illustrative example and verification] No explicit verification of the derived equations against known limiting cases (e.g., unfiltered local APE recovery as filter scale → 0, or global APE conservation in the absence of forcing/dissipation) is provided. The 2D Kelvin-Helmholtz example applies the method but supplies no quantitative check that the filtered quantities satisfy the same conversion-to-KE interpretation as the unfiltered local APE.
minor comments (2)
  1. [Notation and definitions] Notation for the filter kernel and the decomposition into large- and small-scale APE should be introduced with a single consistent symbol set early in the manuscript to avoid later ambiguity when the cross-scale flux is defined.
  2. [Introduction] The abstract states that the results 'can be paired with existing frameworks' but does not cite the specific coarse-grained KE papers that would be used; adding these references would clarify the intended synthesis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the scope and interpretation of our coarse-graining framework for local available potential energy. We respond to each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Derivation of filtered APE equations] The central derivation (abstract and the evolution-equation section) obtains the filtered APE equations by direct application of the filter operator to the local APE expression. This does not address the non-commutativity between spatial filtering and the global reference-state construction (adiabatic rearrangement to minimum potential energy). Because the reference state is determined by a global sorting operation on the buoyancy field, the filtered APE is not in general equal to the APE of the filtered field; the resulting cross-scale flux term may therefore lack the claimed physical meaning as genuine inter-scale transfer of available energy.

    Authors: We acknowledge that the global reference state (obtained by adiabatic rearrangement) does not commute with spatial filtering, so the filtered local APE differs from the APE computed on the filtered buoyancy field. Our derivation applies the filter to the local APE expression as defined in the manuscript, producing a consistent budget that includes an explicit cross-scale flux. This flux represents the scale-to-scale transfer of the filtered APE quantity and, when paired with existing filtered KE equations, still permits examination of the full multi-scale KE-APE cycle. We will revise the manuscript to add an explicit discussion of the non-commutativity, to state clearly that the cross-scale term is the flux of the filtered local APE (rather than of the APE of the filtered field), and to note regimes where the two interpretations approximately coincide. revision: partial

  2. Referee: [Illustrative example and verification] No explicit verification of the derived equations against known limiting cases (e.g., unfiltered local APE recovery as filter scale → 0, or global APE conservation in the absence of forcing/dissipation) is provided. The 2D Kelvin-Helmholtz example applies the method but supplies no quantitative check that the filtered quantities satisfy the same conversion-to-KE interpretation as the unfiltered local APE.

    Authors: We agree that direct verification against limiting cases would strengthen the presentation. In the revised manuscript we will add a dedicated verification subsection (or appendix) showing (i) recovery of the unfiltered local APE evolution equation in the limit of vanishing filter width and (ii) domain-integrated conservation of total (large-scale plus small-scale) APE when forcing and dissipation are absent. For the Kelvin-Helmholtz illustration we will include quantitative diagnostics, such as the integrated APE-to-KE conversion rates at each scale and a closure check on the summed budgets, to confirm consistency with the unfiltered interpretation. revision: yes

Circularity Check

0 steps flagged

Derivation of filtered local APE evolution equations is self-contained with no circular reductions

full rationale

The paper derives evolution equations for coarse-grained local available potential energy by direct spatial filtering of the standard local APE definition, producing terms for large-scale, small-scale, and cross-scale fluxes. This is a straightforward mathematical manipulation of the governing equations and does not involve fitting parameters, re-expressing results in terms of themselves, or relying on load-bearing self-citations for the central claims. The derivation stands independently on the local APE framework and filtering operations without reducing to its inputs by construction, making the result self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard continuum fluid assumptions and the prior definition of local APE; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Boussinesq or similar approximation for stratified incompressible flow remains valid under spatial filtering.
    Implicit in any local APE derivation for geophysical fluids; invoked when defining the reference state and buoyancy.
  • standard math Spatial filtering commutes with the relevant differential operators in the governing equations.
    Required for deriving the filtered evolution equations without additional commutator terms.

pith-pipeline@v0.9.0 · 5655 in / 1407 out tokens · 31949 ms · 2026-05-19T19:38:16.786232+00:00 · methodology

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