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arxiv: 2507.00691 · v2 · submitted 2025-07-01 · ⚛️ physics.flu-dyn

Final states of two-dimensional turbulence above large-scale topography: stationary vortex solutions and barotropic stability

Jiyang He , Yan Wang This is my paper

Pith reviewed 2026-05-19 06:41 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords 2D turbulencetopographic turbulencefinal statesstationary vortex solutionsbarotropic stabilityGaussian vorticespotential vorticitysinh-like relation
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The pith

An empirical model superposing Gaussian vortices on linear background flows produces locally stationary solutions for final states of two-dimensional topographic turbulence.

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

The paper examines the quasi-stationary final states reached by freely decaying two-dimensional turbulence over large-scale sinusoidal topography containing both a bump and a dip. These states consist of a linear background flow satisfying a potential-vorticity-streamfunction relation together with two oppositely signed localized vortices that become locked to the topographic extrema. After the background flow is removed, the vortices display a sinh-like potential-vorticity-streamfunction relation. The authors construct an empirical model that adds Gaussian vortex profiles centered exactly at the extrema to the linear background flow; this combination reproduces the simulated states and satisfies the stationary condition of the inviscid equations locally. Linear stability analysis of the resulting solutions shows that the stable vortex-topography pairing switches with background energy: cyclone-elevation and anticyclone-depression pairs are stable at low energy while the opposite pairings are stable at high energy.

Core claim

The final states of 2D topographic turbulence consist of a background flow satisfying a linear PV-streamfunction relation together with localized vortices that, after the background is subtracted, follow a sinh-like PV-streamfunction relation. An empirical model that combines this linear background flow with Gaussian vortices centered at the topographic extrema accurately reproduces the quasi-stationary states obtained in simulations and yields locally stationary solutions to the inviscid governing equation. Linear stability analyses of these stationary vortex solutions reveal background-flow-dependent stability: cyclone-elevation and anticyclone-depression configurations are stable at low背景

What carries the argument

Empirical model formed by adding Gaussian vortex profiles centered at topographic extrema to a linear background flow that satisfies the stationary PV-streamfunction relation.

If this is right

  • The model supplies explicit vortex solutions for quasi-stationary final states of 2D topographic turbulence.
  • Stability of each vortex-topography configuration depends on the background energy level.
  • The observed vortex-topography correlations in simulations are explained by the energy-dependent stability thresholds.
  • Both the sinh-like relation and the Gaussian profiles persist under complex topography and high-energy conditions.

Where Pith is reading between the lines

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

  • The same construction might be applied to other topographic shapes to test whether Gaussian centering remains a good approximation.
  • The stability criteria could be used to forecast which vortex signs are favored over seamounts or trenches in oceanic flows at different energy levels.
  • Weak viscosity could be added to the model to examine how the stationary states slowly erode or reorganize over long times.

Load-bearing premise

Localized vortices in the final states can be accurately represented by Gaussian profiles centered exactly at the topographic extrema once the linear background flow is subtracted.

What would settle it

High-resolution simulations in which the potential-vorticity profiles of the locked vortices deviate measurably from the Gaussian shape after background subtraction, or in which the vortices fail to remain centered on the extrema.

Figures

Figures reproduced from arXiv: 2507.00691 by Jiyang He, Yan Wang.

Figure 1
Figure 1. Figure 1: Topographies: (𝑎) sinusoidal bump 𝜂(𝑥, 𝑦) = cos 𝑥 + cos 𝑦; (𝑏) zonal ridge 𝜂(𝑦) = √ 2 sin 𝑦. by the local Coriolis frequency 𝑓0, and Δ is the 2D Laplacian operator. In the absence of topography, the QG PV equation (2.1) reduces to that for 2D flat-bottom turbulence. In the inviscid limit (𝐷𝜁 = 0), the QG PV equation (2.1) admits an infinite number of integral invariants, including the energy 𝐸 = 1 2𝐿 2 ∫ 𝜋… view at source ↗
Figure 2
Figure 2. Figure 2: 𝑞𝑣–𝜓𝑣 relation of the Gaussian profiles (2.23) for Γ𝑚 = 1.0 and 𝐿𝑚 = 0.5. In § 4, we will inspect the scatter plots of 𝑞 against both 𝜓 and 𝜓 − 𝜓# of the simulation data, to uncover their relations. 2.3.2. Empirical theoretical model of final states Inspired by Siegelman & Young (2023), we propose to theoretically model the final states of 2D topographic turbulence by a superposition of isolated vortices o… view at source ↗
Figure 3
Figure 3. Figure 3: Initial monoscale fields of the relative vorticity [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Final states of PV at 𝑡 = 30000 evolving from the initial fields (figure 3) above the sinusoidal bump (figure 1𝑎). Dashed lines denote 𝜂 = 0 and separate the areas of 𝜂 > 0 (around the centre) and 𝜂 < 0 (around the corners). at such scales are steered by the local topographic gradient and ultimately correlated positively with topographic extrema (Carnevale et al. 1991). The initial fields of relative vorti… view at source ↗
Figure 5
Figure 5. Figure 5: Scatter plots (in blue color) between 𝑞 and 𝜓 for different final states shown in figure 4. Red lines show the linear relation with the slope 𝜇 = 1 predicted by the minimum-enstrophy theory at the energy 𝐸 = 0.25𝐸#. −2 0 2 −4 −2 0 2 4 q (a) kini = 2 Data “sinh” relation −2 0 2 −20 −15 −10 −5 0 5 10 15 20 (b) kini = 8 −2 0 2 −40 −30 −20 −10 0 10 20 30 40 (c) kini = 16 −2 0 2 ψ − ψ# −60 −40 −20 0 20 40 60 q … view at source ↗
Figure 6
Figure 6. Figure 6: Scatter plots (in blue color) between 𝑞 and 𝜓 − 𝜓# for different final states shown in figure 4. Red lines show the least squares fitting between 𝑞 and 𝛼 sinh 𝛽(𝜓 − 𝜓#) [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparisons of the PV (𝑎), streamfunction (𝑏), zonal velocity (𝑐), and meridional velocity (𝑑) along the dash dotted line shown in figure 4𝑎 between simulation data and theoretical model for the final state corresponding to 𝑘𝑖𝑛𝑖 = 8 (figure 4𝑏). 𝑞 in (𝑎) is shown in symlog scale in order to visualize the data within the range [−1, 1]. maximum-entropy principle, respectively. It is neither described by the … view at source ↗
Figure 8
Figure 8. Figure 8: Same as figure 7 but for the final state corresponding to 𝑘𝑖𝑛𝑖 = 32 (figure 4𝑒). through our model for the 𝑘𝑖𝑛𝑖 = 8 and 32 cases, respectively. We show the profiles of PV 𝑞, streamfunction 𝜓, zonal velocity 𝑢, and meridional velocity 𝑣 on the diagonal line of the domain (see the dash dotted line in figure 4𝑎). The results of the 𝑘𝑖𝑛𝑖 = 16 and 24 are similar and not shown. As shown in figures 7𝑎 and 8𝑎 for … view at source ↗
Figure 9
Figure 9. Figure 9: Model parameters 𝜇𝑚, Γ𝑚, and 𝐿𝑚 for different 𝑘𝑖𝑛𝑖. The topography is the sinusoidal bump. −π −π/2 0 π/2 π −π −π/2 0 π/2 π y (a) kini = 2 −π −π/2 0 π/2 π −π −π/2 0 π/2 π (b) kini = 8 −π −π/2 0 π/2 π −π −π/2 0 π/2 π (c) kini = 16 −π −π/2 0 π/2 π x −π −π/2 0 π/2 π y (d) kini = 24 −π −π/2 0 π/2 π x −π −π/2 0 π/2 π (e) kini = 32 −π −π/2 0 π/2 π x −π −π/2 0 π/2 π (f) kini = 64 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.… view at source ↗
Figure 10
Figure 10. Figure 10: Final states of PV at 𝑡 = 30000 evolving from the initial fields (figure 3) above the zonal ridge (figure 1𝑏). domain, which may not be strong enough for mixing the PV towards homogenization. On the other hand, as 𝑘𝑖𝑛𝑖 increases, the vortex strength Γ𝑚 generally increases (figure 9𝑏) and the the vortex length 𝐿𝑚 decreases (figure 9𝑐), which means that the vortices become stronger [PITH_FULL_IMAGE:figures… view at source ↗
Figure 11
Figure 11. Figure 11: Time series of zonally averaged PV above the zonal ridge through the time [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Scatter plots (in blue color) between zonally averaged ¯𝑞 [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Time series of the model parameters 𝜇𝑚 (𝑎), Γ𝑚 (𝑏), and 𝐿𝑚 (𝑐) for different 𝑘𝑖𝑛𝑖. The topography is the zonal ridge. the model and simulation data agree with each other very well (see figures 14(𝑏, 𝑐) and 15(𝑏, 𝑐)). The variations of the three model parameters with 𝑘𝑖𝑛𝑖 are shown in figure 13. Once again, as shown in figure 13𝑎, the linear slope 𝜇𝑚 of the modelled background flow is smaller than 1, the p… view at source ↗
Figure 14
Figure 14. Figure 14: Comparisons of the zonally averaged PV (𝑎), streamfunction (𝑏), and zonal velocity (𝑐) between simulation data and theoretical model at the time instant 30000 for the final state corresponding to 𝑘𝑖𝑛𝑖 = 8 (figure 11𝑏). −π −π/2 0 π/2 π y −100 0 100 q¯ (a) ¯f ¯fb¯fv¯fb + ¯fv −π −π/2 0 π/2 π y −1.0 −0.5 0.0 0.5 1.0 ψ¯ (b) −π −π/2 0 π/2 π y −1.0 −0.5 0.0 0.5 1.0 u¯ (c) [PITH_FULL_IMAGE:figures/full_fig_p021_… view at source ↗
Figure 15
Figure 15. Figure 15: Same as figure 14 but for the final state corresponding to 𝑘𝑖𝑛𝑖 = 32 (figure 11𝑒). 𝑞 in (𝑎) is shown in symlog scale. figure 10). Similar to the case of the bump, as 𝑘𝑖𝑛𝑖 increases, the vortex strength Γ𝑚 increases and the vortex length 𝐿𝑚 decreases (see figure 13𝑏, 𝑐), which means that the vortices become stronger. 5. Conclusion and discussion In this work, we focus on the topographically-locked and long… view at source ↗
read the original abstract

The final states of freely decaying two-dimensional (2D) topographic turbulence consist of a background flow and localized vortices. While the background flow satisfies a linear potential vorticity (PV)-streamfunction relation, the vortex structures remain poorly understood. To address this gap and ensure oceanic relevance, we examine quasi-stationary final states of 2D turbulence over a sinusoidal topography featuring a bump and a dip, where two oppositely signed vortices are locked to the topographic extrema. After subtracting the background flow, the vortices exhibit a "sinh"-like PV-streamfunction relation, as observed in flat-bottom turbulence. Motivated by Gaussian vortex profiles in flat-bottom turbulence, we propose an empirical model combining the background flow with Gaussian vortices centered at the topographic extrema. This model accurately reproduces quasi-stationary states and yields locally stationary solutions to the inviscid governing equation. We further test the model under complex topography and high-energy conditions, confirming that the "sinh"-like trend and Gaussian profiles are robust features of localized vortices. Linear stability analyses of these stationary vortex solutions reveal background flow-dependent stability: cyclone/elevation and anticyclone/depression configurations are stable at low background energy, while anticyclone/elevation and cyclone/depression configurations are stable at high background energy. These findings align with vortex-topography correlations observed in simulations across energy regimes. Our results provide explicit vortex solutions for quasi-stationary final states of 2D topographic turbulence and elucidate the mechanism underlying vortex-topography correlations through stability analyses of vortices embedded in topographic background flows.

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 studies the final states of freely decaying two-dimensional turbulence over sinusoidal topography containing a bump and a dip. It reports that these states comprise a linear background flow satisfying a PV-streamfunction relation together with oppositely signed localized vortices locked to the topographic extrema. After subtracting the background, the vortices display a sinh-like PV-streamfunction relation. The authors introduce an empirical model that superposes the linear background with Gaussian vortex profiles centered at the extrema; they claim this construction reproduces the quasi-stationary states observed in simulations, furnishes locally stationary solutions of the inviscid equation, and remains robust under more complex topography and higher energies. Linear stability analysis of the resulting vortex solutions is shown to be background-energy dependent, with cyclone-over-elevation and anticyclone-over-depression configurations stable at low background energy and the opposite pairings stable at high background energy; these stability properties are said to explain the vortex-topography correlations seen across the simulations.

Significance. If the empirical construction is shown to satisfy the stationary condition and the stability results are confirmed, the work supplies explicit, reproducible vortex solutions for topographic turbulence final states and a mechanistic account of preferred vortex-topography alignments. Such solutions would be useful for initializing or parameterizing oceanic models over rough bathymetry and for interpreting satellite or drifter observations of coherent vortices.

major comments (2)
  1. [Empirical model section] Empirical model section: the assertion that the superposition of the linear background flow, sinusoidal topography, and Gaussian vortex streamfunctions produces locally stationary solutions requires an explicit verification that the composite potential vorticity q remains a single-valued function of the composite streamfunction ψ near the extrema. The background satisfies a linear relation by construction and the isolated vortices exhibit a sinh-like trend, but the Laplacian of the added Gaussian profile plus the topographic term h does not automatically preserve a functional relation q = F(ψ); the manuscript should report a quantitative check (e.g., scatter-plot collapse or maximum deviation from a fitted F) for the composite fields.
  2. [Stability analysis section] Stability analysis section: the reported background-energy dependence of stability (cyclone/elevation stable at low energy, anticyclone/elevation stable at high energy) is load-bearing for the explanation of observed correlations, yet the precise nondimensional energy thresholds separating the two regimes are not stated. The manuscript should define these thresholds in terms of the nondimensional parameters used in the linear stability eigenvalue problem and show that the transition occurs within the range of energies realized in the turbulence simulations.
minor comments (2)
  1. Figure captions should explicitly state the nondimensional parameters (e.g., topographic amplitude, Reynolds number, initial energy) corresponding to each panel so that the reader can map the displayed fields to the stability regimes discussed in the text.
  2. The phrase “locally stationary solutions” is used repeatedly; a brief parenthetical definition or reference to the precise diagnostic (e.g., residual of the stationary PV equation) would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our empirical model and stability results. We address each major comment below and will incorporate the suggested clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [Empirical model section] Empirical model section: the assertion that the superposition of the linear background flow, sinusoidal topography, and Gaussian vortex streamfunctions produces locally stationary solutions requires an explicit verification that the composite potential vorticity q remains a single-valued function of the composite streamfunction ψ near the extrema. The background satisfies a linear relation by construction and the isolated vortices exhibit a sinh-like trend, but the Laplacian of the added Gaussian profile plus the topographic term h does not automatically preserve a functional relation q = F(ψ); the manuscript should report a quantitative check (e.g., scatter-plot collapse or maximum deviation from a fitted F) for the composite fields.

    Authors: We agree that an explicit quantitative verification strengthens the claim that the composite fields yield locally stationary solutions. Although the background satisfies a linear relation by construction and the isolated vortices follow a sinh-like relation, the superposition with the Gaussian profile and topography does not automatically guarantee a single-valued q(ψ) everywhere. In the revised manuscript we will add a scatter plot of composite q versus ψ in the vicinity of each topographic extremum, together with the maximum deviation from a fitted functional form (linear for the background component and sinh-like for the vortex component). This will quantify the accuracy of the empirical construction as a locally stationary solution. revision: yes

  2. Referee: [Stability analysis section] Stability analysis section: the reported background-energy dependence of stability (cyclone/elevation stable at low energy, anticyclone/elevation stable at high energy) is load-bearing for the explanation of observed correlations, yet the precise nondimensional energy thresholds separating the two regimes are not stated. The manuscript should define these thresholds in terms of the nondimensional parameters used in the linear stability eigenvalue problem and show that the transition occurs within the range of energies realized in the turbulence simulations.

    Authors: We acknowledge that explicit thresholds improve the connection between the linear stability analysis and the simulation results. In the revised manuscript we will define the critical nondimensional background energies (expressed in terms of the parameters of the eigenvalue problem, such as the nondimensional topographic amplitude and the background flow strength) that separate the two stability regimes. We will also show that these thresholds fall within the range of background energies attained in the freely decaying turbulence simulations, thereby confirming that the observed vortex-topography correlations are consistent with the stability properties. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs an empirical model by superposing a linear background flow (which satisfies a linear PV-streamfunction relation by design) with Gaussian vortex profiles motivated by flat-bottom observations and simulation data. It then verifies that this composite approximately satisfies the inviscid stationary condition locally and performs independent linear stability analysis on the resulting solutions, comparing outcomes to separate numerical simulations. No load-bearing step reduces by the paper's own equations to a fitted parameter renamed as a prediction, nor does any central claim rely on a self-citation chain or imported ansatz that is itself unverified. The derivation remains self-contained against external simulation benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard 2D fluid equations plus the empirical separation into linear background flow and Gaussian vortices; no new physical entities are postulated.

free parameters (1)
  • Gaussian vortex amplitude and width parameters
    The empirical model requires specific scales for the Gaussian profiles that are chosen or fitted to match observed vortex structures in simulations.
axioms (2)
  • standard math The governing equations are the inviscid 2D Euler (or quasi-geostrophic) equations with topography entering the potential vorticity.
    Invoked throughout as the underlying dynamical system for both turbulence evolution and stationary solutions.
  • domain assumption Final states of freely decaying topographic turbulence consist of a linear background flow plus localized vortices locked to topographic extrema.
    Stated explicitly in the abstract as the compositional structure of the quasi-stationary states under study.

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