arxiv: 2605.08244 · v1 · submitted 2026-05-07 · ⚛️ physics.ao-ph · nlin.CD· physics.flu-dyn

Recognition: 2 theorem links

· Lean Theorem

Growth of small localized perturbations in Surface Quasi-Geostrophic turbulence

F. De Lillo, G. Boffetta, M. Cencini, S. Musacchio, V.J. Valad\~ao

Pith reviewed 2026-05-12 00:53 UTC · model grok-4.3

classification ⚛️ physics.ao-ph nlin.CDphysics.flu-dyn

keywords surface quasi-geostrophic turbulencelocalized perturbationsbutterfly effectgeophysical flowschaotic systemspredictability

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The pith

In Surface Quasi-Geostrophic turbulence, small localized perturbations decrease in energy for a variable time before growing.

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

The paper studies how tiny, spatially confined disturbances evolve in Surface Quasi-Geostrophic turbulence, a simplified model of mesoscale flows in the ocean and atmosphere under strong rotation and stratification. It establishes that these disturbances do not grow right away as they would in low-dimensional chaos; instead their energy often falls during an initial transient whose length varies widely with the disturbance's starting position and can last many small-scale turnover times. A reader would care because this variability directly shapes how forecast errors spread in large-scale geophysical systems that contain many interacting scales. The finding replaces the simple picture of immediate exponential divergence with a more location-dependent and time-delayed growth process.

Core claim

The evolution of a spatially localized infinitesimal perturbation in SQG turbulence exhibits strong variability, with an initial transient regime in which the perturbation energy decreases. The duration of this transient is broad and can persist for several small-scale characteristic times, depending on the initial location of the perturbation.

What carries the argument

Numerical integration of the Surface Quasi-Geostrophic equations applied to spatially localized infinitesimal perturbations.

If this is right

Error growth rates in mesoscale geophysical turbulence depend on the spatial location where the error is introduced.
Ensemble forecasts must incorporate the possibility of an initial decay phase whose length varies across the domain.
The overall predictability time scale is set by the longest transients rather than by the fastest local growth rates.

Where Pith is reading between the lines

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

The same location-dependent transient may appear in other stratified rotating flows whose equations share the same leading-order balance.
Observations that track the spread of small-scale tracers or model errors in real ocean or atmosphere data could reveal whether such transients occur outside idealised simulations.

Load-bearing premise

The numerical solutions of the SQG equations capture the true physical evolution without being dominated by resolution limits or details of the forcing.

What would settle it

A high-resolution SQG simulation in which the energy of a localized perturbation is tracked from many different starting positions; the claim is falsified if the energy grows immediately from every position.

Figures

Figures reproduced from arXiv: 2605.08244 by F. De Lillo, G. Boffetta, M. Cencini, S. Musacchio, V.J. Valad\~ao.

**Figure 1.** Figure 1: One example of the turbulent scalar fields at 𝑅𝑒 = 25400 at time 𝑡𝜆 = 1.22 after the injection of the small perturbation of size 𝜎 = 0.22𝐿𝑇 in two different locations. The regions where the perturbation exceeds the threshold 10−5 𝜃𝑟𝑚𝑠 are shown in black [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: a) Time evolution of the error energy 𝐸𝛥 normalized with the initial error for the two cases shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Cumulative distribution function 𝐹(𝜏𝑅) of the return time 𝜏𝑅 for the simulation at 𝑅𝑒 = 15900 and a different initial size of the error. Distributions are computed on many realizations ranging from 190 to 375 from smaller to larger 𝜎, respectively. An example of the statistics obtained from these numerical experiments for the case at lower Re is shown in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: a) Mean return time ⟨𝜏𝑅⟩ as a function of the perturbation width 𝜎 for simulations at different Reynolds numbers. b) Mean return time rescaled with the Lyapunov exponent 𝜆 as a function of the perturbation width 𝜎 made dimensionless with 𝐿𝑇 . Colors and symbols as in a). The dashed line represents −1.81 log(𝜎/𝐿𝑇 ) −1.21. Inset: the minimum error energy normalized with the initial one as a function of pertu… view at source ↗

**Figure 5.** Figure 5: Initial error energy spectra 𝐸𝛥(𝑘, 0) (dashed lines) and error energy spectra at the final time 𝐸𝛥(𝑘, 𝑇𝑓 ) (continuous line) for initial error sizes 𝜎 = 0.0015, 𝜎 = 0.0031, 𝜎 = 0.0046, 𝜎 = 0.0077, 𝜎 = 0.012 (from red to black). For clarity, the energy spectra at the final time are multiplied by a factor 103 . The black dotted line represents the energy spectrum 𝐸(𝑘) of the field. universal shape peaked at … view at source ↗

read the original abstract

The ``butterfly effect'', i.e. the growth of a localized infinitesimal perturbation, is the fundamental property of chaotic systems. While the butterfly effect is today an obvious property of low-dimensional chaotic systems, its significance is more nuanced in extended systems with many spatial and temporal scales, such as geophysical flows. In this Letter we explore the butterfly effect, i.e., the fate of infinitesimal localized perturbations, in the Surface-Quasi-Geostrophic turbulence, a minimal model for mesoscale geophysical turbulence in the regime of strong stratification and rotation. We find that the evolution of a spatially localized perturbation exhibits strong variability, with an initial transient regime in which the perturbation energy decreases. The duration of this transient is broad and can persist for several small-scale characteristic times, depending on the initial location of the perturbation.

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.

Desk Editor's Note private letter to a colleague

Localized perturbations in SQG turbulence show an initial energy drop whose duration varies with position, but this may be a numerical artifact from dissipation.

read the letter

The main point is that in surface quasi-geostrophic turbulence a small localized perturbation can lose energy for a while before it grows, and the length of that initial drop depends on where you place it in the flow. The paper documents this through direct simulations of the SQG equations and tracks how perturbation energy evolves from different starting spots. The location dependence and the existence of the transient itself are the concrete observations here, and they refine the usual butterfly-effect picture for this class of geophysical model. That is new enough to note, especially since the abstract ties it to mesoscale predictability questions. The work itself is a straightforward numerical exploration that stays focused on the perturbation dynamics without extra claims. The simulations appear to be set up in a standard way for SQG turbulence studies, and the abstract states the result plainly. The soft spot is the possibility that the early energy decrease is not physical. A localized perturbation carries a broad wavenumber spectrum, and the hyperviscosity or dealiasing routinely used in these runs damps the high wavenumbers at early times. If the paper does not show resolution-doubling tests or explicit checks that the transient survives when the effective Reynolds number is raised, then the central observation rests on untested numerics. The stress-test concern lands directly on the result and cannot be dismissed from the abstract alone. This paper is for people who work on predictability in stratified rotating turbulence, particularly those using reduced models like SQG for ocean or atmosphere applications. A reader already familiar with the equations will see the variability quantified and can judge its implications for error growth. It deserves a serious referee because the topic connects to practical modeling questions and the claim is specific enough to be checked. I would send it to review, but with a request that the authors add clear evidence that the transient is robust to changes in resolution and dissipation.

Referee Report

1 major / 0 minor

Summary. The manuscript explores the butterfly effect in Surface Quasi-Geostrophic (SQG) turbulence by tracking the evolution of infinitesimal, spatially localized perturbations. It reports strong variability in this evolution, including an initial transient regime in which perturbation energy decreases; the duration of the transient is broad, can span several small-scale eddy turnover times, and depends on the initial location of the perturbation within the turbulent flow.

Significance. If the reported transients are physical rather than numerical, the result refines understanding of predictability in multi-scale geophysical turbulence: even in a chaotic system the growth of small errors is not immediate but can be delayed by location-dependent filtering of the perturbation spectrum. This has potential implications for ensemble forecasting and error-growth diagnostics in stratified, rotating flows. The work is a direct numerical exploration of a minimal model, which is a strength when accompanied by appropriate convergence checks.

major comments (1)

The central claim that the initial energy decrease and its location-dependent duration are intrinsic features of SQG dynamics requires that numerical dissipation does not dominate the early evolution. Because a localized perturbation projects onto a broad wavenumber spectrum, hyperviscosity or dealiasing filters can preferentially damp high-k components at early times, producing an apparent energy drop before nonlinear transfer takes over. The manuscript should therefore document resolution-doubling tests, hyperviscosity coefficient sweeps, and explicit checks that the transient duration remains statistically unchanged when the effective Reynolds number is increased.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive comment on numerical robustness. We address the major comment in detail below.

read point-by-point responses

Referee: The central claim that the initial energy decrease and its location-dependent duration are intrinsic features of SQG dynamics requires that numerical dissipation does not dominate the early evolution. Because a localized perturbation projects onto a broad wavenumber spectrum, hyperviscosity or dealiasing filters can preferentially damp high-k components at early times, producing an apparent energy drop before nonlinear transfer takes over. The manuscript should therefore document resolution-doubling tests, hyperviscosity coefficient sweeps, and explicit checks that the transient duration remains statistically unchanged when the effective Reynolds number is increased.

Authors: We agree that it is essential to demonstrate that the reported initial energy decrease is not an artifact of numerical dissipation. In the original simulations the hyperviscosity coefficient was chosen so that dissipation acts only at the smallest resolved scales, and the transient occurs on timescales shorter than the dissipative time for the wavenumbers carrying most of the perturbation energy. Nevertheless, to meet the referee’s request explicitly we will revise the manuscript to include (i) resolution-doubling tests between 1024² and 2048² grids, (ii) a sweep of the hyperviscosity coefficient over a factor of four, and (iii) quantitative statistics showing that the duration of the location-dependent transient remains unchanged within sampling uncertainty. These additional diagnostics will be presented in a new figure and accompanying text, together with a brief discussion of the dealiasing filter’s effect on the early spectrum. The revised results continue to support that the transient is a physical feature of SQG dynamics. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical results on perturbation evolution

full rationale

The paper reports outcomes from direct numerical simulations of the SQG equations, focusing on the time evolution of spatially localized infinitesimal perturbations. The reported initial transient energy decrease and its location-dependent duration are presented as direct simulation outputs rather than as predictions derived from a closed analytical chain. No equations, parameters, or uniqueness claims are introduced that reduce by construction to fitted inputs, self-definitions, or self-citations; the work contains no load-bearing self-citation steps or ansatz smuggling. The derivation chain is therefore self-contained as a computational exploration of chaotic dynamics, with findings grounded in the simulated trajectories themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Since only abstract available, ledger is minimal. The paper relies on numerical simulations of the SQG equations.

axioms (1)

domain assumption The SQG model is a valid minimal model for mesoscale geophysical turbulence
Stated in abstract as 'a minimal model for mesoscale geophysical turbulence in the regime of strong stratification and rotation'

pith-pipeline@v0.9.0 · 5459 in / 1017 out tokens · 34763 ms · 2026-05-12T00:53:20.155258+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
The evolution of a spatially localized perturbation exhibits strong variability, with an initial transient regime in which the perturbation energy decreases... depending on the initial location of the perturbation.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
dE_Δ/dt = −ν⟨(∇δθ)²⟩_x − μ⟨(∇⁻¹δθ)²⟩_x − ⟨δθ δu·∇θ⟩_x

Reference graph

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