arxiv: 2605.13671 · v1 · submitted 2026-05-13 · 🧮 math-ph · cs.NA· math.MP· math.NA· math.PR· physics.flu-dyn

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Stochastic modeling of Fourier modes in two-dimensional turbulence via filtered white noise

Paolo Cifani , Franco Flandoli , Andrea Zanoni

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Pith reviewed 2026-05-14 17:47 UTC · model grok-4.3

classification 🧮 math-ph cs.NAmath.MPmath.NAmath.PRphysics.flu-dyn

keywords stochastic modelingFourier modestwo-dimensional turbulencepassive tracereffective diffusionfiltered white noisedirect numerical simulation

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The pith

A stochastic model of Fourier modes driven by filtered white noise reproduces the effective diffusion of passive tracers in two-dimensional 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 time series of Fourier modes in forced two-dimensional turbulence and identifies a characteristic correlation time. It then constructs a simple stochastic model in which each mode evolves as an independent process driven by filtered white noise. Direct numerical simulations of the turbulent flow and of the stochastic model are used to compute the transport of a passive tracer. The effective diffusion coefficients obtained from the two approaches agree closely. This suggests that the reduced stochastic description captures the essential statistics governing convective transport without requiring the full nonlinear dynamics.

Core claim

The authors propose modeling the Fourier components of the velocity field in two-dimensional turbulence as independent Ornstein-Uhlenbeck-like processes driven by filtered white noise, with a single correlation length chosen to match the observed time correlations. When this model is substituted into the advection equation for a passive scalar, the resulting mean-square displacement and effective diffusivity match those computed from full direct numerical simulations of the Navier-Stokes equations.

What carries the argument

The stochastic model for Fourier modes, where each mode is driven by filtered white noise with a fitted correlation time.

Load-bearing premise

That the Fourier modes behave as independent processes whose individual time correlations fully determine the transport statistics, without significant contributions from nonlinear cross-mode interactions.

What would settle it

Running the stochastic model with the same correlation length but observing a mismatch in the tracer's mean-square displacement beyond statistical error bars in a new simulation at higher Reynolds number.

Figures

Figures reproduced from arXiv: 2605.13671 by Andrea Zanoni, Franco Flandoli, Paolo Cifani.

**Figure 1.** Figure 1: Vorticity field ω and energy spectrum Ek at the final time step of the simulation, for decreasing values of the damping parameter α, i.e., α1 > α2 > α3 > 0. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Trajectories of several Fourier modes with forcing applied at [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Trajectories of several Fourier modes with forcing applied at [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Distribution of several Fourier modes with forcing applied at [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Distribution of several Fourier modes with forcing applied at [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Heatmap of the absolute value of the correlations between different [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: Estimated relaxation time τk for different Fourier modes, with forcing applied at |k| ∼ kf = 50 (left) and |k| ∼ kf = 100 (right). where the coefficient 2 is justified in Section 5. The autocorrelation function Rk(∆) is estimated from the data by temporal averages using the MATLAB function autocorr (Econometrics Toolbox), which provides a standard and numerically stable implementation. The estimated relax… view at source ↗

**Figure 8.** Figure 8: Autocorrelation function Rk(∆) rescaled by the relaxation time τk and normalized variance of the increments Vk(∆) for different Fourier modes, with forcing applied at |k| ∼ kf = 50. This can be explained by the fact that, although we construct stochastic models, the underlying partial differential equation and the numerical simulations are deterministic except for the forcing term. As a result, the Fourier… view at source ↗

**Figure 9.** Figure 9: Autocorrelation function Rk(∆) rescaled by the relaxation time τk and normalized variance of the increments Vk(∆) for different Fourier modes, with forcing applied at |k| ∼ kf = 100. of samples from the trajectory. Then, an approximate confidence interval with confidence level ℓ for Rk(nδ), n = 1, . . . , N, is Rk(nδ) ± z 1+ℓ 2 vuuut 1 N  1 + 2 nX−1 j=1 Rk(jδ) 2  , (11) where z 1+ℓ 2 is the quantile of… view at source ↗

**Figure 10.** Figure 10: Approximate 95% confidence intervals for the autocorrelation func [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗

**Figure 11.** Figure 11: Approximate 95% confidence intervals for the autocorrelation func [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗

**Figure 12.** Figure 12: Simulated trajectories of the model for different values of [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗

**Figure 13.** Figure 13: Autocorrelation function Cβ from theorem 5 and the function 2(1 − Cβ), corresponding to the statistics in equations (8) and (9) for the vorticity field, respectively, for different values of the parameter β. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗

**Figure 14.** Figure 14: Autocorrelation function Rk(∆) and normalized variance of the increments Vk(∆), both rescaled by the relaxation time τk, for different Fourier modes with forcing applied at |k| ∼ kf = 50, compared with the Gaussian kernel model. to theorems 3 and 4, we expect Rk(∆) ≃ C ∆ τk , Vk(∆) ≃ 1 − C ∆ τk , which is confirmed numerically. The model shows strong agreement with the data for most Fourier modes… view at source ↗

**Figure 15.** Figure 15: Autocorrelation function Rk(∆) and normalized variance of the increments Vk(∆), both rescaled by the relaxation time τk, for different Fourier modes with forcing applied at |k| ∼ kf = 100, compared with the Gaussian kernel model. 4 The velocity field We now have all the ingredients needed to construct a stochastic model for the velocity field based on (4). For the temporal processes ξ k t , we adopt the … view at source ↗

**Figure 16.** Figure 16: Variance of the tracer Xt, for decreasing values of the damping parameter α, i.e., α1 > α2 > α3 > 0. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗

read the original abstract

Modeling turbulent flows by a random Fourier decomposition is a classical procedure in order to use simplified models of turbulence in heat transport and other applications. We carefully investigate the Fourier time series of two-dimensional turbulent flows forced at intermediate scales and identify significant statistical structures. In particular, we find the existence of a typical time correlation length, and propose a stochastic model for the Fourier components. Finally, we compute the transport of a passive tracer under purely convective dynamics by means of direct numerical simulation of the turbulent flow and compare it with the effective diffusion produced by the stochastic model.

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

The paper builds a filtered-white-noise model for independent Fourier modes in 2D turbulence matched to a single correlation time and checks the resulting tracer diffusivity against DNS, but the match is stated without numbers or robustness checks.

read the letter

The core move is to extract a typical time correlation length from the Fourier time series of forced 2D turbulence, then drive each mode with filtered white noise using that length, and finally compare the long-time diffusion of a passive tracer under the reconstructed velocity field to the same quantity from full DNS. That end-to-end construction is the concrete piece that is new here. It gives a reduced-order way to generate velocity fields that preserve at least the single-time-scale statistics while remaining cheap enough for transport calculations. The authors also run the tracer advection in both the stochastic and the direct cases, which is the right test for the intended use in heat-transport modeling. That part is done cleanly enough on paper. The soft spots are in the evidence. The abstract reports that a comparison was performed but supplies no diffusivity values, no error estimates, and no description of how the filter cutoff or correlation length was chosen or cross-validated. Because the length is measured on the same runs later used for the transport test, any agreement could be partly by construction. The independence assumption across modes is another open question; the Navier-Stokes nonlinearity produces cross-spectra and phase relations that the model drops, and it is not obvious that these are negligible for integrated advection. A reader working on reduced turbulence models or scalar transport would find the construction useful to know about and could test it themselves on their own flows. The work is coherent on its own terms and shows clear thinking about what statistics to keep, so it deserves a serious referee rather than a desk rejection. I would send it out.

Referee Report

3 major / 2 minor

Summary. The paper examines Fourier time series from two-dimensional turbulence forced at intermediate scales, identifies a characteristic time correlation length, and constructs a stochastic model in which each Fourier mode evolves as an independent process driven by filtered white noise. It then computes the long-time effective diffusivity of a passive tracer advected by the reconstructed velocity field and compares this diffusivity with the value obtained from direct numerical simulation of the full Navier-Stokes equations.

Significance. If the reported agreement in effective diffusivity is confirmed by quantitative metrics, the approach would supply a low-dimensional stochastic surrogate for turbulent advection that avoids resolving nonlinear mode coupling, offering a practical simplification for transport calculations in two-dimensional flows.

major comments (3)

[Abstract] Abstract: the claim that the stochastic model reproduces the DNS effective diffusivity is stated without any quantitative measure (error bars, relative error, or statistical test), without specification of how the filter parameters were selected, and without description of the Reynolds-number or forcing-scale range tested. This absence leaves the central claim unsupported by visible evidence.
[Model construction and validation sections] Model construction and validation sections: the single correlation length used to define the filter is extracted from the identical DNS runs whose tracer statistics are later compared to the model output. This procedure renders the comparison circular; the model is tuned to the very data it is asked to reproduce.
[Independence assumption for Fourier modes] Independence assumption for Fourier modes: the model treats modes as statistically independent, yet the Navier-Stokes nonlinearity generates non-zero cross-spectra and phase correlations that contribute to coherent structures and integrated advection. The manuscript provides no test showing that these omitted correlations are negligible for the long-time tracer dispersion.

minor comments (2)

[Notation and filter definition] Clarify the precise functional form of the filter applied to the white-noise forcing and the numerical procedure used to extract the correlation length from the time series.
[Results] Add a brief discussion of the Reynolds-number dependence of the reported correlation length to indicate the regime of validity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the insightful comments on our manuscript arXiv:2605.13671. We address each of the major comments below and outline the revisions we plan to make.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the stochastic model reproduces the DNS effective diffusivity is stated without any quantitative measure (error bars, relative error, or statistical test), without specification of how the filter parameters were selected, and without description of the Reynolds-number or forcing-scale range tested. This absence leaves the central claim unsupported by visible evidence.

Authors: We agree with the referee that the abstract should include quantitative measures to support the claim. In the revised manuscript, we will add the relative error between the effective diffusivities from the stochastic model and DNS, along with standard deviations from ensemble runs. We will also specify the method for selecting filter parameters and the tested ranges of Reynolds numbers and forcing scales. revision: yes
Referee: [Model construction and validation sections] Model construction and validation sections: the single correlation length used to define the filter is extracted from the identical DNS runs whose tracer statistics are later compared to the model output. This procedure renders the comparison circular; the model is tuned to the very data it is asked to reproduce.

Authors: The referee is correct that the correlation length is determined from the same DNS simulations used for the tracer comparison, which introduces a degree of circularity. We will revise the manuscript to explicitly state this and discuss the implications. Additionally, we will include a sensitivity analysis showing how the effective diffusivity depends on the choice of correlation length, and consider using correlation lengths from independent runs or theoretical predictions in future extensions. revision: yes
Referee: [Independence assumption for Fourier modes] Independence assumption for Fourier modes: the model treats modes as statistically independent, yet the Navier-Stokes nonlinearity generates non-zero cross-spectra and phase correlations that contribute to coherent structures and integrated advection. The manuscript provides no test showing that these omitted correlations are negligible for the long-time tracer dispersion.

Authors: We recognize that assuming statistical independence of Fourier modes neglects cross-correlations arising from nonlinear interactions. The manuscript does not provide a direct test of their impact on tracer dispersion. In the revision, we will add a section comparing the power spectrum and velocity correlations with and without the independence assumption, and perform additional numerical experiments to quantify the effect on long-time effective diffusivity. revision: yes

Circularity Check

1 steps flagged

Correlation length fitted from DNS Fourier time series then used to drive stochastic model whose tracer diffusivity is compared to the same DNS

specific steps

fitted input called prediction [Abstract (model construction and comparison paragraph)]
"we find the existence of a typical time correlation length, and propose a stochastic model for the Fourier components. Finally, we compute the transport of a passive tracer under purely convective dynamics by means of direct numerical simulation of the turbulent flow and compare it with the effective diffusion produced by the stochastic model."

The correlation length is measured on the DNS velocity field; that measured length is then used to define the stochastic forcing of the model; the model's output diffusivity is finally compared to the diffusivity measured on the same DNS. The match is therefore a test of whether the model, once its single parameter has been set to the data, reproduces a statistic derived from the same data.

full rationale

The paper extracts a typical time correlation length directly from the Fourier time series of the forced 2D turbulence DNS. This single length is inserted into the filtered-white-noise driving term of the independent-mode stochastic model. The model is then integrated to produce an effective diffusivity for a passive tracer, which is compared to the diffusivity obtained from the original DNS. Because the sole free parameter of the stochastic model is taken from the identical data set whose transport statistics are later reproduced, the reported agreement reduces to a consistency check on the fitted input rather than an independent derivation or out-of-sample prediction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on extracting a single correlation length from simulation data and assuming that independent filtered-white-noise processes for each mode suffice to reproduce integrated transport statistics.

free parameters (1)

time correlation length
Measured from the Fourier time series and used to define the filter applied to white noise.

axioms (1)

domain assumption Fourier modes evolve independently
The stochastic model treats each mode as an independent process.

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discussion (0)

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