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Richardson dispersion and anomalous dissipation for the random velocity model with controlled sweeping

This paper proves that a random Hölder continuous velocity field with controlled sweeping exhibits Richardson dispersion, characterized by a cubic growth in mean squared particle separation ( $\E R_t^2 \sim t^3$ ), and induces anomalous dissipation for associated tracers.

Original author: Charles Fefferman, Svitlana Mayboroda

Published 2026-11-06

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CC0 1.0

Original author: Charles Fefferman, Svitlana Mayboroda

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Turbulence is the chaotic swirl of fluid that we see in smoke rising from a cigarette, water churning around a boat's propeller, or wind whipping through a canyon. For over a century, scientists have tried to understand how tiny eddies within this chaos interact with one another. A central puzzle is how two particles, starting very close together, drift apart as they are carried by the flow. In 1926, a scientist named Lewis Richardson proposed a specific rule for this separation: as time passes, the distance between the two particles should grow at a rate proportional to the cube of the time. This means that if you wait twice as long, the particles will be eight times farther apart, not just double. While this idea has long been accepted in physics and supported by computer simulations, it has never been rigorously proven with mathematics for a realistic, random flow. The difficulty lies in the fact that real turbulence is not smooth; it is rough and unpredictable, making standard mathematical tools fail.

In a new paper, mathematicians Charles Fefferman and Svitlan Mayboroda have finally provided that rigorous proof for a specific, carefully constructed model of random fluid motion. They did not study a real ocean or atmosphere, but rather a mathematical simulation of a velocity field—a map of how fast and in what direction the fluid moves at every point. Their model is designed to mimic the essential features of real turbulence, particularly a phenomenon known as "controlled sweeping." In real fluids, large swirling motions carry smaller ones along with them, like a giant wave carrying a small ripple. Many previous mathematical models failed to capture this, treating small eddies as if they evolved in isolation, which led to incorrect predictions about how fast particles separate. Fefferman and Mayboroda built a model where the small eddies are indeed transported by the larger ones, ensuring the physics of the simulation matches the physical reality of how turbulence behaves.

The researchers proved that in their model, the average distance between two particles does indeed follow Richardson's cubic law. They showed that the square of the separation distance grows exactly as time cubed, confirming the prediction for a wide range of starting distances. This result is significant because it bridges the gap between physical intuition and mathematical certainty. Furthermore, they demonstrated that this same flow causes a "tracer"—a substance like dye or heat carried by the fluid—to dissipate its energy in a way that defies standard expectations. Usually, as the fluid becomes less viscous (thinner), the rate at which the tracer mixes and loses energy should slow down and eventually stop. However, in this model, the mixing remains strong and persistent even as the fluid becomes perfectly smooth, a phenomenon known as anomalous dissipation.

To achieve this, the authors constructed a flow that is random in nature but follows a strict hierarchy of scales. Imagine the fluid as being made of many layers of eddies, from very large to very small. In their model, the largest eddies move the medium-sized ones, which in turn move the smallest ones. Crucially, they ensured that the time it takes for these eddies to renew or change is linked to their size in a specific way. This "controlled sweeping" ensures that the small-scale fluctuations are not just random noise but are physically transported by the larger flow, a feature that previous models often missed. By proving that this specific structure leads to the cubic growth law, the authors have shown that the mechanism Richardson described is mathematically sound, provided the flow has the right kind of randomness and structure.

The proof also addresses a long-standing question about the smoothness of turbulent flows. The model they used is continuous but not perfectly smooth; it has a specific type of roughness that matches what physicists expect from real turbulence. This roughness is essential for the particles to separate so rapidly. If the flow were too smooth, the particles would drift apart much more slowly. The authors' work confirms that this specific level of roughness is sufficient to generate the explosive separation predicted by Richardson. They also showed that this behavior is robust, holding true regardless of how small the smallest eddies in the model are, as long as the overall structure of the flow remains the same.

This research does not just confirm an old prediction; it provides a new framework for understanding how chaos emerges from order. By constructing a model where the math works out perfectly, the authors have created a benchmark for future studies. Their findings suggest that the key to understanding turbulence lies in how different scales of motion interact and transport one another. The fact that they could prove the cubic law for a random, rough flow means that the mechanism is not an artifact of a specific, simplified scenario but a fundamental property of how fluids mix when they are turbulent. The work stands as a rigorous validation of a century-old idea, turning a physical intuition into a mathematical fact.

Technical Summary: Richardson Dispersion and Anomalous Dissipation for the Random Velocity Model with Controlled Sweeping

Problem Statement
The paper addresses two fundamental problems in the mathematical theory of turbulence: the rigorous derivation of Richardson's dispersion law and the proof of anomalous dissipation for passive scalars. Richardson's 1926 prediction posits that the mean-square separation of a pair of particles in a turbulent flow grows as $t^3$ ( $E R_t^2 \sim t^3$ ) within the inertial range. While this scaling is physically well-motivated by Kolmogorov's $1/3$ -regularity hypothesis, it has historically evaded rigorous mathematical proof for random velocity fields with finite correlation times. Similarly, anomalous dissipation—the phenomenon where a passive scalar dissipates energy at a rate independent of molecular diffusivity as the diffusivity tends to zero—has been linked to "spontaneous stochasticity" (non-uniqueness of particle trajectories in the zero-noise limit). However, previous rigorous constructions either relied on unphysical white-in-time velocity fields (Kraichnan model) or deterministic, engineered flows that lacked time-homogeneity and realistic scale hierarchies.

A critical physical mechanism often missing in synthetic models is "controlled sweeping": the material transport of small eddies by larger scales. Standard kinematic simulations often fail to capture this, leading to incorrect decorrelation times and dispersion exponents. This paper aims to construct a random, time-homogeneous, geometrically dense velocity field that incorporates controlled sweeping and rigorously proves both Richardson dispersion and anomalous dissipation.

Methodology and Model Construction
The authors construct a random velocity field $u$ on the torus $\mathbb{T}^d$ ( $d \ge 2$ ) with prescribed spatial Hölder regularity $C^\alpha$ ( $0 < \alpha < 1$ ). The construction is a multiscale superposition of wavelet-based shear flows, organized into a hierarchy of scales $j \ge j_0$ with length scales $\ell_j = 2^{-j}$ and amplitudes $A_j = \ell_j^\alpha$ .

Key features of the model include:

Controlled Sweeping: The centers of the wavelets at scale $j$ are not fixed but are transported by the flow of all coarser scales ( $< j$ ). This ensures that small eddies are materially advected by larger ones, a feature previously conjectured to be essential for correct turbulent statistics.
Scale-Dependent Renewal: The velocity field is reset at random times $\tau_j \approx \kappa \ell_j^{1-\alpha}$ , where $\kappa$ is the Kubo number. These renewal times are nested, meaning a renewal of a coarser scale triggers a renewal of all finer scales.
Randomness: The field is generated by independent random shifts and coefficients (with zero mean and symmetric distribution) for each scale and window. The randomness is confined to the velocity ensemble; particle trajectories follow deterministic ODEs for a fixed realization.
Structure Function: The model utilizes a specific family of wavelets (e.g., $\psi(x) = (1-|x|^2)_+^5$ ) satisfying a structural hypothesis (ND) ensuring the non-decreasing nature of the longitudinal structure function, which is crucial for the sign of the dispersion gains.

Key Contributions and Results

Richardson Dispersion Law:
The paper establishes a quantitative lower and upper bound for the mean-square separation $E R_t^2$ of two particles starting at distance $r_0$ . For a sufficiently small Kubo number $\kappa$ , and for times $t$ in the admissible range, the authors prove:
$c(r_0^{1-\alpha} + \kappa t)^{2/(1-\alpha)} \le E R_t^2 \le C(r_0^{1-\alpha} + t)^{2/(1-\alpha)}$
Specifically, when $\alpha = 1/3$ (Kolmogorov scaling), this yields the cubic law $c \kappa^3 t^3 \le E R_t^2 \le C t^3$ . The constants are independent of the small-scale cutoff, demonstrating that the Richardson law holds uniformly as the resolution increases.
Anomalous Dissipation:
The authors prove that for the passive scalar equation $\partial_t \theta + u \cdot \nabla \theta = \mu \Delta \theta$ , the mean dissipation remains strictly positive as the diffusivity $\mu \to 0$ . Specifically, for any mean-zero initial data $\theta_0 \in L^2$ , the expected dissipation $E D_\mu(\theta_0)$ satisfies a lower bound proportional to $\|\theta_0\|^2$ uniformly in $\mu$ (for sufficiently small $\mu$ ). Furthermore, they show that for a single realization of the velocity field, the dissipation is anomalous with positive probability.
Deterministic Counterpart:
By selecting a specific realization from the probability space where the dissipation event occurs, the authors deduce the existence of a deterministic, divergence-free vector field $u \in L^\infty_t C^\alpha_x$ that exhibits anomalous dissipation for specific initial data (e.g., $\cos(2\pi x_1)$ ). This provides a deterministic example of non-uniqueness in the zero-viscosity limit for the scalar equation.

Technical Approach
The proofs rely on a "corrected observable" method. Instead of analyzing the raw separation $R_t^\beta$ directly, the authors construct a modified observable $Q$ that includes correction terms involving the time remaining in the current renewal window and the scale operators.

Evolution between Renewals: Between renewal times, the evolution of $Q$ is governed by a deterministic generator. The authors use commutator estimates to show that the "controlled sweeping" (coarse scales transporting fine scales) results in a gain of order $\kappa$ per unit time, while higher-order error terms are controlled.
Jump Identities: At renewal times, the observable jumps. By averaging over the fresh random data (shifts and coefficients), the first-order terms vanish due to symmetry, leaving a positive second-order term proportional to the structure function.
Accumulation of Gains: The authors sum these gains over time and scales. A key lemma shows that at any time, there is a scale comparable to the current separation that contributes a gain of order $\kappa$ , leading to a linear growth in time for the expected value of the observable.
Duality for Dissipation: To prove anomalous dissipation, the authors use a duality argument relating the scalar dissipation to the separation of a pair of diffusing particles evolving backward in time. They establish a similar Richardson-type growth for the diffusing pair, utilizing exponential moment bounds to control the transfer of mass between separation scales.

Significance and Claims
The paper claims to provide the first rigorous proof of Richardson's $t^3$ dispersion law for a random velocity field that is:

Time-homogeneous: Unlike previous deterministic constructions where the complexity was concentrated near $t=0$ .
Geometrically dense: The scales form a dense hierarchy rather than a lacunary (exponentially separated) one.
Physically consistent: It incorporates the "controlled sweeping" mechanism, where small eddies are transported by large ones, a feature argued to be necessary for correct turbulent statistics.

Furthermore, the work establishes a direct link between the Richardson dispersion law and anomalous dissipation in a random setting, showing that the mechanism driving particle separation (spontaneous stochasticity) is sufficient to drive scalar dissipation even in the absence of molecular diffusion. The results hold for a random field with finite, scale-dependent correlation times, bridging the gap between the idealized white-in-time Kraichnan model and physical turbulence. The authors explicitly note that the requirement for a small Kubo number $\kappa$ may be an artifact of the proof technique rather than a physical necessity, as numerical results in their companion physics paper suggest the law holds more broadly.

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