Eddies that ride the flow: a model of turbulence with Richardson dispersion and anomalous dissipation
Abstract
We construct a divergence-free velocity field u(x,t) on 𝕋n that is a genuine function of position and carries both defining properties of turbulent transport: Richardson pair dispersion and a dissipative anomaly. No synthetic field has carried both, because the frame that gives the first destroys the field the second requires. The key is a dichotomy: deforming a spectrally localised pattern smears it across scales, whereas pushing its generating centres forward and re-convolving leaves the spectral envelope exactly invariant --- so transporting the eddy centres preserves both cascade and frame. In the field the finite-size Lyapunov exponent obeys ζ(R)=h(R)−1 threshold by threshold within the scaling range, with a sweeping deficit Δsw=+0.038±0.019, consistent with zero, against +0.091±0.019 for the frozen field on the same data, a gap that doubles at Navier--Stokes intermittency. Companion papers prove that this family shows Richardson dispersion and a dissipative anomaly (Fefferman and Mayboroda 2026), and that a scalar advected by this flow obeys Yaglom's relation (De Lellis 2026), at small Kubo number. Trajectories forget their starting point as the smallest eddy is removed: the median separation of pairs released from one point with independent Brownian kicks varies as κp0.0333±0.0038 over 22 decades of particle diffusivity, against κp1/2 for a smooth control. A scalar in the same field closes the exact scale-by-scale variance budget, four independently measured terms balancing the dissipation to 15\% over 1.2 decades with no adjustable constant, and reaches its inertial--convective asymptote SL=−χr to 0.9995±0.0018 once the measured diffusive transfer is restored, while under an imposed mean gradient χ changes by 14\% over a factor 5.8 in κ.
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BibTeX
@misc{jimenez2026eddies,
title = {{Eddies that ride the flow: a model of turbulence with Richardson dispersion and anomalous dissipation}},
author = {Raul Jimenez and Svitlana Mayboroda and Camillo de Lellis and David N. Spergel and Max Engelstein},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07121},
eprint = {2611.07121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07121},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Eddies that ride the flow: a model of turbulence with Richardson dispersion and anomalous dissipation
AU - Raul Jimenez
AU - Svitlana Mayboroda
AU - Camillo de Lellis
AU - David N. Spergel
AU - Max Engelstein
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07121
UR - https://arxiv.org/abs/2611.07121
AB - We construct a divergence-free velocity field u(x,t) on 𝕋^n that is a genuine function of position and carries both defining properties of turbulent transport: Richardson pair dispersion and a dissipative anomaly. No synthetic field has carried both, because the frame that gives the first destroys the field the second requires. The key is a dichotomy: deforming a spectrally localised pattern smears it across scales, whereas pushing its generating centres forward and re-convolving leaves the spectral envelope exactly invariant --- so transporting the eddy centres preserves both cascade and frame. In the field the finite-size Lyapunov exponent obeys ζ(R)=h(R)−1 threshold by threshold within the scaling range, with a sweeping deficit Δ_sw=+0.038±0.019, consistent with zero, against +0.091±0.019 for the frozen field on the same data, a gap that doubles at Navier--Stokes intermittency. Companion papers prove that this family shows Richardson dispersion and a dissipative anomaly (Fefferman and Mayboroda 2026), and that a scalar advected by this flow obeys Yaglom's relation (De Lellis 2026), at small Kubo number. Trajectories forget their starting point as the smallest eddy is removed: the median separation of pairs released from one point with independent Brownian kicks varies as κ_p^0.0333±0.0038 over 22 decades of particle diffusivity, against κ_p^1/2 for a smooth control. A scalar in the same field closes the exact scale-by-scale variance budget, four independently measured terms balancing the dissipation to 15\% over 1.2 decades with no adjustable constant, and reaches its inertial--convective asymptote S_L=−χr to 0.9995±0.0018 once the measured diffusive transfer is restored, while under an imposed mean gradient χ changes by 14\% over a factor 5.8 in κ.
ER -