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Eddies that ride the flow: a model of turbulence with Richardson dispersion and anomalous dissipation
arXiv cs.NA0 views
arXiv:2611.07121v1 Announce Type: cross Abstract: We construct a divergence-free velocity field $\mathbf{u}(\mathbf{x},t)$ on $\mathbb{T}^{n}$ 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 $\zeta(R)=h(R)-1$ threshold by threshold within the scaling range, with a sweeping deficit $\Delta_{\rm sw}=+0.038\pm0.019$, consistent with zero, against $+0.091\pm0.019$ 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 $\kappa_{p}^{0.0333\pm0.0038}$ over 22 decades of particle diffusivity, against $\kappa_{p}^{1/2}$ 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 $S_{L}=-\chi r$ to $0.9995\pm0.0018$ once the measured diffusive transfer is restored, while under an imposed mean gradient $\chi$ changes by 14\% over a factor 5.8 in $\kappa$.
Source:https://arxiv.org/abs/2611.07121PreviousSupport in the domain of numerical algorithms in the education of future secondary school mathematics teachersNextImproving the accuracy of the momentum-exchange method in lattice Boltzmann methods
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