The Batchelor spectrum of passive scalar turbulence in stochastic fluid mechanics at fixed Reynolds number

Abstract

In 1959, Batchelor predicted that passive scalars advected in fluids at finite Reynolds number with small diffusivity κ should display a |k| −1 power spectrum over a small-scale inertial range in a statistically stationary experiment. This prediction has been experimentally and numerically tested extensively in the physics and engineering literature and is a core prediction of passive scalar turbulence.In this article we provide the first mathematically rigorous proof of Batchelor's prediction on the cumulative power spectrum in the κ → 0 limit. We consider fluids governed by the 2D Navier-Stokes equations and 3D hyperviscous Navier-Stokes equations in T d forced by sufficiently regular, nondegenerate stochastic forcing at fixed (arbitrary) Reynolds number. The scalar is subjected to a smooth-inspace, white-in-time stochastic source, and evolves by advection-diffusion with diffusivity κ > 0. Our results rely on the quantitative understanding of Lagrangian chaos and passive scalar mixing established in our recent works. In the κ → 0 limit, we obtain statistically stationary, weak solutions in H − to the stochastically-forced advection problem without diffusivity. These solutions are almost-surely not locally integrable functions, have a non-vanishing average anomalous flux, and satisfy the Batchelor spectrum at all sufficiently small scales. We also prove an Onsager-type criticality result which shows that no such dissipative, weak solutions with a little more regularity can exist.

Ergodicity and invariant measures for a diffusing passive scalar advected by a random channel shear flow and the connection between the Kraichnan-Majda model and Taylor-Aris Dispersion

“…Additionally they help to provide a basis for understanding problems in fluid turbulence. For example, the k −1 small scale power spectrum a scalar field inherits from a turbulent flow, as predicted by Batchelor [7], has recently been rigorously established in a passive scalar model with velocities taken from randomly driven Navier-Stokes equations [4]. Moreover, they provide insight into intermittency in fluid turbulence whereby higher statistical moments deviate strongly from Gaussianity.…”

Lower bounds on the Lyapunov exponents of stochastic differential equations

“…One can show that this result is essentially optimal up to getting sharper quantitative estimates on µ and D, at least if κ = 0 [18,19]. This uniform, exponential mixing plays the key role in obtaining a proof of Batchelor's power spectrum [12] of passive scalar turbulence in some regimes [15]. Let us simply comment on the Lagrangian chaos statement (5.2), as it is most closely related to the rest of this note.…”

Quantitative spectral gaps and uniform lower bounds in the small noise limit for Markov semigroups generated by hypoelliptic stochastic differential equations

“…It is clear that any results will be deeply tied to the topology, for example, for Batchelor-regime passive scalar turbulence, in fluctuation dissipation scaling as in (1.3), µ ǫ ⇀ δ 0 in H s for s < 1 and µ ǫ ⇀ 0 for H s for s > 1 (losing all mass to infinity); one requires a different scaling to capture non-trivial dynamics [9,10]. In the hypoelliptic setting at least, there is no known, reasonable reference measure with respect to which one can study the stationary density, and even in the case of non-degenerate forcing, it is unclear what estimates could be expected for systems such as (1.3) and the methods for proving any such estimates are essentially non-existent at the current time.…”