A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part I. Flows with no bounding walls

Abstract

An exact relation is derived between scalar dissipation due to molecular diffusivity and the randomness of stochastic Lagrangian trajectories for flows without bounding walls. This "Lagrangian fluctuation-dissipation relation" equates the scalar dissipation for either passive or active scalars to the variance of scalar inputs associated to initial scalar values and internal scalar sources, as those are sampled backward in time by the stochastic Lagrangian trajectories. As an important application, we reconsider the phenomenon of "Lagrangian spontaneous stochasticity" or persistent non-determinism of Lagrangian particle trajectories in the limit of vanishing viscosity and diffusivity. Previous work on the Kraichnan (1968) model of turbulent scalar advection has shown that anomalous scalar dissipation is associated in that model to Lagrangian spontaneous stochasticity. There has been controversy, however, regarding the validity of this mechanism for scalars advected by an actual turbulent flow. We here completely resolve this controversy by exploiting the fluctuation-dissipation relation. For either a passive or active scalar advected by any divergence-free velocity field, including solutions of the incompressible Navier-Stokes equation, and away from walls, we prove that anomalous scalar dissipation requires Lagrangian spontaneous stochasticity. For passive scalars we prove furthermore that spontaneous stochasticity yields anomalous dissipation for suitable initial scalar fields, so that the two phenomena are there completely equivalent. These points are illustrated by numerical results from a database of homogeneous, isotropic turbulence, which provide both additional support to the results and physical insight into the representation of diffusive effects by stochastic Lagrangian particle trajectories. arXiv:1606.00729v3 [physics.flu-dyn]

Circulation and Energy Theorem Preserving Stochastic Fluids

“…We briefly recall some results connected to the formulation (34), (35). First, a different perspective on the Constantin-Iyer-Kelvin theorem was explored by Eyink in [11], where it is shown that (34) arises as a consequence of Noether's theorem via the particle relabelling symmetry of a certain stochastic action 1 Rather than introduce the back-to-labels map, the Constantin-Iyer Kelvin theorem can also be naturally stated in terms of time-reversed Brownian motion and backwards Itô SDEs [12]. For detailed discussions of backward stochastic flows, see [16,17].…”

“…principle for the deterministic incompressible Navier-Stokes equations. See also [12] for a reformulation of Navier-Stokes as a system of stochastic Hamilton's equations, which yield a particularly simple derivation of the statistical Kelvin theorem. This formulation has been since extended to domains with solid boundary [13] and to a Riemannian manifold when the de Rham-Hodge Laplacian is the viscous dissipation operator [14].…”

“…First, a different perspective on the Constantin-Iyer-Kelvin theorem was explored by Eyink in [20], where it is shown that (2.31) arises as a consequence of Noether's theorem via the particle relabeling symmetry of a certain stochastic action principle for the deterministic incompressible Navier-Stokes equations. See also [16] for a reformulation of Navier-Stokes as a system of stochastic Hamilton's equations, which yield a particularly simple derivation of the statistical Kelvin theorem. This formulation has since been extended to domains with solid boundary [10] and to a Riemannian manifold when the de Rham-Hodge Laplacian is the viscous dissipation operator [66].…”

Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence

“…It was shown by those authors in a synthetic model of turbulence that the Lagrangian trajectories become "spontaneously stochastic" in the high Reynoldsnumber limit, with randomness of trajectories persisting even when the initial particle location and the advecting velocity become deterministic and perfectly specified. It has subsequently been shown that such "spontaneous stochasticity" of Lagrangian particle trajectories holds at Burgers shocks [111] and is necessary in incompressible Navier-Stokes turbulence for anomalous dissipation of passive scalars [112,113]. These considerations carry over directly to relativistic fluid world-lines X µ (X 0 , τ ) defined by the equations…”

Stochastic Lagrangian Formulations for Damped Navier-Stokes Equations and Boussinesq System, with Applications

“…if and only if u solves the viscous and forced Burgers' equation (2.9). After this manuscript was completed, the author was pointed out that this also follows from an application of the work by Drivas and Eyink in [17].…”