Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models

Eyink, Gregory L.

doi:10.1063/1.3193681

Cited by 36 publications

(44 citation statements)

References 50 publications

(110 reference statements)

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“…Contrary to the traditional arguments of Taylor & Green (1937), vortex-lines in the ideal limit will not be "frozen-into" the turbulent fluid flow in the usual sense. Similar results holds also for magnetic field-line motion in resistive magnetohydrodynamics (Eyink 2009), and spontaneous stochasticity then implies the possibility of fast magnetic reconnection in astrophysical plasmas for arbitrarily small electrical conductivity (Eyink et al 2013). In our companion papers II, III we extend the derivation of our Lagrangian FDR to wall-bounded flows, and derive similar relations between anomalous scalar dissipation and spontaneous stochasticity, as well as new Lagrangian relations for Nusselt-Rayleigh scaling in turbulent convection.…”

Section: Summary and Discussionsupporting

confidence: 65%

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

Drivas

Eyink

2017

J. Fluid Mech.

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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]

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Section: Summary and Discussionsupporting

confidence: 65%

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

Drivas

Eyink

2017

J. Fluid Mech.

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“…An historical account of the main results obtained between those two works may be found there. Lately, similar results have been derived in the incompressible amplified, magnetohydrodynamic, frame by Eyink [10]. In particular, he showed that viscous and resistive incompressible magnetohydrodynamic (MHD) equations were equivalent to having some stochastic conservation laws, and that similar results could be obtained in more refined non-ideal models, such as the Hall MHD and two-fluid plasma models with incompressible velocities.…”

Section: Non-ideal Effects and Stochasticitysupporting

confidence: 57%

Impact of the Eulerian chaos of magnetic field lines in magnetic reconnection

et al. 2016

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Stochasticity is an ingredient that may allow the breaking of the frozen-in law in the reconnection process. It will first be argued that non-ideal effects may be considered as an implicit way to introduce stochasticity. Yet there also exists an explicit stochasticity that does not require the invocation of non-ideal effects. This comes from the spatial (or Eulerian) chaos of magnetic field lines that can show up only in a truly three-dimensional description of magnetic reconnection since two-dimensional models impose the integrability of the magnetic field lines. Some implications of this magnetic braiding, such as the increased particle finite-time Lyapunov exponents and increased acceleration of charged particles, are discussed in the frame of tokamak sawteeth that form a laboratory prototype of spontaneous magnetic reconnection. A justification for an increased reconnection rate with chaotic vs integrable magnetic field lines is proposed. Moreover, in 3D, the Eulerian chaos of magnetic field lines may coexist with the Eulerian chaos of velocity field lines, that is more commonly named turbulence.

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“…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]. Finally, Eyink [15] extended the work of Constantin and Iyer to nonideal hydromagnetic models. There, a stochastic analogue of the classical Alfvén theorem was proved to be equivalent to smooth solutions of the deterministic, nonideal, incompressible magnetohydrodynamic equations.…”

Section: Resultsmentioning

confidence: 99%

Circulation and Energy Theorem Preserving Stochastic Fluids

Drivas¹,

Holm

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem [1]. Likewise, smooth solutions of Navier-Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin-Iyer (2008) [9]. In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler-Poincaré and stochastic Navier-Stokes-Poincaré equations respectively. The stochastic Euler-Poincaré equations were previously derived from a stochastic variational principle by Holm (2015) [7], which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems, but do not, in general, preserve circulation theorems.

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Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models

Cited by 36 publications

References 50 publications

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

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

Impact of the Eulerian chaos of magnetic field lines in magnetic reconnection

Circulation and Energy Theorem Preserving Stochastic Fluids

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