Coarse-Grained Incompressible Magnetohydrodynamics: analyzing the turbulent cascades

Abstract

We formulate a coarse-graining approach to the dynamics of magnetohydrodynamic (MHD) fluids at a continuum of length-scales. In this methodology, effective equations are derived for the observable velocity and magnetic fields spatially-averaged at an arbitrary scale of resolution. The microscopic equations for the bare velocity and magnetic fields are renormalized by coarse-graining to yield macroscopic effective equations that contain both a subscale stress and a subscale electromotive force (EMF) generated by nonlinear interaction of eliminated fields and plasma motions. At large coarse-graining length-scales, the direct dissipation of invariants by microscopic mechanisms (such as molecular viscosity and Spitzer resistivity) is shown to be negligible. The balance at large scales is dominated instead by the subscale nonlinear terms, which can transfer invariants across scales, and are interpreted in terms of work concepts for energy and in terms of topological flux-linkage for the two helicities. An important application of this approach is to MHD turbulence, where the coarse-graining length $\ell$ lies in the inertial cascade range. We show that in the case of sufficiently rough velocity and/or magnetic fields, the nonlinear inter-scale transfer need not vanish and can persist to arbitrarily small scales. Although closed expressions are not available for subscale stress and subscale EMF, we derive rigorous upper bounds on the effective dissipation they produce in terms of scaling exponents of the velocity and magnetic fields. These bounds provide exact constraints on phenomenological theories of MHD turbulence in order to allow the nonlinear cascade of energy and cross-helicity. On the other hand, we show that the forward cascade of magnetic helicity to asymptotically small scales is impossible unless 3rd-order moments of either velocity or magnetic field become infinite.

33 pages, 2 figures, to appear in the special issue "Focus on Turbulence in Astrophysical and Laboratory Plasmas" of the New Journal of Physics

References in corpus (13)

• A Global Three Dimensional Radiation Magneto-hydrodynamic Simulation of Super-Eddington Accretion Disks
• A Substantial Amount of Hidden Magnetic Energy in the Quiet Sun
• General Relativistic Hydrodynamic Simulation of Accretion Flow from a Stellar Tidal Disruption
• Large scale flow effects, energy transfer, and self-similarity on turbulence
• Kinematic alpha effect in isotropic turbulence simulations
• Small scale structures in three-dimensional magnetohydrodynamic turbulence
• Sub-Kolmogorov-Scale Fluctuations in Fluid Turbulence
• Non-local interactions in hydrodynamic turbulence at high Reynolds numbers: the slow emergence of scaling laws
• Problems with kinematic mean field electrodynamics at high magnetic Reynolds numbers
• Turbulent General Magnetic Reconnection
• Hydrodynamic and hydromagnetic energy spectra from large eddy simulations
• Nonuniversality and finite dissipation in decaying magnetohydrodynamic turbulence
• The Cascade of Circulations in Fluid Turbulence

Cited by in corpus (24)

• Cascades and transitions in turbulent flows
• 3D Turbulent Reconnection: Theory, Tests and Astrophysical Implications
• Mapping the Energy Cascade in the North Atlantic Ocean: The Coarse-graining Approach
• Spatio-temporal coarse-graining decomposition of the global ocean geostrophic kinetic energy
• Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence
• Proof of Taylor's conjecture on magnetic helicity conservation
• Do Androids Dream of Magnetic Fields? Using Neural Networks to Interpret the Turbulent Interstellar Medium
• A local approach to the study of energy transfers in incompressible magnetohydrodynamic turbulence
• Baropycnal Work: A Mechanism for Energy Transfer Across Scales
• Subion Scale Turbulence Driven by Magnetic Reconnection
• Local cascade and dissipation in incompressible Hall magnetohydrodynamic turbulence: the Coarse-Graining approach
• Theoretical analysis of quantum turbulence using the Onsager "ideal turbulence" theory
• Space-filter techniques for quasi-neutral hybrid-kinetic models
• Nanoflare Theory Revisited
• Triad interactions and the bidirectional turbulent cascade of magnetic helicity
• Effects of forcing mechanisms on the multiscale properties of magnetohydrodynamics
• Scaling of Turbulent Viscosity and Resistivity: Extracting a Scale-dependent Turbulent Magnetic Prandtl Number
• Mean fields and fluctuations in compressible magnetohydrodynamic flows
• A-priori study of the subgrid energy transfers for small-scale dynamo in kinematic and saturation regimes
• Normality-based analysis of multiscale velocity gradients and energy transfer in direct and large-eddy simulations of isotropic turbulence
• Inverse transfer of magnetic helicity in direct numerical simulations of compressible isothermal turbulence: scaling laws
• Partial invariants, large-scale dynamo action, and the inverse transfer of magnetic helicity
• Inverse transfer of magnetic helicity in direct numerical simulations of compressible isothermal turbulence: helical transfers
• On Galilean Invariance of Mean Kinetic Helicity