Spectral Energy Dynamics in Magnetohydrodynamic Turbulence

Abstract

Spectral direct numerical simulations of incompressible MHD turbulence at a resolution of up to 1024 3 collocation points are presented for a statistically isotropic system as well as for a setup with an imposed strong mean magnetic field. The spectra of residual energy, E (anisotropic case, perpendicular to the mean field direction). A model of dynamic equilibrium between kinetic and magnetic energy, based on the corresponding evolution equations of the eddy-damped quasi-normal Markovian (EDQNM) closure approximation, explains the findings. The assumed interplay of turbulent dynamo and Alfvén effect yields E R k ∼ kE 2 k which is confirmed by the simulations.The nonlinear behavior of turbulent plasmas gives rise to a variety of dynamical effects such as self-organization of magnetic confinement configurations in laboratory experiments [1], generation of stellar magnetic fields [2] or structure formation in the interstellar medium [3]. The understanding of these phenomena is incomplete as the same is true for many inherent properties of the underlying turbulence.Large-scale low-frequency plasma turbulence is treated in the magnetohydrodynamic (MHD) approximation describing the medium as a viscous and electrically resistive magnetofluid neglecting additional kinetic effects. Incompressiblity of the flow is assumed for the sake of simplicity. In this setting the nature of the turbulent energy cascade is a central and still debated issue with different phenomenologies being proposed [4,5,6,7,8] (cf.[9] for a review). The associated spectral dynamics of kinetic and magnetic energy, in spite of its comparable importance, has received less attention (as an exception see [10]).This Letter reports a spectral relation between residual and total energy,respectively, as well as the influence of an imposed mean magnetic field on the spectra. The proposed physical picture, which is confirmed by accompanying direct numerical simulations, embraces two-dimensional MHD turbulence, globally isotropic three-dimensional systems as well as turbulence permeated by a strong mean magnetic field.In the following reference is made to two highresolution pseudospectral direct numerical simulations of incompressible MHD turbulence which we regard as paradigms for isotropic (I) and anisotropic (II) MHD turbulence. The dimensionless MHD equationsare solved in a 2π-periodic cube with spherical mode truncation to reduce numerical aliasing errors [11]. The equations include the flow vorticity, ω = ∇ × v, the magnetic field expressed in Alfvén speed units, b, as well as dimensionless viscosity, µ, and resistivity, η. In simulation II forcing is applied by freezing the largest spatial scales of velocity and magnetic field. Simulation I evolves globally isotropic freely decaying turbulence represented by 10243 Fourier modes. The initial fields are smooth with random phases and fluctuation amplitudes following exp(−k 2 /(2k 2 0 )) with k 0 = 4. Total kinetic and magnetic energy are initially equal with also decreases during turbulence decay from 0.7...

Solar Wind Turbulence and the Role of Ion Instabilities

“…The difference between magnetic and kinetic energies is usually called in the literature "residual energy". The residual energy has been shown to follow a definite scaling which is related to the scaling of the total energy spectrum (Grappin et al 1983;Müller and Grappin 2005), see also (Boldyrev and Perez 2009;Chen et al 2013b).…”

Recent progress in astrophysical plasma turbulence from solar wind observations

“…The residual energy has also been described as a balance between the Alfvén effect (linear term) leading to equipartition and a turbulent dynamo (nonlinear term) generating the magnetic excess (Grappin, Leorat & Pouquet 1983;Müller & Grappin 2005). Müller & Grappin (2005) used an isotropic closure theory to suggest that E r varies with the total energy spectrum E t , following k −2 for a k −3/2 total energy spectrum and k −7/3 for a k −5/3 total energy spectrum.…”

“…Müller & Grappin (2005) used an isotropic closure theory to suggest that E r varies with the total energy spectrum E t , following k −2 for a k −3/2 total energy spectrum and k −7/3 for a k −5/3 total energy spectrum. Boldyrev, Perez & Wang (2012a) extended this to an anisotropic model in which the perpendicular spectra…”

Solar cycle dependence of scaling in solar wind fluctuations

“…This symmetric case where the fluxes of oppositely directed Alfvén waves are equal does not however strictly apply to the solar wind (Goldreich and Sridhar, 1997), where the fluxes are observed to be asymmetric. Recent numerical simulations (Müller and Grappin, 2005), and analysis (Boldyrev, 2006 hereafter SB) obtain a ∼k −3/2 ⊥ spectrum for the case of a strong local background magnetic field. This −3/2 exponent, combined with the anisotropy of the fluctuations, is in contradiction with WI (either isotropic, −3/2 exponent or anisotropic, −2 exponent) and SA (anisotropic, −5/3 exponent) phenomenologies.…”