Toward an energy transfer and interacting scale analysis of MHD turbulence with implications for space and astrophysical plasmas

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MHD turbulence has numerous applications in space and astrophysical plasmas. In this paper, the eddy-damped quasi-normal Markovian (EDQNM) model is used to perform a preliminary study of the nonlinear transfer process in three-dimensional MHD Both twoand three-dimensional contour plots of the triadic transfer functions are presented for the case of assumed energy spectra corresponding to Kolmogorov inertial subrange scaling. Introduction The magnetohydrodynamic (MHD) approximation has been quite successful in space physics and astrophysics. In particular, the manifestation of turbulence and other nonlinear phenomena in astrophysical plasmas is explainable from an MHD turbulence perspective. The MHD description has been shown to be an excellent starting point for describing plasma motions when the macroscopic level of motions are well separated from the Coulomb collision/particle gyro-scales. * Copyright © 2001 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. The application of MHD turbulence to the solar illustrates how the understanding of basic plasma physics and the universe can be improved. The existence of the solar was deduced in 1896 by Birkeland and later theoretically predicted by Parker. Subsequent observations confirmed the presence of hot. supersonic outflows of electrons, protons, and alpha particles from the upper limits of the corona of the Sun. The solar streams past the magnetosphere of the Earth, and is the means by which mechanical energy is transmitted from the Sun to the Earth. Solar spacecraft observations provide a readily available 'laboratory' for testing theories and assumptions. For example, spacecraft observations demonstrated that the solar can be characterized as a turbulent magnetofluid. Reduced power spectra constructed from Mariner 10 spacecraft magnetometer data revealed a steady power-law spectrum spanning nearly three decades in frequency, with an o;~/ powerlaw, where u; is the spacecraft rest frequency. Relating time measurements to spatial scales using the Taylor frozen-in-flow hypothesis, this translates to a fr~/ wavenumber spectrum, reminiscent of the well-known kinetic energy spectrum in fullydeveloped homogeneous, isotropic fluid (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. Fyfe. Montgomery, and Joyce argued that the original Kolmogorov scaling, and its associated fc~/ power-law, is also applicable to MHD Kraichnan. however, proposed that the usual phenomenological argument should be modified to include magnetic field effects, which leads to a k~/ spectrum. Unquestionably, spacecraft observations have strongly motivated the study of turbulence in MHD models describing the dynamics of the solar wind. Much MHD turbulence research has focused on the spectra of three quadratic, integral 'rugged invariants-. These invariants are deduced from the incompressible, non-dissipative approximation of the MHD equations in the absence of a mean magnetic field. In three-dimensional MHD turbulence, these invariants are the energy (per unit mass), the cross helicity. and the magnetic helicity. Some interesting applications of MHD turbulence to the solar include the evolution of cross helicity. the development of anisotropies. the decay of magnetic helicity with a mean magnetic field, and nearly-incompressible dynamics. While significant progress has been made, some fundamental aspects of MHD turbulence must be investigated and understood regarding the energy transfer and interacting scales. One can appreciate this point by noting that nearly all MHD turbulence, including its applications to the solar wind rely on assumptions regarding the energy transfer process through the inertial range. Spacecraft observations may be able to indicate the total energy at a given scale in the spectrum, and detailed information on the energy transfer and interacting scales can be obtained by an analysis similar to that carried out for fluid turbulence. In this paper, we will develop the basic concepts and equations for the energy transfer and interacting scale analysis. After forming the transfer spectra, we will formulate the principal quantitative measurements for describing the spectral locality, strength, and anisotropies of the nonlinear modal couplings. All of these analyses can be carried out with direct numerical simulation (DNS) databases of MHD The main limitation of DNS data is that the fluid and magnetic Reynolds numbers are restricted to relatively moderate values. Here, we present an alternative method for performing the energy transfer analysis using transfer spectra constructed from the eddydamped quasi-normal Markovian (EDQNM) closure model'. The EDQNM closure can achieve very high Reynolds numbers, and therefore, a wide range of spectral scales for the analysis. The MHD Equations The standard, unforced MHD equations are the Navier-Stokes equation o — i/V ) u = -Vr>+ b V b u Vu (1) dt ) and the magnetic field equation | — _ £ v J b — v x (u x b) (2) / with V • u = V • b = 0. (3) where the kinematic viscosity and magnetic diffusivity are v and £. respectively. For homogeneous turbulence, the MHD equations in wavenumber space are