Transition to Equilibrium and Coherent Structure in Ideal MHD Turbulence, Part 2

Abstract

We continue our study of the transition of ideal, homogeneous, incompressible, magnetohydrodynamic (MHD) turbulence from non-equilibrium initial conditions to equilibrium using long-time numerical simulations on a 1283 periodic grid. A Fourier spectral transform method is used to numerically integrate the dynamical equations forward in time. The six runs that previously went to near equilibrium are here extended into equilibrium. As before, we neglect dissipation as we are primarily concerned with behavior at the largest scale where this behavior has been shown to be essentially the same for ideal and real (forced and dissipative) MHD turbulence. These six runs have various combinations of imposed rotation and mean magnetic field and represent the five cases of ideal, homogeneous, incompressible, and MHD turbulence: Case I (Run 1), with no rotation or mean field; Case II (Runs 2a and 2b), where only rotation is imposed; Case III (Run 3), which has only a mean magnetic field; Case IV (Run 4), where rotation vector and mean magnetic field direction are aligned; and Case V (Run 5), which has non-aligned rotation vector and mean field directions. Statistical mechanics predicts that dynamic Fourier coefficients are zero-mean random variables, but largest-scale coherent magnetic structures emerge and manifest themselves as Fourier coefficients with very large, quasi-steady, mean values compared to their standard deviations, i.e., there is ‘broken ergodicity.’ These magnetic coherent structures appeared in all cases during transition to near equilibrium. Here, we report that, as the runs were continued, these coherent structures remained quasi-steady and energetic only in Cases I and II, while Case IV maintained its coherent structure but at comparatively low energy. The coherent structures that appeared in transition in Cases III and V were seen to collapse as their associated runs extended into equilibrium. The creation of largest-scale, coherent magnetic structure appears to be a dynamo process inherent in ideal MHD turbulence, particularly in Cases I and II, i.e., those cases most pertinent to planets and stars. Furthermore, the statistical theory of ideal MHD turbulence has proven to apply at the largest scale, even when dissipation and forcing are included. This, along with the discovery and explanation of dynamically broken ergodicity, is essentially a solution to the ‘dynamo problem’.

The Statistical Mechanics of Ideal Magnetohydrodynamic Turbulence and a Solution of the Dynamo Problem

“…For each run in Table 3, the quantities that are supposed to be conserved were, in fact, conserved. These runs are fully turbulent and their transition to turbulence was described previously [48,49]; here, the run-times are considerably extended. There are two further sets of runs whose statistics are given in Tables 4 and 5 that will also be discussed.…”

“…In this section, we summarize the results that appear in more detail in previous papers [48,49], which also have further references. With regard to MHD general references, there are several good books available, if needed [51][52][53].…”

“…Here, we will review ideal MHD statistical theory, which predicts [48,49], for Cases I and II of Table 1, a large-scale magnetic field that is quasi-stationary with a 'dipole' energy E D that is related to the magnetic helicity H M and the wavenumber k min of the largest-scale modes, by the expression…”