Transition to Equilibrium and Coherent Structure in Ideal MHD Turbulence

Abstract

Transition of ideal, homogeneous, incompressible, magnetohydrodynamic (MHD) turbulence to near-equilibrium from non-equilibrium initial conditions is examined through new long-time numerical simulations on a 1283 periodic grid. Here, we neglect dissipation because we are primarily concerned with behavior at the largest scale which has been shown to be essentially the same for ideal and real (forced and dissipative) MHD turbulence. A Fourier spectral transform method is used to numerically integrate the dynamical equations forward in time and results from six computer runs are presented with various combinations of imposed rotation and mean magnetic field. There are five separate cases of ideal, homogeneous, incompressible, MHD turbulence: Case I, with no rotation or mean field; Case II, where only rotation is imposed; Case III, which has only a mean magnetic field; Case IV, where rotation vector and mean magnetic field direction are aligned; and Case V, which has nonaligned rotation vector and mean field directions. Dynamic coefficients are predicted by statistical mechanics to be zero-mean random variables, but largest-scale coherent magnetic structures emerge in all cases during transition; this implies dynamo action is inherent in ideal MHD turbulence. These coherent structures are expected to occur in Cases I, II and IV, but not in Cases III and V; future studies will determine whether they persist.

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].…”

“…For purposes of illustration, let us recall some Case I and Case II examples. In recent ideal MHD turbulence simulations [48], Case I Run 1 had k M = 4.86, while Case II Runs 2a and 2b had k M = 8.82 and 4.69, respectively. As for real, i.e., forced and dissipative, MHD turbulence simulations [27], using time averages E, H M , H C for E , H M , H C , in order to determine φ o , we found for Case I run NM02c, k M = 1.10, while for Case II run NM06c, k M = 1.12.…”

“…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…”

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

“…In that regard, our new results come from extending into equilibrium the numerical simulations reported in [13], where relevant background and references are given. In brief, we use a Fourier spectral transform method as a surrogate for a planetary interior that contains a turbulent magnetofluid, such as the Earth's outer core.…”