Relaxation Processes in Turbulent Magnetohydrodynamic Decay

Mechanisms that lead to relaxation of certain global parameters in decaying two-dimensional (2D) and three-dimensional (3D) magnetohydrodynamics (MHD) are studied. The processes investigated include: (a) selective decay, a consequence of an inequality in the decay rates of the magnetic helicity H_{m} and total energy E for 3D MHD and mean square vector potential A and E for 2D MHD, (b) dynamic alignment, a consequence of an inequality in the decay rates of the cross helicity H_{c} and E and (c) a rapid process that in both 2D and 3D involves rapid growth of kinetic or magnetic energy, and adjustments of certain spectral properties of E, H_{c} and A in 2D or H_{m} in 3D that must precede both selective decay and dynamic alignment. These investigations are accomplished by performing many simulations of decaying 2D and 3D MHD in a periodic geometry while varying the appropriate parameters for each run to determine parameter ranges where the different relaxation operate and compete. Also, inviscid models of 2D and 3D MHD are studied both numerically and analytically. Periodic Galerkin representations of inviscid MHD relax to a statistically steady state referred to as absolute equilibrium. A classical statistical mechanics formalism is used to derive expressions for the absolute equilibrium E, H_{m}, H_ {c}, A, kinetic energy E_ {v} and magnetic energy E_ {b}^ectra. Two limiting forms of the spectra are analyzed. In the first the limiting forms of the spectra are determined as the energy becomes minimal, and for 2D MHD as the energy becomes maximal. The second is a thermodynamic like limit, which is believed relevant to high Reynolds number dissipative flows, where the number of degrees of freedom of the system is taken to infinity while certain the global parameters are held fixed. The behavior of the H_{c}^ectrum in this limit is conjectured to give rise to H_{c} back transfer in dissipative flows, providing a new dynamic mechanism for dynamic alignment. The absolute equilibrium values of E_{v}/E_ {b} and properties of the E, H _{c}, H_{m} and A spectra are also studied, and it is proposed that the fast relaxation process, which occurs in dissipative MHD, is driven by the system relaxing towards the absolute equilibrium state.