Statistical properties of ideal three-dimensional magnetohydrodynamics

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Abstract
Classical Gibbs ensemble methods are used to study the spectral structure of three-dimensional ideal magnetohydrodynamics (MHD) in periodic geometry. The intent of this work is to provide further detail and extensions to the work of Frisch et al. [J. Fluid Mech. 68, 769 (1975)], who used equilibrium ensemble methods to predict inverse spectral transfer of magnetic helicity. Here, the equilibrium ensemble incorporates constraints of total energy, magnetic helicity, and cross helicity. Several new results are proven for ensemble averages, including the constraint that magnetic energy equal or exceed kinetic energy, and that cross helicity represents a constant fraction of magnetic energy across the spectral domain, for arbitrary size systems. Two zero temperature limits are considered in detail, emphasizing the role of complete and partial condensation of spectral quantities to the longest wavelength states. The ensemble predictions are compared to direct numerical solution using a low-order truncation Galerkin spectral code. Implications for spectral transfer of nonequilibrium, dissipative turbulent MHD systems are discussed.