Statistical properties of ideal three-dimensional Hall magnetohydrodynamics: The spectral structure of the equilibrium ensemble

Abstract

The nonlinear dynamics of ideal, incompressible Hall magnetohydrodynamics (HMHD) is investigated through classical Gibbs ensemble methods applied to the finite Galerkin representation. The spectral structure of HMHD is derived in a three-dimensional periodic geometry and compared with the MHD case. This provides a general picture of spectral transfer and cascade by the assumption that ideal Galerkin HMHD follows equilibrium statistics as in the case of Euler [U. Frisch et al., J. Fluid Mech. 68, 769 (1975)] and MHD [T. Stribling and W. H. Matthaeus, Phys. Fluids B 2, 1979 (1990)] theories. In HMHD, the equilibrium ensemble is built on the conservation of three quadratic invariants: The total energy, the magnetic helicity, and the generalized helicity. The latter replaces the cross helicity in MHD. In HMHD equilibrium, several differences appear with respect to the MHD case: (i) The generalized helicity (and in a weaker way the energy and the magnetic helicity) tends to condense in the longest wavelength, as in MHD, but also admits the novel feature of spectral enhancement, not a true condensation, at the smallest scales; (ii) equipartition between kinetic and magnetic energy, typical of Alfvénic MHD turbulence, is broken; (iii) modal distributions of energy and helicities show minima due to the presence of the ion skin depth. Ensemble predictions are compared to numerical simulations with a low-order truncation Galerkin spectral code, and good agreement is seen. Implications for general turbulent states are discussed.