Variational principle with singular perturbation of hall magnetohydrodynamics

Abstract

The Hall magnetohydrodynamics (H-MHD) model can describe an intrinsic small scale (ion skin depth ℓi) introduced by the Hall effect. The Hall term appears as a singular perturbation to the conventional magnetohydrodynamics (MHD) model, and hence, the MHD limit (ℓi→0) may be singular. The H-MHD system has three constants of motion, the energy, the magnetic (electron) and ion helicities. The ion helicity is known to be “fragile” with respect to the energy norm of the magnetic and flow fields [Z. Yoshida and S. M. Mahajan, Phys. Rev. Lett. 88, 095001 (2002)]. Under an appropriate ordering of scales, the ion helicity translates as the cross helicity that is a constant of motion of the MHD system. Conservation of the cross helicity is an essential condition to recover the macroscopic MHD picture from the H-MHD framework.