Estimates on hyperdiffusive magnetohydrodynamics

A plasma is hyperdiffusive when the dissipation of the small scales is more efficient than the viscous-resistive dissipation. Mathematically it is represented by the addition of a fourth-order diffusive operator to the standard magnetohydrodynamic (MHD) equations. Since hyperdiffusivity has been widely used in astrophysical numerical models, its consequences upon the behavior of the MHD system must be studied. It is found that, unlike what is known in the classical case, solutions are bounded in the H1-norm for all time; for kinematic dynamos, it provides estimates upon the exponential growth of the magnetic field improving considerably the ones of standard MHD. All this points out that hyperdiffusive MHD evolution may differ substantially from the usual MHD one.