High-frequency magnetohydrodynamics

We consider the case that the displacement current is not neglected in the classical MHD equations, as it is usually done. This amounts to cast them in the relativistic framework of a finite speed of light. We show some consequences in describing magnetic reconnection phenomena and for hydromagnetic waves. In the first case, the equation for the magnetic induction is changed from (formally) parabolic to (formally) hyperbolic, in the second case both, the perturbed magnetic field and the particle velocity, obey to a certain third-order in time partial differential equation, rather than to the classical wave equation. We stress the role of two typically small but nonzero parameters, the magnetic diffusivity, η (corresponding to large values of the Lundquist number).