Magnetohydrodynamic waves in a homogeneous plasma convected uniformly at relativistic speed

Abstract

The plasma in several physical situations such as movement of electrons along the geomagnetic field lines in the magnetosphere, the movement of the ionosphere, propagation of cosmic rays, etc., can be appropriately simulated by a drifting relativistic model. Keeping this in view, a general dispersion relation for magnetohydrodynamic (MHD) waves has been derived in a laboratory stationary coordinate system with respect to which plasma is drifting with a velocity which need not be small compared with the speed of light. This dispersion relation gives several earlier well-known results for MHD waves supported by an ideal relativistic plasma. The characteristic equation for arbitrary direction of propagation with reference to the ambient magnetic field is quite unwieldy. So, the detailed discussion is confined to the special cases when the propagation vector is along or across the magnetic field. However, wherever feasible, approximate solutions for arbitrary direction of propagation have also been discussed.

Linear wave propagation in relativistic magnetohydrodynamics

“…x−z plane (plotted is thus v ph = v ph n by varying n in x−z over 2π), and it should be noted that the suppressed third dimension is no longer a mere revolution about some axis as soon as the velocity v is not aligned with B. For the particular case where v is aligned with B (the case in the study by Kalra and Gebretsadkan 14 ), it is merely to be rotated about the z-axis. In any case, the visual representation of the phase diagram is arguably the most intuitive way to illustrate the combined effects of the inherent anisotropic nature of MHD wave propagation (even in uniform media), with the added complexity brought in by relativistic wave aberration.…”

“…This can indeed be done by linearizing about a moving ͑i.e., stationary͒ uniform plasma, as was pursued in Kalra and Gebretsadkan 14 for the special case where the movement v was aligned with the uniform field B, but this is by far the most algebraically complex means to do so. It must be realized that the more general result ͑for arbitrary orientation between v and B͒ was already given in the textbook by Lichnerowicz 2 and Anile, 3 obtained from the more appropriate covariant form ͑i.e., valid in any reference frame͒.…”

“…3 Here, we revisit these results in a 3 + 1 formalism. The algebra involved is then more cumbersome ͑see, e.g., the results from Kalra and Gebretsadkan 14 ͒, even in the case of linearizing about a stationary, homogeneous plasma. This is partly because of the wave aberration effects, which we mentioned already for the relativistic gas dynamic case, and due to the intrinsic anisotropic nature of the MHD wave families.…”

“…An example of the latter is the discussion of the properties of Alfvén waves for both continuous and discontinuous ͑shock͒ solutions by Komissarov,13 emphasiz-ing the wave polarization properties as viewed in different Lorentzian reference frames. In a more recent publication, 14 a dispersion relation for RMHD linear waves in a homogeneous plasma convected at uniform speed was analyzed in the most tractable special cases ͑aligned velocity and magnetic field vector͒. This study adopted a 3 + 1 formalism, which is the one adhered to here as well.…”

Modification of magnetohydrodynamic waves by the relativistic Hall effect

“…[1,2]), the detailed analysis of HMHD waves was not conducted until Hameiri et al revealed that phase and group diagrams are deformed by the Hall effect [25]. In addition to the non-relativistic models, the dispersion relation for relativistic ideal MHD has been studied [17,18,26,27]. Keppens and Meliani drew phase and group diagrams of the relativistic MHD showing that there is no qualitative difference between non-relativistic and relativistic diagrams in a fluid rest frame, except for the presence of the light limit [27].…”