Resonance absorption of propagating fast waves in a cold plasma

Abstract

Resonance absorption of MHD surface waves has received considerable attention recently, but rather little attention has been paid to the absorption of propagating waves impinging on a “surface” in which the plasma and magnetic field may change. Here we examine in some depth a very simple but instructive problem: the plasma is cold, the magnetic field is uniform, and the density in the “surface” varies linearly from zero at the left end to some finite value at the right end, beyond which the density is constant. We consider two cases: (1) the plasma is a vacuum everywhere to the left of the surface, or (2) the plasma density jumps to a very large value to the left of the surface. Case (1) may correspond to coronal conditions, while case (2) may mimic the magnetosphere with the dense region at the left corresponding to the plasmasphere. The goals of the paper are to study the parametric behavior of the absorption coefficient numerically, and to provide several useful analytical approximations. We find that the parametric dependence of the absorption is far richer than implied by the single curve appearing in Fig. 2 of Kivelson and Southwood (1986, J. geophys. Res. 91, 4345), although we do recover that curve as a limiting case in which the waves are essentially WKB at the right end of the “surface” and a dimensionless parameter K 2 x (defined in Section 5) is moderately large. In case (1) we find that the absorption coefficient is always less than about 50 %, but in case (2) the absorption can approach 100 %. Thus the boundary condition at the left critically affects the results. We also find that the thickness of the surface affects the parametric dependence of the absorption coefficient. For example, a thin surface yields an absorption coefficient scaling as α −2, where α 2 [defined in equation (4)] is a measure of the steepness of the density ramp. On the other hand, if the surface is thick enough so that the waves are essentially WKB as they start down the density ramp, then the absorption scales as α − 8 3 [case (1)] or α − 4 3 [case (2)] for large α 2. Our numerical results are pre in a format which reveals the dependence of the absorption on the propagation direction of an incident wave. The absorption depends on the angle of incidence with respect to the surface, maximizing at moderately large angles of incidence [around 70° in case (1)]. The absorption depends also on the angle between the magnetic field and the plane of incidence [there is a broad maximum around 30° in case (1)]. Finally, along the way we offer two other analytical results: (1) we show that the mathematical discontinuity in the Poynting flux which occurs in the present steady-state analysis is precisely equivalent to the rate at which energy is pumped into the resonant layer as calculated by Hollweg and Yang (1988, J. geophys. Res. 93, 5423) using a simple harmonic oscillator model; (2) we show that a convenient approximation scheme used by us for calculating the absorption of propagating waves in another context (Hollweg, 1988, Astrophys. J. 335, 1005) has a useful domain of validity.