Absolute and convective instabilities of parallel propagating circularly polarized Alfvén waves: Decay instability

Abstract

The absolute and convective instabilities of circularly polarized Alfvén waves propagating along an ambient magnetic field are studied. The approximation of ideal magnetohydrodynamics is used. The analysis is restricted to the decay instability that occurs when the sound speed is smaller than the Alfvén speed. In addition, it is assumed that the amplitude a of an unstable Alfvén wave (pump wave) is small. This assumption allows us to study the problem analytically using expansions in power series with respect to a. It is shown that there are quantities, U l Ͻ 0 and U r Ͼ 0, such that the pump wave is absolutely unstable in a reference frame moving with velocity U with respect to the rest plasma if U l Ͻ U Ͻ U r . If either U Ͻ U l or U Ͼ U r , then the pump wave is convectively unstable. The expressions for U l and U r are found. The signaling problem is studied in a reference frame where the pump wave is convectively unstable. It is shown that spatially amplifying waves exist only when the signaling frequency is in two narrow symmetric frequency bands with the widths of the order of a. The implication of the obtained results on the interpretation of observational data obtained in space missions is discussed. It is shown that circularly polarized Alfvén waves propagating in the solar wind are convectively unstable in a reference frame of any spacecraft moving with the velocity not exceeding a few tens of km/s in the solar reference frame. The spatial amplification scale of these waves is very large, of the order of 1 / 6 a.u. In view of these results it is not surprising at all that evidence of the decay instability of Alfvén waves in the solar wind is sparse.

MHD Waves and Instabilities in Space Plasma Flows

“…When coupling occurs, the perturbations may grow at the expense of the pump. Ruderman and Simpson (2004) studied the absolute and convective instabilities of circularly polarised Alfvén waves propagating along an ambient magnetic field. The analysis was restricted to the parametric instability that occurs when the sound speed is smaller than the Alfvén speed.…”

Absolute and convective instabilities of parallel propagating circularly polarized Alfvén waves: Beat instability

“…Ruderman and Simpson 17 have recently addressed the problem of whether an unstable Alfvén wave appears to give rise to growing modes in a fixed reference frame. This is an important problem from the point of view of observations, as a system with unstable modes will only appear unstable to an observer if the instability grows in time in the observer's reference frame.…”

“…Normal-mode analysis is not enough to study this problem. Ruderman and Simpson 17 used the method formulated by Briggs 18 and Bers 19 to study the absolute and convective natures of the instability. They restricted their analysis to the decay instability which occurs when the sound speed c S in the unperturbed plasma is smaller than the Alfvén speed A .…”

“…The aim of the present paper is to study the absolute and convective beat instabilities of circularly polarized Alfvén waves. Our analysis closely follows the analysis of Ruderman and Simpson, 17 so that we refer the reader to this paper ͑hereafter referred to as "Paper 1"͒ for a detailed method description.…”

Absolute and convective instabilities of parallel propagating circularly polarized Alfvén waves: numerical results

“…Recently Ruderman & Simpson (2004b) and Simpson & Ruderman (2005) (Papers I and II in what follows) studied the absolute and convective instabilities of circularly polarized Alfvén waves analytically using the dimensionless pump wave amplitude a as a small parameter. In this paper we use numerical technics to extend their analysis for arbitrary a.…”