Physics

A. A. Galeev, V. N. Oraevskii

On the Instability of Magnetohydrodynamic Alfvén Waves of Large Amplitude

(Presented by Academician M. A. Leontovich on 28 X 1963)

In a preceding paper \((^{1})\) it was shown that an Alfvén wave of finite but small amplitude \((\delta v \ll v_a\), where \(\delta v\) is the amplitude of the hydrodynamic velocity of the Alfvén wave, \(v_a\) is the Alfvén velocity) is unstable in a compressible medium. The cause of the instability is the presence of positive feedback between small perturbations of the Alfvén and magnetosonic type.

It may be expected that the mechanism indicated above should also lead to the instability of Alfvén waves of not small amplitude. However, natural mathematical difficulties do not allow one to solve the problem for an arbitrary profile of the initial Alfvén wave.* Therefore here we investigate the stability of an Alfvén wave with a sawtooth profile of the magnetic lines of force. The stability problem reduces to finding the frequencies of the natural oscillations of the medium in the periodic field of the initial Alfvén wave. Let us pass to a coordinate system moving together with the initial wave. In our case the equations of ideal magnetic hydrodynamics for small perturbations have coefficients periodic in the coordinates, so that perturbations of the velocity \(\mathbf{v}\), magnetic field \(\mathbf{h}\), and density \(\rho_1\) can be represented in the form:

\[ \psi = \dot{u}(\mathbf{r}) e^{i\mathbf{p}\mathbf{r}-i\omega t}, \tag{1} \]

where \(\psi\) is any of the quantities \(\mathbf{v}, \mathbf{h}, \rho_1\); \(u(\mathbf{r})\) is a solution of the equations periodic (with the period of the initial Alfvén wave); \(\omega\) is the frequency of the perturbations.

In regions where the magnetic field is constant, small perturbations are a superposition of Alfvén, magnetosonic, and entropy waves. In addition, it is necessary to take into account the small displacement of the boundary between regions with a constant magnetic field. At the boundary itself the conditions of continuity of the flux of mass, energy, momentum, of the normal component of the magnetic field, and of the tangential component of the electric field must be satisfied \((^{2})\).

Choose the \(x\)-axis along the direction of the unperturbed magnetic field \(\mathbf{H}_0\), and the \(y\)-axis along the direction of oscillations of the hydrodynamic velocity \(\delta \mathbf{v}\) of the Alfvén wave propagating along \(\mathbf{H}_0\). Then the continuity conditions, linearized with respect to small perturbations, are reduced to the form**

\[ \{\rho_1\} = 0, \]

\[ \left\{ v_{x,z} + \frac{h_{x,z}}{\sqrt{4\pi\rho_0}} \right\} = 0,\quad \left\{ v_y + \frac{h_y}{\sqrt{4\pi\rho_0}} + \frac{1}{2}\frac{\rho_1}{\rho_0}\,\delta v \right\} = 0, \tag{2} \]

\[ \{h_y\delta H + h_x H_0\} = 0; \]

* It is known that Alfvén waves of arbitrary profile and arbitrary amplitude are exact solutions of ideal magnetic hydrodynamics.

** Conditions (2) have been written by us for adiabatic perturbations, which alone are considered below. In addition, in obtaining (2) we used the relation between the perturbation of the velocity of the boundary \(\delta D\) and its displacement \(\xi(y,z)\sim \exp(ik_y y + ik_z z)\):
\(\delta D = -i\omega \xi(y,z)\).

where \(\rho_1, \mathbf{v}, \mathbf{h}\) are, respectively, the perturbations of density, hydrodynamic velocity, and magnetic field; \(\delta H=-\sqrt{4\pi\rho_0}\delta v\) is the magnetic field of the Alfvén wave; \(\delta D\) is the perturbation of the velocity of the boundary between regions with constant magnetic field. By means of braces we have denoted the difference of the values of the perturbed quantities on the two sides of the discontinuity surface.

For simplicity let us consider the case in which the velocity and magnetic field of the perturbations lie in the \((x,y)\) plane (see Fig. 1). Then the solutions in the regions with constant magnetic field may be represented as a superposition of slow and fast magnetosonic waves. (The relation between the quantities \(\rho_1,\mathbf{v},\mathbf{h}\) in the waves is found from the magnetohydrodynamic equations.) Let in the interval \(-b<x<0\) the solution be

Fig. 1

\[ \psi=\left(c_1 e^{ik_1^-x}+c_2 e^{i\widetilde{k}_1^-x}+ c_3 e^{ik_2^-x}+c_4 e^{i\widetilde{k}_2^-x}\right)e^{-i\omega t+ik_y y}, \]

where \(k_{1,2}^-\) are the components of the wave vectors along the \(x\) axis, respectively, for the slow and fast magnetosonic waves; \(k\) and \(\widetilde{k}\) correspond to two linearly independent solutions with the given frequency \(\omega\).

Similarly, in the interval \(0<x<a\) we have:

\[ \psi=\left(c_5 e^{ik_1^+x}+c_6 e^{i\widetilde{k}_1^+x} +c_7 e^{ik_2^+x}+c_8 e^{i\widetilde{k}_2^+x}\right)e^{-i\omega t+ik_y y}. \]

The solution in the region \(a<x<a+b\) is obtained from the periodicity condition (1):

\[ \psi=\left(c_1 e^{ik_1^-(x-a-b)}+c_2 e^{i\widetilde{k}_1^-(x-a-b)} +c_3 e^{ik_2^-(x-a-b)}+c_4 e^{i\widetilde{k}_2^-(x-a-b)}\right) e^{-i\omega t+ip(a+b)+ik_y y}. \]

The boundary conditions (2) constitute a system of 8 linear equations for the 8 coefficients \(c_1,\ldots,c_8\). From the solvability condition for this system we obtain the dispersion relation \(\omega=\omega(p,k_y)\). In the general case it has a rather cumbersome form. We therefore give it for the most interesting case of a large-amplitude wave \((\delta v\gg v_a)\) propagating in a medium of low pressure \((c_s\ll v_a\ll \delta v_a,\ c_s\) is the speed of sound):

\[ \begin{aligned} &\sin\left(\frac{\omega c_s a}{v_a\delta v_a}\right) \sin\left(\frac{\omega c_s b}{v_a\delta v_a}\right) \left[ \cos p(a+b)-\cos k_2^- b\cos k_2^+ a +\frac{1}{2}\left(\frac{k_2^+}{k_2^-}+\frac{k_2^-}{k_2^+}\right) \sin k_2^- b\sin k_2^+ a \right] \\ &= -6\,\frac{c_s v_a}{\omega\delta v_a} \sin\left(\frac{\omega c_s b}{v_a\delta v_a}\right) \times \left\{ \left[k_2^-\cos k_2^+ a\sin k_2^- b +k_2^+\sin k_2^+ a\cos k_2^- b\right] \cos\left(\frac{\omega c_s a}{v_a\delta v_a}\right) \right. \\ &\qquad\left. -\,k_2^+\sin k_2^+ a \cos\left(\frac{\omega+k_y\delta v_a}{v_a}a-p(a+b)\right) -k_2^-\sin k_2^- b \cos\left(\frac{\omega+k_y\delta v_a}{v_a}a\right) \right\} \\ &\quad -6\,\frac{c_s v_a}{\omega\delta v_a} \sin\left(\frac{\omega c_s a}{v_a\delta v_a}\right) \left\{ \left[k_2^-\cos k_2^+ a\sin k_2^- b +k_2^+\sin k_2^+ a\cos k_2^- b\right] \cos\left(\frac{\omega c_s b}{v_a\delta v_a}\right) \right. \\ &\qquad\left. -\,k_2^+\sin k_2^+ a \cos\left(\frac{\omega-k_y\delta v_a}{v_a}b\right) -k_2^-\sin k_2^- b \cos\left(\frac{\omega-k_y\delta v_a}{v_a}b-p(a+b)\right) \right\}, \tag{3} \end{aligned} \]

where

\[ k_2^\pm=\frac{\omega}{\delta v_a} \sqrt{1\pm k_y\frac{\delta v_a}{\omega}},\qquad \delta v_a=-\delta v . \]

Neglecting the right-hand side, equation (3) has real roots. The roots corresponding to the first two factors pertain to the slow magnetosonic wave and have a form reminiscent of the conditions for wave numbers in a resonator \(\left(\dfrac{\omega}{v_a}\dfrac{c_s}{\delta v_a}=\dfrac{\pi n}{a},\ n=0,\pm 1,\pm 2,\ldots\right)\). The third factor pertains to the fast wave and, in form, coincides with the equation for the energy of an electron in a periodic field. Allowance for the right-hand side gives a correction to the frequencies of the zeroth approximation. If, in the zeroth approximation, two roots (corresponding to the slow and fast waves) coincide, then the square of the correction to the frequency proves to be negative. In this case the growth rate of the amplification of small perturbations superposed on the initial Alfvén wave is, in order of magnitude, equal to

\[ \nu \simeq v_a/a. \tag{4} \]

The instability considered here resembles the instability of an Alfvén wave of small amplitude with respect to the simultaneous excitation of slow and fast magnetosonic waves \({}^{1}\).

The authors express their gratitude to Academician M. A. Leontovich for discussion of the results and to R. Z. Sagdeev, whose advice and interest in the work stimulated its completion.

Novosibirsk State
University

Received
21 X 1963

CITED LITERATURE

\({}^{1}\) A. A. Galeev, V. N. Oraevskii, DAN, 147, 71 (1962). \({}^{2}\) L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, Moscow, 1959.