UDC 533.9

PHYSICS

A. B. MIKHAILOVSKII

INSTABILITY OF AN INHOMOGENEOUS HIGH-PRESSURE PLASMA IN A HOMOGENEOUS MAGNETIC FIELD

(Presented by Academician M. A. Leontovich on 29 IX 1969)

1. Let an inhomogeneous collisionless plasma with pressure \(p_0 \gg B_0^2/8\pi\) be in a homogeneous magnetic field \(B_0\), \(\partial B_0/\partial x=0\). (The condition \(\partial B_0/\partial x=0\) is satisfied if \(\partial p_0/\partial x=0\).) Consider plasma perturbations with frequency \(\omega \ll (k_z v_{Te}, \omega_{Bi})\) and transverse wavelength \(\lambda_\perp \gg \rho_i\) (\(v_{Te}\) is the electron thermal velocity, \(\rho_i\) and \(\omega_{Bi}\) are the Larmor radius and cyclotron frequency of the ions, and \(k_z\) is the longitudinal wave number). As follows from work \((^1)\), in the limit of high pressures \(\beta \equiv 8\pi p_0/B_0^2 \to \infty\), such perturbations are described by the local dispersion equation

\[ \frac{1}{T}\hat{l}_i(TxW)+\frac{i\sqrt{\pi}}{4} \left\{2\hat{l}_i\left(\frac{xW}{T}\right)\hat{l}_i(TxW)-[\hat{l}_i(xW)]^2\right\}=0. \tag{1} \]

Here the notation of work \((^1)\) is used:

\[ x=\omega/|k_z|v_{Ti},\qquad v_{Ti}=(2T/m_i)^{1/2}, \]

\[ W=e^{-x^2}\left[1+(2i/\sqrt{\pi})\int_0^x e^{t^2}dt\right], \qquad \hat{l}_i=1-(\omega_T/\omega)\,\partial/\partial\ln T, \]

\[ \omega_T=k_y(\partial T/\partial x)/m_i\omega_{Bi}; \]
\(T\) is the plasma temperature.

For \(x\gg 1\), (1) yields a quadratic equation for \(\omega\), so that

\[ \omega={}^{2}\!/\!_{3}\,\omega_T(1\pm i/\sqrt{2}). \tag{2} \]

It is seen that, with respect to such perturbations, the plasma is unstable. The instability also occurs for \(x\ll 1\), and in this case

\[ \operatorname{Re}\omega=\omega_T/2, \tag{3} \]

\[ \gamma\equiv \operatorname{Im}\omega= \frac{\pi-2}{4\sqrt{\pi}}\, \frac{\omega_T^2}{|k_z|v_{Ti}}. \tag{4} \]

1. If the plasma is described by a system of two-fluid hydrodynamic equations with adiabatic index \(\gamma_0\) \((^2)\), then instead of (2) one obtains

\[ \omega=\omega_T\frac{\gamma_0\pm i\sqrt{\gamma_0}}{1+\gamma_0}. \tag{5} \]

For \(\gamma_0=2\) this coincides with (2), obtained in the approximation of a collisionless plasma. If the frequency of ion-ion collisions exceeds the oscillation frequency, then \(\gamma_0=5/2\), and then

\[ \omega={}^{5}\!/\!_{8}\,\omega_T(1\pm i\sqrt{3/5}). \tag{6} \]

Thus, a collisional plasma with \(\beta\gg 1\) and \(\nabla p_0=0\), like a collisionless one, is unstable.

1. Let us estimate what consequences the instability considered above may lead to. From the pressure-balance equation

\[ -\frac{\partial}{\partial x}\left(p+\frac{B^2}{8\pi}\right)=0 \tag{7} \]

it follows that the perturbed magnetic field \(B_z'\) will become of the order of the equilibrium field \(B_0\) when the perturbed pressure \(p'\) is such that

\[ p'/p_0 \simeq 1/\beta \ll 1. \tag{8} \]

If it is assumed that at large \(p'\) the instability is suppressed, then, using the continuity and motion equations and Ohm’s law, one can obtain an estimate for the plasma relaxation time

\[ \tau \simeq 4\pi\sigma a^2/c^2, \tag{9} \]

where \(\sigma\) is the conductivity, \(a \equiv (\partial \ln T/\partial x)^{-1}\). The quantity \(\tau\) turns out to be, in order of magnitude, equal to the diffusion time of the magnetic field into the plasma \((^3)\).

Thus, one may suppose that if no instabilities other than those considered here develop, then over a time of the order of several inverse increments, a small pressure gradient, \(\partial \ln p/\partial x \simeq 1/a\beta\), must arise in a plasma with \(\nabla p_0=0\), and then, over a time of the order of \(\tau\), plasma decay will occur.

I express my deep gratitude to Academician G. I. Budker, L. V. Mikhailovskaya, and A. M. Fridman for discussions.

Received
26 VIII 1969

CITED LITERATURE

1. A. B. Mikhailovskii, A. M. Fridman, ZhETF, 51, 1430 (1966).
2. S. I. Braginskii, Problems of Plasma Theory, 3, 151 (1963).
3. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, 1957.