PHYSICS

L. V. MIKHAILOVSKAYA, A. B. MIKHAILOVSKII

SAIDEM INSTABILITY FOR A FINITE ION LARMOR RADIUS

(Presented by Academician M. A. Leontovich on 22 XII 1962)

In Suydam’s paper \(^{(1)}\) it was shown that a plasma confined by a helical magnetic field \(\mathbf H=(0,H_\varphi(r),H_z(r))\) is unstable with respect to perturbations of the form \(f=f(r)e^{im\varphi+ikz-i\omega t}\), localized near a certain magnetic surface \(r=r_0\), where

\[ k_\tau(r_0)\equiv \left(\frac{m}{r}\frac{H_\varphi}{H}+k\frac{H_z}{H}\right)_{r=r_0}=0, \]

if

\[ \varkappa^2\equiv -\left.\frac{8\pi p'}{rH_z^2}\left(\frac{\mu}{\mu'}\right)^2\right|_{r=r_0}>\frac14 . \tag{1} \]

Here \(\mu=H_\varphi/rH_z\); \(p=n(T_i+T_e)\) is the equilibrium plasma pressure; \(n\) is the equilibrium density; \(T_i\) and \(T_e\) are the ion and electron temperatures; the prime denotes differentiation with respect to \(r\).

This criterion was obtained in the magnetohydrodynamic approximation (MHD), when the perturbation frequency \(\omega\) is considerably smaller than the ion cyclotron frequency \(\Omega_i=eH/M_i c\), and the ion Larmor radius \(\rho=\sqrt{T_i/M_i\Omega_i^2}\) is much smaller than the perturbation wavelength \(\lambda\sim(m^2/r^2+k^2)^{-1/2}\).

However, if the instability increments are sufficiently small, then the MHD approximation must be refined by taking into account terms small as \((\rho/\lambda)^2\). It is therefore of interest to determine how Suydam’s criterion changes if, following the ideas of the work of Rosenbluth et al. \(^{(2)}\), one takes into account the finiteness of the ratio \((\rho/\lambda)^2\).

For this purpose, starting from the kinetic equation in the absence of collisions, we calculated the currents induced in a plasma of helical geometry by the field of an electromagnetic wave. It was shown that Maxwell’s system of equations can be reduced to the differential equation

\[ \frac{d^2E_n}{dy^2}-k_n^2E_n+\frac{2y}{y^2-b}\frac{dE_n}{dy} +\frac{\varkappa^2}{y^2-b}E_n=0 . \tag{2} \]

Here the following notation has been introduced:

\[ b=\frac{\omega^2}{v_A^2(k_\tau')^2}\left(1-\frac{k_n v_0}{\omega}\right), \]

\[ k_\tau(r_0)=0,\qquad k_\tau(r)=k_\tau'(r_0)y,\qquad y=r-r_0, \tag{3} \]

\[ k_\tau'=-k_n r\mu'\left(\frac{H_z}{H}\right)^2,\qquad k_n=-\frac{m}{r}\frac{H}{H_z},\qquad E_n=\frac{H_\varphi}{H}E_z-\frac{H_z}{H}E_\varphi, \]

\[ v_0=-\frac{1}{\Omega_i n M_i}\frac{d}{dr}(nT_i) \]

is the velocity of the ion Larmor drift,

\[ v_A=\left(\frac{H^2}{4\pi nM_i}\right)^{1/2} \]

is the Alfvén velocity. All coefficients in equation (2) are taken at the point \(r=r_0\), and the prime denotes the derivative with respect to \(r\), taken at \(r=r_0\).

Equation (2) is valid under the following assumptions: \(\omega/\Omega_i\ll 1\); \((\rho/\lambda)^2\ll 1\) (as in the MHD approximation); \(8\pi p\ll H^2\) (the plasma pressure is small in comparison with the magnetic pressure); \(k_\tau\ll k_n\) (the wave vector

almost perpendicular to the magnetic field); \(\omega/k_n \ll v_A\) (the phase velocity of the wave is small compared with the Alfvén velocity), \(\partial E/\partial r \gg E/r\) (the perturbations are strongly localized).

We note that equation (2) differs from the corresponding MHD equation only in that in the MHD approximation \(k_n v_0 \simeq 0\), so that \(b=\omega^2/v_A^2(k'_r)^2\).

In the case \(b<0\) (it is precisely this case that corresponds to instability in the MHD approximation), passing to the new variables \(F=E_n\sqrt{y^2-b}\), \(s^2=-y^2/b\), one can reduce equation (2) to the form

\[ \frac{d^2F}{ds^2}+bk_n^{\prime 2}F+ \left[\frac{\varkappa^2}{s^2+1}-\frac{1}{(s^2+1)^2}\right]F=0. \tag{4} \]

Equation (4) is analogous to the Schrödinger equation for a particle whose energy is

\[ \varepsilon=bk_n^{\prime 2}<0, \tag{5} \]

and the potential is

\[ U=-\frac{\varkappa^2}{s^2+1}+\frac{1}{(s^2+1)^2}. \tag{6} \]

The form of the potential \(U\), and consequently also the eigenvalues of the energy \(\varepsilon=\varepsilon_n\), are the same both in the MHD approximation and in our approximation, which takes into account the finiteness of \((k_n\rho)^2\).

From (3) and (5) we obtain

\[ \omega=\frac{k_n v_0}{2}\pm \sqrt{\left(\frac{k_n v_0}{2}\right)^2+\varepsilon_n \left(\frac{k'_\tau v_A}{k_n}\right)^2}, \tag{7} \]

which in the MHD approximation \((k_n v_0\to 0)\) goes over into the equation

\[ \omega=\pm\frac{k'_\tau v_A}{k_n}\sqrt{\varepsilon_n}. \]

Thus, the MHD approximation gives that the system is unstable if there is a bound state with \(\varepsilon_n<0\). For \(\varkappa^2<1/4\) the potential well does not have sufficient depth for a level to appear, and the plasma is stable \((^{1,3})\).

Let there be instability in the MHD approximation, i.e. \(\varkappa^2>1/4\), and let the well be sufficiently deep for a level with \(\varepsilon_n<0\) to appear. When the finite Larmor radius of the ions is taken into account \((k_n v_0\ne0)\), the plasma will nevertheless be locally stable if the expression under the radical in (7) is positive:

\[ \left(\frac{k_n v_0}{2}\right)^2> |\varepsilon_n|\left(\frac{k'_\tau v_A}{k_n}\right)^2. \tag{8} \]

Since \(|\varepsilon_n|\) in any case does not exceed the magnitude of the minimum of the potential \(U\), it follows from (6) and (8) that the plasma is stable even for large \(\varkappa^2\), if the ion Larmor radius is sufficiently large, so that

\[ \left(\frac{m}{r}\rho\right)^2> \frac{a}{R}\left(1+\frac{T_e}{T_i}\right). \tag{9} \]

Here \(a\) is the characteristic scale of the plasma inhomogeneity, \(a^{-1}=d\ln p/dr\), \(R\) is the radius of curvature of the field line, \(R=r(H/H_\varphi)^2\).

Thus, when the finite ion Larmor radius is taken into account, a stabilizing effect appears, the possibility of which was pointed out in \((^2)\).

We express our gratitude to V. D. Shafranov for posing the problem and for his attention to the work.

Received
14 XII 1962

REFERENCES

1. B. R. Suydam, Proceedings of the 2nd International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958, Selected Reports of Foreign Scientists, 1, Moscow, 1959, p. 89.
2. M. Rosenbluth, N. Krall, N. Rostoker, Nuclear Fusion, Supplements, 1962, book 1, p. 143.
3. B. B. Kadomtsev, JETP, 37, 1646 (1959).