Harris Instability

It is not feasible to give a comprehensive account of velocity-space instabilities in a magnetized plasma, on account of the great number of different instabilities of this type. Rather than trying to analyze the full electromagnetic dispersion relation, we shall concentrate on the stability of electrostatic waves. Instabilities of this type tend to be more important than electromagnetic instabilities, particularly in low-$\beta$ plasmas (Cairns 1985). Our starting point is the Harris dispersion relation, (7.119):

$$\epsilon(k,\omega) =1+ \sum_s \frac{e_s^{2}}{k^2\,\epsilon_0\,m_s}\sum_{n=-\infty,\i... ...}+ k_\parallel\,\frac{\partial f_{0\,s}}{\partial v_\parallel}\right)d^3{\bf v}$$

$$= 0.$$ (7.152)

Making use of the Plemelj formula, (7.26), we can write the previous expression in the form

$$\epsilon(k,\omega)= \epsilon_r(k,\omega)+{\rm i}\,\epsilon_i(k,\omega) = 0,$$ (7.153)

where

$$\epsilon_r(k,\omega)=1+ \sum_s \frac{e_s^{2}}{k^{2}\,\epsilon_0\,... ...+ k_\parallel\,\frac{\partial f_{0\,s}}{\partial v_\parallel}\right)d^3{\bf v},$$ (7.154)

and

$$\epsilon_i(k,\omega) =-\pi \sum_s \frac{e_s^{2}}{k^{2}\,\epsilon_0\,m_s}\sum_{n=-\inft... ...erp/{\mit\Omega}_s)\, \delta(\omega-k_\parallel\,v_\parallel-n\,{\mit\Omega}_s)$$

$$\phantom{\pi \sum_s \frac{e_s^{2}}{k^{2}\,\epsilon_0\,m_s}\sum_{n... ...+ k_\parallel\,\frac{\partial f_{0\,s}}{\partial v_\parallel}\right)d^3{\bf v}.$$ (7.155)

Generally speaking, we expect $\vert\epsilon_i\vert\ll \vert\epsilon_r\vert$. Let us search for an instability whose angular frequency is $\omega=\omega_r + {\rm i}\,\gamma$, where $\omega_r$ is real and positive, $\gamma$ is real, and $\vert\gamma\vert\ll \omega_r$. Thus, $\omega_r$ is the real frequency of the instability, $\omega_r/k$ its phase-velocity, and $\gamma$ its growth-rate. Expanding (7.153) to first-order in $\gamma$, we obtain

$$\epsilon(k,\omega_r+{\rm i}\,\gamma) \simeq \epsilon_r(k,\omega_r... ...ial\epsilon_r(k,\omega_r)}{\partial\omega} + {\rm i}\,\epsilon_i(k,\omega_r)=0.$$ (7.156)

Here, the quantities $\epsilon_r(k,\omega_r)$, $\partial\epsilon_r(k,\omega_r)/\partial\omega$, and $\epsilon_i(k,\omega_r)$ are all real. Thus, the real frequency of the instability is determined from

$$\epsilon_r(k,\omega_r) =0,$$ (7.157)

whereas the growth-rate is given by

$$\gamma = -\frac{\epsilon_i(k,\omega_r)}{\partial\epsilon_r(k,\omega_r)/\partial\omega}.$$ (7.158)

Consider the so-called Harris instability, which occurs for real frequencies close to the ion cyclotron harmonics in a plasma in which the parallel and perpendicular (to the equilibrium magnetic field) temperatures are different (Harris 1970; Cairns 1985). Suppose that the equilibrium velocity distribution functions are two-temperature Maxwellians of the form

$$f_{0\,s} = \frac{n_s}{\left(2\pi\,T_{\perp\,s}^{2/3}\,T_{\paralle... ...{2}}{2\,T_{\perp\,s}}- \frac{m_s\,v_\parallel^{2}}{2\,T_{\parallel\,s}}\right).$$ (7.159)

Here, $T_{\perp\,s}$ and $T_{\parallel\,s}$ are the species-$s$ perpendicular and parallel temperatures, respectively. It follows that

$$\epsilon_r(k,\omega) = 1+\sum_s\frac{{\mit\Pi}_s^{2}}{(k\,v_{\parallel\,s})^2}\sum_{n=... ...ht)I_n\left(-\frac{k_\perp^{2}\,v_{\perp\,s}^{2}}{2\,{\mit\Omega}_s^{2}}\right)$$

$$\phantom{=1+} \times\left[\frac{2\,n\,{\mit\Omega}_s}{k_\parallel... ...( \frac{\omega-n\,{\mit\Omega}_s}{k_\parallel\,v_{\parallel\,s}}\right)\right],$$ (7.160)

and

$$\epsilon_i(k,\omega) =\sum_s\frac{{\mit\Pi}_s^{2}}{(k\,v_{\parallel\,s})^2}\sum_{n=-\i... ...ht)I_n\left(-\frac{k_\perp^{2}\,v_{\perp\,s}^{2}}{2\,{\mit\Omega}_s^{2}}\right)$$

$$\phantom{=}\times \left[\frac{2\,n\,{\mit\Omega}_s}{k_\parallel\,... ...( \frac{\omega-n\,{\mit\Omega}_s}{k_\parallel\,v_{\parallel\,s}}\right)\right],$$ (7.161)

where ${\mit\Pi}_s = (n_s\,e_s^{2}/\epsilon_0\,m_s)^{1/2}$, $v_{\perp\,s} = (2\,T_{\perp\,s}/m_s)^{1/2}$, and $v_{\parallel\,s} = (2\,T_{\parallel\,s}/m_s)^{1/2}$. Moreover, $Z_r$ and ${\rm i}\,Z_i$ denote the principal part and the remainder of the plasma dispersion function, respectively. However, according to Section 7.4,

$$Z_i(\zeta) = \pi^{1/2}\,{\rm e}^{-\zeta^{2}},$$ (7.162)

$$Z_i'(\zeta) = -2\,\pi^{1/2}\,\zeta\,{\rm e}^{-\zeta^{2}}.$$ (7.163)

Hence, we can write

$$\epsilon_i(k,\omega) =2\pi^{1/2}\sum_s\frac{{\mit\Pi}_s^{2}}{(k\,v_{\parallel\,s})^2}\... ...ht)I_n\left(-\frac{k_\perp^{2}\,v_{\perp\,s}^{2}}{2\,{\mit\Omega}_s^{2}}\right)$$

$$\phantom{=} \times\left[\frac{n\,{\mit\Omega}_s}{k_\parallel\,v_{... ...\frac{\omega-n\,{\mit\Omega}_s}{k_\parallel\,v_{\parallel\,s}}\right)^2\right].$$ (7.164)

Suppose, for the sake of simplicity, that the electrons are “cold” but the ions are “hot.” In other words, $v_{\perp\,s},\,v_{\parallel\,s}\rightarrow 0$ for the electrons, but not for the ions. In this situation, Equation (7.160) reduces to

$$\epsilon_r(k,\omega) \simeq 1- \frac{{\mit\Pi}_e^{2}}{\omega^{2}}\,\frac{k_\parallel^{2}}{k^{2}}.$$ (7.165)

The ion contribution to the previous expression is negligible compared to the electron contribution, because of the $m_s^{-1}$ dependence of ${\mit\Pi}_s^{2}$. The real frequency of the instability is determined from

$$\epsilon_r(k,\omega_r) \simeq 1- \frac{{\mit\Pi}_e^{2}}{\omega_r^{2}}\,\frac{k_\parallel^{2}}{k^{2}}=0,$$ (7.166)

which implies that

$$\omega_r \simeq {\mit\Pi}_e\,\frac{k_\parallel}{k}.$$ (7.167)

Furthermore,

$$\frac{\partial\epsilon_r(k,\omega_r)}{\partial\omega}\simeq \frac{2}{\omega_r}.$$ (7.168)

Hence, the growth-rate of the instability is written

$$\gamma = -\frac{\epsilon_i(k,\omega_r)}{\partial\epsilon_r(k,\omega_r)/\partial\omega} = -\frac{\omega_r}{2}\,\epsilon_i(k,\omega_r),$$ (7.169)

or

$$\frac{\gamma}{\omega_r} \simeq\pi^{1/2}\sum_s\frac{{\mit\Pi}_i^{2}}{(k\,v_{\parallel\,i})... ...ht)I_n\left(-\frac{k_\perp^{2}\,v_{\perp\,i}^{2}}{2\,{\mit\Omega}_i^{2}}\right)$$

$$\phantom{==} \times\left[\frac{n\,{\mit\Omega}_i}{k_\parallel\,v_... ...rac{\omega_r-n\,{\mit\Omega}_i}{k_\parallel\,v_{\parallel\,i}}\right)^2\right].$$ (7.170)

The electron contribution to the previous expression is negligible compared to the ion contribution, because $v_{\parallel\,e}\rightarrow 0$.

Figure 7.11: Schematic diagram showing the growth-rate of the Harris instability as a function of its real frequency.

It can be seen, from the previous formula, that if $T_{\parallel\,i}=T_{\perp\,i}$ then $\gamma<0$ for all values of $\omega_r$ (recall that $\omega_r>0$). In other words, there is no instability if the perpendicular and parallel ion temperatures are equal to one another. On the other hand, if $T_{\parallel\,i}< T_{\perp\,i}$ then there is a range of $\omega_r$ values for which each term in the sum on the right-hand side of (7.170) is positive. In other words, there is the possibility of an instability. The variation of $\gamma/\omega_r$ with $\omega_r$ is shown schematically in Figure 7.11 for a case where ${\mit\Omega}_i/(k_\parallel\,v_{\parallel\,i})$ is relatively large. It can be seen that the growth-rate is positive in a narrow range of real frequencies lying on the low frequency side of each harmonic of the ion cyclotron frequency, and negative in a similar range of frequencies on the high frequency side.

According to Equation (7.167), $\omega_r$ varies from zero to ${\mit\Pi}_e$. Thus, a necessary condition for obtaining an instability close to the $n$th ion cyclotron harmonic is ${\mit\Pi}_e> n\,{\mit\Omega}_i$. Now, the positive contribution from the $n$th term in the sum on the right-hand side of Equation (7.170) peaks close to

$$\omega_r= \omega_{r\,n} \equiv n\,{\mit\Omega}_i\left(1-\frac{T_{\parallel\,i}}{T_{\perp\,i}}\right).$$ (7.171)

In order for the net growth-rate to be positive, we require that

$$\omega_{r\,n} > \left(n-\frac{1}{2}\right){\mit\Omega}_i,$$ (7.172)

which implies that (Harris 1970)

$$\frac{T_{\parallel\,i}}{T_{\perp\,e}}< \frac{1}{2\,n}.$$ (7.173)

If this inequality is not satisfied then the negative contribution from the $(n-1)$th term in the sum predominates over the positive contribution from the $n$th term. Observe that the previous inequality becomes harder to satisfy as $n$ increases.