Current-Driven Ion Acoustic Instability

As a second example, consider ion acoustic waves in a plasma with single-charged ions in which the electron velocity distribution function takes the simplified form

$$F_{0\,e}(u_e) = n\,\frac{v_e}{\pi}\,\frac{1}{v_e^{\,2} + (u_e-U_e)^2},$$ (8.134)

and the ion distribution is written

$$F_{0\,i}(u_i) = n\,\frac{v_i}{\pi}\,\frac{1}{v_i^{\,2} + u_i^{\,2}}.$$ (8.135)

Here, $u_e$ and $u_i$ are the parallel (to ${\bf k}$ ) electron and ion velocities, respectively, $n$ is the particle number density, $v_e$ and $v_i$ are the electron and ion thermal spreads, respectively, and $U_e$ is the electron-ion drift velocity. We saw in Section 8.5 that, in the absence of drift, the ion acoustic wave is damped. We now wish to investigate whether the presence of an electron-ion drift (which is associated with a net current flowing in the plasma) can destabilize the mode. The appropriate dispersion relation is Equation (8.49), which, on integration by parts, can be written

$$k^{\,2} = \frac{e^2}{\epsilon_0\,m_e}\int_{-\infty}^{\infty} \fra... ...silon_0\,m_i} \int_{-\infty}^{\infty} \frac{F_{0\,i}}{(u_i-\omega/k)^2}\,d u_i.$$ (8.136)

The previous three equations can be combined together, and the integrals performed as contour integrals in the complex $u_e$ and $u_i$ planes (closed in the lower halves of these planes), making use of the residue theorem (Riley 1974), to give

$$1 = \frac{{\mit\Pi}_e^{\,2}}{(\omega-k\,U_e + {\rm i}\,k\,v_e)^2} + \frac{{\mit\Pi}_i^{\,2}}{(\omega+{\rm i}\,k\,v_i)^2},$$ (8.137)

where we have assumed that $k$ is real and positive, and that $\omega/k$ lies in the upper half of the complex plane. In the limit $k\,\lambda_{D\,e}\ll 1$ , where $\lambda_{D\,e} = v_e/{\mit\Pi}_e$ , the left-hand side of the previous expression is negligible compared to the two terms on the right-hand side, and we obtain

$$\omega + {\rm i}\,k\,v_i \simeq \pm {\rm i}\left(\frac{m_e}{m_i}\right)^{1/2}\,(\omega-k\,U_e+{\rm i}\,k\,v_e).$$ (8.138)

Choosing the negative sign, which ensures that the phase-velocity is in the correct direction, we get

$$\omega\simeq k\left(\frac{m_e}{m_i}\right)^{1/2}v_e + {\rm i}\,k\left[\left(\frac{m_e}{m_i}\right)^{1/2}U_e-v_i\right].$$ (8.139)

If we write $v_e=(T_e/m_e)^{1/2}$ and $v_i=(T_i/m_i)^{1/2}$ , where $T_e$ and $T_i$ are the effective electron and ion temperatures, then the previous expression yields

$$\omega\simeq k\left(\frac{T_e}{m_i}\right)^{1/2} + {\rm i}\,k\lef... ...ac{m_e}{m_i}\right)^{1/2}\left[U_e - \left(\frac{T_i}{m_e}\right)^{1/2}\right].$$ (8.140)

Thus, the phase-velocity of the wave is $(T_e/m_i)^{1/2}$ , whereas the growth-rate is

$$\gamma = k\left(\frac{m_e}{m_i}\right)^{1/2}\left[U_e - \left(\frac{T_i}{m_e}\right)^{1/2}\right].$$ (8.141)

It can be seen that the growth-rate becomes positive (i.e., the mode becomes unstable) when the drift velocity exceeds the critical value

$$U_{e\,c} = \left(\frac{T_i}{m_e}\right)^{1/2} = \left(\frac{T_i}{T_e}\right)^{1/2} v_e.$$ (8.142)

This calculation indicates that if the electron and ion temperatures are similar then the threshold drift velocity is of order the electron thermal speed, which is usually very large. In other words, a significant current is generally required to drive the ion acoustic wave unstable. The instability threshold (relative to the electron thermal speed) is considerably reduced if the electron temperature greatly exceeds the ion temperature.

If we repeat the previous calculation using the more realistic Maxwellian velocity distributions,

$$F_{0\,e}(u_e) = \frac{n}{(2\pi\,T_e/m_e)^{1/2}}\exp\left[-\frac{m_e\,(u_e-U_e)^{\,2}}{2\,T_e}\right],$$ (8.143)

and

$$F_{0\,i}(u_i) = \frac{n}{(2\pi\,T_i/m_i)^{1/2}}\exp\left(-\frac{m_i\,u_i^{\,2}}{2\,T_i}\right),$$ (8.144)

then the dispersion relation (8.136) yields

$$1= \frac{Z'(\zeta_e)}{2\,(k\,\lambda_{D\,e})^{\,2}} + \frac{Z'(\zeta_i)}{2\,(k\,\lambda_{D\,i})^{\,2}},$$ (8.145)

where $\lambda_{D\,s}=(T_s/m_s\,{\mit\Pi}_s^{\,2})^{1/2}$ , $\zeta_e = (m_e/2\,T_e)^{1/2}\,(\omega/k-U_e)$ , and $\zeta_i=(m_i/2\,T_i)\,(\omega/k)$ . As in Section 8.5, we assume that the phase-velocity of the wave is much less than the electron thermal velocity, but much greater than the ion thermal velocity. This implies that $\vert\zeta_e\vert\ll 1$ and $\vert\zeta_i\vert\gg 1$ . Using the small-argument expansion

$$Z'(\zeta_e) \simeq - {\rm i}\,2\!\sqrt{\pi}\,\zeta_e\,{\rm e}^{-\zeta_e^{\,2}} - 2,$$ (8.146)

and the large-argument expansion

$$Z'(\zeta_i)\simeq - {\rm i}\,2\!\sqrt{\pi}\,\zeta_i\,{\rm e}^{-\zeta_i^{\,2}} + \frac{1}{\zeta_i^{\,2}},$$ (8.147)

we obtain

$$2\,(k\,\lambda_{D\,e})^{\,2} \simeq \frac{T_e}{T_i}\,\frac{1}{\ze... ...{\pi}\left( \frac{T_e}{T_i}\,\zeta_i\,{\rm e}^{-\zeta_i^{\,2}} +\zeta_e\right).$$ (8.148)

In the limit $k\,\lambda_{D\,e}\ll 1$ , the previous expression yields $\omega=\omega_r + {\rm i}\,\gamma$ , where

$$\omega_r \simeq k\,c_s,$$ (8.149)

and

$$\frac{\gamma}{\omega_r} \simeq -\sqrt{\frac{\pi}{8}}\left[\left(\... ...\left(\frac{T_e}{T_i}\right)^{3/2} \exp\left(-\frac{T_e}{2\,T_i}\right)\right],$$ (8.150)

Here, $c_s=(T_e/m_i)^{1/2}$ is the phase-velocity of the ion acoustic wave, and it is assumed that $\vert\gamma\vert/\omega_r\ll 1$ . The ion acoustic wave phase-velocity is much less than the electron thermal speed, as previously assumed, but is only much greater than the ion thermal speed if $T_e\gg T_i$ . According to Equation (8.150), the threshold electron-ion drift speed above which the ion acoustic wave is destabilized is

$$U_{e\,c}= c_s\left[1+\left(\frac{m_i}{m_e}\right)^{1/2}\,\left(\frac{T_e}{T_i}\right)^{3/2} \exp\left(-\frac{T_e}{2\,T_i}\right)\right].$$ (8.151)

As before, this formula (which is only accurate when $T_e\gg T_i$ ) indicates that the threshold is strongly reduced (relative to $c_s$ ) as the ratio of the electron to the ion temperature is increased.