Two-stream instability

In Sec. 3, the equilibrium velocity distribution of electrons is chosen to be Maxwellian, where we see that perturbations are damped, which is the well-known Landau damping. In this section, we investigate a case where perturbation grows, instead of being damped. Consider an equilibrium distribution function consisting of two counter-propagating Maxwellian beams of mean speed $v_b$ and thermal spread $v_t$, i.e.,

$$F_0 (v_z) = \frac{n_0}{2} \left[ \frac{1}{v_t \sqrt{\pi}} \exp \l... ...{1}{v_t \sqrt{\pi}} \exp \left( - \frac{(v_z + v_b)^2}{v_t^2} \right) \right] .$$ (91)

Then $\partial F_0 / \partial v_z$ is written

$$\frac{\partial F_0}{\partial v_z} = \frac{n_0}{2} \left[ \frac{1}... ...z + v_b)^2}{v_t^2} \right) \left( - \frac{2 (v_z + v_b)}{v_t^2} \right) \right]$$ (92)

Using the same code discussed in Sec. 3, I solve Equation (75) with $\partial F_0 / \partial v_z$ given by Eq. (92) and the initial perturbation given by Eq. (78). Figure 10 plots the equilibrium distribution function with $v_b / v_t = 4$.

Figure 10: Equilibrium distribution function given by Eq. (91) with $v_b / v_t = 4$.

Figure 11 plots the time evolution of the perturbed electric field, which shows the the electric field grows exponentially in time and thus corresponds to an instability. This instability is called two-stream instability since it happens in the system with two opposite electron beams.

Figure: Comparison of the simulation results with analytical growth rate given by Eq. (93). The parameters are $k v_t / \omega _p = 0.05$ and $v_b / v_t = 4$.

In Fig. 11, the simulation results are also compared with the analytical results in the cold beam approximation ( $v_t \rightarrow 0$), which is given by equation (8.1.35) in Gurnett's book[1], i.e.,

$$\frac{\gamma}{\omega_p} = \sqrt{\sqrt{1 + 4 \frac{k^2 v^2_b}{\omega_p^2}} - \left( 1 + \frac{k^2 v_b^2}{\omega_p^2} \right)},$$ (93)

To make the simulation result able to be compared with the results in the cold beam approximation, the thermal velocity of the beam has been chosen to be a small number $k v_t / \omega _p = 0.05$. The results in Fig. 11 shows that the simulation results agree with the analytical results. The small discrepancy can be attributed to that equation (93) was derived by assuming the electron distribution function is a Dirac $\delta$ function while in the simulation, the distribution function is a Maxwellian distribution with small a thermal spread $(k v_t / \omega_p = 0.05)$. Also note that in this case, the approximate phase velocity of the electron plasma wave is $v_p = \omega_p / k = 20 v_t$ and the beam velocity $v_b = 4 v_t$. Thus, the distribution function is very small at the phase velocity. Therefore the Landau damping in this case is neglectably small. In fact, equation (93) was derived by neglecting the Landau damping.

Figure: Real part (a) and imaginary part (b) of the perturbed distribution function $\hat{F}_1$ at $t \omega _p = 80$. The parameters are $k v_t / \omega _p = 0.05$ and $v_b / v_t = 4$.