Phase mixing and linear Landau damping

Consider the case that the equilibrium distribution $F_0$ is Maxwellian in velocity space:

$$F_0 (v_z) = \frac{n_0}{v_t \sqrt{\pi}} \exp \left( - \frac{v^2_z}{v_t^2} \right),$$ (76)

which satisfies the normalization condition $\int_{- \infty}^{\infty} F_0 (v_z) d v_z = n_0$. The derivative of $F_0$ with respect to $v_z$ is written

$$\frac{\partial F_0}{\partial v_z} = \frac{n_0}{v_t \sqrt{\pi}} \exp \left( - \frac{v^2_z}{v_t^2} \right) \left( - \frac{2 v_z}{v_t^2} \right) .$$ (77)

Using this, Eq. (75) is written

$$\frac{\partial \hat{F}_1}{\partial \overline{t}} = - a \overline{... ...2_z) (- 2 \overline{v}_z) \int_{- \infty}^{\infty} \hat{F}_1 d \overline{v}_z .$$ (78)

Take the initial condition of $\hat{F}_1$ to be

$$\hat{F}_1 (\overline{t} = 0, \overline{v}_z) = \frac{1}{\sqrt{\pi}} \exp (- \overline{v}^2_z) + \frac{i}{\sqrt{\pi}} \exp (- \overline{v}^2_z),$$ (79)

which is a Maxwellian distribution that satisfies the normalization $\int_{- \infty}^{\infty} \hat{F}_1 ( \overline{v}_z) d \overline{v}_z = 1 + i$. Equation (78) with the initial condition Eq. (79) was solved numerically to obtain the time evolution of $\hat{F}_1$ (the code is in /home/yj/project/landau_damping/). Figure 7 compares the velocity distribution function at $\overline {t} = 0$ with that at $\overline{t} = 40$, which shows that the distribution function develops fine structures in velocity space.

Figure: Comparison of $\hat{F}_1$ at $t = 0$ and $t \omega _p = 40$. (a) real part; (b) imaginary part. $k v_t / \omega _p = 0.5$.

It is ready to realize that the fine velocity space structures are partially due to the first term on the right-hand side of Eq. (78). When only this term is retained, Eq. (78) is written

$$\frac{\partial \hat{F}_1}{\partial \overline{t}} = - i \left( \frac{k v_t}{\omega_p} \right) \overline{v}_z \hat{F}_1,$$ (80)

which has the dispersion relation

$$\omega = \left( \frac{k v_t}{\omega_p} \right) \overline{v}_z,$$ (81)

which indicates that Eq. (80) has different eigenfrequencies for different points in velocity space. Thus, an initially rather smooth velocity distribution function will become not so smooth after some time due to the distribution function oscillate with different frequencies at different velocity points. This is the so-called ``phase mixing''. It is obvious that, after some time, the phase mixing will make the velocity distribution function $\hat{F}_1 (v_z)$ rather messy, which poses a great challenge to the numerical resolution of $\hat{F}_1 (v_z)$. Given a fixed velocity grids, the numerical results will become inaccurate when the grids is not fine enough to resolve the fine velocity distribution structures.

Note that the electric field is related to the integration of $F_1$, i.e.

$$\hat{E}_1 = \frac{1}{i k} \frac{q}{\varepsilon_0} \int_{- \infty}^{\infty} \hat{F}_1 d v_z$$ (82)

Then it is fairly obvious that the phase mixing have the possibility of reducing the magnitude of the perturbed electric field. Figure 8 plot the time evolution of $\hat{E}_1$ (the factor $q / (k \varepsilon_0)$ in Eq. (82) is removed), which shows that electric field oscillates with the amplitude decreasing with time. This confirms that the phase mixing reduces the magnitude of the electric field.

Figure: (a) time evolution of real and imaginary parts of $\hat{E}_1$. (b) time evolution of $\vert \hat{E}_1 \vert$. $k v_t / \omega _p = 0.5$.