Collisional damping rates for electron plasma waves reassessed

Collisional damping of electron plasma waves, the primary damping for high phase velocity waves, is proportional to the electron-ion collision rate, ${\nu }_{\mathrm{ei},\mathrm{th}}$. Here, it is shown that the damping rate normalized to ${\nu }_{\mathrm{ei},\mathrm{th}}$ depends on the charge state, $Z$, on the magnitude of ${\nu }_{\mathrm{ei},\mathrm{th}}$ and the wave number $k$ in contrast with the commonly used damping rate in plasma wave research. Only for weak collision rates in low-$Z$ plasmas for which the electron self-collision rate is comparable to the electron-ion collision rate is the damping rate given by the commonly accepted value. The result presented here corrects the result presented in textbooks at least as early as 1973. The complete linear theory requires the inclusion of both electron-ion pitch-angle and electron-electron scattering, which itself contains contributions to both pitch-angle scattering and thermalization.

Figure 1

(a) The frequency difference, $\mathrm{\Delta }\omega ={\omega }_R-{\omega }_{R,\text{kin}}$, and (b) the total EPW damping rate, $\nu$, is shown as a function of ${\nu }_{\mathrm{ei},\mathrm{th}}/{\omega }_{pe}$ for $Z=1,6,16$ and $k{\lambda }_{De}=0.3$. The solid double horizontal line separates (a) and (b). For higher $Z$ values, the electron-ion scattering is the dominant source of collisional damping. The linear Landau damping rate ${\nu }_L$ for $k{\lambda }_{De}=0.3$ in a collisionless plasma is shown in (b) by the horizontal solid red line. The result with pitch-angle collisions (${\nu }_{\text{ei}}\ne 0$) but no self-collisions (${\nu }_{\text{ee}}=0$, i.e., $Z=\infty$) is also shown. For $Z=1$ the damping from a loki simulation restricted to 2V Cartesian velocity coordinates is also shown.

Figure 2

The total ($\nu /{\omega }_{pe}$) and the collisional component of the damping rate of EPWs for (a) $Z=1$ and (b) $Z=16$ for $k{\lambda }_{De}=0.3$. The linear Landau damping rate ${\nu }_{L,\text{kin}}$ for $k{\lambda }_{De}=0.3$ is displayed as a horizontal line. The collisional component of the Vlasov simulation damping rate ${\nu }_{\text{coll}}=\nu -{\nu }_{L,\text{kin}}$ for $k{\lambda }_{De}=0.3$ is shown along with the hydrodynamic damping rate, ${\nu }_{\text{coll}}^{\text{fl}}$ (red dashed). Also displayed in (b) is the total damping rate for $k{\lambda }_{De}=0.2$, which is essentially the same as the collisional component for $k{\lambda }_{De}=0.3$, as the Landau damping for $k{\lambda }_{De}=0.2$ is exceedingly small. The numerical solution to Eqs. (8)–(11) for $k{\lambda }_{De}=0.2$ (${\nu }_{\text{pf}}$) is shown in (b). It agrees very well with the corresponding Vlasov simulation result in (b) and with the Vlasov simulation result for $Z=\infty$, that is, ${\nu }_{\text{ee}}=0$.

Figure 3

The collisional component of the damping normalized to the fluid rate ${\nu }_{\mathrm{coll}}^{\text{fl}}$ as a function of $k{\lambda }_{De}$. For $Z=16$, the collisional damping rate decreases with $k^2{\lambda }_{De}^2$ for ${\nu }_{\mathrm{ei},\mathrm{th}}/{\omega }_{pe}=0.01$ and 0.001. For $Z=1$, the collisional damping rate is apparently independent of $k{\lambda }_{De}$ for ${\nu }_{\mathrm{ei},\mathrm{th}}/{\omega }_{pe}=0.01$ and 0.001. The $Z=16$ damping rate is well approximated by the $Z=\infty$ (${\nu }_{\text{ee}}=0$) results shown by the blue and red lines for ${\nu }_{\mathrm{ei},\mathrm{th}}/{\omega }_{pe}=0.001$ and 0.01, respectively.

Figure 4

The collisional component of the damping normalized to the fluid rate ${\nu }_{\text{coll}}^{\text{fl}}$ as a function of $Z$. For $Z>10$ and $k{\lambda }_{De}=0.3$, the collisional damping rate is ${\nu }_{\text{coll}}\sim 0.7{\nu }_{\text{coll}}^{\text{fl}}$ and ${\nu }_{\text{coll}}\sim 0.8{\nu }_{\text{coll}}^{\text{fl}}$ for ${\nu }_{\mathrm{ei},\mathrm{th}}=0.1$ and 0.01, respectively. For $Z>10$ and $k{\lambda }_{De}=0.2$, the collisional damping rate is ${\nu }_{\text{coll}}\sim 0.8{\nu }_{\text{coll}}^{\text{fl}}$ for ${\nu }_{\mathrm{ei},\mathrm{th}}=0.1$ and ${\nu }_{\text{coll}}\sim 0.9{\nu }_{\text{coll}}^{\text{fl}}$ for ${\nu }_{\mathrm{ei},\mathrm{th}}\lesssim 0.01$.