Friction Force Densities

It is helpful to define the modified collision time [34]

$$\tau_{ss}= \frac{6\!\sqrt{2}\,\pi^{3/2}\,\epsilon_0^{\,2}\,\sqrt{m_s}\,T_s^{3/2}} {e_s^{\,4}\, n_s\,\ln{\mit\Lambda}}.$$ (2.190)

It can be seen, by comparison with Equations (2.20) and (2.21), that $\tau_{ee}=\tau_e$ and $\tau_{ii}= \tau_i/\!\sqrt{2}$.

Our model expressions for the friction force densities, ${\bf F}_{s\,1}$ and ${\bf F}_{s\,2}$, are taken from the moment-based analysis of Hirschman & Sigmar [24,34]. The ion friction force densities are written

$$(F_i) = \frac{n_i\,m_i}{\tau_{ii}}\left\{-[F_{ii}]\,(u_i)\right\} + \frac{n_e\,m_e}{\tau_{ee}}\left\{-[E_{ii}]\,(u_i)+[E_{ie}]\,(u_e)\right\},$$ (2.191)

where

$$(u_s) = \left(\begin{array}{c}{\bf u}_{s\,1}\\ [0.5ex]{\bf u}_{s\,2}\end{array}\right),$$ (2.192)

$$[F_{ii}] = \left[\begin{array}{cc}0,&0\\ [0.5ex]0,&\sqrt{2}\end{array}\right],$$ (2.193)

$$[E_{ii}] = \left[\begin{array}{cc}1,&0\\ [0.5ex]0,&(15/2)\,T_e/T_i\end{array}\right],$$ (2.194)

$$[E_{ie}] = \left[\begin{array}{cc}1,&3/2\\ [0.5ex]0,&0\end{array}\right].$$ (2.195)

The electron friction force densities take the form

$$(F_e) = \frac{n_e\,m_e}{\tau_{ee}}\left\{-[F_{ee}]\,(u_e)+ [F_{ei}]\,(u_i)\right\},$$ (2.196)

where

$$[F_{ee}] = \left[\begin{array}{cc}1,&3/2\\ [0.5ex]3/2,&\sqrt{2}+ 13/4\end{array}\right],$$ (2.197)

$$[F_{ei}] = \left[\begin{array}{cc}1,&0\\ [0.5ex]3/2,&0\end{array}\right].$$ (2.198)

Note that $F_{ii\,1j}=0$, $F_{ee\,1j}=E_{ie\,1j}$, and $F_{ei\,1j}=E_{ii\,1j}$, for $j=1,2$, ensuring that

$${\bf F}_{i\,1}+ {\bf F}_{e\,1} = {\bf0},$$ (2.199)

which is a statement of collisional momentum conservation [18]. Note, further, that $F_{ii\,j1}=0$, $E_{ii\,j1}=E_{ie\,j1}$, and $F_{ee\,j1}=F_{ei\,j1}$, for $j=1,2$, which ensures that ${\bf F}_{s\,1}$ and ${\bf F}_{s\,2}$ are invariant under a Galilean transformation of the coordinate system.