Friction Force Matrices

Let

$$x_{ss'}=\frac{v_{t\,s'}}{v_{t\,s}}.$$ (A.24)

In the following, all quantities that are of order $(m_e/m_i)^{\,1/2}$, $(m_e/m_I)^{\,1/2}$, or smaller, are neglected with respect to unity. The $2\times 2$ dimensionless ion collisional friction force matrices, $[F_{ii}](r)$, $[F_{iI}](r)$, $[F_{Ii}](r)$, and $[F_{II}](r)$, are defined to have the following elements (see Section 2.16) [7,9]:

$$F_{ii\,11} = \frac{\alpha_I\,(1+m_i/m_I)}{(1+x_{iI}^{\,2})^{3/2}},$$ (A.25)

$$F_{ii\,12} =\frac{3}{2}\,\frac{\alpha_I\,(1+m_i/m_I)}{(1+x_{iI}^{\,2})^{5/2}},$$ (A.26)

$$F_{ii\,21} =\frac{3}{2}\,\frac{\alpha_I\,(1+m_i/m_I)}{(1+x_{iI}^{\,2})^{5/2}},$$ (A.27)

$$F_{ii\,22} =\sqrt{2}+ \frac{\alpha_I\,[13/4+4\,x_{iI}^{\,2}+(15/2)\,x_{iI}^{\,4}]}{(1+x_{iI}^{\,2})^{5/2}},$$ (A.28)

$$F_{iI\,11} = \frac{\alpha_I\,(1+m_i/m_I)}{(1+x_{iI}^{\,2})^{3/2}},$$ (A.29)

$$F_{iI\,12} =\frac{3}{2}\,\frac{T_i}{T_I}\,\frac{\alpha_I\,(1+m_I/m_i)}{x_{iI}\,(1+x_{Ii}^{\,2})^{5/2}},$$ (A.30)

$$F_{iI\,21} =\frac{3}{2}\,\frac{\alpha_I\,(1+m_i/m_I)}{(1+x_{iI}^{\,2})^{5/2}},$$ (A.31)

$$F_{iI\,22} =\frac{27}{4}\,\frac{T_i}{T_I}\,\frac{\alpha_I\,x_{iI}^{\,2}}{(1+x_{iI}^{\,2})^{5/2}},$$ (A.32)

$$F_{Ii\,11} = \frac{\alpha_I\,(1+m_i/m_I)}{(1+x_{iI}^{\,2})^{3/2}},$$ (A.33)

$$F_{Ii\,12} =\frac{3}{2}\,\frac{\alpha_I\,(1+m_i/m_I)}{(1+x_{iI}^{\,2})^{5/2}},$$ (A.34)

$$F_{iI\,21} =\frac{3}{2}\,\frac{T_i}{T_I}\,\frac{\alpha_I\,(1+m_I/m_i)}{x_{iI}\,(1+x_{Ii}^{\,2})^{5/2}},$$ (A.35)

$$F_{Ii\,22} =\frac{27}{4}\,\frac{\alpha_I\,x_{iI}^{\,2}}{(1+x_{iI}^{\,2})^{5/2}},$$ (A.36)

$$F_{II\,11} = \frac{\alpha_I\,(1+m_i/m_I)}{(1+x_{iI}^{\,2})^{3/2}},$$ (A.37)

$$F_{II\,12} =\frac{3}{2}\,\frac{T_i}{T_I}\,\frac{\alpha_I\,(1+m_I/m_i)}{x_{iI}\,(1+x_{Ii}^{\,2})^{5/2}},$$ (A.38)

$$F_{II\,21} =\frac{3}{2}\,\frac{T_i}{T_I}\,\frac{\alpha_I\,(1+m_I/m_i)}{x_{iI}\,(1+x_{Ii}^{\,2})^{5/2}},$$ (A.39)

$$F_{II\,22} =\frac{T_i}{T_I}\left\{\sqrt{2}\,\alpha_I^{\,2}\,x_{Ii} + \frac{\... ...,[ 15/2+4\,x_{iI}^{\,2}+(13/4)\,x_{iI}^{\,4}]}{(1+x_{iI}^{\,2})^{5/2}}\right\}.$$ (A.40)

The $2\times 2$ dimensionless electron collisional friction force matrices, $[F_{ee}](r)$, $[F_{ei}](r)$, and $F_{eI}(r)]$ are defined to have the following elements (see Section 2.16) [7,9]:

$$F_{ee\,11} = Z_{\rm eff},$$ (A.41)

$$F_{ee\,12} = \frac{3}{2}\,Z_{\rm eff},$$ (A.42)

$$F_{ee\,21} = \frac{3}{2}\,Z_{\rm eff},$$ (A.43)

$$F_{ee\,22} = \sqrt{2} +\frac{13}{4}\,Z_{\rm eff},$$ (A.44)

$$F_{ei\,11} = Z_{{\rm eff}\,i},$$ (A.45)

$$F_{ei\,12} =0,$$ (A.46)

$$F_{ei\,21} = \frac{3}{2}\,Z_{{\rm eff}\,i},$$ (A.47)

$$F_{ei\,22} =0,$$ (A.48)

$$F_{eI\,11} = Z_{{\rm eff}\,I},$$ (A.49)

$$F_{eI\,12} =0,$$ (A.50)

$$F_{eI\,21} = \frac{3}{2}\,Z_{{\rm eff}\,I},$$ (A.51)

$$F_{eI\,22} = 0.$$ (A.52)