Neoclassical Viscosity Matrices

The $2\times 2$ dimensionless species-$s$ neoclassical viscosity matrix, $[\mu_s](r)$ (see Section 2.17), is defined to have the following elements [7]:

$$\mu_{s\,11} = K_{s\,11},$$ (A.53)

$$\mu_{s\,12} = \frac{5}{2}\,K_{s\,11}- K_{s\,12},$$ (A.54)

$$\mu_{s\,21} = \frac{5}{2}\,K_{s\,11}- K_{s\,12},$$ (A.55)

$$\mu_{s\,22} = K_{s\,22} - 5\,K_{s\,12}+\frac{25}{4}\,K_{s\,11}.$$ (A.56)

(Note that the viscosity matrix elements defined here differ from those defined in Section 2.17 by a factor $g_t$.) Here,

$$K_{e\,jk}(r) = g_t\,\frac{8}{3\!\sqrt{\pi}}\int_0^\infty \frac{e^{-x^{2}}\,x^{... ...D\,e}(x)]\,[x^{4}+(5\pi/8)\,(\omega_{t\,e}\,\tau_{ee})^{\,-1}\,\nu_{T\,e}(x)]},$$ (A.57)

$$\nu_{D\,e}(x) = \frac{3\!\sqrt{\pi}}{4}\left[\left(1-\frac{1}{2\,x^{2}}\right)Y(x)+Y'(x)\right]+\frac{3\!\sqrt{\pi}}{4}\,Z_{\rm eff},$$ (A.58)

$$\nu_{\epsilon\,e}(x) = \frac{3\!\sqrt{\pi}}{2}\left[Y(x)-Y'(x)\right],$$ (A.59)

$$\nu_{T\,s}(x) = 3\,\nu_{D\,s}(x)+\nu_{\epsilon\,s}(x),$$ (A.60)

and

$$Y(x) = \frac{2}{\sqrt{\pi}}\int_0^{x^2}\!\sqrt{t}\,{\rm e}^{-t}\,dt,$$ (A.61)

$$Y'(x) = \frac{2}{\sqrt{\pi}}\,x\,{\rm e}^{-x^{2}}.$$ (A.62)

Furthermore,

$$K_{i\,jk}(r) =g_t\,\frac{8}{3\!\sqrt{\pi}}\int_0^\infty \frac{e^{-x^{2}}\,x^{4... ...D\,i}(x)]\,[x^{3}+(5\pi/8)\,(\omega_{t\,i}\,\tau_{ii})^{\,-1}\,\nu_{T\,i}(x)]},$$ (A.63)

$$\nu_{D\,i}(x) = \frac{3\!\sqrt{\pi}}{4}\left[\left(1-\frac{1}{2\,x^{2}}\right)Y(x)+Y'(x)\right]\frac{1}{x}$$

$$\phantom{===} \phantom{=}+\frac{3\!\sqrt{\pi}}{4}\,\alpha_I\left[\left(1-\frac{... ...left(\frac{x}{x_{iI}}\right) +Y\left(\frac{x}{x_{iI}}\right)\right]\frac{1}{x},$$ (A.64)

$$\nu_{\epsilon\,i}(x) =\frac{3\!\sqrt{\pi}}{2}\left[Y(x)-Y'(x)\right]\frac{1}{x}+\frac{... ...frac{x}{x_{iI}}\right) -\psi'\!\left(\frac{x}{x_{iI}}\right)\right]\frac{1}{x},$$ (A.65)

and, finally,

$$K_{I\,jk}(r) = g_t\,\frac{8}{3\!\sqrt{\pi}}\int_0^\infty \frac{e^{-x^{2}}\,x^{... ...D\,I}(x)]\,[x^{3}+(5\pi/8)\,(\omega_{t\,I}\,\tau_{II})^{\,-1}\,\nu_{T\,I}(x)]},$$ (A.66)

$$\nu_{D\,I}(x) = \frac{3\!\sqrt{\pi}}{4}\left[\left(1-\frac{1}{2\,x^{2}}\right)Y(x)+Y'(x)\right]\frac{1}{x}$$

$$\phantom{===} \phantom{=}+\frac{3\!\sqrt{\pi}}{4}\,\frac{1}{\alpha_I}\left[\lef... ...eft(\frac{x}{x_{Ii}}\right) +Y'\left(\frac{x}{x_{Ii}}\right)\right]\frac{1}{x},$$ (A.67)

$$\nu_{\epsilon\,I}(x) = \frac{3\!\sqrt{\pi}}{2}\left[Y(x)-Y'(x)\right]\frac{1}{x}+\frac... ...eft(\frac{x}{x_{Ii}}\right) -Y'\left(\frac{x}{x_{Ii}}\right)\right]\frac{1}{x}.$$ (A.68)

Note that our expressions for the neoclassical viscosity matrices interpolate in the most accurate manner possible between the three standard neoclassical collsionality regimes (i.e., the banana, plateau, and Pfirsch-Schlüter regimes [8]) [7].