Parallel Viscous Force Densities

It is helpful to define the average fraction of passing particles on a magnetic flux-surface:

$\displaystyle f_p = \frac{3}{4}\,\langle B^2\rangle\int_0^{1/B_{\rm max}} \frac{\lambda\,d\lambda}{\langle
\!\sqrt{1-\lambda\,B}\rangle},$ (2.200)

where $B_{\rm max}$ is the maximum value of $B$ on the surface [34]. For a flux-surface with a circular poloidal cross-section [38],

$\displaystyle f_p = 1-1.46\,\epsilon^{1/2} + 0.46\,\epsilon^{3/2}.$ (2.201)

Thus, the average fraction of trapped particles on the flux-surface is

$\displaystyle f_t \equiv 1-f_p =1.46\sqrt{\epsilon} - 0.46\,\epsilon^{3/2}.$ (2.202)

Note that the previous expression differs slightly from the less exact expression (2.87).

In the so-called banana collisionality regime, $\nu_{\ast\,s}\ll 1$ [see Equation (2.95)], the flux-surface averaged parallel viscous force densities take the form [22,34,43]

$\displaystyle \left\langle {\bf B}\cdot\nabla\cdot ({\mit\pi}_{\parallel\,s})\right\rangle = \frac{n_s\,m_s}{\tau_{ss}}\,f_t\,[\mu_s]\,(u_{\theta\,s}).$ (2.203)

The normalized viscosity coefficients for the ions are

$\displaystyle \mu_{i\,11}$ $\displaystyle = \int_0^\infty {\rm e}^{-x}
\left[\left(1-\frac{1}{2\,x}\right)Y(\!\sqrt{x}) + Y'(\!\sqrt{x})\right]dx=0.533,$ (2.204)
$\displaystyle \mu_{i\,12} =\mu_{i\,21}$ $\displaystyle = \int_0^\infty {\rm e}^{-x}\left(\frac{5}{2}-x\right)\left[\left(1-\frac{1}{2\,x}\right)Y(\!\sqrt{x}) + Y'(\!\sqrt{x})\right]dx=0.625,$ (2.205)
$\displaystyle \mu_{i\,22}$ $\displaystyle = \int_0^\infty {\rm e}^{-x}\left(\frac{5}{2}-x\right)^2\left[\left(1-\frac{1}{2\,x}\right)Y(\!\sqrt{x}) + Y'(\!\sqrt{x})\right]dx=1.386,$ (2.206)


$\displaystyle Y(x)$ $\displaystyle = \frac{2}{\sqrt{\pi}}\int_0^{x^{\,2}}\!\!\!\sqrt{t}\,{\rm e}^{-t}\,dt,$ (2.207)
$\displaystyle Y'(x)$ $\displaystyle = \frac{2}{\sqrt{\pi}}\,x\,{\rm e}^{-x^{\,2}}.$ (2.208)

The normalized viscosity coefficients for the electrons are

$\displaystyle \mu_{e\,11}$ $\displaystyle =\mu_{i\,11} + \int_0^\infty {\rm e}^{-x}\,dx = 1.533,$ (2.209)
$\displaystyle \mu_{e\,12} =\mu_{e\,21}$ $\displaystyle =\mu_{i\,12} +\int_0^\infty \left(\frac{5}{2}-x\right){\rm e}^{-x}\,dx = 2.125,$ (2.210)
$\displaystyle \mu_{e\,22}$ $\displaystyle =\mu_{i\,22}+ \int_0^\infty \left(\frac{5}{2}-x\right)^2{\rm e}^{-x}\,dx = 4.636.$ (2.211)

In the banana collisionality regime, the parallel viscous force arises from collisional drag between passing and trapped particles [22,43]. The origin of this drag is the fact that, while passing particles can drift along magnetic field-lines in one direction, trapped particles are forced to periodically reverse direction. Not surprisingly, the viscous force density is proportional to the fraction of trapped particles, $f_t$. [See Equation (2.203).]