Fluid Velocities

The flux-surface functions $M({\mit\Omega},T)$ and $L({\mit\Omega},T)$ are directly related to the lowest-order perpendicular velocities of the various plasma species in the island rest frame. In fact, as is clear from Sections 4.2, 5.2, and 8.2,

$$V_{\perp\,\theta\,e}(X,\zeta,T) = -\frac{r_s\,\omega_\ast}{m}\,\vert X\vert\,Y_e({\mit\Omega},T),$$ (8.72)

$$V_{\perp\,\theta}(X,\zeta,T) = -\frac{r_s\,\omega_\ast}{m}\,\vert X\vert\,M({\mit\Omega},T),$$ (8.73)

$$V_{\perp\,\theta\,i}(X,\zeta,T) = -\frac{r_s\,\omega_\ast}{m}\,\vert X\vert\,Y_i({\mit\Omega},T),$$ (8.74)

where

$$Y_e({\mit\Omega},T) = M- \left(\frac{\tau}{1+\tau}\right)L,$$ (8.75)

$$Y_i({\mit\Omega},T) = M+\left(\frac{1}{1+\tau}\right)L,$$ (8.76)

Here, $V_{\perp\,\theta\,e}$, $V_{\perp\,\theta}$, and $V_{\perp\,\theta\,i}$ refer to the $\theta$ components of the perpendicular electron, MHD, and ion fluid velocities, respectively. Note that $Y_e$, $M$, and $Y_i$ are all zero inside the magnetic separatrix of the island chain.