Resistive Wall Physics

Outside the plasma, the perturbed electric field induced by the tearing mode satisfies

$\displaystyle \delta {\bf E} = \nabla\times \delta{\bf B},$ (3.91)

which yields

$\displaystyle {\rm i}\,\frac{m}{r}\,\delta E_z + {\rm i}\,\frac{n}{R_0}\,\delta E_\theta$ $\displaystyle = -\frac{\partial \delta B_r}{\partial t},$ (3.92)
$\displaystyle -{\rm i}\,\frac{n}{R_0}\,\delta E_r - \frac{\partial\delta E_z}{\partial r}$ $\displaystyle = -\frac{\partial\delta B_\theta}{\partial t},$ (3.93)
$\displaystyle \frac{1}{r}\,\frac{\partial}{\partial r}\!\left(r\,\delta E_\theta\right) -{\rm i}\,\frac{m}{r}\,\delta E_r$ $\displaystyle =-\frac{\partial \delta B_z}{\partial t}.$ (3.94)

Making use of Equations (3.32)–(3.34), as well as the ordering $n\,\epsilon\ll m$, the previous three equations imply that

$\displaystyle \delta E_r$ $\displaystyle \simeq 0,$ (3.95)
$\displaystyle \delta E_\theta$ $\displaystyle \simeq -\frac{n\,\epsilon_w}{m}\,\frac{\partial\delta\psi}{\partial t},$ (3.96)
$\displaystyle \delta E_z$ $\displaystyle \simeq -\frac{\partial \delta\psi}{\partial t},$ (3.97)

where $\epsilon_w=r_w/R_0$. Here, we have assumed that $\delta_w\ll r_w$, where $\delta_w$ is the radial thickness of the wall. We have also made use of the fact that $j_\theta=j_z=0$ outside the plasma.

Inside the wall,

$\displaystyle \delta\! j_r$ $\displaystyle =0,$ (3.98)
$\displaystyle \delta \! j_\theta$ $\displaystyle =\frac{\delta E_\theta}{\eta_w},$ (3.99)
$\displaystyle \delta \! j_z$ $\displaystyle =\frac{\delta E_z}{\eta_w},$ (3.100)

where $\eta_w$ is the wall's electrical resistivity. Making use of Equations (3.37), (3.38), (3.96), and (3.97), as well as the fact that $\delta_w\ll r_w$, both of the previous two equations reduce to

$\displaystyle \frac{\partial^2\delta\psi}{\partial r^2} \simeq \frac{\mu_0}{\eta_w}\,\frac{\partial\delta\psi}{\partial t}.$ (3.101)

Let us adopt the so-called thin-wall limit, according to which $\delta\psi$ is assumed to vary only weakly in $r$ across the wall. In this limit, integration of the previous equation across the wall yields

$\displaystyle {\mit\Delta\Psi}_w =\tau_w\,\frac{d{\mit\Psi}_w}{dt},$ (3.102)


$\displaystyle \tau_w = \frac{\mu_0\,r_w\,\delta_w}{\eta_w}$ (3.103)

is the so-called wall time-constant [4,9]. Here, use has been made of Equations (3.82) and (3.83). The thin-wall limit is valid as long as the wall thickness is less than the resistive skin depth in the wall material. In other words, provided that

$\displaystyle \frac{r_w}{\delta_w}\gg \tau_w\left\vert\frac{d\ln{\mit\Psi}_w}{dt}\right\vert.$ (3.104)

In the thin wall limit, Equations (3.82) and (3.95)–(3.100) yield

$\displaystyle \delta\! j_r$ $\displaystyle =0,$ (3.105)
$\displaystyle \delta \! j_\theta$ $\displaystyle = -\frac{n\,\epsilon_w}{m\,\eta_w}\,\frac{d{\mit\Psi}_w}{dt},$ (3.106)
$\displaystyle \delta \! j_z$ $\displaystyle = - \frac{1}{\eta_w}\,\frac{d{\mit\Psi}_w}{dt}$ (3.107)

inside the wall. Note that $\nabla\cdot\delta{\bf j}=0$, as is required by charge conservation.