Resistive Wall Physics

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

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

which yields

$${\rm i}\,\frac{m}{r}\,\delta E_z + {\rm i}\,\frac{n}{R_0}\,\delta E_\theta = -\frac{\partial \delta B_r}{\partial t},$$ (3.92)

$$-{\rm i}\,\frac{n}{R_0}\,\delta E_r - \frac{\partial\delta E_z}{\partial r} = -\frac{\partial\delta B_\theta}{\partial t},$$ (3.93)

$$\frac{1}{r}\,\frac{\partial}{\partial r}\!\left(r\,\delta E_\theta\right) -{\rm i}\,\frac{m}{r}\,\delta E_r =-\frac{\partial \delta B_z}{\partial t}.$$ (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

$$\delta E_r \simeq 0,$$ (3.95)

$$\delta E_\theta \simeq -\frac{n\,\epsilon_w}{m}\,\frac{\partial\delta\psi}{\partial t},$$ (3.96)

$$\delta E_z \simeq -\frac{\partial \delta\psi}{\partial t},$$ (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,

$$\delta\! j_r =0,$$ (3.98)

$$\delta \! j_\theta =\frac{\delta E_\theta}{\eta_w},$$ (3.99)

$$\delta \! j_z =\frac{\delta E_z}{\eta_w},$$ (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

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

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

where

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

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

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

$$\delta\! j_r =0,$$ (3.105)

$$\delta \! j_\theta = -\frac{n\,\epsilon_w}{m\,\eta_w}\,\frac{d{\mit\Psi}_w}{dt},$$ (3.106)

$$\delta \! j_z = - \frac{1}{\eta_w}\,\frac{d{\mit\Psi}_w}{dt}$$ (3.107)

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