Solution in Presence of External Magnetic Field-Coil

Suppose that the perfectly conducting wall at $r=r_c$ is replaced by a radially-thin, magnetic field-coil that carries a helical current possessing $m$ periods in the poloidal direction, and $n$ periods in the toroidal direction. Let the current density in the field-coil take the form

$$\delta\! j_r =0,$$ (3.109)

$$\delta \! j_\theta = \frac{n\,\epsilon_c}{m}\,\frac{I_c(t)}{r_c}\,\delta(r-r_c),$$ (3.110)

$$\delta \! j_z = \frac{I_c(t)}{r_c}\,\delta(r-r_c),$$ (3.111)

where $\epsilon_c=r_c/R_0$. Here, the complex quantity $I_c(t)$ specifies the amplitude and phase of the helical current flowing in the field-coil. Note that $\nabla\cdot\delta{\bf j}=0$, as is required by charge conservation.

The most general solution to the cylindrical tearing mode equation, (3.60), in the outer region can now be written

$$\delta\psi(r,t) = {\mit\Psi}_s(t)\,\hat{\psi}_s(r) + {\mit\Psi}_w(t)\,\hat{\psi}_w(r) + {\mit\Psi}_c(t)\,\hat{\psi}_c(r),$$ (3.112)

where the real functions $\hat{\psi}_s(r)$ and $\hat{\psi}_w(r)$ are specified in Sections 3.8 and 3.9, respectively. Moreover, the real function $\hat{\psi}_c(r)$ is a solution of

$$\frac{d^2\hat{\psi}_c}{dr^2} + \frac{1}{r}\,\frac{d\hat{\psi}_c}{dr}-\frac{m^2}{r^2}\,\hat{\psi}_c= 0$$ (3.113)

that satisfies

$$\hat{\psi}_c(r\leq r_w) =0,$$ (3.114)

$$\hat{\psi}_c(r_c) =1,$$ (3.115)

$$\hat{\psi}_c(\infty) = 0.$$ (3.116)

It is easily seen that

$$\hat{\psi}_c(r\leq r_w) =0,$$ (3.117)

$$\hat{\psi}_c(r_w<r \leq r_c) = \frac{(r/r_w)^m-(r/r_w)^{-m}}{(r_c/r_w)^m - (r_c/r_w)^{-m}},$$ (3.118)

$$\hat{\psi}_c(r> r_c) =\left(\frac{r}{r_c}\right)^{-m}.$$ (3.119)

In general, $\delta\psi$ is continuous across the field-coil (in accordance with Maxwell's equations), whereas $\partial\delta\psi/\partial r$ is discontinuous. The discontinuity in $\partial\delta\psi/\partial r$ is caused by the helical current flowing in the field-coil. The complex quantity

$${\mit\Psi}_c(t)=\delta\psi(r_c,t)$$ (3.120)

determines the amplitude and phase of the perturbed magnetic flux at the field-coil. The complex quantity

$${\mit\Delta\Psi}_c(t) = \left[r\,\frac{\partial\delta\psi}{\partial r}\right]_{r_{c-}}^{r_{c+}}$$ (3.121)

parameterizes the amplitude and phase of the helical current sheet flowing in the field-coil. It follows from Equations (3.37), (3.38), (3.110), and (3.111) that

$${\mit\Delta\Psi}_c = -\mu_0\,I_c.$$ (3.122)

Simultaneously matching the outer solution, (3.112), across the rational surface, the resistive wall, and the field-coil, we obtain

$${\mit\Delta\Psi}_s = E_{ss}\,{\mit\Psi}_s + E_{sw}\,{\mit\Psi}_w,$$ (3.123)

$${\mit\Delta\Psi}_w = E_{ws}\,{\mit\Psi}_s + E_{ww}\,{\mit\Psi}_w + E_{wc}\,{\mit\Psi}_c,$$ (3.124)

$${\mit\Delta\Psi}_c = E_{cw}\,{\mit\Psi}_w + E_{cc}\,{\mit\Psi}_c.$$ (3.125)

Here,

$$E_{cc} = \left[r\,\frac{d\hat{\psi}_c}{dr}\right]_{r_{c-}}^{r_{c+}}=- \frac{2m}{1-(r_w/r_c)^{2m}},$$ (3.126)

$$E_{wc} = \left[r\,\frac{d\hat{\psi}_c}{dr}\right]_{r_{w+}} = \frac{2m\,(r_w/r_c)^m}{1-(r_w/r_c)^{2m}},$$ (3.127)

$$E_{cw} =- \left[r\,\frac{d\hat{\psi}_w}{dr}\right]_{r_{c-}} = \frac{2m\,(r_w/r_c)^m}{1-(r_w/r_c)^{2m}},$$ (3.128)

where use has been made of Equations (3.81), (3.118), and (3.119).