Solution in Presence of Perfectly Conducting Wall

Suppose that the plasma occupies the region $0\leq r\leq a$, where $a$ is the plasma minor radius. It follows that $p(r)=j_\theta(r)=j_z(r) =J_z(r)=0$ for $r>a$. Let the plasma be surrounded by a concentric, rigid, radially-thin, perfectly conducting wall of radius $r_w>a$. (In most circumstances, the wall represents the metallic vacuum vessel that surrounds the plasma.) An appropriate physical solution of the cylindrical tearing mode equation, (3.60), takes the separable form

$\displaystyle \delta\psi(r,t)= {\mit\Psi}_s(t)\,\hat{\psi}_s(r),$ (3.63)

where the real function $\hat{\psi}_s(r)$ is a solution of

$\displaystyle \frac{d^2\hat{\psi}_s}{dr^2} + \frac{1}{r}\,\frac{d\hat{\psi}_s}{dr}-\frac{m^2}{r^2}\,\hat{\psi}_s - \frac{J_z'\,\hat{\psi}_s}{r\,(1/q-n/m)}= 0$ (3.64)

that satisfies

$\displaystyle \hat{\psi}_s(0)$ $\displaystyle =0,$ (3.65)
$\displaystyle \hat{\psi}_s(r_s)$ $\displaystyle =1,$ (3.66)
$\displaystyle \hat{\psi}_s(r\geq r_w)$ $\displaystyle = 0.$ (3.67)

Equation (3.65) ensures that the perturbed magnetic field associated with the tearing mode remains finite at the magnetic axis ($r=0$), whereas Equation (3.67) represents the physical constraint that the perturbed magnetic field cannot penetrate a perfectly conducting wall.

Let $\rho=(r-r_s)/r_s$. The solution of Equation (3.64) in the vicinity of the rational surface is

$\displaystyle \hat{\psi}_s(\rho) = 1 + {\mit\Delta}_{s+}\,\rho + \alpha_s\,\rho\,\ln\vert\rho\vert + {\cal O}(\rho^{\,2})$ (3.68)

for $\rho>0$, and

$\displaystyle \hat{\psi}_s(\rho) = 1 + {\mit\Delta}_{s-}\,\rho + \alpha_s\,\rho\,\ln\vert\rho\vert + {\cal O}(\rho^{\,2})$ (3.69)

for $\rho<0$. Here,

$\displaystyle \alpha_s$ $\displaystyle =- \left(\frac{r\,q\,J_z'}{s}\right)_{r=r_s},$ (3.70)
$\displaystyle s(r)$ $\displaystyle = \frac{r\,q'}{q}.$ (3.71)

Moreover, the real parameters ${\mit\Delta}_{s+}$ and ${\mit\Delta}_{s-}$ are fully determined by Equation (3.64) and the boundary conditions (3.65)–(3.67). Note that, in general, $\delta\psi\propto \delta B_r$ is continuous across the rational surface (in accordance with Maxwell's equations), whereas $\partial\delta\psi/\partial r$ is discontinuous. The discontinuity in $\partial\delta\psi/\partial r$ implies the presence of a radially-thin, helical current sheet at the rational surface. This current sheet is resolved in a thin resistive layer that, in principle, can only be described by employing the full set of neoclassical fluid equations, (2.370)–(2.374), rather than the reduced set of marginally-stable ideal-MHD equations, (2.375)–(2.380).

The value of $\delta\psi(r,t)$ at the rational surface,

$\displaystyle {\mit\Psi}_s(t)=\delta\psi(r_s,t),$ (3.72)

is known as the reconnected magnetic flux [4]. Note that ${\mit\Psi}_s$ is, in general, a complex quantity. The complex quantity

$\displaystyle {\mit\Delta\Psi}_s(t) = \left[r\,\frac{\partial \delta\psi}{\partial r}\right]_{r_{s-}}^{r_{s+}}$ (3.73)

parameterizes the amplitude and phase of the current sheet flowing (parallel to the equilibrium magnetic field) at the rational surface. Asymptotically matching the solutions in the so-called inner region (i.e., the region of the plasma in the immediate vicinity of the rational surface) and the so-called outer region (i.e., everywhere in the plasma other than the inner region) with the help of Equations (3.63), (3.68), (3.69), and (3.73), we obtain

$\displaystyle {\mit\Delta\Psi}_s = E_{ss}\,{\mit\Psi}_s,$ (3.74)


$\displaystyle E_{ss}= \left[r\,\frac{d\hat{\psi}_s}{dr}\right]_{r_{s-}}^{r_{s+}}={\mit\Delta}_{s+}-{\mit\Delta}_{s-}$ (3.75)

is a real dimensionless quantity that is known as the tearing stability index [7].