Asymptotic Matching

Consider a tearing-stable, cylindrical, tokamak plasma of minor radius $a$, and simulated major radius $R_0$. In accordance with the analysis of Chapter 3, suppose that the plasma is surrounded by a concentric, radially-thin, rigid, resistive wall of minor radius $r_w>a$. Likewise, suppose that the wall is surrounded by a concentric, radially-thin, magnetic field-coil, of minor radius $r_c>r_w$, that carries a non-rotating (in the laboratory frame) helical current of amplitude $I_c$. Let the current possess $m$ periods in the poloidal direction, and $n$ periods in the toroidal direction. The static magnetic field generated by the field-coil constitutes our error-field. The error-field resonates with the plasma at the rational magnetic flux-surface, minor radius $r_s$, at which the safety-factor (see Section 3.2) takes the value $m/n$.

Setting $d/dt=0$ (because the error-field is static) in Equations (3.187) and (3.188), we obtain

$${\mit\Delta\skew{3}\hat{\Psi}}_s = {\mit\Delta}_{nw}\,\skew{3}\hat{\mit\Psi}_s+\tilde{I}_c.$$ (7.1)

Here,

$$\skew{3}\hat{\mit\Psi}_s = -{\rm i}\,\frac{r_s}{R_0\,B_z}\left[\frac{\delta B_r}{m}\right]_{r_s},$$ (7.2)

$${\mit\Delta\skew{3}\hat{\Psi}}_s = -\frac{r_s}{R_0\,B_z}\,[\delta B_\theta]_{r_{s-}}^{r_{s+}},$$ (7.3)

where $r$, $\theta$, $z$ are conventional cylindrical coordinates, $\delta{\bf B}$ the perturbed magnetic field, and $B_z$ the equilibrium toroidal magnetic field-strength. (See Sections 3.3, 3.8, and 3.15.) Clearly, $\skew{3}\hat{\mit\Psi}_s$ is a measure of the reconnected magnetic flux driven at the rational surface by the error-field, whereas ${\mit\Delta\skew{3}\hat{\Psi}}_s$ is a measure of the helical current sheet that is induced at the surface. Moreover,

$${\mit\Delta}_{nw} = {\mit\Delta}_{pw} + \frac{E_{sw}\,E_{ws}}{(-\tilde{E}_{ww})}$$ (7.4)

is the (real dimensionless) tearing stability index (of a tearing mode with poloidal mode number $m$, and toroidal mode number $n$) in the absence of the wall,

$${\mit\Delta}_{pw}= E_{ss}$$ (7.5)

the (real dimensionless) tearing stability index in the presence of a perfectly conducting wall at $r=r_w$ (see Section 3.8), and

$$\tilde{I}_c =\left(\frac{{\mit\Delta}_{nw} - {\mit\Delta}_{pw}}{E... ...\right)\left( \frac{r_w}{r_c}\right)^m\left(\frac{\mu_0\,I_c}{R_0\,B_z}\right).$$ (7.6)

Here, the (real dimensionless) plasma-wall coupling parameter, $E_{sw}>0$, and the (real dimensionless) wall stability index, $\tilde{E}_{ww}$, are defined in Sections 3.9 and 3.17, respectively. We expect $0>{\mit\Delta}_{nw}> {\mit\Delta}_{pw}$, because the plasma is assumed to be tearing stable (see Chapter 6), and $\tilde{E}_{ww}<0$. In fact, if we make the approximation that the equilibrium plasma current external to the rational magnetic flux-surface is negligible [i.e., $J_z'=0$ in Equation (3.77)] then it is easily demonstrated that

$${\mit\Delta}_{nw} = {\mit\Delta}_{pw} + \frac{2\,m\,(r_s/r_w)^{2\,m}}{1-(r_s/r_w)^{2\,m}},$$ (7.7)

$$\tilde{I}_c = \left(\frac{r_s}{r_c}\right)^m\left(\frac{\mu_0\,I_c}{R_0\,B_z}\right).$$ (7.8)

In the radially-thin, resonant layer that surrounds the rational magnetic flux-surface, Equation (5.70) yields

$${\mit\Delta\skew{3}\hat{\Psi}}_s = S^{1/3}\,\skew{6}\hat{\mit\Delta}\,\,\skew{3}\hat{\mit\Psi}_s,$$ (7.9)

where $S$ is the the Lundquist number at the rational surface, and the complex layer matching parameter, $\skew{6}\hat{\mit\Delta}$, is defined in Equation (5.69). Hence, asymptotic matching between the inner and the outer regions (see Section 4.1) yields

$$\skew{3}\hat{\mit\Psi}_s = \frac{\tilde{I}_c}{(-{\mit\Delta}_{nw}) + S^{1/3}\,\skew{6}\hat{\mit\Delta}},$$ (7.10)

where use has been made of Equations (7.1) and (7.9). The previous equation specifies the (normalized) reconnected magnetic flux, $\skew{3}\hat{\mit\Psi}_s$, driven at the rational surface by the (normalized) error-field coil current, $\tilde{I}_c$. The complex layer parameter, $\skew{6}\hat{\mit\Delta}$, specifies the strength of a shielding current that is induced in the resonant layer, and acts to prevent driven magnetic reconnection. Note that the resistive wall has no influence on $\skew{3}\hat{\mit\Psi}_s$ because no eddy currents are induced in the wall by a static error-field.