Resonant Layer Thickness

According to Equations (3.102) and (3.103), the dispersion relation of a magnetic perturbation interacting with a thin, rigid, resistive wall can be written

$${\mit\Delta\Psi}_w=\gamma\,\tau_r\,\frac{\delta_w}{r_w}\,{\mit\Psi}_w,$$ (6.23)

where ${\mit\Delta\Psi}_w$ is a measure of the current flowing in the wall, ${\mit\Psi}_w$ the perturbed magnetic flux that penetrates the wall, $r_w$ the wall minor radius, $\delta_w$ the wall radial thickness, and $\tau_r= \mu_0\,r_s^{\,2}\,\sigma_w$ the time required for magnetic flux to diffuse a distance $r_w$ through the wall material. Here, $\sigma_w$ is the electrical conductivity of the wall material. By analogy with the previous result, we would expect the dispersion relation of a magnetic perturbation interacting with the thin resistive layer that surrounds the rational surface to take the form

$${\mit\Delta\Psi}_s=\gamma\,\tau_R\,\frac{\delta_s}{r_s}\,{\mit\Psi}_s,$$ (6.24)

where ${\mit\Delta\Psi}_s$ is a measure of the current flowing in the layer, ${\mit\Psi}_s$ the perturbed magnetic flux that penetrates the layer, $\delta _s$ is the radial thickness of the layer, and $\tau _R$ is the time required for magnetic flux to diffuse a distance $r_s$ through the plasma. [See Equation (5.49).] Note that, in general, $\delta _s$ is a complex quantity. In fact, the true layer thickness is $\vert\delta _s\vert$. It follows from Equation (3.74) that

$$\gamma\,\tau_R\,\frac{\delta_s}{r_s} = E_{ss},$$ (6.25)

where $E_{ss}$ is the real tearing stability index. Finally, Equations (5.48), (5.66), (6.2), and (6.4) can be combined with the previous equation to give

$$\frac{\skew{6}\hat{\mit\Delta}}{\hat{\gamma}\,D} = (1+1/\tau)^{1/2}\,\frac{\delta_s}{d_\beta},$$ (6.26)

where $d_\beta$ is the ion sound radius at the rational surface. [See Equation (4.75).]

Table: 6.2 Normalized dimensionless layer parameters, layer thicknesses, and tearing mode growth-rates in a low-field and a high-field tokamak fusion reactor. See Equations (6.31)–(6.33). Here, $\delta _s$ is the complex layer thickness, $\vert\delta _s\vert$ the actual layer thickness, $d_\beta$ the ion sound radius, $\gamma$ the complex growth-rate in a frame of reference that co-rotates with the electron fluid at the rational surface, and $E_{ss}$ the tearing stability index.

	Low-Field	High-Field
$B({\rm T})$	5.0	12.0
$\hat{Q}_\ast$	$2.40\times 10^{-2}$	$5.77\times 10^{-2}$
$\hat{P}_\varphi$	1.15	6.62
$\hat{P}_\perp$	$3.77\times 10^{-1}$	$2.17\times 10^{+0}$
$\delta_s/d_\beta$	$9.60\times 10^{-1}-1.45\times 10^{-2}\,{\rm i}$	$1.44\times 10^{+0}-1.80\times 10^{-2}\,{\rm i}$
$\vert\delta_s\vert({\rm m})$	$4.69\times 10^{-3}$	$2.95\times 10^{-3}$
$\gamma({\rm s}^{-1})/E_{ss}$	$1.93\times 10^{-1}+2.93\times 10^{-3}\,{\rm i}$	$7.36\times 10^{-1}+9.16\times 10^{-3}\,{\rm i}$