Plasma Equilibrium

The unperturbed (by the tearing mode) plasma equilibrium is such that

$\displaystyle {\bf B}$ $\displaystyle = \frac{r\,B_z}{R_0\,q(r)}\,{\bf e}_\theta + B_z\,{\bf e}_z,$ (5.18)
$\displaystyle {\bf V}$ $\displaystyle \simeq V_E(r)\,{\bf e}_\theta + V_{\parallel\,i}(r)\,{\bf e}_z,$ (5.19)
$\displaystyle p$ $\displaystyle = p(r).$ (5.20)

(See Sections 2.11, 2.24, and 3.2.) Here,

$\displaystyle V_E(r)\simeq -\frac{E_r}{B_z}$ (5.21)

is the (dominant $\theta$-component of the) E-cross-B velocity [see Equation (2.138)], and $V_{\parallel\,i}$ the ion parallel fluid velocity.

Now, the resonant layer is assumed to have a radial thickness that is much smaller than $r_s$. (See Section 5.5.) Hence, we only need to evaluate plasma equilibrium quantities in the immediate vicinity of the rational surface. Equations (4.1), (4.23), (4.24), and (5.1)–(5.4) imply that

$\displaystyle \psi(\hat{x})$ $\displaystyle = \frac{\hat{x}^{\,2}}{2\,\hat{L}_s},$ (5.22)
$\displaystyle N(\hat{x})$ $\displaystyle = -\hat{V}_\ast\,\hat{x},$ (5.23)
$\displaystyle \phi(\hat{x})$ $\displaystyle = - \hat{V}_E\,\hat{x},$ (5.24)
$\displaystyle V(\hat{x})$ $\displaystyle = \hat{V}_\parallel,$ (5.25)


$\displaystyle \hat{x}=\frac{r-r_s}{r_s},$ (5.26)


$\displaystyle L_s=\frac{R_0\,q_s}{s_s}$ (5.27)

is the magnetic shear length, $s_s=s(r_s)$,

$\displaystyle s(r) = \frac{d\ln q}{d\ln r}$ (5.28)

the magnetic shear, $\hat{V}_E= V_E(r_s)/V_A$, $\hat{V}_\ast= V_\ast(r_s)/V_A$,

$\displaystyle V_\ast(r) = \frac{1}{e\,n_0\,B_z}\,\frac{dp}{dr}$ (5.29)

the (dominant $\theta$-component of the) diamagnetic velocity [see Equation (4.11)], and $\hat{V}_\parallel=\hat{d}_i\, V_{\parallel\,i}(r_s)/V_A$. We also have

$\displaystyle J(\hat{x})$ $\displaystyle = -\left(\frac{2}{s_s}-1\right)\frac{1}{\hat{L}_s},$ (5.30)
$\displaystyle U(\hat{x})$ $\displaystyle =0,$ (5.31)
$\displaystyle \hat{E}_\parallel(\hat{x})$ $\displaystyle =\left(\frac{2}{s_s}-1\right) \frac{\hat{\eta}_\parallel}{\hat{L}_s}.$ (5.32)

Note that we are neglecting any radial gradients in the equilibrium MHD fluid velocity because, in conventional tokamak plasmas, such gradients do not significantly affect the linear response of a resonant layer, due to the fact that the gradients are comparatively weak (i.e., $\vert dV/dr\vert\ll V_A/r_s$) combined with the fact that a linear resonant layer is very narrow [6]. (See Section 5.5.)