Linear Dispersion Relation

Let us write [see Equation (5.33)]

$$\omega = {\rm i}\,\gamma+\omega_r,$$ (6.1)

where $\gamma$ is the growth-rate of the tearing mode, whereas $\omega_r$ is the real frequency of the mode in the laboratory frame. We shall assume that $\vert\gamma\vert\ll \vert\omega_r\vert$. This assumption can easily be verified a posteriori. (See Table 6.2.) Asymptotic matching (see Section 3.8) between the tearing mode solution in the outer region and that in the thin resistive layer, surrounding the rational surface (see Section 3.7), that constitutes the inner region, yields a linear tearing mode dispersion relation of the form

$$S^{1/3}\,\skew{6}\hat{\mit\Delta}(\gamma,\omega_r) = E_{ss},$$ (6.2)

where use has been made of Equations (3.74) and (5.70). Here, $S$ is the Lundquist number at the rational surface [see Equation (5.48)], $E_{ss}$ is the real tearing stability index (see Section 3.8), whereas the complex layer matching parameter, $\skew{6}\hat{\mit\Delta}$, is defined in Equation (5.69).

In a conventional tokamak fusion reactor [assuming that $m\sim {\cal O}(1)$, where $m$ is the poloidal mode number of the tearing perturbation], $E_{ss}\sim {\cal O}(1)$,^6.1 and $S^{1/3}\gg 1$. (See Table 1.5.) Hence, the previous dispersion relation can only be satisfied if $\omega_r$ takes a value that renders $\vert\skew{6}\hat{\mit\Delta}\vert\ll 1$. It is clear from Equations (5.61)–(5.65), (5.93), (5.94), (5.96), and (5.98) that this goal can be achieved in all of the constant-$\psi$ linear response regimes if

$$\omega_r = \omega_{\perp\,e}\equiv \omega_E+\omega_{\ast\,e}.$$ (6.3)

Here, $\omega_E$ and $\omega_{\ast\,e}$ are the E-cross-B and electron diamagnetic frequencies, respectively, at the rational surface. [See Equations (5.21), (5.29), (5.44), and (5.45).] The previous equation implies that the tearing mode co-rotates with the electron fluid at the rational surface [1]. (See Section 2.24.)