Resistive Kink Modes

Our previous solution of the Fourier-transformed layer equation, (9.45), is only valid in the constant-$\psi$ limit, $Q\ll 1$. In order to determine how this solution is modified when the constant-$\psi$ approximation breaks down, we need to find a solution that is valid as $Q\rightarrow 1$. Now, when expanded to ${\cal O}(p^3)$, the most general small-$p$ asymptotic solution of Equation (9.45) takes the form

$$\skew{3}\bar{\phi}(p)= \frac{a_{-1}}{p}+a_0+a_{-1}\left(\frac{Q^2}{2}-\frac{1}{Q}\right)p + \frac{a_0\,Q^2\,p^2}{6}+{\cal O}(p^3).$$ (9.67)

As before, $a_{-1}$ and $a_0$ are independent of $p$, and it is assumed that $p>0$. Let

$$K(p) = \frac{p^2}{Q+p^2}\,\frac{d\skew{3}\bar{\phi}}{dp}.$$ (9.68)

Equation (9.45) transforms to give

$$\frac{d}{dp}\!\left(\frac{1}{p^2}\,\frac{dK}{dp}\right) - \frac{Q\,(Q+p^2)}{p^2}\,K= 0.$$ (9.69)

As is clear from Equations (9.67) and (9.68), the most general small-$p$ asymptotic solution of the previous equation is

$$Q\,K(p) =-a_{-1} + \frac{a_{-1}\,Q^{2}\,p^2}{2}+\frac{a_0\,Q^2\,p^3}{3}+ {\cal O}(p^4).$$ (9.70)

Let

$$z = Q^{1/2}\,p^2.$$ (9.71)

Equations (9.69) and (9.70) yield

$$z\,\frac{d^2K}{dz^2} -\frac{1}{2}\,\frac{dK}{dz} -\frac{1}{4}\,(Q^{3/2}+z)\,K = 0,$$ (9.72)

and

$$Q\,K(z) = -a_{-1} + \frac{a_{-1}\,Q^{3/2}\,z}{2} +\frac{a_0\,Q^{5/4}\,z^{3/2}}{3} + {\cal O}(z^2),$$ (9.73)

respectively.

Figure: 9.4 The normalized growth-rate, $Q$, as a function of the normalized tearing stability index, ${\mit \Delta }'/S^{1/3}$, according to the resistive-kink dispersion relation, (9.81). The dashed line illustrates the relation ${\mit \Delta }'/S^{1/3}\propto Q^{5/4}$.

Let

$$K(z)= {\rm e}^{-z/2}\,L(z).$$ (9.74)

Equations (9.72) and (9.73) give

$$z\,\frac{d^2 L}{dz^2} + \left(-\frac{1}{2}-z\right)\frac{dL}{dz} - \frac{1}{4}\left(Q^{3/2}-1\right)L = 0,$$ (9.75)

and

$$Q\,L(z)= -a_{-1}+ \frac{a_{-1}\,(Q^{3/2}-1)\,z}{2} + \frac{a_0\,Q^{5/4}\,z^{3/2}}{3} +{\cal O}(z^2),$$ (9.76)

respectively. Equation (9.75) is Kummer's equation (Abramowitz and Stegun 1965). The solution that is well behaved as $z\rightarrow\infty$ is a confluent hypergeometric function of the second kind (Abramowitz and Stegun 1965),

$$L(z) =U\!\left[\frac{1}{4}\,(Q^{3/2}-1),-\frac{1}{2},z\right],$$ (9.77)

which has the small-$z$ asymptotic expansion (Abramowitz and Stegun 1965)

$$L(z) = -\pi\left[\frac{1-(1/2)\,(Q^{3/2}-1)\,z} {{\Gamma}[(5+Q^{3... ...-\frac{z^{3/2}} {{\Gamma}[(Q^{3/2}-1)/4]\,{\Gamma}(5/2)} +{\cal O}(z^2)\right].$$ (9.78)

Comparing Equations (9.49), (9.76), and (9.78), we obtain (Coppi, et al., 1976; Pegoraro and Schep 1986)

$${\mit\Delta} =- \frac{\pi}{8}\,\frac{{\Gamma}(Q^{3/2}/4-1/4)}{{\Gamma}(Q^{3/2}/4+5/4)}\,S^{1/3}\,Q^{5/4},$$ (9.79)

where use has been made of some elementary properties of gamma functions (Abramowitz and Stegun 1965). In the constant-$\psi$ limit, $Q\ll 1$, the previous expression reduces to

$${\mit\Delta} \simeq\frac{2\pi\,{\Gamma}(3/4)}{{\Gamma}(1/4)}\,S^{1/3}\,Q^{5/4},$$ (9.80)

which is consistent with Equation (9.58).

When combined with the matching condition (9.42), Equation (9.79) yields the dispersion relation

$$\frac{{\mit\Delta}'}{S^{1/3}} = - \frac{\pi}{8}\,\frac{{\Gamma}(Q^{3/2}/4-1/4)}{{\Gamma}(Q^{3/2}/4+5/4)}\,Q^{5/4}.$$ (9.81)

This dispersion relation is illustrated in Figure 9.4. It can be seen that $Q^{5/4} \propto {\mit\Delta}'/S^{1/3}$ in the constant-$\psi$ regime, ${\mit\Delta}'/S^{1/3}\ll 1$, in accordance with standard tearing mode theory. (See Section 9.5.) On the other hand, $Q\rightarrow 1$ in the non-constant-$\psi$ regime, ${\mit\Delta}'/S^{1/3}\gg 1$.

The general dispersion relation (9.81) implies that the growth-rate of a tearing mode does not continue to increase indefinitely as the tearing stability index, ${\mit \Delta }'$, becomes larger and larger, which is the prediction of the constant-$\psi$ dispersion relation (9.60). Instead, when ${\mit \Delta }'$ exceeds a critical value that is of order $S^{1/3}$ (implying the breakdown of the constant-$\psi$ approximation), the growth-rate saturates at the value

$$\gamma= \frac{1}{\tau_H^{2/3}\,\tau_R^{1/3}}.$$ (9.82)

In this limit, the tearing mode is usually referred to as a resistive kink mode.