Resistive Kink Modes
Our previous solution of the Fourier-transformed layer equation, (9.45), is only valid in the constant-
limit,
.
In order to determine how this solution is modified when the constant-
approximation breaks down, we need to find a
solution that is valid as
.
Now, when expanded to
, the most
general small-
asymptotic solution of Equation (9.45) takes the form
![$\displaystyle \skew{3}\bar{\phi}(p)= \frac{a_{-1}}{p}+a_0+a_{-1}\left(\frac{Q^2}{2}-\frac{1}{Q}\right)p + \frac{a_0\,Q^2\,p^2}{6}+{\cal O}(p^3).$](img3430.png) |
(9.67) |
As before,
and
are independent of
, and it is assumed that
.
Let
![$\displaystyle K(p) = \frac{p^2}{Q+p^2}\,\frac{d\skew{3}\bar{\phi}}{dp}.$](img3431.png) |
(9.68) |
Equation (9.45) transforms to give
![$\displaystyle \frac{d}{dp}\!\left(\frac{1}{p^2}\,\frac{dK}{dp}\right) - \frac{Q\,(Q+p^2)}{p^2}\,K= 0.$](img3432.png) |
(9.69) |
As is clear from Equations (9.67) and (9.68), the most general small-
asymptotic solution of the
previous equation is
![$\displaystyle Q\,K(p) =-a_{-1} + \frac{a_{-1}\,Q^{2}\,p^2}{2}+\frac{a_0\,Q^2\,p^3}{3}+ {\cal O}(p^4).$](img3433.png) |
(9.70) |
Let
![$\displaystyle z = Q^{1/2}\,p^2.$](img3434.png) |
(9.71) |
Equations (9.69) and (9.70) yield
![$\displaystyle z\,\frac{d^2K}{dz^2} -\frac{1}{2}\,\frac{dK}{dz} -\frac{1}{4}\,(Q^{3/2}+z)\,K = 0,$](img3435.png) |
(9.72) |
and
![$\displaystyle Q\,K(z) = -a_{-1} + \frac{a_{-1}\,Q^{3/2}\,z}{2} +\frac{a_0\,Q^{5/4}\,z^{3/2}}{3} + {\cal O}(z^2),$](img3436.png) |
(9.73) |
respectively.
Figure: 9.4
The normalized growth-rate,
, as a function of the normalized tearing
stability index,
, according to the resistive-kink dispersion relation, (9.81). The
dashed line illustrates the relation
.
|
Let
![$\displaystyle K(z)= {\rm e}^{-z/2}\,L(z).$](img3438.png) |
(9.74) |
Equations (9.72) and (9.73) give
![$\displaystyle z\,\frac{d^2 L}{dz^2} + \left(-\frac{1}{2}-z\right)\frac{dL}{dz} - \frac{1}{4}\left(Q^{3/2}-1\right)L = 0,$](img3439.png) |
(9.75) |
and
![$\displaystyle Q\,L(z)= -a_{-1}+ \frac{a_{-1}\,(Q^{3/2}-1)\,z}{2} + \frac{a_0\,Q^{5/4}\,z^{3/2}}{3} +{\cal O}(z^2),$](img3440.png) |
(9.76) |
respectively. Equation (9.75) is Kummer's equation (Abramowitz and Stegun 1965).
The solution that is well behaved as
is a confluent hypergeometric function of the second kind (Abramowitz and Stegun 1965),
![$\displaystyle L(z) =U\!\left[\frac{1}{4}\,(Q^{3/2}-1),-\frac{1}{2},z\right],$](img3441.png) |
(9.77) |
which has the small-
asymptotic expansion (Abramowitz and Stegun 1965)
![$\displaystyle L(z) = -\pi\left[\frac{1-(1/2)\,(Q^{3/2}-1)\,z}
{{\Gamma}[(5+Q^{3...
...-\frac{z^{3/2}}
{{\Gamma}[(Q^{3/2}-1)/4]\,{\Gamma}(5/2)} +{\cal O}(z^2)\right].$](img3442.png) |
(9.78) |
Comparing Equations (9.49), (9.76), and (9.78), we obtain (Coppi, et al., 1976; Pegoraro and Schep 1986)
![$\displaystyle {\mit\Delta} =- \frac{\pi}{8}\,\frac{{\Gamma}(Q^{3/2}/4-1/4)}{{\Gamma}(Q^{3/2}/4+5/4)}\,S^{1/3}\,Q^{5/4},$](img3443.png) |
(9.79) |
where use has been made of some elementary properties of gamma functions (Abramowitz and Stegun 1965).
In the constant-
limit,
, the previous expression reduces to
![$\displaystyle {\mit\Delta} \simeq\frac{2\pi\,{\Gamma}(3/4)}{{\Gamma}(1/4)}\,S^{1/3}\,Q^{5/4},$](img3444.png) |
(9.80) |
which is consistent with Equation (9.58).
When combined with the matching condition (9.42), Equation (9.79) yields the dispersion relation
![$\displaystyle \frac{{\mit\Delta}'}{S^{1/3}} = - \frac{\pi}{8}\,\frac{{\Gamma}(Q^{3/2}/4-1/4)}{{\Gamma}(Q^{3/2}/4+5/4)}\,Q^{5/4}.$](img3445.png) |
(9.81) |
This dispersion relation is illustrated in Figure 9.4. It can be seen that
in the constant-
regime,
, in accordance with standard tearing mode theory. (See Section 9.5.) On the other hand,
in the
non-constant-
regime,
.
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,
, becomes larger and larger, which is the prediction of the constant-
dispersion relation (9.60). Instead, when
exceeds a critical value that is
of order
(implying the breakdown of the constant-
approximation), the growth-rate saturates at the
value
![$\displaystyle \gamma= \frac{1}{\tau_H^{2/3}\,\tau_R^{1/3}}.$](img3450.png) |
(9.82) |
In this limit, the tearing mode is usually referred to as a resistive kink mode.