Tearing Modes

The layer equations, (9.32) and (9.33), can be solved in a fairly straightforward manner in Fourier space. Let

$$\skew{3}\hat{\phi}(\hat{x}) = \int_{-\infty}^{\infty} \skew{3}\bar{\phi}(p) \,{\rm e}^{\,{\rm i}\,S^{1/3}\, \hat{x}\,p}\, dp,$$ (9.43)

$$\hat{\psi}(\hat{x}) = \int_{-\infty}^{\infty} \bar{\psi}(p)\, {\rm e}^{\,{\rm i}\,S^{1/3}\,\hat{x}\,p}\,dp,$$ (9.44)

where $\skew{3}\bar{\phi}(-p)=-\skew{3}\bar{\phi}(p)$, and $\bar{\psi}(-p)=\bar{\psi}(p)$. Equations (9.32) and (9.33) can be Fourier transformed, and the results combined, to give

$$\frac{d}{dp}\!\left(\frac{p^{2}}{Q+p^{2}}\frac{d\skew{3}\bar{\phi}}{dp}\right) -Q\,p^{2}\,\skew{3}\bar{\phi} = 0,$$ (9.45)

where

$$Q = S^{1/3}\,\hat{\gamma}.$$ (9.46)

The most general small-$p$ asymptotic solution of Equation (9.45) is written

$$\skew{3}\bar{\phi}(p) = \frac{a_{-1}}{p} + a_0 + {\cal O}(p),$$ (9.47)

where $a_{-1}$ and $a_0$ are independent of $p$, and it is assumed that $p>0$. When inverse Fourier transformed, the previous expression leads to the following expression for the asymptotic behavior of $\skew{3}\hat{\phi}$ at the edge of the layer (Erdéyli 1954):

$$\skew{3}\hat{\phi}(\hat{x})= 2\,{\rm i}\left[\frac{a_0}{S^{1/3}\,\hat{x}} + \frac{\pi\,a_{-1}}{2}\,{\rm sgn}(\hat{x}) +{\cal O}(\hat{x}^2)\right].$$ (9.48)

It follows from a comparison with Equations (9.40) and (9.41) that

$${\mit\Delta} = \pi\,\frac{a_{-1}}{a_0}\,S^{1/3}.$$ (9.49)

Thus, the layer matching parameter, ${\mit\Delta}$, is determined from the small-$p$ asymptotic behavior of the Fourier-transformed layer solution.

Let us search for an unstable perturbation characterized by $Q>0$. It is convenient to assume that

$$Q\ll 1.$$ (9.50)

This ordering, which is known as the constant-$\psi$ approximation [because it implies that $\hat{\psi}(\hat{x})$ is approximately constant across the layer] will be justified later on.

In the limit $p\gg Q^{1/2}$, Equation (9.45) reduces to

$$\frac{d^2\skew{3}\bar{\phi}}{d p^2} - Q\,p^{2}\,\skew{3}\bar{\phi} = 0.$$ (9.51)

The solution to this equation that is well behaved in the limit $p\rightarrow \infty$ is written $U(0,\!\sqrt{2}\,Q^{1/4}\,p)$, where $U(a,x)$ is a standard parabolic cylinder function (Abramowitz and Stegun 1965). In the limit

$$Q^{1/2} \ll p \ll Q^{-1/4},$$ (9.52)

we can make use of the standard small-argument expansion of $U(a,x)$ to write the most general solution to Equation (9.45) in the form (Abramowitz and Stegun 1965)

$$\skew{3}\bar{\phi}(p) = A\left[1- \frac{2\,\Gamma(3/4)}{\Gamma(1/4)}\, Q^{1/4}\,p + {\cal O}(p^{2})\right].$$ (9.53)

Here, $A$ is an arbitrary constant, and $\Gamma(z)$ is a gamma function (Abramowitz and Stegun 1965).

In the limit

$$p \ll Q^{-1/4},$$ (9.54)

Equation (9.45) reduces to

$$\frac{d}{dp}\!\left(\frac{p^{2}}{Q+p^{2}}\,\frac{d\skew{3}\bar{\phi}}{dp} \right) = 0.$$ (9.55)

The most general solution to this equation is written

$$\skew{3}\bar{\phi}(p) = B \left(-\frac{Q}{p} + p\right) + C +{\cal O}(p^{2}),$$ (9.56)

where $B$ and $C$ are arbitrary constants. Matching coefficients between Equations (9.53) and (9.56) in the range of $p$ satisfying the inequality (9.52) yields the following expression for the most general solution to Equation (9.45) in the limit $p\ll Q^{1/2}$:

$$\skew{3}\bar{\phi} = A\,\left[\frac{2\,\Gamma(3/4)}{\Gamma(1/4)}\, \frac{Q^{5/4}}{p} + 1 + {\cal O}(p)\right].$$ (9.57)

Finally, a comparison of Equations (9.47), (9.49), and (9.57) gives the result

$${\mit\Delta} =\frac{2\pi\,\Gamma(3/4)}{\Gamma(1/4)}\,S^{1/3}\, Q^{5/4}.$$ (9.58)

The asymptotic matching condition (9.42) can be combined with the previous expression for ${\mit\Delta}$ to give (Furth, Killeen, and Rosenbluth 1963)

$$\hat{\gamma} = \left[\frac{\Gamma(1/4)}{2\pi\,\Gamma(3/4)}\right]^{4/5} {\mit\Delta}'^{\,4/5}\,S^{-3/5},$$ (9.59)

or

$$\gamma = \left[\frac{\Gamma(1/4)}{2\pi\,\Gamma(3/4)}\right]^{4/5} \frac{{\mit\Delta}'^{\,4/5}}{\tau_H^{2/5}\,\tau_R^{3/5}}.$$ (9.60)

Here, use has been made of the definitions of $S$, $Q$, and $\hat{\gamma}$. According to the previous equation, the perturbation, which is known as a tearing mode, is unstable whenever ${\mit\Delta}'> 0$, and grows on the hybrid timescale $\tau_H^{2/5}\,\tau_R^{3/5}$. [This hybrid growth time is consistent with our initial assumption (9.29), provided that $S\gg 1$.] It is easily demonstrated that the tearing mode is stable whenever ${\mit\Delta}'<0$. Thus, we can now appreciate that the solid curve in Figure 9.2, which is indented at the top (because ${\mit\Delta}'> 0$), is the outer solution of an unstable tearing mode, whereas the dashed curve (which is not indented) is the outer solution of a stable tearing mode. Note, finally, that $\hat{\gamma}\rightarrow 0$ as $S\rightarrow \infty$. In other words, the instability of the current sheet when ${\mit\Delta}'> 0$ is only made possible by finite plasma resistivity.

According to Equations (9.42), (9.50), and (9.58), the constant-$\psi$ approximation holds provided that

$${\mit\Delta}' \ll S^{1/3}:$$ (9.61)

that is, provided that the tearing mode does not become too unstable.

Equation (9.51) implies that thickness of the layer in $p$-space is

$$\delta_p \sim \frac{1}{Q^{1/4}}.$$ (9.62)

It follows from Equations (9.43), (9.44), and (9.46) that the thickness of the layer in $\hat{x}$-space is

$$\skew{3}\hat{\delta} \sim \frac{1}{S^{1/3}\,\delta_p} \sim \left( \frac{\hat{\gamma}}{S}\right)^{1/4}.$$ (9.63)

When ${\mit\Delta}'\sim 1$ then $\hat{\gamma}\sim S^{-3/5}$, according to Equation (9.59), giving $\skew{3}\hat{\delta}\sim S^{-2/5}$. It is clear, therefore, that if the Lundquist number, $S$, is very large then the resistive layer centered on the shear-Alfvén resonance, $\hat{x}=0$, is extremely narrow compared to the width of the current sheet.

The timescale for magnetic flux to diffuse across a layer of thickness $\skew{3}\hat{\delta}$ (in $\hat{x}$-space) is [see Equation (9.25)]

$$\tau \sim \tau_R\,\hat{\delta}^{2}.$$ (9.64)

If

$$\gamma\,\tau\ll 1$$ (9.65)

then the tearing mode grows on a timescale that is far longer than the timescale on which magnetic flux diffuses across the layer. In this case, we would expect the normalized flux, $\hat{\psi}$, to be approximately constant across the layer, because any non-uniformities in $\hat{\psi}$ would be smoothed out via resistive diffusion. It follows from Equations (9.63) and (9.64) that the constant-$\psi$ approximation holds provided that

$$\hat{\gamma} \ll S^{-1/3}$$ (9.66)

(i.e., $Q\ll 1$), which is in agreement with Equation (9.50).