Asymptotic Matching

Suppose that the perturbation grows on a timescale that is much less than $\tau_R$, but much greater than $\tau_H$. It follows that

$$\hat{\gamma} \ll 1 \ll S\,\hat{\gamma}.$$ (9.29)

Thus, throughout much of the plasma, we can neglect the right-hand side of Equation (9.27), and the left-hand side of Equation (9.28), which is equivalent to the neglect of plasma resistivity and inertia. In this case, Equations (9.27) and (9.28) reduce to

$$\skew{3}\hat{\phi} = \frac{\hat{\psi}}{F},$$ (9.30)

$$\frac{d^2\hat{\psi}}{d\hat{x}^2} - \hat{k}^2\,\hat{\psi}-\frac{F''}{F}\,\hat{\psi} = 0.$$ (9.31)

Equation (9.30) is simply the flux-freezing constraint that requires the plasma to move with the magnetic field. Equation (9.31) is the linearized, static, force balance criterion: $\nabla\times({\bf j}\times{\bf B}+\nabla p)={\bf0}$. Equations (9.30) and (9.31) are known collectively as the equations of marginally-stable ideal-MHD, and are valid throughout virtually the whole plasma. However, it is clear that these equations break down in the immediate vicinity of the shear-Alfvén resonance, where $F=0$ (i.e., where the equilibrium magnetic field reverses direction). Observe, for instance, that the $x$-component of the plasma velocity, $-\gamma\,a\,\skew{3}\hat{\phi}$, becomes infinite as $F\rightarrow 0$, according to Equation (9.30).

The marginally-stable ideal-MHD equations break down close to the shear-Alfvén resonance because the neglect of plasma resistivity and inertia becomes untenable as $F\equiv\tanh(\hat{x})\rightarrow 0$. Thus, there is a thin (compared to the current sheet thickness $a$) layer, centered on the resonance, $\hat{x}=0$, where the behavior of the plasma is governed by the linearized reduced-MHD equations, (9.27) and (9.28). We can simplify these equations, making use of the fact that $\vert\hat{x}\vert\ll 1$, and $\vert d/d\hat{x}\vert\gg 1$, in a thin layer, to obtain the following layer equations:

$$S\,\hat{\gamma}\,(\hat{\psi}-\hat{x}\,\skew{3}\hat{\phi}) = \frac{d^2\hat{\psi}}{d\hat{x}^2},$$ (9.32)

$$\hat{\gamma}^{2}\,\frac{d^2\skew{3}\hat{\phi}}{d\hat{x}^{2}} = -\hat{x}\,\frac{d^2\hat{\psi}}{d\hat{x}^2}.$$ (9.33)

Here, we have also assumed that $\vert\hat{k}\,\hat{x}\vert\ll 1$ in the layer.

The stability problem reduces to solving the layer equations, (9.32) and (9.33), in the immediate vicinity of the shear-Alfvén resonance, $\hat{x}=0$, solving the marginally-stable ideal-MHD equations, (9.30) and (9.31), everywhere else in the plasma, and matching the two solutions at the edge of the layer. This method of solution, which is known as asymptotic matching, was first described in a classic paper by Furth, Killeen, and Rosenbluth (Furth, Killeen, and Rosenbluth 1963).

Let us consider the solution of the so-called tearing mode equation, (9.31), throughout the bulk of the plasma. We could imagine launching a solution, $\hat{\psi}(\hat{x})$, at large positive $\hat{x}$, which satisfies the physical boundary conditions as $\hat{x}\rightarrow\infty$, and integrating this solution to the right-hand boundary of the layer at $\hat{x}=0_+$. Likewise, we could also launch a solution at large negative $\hat{x}$, which satisfies physical boundary conditions as $\hat{x}\rightarrow-\infty$, and integrate this solution to the left-hand boundary of the layer at $\hat{x}=0_-$. Maxwell's equations demand that $\hat{\psi}$ be continuous on either side of the layer. Hence, we can multiply our two solutions by appropriate factors, so as to ensure that $\hat{\psi}$ matches to the left and to the right of the layer. This leaves the function $\hat{\psi}(\hat{x})$ undetermined to an overall multiplicative constant, just as we would expect in linear problem. In general, $d\hat{\psi}/d\hat{x}$ is not continuous to the left and to the right of the layer. Thus, the marginally-stable ideal-MHD solution can be characterized by the real number

$${\mit\Delta}' =\left[\frac{1}{\hat{\psi}}\,\frac{d\hat{\psi}}{d\hat{x}}\right]_{\hat{x}=0_-}^{\hat{x}=0_+}:$$ (9.34)

that is, the jump in the logarithmic derivative of $\hat{\psi}$ to the right and to the left of the layer. This parameter is known as the tearing stability index, and is solely a property of the plasma equilibrium, the wavenumber, $k$, and the boundary conditions imposed at infinity.

Figure: 9.2 Solutions of the tearing mode equation, (9.31), for a current sheet characterized by $F(\hat{x})=\tanh(\hat{x})$. The solid curve corresponds to $\hat{k}=0.5$, and the dashed curve to $\hat{k}=1.5$.

Let us assume that the current sheet is isolated (i.e., it is not subject to any external magnetic perturbation). In this case, the appropriate boundary conditions at infinity are $\hat{\psi}(\vert\hat{x}\vert\rightarrow\infty)\rightarrow 0$. For the particular plasma equilibrium under consideration, for which $F(\hat{x})=\tanh(\hat{x})$, the tearing mode equation, (9.31), takes the form

$$\frac{d^2\hat{\psi}}{d\hat{x}^2}-\hat{k}^2\,\hat{\psi}+\frac{2}{\cosh^2\hat{x}}\,\hat{\psi} = 0.$$ (9.35)

The previous equation can be solved analytically, subject to the aforementioned boundary conditions, to give (Biskamp 1993)

$$\hat{\psi}(\hat{x}) = {\mit\Psi}\,{\rm e}^{-\hat{k}\,\vert\hat{x}\vert}\left(1+\hat{k}^{-1}\,\tanh \vert\hat{x}\vert\right),$$ (9.36)

where ${\mit\Psi}$ is an arbitrary constant. This solution is illustrated in Figure 9.2. At the edge of the layer, which corresponds to the limit $\vert\hat{x}\vert\rightarrow 0$, the previous expression, in combination with Equation (9.30), yields

$$\hat{\psi}(\hat{x}) \rightarrow {\mit\Psi}\left[1 + \frac{{\mit\Delta}'}{2}\,\vert\hat{x}\vert +{\cal O}(\hat{x}^{\,2})\right],$$ (9.37)

$$\skew{3}\hat{\phi}(\hat{x}) \rightarrow \frac{\hat{\psi}}{\hat{x}},$$ (9.38)

where

$${\mit\Delta}' =\frac{2\,(1-\hat{k}^2)}{\hat{k}},$$ (9.39)

and use has been made of Equation (9.34). As illustrated in Figure 9.3, ${\mit\Delta}'<0$ for $\hat{k}>1$, ${\mit\Delta}'> 0$ for $\hat{k}< 1$, and ${\mit\Delta}'\rightarrow\infty$ as $\hat{k}\rightarrow 0$.

Figure: 9.3 The variation of the tearing stability index, ${\mit \Delta }'$, with the wavenumber, $\hat{k}$, for a current sheet characterized by $F(\hat{x})=\tanh(\hat{x})$.

The layer equations, (9.32) and (9.33), possess the trivial twisting parity solution (Strauss, et al., 1979), $\skew{3}\hat{\phi}=\skew{3}\hat{\phi}_0$, $\hat{\psi}=\hat{x}\,\skew{3}\hat{\phi}_0$, where $\skew{3}\hat{\phi}_0$ is independent of $\hat{x}$. However, this solution cannot be matched to the so-called outer solution, (9.36), which has the opposite parity. Fortunately, the layer equations also possess a nontrivial tearing parity solution, such that $\hat{\psi}(-\hat{x})=\hat{\psi}(\hat{x})$ and $\skew{3}\hat{\phi}(-\hat{x})=-\skew{3}\hat{\phi}(\hat{x})$, which can be matched to the outer solution. The asymptotic behavior of the tearing parity solution at the edge of the layer is

$$\hat{\psi}(\hat{x}) \rightarrow{\mit\Psi}'\left[1+\frac{\mit\Delta}{2}\,\vert\hat{x}\vert+{\cal O}(\hat{x}^{\,2})\right],$$ (9.40)

$$\skew{3}\hat{\phi} \rightarrow \frac{\hat{\psi}}{\hat{x}},$$ (9.41)

where ${\mit\Psi}'$ is an arbitrary constant, and the parameter ${\mit\Delta}(\hat{\gamma},S)$ is determined by solving the layer equations. Matching Equations (9.37), (9.38), (9.40) and (9.41) at the edge of the layer yields ${\mit\Psi}={\mit\Psi}'$, and

$${\mit\Delta}(\hat{\gamma},S)={\mit\Delta}'.$$ (9.42)

The latter matching condition determines the growth-rate of the perturbation.