Linear Tearing Mode Theory

Consider the interface between two plasmas containing magnetic fields of different orientations. The simplest imaginable field configuration is that illustrated in Figure 7.7. Here, the field varies in the $x$ -direction, and is parallel to the $y$ -axis. The field is directed in the $-y$ -direction for $x<0$ , and in the $+y$ -direction for $x>0$ . The interface is situated at $x=0$ . The sudden reversal of the field direction across the interface gives rise to a $z$ -directed current sheet at $x=0$ .

Figure 7.7: A reconnecting magnetic field configuration.

With the neglect of plasma resistivity, the field configuration shown in Figure 7.7 represents a stable equilibrium state, assuming, of course, that we have normal pressure balance across the interface. But, does the field configuration remain stable when we take resistivity into account? If not, then we expect an instability to develop that relaxes the configuration to one possessing lower magnetic energy. As we shall see, this type of relaxation process inevitably entails the breaking and reconnection of magnetic field-lines, and is, therefore, termed magnetic reconnection. The magnetic energy released during the reconnection process eventually appears as plasma thermal energy. Thus, magnetic reconnection also involves plasma heating.

In the following, we shall outline the standard method for determining the linear stability of the type of magnetic field configuration shown in Figure 7.7, taking into account the effect of plasma resistivity. We are particularly interested in plasma instabilities that are stable in the absence of resistivity, and only grow when the resistivity is non-zero. Such instabilities are conventionally termed tearing modes. Because magnetic reconnection is, in fact, a nonlinear process, we shall then proceed to investigate the nonlinear development of tearing modes.

The equilibrium magnetic field is written

$${\bf B}_0 = B_{0\,y}(x)\,{\bf e}_y,$$ (7.171)

where $B_{0\,y}(-x)=-B_{0\,y}(x)$ . There is assumed to be no equilibrium plasma flow. The linearized equations of resistive-MHD [i.e., Equations (7.1)-(7.4), with Equation (7.3) replaced by Equation (7.102)], assuming incompressible flow, take the form

$$\frac{\partial {\bf B}}{\partial t} = \nabla\times({\bf V}\times{\bf B}_0) + \frac{\eta}{\mu_0}\,\nabla^2{\bf B},$$ (7.172)

$$\rho_0\,\frac{\partial{\bf V}}{\partial t} = -\nabla p + \frac{(\nabla\times{\bf B})\times{\bf B}_0}{\mu_0} + \frac{(\nabla\times{\bf B}_0)\times{\bf B}}{\mu_0},$$ (7.173)

$$\nabla\cdot{\bf B} =0,$$ (7.174)

$$\nabla\cdot{\bf V} =0,$$ (7.175)

Here, $\rho_0$ is the equilibrium plasma density, ${\bf B}$ the perturbed magnetic field, ${\bf V}$ the perturbed plasma velocity, $p$ the perturbed plasma pressure, and use has been made of Maxwell's equation. The assumption of incompressible plasma flow is valid provided that the plasma velocity associated with the instability remains significantly smaller than both the Alfvén velocity and the sonic velocity.

Suppose that all perturbed quantities vary like

$$A(x,y,z,t) = A(x)\,\exp\left({\rm i}\,k\,y +\gamma\,t\right),$$ (7.176)

where $\gamma$ is the instability growth-rate. The $x$ -component of Equation (7.172), and the $z$ -component of the curl of Equation (7.173), reduce to

$$\gamma\,B_x = {\rm i}\,k\,B_{0\,y} \,V_x + \frac{\eta}{\mu_0} \left(\frac{d^2}{dx^2} - k^2\right) B_x,$$ (7.177)

$$\gamma\,\rho_0\,\left(\frac{d^2}{dx^2}-k^2\right) V_x = \frac{{\rm i}\,k\,B_{0\,y}}{\mu_0}\left( \frac{d^2}{dx^2} - k^2 - \frac{B_{0\,y}''}{B_{0\,y}}\right) B_x,$$ (7.178)

respectively, where use has been made of Equations (7.174) and (7.175). Here, $'$ denotes $d/dx$ .

It is convenient to normalize Equations (7.177) and (7.178) using a typical magnetic field-strength, $B_0$ , and a typical lengthscale, $a$ . Let us define the Alfvén timescale

$$\tau_A = \frac{a}{V_A},$$ (7.179)

where $V_A = B_0/\!\sqrt{\mu_0\,\rho_0}$ is the Alfvén velocity, and the resistive diffusion timescale

$$\tau_R = \frac{\mu_0\,a^2}{\eta}.$$ (7.180)

The ratio of these two timescales is the Lundquist number:

$$S = \frac{\tau_R}{\tau_A}.$$ (7.181)

Let $\bar{x}=x/a$ , $\psi(\bar{x})= B_x/B_0$ , $\phi(\bar{x}) = {\rm i}\,k\,V_x/\gamma$ , $F(\bar{x})=B_{0\,y}/B_0$ , $F'\equiv dF/d\bar{x}$ , $\bar{\gamma} = \gamma\,\tau_A$ , and $\bar{k} = k\,a$ . It follows that

$$\bar{\gamma}\,(\psi-F\,\phi) = S^{-1}\left(\frac{d^2}{d\bar{x}^{\,2}}-\bar{k}^{\,2}\right) \psi,$$ (7.182)

$$\bar{\gamma}^{\,2}\left(\frac{d^2}{d\bar{x}^{\,2}} -\bar{k}^{\,2}\right)\phi = -\bar{k}^{\,2}\, F \left( \frac{d^2}{d\bar{x}^{\,2}} -\bar{k}^{\,2} - \frac{F''}{F}\right) \psi.$$ (7.183)

The term on the right-hand side of Equation (7.182) represents plasma resistivity, whereas the term on the left-hand side of Equation (7.183) represents plasma inertia.

It is assumed that the tearing instability grows on a hybrid timescale that is much less than $\tau_R$ , but much greater than $\tau_A$ . It follows that

$$\bar{\gamma} \ll 1 \ll S\,\bar{\gamma}.$$ (7.184)

Thus, throughout most of the plasma, we can neglect the right-hand side of Equation (7.182), and the left-hand side of Equation (7.183), which is equivalent to the neglect of plasma resistivity and inertia. In this case, Equations (7.182) and (7.183) reduce to

$$\phi = \frac{\psi}{F},$$ (7.185)

$$\frac{d^2\psi}{d\bar{x}^{\,2}} - \bar{k}^{\,2}\,\psi - \frac{F''}{F}\,\psi = 0.$$ (7.186)

Equation (7.185) is simply the flux-freezing constraint, which requires the plasma to move with the magnetic field. Equation (7.186) is the linearized, static force balance criterion: $\nabla\times({\bf j}\times{\bf B}) = {\bf0}$ . Equations (7.185) and (7.186) are known collectively as the equations of 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 interface, where $F=0$ (i.e., where the magnetic field reverses direction). Observe, for instance, that the normalized ``radial'' velocity, $\phi$ , becomes infinite as $F\rightarrow 0$ , according to Equation (7.185).

The ideal-MHD equations break down close to the interface because the neglect of plasma resistivity and inertia becomes untenable as $F\rightarrow 0$ . Thus, there is a thin layer, in the immediate vicinity of the interface, $\bar{x}=0$ , where the behavior of the plasma is governed by the resistive-MHD equations, (7.182) and (7.183). We can simplify these equations, making use of the fact that $\bar{x}\ll 1$ and $d/d\bar{x} \gg 1$ in a thin layer, to obtain the following layer equations:

$$\bar{\gamma}\,(\psi - \bar{x}\,\phi) = S^{-1}\frac{d^2\psi}{d\bar{x}^{\,2}},$$ (7.187)

$$\bar{\gamma}^{\,2}\,\frac{d^2\phi}{d\bar{x}^{\,2}} = - \bar{x}\,\frac{d^2\psi}{d\bar{x}^{\,2}}.$$ (7.188)

Note that we have redefined the variables $\phi$ , $\bar{\gamma}$ , and $S$ , such that $\phi\rightarrow F'(0)\,\phi$ , $\bar{\gamma}\rightarrow \gamma\,\tau_H$ , and $S\rightarrow \tau_R/\tau_H$ . Here,

$$\tau_H = \frac{\tau_A}{k\,a\,F'(0)}$$ (7.189)

is the so-called hydromagnetic timescale.

The tearing mode stability problem reduces to solving the resistive-MHD layer equations, (7.187) and (7.188), in the immediate vicinity of the interface, $\bar{x}=0$ , solving the ideal-MHD equations, (7.185) and (7.186), everywhere else in the plasma, matching the two solutions at the edge of the layer, and applying physical boundary conditions as $\vert\bar{x}\vert\rightarrow\infty$ . This method of solution was first described in a classic paper by Furth, Killeen, and Rosenbluth (Furth, Killeen, and Rosenbluth 1963).

Let us consider the solution of the ideal-MHD equation (7.186) throughout the bulk of the plasma. We could imagine launching a solution $\psi(\bar{x})$ at large positive $\bar{x}$ , which satisfies physical boundary conditions as $\bar{x}\rightarrow \infty$ , and integrating this solution to the right-hand boundary of the resistive-MHD layer at $\bar{x}=0_+$ . Likewise, we could also launch a solution at large negative $\bar{x}$ , which satisfies physical boundary conditions as $\bar{x}\rightarrow-\infty$ , and integrate this solution to the left-hand boundary of the resistive-MHD layer at $\bar{x}=0_-$ . Maxwell's equations demand that $\psi$ must be continuous on either side of the layer. Hence, we can multiply our two solutions by appropriate factors, so as to ensure that $\psi$ matches to the left and right of the layer. This leaves the function $\psi(\bar{x})$ undetermined to an overall arbitrary multiplicative constant, just as we would expect in a linear problem. In general, $d\psi/d\bar{x}$ is not continuous to the left and right of the layer. Thus, the ideal solution can be characterized by the real number

$${\mit\Delta}' = \left[\frac{1}{\psi}\frac{d\psi}{d\bar{x}}\right]_{\bar{x}=0_-}^{\bar{x}=0_+}:$$ (7.190)

that is, by the jump in the logarithmic derivative of $\psi$ to the left and right 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.

The layer equations, (7.187) and (7.188), possess a trivial solution ( $\phi=\phi_0$ , $\psi=\bar{x}\,\phi_0$ , where $\phi_0$ is independent of $\bar{x}$ ), and a nontrivial solution for which $\psi(-\bar{x})=\psi(\bar{x})$ and $\phi(-\bar{x}) =-\phi(\bar{x})$ . The asymptotic behavior of the nontrivial solution at the edge of the layer is

$$\psi(x) \rightarrow \left(\frac{\mit\Delta}{2}\,\vert\bar{x}\vert + 1\right) {\mit\Psi},$$ (7.191)

$$\phi(x) \rightarrow \frac{\psi}{\bar{x}},$$ (7.192)

where the parameter ${\mit\Delta}(\bar{\gamma}, S)$ is determined by solving the layer equations, subject to the previous boundary conditions. Finally, the growth-rate, $\gamma$ , of the tearing instability is determined by the matching criterion

$${\mit\Delta}(\bar{\gamma}, S) = {\mit\Delta}'.$$ (7.193)

The layer equations, (7.187) and (7.188), can be solved in a fairly straightforward manner in Fourier transform space. Let

$$\phi(\bar{x}) = \int_{-\infty}^{\infty} \skew{3}\hat{\phi}(t) \,{\rm e}^{\,{\rm i}\,S^{1/3}\, \bar{x}\,t}\, dt,$$ (7.194)

$$\psi(\bar{x}) = \int_{-\infty}^{\infty} \hat{\psi}(t)\, {\rm e}^{\,{\rm i}\,S^{1/3}\,\bar{x}\,t}\,dt,$$ (7.195)

where $\skew{3}\hat{\phi}(-t)=-\skew{3}\hat{\phi}(t)$ . Equations (7.187) and (7.188) can be Fourier transformed, and the results combined, to give

$$\frac{d}{dt}\!\left(\frac{t^{\,2}}{Q+t^{\,2}}\frac{d\skew{3}\hat{\phi}}{dt}\right) -Q\,t^{\,2}\,\skew{3}\hat{\phi} = 0,$$ (7.196)

where

$$Q = \gamma\,\tau_H^{2/3}\,\tau_R^{1/3}.$$ (7.197)

The most general small-$t$ asymptotic solution of Equation (7.196) is written

$$\skew{3}\hat{\phi}(t) \rightarrow \frac{a_{-1}}{t} + a_0 + {\cal O}(t),$$ (7.198)

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

$$\phi(\bar{x})\rightarrow a_{-1}\,\frac{\pi}{2}\,S^{1/3}\,{\rm sgn}(x) + \frac{a_0}{\bar{x}} +{\cal O}\left(\frac{1}{\bar{x}^{\,2}}\right).$$ (7.199)

It follows from a comparison with Equations (7.191) and (7.192) that

$${\mit\Delta} = \pi\,\frac{a_{-1}}{a_0}\,S^{1/3}.$$ (7.200)

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

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

$$Q\ll 1.$$ (7.201)

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

In the limit $t\gg Q^{\,1/2}$ , Equation (7.196) reduces to

$$\frac{d^2\skew{3}\hat{\phi}}{d t^2} - Q\,t^{\,2}\,\skew{3}\hat{\phi} = 0.$$ (7.202)

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

$$Q^{\,1/2} \ll t \ll Q^{\,-1/4}$$ (7.203)

we can make use of the standard small argument asymptotic expansion of $U(a,x)$ to write the most general solution to Equation (7.196) in the form (Abramowitz and Stegun 1965e)

$$\skew{3}\hat{\phi}(t) = A\left[ 1- 2 \,\frac{{\mit\Gamma}(3/4)}{{\mit\Gamma}(1/4)} \, Q^{\,1/4}\,t + {\cal O}(t^{\,2})\right].$$ (7.204)

Here, $A$ is an arbitrary constant.

In the limit

$$t \ll Q^{\,-1/4},$$ (7.205)

Equation (7.196) reduces to

$$\frac{d}{dt}\!\left(\frac{t^{\,2}}{Q+t^{\,2}}\,\frac{d\skew{3}\hat{\phi}}{dt} \right) = 0.$$ (7.206)

The most general solution to this equation is written

$$\skew{3}\hat{\phi}(t) = B \left(-\frac{Q}{t} + t\right) + C +{\cal O}(t^{\,2}),$$ (7.207)

where $B$ and $C$ are arbitrary constants. Matching coefficients between Equations (7.204) and (7.207) in the range of $t$ satisfying the inequality (7.203) yields the following expression for the most general solution to Equation (7.196) in the limit $t\ll Q^{\,1/2}$ :

$$\skew{3}\hat{\phi} = A\,\left[ 2\,\frac{{\mit\Gamma}(3/4)}{{\mit\Gamma}(1/4)} \, \frac{Q^{\,5/4}}{t} + 1 + {\cal O}(t)\right].$$ (7.208)

Finally, a comparison of Equations (7.198), (7.200), and (7.208) gives the result

$${\mit\Delta} = 2\pi\,\frac{{\mit\Gamma}(3/4)}{{\mit\Gamma}(1/4)}\,S^{1/3}\, Q^{\,5/4}.$$ (7.209)

The asymptotic matching condition (7.193) can be combined with the previous expression for ${\mit\Delta}$ to give the tearing mode dispersion relation

$$\gamma = \left[\frac{{\mit\Gamma}(1/4)}{2\pi\,{\mit\Gamma}(3/4)}\right]^{\,4/5}\, \frac{({\mit\Delta}')^{\,4/5}}{\tau_H^{\,2/5}\,\tau_R^{\,3/5}}.$$ (7.210)

Here, use has been made of the definitions of $S$ and $Q$ . According to the dispersion relation, the tearing mode is unstable whenever ${\mit\Delta}'>0$ , and grows on the hybrid timescale $\tau_H^{\,2/5}\,\tau_R^{\,3/5}$ . It is easily demonstrated that the tearing mode is stable whenever ${\mit\Delta}'<0$ . According to Equations (7.193), (7.201), and (7.209), the constant-$\psi$ approximation holds provided that

$${\mit\Delta}' \ll S^{1/3}:$$ (7.211)

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

According to Equation (7.202), the thickness of the resistive-MHD layer in $t$ -space is

$$\delta_t \sim \frac{1}{Q^{\,1/4}}.$$ (7.212)

It follows from Equations (7.194) and (7.195) that the thickness of the layer in $\bar{x}$ -space is

$$\bar{\mit\delta} \sim \frac{1}{S^{1/3}\,\delta_t} \sim \left( \frac{\bar{\gamma}}{S}\right)^{1/4}.$$ (7.213)

When ${\mit\Delta}'\sim {\cal O}(1)$ then $\bar{\gamma}\sim S^{-3/5}$ , according to Equation (7.210), giving $\bar{\mit\delta}\sim S^{-2/5}$ . It is clear, therefore, that if the Lundquist number, $S$ , is very large then the resistive-MHD layer centered on the interface, $\bar{x}=0$ , is extremely narrow.

The timescale for magnetic flux to diffuse across a layer of thickness $\bar{\mit\delta}$ (in $\bar{x}$ -space) is [see Equation (7.180)]

$$\tau \sim \tau_R\,\bar{\mit\delta}^{\,2}.$$ (7.214)

If

$$\gamma\,\tau\ll 1$$ (7.215)

then the tearing mode grows on a timescale that is far longer than the timescale on which magnetic flux diffuses across the resistive layer. In this case, we would expect the normalized ``radial'' magnetic field, $\psi$ , to be approximately constant across the layer, because any non-uniformities in $\psi$ would be smoothed out via resistive diffusion. It follows from Equations (7.213) and (7.214) that the constant-$\psi$ approximation holds provided that

$$\bar{\gamma} \ll S^{-1/3}$$ (7.216)

(i.e., $Q\ll 1$ ), which is in agreement with Equation (7.201).