The Tearing Instability of Resistive Magnetohydrodynamics

Abstract

In this chapter we explore the linear onset of one of the most important instabilities of resistive magnetohydrodynamics, the tearing instability. In particular, we focus on two important aspects of the onset of tearing: asymptotic (modal) stability and transient (non-modal) stability. We discuss the theory required to understand these two aspects of stability, both of which have undergone significant development in recent years.

Appendix: Completeness

Since the completeness of eigenfunctions is a vital property for the stability analysis we have discussed, we now say a few words about it here. In the fluid dynamics literature, one popular reference related to proving the completeness of the eigenfunctions of non-self-adjoint eigenvalue problems is Di Prima and Habetler (1969). In the theorem of Di Prima and Habetler (1969), the operators of the eigenvalue problem are written as

$$\displaystyle \begin{aligned} \sigma M\mathbf{v} = L\mathbf{v} = (L_s+B)\mathbf{v}. \end{aligned} $$

(88)

L _s is a self-adjoint operator and B is a “perturbation” (that is, whatever remains). The order of the derivatives in B must be lower than those in L _s since B is a perturbation. If we can express the eigenvalue problem for the onset of the TI in the form of Eq. (88), we can use the theorem of Di Prima and Habetler (1969) to prove that the eigenfunctions are complete.

As they stand, Eqs. (63) and (64) are not in a suitable form for this L _s + B split. In order to achieve a suitable split, we must reconsider how we linearize the MHD equations. Until now, we have linearized the curl of the momentum equation (in order to eliminate the pressure) but have linearized the induction equation directly. If, instead, we also take the curl of the induction equation and linearize this, we find

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \frac{\partial }{\partial t}(D^2-k^2)b &\displaystyle =&\displaystyle ik[B_0(D^2-k^2)+B_0^{\prime\prime}]u - ik[U_0(D^2-k^2)+U_0^{\prime\prime}]b \\ &\displaystyle &\displaystyle +\, 2ik(B_0^{\prime}Du-U_0^{\prime}Db)+\frac{1}{S}(D^2-k^2)^2b. \end{array} \end{aligned} $$

(89)

After some algebraic manipulation, we achieve an eigenvalue problem suitable for the theorem of Di Prima and Habetler (1969):

$$\displaystyle \begin{aligned} \lambda M\mathbf{v} = L\mathbf{v} = (L_s+B)\mathbf{v}, \end{aligned} $$

(90)

where λ = −σ, v = (u, b)^T,

$$\displaystyle \begin{aligned} M=\left(\begin{array}{cc} -{{D}}^2+k^2 & 0\\ 0&-D^2+k^2 \end{array}\right), \end{aligned} $$

(91)

$$\displaystyle \begin{aligned} L_s = \left(\begin{array}{cc} \frac{1}{Re}(-{{D}}^2+k^2)^2 & 0\\ 0&\frac{1}{S}(-D^2+k^2)^2 \end{array}\right), \end{aligned} $$

(92)

and

$$\displaystyle \begin{aligned} B = \left(\begin{array}{cc} {\mathcal{L}}^+_{U} & -{\mathcal{L}}^+_{B}\\ -{\mathcal{L}}^-_{B} + 2ik{B_{0z}}^{\prime}D&{\mathcal{L}}^-_{U}-2ik{U_{0z}}^{\prime}D \end{array}\right), \end{aligned} $$

(93)

where

$$\displaystyle \begin{aligned} {\mathcal{L}}^{\pm}_{U} = ik[{U_{0z}}(-D^2+k^2)\pm U_0^{\prime\prime}] \quad \mbox{and}\quad {\mathcal{L}}^{\pm}_{B} = ik[{B_{0z}}(-D^2+k^2)\pm B_0^{\prime\prime}]. \end{aligned} $$

(94)

Now the operators in B are of lower order compared to those L _s and this representation is suitable for the application of the theorem of Di Prima and Habetler (1969), proving the completeness of the eigenfunctions.

Since we have increased the order of the induction equation, we need to add extra boundary conditions to complete the mathematical description of the problem. The suitable extra boundary conditions in this case are

$$\displaystyle \begin{aligned} Db=0\quad \mbox{at}\quad x=\pm d. \end{aligned} $$

(95)

These conditions derive from the solenoidal constraint (60)₁ in the same way that the Du = 0 conditions derive from the incompressibility condition (60)₂.