Linearized Reduced-MHD Equations

Consider the stability of a current sheet whose equilibrium state is characterized by

$$J_0(x) = -\frac{B_0}{a}\,\cosh^{-2}\left(\frac{x}{a}\right).$$ (9.15)

The corresponding equilibrium magnetic field and current density takes the respective forms

$${\bf B}_0 = B_0\,\tanh\left(\frac{x}{a}\right)\,{\bf e}_y,$$ (9.16)

$${\bf j}_0 =\frac{B_0}{\mu_0\,a}\,\cosh^{-2}\left(\frac{x}{a}\right)\,{\bf e}_z,$$ (9.17)

where ${\bf e}_y$ is a unit vector parallel to the $y$-axis. The equilibrium plasma flow is assumed to be zero. The current sheet consists of filaments that run parallel to the $z$-axis. As illustrated in Figure 9.1, the sheet is centered on the plane $x=0$, and is of thickness $a$ in the $x$-direction. The magnetic field generated by the current sheet is parallel to the $y$-axis, of magnitude $B_0$, and switches direction across the sheet. In other words, $B_{0\,y}=-B_0$ for $x\ll -a$, and $B_{0\,y}=+B_0$ for $x\gg +a$. Note that $B_{0\,y}=0$ at the center of the sheet, $x=0$.

Figure: 9.1 A current sheet. The solid curve shows $B_{0\,y}/B_0$, whereas the dashed curve shows $j_{0\,z}/(B_0/\mu_0\,a)$.

Consider a small perturbation to the aforementioned current sheet that varies periodically in the $y$-direction with wavelength $2\pi/k$. The wavevector of the perturbation is therefore ${\bf k} = (0,\,k,\,0)$. It follows that the perturbation satisfies the shear-Alfvén resonance condition, ${\bf k}\cdot{\bf B}_0=0$, at $x=0$. We can write

$$\psi(x,y,t) = -B_0\,a\,\ln\left[\cosh\left(\frac{x}{a}\right)\right] + \psi_1(x)\,{\rm e}^{\,{\rm i}\,k\,y+\gamma\,t},$$ (9.18)

$$J(x,y,t) = -\frac{B_0}{a}\,\cosh^{-2}\left(\frac{x}{a}\right)+J_1(x)\,{\rm e}^{\,{\rm i}\,k\,y+\gamma\,t},$$ (9.19)

$$\phi(x,y,t) = \phi_1(x)\,{\rm e}^{\,{\rm i}\,k\,y+\gamma\,t},$$ (9.20)

$$U(x,y,t) = U_1(x)\,{\rm e}^{\,{\rm i}\,k\,y+\gamma\,t},$$ (9.21)

where $\gamma$ is the growth-rate of the perturbation, and $\psi_1$, $J_1$, $\phi_1$, and $U_1$ are all considered to be small (compared to equilibrium quantities) quantities.

Substituting Equations (9.18)–(9.21) into the reduced-MHD equations, (9.11)–(9.14), making use of Equation (9.15), and only retaining terms that are first order in small quantities, we obtain the linearized reduced-MHD equations:

$$\gamma\,\psi_1 = {\rm i}\,k\,B_0\,F\,\phi_1 + \frac{\eta}{\mu_0}\left(\frac{d^2}{dx^2}-k^2\right)\psi_1,$$ (9.22)

$$\gamma\,\rho\left(\frac{d^2}{dx^2}-k^2\right)\phi_1 = {\rm i}\,\mu_0^{-1}\,k\,B_0\,F\left( \frac{d^2}{dx^2}-k^2 - \frac{d^2F/dx^2}{F}\right)\psi_1,$$ (9.23)

where $F(x)=\tanh(x/a)$.

It is helpful to define the hydromagnetic timescale,

$$\tau_H = \frac{k^{-1}}{(B_0^{2}/\mu_0\,\rho)^{1/2}},$$ (9.24)

which is the typical time required for a shear-Alfvén wave to propagate a wavelength parallel to the $y$-axis, as well as the resistive diffusion timescale,

$$\tau_R = \frac{\mu_0\,a^2}{\eta},$$ (9.25)

which is the typical time required for magnetic flux to diffuse across the current sheet in the $x$-direction. The effective Lundquist number for the problem is

$$S= \frac{\tau_R}{\tau_H}.$$ (9.26)

Let $x= a\,\hat{x}$, $k=\hat{k}/a$, $\gamma=\hat{\gamma}/\tau_H$, $\psi_1=-a\,B_0\,\hat{\psi}$, and $\phi_1={\rm i}\,(\gamma\,a/k)\,\skew{3}\hat{\phi}$. The dimensionless, normalized versions of the linearized reduced-MHD equations, (9.22) and (9.23), become

$$S\,\hat{\gamma}\left(\hat{\psi}-F\,\skew{3}\hat{\phi}\right) = \left(\frac{d^2}{d\hat{x}^2}-\hat{k}^2\right)\hat{\psi},$$ (9.27)

$$\hat{\gamma}^{\,2}\left(\frac{d^2}{d\hat{x}^2}-\hat{k}^2\right)\skew{3}\hat{\phi} = - F\left(\frac{d^2}{d\hat{x}^2} -\hat{k}^2-\frac{F''}{F}\right)\hat{\psi},$$ (9.28)

where $F(\hat{x})=\tanh(\hat{x})$ and $'\equiv d/d\hat{x}$. Our normalization scheme is designed such that, throughout the bulk of the plasma, $\hat{\psi}\sim \skew{3}\hat{\phi}$, and the only other quantities in the previous two equations whose magnitudes differ substantially from unity are $S\,\hat{\gamma}$ and $\hat{\gamma}^{\,2}$. The term on the right-hand side of Equation (9.27) represents plasma resistivity, whereas the term on the left-hand side of Equation (9.28). represents plasma inertia. The shear-Alfvén resonance condition, ${\bf k}\cdot{\bf B}_0\equiv k\,B_0\,F=0$, reduces to $F=0$.