Reduced-MHD Equations

In order to investigate the stability of the aforementioned current sheets that form at cell boundaries, we shall employ a set of MHD equations that neglects plasma compressibility, but incorporates plasma resistivity [cf. Equations (8.1)–(8.3)]:

$$\nabla\cdot{\bf V} =0,$$ (9.3)

$$\rho\left[\frac{\partial {\bf V}}{\partial t} + ({\bf V}\cdot\nabla) {\bf V}\right]+\nabla p-{\bf j}\times {\bf B} =0,$$ (9.4)

$${\bf E} + {\bf V}\times {\bf B} =\eta\,{\bf j}.$$ (9.5)

Here, the plasma mass density, $\rho$, and resistivity, $\eta$, are both assumed to be spatially uniform, for the sake of simplicity. Compressibility is neglected (i.e., $\nabla\cdot{\bf V}$ is assumed to be zero) in order to decouple the fast and slow magnetosonic waves from the problem. (See Section 8.4.) It turns out that the instabilities that lead to magnetic reconnection in current sheets (so-called “tearing modes") are modified forms of the shear-Alfvén wave (Hazeltine and Meiss 1985), and are not related to either of the magnetosonic waves. Indeed, current sheets that exhibit magnetic reconnection resonate with the shear-Alfvén wave, whose dispersion relation is $\omega={\bf k}\cdot{\bf B}/\!\sqrt{\mu_0\,\rho}$ (see Section 8.4), where ${\bf k}$ is the wavevector. A shear-Alfvén resonance occurs when $\omega=0$ (i.e., when the wave frequency is reduced to zero), which implies that ${\bf k}\cdot{\bf B}=0$ at the resonance.

The three simplified MHD equations, (9.3)–(9.5), form a complete set when combined with Maxwell's equations:

$$\nabla\cdot{\bf B} =0,$$ (9.6)

$$\nabla\times {\bf E} = - \frac{\partial{\bf B}}{\partial t},$$ (9.7)

$$\nabla\times{\bf B} = \mu_0\,{\bf j}.$$ (9.8)

Note that we are justified in neglecting the displacement current because we are dealing with waves whose phase velocities are small compared to the velocity of light in vacuum.

Consider a simplified scenario in which the Cartesian coordinate $z$ is ignorable. In other words, there is no variation in the $z$-direction (i.e., $\partial/\partial z=0$), and no component of the magnetic field or the plasma flow velocity in the $z$-direction (i.e., $B_z=V_z=0$.) We can automatically satisfy Equations (9.3) and (9.6) by writing

$${\bf V} =\nabla\phi\times {\bf e}_z,$$ (9.9)

$${\bf B} = \nabla\psi\times{\bf e}_z,$$ (9.10)

where ${\bf e}_z$ is a unit vector parallel to the $z$-axis. Note that ${\bf V}\cdot\nabla\phi= {\bf B}\cdot\nabla\psi=0$. Thus, $\phi(x,y)$ and $\psi(x,y)$ map out the flow stream-lines and the magnetic field-lines, respectively, in the $x$-$y$ plane. $\psi(x,y)$ is usually referred to as magnetic flux, because the net magnetic flux (per unit length in the $z$-direction) that passes through a surface (whose normal lies in the $x$-$y$ plane) that links points ($x_1$, $y_1$) and ($x_2$, $y_2$) is $\psi(x_1,y_1)-\psi(x_2,y_2)$.

If we take the $z$-component of Equation (9.5), combined with Equations (9.7)–(9.10), and the $z$-component of the curl of Equation (9.4), combined with Equations (9.8)–(9.10), then we obtain the reduced-MHD equations (Strauss 1976):

$$\frac{\partial\psi}{\partial t} = [\phi,\psi]+\frac{\eta}{\mu_0}\,(J-J_0),$$ (9.11)

$$\rho\,\frac{\partial U}{\partial t} = \rho\,[\phi,U] + \mu_0^{-1}\,[J,\psi],$$ (9.12)

$$J =\nabla^2\psi,$$ (9.13)

$$U =\nabla^2\phi,$$ (9.14)

where $[A,B]\equiv \nabla A\times \nabla B\cdot{\bf e}_z$. Here, the current density is written ${\bf j} = -\mu_0^{-1}\,J\,{\bf e}_z$, and $\omega$$=-U\,{\bf e}_z$ is the plasma vorticity. Moreover, $J_0(x)$ parametrizes the $z$-directed inductive electric field that prevents the current in the current sheet from eventually decaying to zero under the action of resistivity. The reduced-MHD equations are so-called because they do not contain the full range of MHD physics (i.e., they do not contain the slow and fast magnetosonic waves).