Introduction

As we saw in Section 4.13, the MHD equations are written

$$\frac{d\rho}{dt} + \rho\,\nabla\cdot{\bf V} =0,$$ (7.1)

$$\rho\,\frac{d{\bf V}}{dt} + \nabla p - {\bf j}\times {\bf B} ={\bf0},$$ (7.2)

$${\bf E} + {\bf V}\times {\bf B} ={\bf0},$$ (7.3)

$$\frac{d}{dt}\!\left(\frac{p}{\rho^{\,{\mit\Gamma}}}\right) =0,$$ (7.4)

where $\rho$ is the plasma mass density, ${\bf V}$ the center of mass velocity, $p$ the pressure, ${\bf E}$ the electric field-strength, ${\bf B}$ the magnetic field-strength, and ${\mit\Gamma}=5/3$ the ratio of specific heats.

It is often remarked that Equations (7.1)-(7.4) are identical to the equations governing the motion of an inviscid, adiabatic, perfectly conducting, electrically neutral, liquid. Indeed, this observation is sometimes used as the sole justification for adopting the MHD equations. After all, a hot, tenuous, quasi-neutral plasma is highly conducting, and if the motion is sufficiently rapid then viscosity and heat conduction can both plausibly be neglected (which implies that the motion is adiabatic). However, as should be clear from the discussion in Section 4.11, this is a highly oversimplified and misleading argument. The problem, of course, is that a weakly coupled plasma is a far more complicated dynamical system than a conducting liquid.

According to the analysis of Section 4.11, the MHD equations are only valid when

$$\delta^{\,-1}\,v_t \gg V \gg \delta\,v_t$$ (7.5)

for both species. Here, $V$ is the typical fluid velocity associated with the plasma dynamics under investigation, $v_t$ is the typical thermal velocity, and $\delta$ is the typical magnetization parameter (i.e., the ratio of a particle gyroradius to the scalelength of the motion). Clearly, the previous inequality is most likely to be satisfied in a highly magnetized (i.e., $\delta\rightarrow 0$ ) plasma.

If the plasma dynamics becomes too rapid (i.e., $V\sim \delta^{\,-1}\,v_t$ ) then resonances occur with the motions of individual particles (e.g., the cyclotron resonances), which invalidate the MHD equations. Furthermore, effects, such as electron inertia and the Hall current, that are not (usually) taken into account in the MHD equations, become important.

MHD is essentially a single-fluid plasma theory. A single-fluid approach is justified because the perpendicular motion is dominated by ${\bf E}\times{\bf B}$ drifts, which are the same for both plasma species. Furthermore, the relative streaming velocity, $U_\parallel$ , of both species parallel to the magnetic field is strongly constrained by the fundamental MHD ordering (see Section 4.11)

$$U \sim \delta \,V.$$ (7.6)

However, if the plasma dynamics become too slow (i.e., $V\sim \delta\,v_t$ ) then the motions of the electron and ion fluids become sufficiently different that a single-fluid approach is no longer tenable. This occurs because the diamagnetic velocities, which are quite different for different plasma species, become comparable to the ${\bf E}\times{\bf B}$ velocity. (See Section 4.14.) Furthermore, effects such as plasma resistivity, viscosity, and thermal conductivity, which are not (usually) taken into account in the MHD equations, become important in this limit.

It follows, from the previous discussion, that the MHD equations describe relatively violent, large-scale motions of highly magnetized plasmas.

Strictly speaking, the MHD equations are only valid in collisional plasmas (i.e., plasmas in which the mean-free-path is much smaller than the typical variation scale-length). However, as is discussed in Section 4.15, the MHD equations also describe the perpendicular (but not the parallel) motions of collisionless plasmas fairly accurately.

Assuming that the MHD equations are valid, let us now investigate their properties.