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Introduction

As we have seen in Sect. 3, the MHD equations take the form
$\displaystyle \frac{d\rho}{dt} + \rho\,\nabla\cdot{\bf V}$ $\textstyle =$ $\displaystyle 0,$ (681)
$\displaystyle \rho\,\frac{d{\bf V}}{dt} + \nabla p - {\bf j}\times {\bf B}$ $\textstyle =$ $\displaystyle {\bf0},$ (682)
$\displaystyle {\bf E} + {\bf V}\times {\bf B}$ $\textstyle =$ $\displaystyle {\bf0},$ (683)
$\displaystyle \frac{d}{dt}\!\left(\frac{p}{\rho^{\Gamma}}\right)$ $\textstyle =$ $\displaystyle 0,$ (684)

where $\rho\simeq m_i\,n$ is the plasma mass density, and ${\Gamma}=5/3$ is the ratio of specific heats.

It is often observed that the above set of equations 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 the MHD equations. After all, a hot, tenuous, quasi-neutral plasma is highly conducting, and if the motion is sufficiently fast then both viscosity and heat conduction can be plausibly neglected. However, we can appreciate, from Sect. 3, that this is a highly oversimplified and misleading argument. The problem is, of course, that a weakly coupled plasma is a far more complicated dynamical system than a conducting liquid.

According to the discussion in Sect. 3, the MHD equations are only valid when

\begin{displaymath}
\delta^{-1}\,v_t \gg V \gg \delta\,v_t.
\end{displaymath} (685)

Here, $V$ is the typical 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 typical ratio of a particle gyro-radius to the scale-length of the motion). Clearly, the above inequality is most likely to be satisfied in a highly magnetized (i.e., $\delta\rightarrow 0$) plasma.

If the plasma dynamics becomes too fast (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 effect, which are not 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 Sect. 3.9)

\begin{displaymath}
U \sim \delta \,V.
\end{displaymath} (686)

Note, however, that if the plasma dynamics becomes 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 whenever the diamagnetic velocities, which are quite different for different plasma species, become comparable to the ${\bf E}\times{\bf B}$ velocity (see Sect. 3.12). Furthermore, effects such as plasma resistivity, viscosity, and thermal conductivity, which are not taken into account in the MHD equations, become important in this limit.

Broadly speaking, 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 discussed in Sect. 3.13, the MHD equations also fairly well describe the perpendicular (but not the parallel!) motions of collisionless plasmas.

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


next up previous
Next: Magnetic Pressure Up: Magnetohydrodynamic Fluids Previous: Magnetohydrodynamic Fluids
Richard Fitzpatrick 2011-03-31