The MHD equations

So far we have applied the arguments of classical fluid dynamics to obtain a closed set of equations for the plasma fluid variables but, except for the introduction of Joule heating, we have taken almost no account of the fact that a plasma is a conducting fluid. This we do now by specifying the force per unit mass F. Except in astrophysical contexts, where gravity is an important influence on the motion of the plasma, electromagnetic forces are dominant. For a fluid element with charge density q and current density j we then have

ρ F = q E + j × B (3.32)

where the fields E and B are determined by Maxwell’s equations (2.2)–(2.5). Equations (2.6) and (2.7) for q and j are not suitable in a fluid model. However, our first objective is to obtain a macroscopic description of the plasma in which the fields are those induced by the plasma motion. Thus, we now introduce the basic assumption of MHD that the fields vary on the same time and length scales as the plasma variables. If the frequency and wavenumber of the fields are ω and k respectively, we have ωτ _H ~1 and kL _H ~1, where τ _H and L _H are the hydrodynamic time and length scales. A dimensional analysis then shows that both the electrostatic force q E and displacement current ε ₀ μ ₀ ∂ E /∂t may be neglected in the non-relativistic approximation ω/k << c. Consequently, (3.32) becomes

ρ F = j × B (3.33)

and (2.3) is replaced by Ampere’s law

j = (1 / μ 0) grad x B (3.34)

Now, Poisson’s equation (2.4) is redundant (except for determining q) and just one further equation for j is required to close the set. Here we run into the main problem with a one-fluid model. Clearly, a current exists only if the ions and electrons have distinct flow velocities and so, at least to this extent, we are forced to recognize that we have two fluids rather than one. For the moment we side-step this difficulty by following usual practice in MHD and adopting Ohm’s law

j = σ( E + u × B ) (3.35)

as the extra equation for j. The usual argument for this particular form of Ohm’s law is that in the non-relativistic approximation the electric field in the frame of a fluid element moving with velocity u is ( E + u × B ). However, this argument is over-simplified, unless u is constant so that the frame is inertial, and later, when we discuss the applicability of the MHD equations, we shall see that the assumption of a scalar conductivity in magnetized plasmas is rarely justified. The status of (3.35) should be regarded, therefore, as that of a ‘model’ equation, adopted for mathematical simplicity.

This closes the set of equations for the variables ρ, u, P, T, E, B and j but before listing them it is useful to reduce the set by eliminating some of the variables. Although in electrodynamics it is customary to think of the magnetic field being generated by the current, in MHD we regard Ampere’s law (3.34) as determining j in terms of B. Then Ohm’s law (3.35) becomes

E = (1/ σμ ₀) grad × B − u × B (3.36)

so determining E. Finally, substituting (3.36) in (2.2), treating σ as a constant, and using (2.5), we get the induction equation for B

∂ B/ ∂t = (1/ σμ ₀) grad² B +grad× ( u × B ) (3.37)

Since we have eliminated j and E, this is now the only equation we need add to the set derived at the end of the last section for the fluid variables.