Although ideal MHD is often a good model for astrophysical and space plasmas and is widely employed in fusion research it is never universally valid, for the reasons discussed in Section 4.1. In this chapter we consider some of the most important effects which arise when allowance is made for finite resistivity and, in the case of shock waves, other dissipative mechanisms. Even though the dissipation may be very weak the changes it introduces are fundamental. For example, finite resistivity enables the plasma to move across field lines, a motion forbidden in ideal MHD. Usually, the effects of this diffusion are concentrated in a boundary layer so that mathematically the problem is one of matching solutions, of the non-ideal equations in the boundary layer and ideal MHD elsewhere. On the length scale of the plasma the boundary layer may be treated as a discontinuity in plasma and field variables and, depending on the strength of the flow velocity, this discontinuity may appear as a shock wave.
A comparison of Tables 3.1 and 3.2 reveals that the difference between resistive and ideal MHD is the appearance of extra terms proportional to the plasma resistivity, η ≡ σ−1, in the evolution equations for P and B.