Excerpt from Courant Institutions of Mathematical Science: Magneto-Fluid Dynamics Division

Thus one turns to macroscopic, i.e. Fluid, descriptions of the plasma. The simplest such model is a one - fluid model in which the plasma is regarded as a perfectly con ducting fluid with an adiabatic pressure. The resulting equations were derived by Lundquist [21] and have yielded a very rich theory which has been extensively studied (see Grad [15] for a survey). However, the model gives no shock structure at all and has little obvious relevance for a collision - free plasma.

Between these two extremes lie many models, one of which is studied in the present work. This is the adiabatic two fluid model proposed and briefly discussed by Gardner et alia The total mass, momentum and energy conservation equa tions are as in the one-fluid model but the individual momentum equations for the two types of particle are combined to give an equation for the current; this eliminates the need for assuming any kind of Ohm's law. The model has the advantage of still being quite a simple fluid model but also, in the important case of a cold plasma, of being completely equivalent to a self - consistent particle formulation. Indeed it serves to relate several disparate elements of collision free shock theory (see Blank and Grad [7] on this subject).

One cannot, of course, expect a proper shock structurefrom such a model for it includes no specific dissipative mechanism. But one can obtain non-trivial compression waves of finite amplitude, and for a considerable range of amplitudes these do not break to form shocks. Even when shocks do form one can hope to localise them and hence those regions in which a more exact theory is necessary. As in classical gas dynamics the canonical problem is that of the compression wave generated by the movement of some sort of piston. It is around such a problem that our study will be built.

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