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  • Print publication year: 2003
  • Online publication date: July 2010

4 - Ideal magnetohydrodynamics

Summary

Introduction

Ideal MHD is used to describe macroscopic behaviour across a wide range of plasmas and in this chapter we consider some of the most important applications. Being dissipationless the ideal MHD equations are conservative and this leads to some powerful theorems and simple physical properties. We begin our discussion by proving the most important theorem, due to Alfvén (1951), that the magnetic field is ‘frozen’ into the plasma so that one carries the other along with it as it moves. This kinematic effect arises entirely from the evolution equation for the magnetic field and represents the conservation of magnetic flux through a fluid element. Of course, any finite resistivity allows some slippage between plasma and field lines but discussion of these effects entails non-ideal behaviour and is postponed until the next chapter.

The concept of field lines frozen into the plasma leads to very useful analogies which aid our understanding of the physics of ideal MHD. It also suggests that one might be able to contain a thermonuclear plasma by suitably configured magnetic fields, although research has shown that this is no easily attainable goal. Further, since the ideal MHD equations are so much more amenable to mathematical analysis they can be used to investigate realistic geometries. The theory has thereby provided a useful and surprisingly accurate description of the macroscopic behaviour of fusion plasmas showing why certain field configurations are more favourable to containment than others.

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