Introduction

In this chapter we turn to a consideration of the physics of inhomogeneous plasmas. Since virtually all plasmas whether in the laboratory or in space are to some degree inhomogeneous, all that can be attempted within the limits of a single chapter is to outline some general points and illustrate these with particular examples. Throughout the book we have dealt in places with plasmas which were inhomogeneous in density or temperature and confined by spatially inhomogeneous magnetic fields. In the case of the Z-pinch the high degree of symmetry allowed us to find analytic solutions in studying the equilibrium. By contrast for a tokamak, even with axi-symmetry, solutions to the Grad-Shafranov equation could only be found numerically. Indeed the only general method of dealing theoretically with problems in inhomogeneous plasmas is by numerical analysis.

Nevertheless useful analytic insights may be gained in two limits. In the first, plasma properties change slowly in the sense that for an inhomogeneity scale length L and wavenumber k, kL ≫ 1 and one can appeal to the WKBJ approximation described in Section 11.2. In this limit we shall draw on illustrations from the physics of wave propagation in inhomogeneous plasmas. If we picture a wave propagating in the direction of a density gradient, at some point on the density profile it may encounter a cut-off or a resonance. As we found in Chapter 6, propagation beyond a cut-off is not possible and the wave is reflected, whereas at a resonance, wave energy is absorbed. The WKBJ approximation breaks down in the neighbourhood of both cut-offs and resonances.