Equilibrium of cylindrical plasmas

We have considered the effects of plasma inhomogeneity on MHD waves and instabilities in Chapter 7 for the model of a plane gravitating plasma slab where inhomogeneity is restricted to the vertical direction. For the description of laboratory and astrophysical plasma dynamics, the concept of magnetic flux tubes is quite central, as we have seen in Chapter 8. This automatically leads to the consideration of cylindrical plasmas where the inhomogeneities are operating in the radial direction. Whereas the model remains one-dimensional, so that most of the analytical techniques developed in Chapter 7 remain valid, the introduction of curvature of the magnetic field brings in qualitatively different physical effects that significantly influence the dynamics of flux tubes. We will now neglect gravity since it plays no role in laboratory plasmas and, for astrophysical plasmas, it is more adequately incorporated in an axi-symmetric model with a central gravitating object. The latter requires a two-dimensional model, which has to be relegated to the more advanced chapters. We will see that curvature of the magnetic field enters the equations in a very similar way to gravity in the plasma slab of Chapter 7.

Diffuse plasmas

For the study of confined plasmas, the diffuse cylindrical plasma column (called ‘diffuse linear pinch’ in the older plasma literature) is one of the most useful models. It is probably the most widely studied model in plasma stability theory. Since we have obtained a basic understanding of the spectrum of inhomogeneous one-dimensional systems, the analysis of the diffuse linear pinch can now be undertaken with more fruit than was possible in the early days of fusion research when this configuration was first investigated.