Incompressible Viscous Magnetohydrodynamics

 The magnetohydrodynamic (MHD) equations are a model for the dynamics of an overall electrically neutral fluid, called a plasma, that is made up of moving charged particles. These moving particles react to magnetic fields and their relative motion, as charged particles (i.e. a current), creates magnetic fields too. Thus, the system is non-linear and a good target for spectral methods. There is interest in the dynamics of a plasma confined in a toroidal geometry such as a fusion reactor. Clearly, it is desirable to be able to also consider plasma in more complicated geometries, thus spectral elements appear to be an ideal tool for this type of simulation.

The model as stated is actually a two fluid problem with a positively charged fluid of ions and a negatively charged fluid of electrons. The mass of an electron is very small compared to the mass of an ion, thus we can represent the system as a single fluid that reacts with magnetic fields.

For this section we will assume that the plasma is incompressible. This approximation greatly reduces the complexity of the system of equations for the model. Using a simplified model that is easily verified provides a good benchmark for the more complex compressible model that we investigate in chapter 8.

The MHD equations have been treated numerically by many including [61] and [62,63] using finite differences and [64] using Fourier collocation. There has been some disagreement about how to deal with the constraint that no magnetic monopoles can exist (i.e. that $\nabla \cdot {\bf B}=0$where B is the magnetic field). However, it has been shown in [65] that the smallest errors in satisfying this constraint can cause a significant cumulative effect as a simulation progresses. We will test a magnetic streamfunction formulation for the incompressible equations as this provides a formulation that fits naturally into the set of operators that we developed in previous chapters. A simple modification was needed to make the incompressible Navier-Stokes code handle the extra field and interactions.