Fundamentals of Plasma Physics 2025, 33 - MHD and Field Configurations
While the least accurate, MHD provides a macroscopic point of view, which is more efficient to compute. Especially for complex geometry. MHD approaches usually ignores plasma viscosity, and limit the viscous damping to those cases, where torques exist, in order to balance it. A vacuum field is a field without local currents, hence the curl of the vacuum field is zero. The vacuum field can then always be expressed by a gradient of a field. This writes the vacuum field potentials into the Laplace equation ∇2ψ = 0. ψ can be expressed as the linear superposition of modes eimθ+ikz with varying choices in m and k. ψ can then be written as a superpositions of linear combinations of the Bessel functions, with the modes. The vacuum field is the lowerst-energy field.
Non-vacuum configurations don't necessarily decay to the vacuum field. In this case, one deals with force-free states, to which a system decay alternatively. These can be understood as local minima of the potential.
The intuition for magnetic pressure can be built by picturing a bundle of parallel wires with currents, running either parallel, or anti-parallel. As the spiral of induced magnetic fields has its direction determined by the current direction, parallel wires attract, and anti-parallel ones repel one another. Plasma can be modeled after the same bundle with the presence of an external field. As in this case, the currents all run parallel, the plasma volume contracts. More complex field configurations can be used to more clearly shape a plasma volume.
The magnetic force then is not uniform across a plasma, and should be formalized as an anisotropic stress tensor.
