Fundamentals of Plasma Physics 2025, 36 - The Energy Principle

The variations of potential energies consists of terms that are either positive-definite, or not. The former are always considered stabilizing, the latter could act destabilizing. Magnetic perturbations interior to the magnetofluid and perpendicular to the equilibirium field are always stabilizing. Incompressible instabilities are stronger than compressible ones, and force-free currents cause internal kink instabilities. At the magnetofluid-vacuum interface, instabilities can occur as the membrane is stretched until δW_int < 0. The current-driven instabilities are helical in nature and driven by gradients in J_0||/B₀.

Finite A₁˙B₁ corresponds to the local helical polarization of the perturbed fields, so the full product is equal to the density of magnetic helicity and the total magnetic helicity is the volume K = ∫_Vd³r AB. This definition is useful primarily due to the gauge-independence of K. From it derives that gradients in J_0||/B₀ can be considered as the free energy for driving kink modes. When this free energy is consumed, the kinks are stabilized. The tendency to coil up or kink increases the plasma inductance, though it kinks toward stable equilibrium (at a local minimum in potential energy). As the energy levels are discretized, one can read it as analogous to a quantum system with the vacuum magnetic field analogous to the ground state, while the various force-free equilibria describe higher energy states.

A Bennet pinch (z-pinch), axially uniform, cylindrical and featuring an axial plasma current with associated azimuthal B-field, with a pressure assumed uniform for r < a and and zero for r > a where a is the plasma radius. At equilibrium, the inward pinch force cancels with the outward force of the pressure gradient, i.e. -J_z0B_θ0 = ∂P₀/∂r, so they are only finite at exactly r = a. Probably thermal noise gives a regular pattern of bulges and constrictions, depending on the relation between the azimuthal magnetic field to its equilibrium value. This instability can be prevented by adding a perfectly conducting wall, through a strong axial vacuum magnetic field. A different kind of instability is produced by the plasma current results in a helical magnetic field, leading to a kinking tube instability. It, too, can be stabilized using a perfectly conducting wall, provided it's fitted close enough to the plasma volume.

Subdividing thusly constructed plasma into concentric regions of an interior, and a thin surface layer, the latter being a product of real experimental constraints. The interior pressure being uniform and interior current density equaling null, so that both J × B and ∇P vanish in the interior, has as a consequence that finite current density exists only in the surface layer and everywhere else the magnetic field is vacuum. By MHD equilibrium, there are no currents in the plasma interior or external vacuum. This enables qualitative discussions of (generic) instabilities via adiabatic pressure and Ampere's law. It emerges, that each of the mentioned instabilities has a characteristic mode.