Fundamentals of Plasma Physics 2025, 22 - Drift Equations and Double Adiabatic MHD

The double adiabatic MHD equations can be derived by summing the currents of the particle drifts, and adding the diamagnetic current. The density of the magnetic dipole moment per unit volume has a magnetization current J_M = ∇ × M. For one particle, the magnetic moment is equal to μ. At large particle numbers, provided a density n_σ and mean magnetic moment, the density is computed as a sum of products nμB. This defines the diamagnetic current J_M in terms of the magnetic field. If a temperature gradient exists, then the density causes a dependency of the net macroscopic current of the pressure gradient. This correspondence is established by the the diamagnetic current. As the dominant cross-field particle motion is the same for both particle species, and does not generate macroscopic currents, all cross-field currents result from other, smaller drift effects.

Adiabatic behavior occurs when the EM-field changes effectively continuously (differentiably) between orbits. Non-adiabatic particle motion occurs otherwise. Descriptions of such motions are non-analytic. Assuming geometrically symmetry of the EM field wrt some coord. Q, some anaylitic description becomes possible. The symmetry causes a exact constant canonical momentum. Sudden reversal of polarity on azimuthally symmetric magnetic field having no azymuthal component will result in spiralling motions around the field lines. When fields change gradually in space relative to the initial gyro-orbit dimensions, but the fields contribute to particle motion so that the gyro-orbit increases to the point that the smallness assumption fails, the particles will experience an E × B drift. For infinitesimal amplitude, the displacements associated with the drift and polarization drift are negligible.