3.6 Relation of Drift Equations to the Double Adiabatic MHD Equations 91where we have usedv∇B∼vc∼v^2 ⊥/ωcr∼ωcr^2 L/r.Thus, if the Larmor radius is much
smaller than the characteristic scale length of the field, the magneticflux term dominates
the action integral and adiabatic invariance corresponds to the particlestaying on a constant
flux surfaceas its orbit evolves following the various curvature and gradBdrifts. This third
adiabatic invariant is much more fragile thanJ, which in turn was more fragile thanμ,
because the analysis here is based on the rather strong assumption that thecurvature and
gradBdrifts are small enough for theδrjto trace out a nearly periodic orbit.
3.6 Relation of Drift Equations to the Double Adiabatic MHD Equations
The derivation of the MHD Ohm’s law involved dropping the Hall term (see p. 48)and
the basis for dropping this term was assuming thatω << ωciwhereωis the characteris-
tic rate of change of the electromagnetic field. The derivation of the singleparticle drift
equations involved essentially the same assumption (i.e., the motion was slow compared to
ωcσ). Thus, if the characteristic rate of change of the electromagnetic fieldis slow com-
pared toωciboth the MHD and the single particle drift equations ought to be equally valid
descriptions of the plasma dynamics. If so, then there also ought to be somesort of a cor-
respondence relation between these two points of view. Some evidence supporting this
hypothesis was the observation that the single particle adiabatic invariantsμandJwere
respectively related to the perpendicular and parallel double adiabaticMHD equations. It
thus seems reasonable to expect additional connections between the drift equations and
the double adiabatic MHD equations.
In fact, an approximate derivation of the double adiabatic MHD equations canbe ob-
tained by summing the currents associated with the various particle drifts — providing one
additional effect,diamagnetic current, is added to this sum. Diamagnetic current is a pe-
culiar concept because it is a consequence of the macroscopic phenomenon of pressure
gradients and so has no meaning in the context of a single particle description.
In order to establish this microscopic-macroscopic relationship we begin by recalling
from electromagnetic theory^2 that a magnetic material with densityMof magnetic dipole
moments per unit volume has an associated magnetization current
JM=∇×M. (3.126)The magnitude of the magnetic moment of a charged particle in a magnetic field was shown
in Sec.3.5.4 to beμ.The magnetic moment of a magnetic dipole is a vector pointing in
the direction of the magnetic field produced by the dipole. The vector magnetic moment
of a charged particle gyrating in a magnetic field ism=−μBˆwhere the minus sign
corresponds to cyclotron motion being diamagnetic, i.e., the magnetic field resulting from
cyclotron rotation opposes the original field in which the particle is rotating. For example,
an individual ion placed in a magnetic fieldB=Bzˆrotates in the negativeθdirection, and
so the current associated with the ion motion creates a magnetic field pointing in the−ˆz
direction inside the ion orbit.
(^2) For example, see p. 192 of (Jackson 1998).