Guiding center

In physics, the motion of an electrically charged particle such as an electron or ion in a plasma in a magnetic field can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relatively slow drift of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.
Gyration.
If the magnetic field is uniform and all other forces are absent, then the Lorentz force will cause a particle to undergo a constant acceleration perpendicular to both the particle velocity and the magnetic field. This does not affect particle motion parallel to the magnetic field, but results in circular motion at constant speed in the plane perpendicular to the magnetic field. This circular motion is known as the gyromotion. For a particle with mass formula_1 and charge formula_2 moving in a magnetic field with strength formula_3, it has a frequency, called the gyrofrequency or cyclotron frequency, of
where formula_5 is the radius of curvature pointing outwards, away from the center of the circular arc which best approximates the curve at that point.
where formula_7 is the unit vector in the direction of the magnetic field. This drift can be decomposed into the sum of the curvature drift and the term
In the important limit of stationary magnetic field and weak electric field, the inertial drift is dominated by the curvature drift term.
Curved vacuum drift.
In the limit of small plasma pressure, Maxwell's equations provide a relationship between gradient and curvature that allows the corresponding drifts to be combined as follows
For a species in thermal equilibrium, formula_10 can be replaced by formula_11 (formula_12 for formula_13 and formula_14 for
formula_15).
The expression for the grad-B drift above can be rewritten for the case when formula_16 is due to the curvature.
This is most easily done by realizing that in a vacuum, Ampere's Law is 
formula_17. In cylindrical coordinates chosen such that the azimuthal direction is parallel to the magnetic field and the radial direction is parallel to the gradient of the field, this becomes
Since formula_19 is a constant, this implies that
and the grad-B drift velocity can be written
Polarization drift.
A time-varying electric field also results in a drift given by
Obviously this drift is different from the others in that it cannot continue indefinitely. Normally an oscillatory electric field results in a polarization drift oscillating 90 degrees out of phase. Because of the mass dependence, this effect is also called the inertia drift. Normally the polarization drift can be neglected for electrons because of their relatively small mass.
Diamagnetic drift.
The diamagnetic drift is not actually a guiding center drift. A pressure gradient does not cause any single particle to drift. Nevertheless, the fluid velocity is defined by counting the particles moving through a reference area, and a pressure gradient results in more particles in one direction than in the other. The net velocity of the fluid is given by
Drift Currents.
With the important exception of the E-cross-B drift, the drift velocities of differently charged particles will be different. This difference in velocities results in a current, while the mass dependence of the drift velocity can result in chemical separation.
References.
T.G. Northrop, The guiding center approximation to charged particle motion, Annals of Physics 15, p.79-101, 1961
H.J. de Blank, Guiding center motion, Fusion Science and Technology / Volume 61 / Number 2T / February 2012 / Pages 61-68
"Cosmic Plasma" (1981), Hannes Alfvén