Drift equations

Approximate equations of motion of a charged particle in electric and magnetic fields, obtained by averaging the rapid motion of the particle under the effect of the magnetic field. Drift equations apply if the magnetic field $ \vec{B} $ is changing slowly in space and in time, while the electric field $ \vec{E} $ is small as compared to the magnetic field:

$$ \tag{1 } \frac{1}{\omega _ {B} B } \frac{\partial R }{\partial t } \sim \epsilon ,\ \ \frac{\rho _ {B} }{B} | \nabla B | \sim \epsilon ,\ \frac{c E }{v B } \sim \epsilon . $$

Here $ \epsilon $ is a small parameter, $ \omega _ {B} = eB / mc $ is the Larmor frequency, $ \rho _ {B} = v _ \perp / | \omega _ {B} | $ is the Larmor radius, $ v $ is the velocity of the particle, and $ v _ \perp $ is the velocity component in the direction normal to the magnetic field. Drift equations are obtained from the complete equations of motion by expanding in powers of $ \epsilon $ with the aid of the averaging method [1]. They have the following form:

$$ \tag{2 } \frac{d \vec{R} }{dt} = V _ {\| } \frac{\vec{B} }{B} + \vec{V} _ {\textrm{ dr } } , $$

$$ \tag{3 } \frac{d}{dt} \left ( \frac{1}{2} m ( V _ \perp ^ {2} + V _ {\| } ^ {2} ) \right ) = e \vec{E} \frac{d \vec{R} }{dt} - \frac{m c V _ \perp ^ {2} }{2 B ^ {2} } \vec{B} \mathop{\rm rot} \vec{E} , $$

$$ \tag{4 } \frac{d}{dt} \left ( \frac{V _ \perp ^ {2} }{B} \right ) = 0 , $$

where

$$ \vec{V} _ { \mathop{\rm dr} } = \frac{c}{B} ^ {2} [ \vec{E} \times \vec{B} ] + \frac{m c V _ {\| } ^ {2} }{c B ^ {4} } [ \vec{B} \times ( \vec{B} \nabla ) \vec{B} ] + \frac{m c V _ \perp ^ {2} }{2 e B ^ {3} } [ \vec{B} \times \nabla B ] . $$

The system (2)–(4), known as the drift system, is written with respect to the auxiliary averaged variables $ \vec{R} $, $ V _ \perp $, $ V _ {\| } $, connected by a certain relation with the initial variables $ \vec{r} $, $ \vec{v} $. The drift rate $ \vec{V} _ { \mathop{\rm dr} } $ in equation (2) describes a slow motion along the averaged trajectory in the direction perpendicular to the magnetic field:

$$ V _ { \mathop{\rm dr} } \sim \epsilon v ,\ \vec{V} _ { \mathop{\rm dr} } \vec{B} = 0 . $$

The equations (3) and (4) have second-order accuracy with respect to $ \epsilon $ and define the magnitudes $ V _ \perp $ and $ V _ {\| } $ up to first order terms in the time interval $ t $ containing many Larmor periods $ t \sim 1 / \epsilon | \omega _ {B} | $. Equation (2) has first-order accuracy with respect to $ \epsilon $.

The magnitude $ \mu = V _ \perp ^ {2} / B $, which is the integral of the drift system (2)–(4), is an approximate integral of the true motion. It is known as the adiabatic invariant. In the static case, when $ \mathop{\rm rot} \vec{E} = 0 $ and $ \vec{E} = - \nabla \phi $, equation (3) admits the energy integral

$$ \frac{1}{2} m ( V _ \perp ^ {2} + V _ {\| } ^ {2} ) + e \phi = \ \textrm{ const } $$

for the averaged motion.

The drift system may be generalized to include the relativistic case [2], [3].

References

[1]	N.N. Bogolyubov, Yu.A. Mitropol'skii, "Asymptotic methods in the theory of non-linear oscillations" , Hindushtan Publ. Comp. , Delhi (1961) (Translated from Russian)
[2]	D.V. Sivukhin, , Problems in the theory of plasma , 1 , Moscow (1963) pp. 7–97 (In Russian)
[3]	A.I. Morozov, L.S. Solov'ev, , Problems in the theory of plasma , 2 , Moscow (1963) pp. 177–261 (In Russian)