Curvature drift

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The curvature drift is a particle drift due to "centrifugal forces" of curving field lines. It is given by:

\mathbf{V}_{curv}=\left({\frac{2W_{||}}{qB^2}}\right)\mathbf{B}\times[(\mathbf{\hat{b}}\cdot \nabla)\mathbf{\hat{b}}]

If there are no internal currents, \nabla\times \mathbf{B}=0 and we can write:

\mathbf{V}_{curv}=\left({\frac{2W_{||}}{qB^3}}\right)\mathbf{B}\times\nabla B

[edit] Derivation

Consider a magnetic field with locally curvature. The guiding centers follow the magnetic field, however the particles do not follow the field lines exactly because the particles feel a 'centrifugal force' as a result of their parallel inertia.

The drift may be examined by picturing the particle on a circular orbit with a radius equal to the local radius of curvature. The particle position and velocity are only valid in this parameterization exactly locally, however we may obtain the force exactly.

\mathbf{r}=R_{curvature} (\cos{\theta(t)} \hat{x} + \sin\theta(t) \hat{y}) + \mathbf{r_0}

\theta(t)= \frac{v_{||} t}{R_c}

Yielding a centrifugal force

F_{cf} = \frac{mv_{||}^2}{r_c} \hat{r} = mv_{||}^2 \frac{\mathbf{R_c}}{R_c^2}

The force, like the gravitational force in the gravitational drift, causes a particle which is on the outside of the curve to move faster, and on the inside to move slower. After each orbit the guiding center then shifts. The general expression, obtained for the gravitational drift, is

\mathbf{v_F} = (\mathbf{F} \times \mathbf{B})/qB^2


\mathbf{V}_{curv}=\left({\frac{mv_{||}^2}{qB^2}}\right)\frac{\mathbf{R_c}\times\mathbf{B}}{R_c^2}=\left({\frac{2W_{||}}{qB^2}}\right)\frac{\mathbf{R_c}\times\mathbf{B}}{R_c^2}

\mathbf{V}_{curv}=\left({\frac{2W_{||}}{qB^2}}\right)\mathbf{B}\times[(\mathbf{\hat{b}}\cdot \nabla)\mathbf{\hat{b}}]


Note: here, the centrifugal force may ALSO act directly, causing a net drift along \mathbf{v_{||}}, which appears to be neglected in Goldston and Rutherford and other texts. I guess I'll look at this later.

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