Test the grad-B drift against the exact orbit

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Plasma Physics · The plasma state and single-particle motion

Plasma Physics, Ch One particle in given fields: gyration and the guiding-centre drifts

Problem

Test the grad-B drift against the exact orbit. A proton moves in the field

\mathbf{B} = B_0(1 + y/L)\hat{\mathbf{z}}

with no electric field. Work in units with

q = m = B_0 = 1

and initial speed

v_\perp = 1

, so that time is measured in units of

1/\Omega_0

and length in units of

r_{L0} = v_\perp/\Omega_0

. Start the proton at

(x, y) = (1, 0)

with velocity

(v_x, v_y) = (0, -1)

and no velocity along

z

; its guiding centre then starts at the origin, on the line

y = 0

where

B = B_0

. Follow the guiding centre through

X = x + v_y/(1 + y/L)

, which removes the fast circle from

x

. ((a)) Use the grad-B drift to predict the guiding-centre velocity, magnitude and direction, as a function of

L

. ((b)) Integrate

\dot{\mathbf{v}} = \mathbf{v}\times\mathbf{B}

numerically for

L = 20

over

0 \le t \le 40\pi

, twenty gyroperiods at

B_0

. Find the ratio of the measured drift

[X(40\pi) - X(0)]/40\pi

to the prediction of part (a). ((c)) Repeat part (b) for

L = 5

, and explain why the agreement is worse.