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Drift Orbit Lab

Learning goal

Separate the two motions that make up every magnetised particle orbit: the fast, boring gyration, and the slow guiding-centre drift that actually decides whether a plasma stays confined. The blue guiding-centre track is what drift theory predicts; the red orbit is what the particle really does.

The integrator is a Boris pusher — the same second-order, phase-space-preserving scheme used in production PIC codes. It conserves energy to one part in 10⁷ over the whole run, which is why the mirror bounces forever instead of slowly leaking. Units are normalised so that \(q/m = 1\) and \(B_0 = 1\), making \(\omega_c = 1\) and the gyroradius numerically equal to \(v_\perp\).

Things to try

  1. Uniform B, flip the charge. The orbit reverses its sense of rotation but not its handedness relative to \(\mathbf{B}\): both signs circulate so that the orbit's own current opposes the applied field. Every plasma is diamagnetic, and this is why.

  2. E × B, flip the charge. Nothing happens to the drift. Same direction, same speed — \(\mathbf{v} = \mathbf{E}\times\mathbf{B}/B^2\) has no \(q\) in it. A drift that moves ions and electrons together carries no current and cannot separate charge.

  3. ∇B, flip the charge. Now the drift reverses. Compare the measured drift for the two signs: they differ slightly in magnitude too, because each species' guiding centre sits on a different side of the starting point and therefore samples a different \(B\). Charge-dependent drift ⇒ current ⇒ this one matters for equilibrium.

  4. Gravitational drift. A force that does not scale with charge produces a drift that does. Push the force up and watch ions and electrons separate — the resulting charge layer makes an \(\mathbf{E}\), the \(\mathbf{E}\times\mathbf{B}\) drift then moves the whole fluid, and you have just built the interchange instability by hand.

  5. Curvature drift. The display switches to the poloidal cross-section of a genuine \(1/R\) toroidal field. Both curvature and ∇B drifts point vertically, and they reverse with charge. This is the single most important negative result in magnetic confinement: a purely toroidal field separates charge vertically, the resulting \(\mathbf{E}\times\mathbf{B}\) pushes the plasma straight out, and no plasma is confined. Twist is not optional — see magnetic confinement.

  6. Magnetic mirror. Set \(v_\parallel\) small and watch the particle bounce. Now raise \(v_\parallel\) until it escapes out the end: you have found the loss cone experimentally. Check the readout — \(\mu\) stays constant to about 1% through the bounce, which is exactly what "adiabatic invariant" means: nearly, not exactly, conserved.

What the readouts mean

Readout Meaning
measured drift Guiding-centre velocity with the streaming along \(\mathbf{B}\) subtracted — a genuine drift, timed over many gyroperiods
predicted drift \(\dfrac{\mathbf{E}\times\mathbf{B}}{B^2} + \dfrac{\mathbf{F}\times\mathbf{B}}{qB^2} + \dfrac{m\left(v_\parallel^2 + \tfrac12 v_\perp^2\right)}{qB^3}\,\mathbf{B}\times\nabla B\), evaluated at the particle's current position
μ \(mv_\perp^2/2B\), the first adiabatic invariant

The two drift columns agree to a couple of percent in every configuration. Where they disagree is instructive: shrink the field scale length \(L\) until the gyroradius is no longer small compared with it, and drift theory visibly breaks down — its whole derivation assumed \(r_L/L \ll 1\).

Drift motions · Larmor radius · Adiabatic invariants · Magnetic mirror · Loss cone · Magnetic confinement