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Grad-B & Curvature Drifts

Equations

Grad-B drift (field strength varies in space):

\[\boxed{\,\mathbf{v}_{\nabla B} = -\frac{m v_{\perp}^2}{2qB^3}\,\nabla B\times\mathbf{B}\,}\]

Curvature drift (field lines bend, curvature vector \(\hat{\mathbf{R}}/R = \hat{\mathbf{B}}\cdot\nabla\hat{\mathbf{B}}\)):

\[\boxed{\,\mathbf{v}_c = \frac{m v_{\parallel}^2}{qB^2}\,\frac{\hat{\mathbf{R}}}{R}\times\mathbf{B}\,}\]

Combined (vacuum-like fields, the tokamak workhorse):

\[\mathbf{v}_{\nabla B} + \mathbf{v}_c = \frac{m}{qB^2}\left(\frac{v_\perp^2}{2} + v_\parallel^2\right)\frac{\hat{\mathbf{R}}}{R}\times\mathbf{B}\]

Physical meaning

  • Grad-B: gyration circles are fatter where \(B\) is weak, tighter where strong; the orbit fails to close and creeps sideways, perpendicular to both \(\nabla B\) and \(\mathbf{B}\).
  • Curvature: streaming along a bent field line costs centripetal force \(mv_\parallel^2/R\); the particle supplies it by drifting so the magnetic force provides the turn.

Variables

\(v_\perp\) — gyration speed · \(v_\parallel\) — speed along \(\mathbf{B}\) · \(\nabla B\) — gradient of field magnitude · \(R\) — field-line curvature radius · \(q\) — signed charge.

The crucial property: charge dependence

Both drifts flip direction with the sign of \(q\) (and scale with energy): ions and electrons separate → currents and space charge. In a purely toroidal field (\(B \propto 1/r\), both drifts vertical) the resulting charge separation creates a vertical E field whose E×B drift pushes the whole plasma outward — the reason tokamaks need poloidal field (rotational transform) to survive. This failure is beautifully visible in the course's toroidal-field orbit exercise.

Assumptions

Guiding-center ordering: \(r_L \ll L_B\), \(R\); drifts slow compared to gyration. Both are first-order corrections in \(r_L/L\) — see guiding-center theory.

Quiz

Q1 (conceptual). In a tokamak, grad-B and curvature drifts point the same way for a given particle. Why?

Answer

For a toroidal field \(B \propto 1/r\): \(\nabla B\) points inward (−R̂) and the curvature vector also points inward, so \(\nabla B\times\mathbf{B}\) and \(\hat{\mathbf{R}}\times\mathbf{B}\) are parallel (both vertical, sign set by charge) — hence the neat combined formula.

Q2 (computational). A 1 keV proton (\(v_\parallel^2 = v_\perp^2 = v^2/2\)) in a \(B = 1\) T field with \(R = 1\) m. Estimate the combined drift speed.

Answer

\(v^2 = 2E/m \approx 1.9\times10^{11}\ \text{m}^2/\text{s}^2\); \(v_d = \frac{m}{qBR}\left(\frac{v_\perp^2}{2} + v_\parallel^2\right) = \frac{1.67\times10^{-27}}{1.6\times10^{-19}\times1\times1}\times1.43\times10^{11} \approx 1.5\times10^{3}\) m/s — a slow creep beside the ~\(4\times10^5\) m/s thermal speed, but relentless.

Q3 (MCQ). Which particle has the larger grad-B drift at equal energy?

  • (a) electron (b) proton (c) equal (d) depends on B
Answer

(c). The drift is \(\frac{mv_\perp^2}{2qB^3}|\nabla B \times \mathbf B|\) — at equal perpendicular energy (\(\frac12 mv_\perp^2\)), mass cancels; magnitudes match (directions oppose).