Drift Motions

Intuition

A charged particle gyrates rapidly around a magnetic field line. If the field has a gradient, or if there is an electric field perpendicular to B, the center of the gyration orbit moves slowly across B. These drift motions govern particle transport in non-uniform magnetized plasmas.

Formal Definition

A drift is the guiding-center motion perpendicular to the local magnetic field. For slowly varying fields ($\omega/\Omega_c \ll 1$) and $v_\perp, v_\parallel$ constant over a gyroperiod, the average velocity is:

\[\mathbf{v}_d = \frac{\mathbf{F} \times \mathbf{B}}{q B^2}\]

where $\mathbf{F}$ is any external force.

Mathematical Formulation

$\mathbf{E}\times\mathbf{B}$ drift:

$$\mathbf{v}_{E\times B} = \frac{\mathbf{E}\times\mathbf{B}}{B^2}$$

Independent of charge and mass.

Gradient drift:

$$\mathbf{v}_{\nabla B} = \frac{m v_\perp^2}{2 q B^3} (\mathbf{B} \times \nabla B)$$

Curvature drift:

$$\mathbf{v}_c = \frac{m v_\parallel^2}{q B^3} \bigl[\mathbf{b} \times (\mathbf{b}\cdot\nabla)\mathbf{b}\bigr]$$

Gravity drift:

$$\mathbf{v}_g = \frac{m \mathbf{g} \times \mathbf{B}}{q B^2}$$

Derivation

Use the guiding-center expansion:

Decompose $\mathbf{r} = \mathbf{R} + \boldsymbol{\rho}$, where $\mathbf{R}$ is the guiding center and $\boldsymbol{\rho}$ is the gyration vector.
Slow fields: $\mathbf{E}(\mathbf{R}), \mathbf{B}(\mathbf{R})$.
Average the Lorentz force over one cyclotron period.
The drift velocity emerges from the $\mathbf{E}\times\mathbf{B}$ advection and the polarization drift from time-varying E.

For the gradient drift, note that the orbital radius $\rho_L = m v_\perp/|q|B$ changes as $B$ varies, giving an net orbital excursion proportional to $\nabla B$.

Worked Example

Tokamak ion gradient drift: In a tokamak with circular cross-section, the toroidal field $B_\phi \sim 1/R$ has a gradient toward the center. Compute the drift direction for a co-going ion ($v_\parallel$ along $\hat{z}$). Using $\mathbf{v}_c \propto m v_\parallel^2 B^{-3}(\mathbf{b}\times\kappa)$, the curvature vector points outward ($+\hat{r}$), so the drift is vertically upward for ions ($q>0$).

Common Mistakes

Drift speed is set by $v_\perp$ or $v_\parallel$. Confusing them gives wrong scaling.
Drifts are NOT caused by $\mathbf{E}$ alone. The gradient and curvature drifts persist even in vacuum B-fields.
Polarization drift is order $\omega/\Omega_c$. It is often small but becomes important in time-dependent problems.

Quiz Questions

Conceptual: Why do electrons and ions drift in the same direction under $\mathbf{E}\times\mathbf{B}$ but in opposite directions under $\nabla B$?
Computational: For a $\nabla B$ drift in $B = 1$ T with gradient $\nabla B/B = 0.1/m$, find $v_{\nabla B}$ for 1 keV protons with $v_\perp = v_\parallel$.
MCQ: Which drift is independent of the particle’s charge?
A) $\mathbf{E}\times\mathbf{B}$
B) Gradient
C) Curvature
D) Gravity