Guiding Center Drifts in Inhomogeneous Fields & Adiabatic Invariants

1. Spatial Inhomogeneity and Scale Separation

In realistic astrophysical and laboratory magnetic geometries (such as tokamaks, stellarators, and planetary dipoles), magnetic fields are non-uniform in space. Exact particle trajectories cannot be integrated analytically. However, when the magnetic field varies slowly across the dimensions of a single Larmor orbit:

$$\epsilon \equiv \frac{r_L}{L_B} = \frac{r_L}{|\nabla B / B|} \ll 1$$

we can employ the guiding center approximation. The particle motion separates into two disparate spatial and temporal scales:

• A fast, quasi-periodic circular gyration at frequency $\omega_c$ with radius $r_L$.
• A slow, secular drift of the gyration center (the guiding center $\vec{R}$) across and along magnetic field lines.

1. Physical Mechanism of Gradient Drift

Consider a magnetic field directed along $\hat{z}$ whose strength increases along $\hat{y}$: $\vec{B} = B(y)\hat{z}$, with $\nabla B = \frac{dB}{dy}\hat{y}$.

As a charged particle gyrates in the $xy$-plane, its instantaneous Larmor radius $r_L(y) = \frac{m v_\perp}{|q| B(y)}$ is smaller at larger $y$ (stronger $B$) and larger at smaller $y$ (weaker $B$). This asymmetric curvature prevents the orbit from closing into a circle, creating a steady lateral drift perpendicular to both $\vec{B}$ and $\nabla B$.

2. Mathematical Derivation

Taylor expand the magnetic field around the guiding center $\vec{R}_0 = (x_0, y_0, z_0)$:

$$\vec{B}(\vec{r}) \approx \vec{B}_0 + (\vec{r} - \vec{R}_0)\cdot\nabla\vec{B} = \left( B_0 + y \frac{\partial B}{\partial y} \right)\hat{z}$$

The particle's unperturbed circular orbit around the guiding center is:

$$x(t) = r_L \sin(\omega_c t), \quad y(t) = \pm r_L \cos(\omega_c t)$$ $$v_x(t) = v_\perp \cos(\omega_c t), \quad v_y(t) = \mp v_\perp \sin(\omega_c t)$$

The instantaneous Lorentz force along $y$ is $F_y = -q v_x B(y)$. Taking the time average of $F_y$ over one complete gyration period $\tau_c = 2\pi / \omega_c$:

$$\langle F_y \rangle = -q \langle v_x B(y) \rangle = -q \left\langle v_\perp \cos(\omega_c t) \left[ B_0 + \left(\pm r_L \cos(\omega_c t)\right)\frac{\partial B}{\partial y} \right] \right\rangle$$

Noting that $\langle \cos(\omega_c t) \rangle = 0$ and $\langle \cos^2(\omega_c t) \rangle = \frac{1}{2}$:

$$\langle F_y \rangle = \mp q v_\perp r_L \frac{1}{2} \frac{\partial B}{\partial y} = \mp \frac{1}{2} q v_\perp \left( \frac{m v_\perp}{|q| B_0} \right) \frac{\partial B}{\partial y} = -\frac{1}{2} \frac{m v_\perp^2}{B_0} \frac{\partial B}{\partial y}$$

In coordinate-free vector notation, the net time-averaged transverse force is:

$$\langle \vec{F}_{\nabla B} \rangle = -\mu \nabla B = -\frac{m v_\perp^2}{2 B} \nabla B$$

where $\mu \equiv \frac{m v_\perp^2}{2 B}$ is the particle's magnetic dipole moment. Substituting this force into the general guiding center force drift formula $\vec{v}_F = \frac{1}{q}\frac{\vec{F}\times\vec{B}}{B^2}$:

$$\vec{v}_{\nabla B} = \frac{1}{q} \frac{(-\mu \nabla B) \times \vec{B}}{B^2} = \frac{\mu}{q B^2} (\vec{B} \times \nabla B) = \frac{m v_\perp^2}{2 q B^3} (\vec{B} \times \nabla B)$$

Because of the explicit factor of $q$, ions and electrons drift in opposite directions across magnetic gradients!

1. Centrifugal Acceleration in Curved Geometry

Consider magnetic field lines with local radius of curvature $\vec{R}_c$, pointing from the field line toward the center of curvature:

$$\frac{\vec{R}_c}{R_c^2} = -(\hat{b}\cdot\nabla)\hat{b}$$

where $\hat{b} = \vec{B} / B$ is the unit vector along the field. A particle moving along the field with parallel velocity $v_\parallel$ experiences a centrifugal force directed outward:

$$\vec{F}_c = \frac{m v_\parallel^2}{R_c^2} \vec{R}_c$$

2. The Curvature Drift Velocity

Substituting this centrifugal force into the general guiding center force drift formula:

$$\vec{v}_c = \frac{1}{q} \frac{\vec{F}_c \times \vec{B}}{B^2} = \frac{m v_\parallel^2}{q B^2} \frac{\vec{R}_c \times \vec{B}}{R_c^2}$$

Key physical properties of curvature drift:

• $\vec{v}_c$ depends quadratically on parallel velocity $v_\parallel^2$.
• $\vec{v}_c$ is inversely proportional to charge $q$: ions and electrons drift in opposite directions, driving a net electric current.

1. The Vacuum Field Relation

In a current-free vacuum magnetic field, Ampère's law demands $\nabla \times \vec{B} = 0$. In curvilinear coordinates with cylindrical symmetry ($\vec{B} = B_\phi(r)\hat{\phi}$):

$$(\nabla \times \vec{B})_z = \frac{1}{r}\frac{\partial(r B_\phi)}{\partial r} = 0 \implies B_\phi(r) \propto \frac{1}{r}$$

Consequently:

$$\frac{\nabla B}{B} = -\frac{\vec{R}_c}{R_c^2} \implies \frac{\vec{R}_c \times \vec{B}}{R_c^2} = \frac{\vec{B} \times \nabla B}{B}$$

2. Total Combined Guiding Center Drift

Summing the gradient-B drift and curvature drift in any vacuum magnetic field:

$$\vec{v}_D = \vec{v}_{\nabla B} + \vec{v}_c = \frac{m v_\perp^2}{2 q B^3}(\vec{B}\times\nabla B) + \frac{m v_\parallel^2}{q B^3}(\vec{B}\times\nabla B)$$ $$\vec{v}_D = \frac{m}{q B^3} \left( v_\parallel^2 + \frac{1}{2}v_\perp^2 \right) (\vec{B} \times \nabla B)$$

In a toroidal magnetic field (such as a simple torus without rotational transform), this combined drift forces ions vertically upward and electrons vertically downward. This catastrophic charge separation sets up a vertical electric field $\vec{E}$, which subsequently drives an outward $\vec{E}\times\vec{B}$ drift that dumps the entire plasma into the outer chamber wall—necessitating helical twisted magnetic fields (tokamaks/stellarators) to short-circuit the charge buildup.

1. Magnetic Convergence & Gauss's Law

Consider a cylindrically symmetric magnetic field whose strength increases along the axis of symmetry ($z$-axis): $\frac{\partial B_z}{\partial z} > 0$. Because magnetic field lines must converge as $B$ intensifies, Gauss's law for magnetism $\nabla \cdot \vec{B} = 0$ requires a non-zero radial magnetic field component $B_r$:

$$\nabla \cdot \vec{B} = \frac{1}{r}\frac{\partial(r B_r)}{\partial r} + \frac{\partial B_z}{\partial z} = 0$$

Near the axis of symmetry where $\frac{\partial B_z}{\partial z}$ is approximately independent of $r$:

$$\frac{\partial(r B_r)}{\partial r} \approx -r \frac{\partial B_z}{\partial z} \implies r B_r \approx -\frac{r^2}{2}\frac{\partial B_z}{\partial z} \implies B_r(r, z) \approx -\frac{r}{2}\frac{\partial B_z}{\partial z}$$

2. The Longitudinal Mirror Restoring Force

The particle's azimuthal cyclotron gyration velocity $v_\theta = \mp v_\perp$ couples with this radial magnetic field $B_r$ to produce a Lorentz force along the $z$-axis:

$$F_z = q(\vec{v}\times\vec{B})_z = q(v_\theta B_r - v_r B_\theta) = q v_\theta B_r$$

Substituting $B_r = -\frac{r_L}{2}\frac{\partial B_z}{\partial z}$ and $v_\theta = -\frac{q}{|q|}v_\perp$:

$$F_z = q \left( -\frac{q}{|q|}v_\perp \right) \left( -\frac{r_L}{2}\frac{\partial B_z}{\partial z} \right) = -\frac{1}{2} |q| v_\perp \left( \frac{m v_\perp}{|q| B_z} \right) \frac{\partial B_z}{\partial z} = -\frac{m v_\perp^2}{2 B_z} \frac{\partial B_z}{\partial z}$$

Recalling the magnetic moment $\mu = \frac{m v_\perp^2}{2 B}$, the longitudinal force is:

$$F_\parallel = -\mu \frac{\partial B}{\partial s} = -\mu \nabla_\parallel B$$

This is the fundamental magnetic mirror force. Because $\mu > 0$ and the negative sign is universal, the mirror force always repels charged particles away from regions of stronger magnetic field back toward regions of weaker magnetic field.

1. Invariance of the Magnetic Moment

The total kinetic energy of a charged particle in a static magnetic field is strictly conserved:

$$E = E_\parallel + E_\perp = \frac{1}{2}m v_\parallel^2 + \frac{1}{2}m v_\perp^2 = \frac{1}{2}m v_\parallel^2 + \mu B = \text{const}$$

Differentiating total energy with respect to time along the trajectory:

$$\frac{dE}{dt} = m v_\parallel \frac{dv_\parallel}{dt} + \frac{d(\mu B)}{dt} = v_\parallel F_\parallel + \mu \frac{dB}{dt} + B \frac{d\mu}{dt} = 0$$

Using $F_\parallel = -\mu \frac{\partial B}{\partial s}$ and noting that along the orbit $\frac{dB}{dt} = v_\parallel \frac{\partial B}{\partial s}$:

$$v_\parallel \left(-\mu \frac{\partial B}{\partial s}\right) + \mu \left(v_\parallel \frac{\partial B}{\partial s}\right) + B \frac{d\mu}{dt} = 0 \implies B \frac{d\mu}{dt} = 0$$

Therefore, the magnetic moment $\mu$ is an exact adiabatic invariant:

$$\mu \equiv \frac{m v_\perp^2}{2 B} = \text{const}$$

2. The Magnetic Loss Cone

As a particle travels into a converging magnetic mirror with minimum field $B_{\text{min}}$ and maximum field $B_{\text{max}}$ (mirror ratio $R_m \equiv B_{\text{max}} / B_{\text{min}}$):

$$v_\perp^2(s) = v_{\perp,0}^2 \frac{B(s)}{B_{\text{min}}}$$

By conservation of total energy $v^2 = v_\parallel^2(s) + v_\perp^2(s) = v_0^2$:

$$v_\parallel^2(s) = v_0^2 - v_{\perp,0}^2 \frac{B(s)}{B_{\text{min}}}$$

A particle will be reflected at a turning point ($v_\parallel = 0$) if and only if $B(s)$ reaches a value where $v_\perp^2 = v_0^2$ before reaching $B_{\text{max}}$. The threshold pitch angle $\theta$ at the midplane ($B = B_{\text{min}}$) defining the boundary between trapped and lost particles is the loss cone angle $\theta_m$:

$$\sin^2 \theta_m = \frac{v_{\perp,0}^2}{v_0^2} = \frac{B_{\text{min}}}{B_{\text{max}}} = \frac{1}{R_m} \implies \theta_m = \arcsin\left( \frac{1}{\sqrt{R_m}} \right)$$

Particles with initial pitch angles $\theta < \theta_m$ lie inside the loss cone and escape out the ends of the mirror.

1. The Hierarchy of Adiabatic Invariants

Hamiltonian mechanics dictates that whenever a periodic motion has action integral $J = \oint p\,dq$, the action is an adiabatic invariant under slow perturbations:

1. First Invariant $\mu$ (Magnetic Moment): Associated with fast cyclotron gyration (period $\tau_c \sim 10^{-6}\text{ s}$): $$\mu = \frac{m v_\perp^2}{2 B} = \text{const}$$
2. Second Invariant $J$ (Longitudinal Action): Associated with periodic bounce motion between mirror points (period $\tau_b \sim 10^{-2}\text{ s}$): $$J = \oint v_\parallel ds = \text{const}$$
3. Third Invariant $\Phi$ (Magnetic Flux Drift Invariant): Associated with the slow azimuthal drift of guiding centers around a closed magnetic surface (period $\tau_d \sim 10^2\text{ s}$): $$\Phi = \oint \vec{A} \cdot d\vec{l} = \int \vec{B} \cdot d\vec{S} = \text{const}$$

2. Planetary Magnetospheres & The Van Allen Belts

The Earth's dipolar magnetic field naturally traps energetic protons and electrons in the Van Allen radiation belts. Trapped particles simultaneously execute:

• Fast cyclotron gyration around dipole field lines ($\tau_c \sim 1\;\mu\text{s}$).
• North-South bouncing between northern and southern auroral mirror points ($\tau_b \sim 0.1\text{ s}$).
• Slow longitudinal drift around the Earth (electrons east, ions west) creating the geomagnetic ring current ($\tau_d \sim 10\text{ minutes}$).