Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part I, Chapter 6

Adiabatic Invariants

Three fundamental invariants of particle motion in slowly varying fields

6.1 Adiabatic Invariants in Physics

An adiabatic invariant is a quantity that remains approximately constant when a system's parameters change slowly compared to the system's natural period. In plasma physics, three adiabatic invariants govern particle motion across different timescales.

The Three Adiabatic Invariants

  • • First invariant (μ): Magnetic moment (gyration timescale ∼ ωc−1)
  • • Second invariant (J): Longitudinal invariant (bounce timescale ∼ τb)
  • • Third invariant (Φ): Flux invariant (drift timescale ∼ τd)

These invariants are ordered by timescale: ωc−1 << τb << τd. Each is conserved when fields vary slowly compared to its associated timescale.

6.2 First Adiabatic Invariant: Magnetic Moment

The magnetic moment is conserved when B varies slowly on the gyration timescale:

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

Proof of Conservation

Using the equation of motion in the guiding center approximation and averaging over a gyroperiod, one finds:

$$\frac{d\mu}{dt} = \frac{d}{dt}\left(\frac{mv_\perp^2}{2B}\right) = 0 \quad \text{(to lowest order)}$$

This requires: δB/δt << ωcB and ∇B · rL << B.

Applications

  • • Magnetic mirrors: As particle enters stronger B, v⊥ increases, v∥ decreases → reflection
  • • Adiabatic heating: Slowly increasing B heats perpendicular motion
  • • Magnetic pumping: Oscillating B field transfers energy
  • • Radiation belts: Particles conserve μ while bouncing

6.3 Second Adiabatic Invariant: Longitudinal Invariant

For particles bouncing between mirror points (e.g., in Earth's radiation belts or tokamaks), the action integral along the field line is conserved:

$$J = \oint p_\parallel ds = \oint m v_\parallel ds = \text{const}$$

where the integral is taken over one complete bounce motion between mirror points.

Derivation

From Hamilton-Jacobi theory, J is an action variable. For the parallel motion:

$$m\frac{dv_\parallel}{dt} = -\mu \frac{\partial B}{\partial s}$$

Using conservation of energy E = ½m(v∥² + v⊥²) and μ = mv⊥²/2B:

$$v_\parallel = \sqrt{\frac{2}{m}(E - \mu B)}$$

Substituting into J and showing dJ/dt = 0 requires δB/δt << ωbB, where ωb = 2π/τb is the bounce frequency.

Physical Consequences

  • • Particles maintain constant bounce motion as fields slowly evolve
  • • In tokamak sawteeth, J is conserved during slow field changes
  • • Radiation belt particles conserve J during magnetic storms

6.4 Third Adiabatic Invariant: Flux Invariant

For toroidal confinement (tokamaks, planetary magnetospheres), particles drift around the torus. The enclosed magnetic flux is conserved:

$$\Phi = \int \vec{B} \cdot d\vec{A} = \text{const}$$

where the integral is over any surface bounded by the particle's drift orbit.

Consequences for Confinement

Conservation of Φ means particles remain on nested flux surfaces. If a perturbation breaks flux surfaces slowly (τ >> τdrift), particles readjust adiabatically.

However, Φ is the most easily broken invariant:

  • • Magnetic reconnection violates Φ on Alfvén timescale
  • • Turbulent fluctuations with ω ∼ ωdrift break Φ
  • • Large perturbations (ELMs, sawteeth) destroy Φ

Example: Earth's Radiation Belts

Van Allen belt particles drift azimuthally around Earth, conserving all three invariants during quiet times. Magnetic storms can violate the third invariant, allowing particles to access new drift shells and potentially be lost.

6.5 Violations of Adiabatic Invariants

Adiabatic invariants are conserved only when fields vary slowly. When this condition is violated:

Breaking μ

  • • Rapid field variations: ωB ∼ ωc
  • • Sharp field gradients: ∇B · rL ∼ B
  • • Example: Magnetic pumping with ω ∼ ωc (stochastic heating)

Breaking J

  • • Field changes on bounce timescale: τ ∼ τbounce
  • • Example: Fast sawtooth crashes in tokamaks

Breaking Φ

  • • Magnetic reconnection events
  • • Edge localized modes (ELMs)
  • • Example: Solar flares, magnetospheric substorms

6.6 Applications to Particle Confinement

Adiabatic invariants provide the foundation for understanding particle confinement:

Tokamak Confinement

  • • Particles conserve μ → remain on flux surfaces radially
  • • Trapped particles conserve J → bounce poloidally
  • • All particles conserve Φ → drift toroidally on flux surfaces
  • • Result: Excellent confinement when turbulence is weak

Magnetic Bottles

  • • μ conservation → mirror confinement
  • • Loss cone → particles with low μ escape
  • • Applications: Fusion mirrors, plasma thrusters

Radiation Belts

  • • All three invariants conserved during quiet periods
  • • Particles trapped for years
  • • Storms violate invariants → particle acceleration/loss

Summary: Part I Complete

We've covered the fundamentals of plasma physics:

  • ✓ Plasma definition and Debye shielding
  • ✓ Single particle motion and drift velocities
  • ✓ Coulomb collisions and transport
  • ✓ Plasma parameters and classification
  • ✓ Magnetized plasma dynamics
  • ✓ Three adiabatic invariants

These concepts form the foundation for kinetic theory (Part II), fluid theory (Part III), and all subsequent advanced topics.

← Magnetized PlasmasContinue to Part II →