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. This concept, fundamental across all of physics, provides the foundation for understanding particle confinement in magnetic fields.

6.1.1 Definition and General Theory

Consider a periodic system with Hamiltonian H(q, p, λ), where λ(t) is a slowly varying external parameter. The action variable:

$$I = \oint p \, dq$$

is an adiabatic invariant when:

$$\frac{1}{\omega}\frac{d\lambda}{dt} \ll 1$$

where ω is the natural frequency of oscillation. The invariance is not exact but holds to exponential accuracy:

$$\frac{\Delta I}{I} \sim \exp\left(-\frac{c\omega}{\dot{\lambda}/\lambda}\right)$$

where c is a constant of order unity. This exponential accuracy is remarkable — adiabatic invariants are conserved to parts in 10¹⁰ or better in typical plasma situations!

6.1.2 Three Plasma Adiabatic Invariants

In magnetized plasmas, three adiabatic invariants emerge from three distinct periodic motions, each with its own timescale:

Invariant	Definition	Motion	Timescale
μ (First)	mv_⊥²/(2B)	Gyration	τ_c = 2π/ω_c
J (Second)	∮ mv_∥ ds	Bounce	τ_b ∼ L/v_∥
Φ (Third)	∫ B · dA	Drift	τ_d ∼ R/v_d

These timescales form a hierarchy:

$$\tau_c \ll \tau_b \ll \tau_d$$

For typical tokamak parameters:

τ_c ∼ 10⁻¹⁰ s (electrons) to 10⁻⁸ s (ions)
τ_b ∼ 10⁻⁶ s (trapped particles bouncing poloidally)
τ_d ∼ 10⁻³ s (toroidal drift period)

6.1.3 Nested Hierarchy

The invariants form a nested hierarchy:

Averaging Procedure

• μ conserved: when fields vary slowly over τ_c
• J conserved: when fields vary slowly over τ_b (and μ conserved)
• Φ conserved: when fields vary slowly over τ_d (and μ, J conserved)

Each invariant is progressively easier to violate: rapid perturbations first break Φ, then J, then μ. Conversely, μ is the most robust — even violent processes like sawteeth typically conserve μ.

Historical Note

The three adiabatic invariants were discovered progressively: μ by Alfvén (1940s), J by Northrop & Teller (1960), and Φ formalized by Rosenbluth (1960s). Their conservation explains particle trapping in radiation belts, tokamak confinement, and magnetic mirror behavior.

6.2 First Adiabatic Invariant: Magnetic Moment

The magnetic moment is the most fundamental and robust adiabatic invariant:

$$\boxed{\mu = \frac{m v_\perp^2}{2B} = \frac{E_\perp}{B} = \frac{p_\perp^2}{2mB}}$$

It represents the magnetic dipole moment of the current loop formed by the gyrating particle:

$$\mu = IA = \frac{q\omega_c}{2\pi} \pi r_L^2 = \frac{q}{2\pi}\frac{qB}{m}\frac{m^2 v_\perp^2}{q^2 B^2} = \frac{mv_\perp^2}{2B}$$

6.2.1 Rigorous Derivation

Start with the perpendicular energy equation. From the guiding center equations:

$$\frac{dE_\perp}{dt} = \frac{d}{dt}\left(\frac{1}{2}m v_\perp^2\right) = m v_\perp \frac{dv_\perp}{dt}$$

The magnetic field at the particle position changes due to motion through space and time variation:

$$\frac{dB}{dt} = \frac{\partial B}{\partial t} + \mathbf{v} \cdot \nabla B$$

For slowly varying fields, the parallel equation of motion gives:

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

Energy conservation E = ½m(v_∥² + v_⊥²) + qφ implies:

$$\frac{dE_\perp}{dt} + \frac{dE_\parallel}{dt} = q E_\parallel v_\parallel$$

Computing dE_∥/dt from the parallel equation and using v ≈ v_∥b̂ + v_d:

$$\frac{dE_\perp}{dt} = \mu v_\parallel \frac{\partial B}{\partial s} + \mu \mathbf{v}_d \cdot \nabla B + \mu \frac{\partial B}{\partial t}$$

But this equals E_⊥ (dB/dt)/B when μ = E_⊥/B. Thus:

$$\frac{d\mu}{dt} = \frac{d}{dt}\left(\frac{E_\perp}{B}\right) = \frac{1}{B}\frac{dE_\perp}{dt} - \frac{E_\perp}{B^2}\frac{dB}{dt} = 0$$

6.2.2 Validity Conditions

The magnetic moment is conserved when fields vary slowly compared to the gyroperiod:

1. Temporal Condition

$$\frac{1}{\omega_c}\left|\frac{1}{B}\frac{\partial B}{\partial t}\right| \ll 1$$

Field changes slowly compared to cyclotron period.

2. Spatial Condition

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

Larmor radius much smaller than field gradient scale length L_B = B/|∇B|.

3. Curvature Condition

$$\frac{r_L}{R_c} \ll 1$$

Larmor radius much smaller than field line radius of curvature.

6.2.3 Accuracy of Conservation

The fractional change in μ per gyroperiod is exponentially small:

$$\frac{\Delta \mu}{\mu} \sim \epsilon^2 \exp(-c/\epsilon)$$

where ε = r_L/L_B ≪ 1 and c ∼ 1. For typical fusion plasmas with ε ∼ 10⁻³:

$$\frac{\Delta \mu}{\mu} \sim 10^{-6} \exp(-1000) \approx 10^{-440}$$

This incredible accuracy explains why μ is conserved even during violent events like sawteeth crashes!

6.2.4 Mirror Force and Parallel Dynamics

Conservation of μ leads to the mirror force. Using E = E_∥ + μB = const:

$$\frac{1}{2}m v_\parallel^2 = E - \mu B \quad \Rightarrow \quad v_\parallel = \pm\sqrt{\frac{2}{m}(E - \mu B)}$$

As B increases along the field line, v_∥ decreases. At the mirror point where B = B_m = E/μ, the parallel velocity vanishes and the particle reflects:

$$v_\parallel(B_m) = 0 \quad \Rightarrow \quad B_m = \frac{E}{\mu} = \frac{E}{E_\perp/B_0} = B_0\frac{E}{E_\perp}$$

The parallel equation of motion is:

$$m\frac{dv_\parallel}{dt} = -\mu \nabla_\parallel B = -\frac{mv_\perp^2}{2B}\frac{\partial B}{\partial s}$$

This force points opposite to the gradient of B, pushing particles away from regions of strong field.

6.2.5 Applications and Examples

1. Magnetic Mirrors

A particle with pitch angle α₀ at B₀ reflects at B_m given by:

$$\sin^2\alpha_0 = \frac{v_{\perp 0}^2}{v^2} = \frac{\mu B_0}{E} = \frac{B_0}{B_m} \quad \Rightarrow \quad B_m = \frac{B_0}{\sin^2\alpha_0}$$

Particles with α₀ < α_lc = arcsin(√(B_min/B_max)) escape through theloss cone. For mirror ratio R_m = B_max/B_min = 10, α_lc ≈ 18°.

2. Adiabatic Compression

If B increases from B₀ to B₁ slowly (τ ≫ τ_c), energy increases:

$$\frac{E_{\perp 1}}{E_{\perp 0}} = \frac{\mu B_1}{\mu B_0} = \frac{B_1}{B_0}$$

This adiabatic heating mechanism is used in magnetic compression experiments. For fast compression (τ ∼ τ_c), μ is violated and stochastic heating occurs.

3. Magnetic Pumping

Oscillating B(t) = B₀(1 + ε sin ωt) with ω ≪ ω_c conserves μ, producing oscillating E_⊥. With collisions at frequency ν, energy is irreversibly transferred, giving heating rate:

$$\frac{dE_\perp}{dt} \approx \nu \epsilon^2 E_\perp \omega^2/\omega_c^2$$

This weak heating mechanism is inefficient compared to RF heating but is important for understanding instability saturation.

4. Van Allen Radiation Belts

Energetic particles (E ∼ MeV) trapped in Earth's dipole field conserve μ as they spiral along field lines. For a particle at equatorial pitch angle α_eq, the mirror latitude λ_m satisfies:

$$\cos^6\lambda_m = \sin^2\alpha_{eq}$$

For α_eq = 45°, λ_m ≈ 55°. Particles with α_eq < 5-10° are lost to the atmosphere via pitch-angle scattering.

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 << ω_bB, where ω_b = 2π/τ_b 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 >> τ_drift), particles readjust adiabatically.

However, Φ is the most easily broken invariant:

• Magnetic reconnection violates Φ on Alfvén timescale
• Turbulent fluctuations with ω ∼ ω_drift 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 ∼ ω_c
• Sharp field gradients: ∇B · r_L ∼ B
• Example: Magnetic pumping with ω ∼ ω_c (stochastic heating)

Breaking J

• Field changes on bounce timescale: τ ∼ τ_bounce
• 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 Plasmas Continue to Part II →

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