Adiabatic invariant

An adiabatic invariant in general is a property of motion which is conserved to exponential accuracy in the small parameter representing the typical rate of change of the gross properties of the body. For periodic motion, the adiabatic invariants are the action integrals taken over a period of the motion. These are constants of the motion and remain so even when changes are made in the system, as long as the changes are slow compared to the period of motion.

In plasma physics there are three adiabatic invariants of charged particle motion.

The first adiabatic invariant, μ

The magnetic moment of a gyrating particle,

,

is a constant of the motion (as long as q/m does not change). In fact, it is invariant to all orders in an expansion in ω / ω_c, so the magnetic moment remains nearly constant even for changes at rates approaching the gyrofrequency.

There are some important situations in which the magnetic moment is not invariant:

• Magnetic pumping: When μ is constant, the perpendicular particle energy is proportional to B, so the particles can be heated by increasing B, but this is a 'one shot' deal because the field cannot be increased indefinitely. On the other hand, if the collision frequency is larger than the pump frequency, μ is no longer conserved. In particular, collisions allow net heating by transfering some of the perpendicular energy to parallel energy.
• Cyclotron heating: If B is oscillated at the cyclotron frequency, the condition for adiabatic invariance is violated and heating is poosible. In particular, the induced electric field rotates in phase with some of the particles and continuously accelerates them.
• Magnetic cusps: The magnetic field at the center of a cusp vanishes, so the cyclotron frequency is automatically smaller than the rate of any changes. Thus the magnetic moment is not conserved and particles are scattered relatively easily into the loss cone.

The second adiabatic invariant, J

The longitudinal invariant of a particle trapped in a magnetic mirror,

,

where the integral is between the two turning points, is also an adiabatic invariant. This guarantees, for example, that a particle in the ionosphere moving around the Earth will always return to the same line of force. The adiabatic condition is violated in transit-time magnetic pumping, where the length of a magnetic mirror is oscillated at the bounce frequency, resulting in net heating.

The third adiabatic invariant, Φ

The total magnetic flux Φ enclosed by a drift surface is the third adiabatic invariant, associated with the periodic motion of mirror-trapped particles drifting around the axis of the system. Because this drift motion is relatively slow, Φ is often not conserved in practical applications.

External links

• lecture notes on the second adiabatic invariant
• lecture notes on the third adiabatic invariant