Fundamentals of Plasma Physics

88 Chapter 3. Motion of a single plasma particle

of adiabatic invariance presented in Sec.3.4.1, the quantity

J=

∮

P‖ds=

∮

(mv‖+qA‖)ds (3.117)

will be an invariant if

1. any time dependence of the well shape is slow compared to the bounce frequency of
   the trapped particle,


2. any spatial inhomogeneities of the well magnetic field are so gradual that the particle’s
   bounce trajectory changes by only a small amount from one bounce to the next.
   To determine the circumstances whereA‖= 0,we use Coulomb gauge (i.e., assume
   ∇·A=0)and at any given location define a local Cartesian coordinate system withzaxis
   parallel to the local field. From Ampere’s law it is seen that

[∇×(∇×A)]z=−∇^2 Az=μ 0 Jz (3.118)

soAzis finite only if there is acurrent parallel to the magnetic field. BecauseJzacts as
the source term in a Poisson-like partial differential equation forAz,the parallel current
need not be at the same location asAz. If there are no currents parallel to the magnetic
field anywhere thenA‖= 0,and in this case the second adiabatic invariant reduces to

J=m

∮

v‖ds. (3.119)

Having a currentflow along the magnetic field corresponds to a more complicated magnetic
topology. The axial current produces an associated azimuthal magnetic fieldwhich links
the axial magnetic field resulting in a helical twist. This more complicated situation of
finite magnetic helicity will be discussed in a later chapter.

3.5.7 Consequences ofJ-invariance

Just asμinvariance was related to the perpendicular CGL adiabatic invariant discussed
in Sec.(2.101),J-invariance is closely related to the parallel CGL adiabatic invariant also
discussed in Sec.(2.101). To see this relation, recall that density in a one dimensional
system has dimensions of particles per unit length, i.e.,n 1 D∼ 1 /L, and pressure in a one
dimensional system has dimensions of kinetic energy per unit length, i.e.,P 1 D∼v^2 ‖/L.

For parallel motion the number of dimensions isN= 1so thatγ= (N+ 2)/N= 3and
thefluid adiabatic relation is

const.∼

P 1 D

n^31 D

∼

v^2 ‖/L

L−^3

∼

(

v‖L

) 2

(3.120)

which is a simplified form of Eq.(3.119) since Eq.(3.119) has the scalingJ∼v‖L=const.
J-invariance combined with mirror trapping/detrapping is the basis of an acceleration
mechanism proposed by Fermi (1954) as a means for accelerating cosmic ray particles to
ultra-relativistic velocities. The Fermi mechanism works as follows: Consider a particle
initially trapped in a magnetic mirror. This particle has an initial angle in velocity space
θ >θtrap;bothθandθtrapare measured when the particle is at the mirror minimum. Now
suppose the distance between the magnetic mirrors is slowly reduced so thatthe bounce