Fundamentals of Plasma Physics

90 Chapter 3. Motion of a single plasma particle

coordinate system(r,θ,z)withzaxis coaxial with the coils, it is seen that in the region
between the two coils where the field bulges out, the field strength is a decreasing function
ofr,i.e.∂B/∂r < 0 , whereas in the plane of each coil the opposite is true. Thus, in the
mirror minimum, both the centrifugal and gradBforces are radially outward, whereas the
opposite is true near the coils.
In both Figs. 3.10(a) and (b) a particle moving along the field line can be mirror-
trapped because in both cases the field has a minimumflanked by two maxima. However,
for Fig.3.10(a), the particle will have gradBand curvature drifts always in the same az-
imuthal sense, whereas for Fig.3.10(b) the azimuthal direction of these drifts will depend
on whether the particle is in a region of concave or convex curvature. Thus, in addition to
the mirror bouncing motion, much slower curvature and gradBdrifts also exist, directed
along the field binormal (i.e. the direction orthogonal to both the field and its radius of
curvature). These higher-order drifts may alternate sign during the mirror bouncing. The
binormally directed displacement made by a particle during itsithcomplete periodτof
mirror bouncing is

δrj=

∫τ

0

vdt (3.121)

whereτis the mirror bounce period andvis the sum of the curvature and gradBdrifts
experienced in the course of a mirror bounce. This displacement is due to thecumulative
effect of the curvature and gradBdrifts experienced during one complete period of bounc-
ing between the magnetic mirrors. The average velocity associated with this slow drifting
may be defined as

〈v〉=

1

τ

∫τ

0

vdt. (3.122)

Let us calculate the action associated with a sequence ofδrj.This action is

S=

∑

j

[m〈v〉+qA]j·δrj (3.123)

where the quantity in square brackets is evaluated on the line segmentδrj.If theδrjare
small then this can be converted into an action ‘integral’ for the pathtraced out by the
δrj. If theδrjare sufficiently small to behave as differentials, then we may writethem as
drbounceand express the summation as an action integral

S=

∫

[m〈v〉+qA]·drbounce (3.124)

where it must be remembered that〈v〉is thebounce-averagedvelocity. The quantity
m〈v〉+qAis just the canonical momentum associated with the effective motion along
the sequence of line segmentsδrj.The vectorrbounceis a vector pointing from the origin
to the particle’s location at successive bounces and so is the generalized coordinate asso-
ciated with the bounce averaged velocity. If the motion resulting from〈v〉is periodic, we
expectS to be an adiabatic invariant. The first term in Eq.(3.124) will be of the order of
mvdrift 2 πrwhereris the radius of the trajectory described by theδrj.The second term is
justqΦwhereΦis the magneticflux enclosed by the trajectory. Let us compare the ratio
of these two terms
∫
m〈v〉·dr
∫
qA·dr

∼

mvdrift 2 πr

qBπr^2

∼

vdrift

ωcr

∼

r^2 L

r^2

(3.125)