3.7 Non-adiabatic motion in symmetric geometry 103Becauseψ(r(t),z(t),t)is theflux measured in the frame of a particle moving with trajec-
toryr(t)andz(t),Eq.(3.169) shows that theE×Bdrift maintains the particle on a surface
of constantψ,i.e., theE×Bdrift is such as to maintaindψ/dt= 0whered/dtmeans
time derivative as measured in the particle frame.
The implication of this attachment of the particle to a surface of constantψcan be
appreciated by making an analogy to the motion of people initially located on the beach
of a volcanic island which is slowly sinking into the sea. In order toavoid being drowned
as the island sinks, the people will move towards the mountain top to stay at aconstant
height above the sea. The location ofψmaxhere corresponds to the mountain top and the
particles trying to stay on surfaces of constantψcorrespond to people trying to stay at
constant altitude. A particle initially located at some location away from the “mountain
top”ψmaxmoves towardsψmaxif the overall level of all theψsurfaces is sinking. The
reduction ofψas measured at a fixed position will create the azimuthal electric field given
by Eq.(3.167) and this electric field will, as shown by Eqs.(3.168) and (3.169), causean
E×Bdrift which convects each particle in just such a way as to stay on a constantψ
contour.
TheE×Bdrift approximation breaks down whenBbecomes zero, i.e., whenψchanges
polarity. This breakdown corresponds to a breakdown of the adiabatic approximation. If
ψchanges polarity before a particle reachesψmax, the particle becomes axis-encircling.
The extra energy associated with being axis-encircling is obtained whenψ≃ 0 but∂ψ/∂t=
0 so that there is an electric fieldEθ, but no magnetic field. FiniteEθand no magnetic
field results in a simple theta acceleration of the particle. Thus, whenψreverses polarity the
particle is accelerated azimuthally and develops finite kinetic energy. Afterψhas changed
polarity the magnitude ofψincreases and the adiabatic approximation again becomes valid.
Because the polarity is reversed, increase of the magnitude ofψis now analogous to creat-
ing an ever deepening crater. Particles again try to stay on constantflux surfaces as dictated
by Eq.(3.169) and as the crater deepens, the particles have to move awayfromψminto
stay at the same altitude. When the reversedflux attains the same magnitude as the origi-
nalflux, theflux surfaces have the same shape as before. However, the particles arenow
axis-encircling and have the extra kinetic energy obtained at field reversal.
3.7.2 Spatial reversal of field - cusps
Suppose two solenoids with constant currents are arranged coaxially withtheir magnetic
fields opposing each other as shown in Fig.3.16(a). Since the solenoid currents are constant,
the Lagrangian does not depend explicitly on time in which case energy is a constant of the
motion. Because of the geometrical arrangement, theflux function is anti-symmetric inz
wherez= 0defines the midplane between the two solenoids.
Consider a particle injected with initial velocityv=vz 0 zˆatz=−L,r=a. Since
this particle has no initialv⊥, it simply streams along a magnetic field line. However,
when the particle approaches the cusp region, the magnetic field lines startto curve causing
the particle to develop both curvature and gradBdrifts perpendicular to the magnetic
field. When the particle approaches thez= 0plane, the drift approximation breaks down
becauseB→ 0 and so the particle’s motion becomes non-adiabatic [cf. Fig.3.16(a)].
Although the particle trajectory is very complex in the vicinity of the cusp, it is still
possible to determine whether the particle will cross into the positivezhalf-plane, i.e.,