102 Chapter 3. Motion of a single plasma particle
Eq.(3.165) can only be satisfied if|Pθ|is not too large, because the right hand side of
Eq.(3.165) has a maximum value. If all these conditions are satisfied, thenχwill have
a non-zero minimum as shown in the bottom plot of Fig.3.14.
A particularly simple example of this behavior occurs if Eq.(3.165) is satisfied near the
r= 0axis (i.e., whereψ∼r^2 ) so that this equation becomes simply
Pθ=−q
2 πψ (3.166)which is just the opposite of Eq.(3.161). Substituting in Eq.(3.162) we see thatθ ̇now
neverchanges sign;i.e., the particle isaxis-encircling. The Larmor radius of this axis-
encircling particle is just the radius of the minimum of the potential well, the radius where
Eq.(3.165) holds. The azimuthal kinetic energy of the particle corresponds to the height of
the minimum ofχin the bottom plot of Fig.3.14.
3.7.1 Temporal Reversal of Magnetic Field - Energy Gain
Armed with this information about axis-encircling and non-axis encircling particles, we
now examine the strongly non-adiabatic situation where a coil current startsatI =I 0 ,
is reduced to zero, and then becomesI=−I 0 ,so that all fields andfluxes reverse sign.
The particle energy will not stay constant for this situation because the Lagrangian depends
explicitly on time. However, since symmetry is maintained,Pθmustremain constant. Thus,
a non-axis encircling particle (with radial location determined by Eq. (3.161)) will change
to an axis-encircling particle if a minimum exists forχwhen the sign ofψis reversed. If
such a minimum does exist and if the initial radius was near the axis whereψ∼r^2 ,then
comparison of Eqs.(3.161) and (3.166) shows that the particle will have the sameradius
after the change of sign as before. The particle will gain energy during thefield reversal by
an amount corresponding to the finite value of the minimum ofχfor the axis-encircling
case.
This process can also be considered from the point of view of particle drifts: Initially,
the non-axis-encircling particle is frozen to a constantψsurface (flux surface). When the
coil current starts to decrease, the maximum value of theflux correspondingly decreases.
The constantψcontours on theinsideofψmaxmove outwards towards the location of
ψmax where they are annihilated. Likewise, the contours outside ofψmaxmove inwards
toψmaxwhere they are also annihilated.
To the extent that theE×Bdrift is a valid approximation, its effect is to keep the
particle attached to a surface of constantflux. This can be seen by integrating Faraday’s
law over the area of a circle of radiusrto obtain
∫
ds·∇×E=−∫
ds·∂B/∂tand then
invoking Stoke’s theorem to give
Eθ 2 πr=−∂ψ
∂t. (3.167)
The theta component ofE+v×B= 0is
Eθ+vzBr−vrBz= 0 (3.168)and from (3.146),Br=−(2πr)−^1 ∂ψ/∂zandBz=−(2πr)−^1 ∂ψ/∂r.Combination of
Eqs.(3.167) and (3.168) thus gives
∂ψ
∂t
+vr∂ψ
∂r
+vz∂ψ
∂z