3.5 Drift equations 83
- Conservation of magneticflux enclosed by gyro-orbit- Because the magneticfluxΦ
enclosed by the gyro-orbit is
Φ =BπrL^2 =
(
2 mπ
q
)
μ, (3.103)
μconservation further implies conservation of the magneticflux enclosed by a gyro-
orbit. This is consistent with the concept that the magneticflux is frozen into the
plasma, since if the field is made stronger, the field lines squeeze together such that
the density of field lines per area increases proportional to the field strength.As shown
in Fig.3.6, the particle orbit area contracts in inverse proportion to the field strength so
that after a compression of field, the particle orbit links the same numberof field lines
as before the compression.
- Hamiltonian point of view (cylindrical geometry with azimuthal symmetry)- Define
a cylindrical coordinate system(r,θ,z)withzaxis along the axis of rotation of the
gyrating particle. SinceBz=r−^1 ∂(rAθ)/∂rthe vector potential isAθ=rBz/ 2.
The velocity vector isv= ̇rrˆ+rθ ̇ˆθ+ ̇zzˆand the Lagrangian is
L=
m
2
(
r ̇^2 +r^2 θ ̇
2
+ ̇z^2
)
+qrθA ̇ θ−qφ (3.104)
so that the canonical angular momentum is
Pθ=mr^2 θ ̇+qrAθ=mr^2 θ ̇+qr^2 Bz/ 2. (3.105)
Since particles are diamagnetic,θ ̇=−ωc.Because of the azimuthal symmetry,Pθ
will be a constant of the motion and so
const.=Pθ=−mr^2 ωc+qr^2 B/2 =−
mvθ^2
2 ωc
=−
m
q
μ. (3.106)
This shows that constancy of canonical angular momentum is equivalent toμconser-
vation. It is important to realize that constancy of angular momentum due to perfect
axisymmetry is a much more restrictive assumption than the slowness assumption
used for adiabatic invariance.
- Adiabatic gas law- The pressure associated with gyrating particles has dimensionality
N= 2,i.e.,P= (m/2)
∫
v′·v′fd^2 vwherev′=vxxˆ+vyˆyand thex−yplane is
the plane of the gyration. Also the density for a two dimensional system has units of
particles/area, i.e.n∼ 1 /A. Hence, the pressure will scale asP∼v^2 T⊥/A. Since
γ= (N+ 2)/N= 2,the adiabatic law, Eq.(2.37), gives
const.∼
P
n^2
∼
v^2 T⊥
A
A^2 ; (3.107)
but from theflux conservation property of orbitsA∼ 1 /Bso Eq.(3.107) becomes
P
n^2
∼
vT^2 ⊥
B
(3.108)
which is again proportional toμsincev^2 T⊥is proportional to the mean perpendicular
thermal energy, i.e., the average of the gyrational energies of the individualparticles
making up thefluid.
