Fundamentals of Plasma Physics

50 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD

This means that, as shown in Fig.2.5, a tube of plasma initially occupying the same vol-
ume as a magneticflux tube is constrained to evolve in such a way that

∫

B·dsstays con-
stant over the plasma tube cross-section. For aflux tube of infinitesimal cross-section, the
magnetic field is approximately uniform over the cross-section and we may write this as
BA=const.whereAis the cross-sectional area. Let us define two temperatures for this
magnetized plasma, namelyT⊥the temperature corresponding to motions perpendicular to
the magnetic field, andT‖the temperature corresponding to motions parallel to the mag-
netic field. If for some reason (e.g., anisotropic heating or compression) the temperature
develops an anisotropy such thatT⊥=T‖and if collisions are infrequent, this anisotropy
will persist for a long time, since collisions are the means by which thetwo temperatures
equilibrate. Thus, rather than assuming that the MHD pressure is fullyisotropic, we con-
sider the less restrictive situation where the MHD pressure tensoris given by

←→

PMHD=





P⊥ 0 0

0 P⊥ 0

0 0 P‖



=P⊥←→I + (P‖−P⊥)B̂B̂. (2.84)

The first two coordinates(x,y-like) in the above matrix refer to the directions perpendic-
ular to the local magnetic fieldBand the third coordinate(z-like) refers to the direction

parallel toB. The tensor expression on the right hand side is equivalent (here

←→

I is the unit
tensor) but allows for arbitrary, curvilinear geometry. We now developseparate adiabatic
relations for the perpendicular and parallel directions:

• Parallel direction- Here the number of dimensions isN= 1so thatγ= 3and so
  the adiabatic law gives
  P‖^1 D
  ρ^31 D

=const. (2.85)

whereρ 1 D is theone-dimensionalmass density;i.e.,ρ 1 D ∼ 1 /LwhereLis the

length along the one-dimension, e.g. along the length of theflux tube in Fig.2.5. The

three-dimensional mass densityρ,which has been used implicitly until now has the

proportionalityρ∼ 1 /LAwhereAis the cross-section of theflux tube;similarly

the three dimensional pressure has the proportionalityP‖∼ρT‖.However, we must

be careful to realize thatP‖^1 D∼ρ 1 DT‖so, usingBA=const.,Eq. (2.85) can be

recast as

const.=

P‖^1 D

ρ^31 D

∼

ρ 1 DT‖

ρ^31 D

∼T‖L^2 ∼

(

1

LA

)

T‖

︸ ︷︷ ︸

P‖

(LA)^3

︸︷︷︸

ρ−^3

B^2. (2.86)

or

P‖B^2

ρ^3

=const. (2.87)

• Perpendicular direction- Here the number of dimensions isN= 2so thatγ= 2
  and the adiabatic law gives

P⊥^2 D

ρ^22 D

=const. (2.88)