Fundamentals of Plasma Physics

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

Normalize the collision frequency, thermal velocity, and mean free paths to their

values atx =x 0 whereT = T 0 ;e.g. vTe(T) = vTe 0 (T/T 0 )^1 /^2 .By writing

∂/∂x= (∂T/∂x)∂/∂Tand using these normalized values show that

Rthermal=− 2 neκ∇Te.

A more accurate treatment which does a proper averaging over velocities gives

Rthermal=− 0. 71 neκ∇Te.

8. MHD with neutrals- Suppose a plasma is partially (perhaps weakly) ionized so that
   besides moment equations for ions and electrons there will also be moment equations
   for neutrals. Now the constraints will be different since ionization andrecombination
   will genuinely produce creation of plasma particles and also of neutrals.Construct a
   set of constraint equations on the collision operators which will now include ionization
   and recombination as well as scattering. Take the zeroth and first moment of the
   three Vlasov equations for ions, electrons, and neutrals and show that the continuity
   equation is formally the same as before, i.e.,

∂ρ

∂t

+∇·(ρU) = 0

providingρrefers to the total mass density of the entirefluid (electrons, ions and

neutrals) andUrefers to the center of mass velocity of the entirefluid. Show also that

the equation of motion is formally the same as before, provided the pressurerefers to

the pressure of the entire configuration:

ρ

DU

Dt

=−∇P+J×B.

Show that Ohm’s law will be the same as before, providing the plasma is sufficiently

collisional so that the Hall term can be dropped, and so Ohm’s law is

E+U×B=ηJ

Explain how the neutral component of the plasma gets accelerated by theJ×Bforce

— this must happen since the inertial part of the equation of motion (i.e.,ρDU/Dt)

includes the acceleration on neutrals. Assume that electron temperature gradients are

parallel to electron density gradients so that the electro-thermalforce can be ignored.

9. MHD Heat Transport Equation- Define the MHDN−dimensional pressure

P=

1

N

∑

σ

mσ

∫

v′·v′fσdNv

wherev′=v−UandNis the number of dimensions of motion (e.g., if only motion

in one dimensions is considered, thenN= 1,andv,Uare one dimensional, etc.).

Also define the isotropic MHD heatflux

q=

∑

σ

mσ

∫

v′

v′^2

2

fdNv