Fundamentals of Plasma Physics

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

2.5 Magnetohydrodynamic equations

Particle motion in the two-fluid system was described by the individual species mean veloc-

itiesue,uiand by the pressures

←→

Pe,

←→

Piwhich gave information on the random deviation
of the velocity from its mean value. Magnetohydrodynamics is an alternate description of
the plasma where instead of usingue,uito describe mean motion, two new velocity vari-
ables that are a linear combination ofue,uiare used. As will be seen below, this means a
slightly different definition for pressure must also be used.
The new velocity-like variables are (i) the current density

J=

∑

σ

nσqσuσ (2.62)

which is essentially the relative velocity between ions and electrons, and (ii) the center of
mass velocity

U=

1

ρ

∑

σ

mσnσuσ. (2.63)

where

ρ=

∑

σ

mσnσ (2.64)

is the total mass density. Magnetohydrodynamics is primarily concerned with low fre-
quency, long wavelength,magneticbehavior of the plasma.

2.5.1 MHD continuity equation

Multiplying Eq.(2.19) bymσand summing over species gives the MHD continuity equa-
tion
∂ρ
∂t

+∇·(ρU) = 0. (2.65)

2.5.2 MHD equation of motion

To obtain an equation of motion, we take the first moment of the Vlasov equation, then
multiply bymσand sum over species to obtain

∂

∂t

∑

σ

mσ

∫

vfσdv+

∂

∂x

·

∑

σ

∫

mσvvfσdv+

∑

σ

qσ

∫

v

∂

∂v

·[(E+v×B)fσ] = 0;

(2.66)
the right hand side is zero sinceRei+Rie= 0,i.e., the total plasma cannot exert drag on
itself. We now define random velocities relative toU(rather than touσas was the case for
the two-fluid equations) so that the second term can be written as

∑

σ

∫

mσvvfσdv=

∑

σ

∫

mσ(v′+U)(v′+U)fσdv=

∑

σ

∫

mσv′v′fσdv+ρUU

(2.67)
where

∑

σ

∫

mσv′fσdv= 0has been used to eliminate terms linear inv′. The MHD
pressure tensor is now defined in terms of the random velocities relative toUand is given