Fundamentals of Plasma Physics

2.5 Magnetohydrodynamic equations 47

by
←→
PMHD=

∑

σ

∫

mσv′v′fσdv. (2.68)

We insert Eqs.(2.67) and (2.68) in Eq.(2.66), integrate by parts on the acceleration term,
and perform the summation over species to obtain the MHD equation of motion

∂(ρU)

∂t

+∇·(ρUU) =

(

∑

σ

nσqσ

)

E+J×B−∇·

←→

PMHD. (2.69)

MHD is typically used to describe phenomena with large spatial scales where the plasma
is essentially neutral, so that

∑

σnσqσ≈^0 .Just as in the two-fluid situation, the left hand
side of Eq.(2.69) contains a factor times the MHD continuity equation,

∂(ρU)

∂t

+∇·(ρUU) =

[

∂ρ

∂t

+∇·(ρU)

]

U+ρ

∂U

∂t

+ρU∇·U. (2.70)

Using Eq.(2.65) leads to the standard form for the MHD equation of motion

ρ

DU

Dt

=J×B−∇·

←→

PMHD (2.71)

where
D
Dt

=

∂

∂t

+U·∇ (2.72)

is the convective derivative using the MHD center of mass velocity. Scalar approximations
of the MHD pressure tensor will be postponed until after discussing implications of the
MHD Ohm’s law.

2.5.3 MHD Ohm’s law

Equation (2.71) provides one equation relatingJandU;let us now find the other one. In
order to find this second relation betweenJandUconsider the two-fluid electron equation
of motion:

me

due

dt

=−e(E+ue×B)−

1

ne

∇(neκTe)−υeime(ue−ui). (2.73)

In MHD we are interested in low frequency phenomena with large spatial scales. If the
characteristic time scale of the phenomenon is long compared to the electron cyclotron
motion, then the electron inertia termmedue/dtcan be dropped since it is small compared
to the magnetic force term−e(ue×B).This assumption is reasonable for velocities per-
pendicular toB, but can be a poor approximation for the velocity component parallel toB,
since parallel velocities do not provide a magnetic force. Sinceue−ui=−J/neeand
ui≃U,Eq.(2.73) reduces to the generalized Ohm’s law

E+U×B−

1

nee

J×B+

1

nee

∇(neκTe) =ηJ. (2.74)

The term−J×B/neeon the left hand side of Eq.(2.74) is called theHall termand can be
neglected in either of the following two cases: