48 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD
- The pressure term in the MHD equation of motion, Eq.(2.71) is negligible compared
to the other two terms which therefore must balance giving
|J|∼ωρ|U|/|B|;hereω∼D/Dtis the characteristic frequency of the phenomenon. In this case com-
parison of the Hall term with theU×Bterm shows that the Hall term is small by
a factor∼ω/ωciwhereωci =qiB/miis the ion cyclotron frequency. Thus drop-
ping the Hall term is justified for phenomena having characteristic frequencies small
compared toωci.- The electron-ion collision frequency is large compared to the electron cyclotron fre-
quencyωce=qeB/me in which case the Hall term may be dropped since it is small
by a factorωce/υeicompared to the right hand side resistive termηJ=(meνei/nee^2 )J.
From now on, when using MHD it will be assumed that one of these conditions is true
and Hall terms will be dropped (if Hall terms are retained, the systemis called Hall MHD).
Typically, Eq. (2.74) will not be used directly;instead its curl will be used to provide the
inductionequation
−
∂B
∂t+∇×(U×B)−
1
nee∇ne×∇κTe=∇×(
η
μ 0∇×B
)
. (2.75)
Usually the density gradient is parallel to the temperature gradient so that the thermal elec-
tromotive force term(nee)−^1 ∇ne×∇κTecan be dropped, in which case the induction
equation reduces to
−
∂B
∂t+∇×(U×B) =∇×
(
η
μ 0∇×B
)
. (2.76)
The thermal term is often simply ignored in the MHD Ohm’s law, which is written as
E+U×B=ηJ; (2.77)this is only acceptable providing we intend to take the curl and providing∇ne×∇κTe≃ 0.
2.5.4 Ideal MHD and frozen-influx
If the resistive termηJis small compared to the other terms in Eq.(2.77), then the plasma
is said to beidealorperfectly conducting. From the Lorentz transformation of electromag-
netic theory we realize thatE+U×B=E′whereE′ is the electric field observed in
the frame moving with velocityU.This implies that the magneticflux in ideal plasmas
is time-invariant in the frame moving with velocityU, because otherwise Faraday’s law
would imply the existence of an electric field in the moving frame. In order to have the
magneticflux invariant in the moving frame, the magnetic field lines must convectwith
the velocityU, i.e., the magnetic field lines arefrozeninto the plasma andmove with the
plasma. The frozen-in field concept is the essential “bed-rock” concept underlying ideal
MHD. While this concept is often an excellent approximation, it must be keptin mind
that the concept becomes invalid in situations when any one of the electron inertia, electron
pressure, or Hall terms become important and lead to different, more complex behavior.