Magnetospheric MHD Resonances and ULF Pulsations - Ultra and Extremely Low Frequency Electromagnetic Fields - page 206

The neutral gas is so rarefied at magnetospheric heights that the number density
of ions is much greater than that of neutrals, so that the plasma can be considered
fully ionized in this region. Furthermore, the plasma itself is so tenuous that ei
i and thus the medium can be treated as a collisionless magnetized plasma. In such
a case we can use the MHD approach, in which, as we have noted in Sect. 1.2.2 ,the
plasma is considered as a single fluid having infinite conductivity. This means that
in a reference frame moving at plasma velocity V the electrical field E 0 D E C V B
vanishes in both directions parallel and perpendicular to B , whence it follows that
in a reference frame fixed to the Earth E D B V .
The magnetospheric plasma dynamics is described by Eq. ( 1.13 ), which relates
the plasma velocity to the forces acting on the plasma. This equation in its general
form contains the pressure gradient, the terms describing the gravitational and
“viscous” forces, and the magnetic/Ampere's force given by j B . In the reference
frame fixed to the Earth this equation includes all the inertial forces acting on
the plasma due to the Earth spin. The pressure gradient and the magnetic force
dominate if the typical frequencies are smaller than 0:1 Hz. In this frequency range
the gravity, viscosity, and inertial terms in Eq. ( 1.13 ) can be neglected.
Following Dungey ( 1954 , 1963 ) we first assume that the geomagnetic field and
electric currents in the magnetosphere are large enough so that the magnetic force
in Eq. ( 1.13 ) is much greater than the pressure gradient. It should be noted that
the plasma motion parallel to the magnetic field lines must be due to only the
pressure gradient r P since the magnetic force j B is always perpendicular to B .On
the other hand the plasma motion parallel to B does not greatly affect the magnetic
field. In this picture the cancel of r P in Eq. ( 1.13 ) is not so a burdensome condition.
Finally, we have

d V =dt D j B ; (6.1)

where is the plasma mass density, and the total time-derivative d=dt is given by
Eq. ( 1.12 ). To treat the plasma dynamics, Maxwell's equations are required, whose
full forms are given by Eqs. ( 1.1 )-( 1.4 ). Since the vacuum displacement current
@ t D can be ignored due to the high plasma conductivity, Eq. ( 1.1 ) is simplified to
the form in which the curl of magnetic field is related to the conduction current j
through Eq. ( 1.5 ).
Substituting E D B V into Eq. ( 1.2 )givesEq.( 1.18 ). The meaning of this
equation is that the magnetic field is frozen to the conducting plasma and thus can
be considered to move with the plasma. The concept of “frozen-in” magnetic field
lines has been discussed in more detail in Sect. 1.2.1 .
Let ı B be the small perturbation of the ambient/geomagnetic field B 0 , so that
B D B 0 C ı B , and j ı B j j B 0 j . The unperturbed geomagnetic field B 0 is not
a function of time. In the first approximation, one can replace the term V B
in Eq. ( 1.18 ) with V B 0 . After these simplifications we come to the following
equation:

@ t ı B Dr . V B 0 /: (6.2)