High-Latitude Electrodynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 379

and

ρ d V / dt =−∇ p + ρ g + J × B (8.3)

Taking the cross product of (8.3) with B , the current density perpendicular to
B is given by

B 2

J

⊥ = (

1 / )(ρ B × d V / dt + ρ g × B + B ×∇ p ) (8.4)

The gravitational term in (8.4) can usually be neglected, so we note that in
the high-altitude magnetosphere the cross-field current is controlled by pressure
gradients and space or time-dependent flow and not directly by the electric field.
Electrical coupling between the regions is described by the equation for current
continuity

∇· J = 0 (8.5)

and Faraday's law

∇ × E + ∂ B /∂ t = 0 (8.6)

which must apply throughout the system. The boundaries of these regions can
be defined rather loosely for our purposes, since only a qualitative treatment of
magnetosphere-ionosphere coupling is being undertaken. The region between
200 and 2000 km is a transition zone, but for most purposes the plasma motion
there is also given by (8.2). Throughout this chapter we assume that the magnetic
field lines connecting these regions are electric equipotentials. This assumption
breaks down drastically in the lower magnetosphere on auroral zone field lines
where electrons and ions are accelerated by parallel electric fields. This source
region for auroral arcs is discussed in Chapter 9.
The space above 2000 km can be divided into two topologically different
regions by the magnetic field geometry described in Chapter 1 (see Fig. 1.13).
At the highest latitudes, the magnetic field lines extend either to the magne-
topause and subsequently into the magnetosheath and solar wind, or far down
the magnetotail into the boundary layer that lies just inside the magnetopause.
Field lines extending to the magnetosheath are “open”—that is, they have one
foot on the earth and the other connected to the interplanetary magnetic field
(IMF). Field lines extending to the boundary layer and inner magnetosphere are
“closed”—that is, they have both feet on the earth even though the field line may
be extremely long. In the magnetosheath and in the boundary layer, the plasma
is flowing rapidly antisunward and is driven by the expanding solar atmosphere.
In the inner magnetosphere the plasma flow is dependent on the plasma pressure
and upon both internally and externally applied electric fields.