Fundamentals of Atmospheric, Ionospheric, and Magnetospheric Plasma Dynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 38

ionospheric heights are such that 5%, we take (2.23c) as a diagnostic

equation only. That is, given some current J , we can find

δ B / B ≤

B , the perturbation to
the geomagnetic field, from (2.23c), but the B we use in the rest of the dynamical
equations is the earth's field, so we can take

δ

0. This approximation thus
rules out induced electric fields from magnetic field changes. Of course, when
we include electromagnetic waves such as Alfvén waves, we include both electric
and magnetic field fluctuations. We are thus left with the set

∂ B /∂ t =

∇× =

E 0 (2.25a)

∇· = ρ /ε 0 (2.25b)

E c

∇× B = μ 0 J (2.25c)

∇· B = 0 (2.25d)

∇· J =− ∂ρ c /∂ t (2.25e)

Equation (2.25a) shows that the electric field is derivable from a potential func-
tion . Finally, one further important simplification is

possible. In an ionized medium, very small charge differences create large electric
fields. Thus, a plasma must exhibit nearly perfect charge neutrality. This implies

φ( r , t ) through E =−∇ φ

t =

∇· J =− ∂ρ c /∂ 0 (2.26)

That is, the divergence of the electric current on any macroscopic time scale must
be zero. As we shall see in Section 2.4, in most cases we use 0 to calculate

the electric field rather than (2.25b). Poisson's equation [i.e., (2.25b)] is, of
course, still valid, but it is not very useful, since the charge difference associated
with geophysically important electric fields is very small. This situation is similar
to the problem of calculating the vertical velocity in atmospheric dynamics.
The vertical component of the momentum equation is not very useful, since the
vertical pressure gradient and the gravitational terms nearly cancel each other
out. In practice, the vertical velocity is often found from the divergence of the
horizontal wind, much as we use

∇· J =

∇· =

0 to find the electric field. The following
combined set of dynamic and electrodynamic equations for the plasma (plus
the equation of state) remains after these simplifications:

J

q j ρ j
M j

E

B

ρ j d V j / dt =−∇ p j + ρ j g + + V j ×

k ρ j v jk V j −

V k − k

j

(2.27a)

=

∂ρ j
∂

+∇· ρ j V j = P j − L j M j

(2.27b)

t

p j = ρ j k B T j / M j = n j k B T j (2.27c)