Inhomogeneous Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 274

yields

X 1 ( x, y )

∇ ⊥ Φ 0 ( x, y ) =2 √ ε m ∇

2
⊥

Φ ( i ) ( x, y ) ,

∇ ⊥ · (9.10)

where

c Σ + √ ε m 1 .
1 is a unit diagonal matrix. The integration of (9.10 ) results in

X 1 ( x, y )= 4 π

∇ ⊥ Φ 0 ( x, y )= 4 π 2 √ ε m ∇ ⊥ Φ ( i ) ( x, y ) .

(9.11)

− X 1 ( x, y )

c ∇ ⊥ ×

( Ψ ( x, y ) z ) −

where Ψ is a function defined by the boundary conditions.
Equations (9.7) and (9.11) enable us to find the ionospheric current and
from it the magnetic field under the ionosphere ([4], [5]). These equations have
been generalized for a more realistic model of the magnetosphere (see e.g. [9]).
Equation (9.11) may also be used to investigate relations between the ground
and magnetospheric magnetic fields in the presence of stochastic ionospheric
irregularities (see Chapter 10).

Explicit Solutions

Contact of two Half-Planes

In a number of important cases (9.11) is integrable. For example, such solu-
tions can be found if the ionosphere consists of strips with piecewise constant
tensor integral conductivities Σ n . In the n -th strip the potential is of the form
Φ 0 ( x, y )= 2 √ ε m Φ ( i ) ( x, y )+ Φ n ( x, y ) .

(9.12)

X 1 n

Here

X 1 n ( x, y )= 4 π
c

Σ Pn + √ ε m .

Substituting (9.12) into (9.11), we obtain

− X 1 n ∇ ⊥ Φ n = ∇ ⊥ × Ψ n z , (9.13)

where

Ψ n ( x, y )= 4 π
c

Ψ n +2 √ ε m

Y n
X 1 n

Φ ( i ) .

Equation (9.13) is to be supplemented by the boundary conditions between
adjacent regions. The first condition follows from the continuity of the tan-
gential component of the electric field. The second follows from the continuity
equation for the electric current (9.6). The normal component of

− X 1 n ( x, y ) ∇ ⊥ Φ n ( x, y )