Electrodynamic Properties of Space - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 35

j y = − σ H E x sin I + σ P E y + σ H E z cos I, (2.5)

j z =( σ − σ H E y cos I

+( σ sin 2 I + σ P cos 2 I ) E z .

σ P ) E x cos I sin I −

(2.6)

Suppose the ionosphere is an anisotropic sheet with non-conductive upper and
lower boundaries. Then vanishing of the vertical component of the current j z
enables us to find E z from (2.6) in the form

E z = ( σ P −

σ ) E x cos I sin I + σ H E y cos I
σ sin 2 I + σ P cos 2 I

. (2.7)

Substitution of ( 2.7) into (2.4) and (2.6) yields [5]:

j x = σ xx E x + σ xy E y ,
j y = − σ xy E x + σ yy E y ,

with components of the so-called 'layered conductivity'

σ σ P
σ sin 2 I + σ P cos 2 I ,

σ xx =

σ σ H sin I
σ sin 2 I + σ P cos 2 I

σ xy = = − σ yx ,

σ yy = σ P σ sin 2 I +( σ P + σ 2 H )cos 2 I
σ sin 2 I + σ P cos 2 I

. (2.8)

These formulae are valid just for large-scale low frequency fields. Since
σ σ P ,σ H the field-lines can be considered equipotential for the ULF

large-scale perturbations, i.e. the E x and E y electric components are height-
independent. Integration the equations for the currents j x ,j y over height gives

J x = Σ xx E x + Σ xy E y ,
J y =

− Σ xy E x + Σ yy E y ,

where

Σ xx = σ xx dz, Σ xy = σ xy dz, Σ yy = σ yy dz

(2.9)

are the height integrated conductivities and
J x = j x dz,

J y = j y dz

are components of the surface ionospheric current.