Ionospheric Alfvén Resonator (IAR) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 153

electromagnetic perturbations (Surkov et al. 2004 ). For clarity we consider that the
stationary convective electric field is absent, and the electromagnetic perturbations
are solely due to the neutral wind and external sources in the magnetosphere or
atmosphere.
Let l be a typical thickness of the conductive E layer of the ionosphere. Taking
a maximum of P in the ionosphere as a representative conductivity of the plasma,
the skin-depth in the E layer of the ionosphere is estimated as l s . 0 P !/ 1=2 .If
l l s or ! 0 P l 2 1 , the electromagnetic fields are slowly varying functions
of height inside the E layer and thus “thin” ionosphere approximation can be applied
(Lysak 1991 ; Pokhotelov et al. 2000 ). Using the following numerical parameters
l D 30 km, P D 10 4 S/m, we find that the above approach is valid if ! 10 Hz.
In what follows we consider the E layer in the “thin” ionosphere approximation,
that is l l s , because the IAR eigenfrequencies typically lie in the range of
0.5-3 Hz. In this notation we shall use the so-called height-integrated Pedersen and
Hall conductivities

Z

l

† P;H D

P;H . z /d z ; (5.25)

0

which are measured in 1 . Certainly, the boundaries of the actual E-layer cannot
be determined exactly and the peaks of the Hall and Pedersen conductivities are
situated at different altitudes due to the inhomogeneity of the actual ionosphere. This
inaccuracy, however, is of no practical importance on account of exponential fall off
of both conductivities with distance from the conducting E-layer. It is customary
to use the normalized parameters Ǜ P D † P =† w and Ǜ H D † H =† w , which are
the ratios of the height-integrated Pedersen and Hall conductivities to the so-called
wave conductivity † w D . 0 V AI / 1 , where V AI is the Alfvén speed inside the IAR.
Substituting Eq. ( 5.24 )for j into Eq. ( 5.23 ) we obtain the equation which can be
integrated across the E-layer with respect to z from z D 0 to z D l. Taking only the
projection of j ? perpendicular to B 0 and making formally l ! 0 gives the boundary
conditions at z D 0

V AI . O z Œ b ? / D Ǜ P e 0 ? C Ǜ H O z e 0 ? :

(5.26)

Here the square brackets denote the jump of magnetic field b ? across the E-layer,
that is Πb ? D b ? .0 C / b ? .0 /. In the thin slab approximation the electric field
e 0 D e C v B 0 in Eq. ( 5.26 ) should be taken at z D 0. Besides, for the sake of
simplicity, the Fourier transform of the wind velocity is assumed to be independent
of the z -coordinate, that is v D v .!; k ? /.
The appearance of the term Πb ? in Eq. ( 5.26 ) follows the general laws of the
electrodynamics according to which the horizontal component of magnetic field
must be discontinuous across the current flowing on an infinitely thin sheet (e.g., see
monograph by Jackson ( 2001 )). In our approach the jump of the horizontal magnetic
field, i.e., the term O z Πb ? , arises just due to the presence of the surface Pedersen
and Hall currents flowing in the thin E-layer.