Magnetospheric MHD Resonances and ULF Pulsations - Ultra and Extremely Low Frequency Electromagnetic Fields - page 241

In order to eliminate the nonzero parallel current k E k from Eqs. ( 6.61 )
and ( 6.63 ) one should slightly rearrange these equations. Equation ( 6.61 ) multiplied
by sin plus equation ( 6.63 ) multiplied by cos gives
@ y ıB z @ z ıB y sin C @ x ıB y @ y ıB x cos D 0 J ? C J . w x sin C J . w /

cos :

z

(6.68)

In the frequency range f<0:1Hz the thickness, l,oftheE layer is much
smaller than the skin-depth in the ionosphere. In this notation the “thin” layer
approximation can be used in order to derive the boundary conditions at the E
layer of the ionosphere. This approximation is described in more detail in Sect. 5.
Integrating of Eq. ( 6.68 ) with respect to z across the E-layer, making formally
l ! 0, and taking into account Eq. ( 6.67 ), gives the boundary conditions at z D 0
sin ıB y D 0 † P E x = sin C † H E y C I . w / ;

(6.69)

where the square brackets denote the jump of magnetic field across the E-layer,
† P and † H are the height-integrated Pedersen and Hall conductivities given by
Eq. ( 5.25 ). Here I . w /

D I . w /
x

sin C I . w /
z

cos stands for the height-integrated wind-

driven currents, i.e.

Z Z

l l

I . w x D J . w x d z and I . w z D J . w /

z

d z : (6.70)

0 0

Similarly, integrating of Eq. ( 6.62 ) with respect to z across the E-layer yields
ŒıB x D 0 † P E y † H E x = sin C I . w y ;

(6.71)

where I . w /
y

is another component of the height-integrated wind-driven current, i.e.

Z

l

I . w y D J . w y d z :

(6.72)

0

In the framework of our model the region above the E-layer is supposed to be
the area consisting solely of a cold collisionless plasma, which is described by
Eq. ( 5.2 ). In the ULF frequency range the absolute value of parallel components
of the plasma dielectric permittivity, " k , is much greater than perpendicular ones
and thus can be assumed to be infinite. This means that the parallel electric field E k
equals approximately zero, and we come to Eq. ( 6.67 ). Thus, we can eliminate the
parallel current from Eq. ( 5.2 ) in analogy to the procedure used for the derivation of
Eq. ( 6.68 ). Whence, we get