Magnetospheric MHD Resonances and ULF Pulsations - Ultra and Extremely Low Frequency Electromagnetic Fields - page 245

Considering the ground-based observation we first study the spectral density of
correlation matrix of the magnetic perturbations given by

nm !; k ;! 0 ; k 0 D ǝ ıb n .!; k /ıb m ! 0 ; k 0 Ǜ ;

. B /

(6.80)

where the symbols in the brackets denote the spectral amplitudes of the ground-
based field fluctuations.
Substituting Eq. ( 6.75 )forı b into Eq. ( 6.80 ) and rearranging yields

M . w /

lp ;

X

3

X

3

nl M . w / mp . w / lp C M .m/ nl M .m/

.m/ .B/

nm D

(6.81)

mp

l D 1 pD1

where

D ıI . w /
l

.!; k /ıI . w / p ! 0 ; k 0 E ; . w /

lp D

(6.82)

lp D D ıb .m/

.!; k /ıb .m/ p ! 0 ; k 0 E : .m/

(6.83)

l

Here we have assumed that the forcing functions, ı b .m/ .!; k / and ı I . w / .!; k /,are
statistically independent of each other.
In what follows we focus on the correlation matrix, which describes the
contribution of the ionospheric wind-driven currents to the natural electromagnetic
noise observed on the ground surface. We choose first to study the case of vertical
Earth's magnetic field. The fluctuations of height-integrated ionospheric current,
ı I . w / . R ;t/, is considered as a 2D random field of R D .x;y/. This random field is
assumed to be uniform in time so that shift of the initial time has no effect on the
random process. In this notation the spatiotemporal correlation functions, ‰ . w /

lp , and
their linear combination in Eq. ( 6.81 ) depend on the time difference D t t 0 .Ifthe
random field is uniform in space, the shift of the origin of coordinate system O is
insignificant, so that the correlation functions must depend on only relative distance

L D j L j D LJ LJ R R 0 LJ LJ . Since the plasma conductivity is anisotropic in the E layer,
‰ . w /
lp

cannot depend only on relative distance. So we assume that ‰ . w /

lp is a function
of both L x Dj x x 0 j and L y Dj y y 0 j . In such a case the spectral density of this
random process is delta-correlated both over k x , k y and !

lp k ;!; k 0 ;! 0 D ı ! ! 0 ı k x k 0 x ı k y k 0 y G lp .!; k /: (6.84)

. w /

Here the function G lp .!;k/ is derivable through the spatial distribution of the
correlation function ‰ . w /

lp . L ;!/

Z Z

1
4 2

‰ . w / G lp . !; k / D

lp . L ;! / exp . i k L / dL x dL y :

(6.85)

1 1