Ci a ion: Jandie i, G.; Tugushi, N.
T ans o ma ion o he Spa ial
Spec um o Sca e ed Radio Wa es
in he Conduc i e Equa o ial
Ionosphe e. Elec onics 2023,12, 2759.
h ps://doi.o g/10.3390/
elec onics12132759
Academic Edi o : Roald M. Tiggelaa
Recei ed: 29 Ap il 2023
Re ised: 29 May 2023
Accep ed: 12 June 2023
Published: 21 June 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
elec onics
A icle
T ans o ma ion o he Spa ial Spec um o Sca e ed Radio
Wa es in he Conduc i e Equa o ial Ionosphe e
Gio gi Jandie i 1,2 and Nika Tugushi 3,*
1Depa men o S ochas ic Analysis and Ma hema ical Simula ion, Geo gian Technical Uni e si y,
0175 Tbilisi, Geo gia; geo [email p o ec ed]
2Nano echnology Cen e, VSB—Technical Uni e si y o Os a a, 701 03 Os a a, Czech Republic
3Depa men o Physics, Tbilisi S a e Uni e si y, 0179 Tbilisi, Geo gia
*Co espondence: [email p o ec ed]; Tel.: +99-5598-228-086
Abs ac :
The oblique adio wa e incidence on a u bulen equa o ial conduc i e collision plasma
laye is conside ed. The “Compensa ion E ec ” has been disco e ed by us. A complex e ac i e
index o he equa o ial e es ial ionosphe e has been de i ed o he i s ime. Second-o de
s a is ical momen s o he spa ial powe spec um (SPS) o sca e ed adio wa es a e ob ained o he
i s ime using he WKB me hod, aking in o accoun he asymme y o he p oblem: he inclined
incidence o he wa e on a plasma bounda y and he asymme y o he magne o-ionic pa ame e s. I
was es ablished o he i s ime ha a ce ain di ec ion exis s along which he inclined incidence
adio wa e on a plasma laye and he aniso opy pa ame e s o a magne oplasma compensa e each
o he . This esul will ha e g ea p ac ical applica ion in communica ion. In his case, he SPS o
sca e ed adio wa es nei he widens no is i s maximum displaced. The beha io o his spec um
e sus dis ance p opaga ed by adio wa es in he conduc i e equa o ial ionosphe e is analyzed
nume ically o di e en pene a ion angles and aniso opy ac o s o asymme ic aniso opy elec on
densi y i egula i ies. I was shown ha he aniso opy ac o o elonga ed plasmonic s uc u es
has a subs an ial in luence on he “Compensa ion E ec ” o sca e ed o dina y and ex ao dina y
wa es pene a ing in he conduc i e collision ionosphe ic plasma he slab. Nume ical calcula ions
a e ca ied ou o he aniso opic Gaussian co ela ion unc ion applying IRI expe imen al da a.
Keywo ds:
u bulence; s a is ical momen s; ionosphe ic plasma; conduc i i y; adio wa es; o dina y
and ex ao dina y wa es; i egula i ies
1. In oduc ion
Randomly a ying magne o-ionic plasma pa ame e s con aining elec on concen a-
ion inhomogenei ies ha e a signi ican impac on communica ion accu acy and na iga ion
in a wide equency band. Randomly a ying plasma pa ame e s ha e a subs an ial in lu-
ence on he adio wa es. Conduc i i y, collision be ween he plasma pa icles, complex
e ac i e index, and i s imagina y pa cause signi ican impac on he s a is ical cha -
ac e is ics o sca e ed adio wa es p opaga ing in he conduc i e collision ionosphe ic
magne ized (COCOIMA) plasma, pa icula ly in he equa o ial egion o he e es ial
a mosphe e. Asymme y o he ask can also lead o he inc ease in u bulence in magne o-
plasma. The di ec ion o bo h he geomagne ic ield and an inclined inciden o adio wa e
on a conduc i e u bulen collision magne oplasma is a eason o he asymme y o he
p oblem. Inclined inciden plane adio wa es on he bounda y o abso p i e plasma laye
pene a ing in o he u bulen plasma become inhomogeneous.
In many pape s [
1
–
4
], second-o de s a is ical cha ac e is ics o he mul iple sca e ed
adio wa es in he e es ial ionosphe e ha e been in es iga ed analy ically and nume i-
cally using expe imen ally obse ed da a. I was shown [
5
] ha a small angles o incidence
wa e on a su ace be ween acuum and iso opic collisional u bulen plasma, he spa-
ial (angula ) powe spec um (SPS) o a mul iple sca e ed adio wa e mono onically
Elec onics 2023,12, 2759. h ps://doi.o g/10.3390/elec onics12132759 h ps://www.mdpi.com/jou nal/elec onics
Elec onics 2023,12, 2759 2 o 11
b oadens wi h dis ance app oaching a ce ain asymp o ic alue [
5
]. New phenomena a ise
inc easing a inc easing asymme y o he p oblem: he incline incidence o adio wa es on
he aniso opic collision plasma slab. Di e en spec al componen s o he SPS a enua e
a iously; he wid h o he SPS anomalously b oadens and becomes asymme ic wi h
espec o i s maximum. The peak o he SPS maximum is non-mono onically shi ed o
he no mal ela i e o he in e ace be ween acuum and collisional magne oplasma. This
is a consequence o he asymme y o he ask. Simila phenomena a ise no only a he
inclined inciden wa e on a su ace; such aniso opy can also be an in e nal p ope y o a
u bulen plasma i sel .
Tu bulen conduc i e collisional magne oplasma is a andomly inhomogeneous ab-
so p i e medium. The inclined incidence o a plane adio wa e on a semi-in ini e collision
magne oplasma laye wi h elec on concen a ion luc ua ions was conside ed [
6
–
8
] in
a small-angle sca e ing app oxima ion. Spec um wid h a ies nonmono onically wi h
dis ance om a plasma bounda y. The widening o he SPS o sca e ed adio wa es and
shi o i s maximum o bo h aniso opic Gaussian and powe –law co ela ion spec al
unc ions o elec on concen a ion luc ua ions in he u bulen collisional magne oplasma
we e analyzed [
7
–
9
], applying he Wen zel–K ame s–B illuen me hod. In his case, he
wid h o he spec um o a sca e ed elec omagne ic wa e subs an ially exceeds he wid h
o he collisionless plasma. The “Compensa ion E ec ” in he pola e es ial ionosphe e
was conside ed in [10].
Physical p ocesses in he pola and equa o ial ionosphe es a e e y di e en . S a is ical
momen s o sca e ed adio wa es p opaga ing in he equa o ial e es ial ionosphe e
a e in es iga ed o he i s ime. They include he ollowing aniso opy pa ame e s:
aniso opy o he ionosphe e, aking in o accoun Hall’s, Pede sen, and longi udinal con-
duc i i y luc ua ions, he di ec ion o an ex e nal magne ic ield, aniso opy o elonga ed
plasmonic s uc u es due o he di usion p ocesses in he iled aligned and ield pe pen-
dicula di ec ions con aining aniso opy ac o and he inclina ion angle o aniso opic
ionosphe ic i egula i ies wi h espec o he geomagne ic lines o o ces, incline incidence
o wa e on a u bulen plasma laye . A Complex e ac i e index o he equa o ial e es-
ial ionosphe e has been ob ained o he i s ime. E alua ion o he SPS o sca e ed
elec omagne ic wa es p opaga ing in he COCOIMA plasma is in es iga ed in his pape
using he complex ay-(op ics) app oxima ion. Compensa ion condi ions a e es ablished o
he expe imen ally obse ed powe –law spec al unc ion o elec on densi y i egula i ies
applying he s a is ical ionosphe ic model IRI.
2. Radio Wa e P opaga ion in a Homogeneous Conduc i e Magne ized Plasma
The elec ic ield sa is ies he wa e equa ion:
(∇i∇j−∆δi j −k2
0eεi j )Ej( ) = 0 (1)
whe e:
k0=ω/c
is he wa enumbe o an inciden wa e ha ing equency
ω
;
∆
is he
Laplacian,
δij
is he K onecke symbol,
eεij =εij −ieσij
,
eσij ≡σij (
4
π/k0c)
a e he second
ank pe mi i i y and conduc i i y enso s o he COCOIMA plasma, espec i ely.
Le a plane adio wa e be inciden om acuum on a semi-in ini e homogeneous
COCOIMA magne oplasma laye ; Y axis coincides wi h an ex e nal homogeneous mag-
ne ic ield
H0
. Conduc i e homogeneous collision magne oplasma ha e pe mi i i y
enso componen s:
eεxx =eεzz =ε⊥−i(σ⊥+s g ),eεxz =−eεzx =−sæδ+i(eσH+æ),
eεyy =ε|| −iheσ|| +s(g−p0u g1)i,eεxy =−eεyx =eεyz =eεzy =0(2)
whe e
ε⊥=
1
−p0
,
p0= /(
1
−u)
,
æ=p0√u
,
ε|| =
1
−
,
g=p0(
1
+
u)/(
1
−u)
,
δ=
2
/(
1
−u)
,
g1= (
3
−u)/(
1
−u)
,
s=νe /ω
,
u= (e H0/mecω)2
,
ωp( ) = 4πne( )e2/me1/2
is he plasma equency,
( ) = ω2
p( )/ω2
,
νe
is he e -
Elec onics 2023,12, 2759 3 o 11
ec i e collision equency o elec ons wi h o he plasma pa icles,
ne( ) = n0+n1( )
is
he elec on densi y,
n0
is a homogeneous backg ound e m,
n1( )
is a andom unc ion o
posi ion, n1<< n0;eand mea e cha ge and mass o an elec on, espec i ely.
σ⊥=e2N νe
me(ν2
e+ω2
e)+νi
mi(ν2
i+ω2
i)!,
σH=e2N ωe
me(ν2
e+ω2
e)−ωi
mi(ν2
i+ω2
i)!,
σ|| =e2N1
meνe
+1
miνi,
σ⊥
,
σH
, and
σ||
a e he Pede sen, Hall’s, and longi udinal conduc i i ies, espec i ely,
νe,i
is he elec on o ion collision equency wi h he neu al molecules,
ωe
and
ωi
a e an
elec onic and an ionic angula gy o equencies, espec i ely;
me
and
mi
a e he mass o
an elec on and an ion, espec i ely.
Equa ion (1) yields he se o equa ions o he COCOIMA plasma
(k2sin2θcos2ϕ+k2cos2θ−k2
0eεxx)Ex−(k2sin2θsin ϕcos ϕ)Ey−
−(k2sin θcos θsin ϕ+k2
0eεxz)Ey=0,
(k2sin2θsin ϕcos ϕ)Ex−(k2sin2θsin2ϕ+k2cos2θ−k2
0eεyy)Ey+
+ (k2sin θcos θcos ϕ)Ez=0,
(k2sin θcos θsin ϕ−k2
0eεxz)Ex+ (k2sin θcos θcos ϕ)Ey−
−(k2sin2θsin2ϕ+k2sin2θcos2ϕ−k2
0eεxx)Ez=0(3)
He e, ϕis he pola angle be ween o he wa e ec o kon he XOZ plane.
I he di ec ions o adio wa e p opaga ion and he homogeneous imposed magne ic
ield a e pe pendicula (θ=900)and ϕ=00, we ob ain wo oo s
kI=k0[(ε⊥+æ+σH)−iσ⊥]1/2 and kII =k0[(ε⊥−æ−σH)−iσ⊥]1/2 (4)
only he Hall’s and Pede sen conduc i i ies gi e he con ibu ion in Equa ion (3),
Ey=
0 .
Subs i u ion (4) in o (3) gi es
(Ez/Ex) = ∓i
. Consequen ly, we ha e a igh -hand ci cula ly
pola ized (RHCP) wa e (uppe sign) and a le -hand ci cula ly pola ized (LHCP) wa e
(lowe sign) p opaga ing along he homogeneous magne ic ield. Analyses show ha
kII <kI
, he o a ion is clockwise and, hence, he Fa aday o a ion angle is
θF= (kII −kI)/
2.
Fo an inciden 3 MHz equency wa e, θF=0.01.
A quasi-longi udinal p opaga ion o wa e
(θ=
0
0)
, we ha e he ollowing oo s:
kI=k0√m1+i m2
,
kII =k0√εyy
, whe e
m1= (ε⊥ 0+σ⊥ 1)/(ε2
⊥+σ2
⊥)
,
m2=
(σ⊥ 0−ε⊥ 1)/(ε2
⊥+σ2
⊥)
,
1=
2
ε⊥σ⊥
,
0=ε2
⊥−σ2
⊥−(æ+σH)2
. Con a y o he
p e ious case, he second oo con ains he longi udinal conduc i i y. Fo nonconduc i e
plasma, kI=k0q(ε2
⊥−æ2)/ε⊥. The second oo is he same.
Ionosphe ic plasma is an impo an subjec o esea ch in he ield o adio physics
o adio wa e p opaga ion. I
θ
is an angle be ween
H0||Y
and
k0⊂YOZ
ec o s,
s<< εi j
,
eσi j
, complex e ac i e index o he conduc i e collision equa o ial ionosphe ic
plasma was ob ained in [11].
A e some ans o ma ions om Equa ion (1), we ob ain
g2θ=−εyy (N2−εL) (N2−εR)
(N2−εyy)(εxx N2−εRεL)(5)
Elec onics 2023,12, 2759 4 o 11
whe e:
εR=εxx +iεxz
,
εL=εxx −iεxz
. Equa ion (5) has wo ze o and wo poles, whe ein
each ea u e co esponds o one o ou majo wa es (modes) ha may exis in he plasma.
These modes in he conduc i e equa o ial plasma can be classi ied as ollows:
(1)
P opaga ion o wa e pe pendicula o he ex e nal magne ic ield (θ=π/2 )
(a)N2=1−( +σ||); O-wa e. (6)
(b)N2=εRεL
εxx
=ε⊥"1−(æ+σH)
ε2
⊥−σ2
⊥#−iσ⊥"1+(æ+σH)
ε2
⊥−σ2
⊥#; E-wa e. (7)
(2)
Wa e p opaga ion along he ex e nal magne ic ield (θ=0)
(a)N2=εR=1−
1−√u−σH−iσ⊥; a RHCP spi al wa e (8)
(b)N2=εL=1−
1+√u+σH−iσ⊥; a LHCP spi al wa e (9)
hese a e weakly dumping adio wa es p opaga ing in he equa o ial e es ial ionosphe e.
In he absence o an ex e nal magne ic ield and in a non-conduc i e plasma nonconduc i -
i ies
(σij =
0
)
, all wa es a e educed o he o dina y wa e. A in e media e angles be ween
0 and π/2, p opaga ing wa es will ep esen some combina ions o hese majo wa es.
G oup eloci ies
g
o bo h o dina y and ex ao dina y (O
−
and E
−
) wa es gene ally
do no coincide wi h he phase eloci y
ph =ωk/k2
. The angle
Θ
be ween
g
and
k
in a
homogeneous medium, as is known, is de e mined by he ela ion [1–3]
g Θ1,2 =−1
2N2
1,2
∂N2
1,2
∂ θ (10)
A small angles θ(quasi-longi udinal p opaga ion) N1,2 has a simple o m
N1,2 =1−
1∓√ucos θ,
g Θ1,2 =∓ √usin θ
2N2
1,2 (1∓√ucos θ)2
(11)
Uppe sign co esponds o he o dina y wa e (O-wa e), lowe sign— o he ex ao di-
na y wa e (E-wa e). I is easy o make su e ha he a io
g Θ1/ g Θ2
de e mined by he
las a io less han ze o, i.e., he g oup eloci ies o he O
−
wa e and E
−
wa e de ia e om
he phase in opposi e di ec ions.
I can be concluded ha a e ical sounding o he ionosphe e, he a eas o e lec ion
o O
−
wa e and E
−
wa e a e spaced apa in he ho izon al di ec ion. Calcula ions show
ha a e ical sounding o he ionosphe e in bo h he sou he n and no he n hemisphe es,
he ajec o y o O
−
wa e de ia e om he e ical o he equa o , and he E
−
wa e in he
opposi e di ec ion.
I adio wa es p opaga e along he ex e nal magne ic ield in he equa o ial ionosphe e,
om Equa ions (8) and (9) we ob ain
ω(e)
1= (1−σH)(ωps1+1−σH
4
Ω2
e
ω2
p−Ωe
2)<ωp, (12)
ω(e)
2= (1+σH)(ωps1+1−σH
4
Ω2
e
ω2
p
+Ωe
2)>ωp. (13)
Elec onics 2023,12, 2759 5 o 11
he RHCP wa es a e e lec ed in he ionosphe e, while LHCP wa es pene a ing he
ionosphe ic laye s will p opaga e in he uppe ionosphe e;
Ωe=e H0/mec
is he elec-
on gy o equency.
3. S a is ical Momen s o he Phase Fluc ua ions
The main heo e ical ool desc ibing sho wa eleng h elec omagne ic wa es p opaga-
ion in he e es ial a mosphe e is he ay heo y. I is one o he mos impo an me hods
among o he asymp o ic me hods o a numbe o easons: one o hem is i s simplici y and
he possibili y o ob aining an analy ical solu ion o a wide ange o p oblems ha canno
be in es iga ed by accu a e o o he asymp o ic me hods.
Le us in oduce he Ca esian sys em o coo dina es; a plane laye o a u bulen
COCOIMA plasma is adia ed by a plane adio wa e, e ac ing a an angle
θ1
ela i e o
he no mal o his laye (i is ela ed o he angle o incidence
θi
by Snell’s law). This is he
asymme y o he p oblem. The hickness o a plasma laye is L. A he equa o ial la i udes,
he geomagne ic ield is nea ly ho izon al; he e o e, we will assume ha he homogeneous
magne ic ield ec o
H0
is di ec ed along he y-axis and he wa e ec o o an inciden
wa e
k0
; bo h a e loca ed in he YOZ plane ( he main plane). Rec angula componen s
o a complex wa e ec o on he Y and Z coo dina e axes in acuum a e pu ely eal
alues:
k0z=k0sin θi
,
ky1=k0cos θi=qk2
0−k2
0z
, pa icula ly
∂ky1/∂k0z=− g θi
.
Re ac ing a he in e ace acuum–plasma, his wa e becomes inhomogeneous, as i
obeys he bounda y condi ions, he Z-p ojec ion o i s wa e ec o emains eal, bu he
Y-p ojec ion becomes complex
ky2=qk2
0N2−k2
0z
. We assume ha he cha ac e is ic
spa ial scale o elec on concen a ion i egula i ies exceeds he wa eleng h o an inciden
wa e
l>> λ
and
L>> l
. This allows he use he ay-(op ics) app oxima ion o he
in es iga ion o s a is ical momen s o a sca e ed adio wa e ield.
Elec on concen a ion luc ua ions in a plasma laye cause luc ua ion o a sca e ed
ield in he obse a ion poin . A small-angle sca e ing in he ay-(op ics) app oxima ion,
second-o de s a is ical cha ac e is ics a e de e mined by luc ua ion o a complex phase
ϕ( )
[
3
] sa is ying he eikonal equa ion
k2=k2
0N2
,
k( ) = − ∇ ϕ
complex wa e ec o
is a unc ion o posi ion. Complex e ac i e index o he COCOIMA plasma con ains
componen s o he wa e ec o
N2( ) = N2(n( )
,
kx
,
kz
,
ω)
. The eikonal equa ion can be
ew i en as [8]
(k·∇k)−1
2k2
0
∂N2
∂k⊥∇k=1
2k2
0
∂N2
∂n∇n, (14)
whe e,
k⊥=k⊥(kx
,
kz)
. We can use he ollowing se ies expansions o he wa e
ec o and he ollowing phase:
k=k0+k1( ) + ···
,
ϕ=ϕ0+ϕ1+···
leading o he
ela ionship o he phase luc ua ion conside ing only i s o de small e ms o elec on
densi y luc ua ions n1/n0. A e some algeb aic ans o ma ions, we ob ain
k0y
∂ ϕ1
∂y+
k0z−1
2k2
0
∂N2
0
∂k0zk0x=0
∂ ϕ1
∂z=−1
2k2
0
∂N2
0
∂n0
n1. (15)
in eg a ing Equa ion (15) along he complex cha ac e is ics, aking in o accoun ha
∂ky2
∂k0z
=−1
ky2 k0z−k2
0
2
∂N2
∂k0z!≡Υ1+iΥ2. (16)
Υ1= g θ−1
2 cos2θ(Γ2
0+Γ2
1)Γ0∂Γ0
∂ θ +Γ1∂Γ1
∂ θ ,
Υ2=1
2 cos2θ(Γ2
0+Γ2
1)Γ1
∂Γ0
∂ θ −Γ0∂Γ1
∂ θ (17)
He e, θis an angle be ween H0and he wa e ec o o a e ac ed wa e.

Elec onics 2023,12, 2759 6 o 11
I egula i ies in he equa o ial ionosphe e a e highly ield aligned, and he e o e he
sp ead o F u bulence is 2D. A some scale leng h he spec um is aniso opic, i.e., he
ionosphe e is much mo e na owly bounded e ically han ho izon ally.
Taking in o accoun he bounda y condi ion,
ϕ1(z=
0
) =
0 and expand he unc ion
ϕ1in a wo-dimensional Fou ie in eg al; o he phase luc ua ions, we ob ain
ϕ1(x,L,z) = α
ky
∞
Z
−∞
dkx
∞
Z
−∞
dkzexp(ikxx+ikzz)
L
Z0
dξn1(kx,ξ,kz)exp−ikz∂ky2
∂k0z0
(L−ξ)(18)
whe e, α≡ − 1
2k2
0
∂N2
0
∂n0; o b e i y, he index ‘00will be omi ed e e ywhe e.
Phase luc ua ions allow o he in es iga ion o second o de s a is ical cha ac e is ics
o sca e ed adio wa es. A ans e se co ela ion unc ion o he phase luc ua ion may be
easily ob ained
Vϕ(ρx,L,ρz) = 2πα2
k2
y
∞
R
−∞
dkx
∞
R
−∞
dkzWn(kx,Λ2kz,kz)1
2Λ1kz[1−exp (−2Λ1kzL)]·
·exp (ikxρx+ikzρz)
(19)
whe e,
ρz
and
ρx
a e dis ances be ween obse a ion poin s spaced apa a small dis ances
in he main and pe pendicula planes,
Vn(k)
is he a bi a y co ela ion unc ion o elec on
concen a ion luc ua ions.
A s ong luc ua ions o he phase
<ϕ1ϕ∗
1> >>
1, i can be assumed, as usual [
3
],
ha hey a e no mally dis ibu ed. I he egula phase di e ence is negligible, he ans-
e se co ela ion unc ion o he complex ield is desc ibed by he o mula
VE(ρx,L,ρz) = E2
0expikzρz−2(Im ky2)Lexp ∂Vϕ
∂ ρz
ρz+1
2
∂2Vϕ
∂ ρ2
z
ρ2
z+1
2
∂2Vϕ
∂ ρ2
x
ρ2
x!(20)
all de i a i es o he co ela ion unc ion o he phase a e aken a
ρx=ρz=
0. Using
he 2D Fou ie ans o ma ion, we ob ain he APS ha ing g ea p ac ical impo ance.
This s a is ical cha ac e is ic is he same as he ay in ensi y (b igh ness) in he adia ion
anspo equa ion [3]
S(kx,L,kz) = S0exp(−(kz−∆kz)2
2<k2
z>−k2
x
2<k2
x>), (21)
∆kz=2
i
∂Vϕ
∂ρz,<k2
z>=−∂2Vϕ
∂ρ2
z
,<k2
x>=−∂2Vϕ
∂ρ2
x
, (22)
Second-o de s a is ical cha ac e is ics
∆z≡∆kz
desc ibe he displacemen o maximum
o he SPS due o he andom a ia ion o elec on densi y,
Σz≡<k2
z>
and
Σx≡<k2
x>
de e mines he b oadening o he SPS in he YOZ and XOY planes, espec i ely.
Subs i u ing (18) in o Equa ion (22), we yield
Σx=
∞
R
−∞
dkx
∞
R
−∞
dkzk2
x
kzWn(kx,Υ1kz,kz)Ξ(kz),
Σz=
∞
R
−∞
dkx
∞
R
−∞
dkzkzWn(kx,Υ1kz,kz)Ξ(kz),
∆z=
∞
R
−∞
dkx
∞
R
−∞
dkzWn(kx,Υ1kz,kz)Ξ(kz),
Ξ(kz) = 2π
Υ2
α2
k2
y[1−exp (−2Υ2kzL)].
(23)
he double in eg als depend only on he spa ial spec um o elec on concen a ion luc ua-
ions bu no on he s eng h o he luc ua ions. As i was men ioned abo e, he asymme y
o he p oblem includes all aniso opic pa ame e s o he u bulen conduc i e magne o-
Elec onics 2023,12, 2759 7 o 11
plasma and oblique incidence o he wa e. Pa icula ly (a) a inclined incidence wa e on
he iso opic abso p i e plasma
∂ky2/∂k0z=−k0z/ky2
,
Υ2=
0; (b) a oblique incidence
o he wa e on a magne oplasma slab, we can apply Fo mula (19). F om hese o mulas,
ollow ha anomalous b oadening o he APS and shi o i s maximum a e clea ly man-
i es ed a
Υ2kzL>
1. Taking in o accoun ha
∂N2/∂kz∼∂N2/∂ θ
, we can di ec ly
apply o mula (5) di e en ia ing by angle
θ
. F om he Equa ions (18) and (24), ollow
ha a
Υ2=
0 exis s he compensa ion di ec ion, along which he SPS nei he b oadens
no does i s maximum shi . A di e en inciden angles on a plasma laye , he SPS will
anomalously b oaden, bu he mean alue o he SPS will displace o he compensa ion
di ec ion, inc easing dis ance a elling by adio wa e in he equa o ial egion o he
COCOIMA plasma.
4. Nume ical Calcula ions
A i icial ionosphe ic inhomogenei ies gene a ing ac oss he geomagne ic ield ha e
scales anging om a me e up o mo e kilome e s. The s a is ical ionosphe ic model
IRI ecommended by URSI as a basic model in he s udy and p edic ion o ionosphe ic
p opaga ion o adio wa es [
12
,
13
] is he mos p o en echnology o desc ibing he spa ial
dis ibu ion o elec on densi y in he equa o ial ionosphe e.
Obse ing he cha ac e is ic linea scale o elec on densi y i egula i ies in F egion
o he equa o ial ionosphe e is in he ange om 100 m o 10 km. Sa elli e expe imen s
consis en ly obse e a powe law spec um o plasmonic s uc u es, wi h he spec al index
in he in e al −1 o −3.
Expe imen al obse a ion o he equa o ial ionosphe ic i egula i ies dis ibu ion was
analyzed by GPS signals egis e ing by g ound-based ada s a ions and by Ionosphe ic
Scin illa ion Moni o s (ISM). I egula i ies in he equa o ial ionosphe e ha e a bubbles
o m, c ea ed nea he magne ic equa o on he bo om side o he F2 laye , ising along
he geomagne ic lines o o ces. These bubbles a e gene a ed due o a Rayleigh–Taylo -like
ins abili y. Expe imen ally obse ing he phase spec al index p desc ibes he s eng h o
elec on densi y i egula i ies. Acco ding o he ISM obse a ions, index p is in he in e al
2.5 ≤p≤3.5. Expe imen al obse a ions o he aniso opic elec on densi y i egula i ies
in he F- egion o he ionosphe e show ha hey ha e powe –law spec um wi h a powe
index p [14,15]. The 3D spec al unc ion can be w i en as ollows:
Wn(k) = C2
n
(2π)3/2
3
0(k0 0)(p−3)/2
 0qk2+k2
0p/2
Kp/2 0qk2+k2
0
K(p−3)/2(k0 0),
whe e,
C2
n
is he mean-squa e de ia ion o elec on densi y,
Kν(x)
is McDonald unc ion,
0
is he inne scale o u bulence,
L0=
2
π/k0
is he ou e scale. In he in e al
k0 0<<
k 0<< 1 co ela ion unc ion is desc ibed by he o mula [14,15]:
Wn(k) = σ2
n
(2π)3/2
Γ(p/2)
Γ[ (p−3)/2 ]
kp−3
0
(k2+k2
0)p/2 , (24)
whe e Γ(x)is he gamma unc ion.
In analy ical calcula ions, we apply he powe –law spec al unc ion:
Wn(k) = C2
n
8π5/2
Apl3
||
χ2h1+l2
⊥(k2
x+k2
y) + l2
|| k2
zip/2 . (25)
whe e,
Ap=Γ(p/
2
)Γ[(5−p)/2 ]sin[(p−3)π/2]
,
χ=l||/l⊥
is he aniso opy ac o —
he a io o longi udinal and ans e se cha ac e is ic linea sizes o plasma i egula i ies.
The shape o elec on densi y i egula i ies has a sphe oidal o m due o di usion p ocesses
Elec onics 2023,12, 2759 8 o 11
in he ield align and ield pe pendicula di ec ions. Small-scale plasmonic s uc u es o he
ionosphe ic F egion a e mainly ield-aligned [16–19].
In nume ical calcula ions, we used expe imen al da a on ionosphe ic pa ame e s o
an al i ude o 300 km [12,13].
Figu e 1illus a es he de ia ion o ay ajec o ies o bo h O- and E-wa es (
∆h
in
me e s) ela i e o he di ec ion connec ing he sou ce and he ecei e e sus he dis-
ance be ween obse a ion poin s (in km-s) (see o mula (12). This igu e illus a es he
phenomenon o he “ oun ain e ec ” o he O- and E-wa es, which was e ealed in he
app oxima ion o geome ic op ics. In he a ea o he geomagne ic equa o , he geomagne ic
ield is almos pa allel o he Ea h’s su ace. A global empi ical ionosphe ic model IRI
(In e na ional Re e ence Ionosphe e) was used in nume ical calcula ions o he equa o ial
la i udes. The beha io o O- and E-wa es in he equa o ial egion is di e en . In his egion
o he ionosphe e, he plasma ises and g adually u ns no h in he no he n hemisphe e
and sou h in he sou he n hemisphe e, which is caused by an inc ease in he inclina ion o
geomagne ic lines o o ces on bo h sides o he geomagne ic equa o . As a esul , maxima
(o c es ) o elec on concen a ion a e o med on bo h sides o he geomagne ic equa o ,
i.e., no h and sou h c es s o equa o ial anomaly o c es s co esponding o o dina y and
unusual wa es. This e ec signi ican ly a ec s he ope a ion o adio communica ion, adio
na iga ion, loca ion, e c.
Figu e 1. Rela ion o O- and E-wa es ay pa hs de ia ion o obse a ion poin s.
Figu e 2demons a es he dependence o oo mean squa e (RMS) de ia ion o he
phase andom a ia ion e sus he RMS de ia ion o elec on concen a ion luc ua ions.
The phase spec al index p depends on he s a e o he ionosphe e. In a calm ionosphe e,
pa ame e p a ies wi hin in e al 1.7
≤p≤
2, while in a highly pe u bed ionosphe e, he
p alue can inc ease o 4 and e en o 6. A spec al index
p=
2, he a iance o he phase
luc ua ions o he ecei ed na iga ion adio signal is signi ican ly less han a spec al
index
p≥
3, and hei di e ence inc eases in p opo ion o he RMS de ia ion o elec onic
concen a ion luc ua ions in small-scale i egula i ies.
Figu e 3shows he 3D co ela ion unc ion o he phase luc ua ions as a unc ion
o dis ances be ween he obse a ion poin s o hogonal
(ηx)
and pa allel
(ηy)
o he
magne ic ield lines. Figu e 4illus a es he displacemen o he maximum o he SPS o he
O- (blue cu es) and E-wa es ( ed cu es) in he conduc i e collision magne ized plasma
a di e en pene a ion angle
θ
and aniso opy ac o
χ=
2. SPS o he O-wa e eaches
maximums a
L/l|| =
49,
θ=
5
0
; and
L/l|| =
71,
θ=
15
0
. In his case, O-wa es end o he
compensa ion di ec ion a
L/l|| ≈
260. SPS o E-wa e has maxima a
L/l|| =
60,
θ=
10
0
;
and
L/l|| =
51,
θ=
23
0
. The cu es co esponding o E-wa e each he compensa ion
di ec ion a
L/l|| ≈
210. The “Compensa ion E ec ” o he displacemen o he SPS o bo h
wa es akes place a a dis ance L/l|| ≈800.
Elec onics 2023,12, 2759 9 o 11
Figu e 2. RMS de ia ion o σϕs e sus RMS de ia ion o σn o di e en phase spec al index p.
Figu e 3. Th ee-dimensional co ela ion unc ion o he phase luc ua ions.
Figu e 4. Shi o maximum o he SPS a non-dimensional space pa ame e .
Figu e 5shows he “Compensa ion E ec ” o he b oadening o bo h wa es in he
conduc i e equa o ial e es ial ionosphe e a
θ=
5
◦
. The b oadening o he SPS o O-
wa e in he main plane has a maximum a
χ=
6, p opaga ing dis ance
L/l|| ≈
3.4
·
10
3
; he
b oadening ends o he compensa ion di ec ion a a dis ance
L/l|| ≈
8
·
10
3
. Fo E-wa e,
he b oadening o he SPS a he aniso opy ac o
χ=
6 has a maximum a
L/l|| ≈
2.2
·
10
3
and ends o he compensa ion di ec ion a
L/l|| ≈
10
4
. The “Compensa ion E ec ” o he
b oadening o he SPS o E-wa es is e ealed ea lie han o he O-wa es. Figu e 6depic s
he “Compensa ion E ec ” o he b oadening o bo h wa es in he conduc i e equa o ial
egion o he e es ial ionosphe e a
θ=
20
◦
. B oadening o he SPS co esponding o he
O-wa e eaches o he compensa ion di ec ion a small aniso opy ac o o elec on densi y
i egula i ies. The “Compensa ion E ec ” o
Σx
is e ealed a dis ance
L/l|| ≈
5
·
10
3
o
O-wa e. A a ying aniso opy ac o in he in e al
χ=
22
÷
25, he “Compensa ion E ec ”
is obse ed a dis ances L/l|| ≈2·10 4.