Transformation of the spatial spectrum of scattered radio waves in the conductive equatorial ionosphere

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.