Electromagnetic Wave Propagation in Ionospheric Plasma

Transcription

1 10 Elctromagntic Wav Propagation in Ionosphric Plasma Ali Yşil 1 and İbrahim Ünal 1 Fırat Univrsity, Elazığ İnönü Univrsity, Malatya Turky 1. Introduction Ionosphr physics is rlatd to plasma physics bcaus th ionosphr is, of cours, a wak natural plasma: an lctrically nutral assmbly of ions and lctrons [1]. Th ionosphr plays a uniqu rol in th Earth s nvironmnt bcaus of strong coupling procss to rgions blow and abov []. Th ionosphr is an xampl of naturally occurring plasma formd by solar photo-ionization and soft x-ray radiation. Th most important fatur of th ionosphr is to rflct th radio wavs up to 30 MHz. Espcially, th propagation of ths radio wavs on th HF band maks th ncssary to know th faturs and th charactristics of th ionosphric plasma mdia. Bcaus, whn th radio wavs rflct in this mdia, thy ar rflctd and rfractd dpnding on thir frquncy, th frquncy of th lctrons in th plasma and th rfractiv indx of th mdia and thus, thy ar absorbd and rflctd by th mdia. Th undrstanding of th xistnc of a conductiv layr in th uppr atmosphr has bn mrgd in a cntury ago. Th ida of a conductiv layr affctd by th variations of th magntic fild in th atmosphr has bn put forward by th Gauss in 1839 and Klvin in Nwfoundland radio signal from Cornwall to b issud by Marconi in 1901, at th first xprimntal vidnc of th xistnc of ionosphric, rspctivly. In 190, Knnly and Havisid indicatd that th wavs ar rflctd from a conductiv layr on th uppr parts of th atmosphr. In 190, Marconi statd that changing th conditions of night and day sprad. In 1918, high-frquncy band has bn usd by aircraft and ships. HF band in th 190s has incrasd th importanc of xpansion. Thn, put forward th thory of rflctiv conductiv rgion, has bn shown by xprimnts mad by Applton and Barnt. Th data from th 1930s startd to gt clarr about th ionosphr and Th Radio Rsarch Station, Cavndish Laboratory, th National Braun of Standards, th various agncis such as th Carngi Institution bgan to dal with th issu. In th scond half of th 0th cntury, th work of th HF lctromagntic wav has bn studid by dividd into thr as th fullwav thory, gomtrical optics and conductivity. Dspit initiation of widsprad us of satllit-earth communication systms, th us of HF radio spctrum for civilian and military purposs is incrasing. Collaps of communication systms, spcially in cas of mrgncy situations, communication is vital in this band. In th ionosphr, a balanc btwn photo-ionization and various loss mchanisms givs ris to an quilibrium dnsity of fr lctrons and ions with a horizontal stratifid

2 190 Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs structur. Th dnsity of ths lctrons is a function of th hight abov th arth s surfac and is dramatically affctd by th ffcts of sunris and sunst, spcially at th lowr altituds. Also, th many paramtrs in th ionosphr ar th function of th lctron dnsity. Th ionosphr is convntionally dividd into th D, E, and F-rgions. Th D-rgion lis btwn 60 and 95 km, th E-rgion btwn 95 and 150 km, and th F- rgion lis abov 150 km. During daylight, it is possibl to distinguish two sparat layrs within th F-rgion, th F1 (lowr) and th F (uppr) layrs. During nighttim, ths two layrs combin into on singl layr. Th combind ffct of gravitationally dcrasing dnsitis of nutral atoms and molculs and incrasing intnsity of ionizing solar ultraviolt radiation with incrasing altituds, givs a maximum plasma dnsity during daytim in th F-rgion at a fw hundrd kilomtrs altitud. During daytim, th ratio of chargd particls to nutral particls concntration can vary from10-8 at 100 km to 10-4 at 300 km and 10-1 at 1000 km altitud. Th main proprty of F-rgion consists of th fr lctrons. As known, th prmittivity and prmability paramtrs ar rlatd to lctric and magntic suscptibilitis of matrial, on account of mdium and morovr, th spd of lctromagntic wav and th charactristic impdanc dpnd on any mdium and th rfractiv indx of mdium givs dtail information about any mdium. Bcaus of all of ths rasons, ε is a masur of rfractiv indx, rflction, volum and wav polarization of lctromagntic, impdanc of mdium. Propagation of lctromagntic wavs in th atmosphr is influncd by th spatial distribution of th rfractiv indx of th ionosphr [3]. Th thory of ionosphric conductivity was dvlopd by many scintists and is now quit wll undrstood, though rfinmnts ar still mad from tim to tim. Th ionosphr carris lctric currnts bcaus winds and lctric filds driv ions and lctrons. Th dirction of th drift is at right angls to th gomagntic fild [4]. Furthrmor, lctrical conductivity is an important cntral concpt in spac scinc, bcaus it dtrmins how driving forcs, such as lctric filds and thrmosphr winds, coupl to plasma motions and th rsulting lctric currnts. Th tnsor of lctrical conductivity finds application in all th aras of ionosphric lctrodynamics and at all th latituds [5]. On th othr hand, th most important paramtr dtrmining th bhavior of any mdium is th dilctric constant, which at any frquncy dtrmins th rfractiv indx, th form of wav in mdium, to b polarizd, th stat of wav nrgy and th propagation of wav. In this chaptr, th bhavior of lctromagntic wavs mittd from within th ionosphric plasma and th analytical solutions ar ncssary to undrstand th charactristics of th nvironmnt will b dfind. Problms in plasma physics at th conductivity, dilctric constants and rfractiv indx will b dfind according to th mdia paramtrs. Whn ths xprssions, using Maxwll's quations xprssd in th wav disprsion quation, wav propagation, dpnding on th paramtrs of th nvironmnt will b xamind. Ths statmnts ar xprssd in trms of Maxwll's quations using th wav disprsion quation, wav propagation, dpnding on th paramtrs of th nvironmnt will b xamind. By xamining of th disprsion rlation, th typs of wav occurrd in th mdia and rlaxation mchanisms, polarizations and conflicts causd by ionosphric amplitud attnuation of ths wavs will b obtaind analytically. Thus, rsolving problms of ionosphric plasma in th mittd radio wavs, th basic information that will b undrstood.

3 Elctromagntic Wav Propagation n Ionosphric Plasma 191. Conductivity for ionosphric plasma On of th main paramtrs affcting th progrssion of th lctromagntic wav in a mdium is th conductivity of th mdia. Conductivity of th mdia statmnt is obtaind from motion quation of th chargd particl that is takn account in th total forc acting on chargd particl in ionosphric plasma. Accordingly, th forcs acting on chargd particl is givn as follows [6]. Mass xacclration = Elctrical forcs + Magntic forcs + Shooting gravitational forcs + Prssur changing forcs (1) + Collision forcs Th plasma approach, which th thrmal motion du to tmpraturs of particls, ar nglctd, is calld cold plasma. In this study, bcaus of th cold plasma approximation, any forc dos not ffct on chargd particls du to th tmpratur and thrfor th trm of prssur changs is ignord [7]. Likwis, th gravitational forc from th gravity is ngligibl du to so small according to th lctric and magntic forcs. In addition, du to th lctron mass (m ) is vry small according to th mass of th ion (m i ) (m <<m i ) and thus th lctron collision frquncy is gratr than th ion collision frquncy, th movmnt of ions in th plasma is ignord nar th lctron movmnt. Accordingly, Equation (1) is writtn as th following quation for th lctron: m dv dt ( V ) = E+ B m ν V () This xprssion is calld th Langvin quation [8]. Whr, th lctron collision frquncy ν is sum of th lctron-ion collision frquncy ( ν i ) and lctron-nutral particl collision ν frquncy. ( n ).1 DC conductivity Th first application of th Langvin quation is stady flow causd by of th lctric fild. For this, V = 0. Accordingly, th Langvin quation for lctrons can b dfind as follows: 0= E+ V B m νv (3) Taking into considration that th currnt dnsity is J = NV, in ordr to obtaind th currnt dnsity, both sids of th xprssion is multiplid by th trm of N. In this cas, and th xprssion is transformd as follows: N + = mn ν E V B V (4) N + =m E V B J (5)

4 19 Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs For th currnt dnsity, th following quation is obtaind: N ( ) J = E+ V B (6) m ν Taking into considration that J = σ E, th gnralizd Ohm Law can b dfind by, J = E+ V B (7) σ 0 N Whr, σ0 = is th dc conductivity of t plasma. If ν 0, th trm σ 0 is bing mν σ0. In this cas, if th E+ V B = 0, th currnt is finit. For this rason, th cas of σ 0 =, th stat E+ V B = 0 is calld OHM law for th plasma. Th gomtry of th vlocity and lctric fild and magntic fild is as shown in Figur 1, namly V = xv ˆ x + yv ˆ y + zv ˆ z, E = xe ˆ x + ye ˆ y + ze ˆ z and B = ẑb, th V B trm in Equation (7) is bing as follows: xˆ yˆ zˆ V B = V V V = xˆ V B y V B (8) ˆ x y z y x 0 0 B Fig. 1. Th gomtry of th vlocity, lctric fild and magntic fild Hr, by th ncssary mathmatical oprations ar mad; th following xprssions ar obtaind for thr componnts of currnt dnsity: N N NVB J E V B E y x = ( x + y ) = x + mν mν mν N N NVB J E -V B E - x y = ( y x ) = y mν mν mν (9) (10)

5 Elctromagntic Wav Propagation n Ionosphric Plasma 193 N z = z = 0 z mν J E σ E (11) B Taking into considration that cyclotron frquncy ωc = for th lctron, if th scond m trms on th right sid of this xprssion is writtn in trms of currnt dnsity J and th ncssary procdurs ar mad, x and y componnts of currnt dnsity ar obtaind as follows: Nν Nω J E E m ω m ω c x = x + y ( ν + ) c ν + c (1) Nω Nν J E E m ω m ω c y = x + y ( ν + ) c ν + c Th xprssions J x, J y and J z in th tnsor form of ( = σ ) follows: Nν Nω c m ω m ω ( ν + ) ( ν + ) c c Jx x E Nω c N J ν y - 0 y = E m( ωc ) m( ωc J ) ν + ν + z Ez N 0 0 (13) J E can b writtn as 0 mν Hr, th conductivity tnsor is calld as dc conductivity tnsor. As can b sn in this N xprssion, σ0 = trm provids th flow of th currnt in B dirction. If th mν conductivity tnsor is dfind as follows: (14) σ1 σ 0 σ= σ σ σ0 (15) Nυ σ 1 = Conductivity is calld as prpndicular conductivity ( σ ). Whn th E m( ν + ωc ) is prpndicular to th B, this conductivity provids to flow th currnt in th dirction of E. Nω c Th conductivity dfind by σ = m( ν + ωc ) provids th currnt, which is prpndicular to th both of th E and B (Figur ). is calld as Hall conductivity ( σ H ) and it

6 194 Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs Fig.. Th dirction of th currnts providd by th conductivitis σ0, σ1and σ according to th gomtry of E and B If th σ σ ν σ = = ν + ω c ωc 1 + ν and σ ω c σ0 σ0νωc = = ν ν + ω c ωc 1 + ν, th variation of th ωc conductivitis is bing as shown in Figur 3. As shown in Figur 3, aftr th 1 σ 1 trm ν dcrass rapidly. Thus, th currnt prpndicular to th magntic fild is vry small [8]. Fig. 3. Th variation of σ 1and σ conductivitis. AC conductivity Thr is no stady-stat in th ac conductivity. Taking into considration that th filds vary i t as ω, th Equation () is transformd as follows: -m iωv = - E+ V B m ν V (16)

7 Elctromagntic Wav Propagation n Ionosphric Plasma 195 According to th gomtry givn in Figur 1, this xprssion in trms of currnt dnsity can b dfind as follows: N N J( ν iω) = E m V m B (17) Thus, currnt dnsity componnts ar obtaind as follows: J J N N x = Ex + y m i m i ( ν ω) ( ν ω) N N ( ν ω) ( ν ω) V B y = Ey x m i m i J z = m N ( ν E z V B (18) (19) (0) Considring th ω c cyclotron frquncy, th componnts J x and J y can b obtaind as follows: ( ν ) + ( ν ) + ( ν ) N iω Nω J E E c x = x + y m ωc iω m ωc iω (1) ( ν Nω c N y = x + y m ωc + ν iω m ωc + ν iω J E E If th xprssions (0)-() in th tnsor form and th conductivity tnsor σ ar dfind as follows, () σ 1 σ J 0 x Ex J = σ σ 0 E y 1 y Jz 0 0 σ Ez 0 (3) σ 1 σ 0 σ = σ σ σ 0 Th conductivitis σ 0, σ 1 and σ will b N ( ν + ( ν ) (4) N 1 = m ωc iω σ 0 =, σ, m( ν Nω c σ =. Considring th oscillations frquncy of th plasma for th m ωc + ( ν

8 196 Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs lctron as ω p ( ν ( iω) ωp σ ε0 1 =, σ ωc + ν N =, ths quations can b writtn as mε0 ω ω p ε 0 c = ωc + ν ( iω) ωp ε0 σ =, ν 0 ( iω), rspctivly. Whr, as in dc conductivity σ 0 is th conductivity of th magntic fild dirction, σ 1 is th conductivity prpndicular to th magntic fild and σ is th conductivity prpndicular to both of th lctrical and th magntic filds, rspctivly. If σ 0 > σ 1 > σ and ( ν iω), th ac conductivity tnsor transforms to th dc conductivity tnsor. Exampl: ac conductivity tnsor according to th gomtry givn in Figur 4 is obtaind as in Equation (5): 0 ωp ε0 ν iω ωp ε ω 0 c ωc + ν iω ωc + ν iω ωp ε 0 σ= 0 0 ( ν ωp ε0ωc ωp ε0 ( ν 0 ωc + ( ν iω) ωc + ( ν If th magntic fild is only B = xb ˆ dirction, ac conductivity tnsor bcoms as follows: (5) ω p 0 ε 0 0 ( ν ωp ε0 ( ν iω ) ωp ε0ωc σ= 0 ωc + ( ν iω) ωc + ( ν ωp ε0ωc ωp ε0 ( ν 0 ωc + ( ν iω) ωc + ( ν (6) Fig. 4. Th gomtry of th vlocity and lctric fild and magntic fild

9 Elctromagntic Wav Propagation n Ionosphric Plasma 197 Exampl: If th magntic fild has two-dimnsional gomtry as givn in Figur 5 B = yb ˆ + zb ˆ = y ˆ BCosθ + z ˆ BSinθ, th conductivity tnsor ar dfind as in Equation (7): ( y z ) Fig. 5. Th gomtry of th vlocity and lctric fild and magntic fild σ 1 σ Sin θ σ Cosθ σ= σ Sin θ σ 0 + ( σ 0 σ 1) Sin θ σ ( 0 σ 1) CosθSinθ σ Cos θ ( σ 0 σ 1) CosθSin θ σ 0 + ( σ 0 σ 1) Cos θ (7).3 Earth s magntic fild and ionosphric conductivity In this sction, th Langvin quation dfind by Equation (16) for ac conductivity will b discussd. Howvr, th tru magntic fild in th Earth s northrn half-sphr givn in Figur 6 will b dalt with. Accordingly, in th slctd Cartsian coordinat systm, th x- axis rprsnts th gographic ast, th y-axis rprsnts th th gographic North and z-axis rprsnts th up in th vrtical dirction, th magntic fild can b dfind as follows [9]: B = x ˆ BCosISinD + y ˆ BCosICosD - z ˆ BSinI (8) Whr, I is th dip angl and D is th dclination angl (btwn th magntic north and th gographic north). Thus, th trm of V B in Equation (16) can b obtaind as in Equation (9). xˆ yˆ zˆ V B = V V V = xˆ -V BSinI - V BCosICosD x y z y z BCosISinD BCosICosD -BSinI ( x z ) ( VxBCosICosD - VyBCosISinD) yˆ -VBSinI-VBCosISinD If this xprssion is writtn in Equation (16) and th ncssary mathmatical manipulations ar mad, thr quations ar obtaind for x, y and z dirctions as follows: + ẑ (9) ω SinI ω CosICosD = + + m iω iω iω c c Ex Vx Vy Vz ( ν ) ( ν ) ( ν ) (30)

10 198 Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs ω SinI ω CosISinD = m iω iω iω c c Ey Vy Vx Vz ( ν ) ( ν ) ( ν ) ω CosICosD ω CosISinD = + m iω iω iω c c Ez Vz Vx Vy ( ν ) ( ν ) ( ν ) (31) (3) Fig. 6. Earth s magntic fild on north hmisphr In ordr to obtain th currnt dnsity, if th both sids of th xprssion is multiplid by N, th Equations (30)-(3) transform to Equations (33)-(35). N ω c SinI ω c CosICosD Ex = Jx + Jy + Jz ( ν ) ( ν ) ( ν ) m iω iω iω N ω c SinI ω c CosISinD Ey = Jx + Jy Jz ( ν ) ( ν ) ( ν ) m iω iω iω (33) (34) ω CosICosD ω CosISinD m iω iω iω N c c Ez = Jx + Jy + Jz ( ν ) ( ν ) ( ν ) Ths quations ar first ordr linar quation in thr unknowns. It is impossibl to obtain this xprssion in th solution of ach othr. Thrfor, whthr or not th solution, it is ncssary to xprss by using, Cramr s Mthod. This mthod givs th solution of linar (35)

11 Elctromagntic Wav Propagation n Ionosphric Plasma 199 quation systm which has cofficints matrix as squar matrix. According to this mthod, if th dtrminant of th cofficints matrix of th systm is non-zro, th quation has a singl N ω c SinI solution. Lt accpt th =σ 0, m ν iω ν iω = a ω c CosICosD, = b and ν iω ( ν ( ) ( ) ( ) ω c CosISinD = c in Equations (33)-(35). Thus, ths tr quations can b dfind as follows: σ 0E x 1 a b Jx 0Ey a 1 c Jy σ = σ 0Ez b c 1 Jz Hr, th cofficints matrix is namd as A, by taking th dtrminant of this matrix, th Equation (37) can b obtaind. (36) 1 a b dta = a 1 c = 1 + a + b + c 0 b c 1 (37) This tlls us that on solution of th quation. According to Cramr solution is as follows: dta1 dta dta3 J x =, J y =, Jz = (38) dta dta dta Hr, th trms of A, 1 A and A 3 ar dfind as follows: σ E a b 1 σ E b 1 a σ E dta =σ E 1 c, dta = a σ E c, dta = a 1 σ E σ E c 1 b σ E 1 b c σ E 0 x 0 x 0 x 1 0 y 0 y 3 0 y 0 z 0 z 0 z (39) In this statmnt, th x dirction currnt dnsity is rsolvd as in Equation (40). J x ( 1 c ) σ0( bc a) σ0( ac b) dta σ = = E + E + E dta 1+ a + b + c 1+ a + b + c 1+ a + b + c x y z For th a, b and c, if th instad of th xprssions givn abov ar writtn, th J x trm is obtaind as follows: J σ 0 ( ν iω ) +ω c Cos ISin D 0 c Cos ICosDSinD - ( i ) csini σ ω ν ω ω = E + E ( v ω i ) +ω c ( ν ω i ) +ω c σ 0 -ωc CosISinISinD - ( ν ωccosisind + Ez ( ν iω ) +ωc x x y (40) (41)

12 00 Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs By th sam way, y dirction th currnt dnsity is rsolvd as follows: dta σ 0 0 a+ bc σ ( 1+ b ) σ 0( c ab) J y = = Ex + Ey + Ez dta 1+ a + b + c 1+ a + b + c 1+ a + b + c and th J y trm is obtaind as follows: ( iω) ω Cos ICos D σ ν + J E E ( ν iω) + ω c ( ν iω) + ω c σ 0 ( ν iω ) ωccosisind ω c CosISinICosD + Ez ( ν iω) + ω c Likly, z dirction th currnt dnsity is rsolvd as follows: σ0 ν iω ωcsini + ω c Cos ICosDSinD 0 c y = x + y ( b ac) ( c ab) σ 0 ( 1+ a ) dta3 σ 0 σ 0 Jz = = Ex + Ey + Ez dta 1+ a + b + c 1+ a + b + c 1+ a + b + c and th J z trm is obtaind as follows: (4) (43) (44) σ 0 ν iω ωccosicosd ω c CosISinISinD Jz = Ex ( ν iω) + ω c 0 ( ) c c 0 ( iω) ω csin I σ ν iω ωcosisind ω CosISinICosD σ ν + + Ey + E ( v iω) + ω c ( ν iω) + ω c Aftr bing currnt dnsitis obtaind in this mannr, J x, J y and J z trms can b ditd again by considring th σ 1 an σ conductivitis. Also, J x, J y and J z trms can b writtn in th form of tnsor lik th following: z (45) Jx σ11 σ1 σ13 Ex Jy 1 3 Ey = σ σ σ Jz σ31 σ3 σ33 Ez Th conductivity tnsor can dfind as in Equation (47): σ11 σ1 σ13 σ = σ1 σ σ3 σ31 σ3 σ33 and tnsor componnts can b achivd in a simplr as follows [9, 10]: (46) (47)

13 Elctromagntic Wav Propagation n Ionosphric Plasma 01 σ = σ + σ σ Cos I Sin D σ = σ SinI + σ σ Cos ICosDSinD σ = σ CosICosD σ σ CosISinISinD σ = σ SinI + σ σ Cos ICosDSinD σ = σ + σ σ Cos ICos D σ = σ CosISinD σ σ CosISinICosD σ = σ CosICosD σ σ CosISinISinD σ = σ CosISinD σ σ CosISinICosD σ = σ + σ σ Sin I Dilctric constant for ionosphric plasma Dilctric constant for ionosphric plasma could b foundd by using Maxwll quations. 1) ρ E = 4π ρ = (48) ε 0 ) B = 0 (49) 3) B E = (50) t 1 E 4) B= + 0J (51) c t 1 Whr c= = 3 10 ε is th light spd. According to fourth Maxwll quation, and if th lctric fild is accptd th form chang ω i t E B= 0ε 0 + 0J (5) t thn, From hr i t ( 0 ω ) B= 0ε 0 E + 0J (53) t

14 0 it is obtaind as follow or Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs σe B= iωµ 0ε0E iωε0 (54) σ B= iωµ 0ε01 E iω ε0 (55) σ B= iωµ 0Eε0 1 iω ε (56) 0 According to th latst s quation, th dilctric constant of any mdium σ ε=ε0 1 iω ε0 In which, bcaus of ( σ ) th tnsorial form 1 is th unit tnsor. (57) = Gnrally, th xprssion of ionosphric conductivity σ givn in Equation (47) by using Equation (57), th dilctric structur of ionosphric plasma is shown by (58) σ11 σ1 σ13 1 ε=ε iω σ σ σ ε σ31 σ3 σ33 (59) 4. Th rfractiv indx of th cold plasma Th rfractiv indx (n) dtrmins th bhavior of lctromagntic wav in a mdium and of rfractiv indx of th mdium is foundd by using Maxwll quations. If th curl th Equation (50) is takn, If ( B) trm in this xprssion is r-writtn in (60), E= ( B) (60) t iσ E=µ 0εω 0 1+ E (61) εω 0

15 Elctromagntic Wav Propagation n Ionosphric Plasma 03 i( t) Sinc th lctric fild E varis as kr ω, it can b assumd that = i k. In this cas, th lft sid of th Equation (61) can b dfind as follows: E= k E-k k E (6) If th Equations (61) and (6) ar rarrangd, thn Equation (63) can b dfind as follows: ω iσ k E-k( k E) = 1+ E (63) c εω 0 If th wav vctor k in this quation is writtn in trms of rfractiv indx n, th Equation (63) can b dfind in trms of rfractiv indx as follows: ω k = n (64) c iσ n E-n( n E) = 1+ E (65) εω 0 If th ncssary procdurs ar usd, th Equation (66) is obtaind as follows: i 0 n 0 E = σ E n 0 0 Ex Ex y y εω Ez Ez (66) By writing th conductivity tnsor σ in th Equation (47) instad of th conductivity in th Equation (66), a rlation for th cold plasma is obtaind as follows: iσ11 iσ1 iσ13 n 1 εω εω εω Ex iσ1 iσ iσ 3 y n 1 E = 0 εω 0 εω 0 εω 0 Ez iσ31 iσ3 iσ 33 1 εω 0 εω 0 εω 0 Hr, sinc th lctric filds do not qual to zro, th dtrminant of th matrix of th cofficints quals to zro. So: (67) iσ11 iσ1 iσ13 n 1 εω 0 εω 0 εω 0 iσ1 iσ iσ 3 n 1 = 0 εω 0 εω 0 εω 0 iσ31 iσ3 iσ 33 1 εω 0 εω 0 εω 0 (68)

16 04 Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs Th matrix givn by Equation (68) includs th information about th propagation of th wav. Sinc th matrix has a complx structur, it is impossibl to solv th matrix in gnral. Howvr, th numrical analysis can b don for th matrix. Thrfor, solutions should b mad to crtain conditions, in trms of convninc. 4.1 k//b condition: plasma oscillation and polarizd wavs Whn th progrss vctor of th wav (k) is paralll or anti-paralll to th arth s magntic fild or th any componnts of th arth s magntic fild, two cass can b obsrvd for th ionosphric plasma dpnding on th rfractiv indx of th mdium. For xampl, if th wav propagats in th z dirction as in th vrtical ionosondas, th vrtical componnt of th arth s magntic fild ffcts to th propagation of th wav. In this cas, two wavs occur from th solution of th dtrminant givn by Equation (68). Th first on is th vibration of th plasma dfind as follows: Th scond on is th polarizd wav givn by Equation (70) [9]. ω p =ω 1 n (69) ( ) ( ) ( ) X1 Y X n p =1- +iz 1 Y +Z 1 Y +Z (70) Th signs ± in th Equation (70) rprsnts th right-polarizd (-) and lft-polarizd (+) p wavs, rspctivly. In this quation, X = ω, Y = ωc and Z = ν. Th magntic fild in ω ω ω th xprssion Y is th cyclotron frquncy causd by th z componnt of th arth s ωc magntic fild Y= SinI. This quation is th complx xprssion. Sinc th rfractiv ω indx of th mdium dtrmins th rsonanc (n ) and th cut-off (n =0) conditions of th wav, it is th most important paramtr in th wav studis. If it is takn that Z=0, th quation bcoms simpl. Th rsonanc and th rflction conditions of th polarizd wav bcom diffrnt from th cas of Z k B condition: ordinary and xtra-ordinary wavs Whn th propagation vctor of th wav (k) is prpndicular to th arth s magntic fild, two wavs occur in th ionosphric plasma [9]. Th first wav is th ordinary wav givn as follows: X X no = 1 + iz 1+ Z 1+ Z This wav dos not dpnd on Earth's magntic fild. Howvr it dpnds on th collisions. Th collisions can chang th rsonanc and th rflction frquncis of th wav. Th scond wav is th xtra ordinary wav dpnding on th magntic fild. Th rfractiv indx of th xtra ordinary wav can b dfind as follows: (71)

17 Elctromagntic Wav Propagation n Ionosphric Plasma 05 ax 1 X + Z X X X 1 X X ax nx = 1 + iz a + b a + b Whr a 1 X Y Z b= Z X. This rlation is valid for th y dirction. So, th magntic fild in th Y is th cyclotron frquncy causd by th y componnt of th arth s magntic fild. By th sam way, th wav dfind by this rlation is also obsrvd in th x dirction. For th xtra ordinary wav in th x dirction, th cyclotron frquncy, that is containd in Y, is th cyclotron frquncy causd by th x componnt of th arth s magntic fild. = and 4.3 Th Binom xpansion Z 1- Z and th rfractiv indics Th solutions of th rfractiv indics mntiond abov ar complx and difficult. Th solutions can b obtaind by using th binom xpansion. Accordingly, th rfractiv indics can b obtaind by using th Binom xpansion and th ral part of rfractiv indics as follows: 1. For th right-polarizd wav givn by Equation (70): (7) ( ) ( ) p X 1 X Z 4 3X µ + for X 1 41 X X µ p Z for X 1 4X 1 ( ) (73) (74) X Z Whr, X = and Z =. 1- YSinI 1- YSinI. For th xtra ordinary wav givn by Equation (71): ( 3X) ( ) X 4 µ o ( 1 X) + Z for X1 41 X X µ o Z for X1 4 X 1 ( ) (75) (76) 3. For th xtra ordinary wav givn by Equation (7): µ x 1 X Y Cos I Cos D 1 X Y Cos I Cos D ( ) X ( 1 X) + Y Cos I Cos D + Z X Y Cos I Cos D ( 1 X) Y Cos I Cos D (77)

18 06 Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs 5. Rlaxation mchanism of cold ionosphric plasma 5.1 Charg consrvation Maxwll addd th displacmnt currnt to Ampr law in ordr to guarant charg convrsation. Indd, talking th divrgnc of both sids of Ampr s law and using Gauss s law D =ρ, w gt: D ρ H= J+ = J+ D= J+ (78) t t t Using th vctor idntity H = 0, w obtain th diffrntial from of th charg convrsation law: ρ J+ = 0 (charg consrvation) (79) t Intgrating both sids ovr a closd volum V surroundd by th surfac S, and using th divrgnc thorm, w obtain th intgratd d J d S = ρ dv dt (80) S Th lft-hand rprsnts th total amount of charg flowing outwards through th surfac S pr unit tim. Th right-hand sid rprsnts th amount by which th charg is dcrasing insid th volum V pr unit tim. In othr words, charg dos not disappar into (or gt cratd out of) nothingnss-it dcrass in a rgion of spac only bcaus it flows into othr rgions. V n J Fig. 7. Flux outwards through surfac Anothr consqunc is that in good conductors, thr cannot b any accumulatd volum charg. Any such charg will quickly mov to th conductor s surfac and distribut itslf such that to mak th surfac into an quipotntial surfac. Assuming that insid th conductor w hav D=ε E and J=σ E, w obtain σ σ J=σ E= D= ρ (81) ε ε

19 Elctromagntic Wav Propagation n Ionosphric Plasma 07 Thrfor, with solution: dρ dt σ + ρ = ε 0 (8) t 0() σ τ ρ (,t) r =ρ r (83) Whr, ρ0( r ) is th initial volum charg distribution. Th solution shows that th volum charg disappars from insid and thrfor it must accumulat on th surfac of th conductors. For xampl, in coppr, 1 ε τ rl = = = 7 σ sc By contrast, τ rl is of th ordr of days in a good dilctric. For good conductors, th abov argumnt is not quit corrct bcaus it is basd on th stady-stat vrsion of Ohm s law, which must b modifid to tak into account th transint dynamics of th conduction chargs. It turn out that th rlaxation tim τ rl is of th collision tim, which is typically sc. (84) 5. Charg rlaxation in conductors W discuss th issu of charg rlaxation in good conductors. Writing thr-dimnsionally and using, Ohm s law rads in th tim domain: t α(t t ) p 0 J(r,t) =ω ε E(r,t )dt (85) Taking th divrgnc of both sids and using charg convrsation, J+ρ = 0, and Gauss s law, ε0 E =ρ,w obtain th following intgro-diffrntial quation fort h charg dnsity ρ( r,t): t t α(t t ) α(t t ) p 0 p ρ ( r,t) = Jr (,t) =ω ε E ( r,t )dt =ω ρ( r,t )dt (86) Diffrntiating both sids with rspct to, w find that ρ satisfis th scond-ordr diffrntial quation: whos solution is asily vrifid to b a linar combination of: ρ ( r,t) +αρ ( r,t) +ω ρ ( r,t) = 0 (87) p ω, αt Cos( rlt) αt Sin( rlt) ω (88) α Whr ω rl = ωp. Thus th charg dnsity is an xponntially dcaying sinusoid with 4 a rlaxation tim constant that is twic th collision tim τ= 1 α:

20 08 Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs τ rl = = τ (rlaxation tim constant) (89) α Typically, ωpα, so that ω rl is practically qual to ω p. For xampl, using th numrical data of xampl, w find for coppr τ rl = τ= sc. W calculat also: 15 fl =ωrl π=.6 10 Hz. In th limit α or τ 0, rducs to th naiv rlaxation. In addition to charg rlaxation, th total tim dpnds on th tim it taks fort h lctric and magntic filds to b xtinguishd from th insid of th conductor, as wll as th tim it taks fort h accumulatd surfac charg dnsitis to stl, th motion of th surfac chargs bing dampd bcaus of ohmic losss. Both of ths tims dpnd on th gomtry and siz of th conductor. Th componnts of conductivity hav givn by Equation (47) and th prmittivity of plasma has givn by Equation (59). As th conductivity of plasma has a tnsor form, th prmittivity of plasma is also a tnsor. Morovr, th prmittivity of plasma is frquncy dpndnt and ach componnt of th prmittivity tnsor has ral and imaginr part. Consquntly, th prmittivity of plasma dpnds on th conductivity of plasma. Th prmittivity of plasma could b xprssd as follow. ε ε ε ε= ε ε ε ε ε ε Th ral and imaginary part of th prmittivity is an indication of th ohmic powr loss. Anisotropy is an inhrnt proprty of th atomic/molcular structur of th dilctric. It may also b causd by th application of xtrnal filds. For xampl, conductors and plasmas in th prsnc of a constant magntic fild -such as th ionosphr in th prsnc of th Earth s magntic fild- bcom anisotropic. Th rlaxation tim in th ionosphric plasma has tnsorial form as follow. τ τ τ τ= τ τ τ τ τ τ Th rlaxation tim in th ionosphric plasma is diffrnt at vry dirction and hav ral and imaginary parts [11]. (90) (91) 6. Ionosphric absorption and attnuation of radio wav Whn th radio wavs propagat in th ionosphric plasma, thy xhibit diffrnt bhaviors rlatd to thir wav frquncy, oscillation frquncy of th lctrons in th plasma mdium and th rfractiv indx of th mdium. Dpnding on ths bhaviors, th wav is rfractd, rflctd or attnuatd by absorption from mdium. Radio-wav damping is du to movmnts in th ionosphr of lctrons and ions ar causd by collisions with othr particls [1]. Du to th incras of collisions, absorption incrass and fild strngth of th radio-wav dcrass. As a rsult of this, amplitud of th radio-wav propagatd in th ionosphr will dcras bcaus of th absorption. Thus, th ral part of th rfractiv

21 Elctromagntic Wav Propagation n Ionosphric Plasma 09 indx ffcts to th phas vlocity and th imaginary part of th rfractiv indx is associatd with spatial attnuation of th wav. In accordanc with th abov information, in ordr to obtain th rlation of th attnuation, amplitud of th wav, such as th lctric fild strngth, is nd to b xprssd dpnding on th rfractiv indx of th mdium. Accordingly, th lctric fild strngth of th wav can b dfind as follows: E= i( ωt) 0 kr E Hr, th wav vctor k is writtn in trms of rfractiv indx n, Equation (9) bcoms as follows: (9) ω ω c i t E= E nr 0 (93) Th rfractiv indx in th ionosphric plasma is also dfind as in quation (93). n = µ + iχ (94) whr, µ and χ rprsnt th ral and th imaginary part of th rfractiv indx, rspctivly. Th collision of th lctron with th othr particls ffcts to th ral and th imaginary part of th rfractiv indx. In th High Frquncy (HF) wavs, Z is vry smallr than 1 ( Z 1). Thrfor, Z can b dfind as 1 1+ Z 1 Z. Th ral and th imaginary part of th rfractiv indx and th phas vlocity of th polarizd ordinary and xtraordinary wav can b dtrmind by using this volution. Accordingly, if th rfractiv indx givn by Equation (94) is writtn in Equation (93), th lctric fild strngth can b obtaind as follows: ω r ω ω χr c c i t µ E= E 0 (95) Th part of th damping in th lctric fild strngth in Equation (95) is th xponntial trm rlatd to scond χ in th right sid of th quation. Thus, th attnuation of th wav is rprsntd by χ. According to this, th rfractiv indxs of th polarizd, ordinary and xtra-ordinary wavs givn (70)-(7) quations for cold plasma could b showd as th ral and imaginary parts as follow. n = F+ ig (96) Accordingly, th lctric fild strngth givn in Equation (95) can b dfind as follows: 1 1 F G F ω ± + r c ω i r ω t µ c E= E0 (97) In th Equation (97), signs (+) and (-) in front of th first xponntial xprssion rprsnts th wav propagatd to upward ( + ẑ ) and downward ( ẑ ) aftr th rflction, rspctivly.

22 10 Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs Hr, by considring th lctromagntic wav that propagatd to upward ω χr c ω χz c + ẑ, th trm can b dfind as. Thrfor, th damping trm of th wav amplitud is dfind with th hight as follows: 1 1 F G F ω ± + z c ω i z ω t µ c E= E0 (98) Th scond trm in quation (98) rprsnts how much th attnuation of wav in ionosphric plasma. 7. Conclusion Ionosphric plasma has doubl-rfractiv indx and gnrally wak conductivity in vry dirction, sason and local tim. Du to ths, th ionosphric paramtrs such as conductivity, dilctric constant and th rfractiv indx ar diffrnt in vry dirction and hav th complx structur (both th ral and imaginary part). This shows that th diagonal lmnts of conductivity, rfractiv indx and dilctric tnsor which th conductivitis dominat hav gnrally highr conductivity than othr lmnts; bsids any lctromagntic wav propagating in ionosphric plasma could b sustaind attnuation in vry dirction. So this attnuation rsults from th imaginary part of th rfractiv indx. 8. Symbol list m : Mass of lctron m i : Mass of ion V : Vlocity of lctron t : Tim - : Charg of lctron E : Elctric fild intnsity B : Magntic induction : Elctron collision frquncy i : Elctron-ion collision frquncy n : Elctron-nutral particl collision frquncy J : Currnt dnsity N : Elctron dnsity σ : Conductivity ω c : Elctron cyclotron frquncy ω : Wav angular frquncy ω : Elctron plasma frquncy ε p 0 I D : Dilctric constant of fr spac : Dip angl : Dclination angl : Dll oprator

23 Elctromagntic Wav Propagation n Ionosphric Plasma 11 ρ : Charg dnsity µ 0 : Magntic prmability of fr spac c : Spd of light ε : Dilctric constant of mdium 1 : Unit tnsor k : Elctromagntic wav vctor r : Displacmnt vctor n : Rfractiv indx µ : Ral part of rfractiv indx µ : Ral part of rfractiv indx of polarizd wav p µ o : Ral part of rfractiv indx of ordinary wav µ x : Ral part of rfractiv indx of xtra-ordinary wav D : Displacmnt currnt H : Magntic fild intnsity S : Surfac V : Volum τ : Collision tim τ rl : Rlaxation tim α : Th masur of th rat of collisions pr unit tim f : Linar frquncy χ : Imaginary part of rfractiv indx 9. Rfrncs [1] H. Rishbth, A rviv of ionosphric F rgion thory, Proc. of th IEEE., Vol.55(1), pp.16-35, [] H. Rishbth and M. Mndillo, Pattrns of F-layr variability, J. Atmos. Solar-Trr. Phys., Vol.63(15), pp , 001. [3] M. Grabnr and V. Kvicra, Rfractiv indx masurmnts in th lowst troposphr in th Czch Rpublic, J. Atmos. Solar-Trr. Phys., Vol.68(1), pp , 006. [4] H. Rishbth, Rviw papr:th ionosphric E layr and F layr dynamos a tutorial rviw, J. Atmos. Solar-Trr. Phys., Vol.59(15), pp , [5] R. Dlla and J. Dvin, A graphical intrprtation of th lctrical conductivity tnsor, J. Atmos. Solar-Trr. Phys., Vol.67(4), pp , 005. [6] H. G. Bookr, Cold Plasma Wavs. Martinus Nijhoff Publishrs, Dordrcht-Nthrlands, [7] R. O. Dndy, Plasma Dynamics. Clarndon Prss, Oxford, [8] B. S. Tannbaum, Plasma Physics. McGraw-Hill Book Company, Nw York [9] M. Aydoğdu, A. Yşil and E. Güzl, Th group rfractiv indics of HF wavs in th ionosphr and dpartur from th magnitud without collisions, J. Atmos. and Solar-Trr. Phys. Vol. 66(5), pp , 004. [10] M. Aydoğdu, E. Güzl, A. Yşil, O. Özcan and M. Canyılmaz, Comparison of th calculatd absorption and th masurd fild strngth of HF wavs rflctd from th ionosphr, Il Nouovo Cimnto, Vol.30C, No.3, pp.43-53, 007.

24 1 Bhaviour of Elctromagntic Wavs in Diffrnt Mdia and Structurs [11] A. Yşil, and İ. Ünal, Th ffct of altitud and sason dilctrical rlaxation mchanism of ionosphric plasma, Il Nouovo Cimnto, Vol.14B, No.7, pp , 009. [1] J. A. Ratcliff, Th Magnto-Ionic Thory and its Applications to th Ionosphr. Cambridg Univrsity Prss, 1959.

25 InTch Europ Univrsity Campus STP Ri Slavka Krautzka 83/A Rijka, Croatia Phon: +385 (51) Fax: +385 (51) InTch China Unit 405, Offic Block, Hotl Equatorial Shanghai No.65, Yan An Road (Wst), Shanghai, 00040, China Phon: Fax: