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ARTICLE IN PRESS

Journal of Atmospheric and Solar-Terrestrial Physics 67 (2005) 921- 930 www.elsevier.com/locate/jastp

Gyrotropic waves in the mid-latitude ionosphere
V.M. Sorokina,Ã, O.A. Pokhotelov
a

b

Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation, Troitsk, 142190 Moscow Region, Russia b Institute of Physics of the Earth, 123995 Moscow, Russia Received 16 November 2004; received in revised form 16 November 2004; accepted 25 February 2005 Available online 24 May 2005

Abstract The propagation of electromagnetic ULF perturbations in the thin conductive ionospheric layer in the magnetic meridian plane is considered. The dispersion relation for the waves propagating in the horizontal direction is obtained accounting for the influence of the conjugate ionosphere. The dependence of the phase velocity and absorption coefficient as the function of the wave frequency and magnetic field inclination are found. It is shown that in the frequency range 0.001-1 Hz the phase velocity increases from a few units to a few tenths of km/s depending on the frequency and so far follows the approximate dependence o1=3 . The phase velocity decreases with the increase in the magnetic field inclination. The propagation of quasi-harmonic and unipolar impulses of gyrotropic waves in the horizontal direction is analyzed. The dependence of their characteristics on the values of conductivity tensor components and on the magnetic field inclination is evaluated. The generation of gyrotropic waves by the electromagnetic field of atmospheric origin in the presence of ionospheric inhomogeneities is considered. The calculation of the power spectra of the geomagnetic field oscillations at the ground level is carried out. The dependence of the spectrum on the magnetic field inclination is obtained. Some features of geophysical phenomena, associated with propagation of gyrotropic waves in the low ionosphere are discussed. r 2005 Elsevier Ltd. All rights reserved.
Keywords: Gyrotropic waves; Mid-latitude ionosphere; Electric fields in the ionosphere

1. Introduction A long time ago Herron (1965) using the data of midlatitude geophysical observatories pointed out that Pi2 and Pc3 geomagnetic pulsations can propagate along the Earth's surface with the phase velocity of the order of a few tenths of km/s. The peculiarities of spatiotemporal distribution of the Pi2 pulsations at midlatitudes have been extensively studied by Gokhberg
ÃCorresponding author. Tel.: +7 95 330 9902; fax:+7 95 334 0124. E-mail address: sova@izmiran.ru (V.M. Sorokin).

et al. (1973). These authors have estimated the pulsation group velocity along the Earth's surface and found the value of the 40-300 km/s. Gogatishvili (1979) provided the analyses of the spatio-temporal distribution of Pc5, Pc6 pulsations with periods 5-8 min. He also pointed out the phase shift that was of the order of 10-15 km/s. A possible influence of the ionosphere on the propagation of geomagnetic pulsations has been discussed by Piddington (1959). This author came to conclusion that strong atmospheric activity can serve as source for generation of specific ionospheric perturbations and geomagnetic pulsations propagating with the phase velocities of the same order. Analyses of magnetograms

1364-6826/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jastp.2005.02.015


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carried out by some researches (Sorokin and Fedorovich, 1982a; Alperovich et al., 1982; Alperovich and Zheludev, 1999; Ismaguilov et al., 2001) provided the evidence for the geomagnetic pulsation generation by earthquakes propagating from the epicenters along the Earth's surface with the velocities from a few units to a few tenths of km/s. The greater the signal period the smaller the velocities. Similar velocities of the ionospheric perturbations have been observed in the ``MASSA'' experiment (e.g., Alperovich et al., 1985) and during rocket launches (Karlov et al., 1984). Sorokin and Fedorovich (1982a, b) and Sorokin (1988) have attributed such ionospheric and geomagnetic perturbations to the so-called gyrotropic waves. These waves (with frequencies 0.001-1 Hz) can propagate in the conductive ionospheric layer with velocities 10-100 km/s possessing small damping and dispersion. Outside the conductive layer the wave electric field produces a strong electron motion and thus the waves are observed in the form of the ionospheric perturbations. Recently Sorokin et al. (2003) developed a model for the formation of the narrow band ULF spectrum near the Earth's surface by gyrotropic waves driven by thunderstorm activity. Similar effects can display themselves in seismoactive regions as the result of the increase in the electric field (Sorokin et al., 1998; Chmyrev et al., 1999) or due to acoustic-gravity waves generated by seismic sources (Mareev et al., 2002). The mechanism of formation of the geomagnetic pulsation spectra can be used for monitoring of seismic activity (Kopytenko et al., 2002; Hayakawa et al., 1996). All the above-mentioned geophysical phenomena are associated with generation and propagation of the gyrotropic waves which may be used for relevant geophysical interpretation. The present paper represents the generalization of the theory of gyrotropic waves accounting for the effects of oblique external magnetic field and the effects due the influence of the conjugate ionosphere. The paper is organized in the following fashion: Section 2 is devoted to derivation of general dispersion relation for the gyrotropic waves. The spatio-temporal characteristics of these waves are analyzed in Section 3. In Section 4 the role of the magnetic field inclination in the formation of geomagnetic pulsation spectra is investigated. The Appendix gives the details of our calculations. Our discussion and conclusions are found in Section 5.

by (cf. Sorokin et al., 2003)
u vp ? 2u cos ju vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u oð1 þ g2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , t oi 2 cos j 1 þ g2 þ 4 cos2 j À g2 sin4 j

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ? B= 4pMN , gð1 þ cos2 jÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 cos j 1 þ g2 þ 4 cos2 j À g2 sin4 j ne oi þ, g? oe nin ? l ? 2pd where u stands for the Alfven velocity, M and N are the mass and the ion number density, respectively, oe and oi the electron and the ion gyrofrequencies, ne and nin the electron and the ion collision frequencies, l the wavelength, d the distance at which the field decreases in e-times, j the angle between the direction of the wave propagation and the external magnetic field B. This expression is valid when o5oi N =N m , where N m is the neutral particle number density. In the E-layer go1 and thus the gyrotropic wave phase velocity is much smaller than the Alfven velocity. Moreover, the waves possess a small damping. Contrary to the Alfven waves, the phase velocity of these waves strongly depends on the collision frequencies. These specific waves are localized in the region where the electrons are magnetized whereas the ions not. The gyrotropic waves can be described in terms of the conductivity tensor of partly ionized plasma (Sorokin et al., 2003). They possess small damping if the off-diagonal elements of the dielectric tensor are greater than the diagonal terms. For the sake of simplicity we obtain the dispersion relation for these waves in a limiting case of a thin conductive E-layer when the wavelength is greater than the depth of the layer. The electric E and magnetic field b perturbations can be found from Maxwell equations r E ? À 1 qb 4p 1 qE ; r b ? j þ , c qt c c qt rÁ E ? 4pq; rÁ b ? 0,

ð1Þ

where q is the electric charge density, j is the electric current. Eqs. (1) should be supplemented by the generalized Ohm's law given by & ' BðE Á BÞ BðE Á BÞ BÂE j ? s11 . (2) þ sH þ sP E À B B2 B2 Here s11 stands for the field-aligned conductivity, sP and sH are the Pedersen and Hall conductivities. Furthermore, we introduce a local Cartezian system of coordinates (x,y,z) with the longitudinal x and latitudinal y coordinates. The z-axis of our system coincides with local vertical direction. A uniform external magnetic field B lies in the (x,z) plane and directed under the angle j to the x-axis. The ionosphere is

2. Dispersion and damping of gyrotropic waves in the mid-latitude ionosphere In a uniform partly ionized plasma the phase velocity vp of the gyrotropic waves and their damping are given


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assumed to be plane with the electric conductivity that might depend on the z-coordinate. Let us consider the propagation of the electromagnetic field in the (x,z) plane assuming q=qy ? 0. Due to the high mobility of the electrons along the external magnetic field s11 5sP ; sH . Thus, setting in Eq. (2) s11 ! 1 we find ðE Á BÞ ? 0. The latter gives E x ? ÀE z tan j From Eqs. (1) and (2) one perpendicular components conductive ionosphere 1 q2 D À 2 2 ðB Â EÞÀ B Â c qt & 4p qE ? 2 sP B Â À sH B c qt (3) finds the equation for the of the electric field in the

thin ionosphere in the form (see Appendix, Eq. (A.5)) & ' dE y o2 þ inok2 cos2 j þ E y ð0Þ dz z?0 k2 a2 l cos2 j 2o sin j dE z ðz ? À0Þ ?À ; fE y gz?0 ? 0; ka cos2 j dz & ' dE z no ? Ài E z ðz ? þ0Þ, dz z?0 a2 l sin2 j o E y ð0Þ. ð6Þ fE z gz?0 ? À 2ka sin j Here the following abbreviations are , sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z1 2 dzs2 ðzÞ; l s2 ð0Þ ? 4p l a?c H H
À1

rðr Á EÞ ' qE . qt

1

dzs2 ðzÞ, H ð7Þ

À1

ð4Þ

n ? a2 l =aP .

With the help of Eqs. (4) and (3) one finds the system of equations for the electric field components 2 qE y q q2 q2 4p sH qE z À þ 2 À 2 2 E y ? 2 sP c qx2 qz c qt qt cos j qt " # 2 2 q q q À 2 2 Ez sin j þ cos j qz qx c qt qE y 4p qE z þ sH cos j . ð5Þ ? 2 sP c qt qt The altitude distribution of sP and sH is depicted in Fig. 1. Since the wavelength is much greater than the typical scale of the conductivity inhomogeneity one can replace these equations by the relevant boundary conditions at the E-layer and obtain the solution of the system (5). Outside the layer we set the conductivities to zero. Let the thin E-layer to coincide with the z ? 0 plane. Assuming all perturbed values to vary as expðikx À iotÞ, we write the boundary conditions for the components of the electric field at the conductive
250 h,
KM

Let us consider the dispersion properties of the wave propagating in the z ? 0 plane. For that purpose we substitute the solution of Eqs. (5) into the boundary conditions (6) for the field above and below this plane. Above the conductive ionosphere in the semi-plane, where sP ? sH ? 0, we have d2 E y À k2 E y ? 0, dz2 d2 E z dE z À k2 cot2 jE z ? 0. þ 2ik cot j dz2 dz

ð8Þ

The solution for Ey, decaying at the infinity, and oscillatory solution for Ez take the form E y ? a2 expðÀkzÞ; E z ? b2 expðÀikz cot jÞ (9) In the region below the ionosphere zo0 corresponding to the insulated atmosphere we have DE ? 0. The solutions for the electric field in this region reduce to E y ? a1 expðkzÞ; E z ? b1 expðkzÞ (10)

200
P

150

H

Substituting Eqs. (9) and (10) into the boundary conditions (6) one obtains the system of linear equations for the coefficients a1, a2, b1 and b2. Equating the determinant of this system to zero one finds the dispersion relation for the spectrum of eigen-oscillations given by o2 no 2k3 l cos2 j À 2 À i 2 k2 cos2 j a a ! no  kl þ i kl cot j À a2 sin2 j 2 o no ?0 ð11Þ À i 2 kl kl cot j À a a2 sin2 j Eq. (11) represents the dispersion relation of gyrotropic waves propagating in z ? 0 plane. Let us now consider the influence of the conjugate ionosphere on the properties of the gyrotropic waves. For that purpose we use the model of the thin conjugate

100 0 5

6 , 10 s-1

10

Fig. 1. Altitude dependence of the Hall and Pedersen conductivities.


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ionosphere (e.g., Lyatskii and Maltsev, 1983). According to this model the conjugate ionosphere can be considered as a thin layer at the altitude z ? L, as it is insured in the Fig. 2, where L corresponds to the distance between the conjugate ionospheres. We assume that at z ? 0 the boundary condition (6) holds. At the conjugate plane z ? L the external magnetic field is directed under the angle c ? p À j to the x-axis. Similar to Eqs. (6) one finds & ' dE y o2 þ inok2 cos2 c þ E y ðLÞ dz z?L k2 a2 l cos2 c 2o sin c dE z ðz ? L þ 0Þ ; fE y gz?L ? 0, ?À ka cos2 c dz & ' dE z no ? Ài E z ðz ? L À 0Þ, dz z?L a2 l sin2 c o E y ðLÞ. ð12Þ fE z gz?L ? À 2ka sin c In the region zo0 the electric field components are defined by Eq. (10). In the region 0ozoL, coinciding with the uniform magnetosphere, Ez yields to Eq. (11), the solution of which is E z ? ðb2 þ b3 zÞ expðÀikz cot jÞ. (13)

The Ey component in the region 0ozoL satisfies the first equation in Eq. (11), the solution of which is z40; zoL; E y ? a2 expðÀkzÞ, E y ? a3 exp?kðz À LÞ. ð15Þ

Substituting Eqs. (10) and (13)-(15) into the boundary conditions (6) and (12) one finds the system of linear equations for the coefficients a1 2a4 and b1 2b4 . By setting the determinant of this system to zero one finds the dispersion relation o2 no 2k3 l cos2 j À 2 À i 2 k2 cos2 j a a ! 2l no  kl þ þ i kl cot j À L a2 sin2 j ! o2 2l no þ i kl cot j À ? 0. ð16Þ À 2 kl L a a2 sin2 j Eq. (16) is the dispersion relation for gyrotropic waves accounting for the effect of the conjugate ionosphere. In the limiting case kl 51 and kLb1 this relation reduces to o2 þ iok2 n cos2 j À 2k3 a2 l cos2 j ? 0, (17)

For z4L, i.e. in the conjugate atmosphere, one obtains E y ? a4 exp?Àkðz À LÞ; E z ? b4 exp?Àkðz À LÞ. (14)

z

where the quantity n (it is proportional to Pedersen conductivity) describes the damping of the gyrotropic waves. ~ Let us introduce the complex refractive index n ? n þ ik according to the formula k ? ðn þ ikÞo=a. Substituting this equality into Eq. (17) and assuming that the waves are weakly damping one finds the dependence of phase velocity vp ? a=nðoÞ and damping rate ? l=2pd ? k=n as the function of frequency vp ? ð2la2 Þo
1=3

cos

2=3

j,
1=3

? k=n ? ðn=6Þð2=l 2 a4 Þ1=3 o

cos2=3 j.

ð18Þ

L

B 0

x

Fig. 3 shows the dependence of vp ? vp ðoÞ versus frequency for different magnetic inclination j. For numerical calculations the following parameters have been selected: l ? 3 Â 106 cm, sH ð0Þ ? 8 Â 106 sÀ1 and sP ð0Þ ? 2 Â 106 sÀ1 . With these parameters in mind one finds: a % c2 =4pl sH ð0Þ ? 3 Â 106 cm=s and n ? 2 Â 1012 cm2 =s. Eq. (18) shows that the phase velocity of the gyrotropic waves and their specific damping increases with the growth of frequency as o1=3 . With the increase in the magnetic field inclination the wave phase velocity decreases. In the frequency range from 0.001 to 1 Hz the phase velocity varies from a few units to a few tenths of km /s.

Fig. 2. Schematic drowing of the model. The planes x ? 0 and x ? L coinside with the thin conjugate conductive layers. j is the magnetic field inclination and c corresponds to the conjugate ionosphere.

3. Spatio-temporal characteristics of gyrotropic waves Now we consider the spatio-temporal properties of geomagnetic pulsations generated by the gyrotropic


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Fig. 3. Phase velocity of gyrotropic wave as a function of frequency for different values of j. (1) j ? 0, (2) j ? p=6, (3) j ? p=4 and (4) j ? p=3. The Hall and Pedersen conductivities are: sH ð0Þ ? 8 Â 106 sÀ1 and sP ð0Þ ? 2 Â 106 sÀ1 . The depth of the conductive layer l ? 3 Â 106 cm, a % c2 =4pl sH ð0Þ ? 3 Â 106 cm=s and n ? 2 Â 1012 cm2 =s.

wave in the ionospheric E-layer. The Fourier-components of the magnetic and electric fields are connected through Faradey's law c dE y ðk; o; zÞ bx ðk; oÞ ? io dz .
z?0

dence be bx ðx1 ; tÞ ? b0 ðtÞ, the Fourier-component of b0(t) is Z b0 ðoÞ ?
À1 1

dtb0 ðtÞ expðiotÞ.

(20)

The magnetic field can be written as bx ðk; tÞ ? DðkÞ expfÀio1 ðkÞtg, where the amplitude D(k) should be determined from the boundary condition and o1 ðkÞ is the solution of dispersion relation (17) o1 ðkÞ ? À ði=2Þnk2 cos2 j þ o0 ðkÞ cos j, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o0 ðkÞ ? 2jkj3 a2 l . The spatio-temporal distribution of the magnetic field is Z bx ðx; tÞ ?
À1 1

Setting in Eq. (19) x ? x0 and making a Fourier transform one obtains Z b0 ðoÞ ?
À1 1

dkDðkÞ expðikx0 Þdðo1 ðkÞÀ oÞ.

From this expression one can obtain D(k). Substituting it into Eq. (19) we have Z bx ðx; tÞ ? dk qo1 ðkÞ b0 ðo1 ðkÞÞ expfikðx À x0 Þ 2p q k À io1 ðkÞtg.
À1 1

ð21Þ

dk DðkÞ expfikx À io1 ðkÞtg. 2p

(19)

This expression describes the wave propagating in the positive x-direction. Let at x ? x0 the temporal depen-

Expression (21) describes propagation of the magnetic field along the x-axis from the point x0. Let us consider the propagation of the wave packet along the x-axis. Let at x ? x0 the magnetic field be b0 ðtÞ ? AðtÞ sin ðOtÞ,


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where the amplitude A(t) slowly varies in time. The Fourier transform of Eq. (20) gives b0 ðoÞ ? fAðo þ OÞþ Aðo À OÞg=2i. The spatio-temporal distribution (21) in this case represents the sum over positive and negative values of k bx ðx; tÞ ?
1 4pi &Z
1
0

nonzero we have xÀx x À x0 0 bx ðx; tÞ ? A t À exp À vg d ! x À x0 , Â sin O t À vp

ð22Þ

dk
À1

Z
0

À

dk

' qo1 Aðo1 À OÞ exp?ikðx À x0 ÞÀ io1 t . qk

qo1 Aðo1 þ OÞ exp?ikðx À x0 ÞÀ io1 t qk

where vp ? O=x0 , d ? 2vg =nx2 cos2 j. The phase and 0 group velocities can be expressed in terms of O according to vp ? ð2a2 l OÞ1=3 cos
2=3

j;

vg ? ð3=2Þvp .

(23)

Let us consider the second integral in the complex plane k ? x þ iB. Rapidly varying integrand A attains the maximum value at k ? k0 that is determined from the equation o1 ðk0 Þ ? O. For small damping, B0 5x0 , the latter condition gives qffiffiffiffiffiffiffiffiffiffiffiffiffi O ? o0 ðx0 Þ cos j ? 2a2 l x3 , 0 B0 ? nx2 cos2 j=2vg , 0 where vg ? qo1 ðkÞ=qk
k%x0

Fig. 4 illustrates the propagation of the wave packet described by Eqs. (22) and (23). The function AðtÞatx ? x0 is taken in the form AðtÞ ? b0 exp?ðt=T Þ2 . The duration of the wave packet is T ? 15 s and the wave packet central frequency is O=2p ? 0:1Hz. The curves 2 and 3 shows the perturbation of the magnetic field at the distances 1000 and 2000 km for the magnetic field inclinations p/6 and p/3, respectively. Now we consider the propagation of the impulse of the gyrotropic waves. We assume that at x ? 0 the magnetic field is given by b0 ðtÞ ? b0 dðtÞ, where dðtÞ is the Dirac delta function. Substituting this condition into Eqs. (20) and (21) reduces to pffiffi Z nt b0 3a l cos j 1 pffiffiffiffiffiffi pffiffiffi bx ðx; tÞ ? dk jkj exp À k2 cos2 j 2 2p 0 h i pffiffiffiffiffiffiffiffiffiffi  cos k x À at 2l jkj cos j . ð24Þ

? ð3=2Þ cos j

pffiffiffiffiffiffiffiffiffiffiffiffiffi 2a2 l x0 .

The damping is small if nx0 cos2 j52vg . Taking this into account we expand o1 in the power series on x À x0 . In the leading order one obtains o1 % O þ vg ðx À x0 Þ. Similar expansion one can made in the first integral. Taking into account that in the vicinity of x0 A is

Fig. 4. Time dependence of the magnetic field perturbation generated by the gyrotropic wave packet at different distances for magnetic field inclinations j ? p=6 and p=3 and a % c2 =4pl sH ð0Þ ? 3 Â 106 cm=s and n ? 2 Â 1012 cm2 =s. (1) x ? 1000 km, (2) x ? 2000 km.


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The condition of small damping allows us to use the stationary phase method for the estimation of the integral. pffiffiffiffiffiffiffiffiffiffi us Á expand the function f ? Let À k x À at 2l jkj cos j in power series on k À k0 accounting for the small terms up to the second order f ðkÞ ? f ðk0 Þþ 1f 00 ðk0 Þðk À k0 Þ2 . 2 (25)

The point of stationary phase k0 and the expansion terms (25) are defined by df ðkÞ ? 0; dkk?k0 f ðk0 Þ ?
2 a2

k0 ?

9t2 a2

2x2 , l cos2 j f 00 ðk0 Þ ? À 9t2 a2 l cos2 j . 4x ð26Þ

2x3 ; 27t l cos2 j

Substituting Eqs. (25) and (26) into Eq. (24), one obtains pffiffiffi 2 x3=2 bx ðx; tÞ ? b0 pffiffiffiffiffi 3a pl cos j t2 2nx4 2x3 p . ð27Þ Â exp À 4 cos 3 2 2 À 3 t a l cos2 j 4 3 t3 a4 l 2 cos2 j Fig. 5 shows the time dependence of the magnetic field perturbations at the distances 1000 and 2000 km, respectively. The magnetic field inclinations are p/6 and p/3, respectively.

background electromagnetic field on horizontal inhomogeneities of the ionospheric conductivity. Recently Sorokin et al. (2003) considered the formation of such spectra on the Earth's surface at low latitudes. It has been shown that this mechanism is due to generation of the gyrotropic waves in the low ionosphere by polarization currents that arise in the horizontal inhomogeneities of the conductivity under the action of the background electromagnetic field from the atmospheric sources. The calculations of the spectra were carried out with the use of the model of the wave propagation in the horizontal magnetic field. Below we assume that the waves propagate along the x-axis. The magnetic field is assumed to be in the (x, z) plane. The conductivity _ _ _ tensor is taken in the form sðx; zÞ ? sðzÞþ sðx; zÞ, where 0 1 the subscripts 0 and 1 correspond to the unperturbed and perturbed values, respectively. We decompose the electric field as E ? E0 þ E1 , where E0 stands for the _ unperturbed electric filed (s ? 0) and E1 is the 1 perturbation caused by the presence of the ionosphere inhomogeneities. Assuming the perturbations to be _ _ small, s5s and neglecting the small terms of the 1 0 second order from Eq. (5) one finds the electric field Fourier components as E y1 ðo; kÞ ? ÀE y0 ðoÞ 2o2 o2 À o2 ðk; jÞþ inok2 cos2 j 0

4. The role of magnetic inclination in the formation geomagnetic pulsation spectra by gyrotropic waves Let us consider the influence of the magnetic field inclination on the characteristics of the spectra of geomagnetic pulsations in the ULF frequency range. The latter can arise as the result of scattering of the

Z Â

1

dx expðikxÞhðxÞ; o2 ðk; jÞ ? 2la2 jkj3 cos2 j, 0
À1

hðxÞ ? sH1 ðx; zÞ=sH0 ðzÞ.

ð28Þ

Fig. 5. Time dependence of the magnetic field perturbation generated by narrow unipolar impulse of the gyrotropic waves at different distances for magnetic field inclinations j ? p=6 and p=3. Other parameters are same as in Fig. 4.

Fig. 6. Power spectra of geomagnetic pulsations on the Earth's surface generated by the gyrotropic waves in the presence of horizontal inhomogeneities of the ionospheric conductivity. (1) j ? p=6 and (2) j ? p=3, respectively. The parameters used: horizontal scale of the inhomogeneities l ? 2p=k0 ? 100 km, horizontal dimension of the perturbed region L ? 500 km, a % c2 =4pl sH ð0Þ ? 3 Â 106 cm=s and n ? 2 Â 1012 cm2 =s.


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Let us take the perturbation of the Hall conductivity in the form hðxÞ ? A expðÀjxj=LÞ cosðk0 xÞ, where Lbl ? 2p=k0 . Applying the inverse Fourier transform over k with the help of Faradey's law one obtains b1x ðx ? 0; oÞ o2 b ðoÞ ? Ao2 À o2 ðk ; jÞþ iF2 ðo; jÞ cos2 j, 0x 0 0 F2 ðo; jÞ ? nk2 o þ 6la2 k2 =L. 0 0 ð29Þ Fig. 6 shows the power spectra for the angles p=6 and p=3, calculated with the help of Eq. (29). The parameters used are: l ? 2p=k0 ? 100 km and L ? 500 km. The frequencies corresponding to the maxima of the spectra lies in the frequency range 0.1-1 Hz. With the increase in the magnetic field inclination and horizontal scale of the inhomogeneity the frequencies of the maxima become smaller.

5. Discussion and conclusions In the present paper it has been shown that the Earth's ionosphere can support a specific class of electromagnetic wave modes termed the gyrotropic waves. They represent the weakly damping oscillations of the electric current and the electric field and propagate in the thin ionospheric layer. The magnetic field induced by this current can be observed on the Earth's surface as geomagnetic pulsations whereas the electric field forms the ionospheric perturbations in the F-layer. The phase velocity of these waves lies in the frequency range 0.001-1 Hz and amounts for the values stretching from a few units to a few tenths of km/s. It decreases as cos2=3 j with the increase in the magnetic field inclination. In the mid- and high-latitudes the phase velocity is smaller than in the low latitudes. The damping of these waves possess a similar latitudinal dependence. The frequency dependence of the phase velocity varies as o1=3 . The high-frequency harmonics of the impulse take the lead over the low-frequency harmonics. The wave phase velocity decreases with the growth of Hall conductivity. Thus, it is greater in the nighttime ionosphere. The wave damping is controlled by Pedersen conductivity. The analysis of the influence of the magnetic field inclination and Pedersen conductivity on propagation of the wave packet has shown that at the distance of 1000 km its amplitude is two times smaller whereas at the distance 2000 km it is three times smaller. The wave damping constitutes the value k=n ? l=2pd ? 10À1 210À2 . The phase velocity decreases with the growth of the wave period and magnetic field inclination. The wave packet propagates with the group velocity exponentially decreasing with the amplitude without significant distortion of the form. The characteristic spatial scale of the amplitude variation

constitutes the value of 2000-3000 km. The phase velocity exceeds the group velocity. The obtained propagation characteristics of the quasi-harmonic wave packet agree with observations of the phase shift of geomagnetic pulsations observed at the array of spatially displaced magnetometers. In the course of propagation of the short unipolar impulse it has been observed the enhancement of its duration and appearance of oscillations. Their frequency decreases with time. The signal duration increases with the magnetic field inclination. The formation of the ionospheric horizontal periodic inhomogeneities of the conductivity leads to the appearance of geomagnetic pulsations on the Earth's surface. These oscillations are associated with the gyrotropic waves. The waves are generated by the electric current that arises in the inhomogeneities of the ionospheric conductivity under the action of the external electric field from the atmospheric sources. The spectrum of oscillations possesses the maximum in the ULF frequency range. The inhomogeneities can arise, for example, during upward propagation of gravitational waves from seismic sources. If the inhomogeneities are elongated in the latitudinal direction and their horizontal scale is of the order of 100 km the central spectrum frequency lies in the range 0.1-1 Hz. The central frequency decreases with the increase in magnetic field inclination and spatial scale of the inhomogeneity. The considered mechanism can be used for the interpretation of the observed phase shift at the Earth's surface and dispersion of the ULF waves, connected with the growth of seismic activity. The horizontal phase velocity of these oscillations constitutes the values 20-30 km/s. The features of the gyrotropic waves can be used for the interpretation of the electromagnetic effects that arise in the course of natural and man-made action on the ionosphere.

Acknowledgments This research was partially supported by ISTC under Research Grant No. 2990, by the Russian Fund for Basic Research through the Grants No. 04-05-64657 and 03-05-64553 and by the Russian Academy of Sciences through the Grant ``Physics of the Atmosphere: electrical processes and radiophysical methods''.

Appendix Fig. 1 shows the variation of sP and sH conductivities versus altitude. One sees that both conductivities attain the maximum values at different altitudes. The maximum of Hall conductivity lies higher than that for Pedersen conductivity. The maximum value of Hall conductivity exceeds that for Pedersen conductivity by


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one-two orders in value. At the higher altitudes the Pedersen conductivity substantially exceeds Hall conductivity. Such an altitude profile allows us to represent the ionosphere in the form of two distinct thin layers. Let us assume that in the low layer sP ? 0 whereas in the upper layer sH ? 0. In the low frequency limit, o5sP;H , we assume all perturbed values to vary as expðikx À iotÞ. The system of Eqs. (5) can be written independently in each of these layers. In the low layer, in which Hall conductivity is nonzero, one has d2 E y 4posH Ez, À k2 E y ? i 2 dz2 c cos j sin2 j d Ez dE z À k2 cos2 jE þ 2ik sin j cos j 2 dz dz 4posH cos j ? Ài Ey. c2
2 z

should be satisfied & dE y fE y gz?0 ? 0; dz 2o sin ?À ka cos2 o fE z gz?0 ? À 2ka sin

'

o2 E y ð0Þ k l cos2 j z?0 & ' j dE z ð0Þ dE z ? 0, j dz dz z?0 þ
22 a

j

E y ð0Þ.

ðA:3Þ

In Eq. (A.3) the following abbreviations are introduced , sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z
1 1

a ? c2

4p l
À1

dzs2 ðzÞ; H

l s2 ð0Þ ? H

dzs2 ðzÞ. H

À1

ðA:1Þ

The boundary condition (A.3) for Ez one can obtain from the solution of the second Eq. (A.1) in the layer. Letting E z ðzÞ ? AðzÞ expðÀikz cot jÞ, one finds d2 A 4po cos j ? Ài sH E y expðikz cot jÞ. dz2 c2 sin2 j Inside the layer one has ! l kl l exp Ài cot j þ z þ C E z ðzÞ ? E z À 2 2 2 Zz 4po cos j dz0  expðÀikz cot jÞÀ i c2 sin2 j Àl =2 Z z0 dz00 sH ðz00 ÞE y ðz00 Þ exp?ikðz00 À zÞ cot j.
Àl =2

In the upper layer, where Pedersen conductivity is nonzero, one obtains d2 E y 4posP À k2 E y ? Ài Ey, dz2 c2 d 2Ez dE z À k2 cos2 jE sin2 j 2 þ 2ik sin j cos j dz dz 4posP ? Ài Ez. c2

z

ðA:2Þ

Let us assume that the horizontal scale of the wave field variation is much greater than the depth of the conductive layer l, i.e. kl 51. In this case Eqs. (A.1) and (A.2) can be replaced by the boundary conditions for the electric field components and their derivatives. Let us find the boundary conditions in the lower layer coinciding with the z ? 0 plane. Substituting Ez from the second Eq. of the system (A.1) into the second equation and integrating over the layer we obtain & ' 2 Z l =2 Z l =2 dE y 4po À k2 dzE y þ 2 dzs2 E y H c k cos j dz z?0 Àl =2 Àl =2 Z l =2 8po sin j dE z ?À 2 dzsH dz c k cos2 j Àl =2 Z l =2 2 4po sin j d2 E z dzsH , þi 2 k2 cos3 j dz2 c Àl =2 & ' dE z sin2 j dz z Z À k2 cos2 j þ 2ik sin j cos jfE z gz
?0 l =2

The constant C is defined from the condition of continuity of the derivative " # & ' dE z dE z dE z ? 0. ? liml !0 À dz z?0 dzjz?1=2 dzjz?À1=2 Defining the constant, for kl cot j1, one obtains E Z l =2 l l 2po dzsH ðzÞE y ðzÞ À Ez À ?À 2 z 2 2 c k sin j Àl =2 ! ! l l  ik À z cot j þ 1 exp Àik À z cot j . 2 2

?0 l =2

dzE z ? Ài

Àl =2

4po cos j c2

Z

dzsH E y ,
Àl =2

where the parentheses denote the difference between the values above and below the layer. The integrand denotes the multiplication of rapidly varying function sH that attains the maximum value at z ? 0 and slowly varying along the layer function. From this system one sees that in the plane z ? 0 the following conditions

Substituting sH ðzÞ ? sH ð0Þl dðzÞ, where dðzÞ is the Dirac delta function, for kl cot j51, one obtains 2po l fE z gz?0 ? À 2 sH ð0ÞlE y ð0Þ ik cot j þ 1 c k sin j 2 l o E y ð0Þ Â exp Àik cot j % À 2 2ka sin j This expression coincides with Eq. (A.3). Thus, the boundary condition for the vertical component of the electric field is valid if the condition sin jbkl is satisfied. Let the layer in which Pedersen conductivity is nonzero lie in the z ? z0 plane. Integrating Eq. (A.2)


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930 V.M. Sorokin, O.A. Pokhotelov / Journal of Atmospheric and Solar-Terrestrial Physics 67 (2005) 921 -930

over the layer, one & ' dE y o ? Ài dz z?z0 aP & ' dE z ? Ài dz z?z0 aP ? 0; Z aP ? c2 =4p fE y gz
?z
0

obtains E y ðz0 Þ, o E z ðz0 Þ, sin2 j
?z
0

fE z gz
1

? 0, ðA:4Þ and the thin

dzsP ðzÞ.

À1

Setting z0 to zero and summing up the Eqs. (A.3) (A.4), one obtains the boundary conditions for components of the electric field on the conductive ionosphere & ' dE y o2 þ inok2 cos2 j þ E y ð0Þ dz z?0 k2 a2 l cos2 j 2o sin j dE z ðz ? À0Þ ; fE y gz?0 ? 0, ?À ka cos2 j dz & ' dE z no ? Ài E z ðz ? þ0Þ, 2 l sin2 j dz z?0 a o E y ð0Þ, ð fE z gz?0 ? À 2ka sin j where n ? a2 l =aP .

A:5Þ

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Gogatishvili, Y.M., 1979. Peculiarities of appearance of long period pulsations in middle latitudes. Geomagnetism and Aeronomy 19 (2), 382-384. Gokhberg, M.B., Kocharyants, E.G., Kopytenko, Y.G., et al., 1973. Peculiarities of spatio-temporal distribution of Pi-2 pulsations. Izvestia Akademii Nauk SSSR, Fizika Zhemli 2, 62-69. Hayakawa, M., Kawate, R., Molchanov, O.A., et al., 1996. Results of ultra-low-frequency magnetic field measurements during the guam earthquake of 8 August 1993. Geophysical Research Letters 23 (3), 241-244. Herron, T.J., 1965. Journal of Geophysical Research 71, 834. Ismaguilov, V.S., Kopytenko, Y.A., Hattori, K., Voronov, P.M., Molchanov, O.A., Hayakawa, M., 2001. ULF magnetic emissions connected with under sea bottom earthquakes. Natural Hazards and Earth System Sciences 1, 23-31. Karlov, V.D., Kozlov, S.I., Kudryavtsev, V.P., et al., 1984. On one type of large-scale perturbations in the Ionosphere. Geomagnetism and Aeronomy 24, 319-324. Kopytenko, Y.A., Ismaguilov, V.S., Hattori, K., et al., 2002. Monitoring of the ULF electromagnetic disturbances at the station network before EQ in seismic zones of Izu and Chiba peninsulas (Japan). In: Hayakawa, M., Molchanov, O.A. (Eds.), Lithosphere-Atmosphere-Ionosphere Coupling. TERRAPUB, Tokyo, pp. 11-18. Lyatskii, V.B., Maltsev, Y.P., 1983. Magnetosphere-Ionosphere Interaction. Nauka, Moscow. Mareev, E.A., Iudin, D.I., Molchanov, O.A., 2002. Mosaic source of internal gravity waves associated with seismic activity. In: Hayakawa, M., Molchanov, O.A. (Eds.), Seismo Electromagnetics, Lithosphere-Atmosphere-Ionosphere Coupling. TERRAPUB, Tokyo, pp. 335-342. Piddington, J.H., 1959. The Transmission of Geomagnetic Disturbances through the Atmosphere and Interplanetary Space. Geophysical Journal 2, 173-189. Sorokin, V.M., 1988. Wave processes in the ionosphere related to the geomagnetic field. Izvestia Vuzov, Radiophysics 31 (10), 1169. Sorokin, V.M., Fedorovich, G.V., 1982a. Propagation of shortperiod waves in the ionosphere. Izvestia Vuzov, Radiophysica 25 (5), 495-507. Sorokin, V.M., Fedorovich, G.V., 1982b. Physics of Slow MHD Waves in the Ionospheric Plasmas. Energoizdat, Moscow. Sorokin, V.M., Chmyrev, V.M., Isaev, N.V., 1998. A generation model of small-scale geomagnetic field-aligned plasma inhomogeneities in the ionosphere. Journal of Atmospheric and Solar-Terrestrial Physics 60, 1331-1342. Sorokin, V.M., Chmyrev, V.M., Yaschenko, A.K., 2003. Ionospheric generation mechanism of geomagnetic pulsations observed on the Earth's surface before earthquake. Journal of Atmospheric and Solar-Terrestrial Physics 65, 21-29.