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Journal of Atmospheric and Solar-Terrestrial Physics 64 (2003) 21 ­29

www.elsevier.com/locate/jastp

Ionospheric generation mechanism of geomagnetic pulsations observed on the Earth's surface before earthquake
V.M. Sorokin , V.M. Chmyrev, A.K. Yaschenko
Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation, Russian Academy of Sciences, Moscow Region, 142092, Troitsk, Russia Received 23 January 2001; accepted 1 March 2002

Abstract New generation mechanism of ULF geomagnetic oscillations observed on the Earth's surface in seismic zones is presented. This mechanism is based on the formation of periodic structure of ionospheric conductivity due to acoustic-gravity wave instability stimulated by DC electric Úeld enhancement in the ionosphere. Interaction of the background electromagnetic ULF waves with such structure leads to an excitation of polarization currents and generation of narrow band gyrotropic waves at the 0.1­10 Hz frequency range in the ionosphere. The magnetic Úeld of these waves can be observed on the ground. Since the growth of seismic activity is often accompanied by DC electric Úeld enhancement on the ground and in the ionosphere, the suggested mechanism can be considered as a possible source of the seismogenic geomagnetic pulsations generated before and during earthquakes. c 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Ionosphere; Irregularities; Gyrotropic waves; Earthquakes

1. Introduction According to numerous publications, an enhancement of the geomagnetic oscillation intensity at the frequencies 0.01­10 Hz is often observed in the near epicenter zones of strong earthquakes before the main shock (Fraser-Smith et al., 1990; Kopytenko et al., 1994; Hayakawa et al., 1996). Practically, all the mechanisms suggested for explanation of this phenomenon were connected with di erent radiation sources situated within the lithosphere (Draganov et al., 1991; Fenoglio et al., 1994; Johnston et al., 1994; Molchanov and Hayakawa, 1995). Besides, Alperovich and Zheludev (1999) found the geomagnetic pulsations associated with ionospheric sources above the earthquake preparation zone. In this paper, new generation mechanism of the seismogenic geomagnetic pulsations connected with the radiation source in the ionosphere is presented. This mechanism is


Corresponding author. E-mail address: sova@izmiran.rssi.ru (V.M. Sorokin).

based on the excitation of gyrotropic waves (GW) in the presence of periodic horizontal inhomogeneities of electric conductivity in the lower ionosphere. These waves Úrst reported by Sorokin and Fedorovich (1982) propagate within thin layer of lower ionosphere along the Earth's surface in low and middle latitudes with small attenuation and with phase velocities, tens to hundreds of km= s. Some geophysical e ects of GW are analyzed by Sorokin (1986, 1988) and Sorokin and Yaschenko (1988). Chmyrev et al. (1989) found the seismic-related DC electric Úeld disturbances in the ionosphere over the earthquake zone. The data of electric Úeld enhancement up to 20 mV= m that occurred in the ionosphere are presented in Isaev et al. (2000). Discussion about possible mechanism of the observable electric Úeld enhancement in the ionosphere was carried out by Sorokin et al. (2001). It has been shown that an enhancement of the DC electric Úeld at deÚnite conditions is accompanied by the generation of periodic inhomogeneous structure of electric conductivity in the lower ionosphere and the formation of geomagnetic Úeld, which aligned with plasma layers in the upper ionosphere with the

1364-6826/03/$ - see front matter c 2002 Elsevier Science Ltd. All rights reserved. PII: S1364-6826(02)00082-2


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V.M. Sorokin et al. / Journal of Atmospheric and Solar-Terrestrial Physics 64 (2003) 21 ­ 29

characteristic transverse spatial scale 10 km (Chmyrev et al., 1999; Sorokin et al., 1998; Sorokin and Chmyrev, 1999). Such layers observed by satellite as the variations of plasma density were reported by Chmyrev et al. (1997). So a growth of DC electric Úeld in the ionosphere before earthquake leads to the formation of the quasi-periodic horizontal inhomogeneities of ionospheric conductivity. Various sources generate the electromagnetic noise at ULF range. The most powerful are thunderstorms. Oscillating noise electric Úeld forms the polarization currents on the inhomogeneities of conductivity in the ionosphere. These horizontal currents with the spatial scale 10­100 km are considered as a source of GW. Generation and propagation of these waves lead to the excitation of narrow band electromagnetic radiation on the ground at the ULF range. According to the electrodynamic model of the seismic related lower atmosphere and the ionosphere coupling (Sorokin et al., 1999, 2001; Sorokin and Yaschenko, 1999), the main cause of the ionosphere modiÚcation processes is the injection of radioactive substances and charged aerosols into the atmosphere. This leads to: 1. the formation of an external electric current near ground layer and the electric Úeld increase in the ionosphere; 2. acoustic-gravity waves instability and spatial modulation of the conductivity in the ionosphere E layer; 3. appearance of polarization electric Úelds, which propagate into the upper ionosphere and generates the plasma density variations and the Úeld-aligned currents at these altitudes; and 4. generation of seismic related geomagnetic pulsations that must be observed on the ground in epicenter region. Our paper is devoted to question 4. It addresses theoretically the generation of GW in the 0.1­10 Hz range and focusses on the reaction of the lower ionosphere and the creation of geomagnetic pulsations registered on the ground before the earthquake. 2. Ultra-low-frequency electromagnetic waves in the lower ionosphere The electric Úeld components E of monochromatic wave with frequency ! satisfy Maxwell's equation: @EÚ @ @E 4! =i 2 jÚ : - (1) @x @xÚ @x c The equations of quasi-hydrodynamics for electrons, ions and molecules in the frequency range 0.01­10 Hz (Ginzburg, 1967) give Ohm's law in the following form (Sorokin,1988): !i c 2 j= 4 u2 !e E +[G ve
1 Ú

n is the unit vector directed along the geomagnetic Úeld, u = B(4 MN )-1=2 is the Alfven velocity, G1 = g(1 + g2 )-1 ; G2 = (1 + g2 )-1 ; g = ve =!e + !i = (vin - i!); Ú is completely antisymmetric unit tensor, Ú is the unit tensor, each of the tensor indexes notes the coordinates x; y and z . A product of the same index tensors denotes a sum of their components over this index (for example, Ú n = Úx nx + Úy ny + Úz nz ). The Úrst term in right-hand side of Eq. (2) corresponds to the current caused by the magnetic Úeld-aligned electric Úeld, and the second one--to transverse electric Úeld. Since ve =!e exceeds G1 and G2 , more than four orders we neglect the parallel electric Úeld in the wave. To obtain the equation for transverse components of the wave electric Úeld let us substitute Eq. (2) into Eq. (1) and multiply the equation by the tensor Ú nÚ . As a result we Únd
Ú

n

Ú

@ @x

@E @E - @x @x
1 v

-i E =(

!!i (G u2
Ú

n+

v

G2 )Ev =0;

- n nÚ )EÚ :

(3)

Eq. (3) describes the propagation of the transverse low-frequency electromagnetic waves in weakly ionized ionospheric plasma with the ideal longitudinal conductivity. We assume that homogeneous external magnetic Úeld is directed along the x-axis, the wave vector k lies in xy-plane, and ' is the angle between k and x. System (3) gives the dispersion equation for complex refraction index (n +iä)
2 2

= c2 !2 =k =

(c2 !i =u2 !)[ig(1+cos2 ')±(4 cos2 '-g2 sin4 ')1=2 ] : 2(1+g2 ) cos2 ' (4)

In the case of a lower sign in Eq. (4), it determines the characteristics of the fast magnetosonic wave propagating with phase velocity v = u. This wave dissipates in the lower ionosphere. Let us consider the wave properties at the Alfven branch corresponding to upper sign in Eq. (4). Fig. 1 shows the plots for altitude dependence of the wave phase velocity and the = ratio, where is the wavelength and is a depth of the wave penetration calculated according to Eq. (4). In the upper ionosphere, where vin !i , we obtain the following: v=u =2 !+ = 2! cos '; 2 !2 + vin
2 !2 + vin - !

+G

2

Ú

n ]E

Ú

;

(2)

where !e ;!i are the gyrofrequencies of electrons and ions, vab is the collision rate of a and b type particles, ve = vei + ven ;

vin :


V.M. Sorokin et al. / Journal of Atmospheric and Solar-Terrestrial Physics 64 (2003) 21 ­ 29
KM 600 500 400 300 200 100
100 150 250

23

h, kM

/2

v/ u
200



P



H

, 10 c

6

-1

0

0.5

1 .0

0

5

10

Fig. 1. Altitude dependence of the relative wave phase velocity v=u and relative dissipation = 2 . (u--Alfven velocity, --wavelength, and --depth of the wave penetration).

Fig. 2. Altitude distribution of the ionosphere Pedersen ( Hall ( H ) conductivities.

P

) and

Note, that Alfven wave propagates with weak dissipation in the F2-layer, where !vin ; v = u cos '; = 2 = vin = 2! 1. At the lower altitudes, where ! vin ; v = u(2!=vin )1=2 cos '; = 2 = 1, the phase velocity decreases and dissipation grows. In E -region of the ionosphere, where !i vin , we Únd that v =2u cos ' !(1 + g2 ) !i (2 cos ' = 1+ g2 + 4 cos2 ' - g2 sin4 ') ; ;

Taking into account this notion we can rewrite Eq. (1) in the following vector form: 4 (5) rotrot E - i! 2 E =0; c where ^ is the tensor of ionosphere conductivity. 3. GW propagation in the horizontal homogeneous ionosphere For example, let us analyze the basic GW properties in the ionosphere considering their propagation along the horizontal magnetic Úeld directed along x-axis in the Cartesian coordinates, here z -axis is directed upwards. Therefore, we obtain from Eq. (5) the equation for Ey and Ez components: @2 @2 +2 2 @x @z = i 4! c2 4! c2
P 2

g(1 + cos2 ') 2 cos ' 1+ g2 + 4 cos2 ' - g2 sin4 '

2 g=

i (

!i ve + : !e vin

4! c2
2 H

P

+

@2 @x2

Ey +i

4! c2

P

@2 E y @x2

In the case of g¡ 1, the wave phase velocity is smaller than that of Alfven wave, and even their absorption is small. These waves propagate in the lower ionosphere with small attenuation and they have been Úrst investigated by Sorokin and Fedorovich (1982) and called as gyrotropic waves (GW). They exist in weakly ionized plasma with magnetized electrons and non-magnetized ions. It is shown that GW propagate with small absorption along the Earth surface in the thin layer of the lower ionosphere at the heights 100 ­120 km. The wave properties in homogeneous and non-homogeneous environments are essentially di erent. Below, we analyze the dispersion characteristics of GW with the wavelengths considerably exceeding the width of the layer. In the lower ionosphere ! vin , the quantities G1 and G2 are expressed through Pedersen ( P ) and Hall ( H ) conductivities, i.e. G1 =4 u
2 P

+

2 P

)Ey ; 4! c2
H

+

@2 @x2

Ez =i

Ey :

(6)

The characteristic altitude distribution of the ionosphere conductivity, as shown in Fig. 2 displays two maximums, which correspond to two layers. At the upper layer, the value P is greater than H . At lower layer (the Hall one), the value H exceeds P by two orders approximately. Therefore, we can assume that in Eq. (6) P = 0, for Hall layer. Assuming a dependence of unknown functions on x as exp(i!x), we obtain d 2 Ey - k2 - d z2 Ez = - 4! c2 k
2 2 H

(z ) Ey =0; (7)

=c2 !i ;

G2 =4 u

2

H

=c2 !i :

i4 ! H (z ) Ey : c2 k 2


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V.M. Sorokin et al. / Journal of Atmospheric and Solar-Terrestrial Physics 64 (2003) 21 ­ 29

Let us choose the origin of coordinates at a height point of the maximum of Hall conductivity. Interpolate the altitude distribution of Hall conductivity by function H (z ) = H0 = cosh (z=l) and introduce according to Appendix A new independent function w = Ey [cosh(z=l)]s in Eq. (7). We Únd that -s + kl -s - kl 1 w=F ; ; ;- : (8) 2 2 2 Since the Úeld magnitude Ey = E0 w(1 + )s=2 tends to zero at , the quantity (-s + kl)= 2 should be the integer negative number (including zero). It determines a spectrum of normal waves in the Hall layer: Sn - kn l =2n; n =0; 1; 2 ::: : a = c2 = 4 l: (9) Substituting s from Eq. (A.1), we obtain the following: ! = akn [(kn l +2n)2 + kn l +2n]1=2 ;
H0

2 1 0 -1
(a)

Eyn/E0

3 1 2

-2 2 1 0 -1

Eyn/E0

3 1 2

(b)

-2

Relation (9) gives the connection between frequency and wave number for each mode of GW. It follows from Eq. (9) that the phase velocity of GW is determined by the Hall conductivity. These waves exist in an interval of heights, where the Hall conductivity exceeds the Pedersen conductivity, and the weakly ionized plasma is essentially gyrotropical. The phase velocity vn of normal waves is determined as vn = a[(kn l +2n)2 + kn l +2n]1=2 : Phase velocity of the basic mode n = 0 approaches zero with !0, while the velocities of waves with n =1; 2 :::, remain Únite. In a high-frequency limit !l=a1, the basic mode (n = 0) has dispersion v0 =(a!l)1=2 . This corresponds to the transition to the homogeneous media. In the opposite case !l=a 1, we have v0 =(a2 !l)1=3 . In a low-frequency limit of the basic mode, the frequency dependence of the phase velocity coincides with the dispersion law of the waves propagated in an inÚnitely thin layer with Hall conductivity. The group velocity Vn is expressed through the phase by
2 Vn =2vn + a2 [1 - (4n +1)(1+4vn =a2 )1=2 ]= 4vn :

1 H /

0

-4

-2

0

2

4

z/l
Fig. 3. Altitude dependence of the relative electric Úeld component Eyn =E0 for three wave modes. The bottom panel shows the model of the Hall conductivity altitude distribution: (a) !l=a = 1:0; (b) !l=a =0:2; (1) n = 0; (2) n = 1; (3) n =2.

For a wave with n = 0 at !l=a 1, we have V0 = (3= 2)v0 , i.e. the group velocity exceeds that of the phase. Let us make the numerical estimations of the parameters. Assuming the maximal value of Hall conductivity in the layer 6 -1 and their semi-thickness l = 15 km, we H0 = 6 â 10 c obtain a = 100 km= s. For example, the waves with the period T =60 s (!l=a =1:5 â 10-2 ) have phase speeds: v0 = 0:24a =24 km= s; v1 =2:5a = 250 km= s; v2 = 450 km= s. It is necessary to notice that the considered layer di ers in properties from wave guide, as Eq. (7) to which a Úeld satisÚes is not the wave equation. Nevertheless, the waves within the layer propagate without attenuation, but with frequency dispersion, and the modes with large numbers have the large phase velocity. The vertical distribution of normal wave amplitudes is determined by formula (8). In Fig. 3 the dependence of Eyn on z for three normal waves are given. It is seen from Fig. 3

that within the conducted layer the Úeld amplitude of the basic mode of the wave is practically homogeneous. The Úeld of the waves with higher numbers signiÚcantly changes within the layer. Outside the layer, the Úeld exponentially increases with distance over the wavelength of the appropriate mode. For the basic mode at n = 0, we have E
y0

= E0 [cosh(z=l)]

-(!l=a)2= 3

:

(10)

The slow change of the basic mode Úeld across the conducting layer allows to apply the approached method for its accounts. This method consists of the replacement of real distribution of conductivity by inÚnitely thin conducting 2 2 layer H (z )= H0 (z= 2l) under the condition kl 1 (where (z= 2l) is Dirac delta-function, and l is the characteristic thickness of a conducting layer). Using this method, we shall estimate the accuracy of the approached solution, comparing it with exact one (10). Let us integrate Eq. (7) on z and tend l to zero, then we obtain l d Ey dz +2 ! ak
2

Ey (0)=0;

{Ey } =0;


V.M. Sorokin et al. / Journal of Atmospheric and Solar-Terrestrial Physics 64 (2003) 21 ­ 29

25

where brackets {:::} designate the di erence of quantities above and below the conducting layer. The electric Úeld is determined from Laplace equation outside this layer. Its solution is z¿ 0 Ey = E0 exp(-kz ); z ¡ 0 Ey = E0 exp(kz ): Substituting this solution in the boundary condition, we obtain dispersion relation for a wave in the layer k 3 = !2 =la2 . Taking into account this relation we shall write down the electric Úeld distribution in the upper semi-area: z¿ 0 Ey = E0 exp(-kz )= E0 exp - z l !l a
2= 3

Úeld and its vertical derivative above and below the ionosphere d Ey dz +i + 4! c2 4! kc2
- P 2 - 2 H

(z )d z {Ey } =0: (13)

(z )d z Ey =0;

:

As the velocity of electromagnetic wave in the upper ionosphere and in the Earth-ionosphere layer is much higher than the velocity of GW, the Úelds in these areas determined from the Laplace equation have the following form: Ey = Ey0 exp(-kz ); z ¿ 0; -z1 ¡z ¡ 0:

Let us compare the obtained approached solution with the exact one, which follows from Exp. (10) at z=l1: E y = E0 2
(!l=a)2= 3

Ey = Ey0 sinh[k (z + z1 )]= sinh(kz1 );

exp -

z l

!l a

2= 3

:

Substituting this solution in the boundary conditions (13), we obtain !2 +iv!k 2 - k 3 a2 l[1 + coth(kz1 )]=0; where a= c 4 c 4
2 2

Di erence between these solutions is determined by the fac2= 3 tor 2(!l=a) 1+(kl) ln 2, which is of the order of unit in the considered approximation kl 1. Thus, the approached method with a su cient accuracy allows to analyze the effects of the wave generation and propagation in the thin layers by introducing the appropriate boundary conditions. Below, we take advantage of this method for the analysis of absorption of GW. We believe that the conductivity of ionosphere is concentrated within two at layers near the plane z = 0. We have non-zero Hall conductivity in the bottom layer and non-zero Pedersen conductivity in the top layer. The ideally conducting Earth coincides with the plane z = -zl . In the Earth-ionosphere layer, the conductivity is assumed to be zero. From Eq. (6) we obtain the equations for Ey in the layer with Hall conductivity @2 @x2 @2 @2 +2 @x2 @z Ey - 4! c2
2 2 H

(14)

l

dz

2 H

(z )

=

c 4l

2 H0

; (z )

v=

d z P (z ) ; 2 d z H (z )

l=

dz

2 H 2 H0

:

Expression (14) is the dispersion equation for GW in the lower ionosphere accounting absorption and in uence of the ideally conducting Earth. 4. Calculation of ULF Úeld oscillations on the ground during the occurrence of periodic conductivity inhomogeneities in the ionosphere We will consider the generation of GW propagated along an x-axis by inhomogeneities of ionospheric conductivity. Let the coordinate dependence of conductivity have a form
H

(z )Ey = 0

(11)

and in the layer with Pedersen conductivity @2 @2 +2 2 @x @z Ey +i 4! c2
P

(z )Ey =0:

(12)

=

H0

(z )+

H1

(x; z );

P

=

P0

(z )+

P1

(x; z );

Integration of the equations on z at the condition that the width of the layers will tend to zero leads to boundary conditions for components Ey and its vertical derivative at transition through each of these layers. Assuming the Úeld dependence on x as exp(ikx) we obtain a boundary condition for the layer with Hall conductivity d Ey dz + 4! kc2
2 - 2 H

(z )d zEy ;

{Ey } =0

and for a layer with Pedersen conductivity d Ey 4! {Ey } =0: +i 2 P (z )d zEy =0; dz c - Equating the distance between the layers to zero and putting these expressions, we Únd the boundary conditions connecting mean values of tangential components of an electrical

where the index 0 designates the undisturbed conductivity, and index 1 is its disturbance. Let us present the electric Úeld as a sum Ey = Ey0 + Ey1 , where E0 is background electric and E1 represents a Úeld arising during the occurrence of ionospheric conductivity inhomogeneities. Let us substitute these expressions to equalities (11) and (12). Assuming that the conductivity disturbances are small, and taking into account the Úrst order disturbances of the electric Úeld we obtain the equations for Eyl within the layers with Hall conductivity: @2 @x2 = @2 @2 +2 2 @x @z 4! c2
2 2 H0

E

y1

-

4! c2
y0

2

2 H0

(z )E

y1

(z )h(x)E


26

V.M. Sorokin et al. / Journal of Atmospheric and Solar-Terrestrial Physics 64 (2003) 21 ­ 29

and Pedersen conductivity @ @ + @x2 @z =-i
2 2 2

E

y1

+i

4! c2

P

(z )E

y1

is possible to Únd the approximate expression for inverse Fourier transform in k , having obtained the spatial structure of a spectrum of GW Ey1 (!; x) i !2 = e- Ey0 (!) 4!0 u +e
i k0 x 0 i k0 x 0

4! P0 (z )p(x )Ey0 : c2 In these equations, the relative disturbances of a Hall and Pedersen conductivities are entered H1 (x; z ) P1 (x; z ) h(x)= ; p(x)= : H0 (z ) P0 (z ) Let the unknown values depend on coordinate x as exp(ikx). Using a method stated above, we shall obtain the boundary conditions connecting the tangential components of the electrical Úeld and its vertical derivative above and below the ionosphere d Ey 1 !2 1 +i!v Ey1 +2 dz la k 1 =- 2 la {Ey1 } =0: In Eq. (15) the following designations are introduced: fH (k; z )= fP (k; z )=
- -

C (x + s)eisq d s (17)



C (x - s)eisq d s ;

2 2 where u =d !0 (k0 )= d k0 ; q = !2 - !0 +iv!k0 = 2!0 u. The electric Úeld disturbance Ey1 generates an additional current in the ionosphere. The magnetic Úeld of this current can be observed on the Earth's surface. Using the Maxwell equation rot E =i!B=c, and the expression determining the electric Úeld in the Earth-ionosphere layer, it is easy to show that the magnetic disturbance on the ground is also deÚned by Eq. (17):

! k

2

fH (k; 0)+ifP (k; 0) ;

Bx1 (x; !) Bx0 (!)

=
z =-z1

Ey1 (x; !) Ey0 (!)

:
z =0

Let us choose the model distribution C (x)= A exp(-|x|=L). Then integrals (17) are expressed in elementary functions (15) Bx1 (x; !) (!= )2 {e =A Bx0 (!) 1+(qL)2
-|x|=L

[sin(k0 |x|)

eikx h(x)Ey0 (x; z; !)d x; eikx p(x)Ey0 (x; z; !)d x:

-(qL) cos(k0 |x|)] + i exp[i(k0 + q)|x|]}; (18) where = 2!0 u=L. The estimates show (Fatkullin et al., 1981) that a = 4 â 104 m= s; v = 2 â 108 m2 = s. According to the satellite data (Chmyrev et al., 1997), the characteristic sizes of the disturbed ionosphere area 2L is few hundred kilometers, and spatial scale of disturbance d = l= 2 = =k0 is tens of kilometers. For accounts we assume the size of the relative disturbance of Hall conductivity A =0:1 and L =105 m. The diagram of relative disturbance of the geomagnetic uctuations spectrum |Bx1 (x; !)=Bx0 (!)|, found from formula (18) for epicenter, is given in Fig. 4. It is seen that the relative disturbance is maximal at the !m 2 and 5 Hz frequencies for two spatial scales of disturbance d = l= 2 = =k0 , and its magnitude in epicenter reaches approximately 20 ­25% of the undisturbed value. It has been calculated by formula (18) that the average value of the spectrum maximum amplitude B1 (x; !m ) =B0 (!m ) depends on the x distance from epicenter of the disturbed region. The result of this calculation presented in Fig. 5 enables one to Únd a spatial scale of the region, where ULF pulsations can be observed. As follows from Fig. 5, this spatial scale is of the order of 200 ­300 km.

During the performance of an inequality (!=k )2 iv! absorption of GW is weak. Let us assume that the relative disturbance of Hall conductivity is not less than the relative disturbance of Pedersen conductivity. This condition is not basic, however it allows one to simplify the calculations. Thus, the second term in the right-hand side of Eq. (15) can be neglected. In such a case, when horizontal scale of E0 (x; z; !) variation, representing the undisturbed Úeld, exceeds the horizontal scale of disturbed area, it can be taken from a sign of integral: fH (k; z ) = Ey0 (z; !)H (k ), where H (k ) is Fourier image of h(x) H (k )=
-

d x eikx h(x):

Substituting the solution for Úelds above the ionosphere and in the Earth-ionosphere layer into the boundary conditions (15), we obtain Ey1 (!; k )= -Ey0 (!) !2 H (k ) ; 2 ! - !0 (k )+iv!k 2
2

(16)

2 where !0 (k )=2la2 |k |3 . As an example, we choose the relative disturbance of Hall conductivity as h(x) = C (x) cos(k0 x), where C (x) slowly varies on scale l0 = 2 =k0 . Taking into account the quasi-periodic character of h(x) variation in Eq. (16) it

5. Conclusion It is shown that the interaction of background electromagnetic noise with periodic horizontal inhomogeneities


V.M. Sorokin et al. / Journal of Atmospheric and Solar-Terrestrial Physics 64 (2003) 21 ­ 29

27

0.25

|Bx1 ( x, ) / Bx0 ()| 1

0.20

0.15

2
0.10

0.05

1

2

3

4

5

6

/2, H z
Fig. 4. The calculated normalized spectrum of geomagnetic oscillations for di erent spatial scales of inhomogeneities. (1) d =15 km; (2) d =30 km.

< B1( x. m )>/B0 (m )
0.14 0.12 0.10 0.08 0.06 0.04 0.02

epicenter reaches approximately 20 ­25% above the undisturbed background level. The spectral maximum amplitude decreases dependence on a distance from epicenter. The spectral maximum frequency decreases monotonously the dependence on the spatial scale of inhomogeneities. The calculations show that the characteristic frequency of geomagnetic pulsations connected with GW belongs to ULF range 0.1­10 Hz of electromagnetic emissions registered, for example, by Kopytenko et al. (2001) before earthquakes. The electromagnetic emissions with the maximal frequency over units of Hz during seismic activity and volcanic eruptions have been observed by Rauscher and Van Bise (1999). From the presented model, it follows that if some irregularities of the ionospheric conductivity with various spatial scales exist simultaneously, then perturbations on the Earth surface can be observed in several, connected to them by spectral bands. Note that the spectrum maximum frequency depends essentially on the volume of Hall and Pedersen ionospheric conductivity, angle of magnetic Úeld inclination, azimuth of direction of the wave propagation, and spatial scale of conductivity irregularities. The variations of these parameters lead to the change of the spectrum maximum frequency over a wide range. Using a simple model, we illustrate the generation mechanism of geomagnetic pulsations. For analysis of the experimental data received during seismic activity on the basis of this mechanism, it is necessary to develop a submitted model. Therefore, in the present work, we are limited by the estimations of the geomagnetic pulsation characteristics. For discovering ULF pulsations connected with GW in the ionosphere, it is necessary to provide the precise measurements of the wave phase delay in some points of the Earth's surface. The horizontal velocities of the ULF waves propagating from the epicenter must be of the order of 10 ­100 km= s. Those velocities should be increased depending on the frequency growth.

0

50

100

150

200

250

x . km

Fig. 5. Dependence of average value of the spectrum maximum amplitude on the x distance from the epicenter of the disturbed region.

Acknowledgements This work is executed with the partial support of RFBR (Grants 99-05-65650 and 00-05-64503).

of ionospheric conductivity with the spatial scale over 10 km results in the generation of electromagnetic waves with a narrow-band spectrum. Various sources generate background electromagnetic noise in ULF range. The most powerful are thunderstorms. Oscillating noise of the electric Úeld forms the polarization currents by conductivity inhomogeneities in the ionosphere. These horizontal periodic electric currents with the spatial scale 10 km are considered as a source of GW. Generation and propagation of these waves lead to the excitation of the narrow band magnetic oscillations at 0.1­10 Hz frequencies on the ground, which result in interference e ect. Its value in

Appendix A Below we Únd the solution of Eq. (7). For the function w we obtain d2 w 2s z - tanh d z2 l l - s + l2 dw + dz
2

s(s + 1)[sinh (z=l)] l2 [cosh (z=l)]2 -k
2

2

4 ! H0 c2 k cosh(z=l)

w =0:


28

V.M. Sorokin et al. / Journal of Atmospheric and Solar-Terrestrial Physics 64 (2003) 21 ­ 29 Fatkullin, M.N., Zelenova, T.N., Kozlov, Z.K., Legenka, A.D., Soboleva, T.N., 1981. Empirical Models of Middle­Latitude Ionosphere. Science, Moscow, p. 256. Fenoglio, M.A., Johnston, M.J.S., Byerllee, J.D., 1994. Magnetic and electric Úelds associated with changes in high pore pressure in fault zone--application to the Loma Prieta ULF emissions. Proceedings of Workshop LXIII, Menlo Park, CA, pp. 262­278. Fraser-Smith, A.C., Bernardi, A., McGill, P.R., Ladd, M.E., Helliwell, R.A., Villard Jr., O.G., 1990. Low-frequency magnetic Úeld measurements near epicenter of the MS 7.1 Loma Prieta earthquake. Geophysical Research Letters 17, 1465­1468. Ginzburg, V.L., 1967. Propagation of Electromagnetic Waves in Plasma. Nauka, Moscow, p. 683. Hayakawa, M., Kawate, R., Molchanov, O.A., Yumoto, K., 1996. Results of ultra-low-frequency magnetic Úeld measurements during the Guam earthquake of August 1993. Geophysical Research Letters 23, 241­244. Isaev, N., Sorokin, V., Chmyrev, V., 2000. Sea storm electrodynamic e ects in the ionosphere. International Workshop on Seismo--Electromagnetics of NASDA. Tokyo, Japan, p. 43. Johnston, M.J.S., Muller, R.J., Sasai, Y., 1994. Magnetic Úeld observations in the near-Úeld: the 28 June 1992 Mw 7.3 Landers, California. Earthquake. Bulletin of the Seismological Society of America 84, 792­798. Kopytenko, Y.A., Matiashvili, T.G, Voronov, P.M., Kopytenko, E.A., 1994. Observation of electromagnetic ultralow-frequency litospheric emissions in the Caucasian seismically active zone and their connection with earthquakes. In: Hayakawa, M. (Ed.), Electromagnetic Phenomena Related to Earthquake Prediction. Terra ScientiÚc Publishing Company, Tokyo, pp. 175 ­180. Kopytenko, Y., Ismagilov, V., Hayakawa, M., Smirnova, N., Troyan, V., Peterson, T., 2001. Investigation of the ULF electromagnetic phenomena elated to earthquakes: contemporary achievements and the perspectives. Ann. Geo. Phys. 44, 325. Molchanov, O.A., Hayakawa, M., 1995. Generation of ULF electromagnetic emissions by microfracturing. Geophysical Research Letters 22, 3091­3094. Rauscher, E.A., Van Bise, W.L., 1999. The relationship of extremely low frequency electromagnetic and magnetic Úelds associated with seismic and volcanic natural activity and artiÚcial ionospheric disturbances. In: Hayakawa, M. (Ed.), Atmospheric and Ionospheric Electromagnetic Phenomena Associated with Earthquakes. Terra ScientiÚc Publishing Company, Tokyo, pp. 459­487. Sorokin, V.M., 1986. On a role of the ionosphere in propagation of the geomagnetic pulsations. Geomagnetism and Aeronomy 26, 646­652. Sorokin, V.M., 1988. Wave processes in the ionosphere, connected with a geomagnetic Úeld. Radiophysics 31, 1169­1179. Sorokin, V.M., Chmyrev, V.M., 1999. Instability of acoustic-gravity waves in the ionosphere connected with in uence of an electric Úeld. Geomagnetism and Aeronomy 39, 38­45. Sorokin, V.M., Fedorovich, G.V., 1982. Propagation of short-period waves in the ionosphere. Radiophysics 25, 495­501. Sorokin, V.M., Yaschenko, A.K., 1988. Propagation of Pi2 pulsations in the lower ionosphere. Geomagnetism and Aeronomy 28, 655­660. Sorokin, V.M., Yaschenko, A.K., 1999. Disturbance of conductivity and electrical Úeld in the Earth-ionosphere layer above the preparing earthquake epicenter. Geomagnetism and Aeronomy 39, 100­106.

Parameter s is determined from a condition of the constancy of coe cient in the bracket at unknown w s(s + 1)[sinh (z=l)]2 +(4 l2 [cosh (z=l)] Hence, we obtain 1 s= -1+ 1+4 2
H0 2

!l=c2 k )

2

=

s(s +1) : l2

4 c

H0 2k

!l

2



:

(A.1)

The equation for w with parameter s determined from Eq. (A.1) has the following form: d2 w 2s z - th d z2 l l dw + dz s2 -k l2
2

w =0:

The variable replacement according to = [sinh (z=l)]2 transforms this equality to the hypergeometric equation (Bateman and Erdelyi, 1953) ( +1)w + (1 - s) + 1 1 w + (s2 - k 2 l2 )w =0: 2 4

The solution of this equation is the hypergeometric function: w=F -s + kl -s - kl 1 ; ; ;- 2 2 2 : (A.2)

References
Alperovich, L.S., Zheludev, V.A., 1999. Long-period geomagnetic precursors of the Loma-Prieta earthquake discovered by wavelet method. In: Hayakawa, M. (Ed.), Atmospheric and Ionospheric Electromagnetic Phenomena Associated with Earthquakes. Terra ScientiÚc Publishing Company, Tokyo, pp. 123­136. Bateman, H., Erdelyi, A., 1953. Higher Transcendental Functions. McGraw-Hill, New York, Toronto, London, p. 296. Chmyrev, V.M., Isaev, N.V., Bilichenko, S.V., Stanev, G.A., 1989. Observation by space-borne detectors of electric Úelds and hydromagnetic waves in the ionosphere over on earthquake center. Physics of Earth and Planet Interactions 57, 110­114. Chmyrev, V.M., Isaev, N.V., Serebryakova, O.N., Sorokin, V.M., Sobolev, Ya.P., 1997. Small-scale plasma inhomogeneities and correlated ULF emissions in the ionosphere over an earthquake region. Journal of Atmospheric and Solar-Terrestrial Physics 59, 967­974. Chmyrev, V.M., Sorokin, V.M., Pokhotelov, O.A., 1999. Theory of small scale plasma density inhomogeneities and ULF= ULF magnetic Úeld oscillations excited in the ionosphere prior to earthquakes. In: Hayakawa, M. (Ed.), Atmospheric and Ionospheric Electromagnetic Phenomena Associated with Earthquakes. Terra ScientiÚc Publishing Company, Tokyo, pp. 759­776. Draganov, A.B., Inan, U.S., Taranenko, Yu.T., 1991. ULF magnetic signatures at the Earth's surface due to ground water ow. Geophysical Research Letters 18, 1127­1130.


V.M. Sorokin et al. / Journal of Atmospheric and Solar-Terrestrial Physics 64 (2003) 21 ­ 29 Sorokin, V.M., Chmyrev, V.M., Isaev, N.V., 1998. A generation model of small-scale geomagnetic Úeld-aligned plasma inhomogeneities in the ionosphere. Journal of Atmospheric and Solar-Terrestrial Physics 60, 1331­1342. Sorokin, V.M., Chmyrev, V.M., Yaschenko, A.K., 1999. Electrodynamic model of the atmosphere--ionosphere coupling related to seismic activity. Workshop on the Micro-satellite

29

DEMETER. Detection of Electro-Magnetic Emission Transmitted from Earthquake Regions, Orleans, France, Abstracts, pp. 46 ­47. Sorokin, V.M., Chmyrev, V.M., Yaschenko, A.K., 2001. Electrodynamic model of the lower atmosphere and the ionosphere coupling. Journal of Atmospheric and Solar-Terrestrial Physics 63, 1681.