Depression of the ULF geomagnetic pulsation related to ionospheric irregularities

We consider a depression in intensity of ULF magnetic pulsations, which is observed on the ground surface due to appearance of the irregularities in the ionosphere. It is supposed that oblique Alfven waves in the ULF frequency range are downgoing from the magnetosphere and the horizontal irregularities of ionospheric conductivity are created by upgoing atmospheric gravity waves from seismic source. Unlike the companion paper by Molchanov et al. (2003), we used a simple model of the ionospheric layer but took into consideration the lateral inhomogeneity of the perturbation region in the ionosphere. It is shown that ULF intensity could be essentially decreased for frequencies f = 0.001-0.1 Hz at nighttime but the change is negligible at daytime in coincidence with observational results. Mailing address: Dr. Valery M. Sorokin, Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation (IZMIRAN), Russian Academy of Science, Troitsk (Moscow Region), 142092 Russia; e-mail: sova@izmiran.rssi.ru 192 Valery M. Sorokin, Evgeny N. Fedorov, Alexander Yu. Schekotov, Oleg A. Molchanov and Masashi Hayakawa sumption of atmospheric gravity wave intensification induced by changes in temperature and pressure near the ground due to gas and water release in a course of earthquake preparation. Mareev et al. (2002) considered gravity waves intensification in the ionosphere. An appearance of gravity waves in the ionosphere leads to depression of downgoing from magnetosphere ULF waves due to loss of coherency along the wave front (like scattering) and due to a change in effective ionospheric conductivity. Here we are going to investigate the latter process theoretically. Some results on this subject are obtained in a companion paper by Molchanov et al. (2003) in assumption of vertical stratification of the ionosphere. Here we consider the same effect in assumption of lateral inhomogeneity but for thin ionospheric layer. 2. Electromagnetic ULF field in horizontally inhomogeneous ionosphere We use a simple model, which includes the following: i) Source of the magnetic field ULF pulsations is situated in the magnetosphere and it generates downgoing Alfven waves. They propagate in a homogeneous magnetospheric medium above ionosphere along z-axis, which is coincident with the direction of the external magnetic field. The frequency of the waves ω << ωci, where ωci is ion cyclotron frequency, therefore they propagate with characteristic Alfven wave velocity CA and their vertical wave number kz = = kA = ω /CA. In addition we suppose their horizontal wave number k >> kA, another words we consider oblique Alfven waves. ii) Ionosphere is a thin layer at z = 0 with integrated Pedersen and Hall conductivities ΣP ⋅ ⋅ (x, t), ΣH (x, t) respectively, which are time-dependent and inhomogeneous along the horizontal x-axis (see fig. 1). For simplicity, we neglect input due to ionosphere thickness ∆h, because of kA∆h << 1 and present ( , ) ( , ) x t x t , , , P H P H P H 0 = + Σ Σ ∆Σ . (2.1) iii) Atmosphere below ionospheric layer is nonconductive and current-free, but the ground medium is completely conductive and tangential electric field disappears at the ground surface z = h. iv) For simplicity, we assume independence of the all field components on y-coordinate, i.e. ∂/∂y = 0, and suppose large conductivity along z-axis for magnetosphere and ionosphere that leads to disappearance of electric field component Ez ∼ 0. Our model is about the same as was discussed in many other papers (e.g., Lyatsky and Maltsev, 1983). The only difference is assumption on lateral ionospheric inhomogeneity. Alfven wave can reflect from the ionospheric layer and transform in the isotropic mode wave (so-called fast magneto-sonic wave) inside ionosphere. Then both waves penetrate into the Earth-ionospheric cavity. As usual we consider Fourier expansion ) i t ~ ( , , ) ( , , ) E x z t dk d E k z e 2 2 ( r r ikx = r r ~ ~ 3 3

sumption of atmospheric gravity wave intensification induced by changes in temperature and pressure near the ground due to gas and water release in a course of earthquake preparation.Mareev et al. (2002) considered gravity waves intensification in the ionosphere.An appearance of gravity waves in the ionosphere leads to depression of downgoing from magnetosphere ULF waves due to loss of coherency along the wave front (like scattering) and due to a change in effective ionospheric conductivity.Here we are going to investigate the latter process theoretically.Some results on this subject are obtained in a companion paper by Molchanov et al. (2003) in assumption of vertical stratification of the ionosphere.Here we consider the same effect in assumption of lateral inhomogeneity but for thin ionospheric layer.

Electromagnetic ULF field in horizontally inhomogeneous ionosphere
We use a simple model, which includes the following: i) Source of the magnetic field ULF pulsations is situated in the magnetosphere and it generates downgoing Alfven waves.They propagate in a homogeneous magnetospheric medium above ionosphere along z-axis, which is coincident with the direction of the external magnetic field.The frequency of the waves ω << ωci, where ωci is ion cyclotron frequency, therefore they propagate with characteristic Alfven wave velocity CA and their vertical wave number kz = = kA = ω /CA.In addition we suppose their horizontal wave number k >> kA, another words we consider oblique Alfven waves.
ii) Ionosphere is a thin layer at z = 0 with integrated Pedersen and Hall conductivities ΣP⋅ ⋅ (x, t), ΣH (x, t) respectively, which are time-dependent and inhomogeneous along the horizontal x-axis (see fig. 1).For simplicity, we neglect input due to ionosphere thickness ∆h, because of kA∆h << 1 and present iii) Atmosphere below ionospheric layer is nonconductive and current-free, but the ground medi-um is completely conductive and tangential electric field disappears at the ground surface z = h.iv) For simplicity, we assume independence of the all field components on y-coordinate, i.e. ∂/∂y = 0, and suppose large conductivity along z-axis for magnetosphere and ionosphere that leads to disappearance of electric field component Ez ∼ 0.
Our model is about the same as was discussed in many other papers (e.g., Lyatsky and Maltsev, 1983).The only difference is assumption on lateral ionospheric inhomogeneity.
Alfven wave can reflect from the ionospheric layer and transform in the isotropic mode wave (so-called fast magneto-sonic wave) inside ionosphere.Then both waves penetrate into the Earth-ionospheric cavity.As usual we consider Fourier expansion where index r = x, y, and the same for magnetic components br.In our model for Alfven wave in the magnetosphere only Ex and by exist and these spectral component are described by following equations: ( , , ) ( , , ) , .
In opposite only component E y, bx, bz keep in the isotropic wave

2.4)
As concerned the situation below ionosphere (at the atmosphere) we have well-known relationships where k 0 2 << k 2 and k z Then after matching of the magnetospheric and atmospheric fields in the ionosphere layer we obtain integral equation for the fields inside ionosphere (see Appendix for details) Its connection with ULF magnetic field at the ground is also analyzed in Appendix, where shown bx (ω, h) >> by (ω, h) and

Change in the ULF spectrum on the ground induced by ionospheric perturbations
Using conventional approach of perturbation theory we transform (2.6) as following: where inverse tensor K After substitution in (2.7) we have where ULF magnetic field at the ground we find as the inverse Fourier transform of (3.2) In (3.3) the second term b x1 is due to presence of perturbations in the ionosphere and could be compared with bx0.As example for solitary initial wave where k x = 2π /λx and λx is horizontal scale of downgoing Alfven wave.Let us consider influence a moving density variation in the ionosphere (e.g., gravity waves).Alperovich et al. (2002) showed that such a variation leads to mainly perturbation of Pedersen conductivity.So in (3.4) we leave only the first term in the integrand and present the perturbation as following: where ∆ΣP0 is averaged amplitude of the perturbation induced by gravity wave, ϕ is random phase of the gravity wave, L is spatial scale of the seismic region.Substituting (3.5) in (3.4) we obtain the relative change of ULF magnetic field at the ground (3.6) Finally after averaging we have for the value the following resultant relation: (3.8) In the center of the zone, ξ = 0 relation (3.8) reduced to the following:

Appendix
Let us represent the solution of (2.3) as the sum of downgoing and reflecting waves where E * (x, ω) is amplitude distribution on the wave front and R a is reflection coefficient.
It is evident that 1)/(a2 + 1) and β (ω) decreases on frequency.At night-time ionosphere when ΣP0 << ΣA we have ε ≈ ∆ΣP0/ΣA, a1≈1, and a2<<1.At day-time ionosphere when ΣP0 >> ΣA we have ε ≈ ∆ΣP0 / ΣA, a1 ≈ a2 >> 1. Dependence of β on frequency under the center of inhomogeneity is shown in fig. 2 for the night and day hours.The values of integrated Pedersen and Hall conductivities are given in the figure.For comparison, these dependences calculated for laterally homogeneous ionosphere of the finite thickness are also shown in fig. 2 for the same values of integrated ionospheric conductivities.The values of β (ω) at relatively low frequencies (f < 0.05 Hz) are about 0.6-0.7 for nighttime and 0.9-0.95 at daytime for both models.At higher frequencies (0.2-0.3 Hz) β (ω) calculated with the use of IRI model grows almost up to unity.Several extrema seen at the curve are caused by the ionospheric Alfven resonance and ionospheric MHD waveguide.This effect is not obtained in the thin ionosphere model that gives strong monotonous decrease of β with frequency at f > 0.3 Hz.At daytime β weakly depends on frequency and is about unity in all the frequency range considered.Spatial distribution of β value at the ground found with the relation (3.7) is presented in fig. 3. Note that the size of the depression area is larger than the perturbation scale in the ionosphere.

Discussion and conclusions
It seems that theoretical results here coincide with observational data reported by Molchanov et al. (2003).Recently Sorokin et al. (2002Sorokin et al. ( , 2003) ) investigated the possibility of generating seismoinduced geomagnetic pulsations due to preseismic changes in the background electric field.
Here we try to estimate another possibility in connection with intensification of the atmospheric gravity waves before earthquakes and following the appearance of the ionospheric inhomogeneities.While influence of the horizontal ionospheric irregularities on propagation of the VLF waves is known (e.g., Shklyar and Nagano, 1998), the same influence on the Alfven waves is a rather original problem and we are going to continue this research for a more complicated model taking into consideration both vertical and horizontal stratifications of the ionosphere.
The similar relationship for isotropic wave as follows: where R i is reflection coefficient for isotropic waves.As concerned solution of (2.5) we need to take into consideration that Ex,y (z = h) = 0, and to match with above-mentioned solutions at the lower boundary of the ionosphere, hence , , ( ) ( , ) ( , , ) , , sinh cosh coth where T a is transmission coefficient for Alfven wave and T i is coefficient of transformation the Alfven wave into isotropic wave in the ionosphere.It is evident that by/bx = (ω /c) 2 T a /(k 2 T i ) << 1, if the transformation is essential.It means that geomagnetic pulsations observed at the ground surface are mainly related to isotropic wave, which is originated from Alfven wave inside ionosphere.
The electric fields at the upper boundary of the ionosphere are continuous, hence However discontinuity of the magnetic fields equals to surface currents at the boundary   Suppose now that time variation of the ionospheric conductivity is slow in comparison with geomagnetic pulsations.For example characteristic frequencies of the atmospheric gravity waves are ω0 ∼ (10 −3 − 10 −4 )c −1 << ω.If so, we can simplify time integration in (A.5).Using now (2.1), (A.1-A.4) and obvious relation

Fig. 1 .
Fig. 1.The model used to calculate the depression of the ULF geomagnetic pulsations: 1 -conducting ionosphere; 2 -ionospheric inhomogeneities; 3 -Earth surface.h is the height of the lower boundary of the ionosphere; L is the spatial scale of the seismic region.
we denote undisturbed integrated Pedersen and Hall conductivities Σ P0, ΣH0 respectively.In a case of the large perturbation zone, α >> 1 after simple calculations we have

Fig. 2 .
Fig. 2. Dependence of the relative magnetic field β at the ground on frequency for night and day hours.Thin lines show the results of calculations in the thin film ionospheric model, bold lines correspond to full-wave calculations in the IRI-90 model.kx = 0.01 km −1 and other parameters are shown on the picture.

Fig. 3 .
Fig. 3. Spatial distribution of the relative magnetic field β dependence on distance from projection of the perturbation center in the ionosphere, h is height of lower boundary of the ionosphere (here h = 100 km).