On the acoustic model of lithosphere-atmosphere-ionosphere coupling before earthquakes

In this work, the many-fluid magnetohydrodynamic theory is applied to describe the modification of the electromagnetic field of the ionospheric E-layer by acoustictype waves. There, altitudinal profiles of the electromagnetic field and the plasma parameters of the atmosphere and ionosphere are taken into account. It is concluded that at Eregion altitudes above seismo-active regions, magnetohydrodynamic waves as Alfv́ en and magnetoacoustic ones might change their amplitude and direction of propagation. Waves of the Farley-Buneman type might also be excited a few days before very strong earthquakes. The collisions between the neutral and charged particles of the E-layer also cause diffusion and heating processes. Thus, changes of the characteristic foE-frequency might be obtained.


Introduction
Acoustic and acoustic-gravity waves were indeed observed at subionospheric altitudes during earthquake preparation times (Cook, 1971;Koshevaya et al., 2002;Rozhnoi et al., 2007;Pulinets and Boyarchuk, 2004;Hayakawa, 2011).Thus, in some of the models of lithosphere-atmosphere-ionosphere coupling before earthquakes, it is assumed that atmospheric acoustic and acoustic-gravity waves are generated several days before earthquakes in earthquake preparation zones and that they propagate from the Earth's surface through the atmosphere up to ionospheric altitudes (Shalimov and Gokhberg, 1998;Pulinets and Boyarchuk, 2004;Liperovsky et al., 2008).There, due to collisions between the neutral and charged particles, disturbances of the charged particle densities are possible.In ionospheric E-layers, atmospheric acous-Correspondence to: C.-V. Meister (c.v.meister@skmail.ikp.phyik.tudarmstadt.de)tic and acoustic-gravity waves interact in particular with sporadic layers and cause nonlinear current systems and diffusion processes.Liperovskaya et al. (1994) found that acoustic-gravity waves with periods of 2-3 h can apparently cause the diffusion of sporadic E-layers at distances of about 1000 km from the wave generation region.Further, upon investigating the interaction of infrasound waves of seismic origin with sporadic layers, it was found that waves observable by groundbased radar stations of the Farley-Buneman type, may be excited (Liperovsky et al., 1997;Meister, 1995).
Besides this, in (Koshevaya et al., 2002) it was concluded that at E-region altitudes, the conversion of sound waves into Alfvén waves is possible.
The theoretical description of the wave conversion in a stratified magnetized plasma has yet to be further developed.Solutions for acoustic waves are well-described for non-magnetic systems using one-fluid magnetohydrodynamics; when studying electromagnetic waves one usually neglects the stratification of the medium and the finite electrical conductivity values (Koshevaya et al., 2002;Sturrock, 1994) or the collisions between neutral and charged particles (Axelsson, 1998).Nonetheless it is of interest to analyse the modification of electromagnetic waves that have been locally excited in the atmosphere by acoustic-type ones.In the work of (Meister et al., 2010), some first steps were made to consider within the frame of multi-fluid magnetohydrodynamics, altitudinal profiles of particle velocities and electromagnetic fields in an isothermal system.Thus, in the present paper it is attempted to consider the modulation of magnetoacoustic waves by acoustic-type ones within the frame of multi-fluid magnetohydrodynamics, taking into account altitudinal profiles of the non-isothermal, isotropic ionospheric E-layer.

Variation of electromagnetic waves by acoustic ones
In the proposed multi-fluid magnetohydrodynamic model, the variation and excitation of electromagnetic waves by acoustic-type ones is studied starting with the continuity equations of the charged (a = e -electron, a = i -ion), and neutral (a = n) particles, the momentum balances (2) and the equation of state v a , m a , n a , ω a , and p a describe the velocities, masses, number densities, cyclotron frequencies, and the partial pressure of the particles of type a, respectively.m ab = m a m b /(m a + m b ), γ is the polytropic coefficient, and ν ab are the frequencies of the collisions between particles of kinds a and b. b designates charged and neutral particles too.Besides, the Maxwell equations are taken into account (replacement currents are neglected), rotB(r,t) = µ o j (r,t) = µ o q e (n e v e − n i v i ), ( 5) Under equilibrium conditions (index "o"), when all particle velocities equal zero, one obtains for the momentum balance is the altitudinal-dependent scale height of the pressure of the Earth's atmosphere and ionosphere.Further, the background vertical electric field is approximated by a linear expression (Pfaff et al., 2005).z = 0 corresponds to an altitude of 80 km.Values of the background electric field at altitudes between 80 km and 130 km are presented in Fig. 1.It has to be noted, that the direction of E considered in (Pfaff et al., 2005) is unclear.Here it is assumed that the presented data are those of the vertical electric field.
Further, taking into account that the height scale of the density of the oxygen molecules O 2 with mass m n may be described by et al., 2007), one finds for the temperature T of the E-layer, which is equal for all particles Then for the altitudinal scale height of the pressure of neutral and ionized particles of mean mass m n ≈ m i and charge Temperature profiles of the E-layer according to Eq. ( 12) are presented in Fig. 1 for α equal to 0.1 and 0.3.The temperature values are compared with experimental data taken from the MSIS-E-90 Atmospheric Model of the Virtual Ionosphere, Thermosphere, Mesosphere Observatory (VITMO) (http://omniweb.gsfc.nasa.gov/vitmo/msisvitmo.html).
Next, it is supposed that the plasma system is perturbed by an acoustic-gravity or infrasound wave, expressed by the velocity of the neutral particles v n .In such a case, also the partial pressures and densities of the plasma particles as well as the electromagnetic field show deviations from the equilibrium values, v a = δv a , n a = n ao + δn a , p a = p ao + δp a , ( 14) The index "o" designates the unperturbed values of the parameters.In the following it is assumed that the unperturbed electric and magnetic fields are directed along the z-axis.
The mean electric field E o is altitudinal dependent, and the mean magnetic field B o is constant.Viscosity effects are neglected.Substituting Eq. ( 14) into Eq.( 8) and averaging over the deviations, one finds Then the altitudinal profile of the background electron pressure may be presented by Further, putting Eq. ( 14) into Eqs.(1-3), subtracting the equilibrium values, and retaining only terms of first order in the perturbations, one arrives at the following equations (ω ca = q a B/m a ): n ao ∂δv a ∂t = − 1 m a gradδp a + q a m a δn a E oz + q a m a n ao δE ( 18) From the Maxwell equations follows for the electromagnetic field of the plasma disturbances rotδB = µ o δj = µ o q e (n eo δv e − n io δv i ), ( 22) Usually, to obtain the relations for the waves excited in the non-stratified plasma, one introduces the Fourier transformation of the plasma parameters and of the electromagnetic field.But, since the background plasma pressure and particle densities show an exponential decrease with the altitude, solving the system of Eqs.(17-23), for the plasma and electromagnetic field parameters here the following expressions are assumed: It is of importance that, omitting horizontal profiles of the electromagnetic field, the altitudinal scales of this field (H x and H y ) have to be chosen differently in the x-and ydirections.Otherwise, no magnetic field disturbances will occur.Thus, the model presented here is valid only locally, and there have to be experimental data used for the horizontal scales.Substituting Eq. ( 24) in the continuity Eq. ( 17), one gets (a = i,n) The index i in Eq. ( 29) describes the ions.For the linearized equation of state Eq. ( 19) follows From the linearized momentum equation Eq. ( 18) follows a=e,i m a n ao δv ao /(m e n eo + m i n io ).
The y-component of this relation reads If one neglects particle collisions, the electric background field (considering infinite electrical conductivity values) and the stratification of the atmosphere, the ycomponent of Eq. ( 42) coincides with the dispersion relation of Alfvén waves usually considered (see, e.g.Sturrock, 1994, Eqs. 14.1.25, 14.1.29).For wave vectors in the x-zplane (this assumption does not restrict the general solution), δv oy is proportional to δB oy .One finds a dispersion relation for the waves which are transverse to the background magnetic field.
Collisions of charged particles with the neutral ones change the growth rate of the Alfvén waves, as follows from Eq. ( 42).In case of fluctuations of the velocities of the neutral particles larger than the fluctuations of the speeds of the charged particles (δv noy > δv eoy ,δv ioy ), for instance by infrasound or acoustic-gravity waves, the wave amplitudes should grow.Further, taking into account a finite electrical conductivity of the atmosphere, δv oy is also correlated to magnetic field fluctuations δB oz parallel to the background field B o as Thus, we have found that in the ionospheric E-layer, normal Alfvén waves may be generated.The amplitudes and propagation directions of the waves are modified by upstreaming acoustic waves.This phenomenon is possible when the frequency of the acoustic waves is larger than the cut-off frequency = c s /2H i , so that the waves penetrate into the E-layer (Liperovsky et al., 1997;Meister, 1995;Koshevaya et al., 2002).Via collisions of the neutral particles with the charged ones, the momentum of the neutral particles is transferred to the plasma components.Then the variations of the velocities of the charged particles generate an electromagnetic wave field.The waves may be Alfvén ones, but magnetoacoustic waves or the so-called Farley-Buneman waves may also occur.The wave generation is influenced by the background electromagnetic field and by the altitudinal profiles of the plasma parameters.
In Liperovsky et al. (1997), considering acoustic waves with frequencies of 5×10 −3 −20 Hz and neglecting viscosity effects, the Farley-Buneman waves were found to be caused by the y-components of the electron velocities.Neglecting altitudinal profiles of the plasma and magnetic wave fields δB in the E-layer, it was already found by (Meister, 1995) that always three electrostatic waves of the Farley-Buneman type (described by Eqs.39, 40) exist.One wave possesses a growing amplitude, and the other two waves are damped.The non-damped wave has in the E-layer (T = 250 K, ν en = 4 × 10 4 Hz, ν en = 1.4 × 10 3 Hz, ω ce = 5.3 × 10 6 Hz) wave numbers of the order of 1-70 m −1 and phase velocities of about 500 ms −1 .The Farley-Buneman waves are driven by j +k y V eyo , γ i describes the growth rates, V eyo -the amplitude of the electron Hall velocity exciting the waves, and k is the wave number.j = {1,2,3} (Meister, 1995).
the Hall velocity of the electrons, which is proportional to the neutral gas velocity, v ey ≈ 3-6 v n .The waves are excited under the condition that The Hall currents cause fluctuations of the magnetic field in the directions of the neutral gas motion and along the mean magnetic field.
The electron velocities parallel to the earth's magnetic field are of the order of 0.025 v ez a ≈ 2.5-5 v ey , where a is the horizontal dimension of the sporadic E-layer amounting to a few hundreds of meters.Thus v ez may reach 720-1900 ms −1 at neutral gas velocities larger than 50 ms −1 , and it may generate rather strong variations of the magnetic field.In addition, the plasma will be heated (Liperovsky and Meister, 1996).
3 Discussion of the results and conclusions 1.First steps have been performed to investigate the modification of the electromagnetic field of the ionospheric E-layer by acoustic-type waves, which might be generated by earthquake precursors or for meteorological reasons.
2. Using many-fluid magnetohydrodynamics, a full system of equations describing electromagnetic waves in the E-layer has been derived analytically.Influences of the background electromagnetic field as well as the stratification of the atmosphere have been taken into account.
3. Concerning electromagnetic waves in the E-layer, one may conclude that usual (according to the work of Sturrock, 1994) magnetohydrodynamic waves like Alfvén and magnetoacoustic ones are generated in the E-layer.These waves are modified with respect to amplitude and the direction of motion by acoustic-type waves propagating into the E-layer from lower altitudes.Such acoustic-type waves might be of seismic origin.But they also occur for meteorological reasons.
4. Almost electrostatic waves of the Farley-Buneman type might also be generated in seismo-active regions at Elayer altitudes a few days before earthquakes.Solutions for the dispersion relation and excitation conditions of the waves are discussed.
5. Wave turbulence in the ionospheric E-layer, and especially in sporadic E-layers, should influence the ionograms observed by ionospheric vertical radar stations.And indeed, some tendencies of variations of Es-spread effects have been observed (Liperovsky et al., 1999a(Liperovsky et al., , b, 2000(Liperovsky et al., , 2005)).
6.Additional Joule heating caused by acoustic wavesand the corresponding increased intensities of the vertical atmospheric currents -result also in modifications of the charged particle densities of the ionosphere (Meister, 1995;Liperovsky and Meister, 1996;Liperovsky et al., 1996;Meister et al., 1998).Thus, changes in the characteristic frequencies foE might be obtained.

Fig. 2 .
Fig. 2. Three solutions of the dispersion relation of Farley-Buneman waves.The wave frequencies equal ω j = ω *j +k y V eyo , γ i describes the growth rates, V eyo -the amplitude of the electron Hall velocity exciting the waves, and k is the wave number.j = {1,2,3}(Meister, 1995).
Meister et al.: Possible MHD wave modulation in seismic regions Expressing the plasma pressure p by the equation of state of the ideal plasma, p a = k B T n a , one obtains with δp a = k B T o δn a + n ao k B δT (33) 2H i (z)−ikr+iωt δE + ω ce δv eoy n x − ω ce δv eox n y +gδn eo /n eo − ν ei (δv eo − δv io ) − ν en (δv eo − δv no ), Thus, expressing in the equations Eqs.(38)(39)(40)δp ao by the formulae Eqs.(33, 34), δE o by Eqs.(35-7), as well as δn ao and δn eo by Eqs.(27-29), one finds a dispersion relation of magnetohydrodynamic waves in the ionospheric E-layer caused by the motion of neutral and ionized particles.The expression for the dispersion relation is rather large.Thus it will not be given here.It has to be solved numerically.Here, only a first discussions of the special solution of Alfvén waves modified by the acoustic ones is presented.If one forms the sum of Eqs.(39, 40) describing the momentum balance of the plasma components of the system, one obtains (n eo ν ei