Natural Hazards and Earth System Sciences Ionospheric conductivity effects on electrostatic field penetration into the ionosphere

The classic approach to calculate the electrostatic field penetration, from the Earth’s surface into the ionosphere, is to consider the following equation ∇· ( σ̂ ·∇8 ) =0 whereσ̂ and8 are the electric conductivity and the potential of the electric field, respectively. The penetration characteristics strongly depend on the conductivities of atmosphere and ionosphere. To estimate the electrostatic field penetration up to the orbital height of DEMETER satellite (about 700 km) the role of the ionosphere must be analyzed. It is done with help of a special upper boundary condition for the atmospheric electric field. In this paper, we investigate the influence of the ionospheric conductivity on the electrostatic field penetration from the Earth’s surface into the ionosphere. We show that the magnitude of the ionospheric electric field penetrated from the ground is inverse proportional to the value of the ionospheric Pedersen conductance. So its typical value in day-time is about hundred times less than in night-time.


Introduction
First unusual disturbances of the vertical component E z of the electrostatic field were observed prior to earthquakes in the epicentral zones on the Earth's surface by Kondo (1968).Electrostatic potential fluctuations were measured onboard a satellite at an altitude of 400 km (Kelley and Mozer, 1972).A systematic research started to develop in 80th (Gokhberg Correspondence to: M. Y. Boudjada (mohammed.boudjada@oeaw.ac.at) et al., 1983).Response of the atmosphere and the ionosphere to intense earthquakes were reported by several studies (Kelley et al., 1985;Kingsley, 1989;Pogorel'tsev, 1989;Depuev and Zelenova, 1996;Ruzhin and Depueva, 1996).Also disturbances related to nuclear accidents were observed (Ruzhin et al., 1995;Martynenko et al., 1996) and associated to the propagation of acoustic-gravity waves (Row, 1967).More details about these investigations and others are reviewed by Parrot (1995).

Models of seismic precursory phenomena
According to the sources of precursory phenomena, models are generally based on the formation of micro-cracks in the days to weeks before the event (Molchanov andHayakawa, 1995, 1998).Due to mechanical forces on a specific part of the lithosphere, micro-cracks appear, which increase in their number density and finally end in the earthquake itself.Part of the mechanical energy, which is released due to the cracks, is transformed into electromagnetic energy (Mastov and Lasukov, 1989;Molchanov et al., 1995).The resulting electromagnetic emission is in the same frequency range as the mechanical disturbances and is in the low frequency parts (kHz and below).Since parts of the lithosphere are saturated with water, cracks also result in changes of the pore water pressure, which leads to electrokinetic effects (Gershenzon et al., 1993).Movement of crustal material (e.g.due to seismic waves) may result in inductive effects due to a relative movement in the geomagnetic field.Additionally, the Earth's crust involves piezomagnetic and electric features.
V. V. Denisenko et al.: Electrostatic field penetration into the ionosphere All these effects or changes in the electromagnetic environment of the near Earth atmosphere due to chemical mechanisms may lead to the rise of electromagnetic fields on the Earth's surface (Gershenzon and Bambakidis, 2001).Several models have been proposed to summarize the main physical mechanisms and the corresponding effects starting from the ground up to the ionosphere/magnetosphere (Gokhberg et al., 1985;Pulinets et al., 2000Pulinets et al., , 2002;;Sorokin et al., 2001;Molchanov et al., 2004).Characteristic variations in the critical frequency foF2 (Silina et al., 2001), the total electron content (Liu et al., 2004), the ion temperature (Sharma et al., 2006) or the local ion and electron density (Parrot et al., 2006) were found.

Electric field penetration in the ionosphere
In the frame of ionospheric precursors, it is important to know what kind of effect could be observable to maintain proper satellite missions.In this connection it is essential to understand how electrostatic fields from lithospheric origin penetrate into higher altitudes of the atmosphere.Theoretical investigations are generally based on the work of Park and Dejnakarintra (1973).This approach was mainly applied to study the penetration of thundercloud electric field into the ionosphere and the magnetosphere.First this model has been used and developed towards a better description of physical phenomena occurring during lightning (Roble and Hays, 1979;Nisbet, 1983;Makino and Ogawa, 1984;Tsur and Roble, 1985;Baginski et al., 1988;Khegay et al., 1990;Ma et al., 1998;Velinov and Tonov, 1995;Rodger et al., 1998).The conductivity of the atmosphere is found to be a key physical parameters in the processes which occurred between the ground and the other layers, in particular the ionosphere (James, 1985;Kim and Khegay, 1985;Kamra and Ravichandran, 1993).A second application of the model of Park and Dejnakarintra (1973) is discussed and investigated in the frame of the studies of the seismic precursor emissions.In this case the electric field of seismic origin is analyzed in the way to estimate the effect of such field within the ionosphere (Gokhberg et al., 1984;Kim et al., 1994;Pulinets et al., 1998).One has also to report models based on enhancement of radioastivity and charged aerosols in the atmosphere before earthquakes (Pierce, 1976;Boyarchuk et al., 1998;Sorokin et al., 2000Sorokin et al., , 2005)).The general conclusion is that electric fields can effectively penetrate into the ionosphere and disturb the ionospheric plasma under certain circumstances.
The field penetration is more effectively at night and the field intensity value critically depends on the characteristic source dimension, which could be described by the earthquake preparation area (Dobrovolsky et al., 1979).Pulinets et al. (2003) concluded that the electrostatic field effectively penetrates into the ionosphere when the source area is greater than 100 to 200 km.This corresponds to a magnitude greater than 4.6 to 5.3, which is some kind of threshold value for the ionospheric sensibility.Grimalsky et al. (2003) estimated a critical value for the ground electrostatic field of the order of 1 to 3 kV/m.These values were also discussed in the frame of in-situ measurements (Mikhailov et al., 2007).
The effect of a near Earth precursory sign on the ionosphere can be determined by the projection along geomagnetic field lines.In a simplified approach, the inclination of the geomagnetic field lines is taken to be I =90 • (which is nearly the case at the geomagnetic poles) and the atmospheric conductivity is assumed to be isotropic.The atmospheric (and ionospheric) conductivity has the most important influence on the penetration of an electrostatic field through the atmosphere.In altitude regions above 80 km the conductivity can no longer be taken as isotropic, since the influence of the rotation of the ionized particles around magnetic field lines becomes more important.The Ohm's law that relates electric field E and the current density j , involves the conductivity tensor σ .Let us split the vector E into fieldaligned component E parallel to the magnetic field and the normal components E ⊥ , where quantities σ P , σ H , σ are the Pedersen, Hall, and fieldaligned conductivities (Hargreaves, 1979).
In the atmosphere below 80 km the conductivity tensor is isotropic one, so that σ H =0, σ P =σ =σ .Values of the near Earth atmospheric conductivity σ are in the range 10 −14 S/m to 10 −13 S/m (Molchanov and Hayakawa, 1994).Huge variations are possible due to solar and geochemical influences (Prölss, 2003).Grimalsky et al. (2002) concluded that an increase of the near-Earth atmospheric conductivity leads to a modification of the electric environment of the ionosphere due to changed conditions for the electrostatic field penetration.
In this paper we are interested to study the possible influences of conductivity variations on the penetration of an electrostatic field through the atmosphere.Obviously, it is complicated to calculate regions where the conductivity is anisotropic.Thus it is necessary to obtain a proper upper boundary condition, which makes possible to estimate electrostatic fields at higher altitudes.In the section below, a model was obtained, where the atmospheric conductivity has the most important influence on the electrostatic field penetration.

Basic equations
Since we are interested in the steady state case the basic equations for the atmospheric electric field are the Faraday law, the charge conservation law and the Ohm's law where the Ohm's law (Eqs. 1 and 2) is written in the short form.Because of (Eq. 3) the electric potential can be introduced so that Then the system of the equations (Eqs.3-5) is reduced to the equation that would be the Laplace equation if the conductivity tensor σ is isotropic and constant.

Conductivity
Clearly one can see from Eq. ( 7) that the conductivity tensor σ takes a major part in our investigations.The conductivity is approximately isotropic in the atmosphere.It is regarded here as altitude range 0≤z≤z up , where z up =80 km is the upper altitude range.The height distributions of the atmospheric conductivity can be approximated with an exponential function where the conductivity only depends on an initial value σ and on the conductivity scale-height h.
In accordance with (Handbook of Geophysics, 1960) the typical values are σ =10 −13 S/m, h=6 km for the main part of the atmosphere and σ =3•10 −14 S/m, h=3 km for the region near the ground.We mainly use the first set of the parameters and describe possible differences in the results due to this choice.Of course more detailed approximation than Eq. ( 8) can be used to present real conductivity distribution.However numerical solution of differential equations is necessary in such a model with no principal results in comparison with this simplified model, that permits to get the solutions analytically.

Model geometry
The geometry of the model is shown in Fig. 1.The origin of the coordinate system is exactly at the epicenter, the y-axis is directed along the fault and the x-axis is directed perpendicular to the fault.The z-axis is directed upward, from the ground to the ionosphere.The distribution only depends on the distance perpendicular to the tectonic fault x, but along the fault y the field is considered to be not variable.This simple approach is consistent with earlier work (Pulinets et al., 1998).As a tectonic fault is elongated, and some hundreds of kilometers long, one can also assume an elongated electrostatic field distribution on the Earth's surface.So we analyze the two-dimensional model in which no parameter depends on y.In such a model the equation (Eq.7) has the form In our model the atmosphere is considered to be a horizontal layer between ground and some height z up .
The general solution of this equation in such a domain can be obtained by separation of variables, i.e. superposition of exponential functions.To solve the differential equation, we need a lower and upper boundary conditions.

Lower boundary condition
As the lower boundary condition, the vertical component of the electric field on the Earth's surface is given The model distribution for the vertical electrostatic field can be written as (see Fig. 1): where a indicates the size of the affected area on the Earth's surface, so that E 0 (x)=0 outsides, and Ē0 is the maximal value of the electrostatic field at z=0 km.Quantity a can be interpreted as the earthquake preparation area.Negative vertical electrostatic fields on the ground means an increase of the fair weather electric field of the atmosphere.These values are in agreement with estimations of the electrostatic source in connection with an earthquake preparation process (Mikhailov et al., 2007;Pulinets and Boyarchuk, 2004).In our model Ē0 =100 V/m and a=200 km were chosen.This value of a makes the function (Eq.11) close to − Ē0 / cosh(2x/c), that was used in the mentioned models with c=150 km.We opt for the function (Eq.11) because it is equal to zero outside the domain of interest and stays smooth.

Upper boundary condition
The most significant results concerning the penetration of an electrostatic field into the ionosphere are related to the upper boundary condition (Grimalsky et al., 2003).
In our model the ionosphere is considered to be a horizontally stratified in the height more than some z up above ground.It could be shown that because of the high fieldaligned conductivity in the ionosphere our model of the ionospheric electric field is not sensitive to the value of z up .We choose z up =80 km.Our test calculations show no significant change if z up is chosen in the range 80-90 km.
The inclination of geomagnetic field lines is assumed to be I =90 • , which means vertical magnetic field lines that is nearly fulfilled in polar regions.The boundary z=z up separates the region below it with isotropic finite conductivity, from the region above it with nearly infinite field-aligned conductivity.Below z up , the value of the conductivity increases by many orders of magnitude with increasing altitude.Above z up , the conductivity along geomagnetic field is considered to be huge (practically infinite).It means that the magnetic field lines are equipotential and E ⊥ is independent of the height for z>z up .Therefore the Ohm's law (Eq.2) can be integrated over z where which are referred to as Pedersen -P -and Hall conductances -H -(Hargreaves, 1979).Since j z from the atmosphere at z=z up is closed with this J , the charge conservation law means Using the Ohm's law in the atmosphere with scalar conductivity σ this equation can be written as the upper boundary condition for the equation (Eq.9) where the derivatives over y are omitted because no function depends on y.Moreover we use only constant P since the horizontal scale of interest is much less than the horizontal scale of the ionosphere.

Model calculation
To study the influence of ionospheric conductivity variations on the electrostatic field propagation through the atmosphere we use typical values of the ground value of the conductivity σ and the parameter h, and consider different values of P .
The integrated conductivity P is assumed to be P =10 S in day-time (and in auroral zone) and P =0.1 S in night-time.
The boundary value problem (Eqs.9, 10 and 14) aught to be solved in the two-dimensional domain 0<z<z up , that is infinite in x direction.We can simplify the problem by addition some sources at large distances b a in such a manner that the solution of the original problem is not disturbed in the domain of interest, |x|<2a for example.Namely we add Ē0 (1+ cos((x−b)π/a))/2, |x−b|<a, and continue this function to the whole x-axis with period 2b.The large parameter b a is chosen by test calculations so that such a modification of Eq. ( 11) has no influence on the results in the domain of interest |x|<2a.
After that we have the boundary value problem (Eqs.9, 10 and 14) with additional condition of periodicity (x+2b, z)= (x, z). (15) It is much more simple to deal with periodical functions since such a function can be presented with the Fourier Series, while the Fourier Integral is necessary (Korn and Korn, 1968) to present a function that is defined at the infinite axe and equals zero outside of some finite domain, that is |x|<a for the original function E 0 (x) (Eq.11).
Such a modification of the problem can be described by other words in the manner that is usual for numerical methods.The boundary value problem (Eqs.9, 10 and 14) has a particular solution (x, z)=αx+β with arbitrary constants α, β.It can be used as the asymptotic at x→∞ and it means that (x, z) is constant at vertical lines, when x→∞.
To avoid infinite domain it is possible to chose some large parameter b and approximately use this condition at x=b/2: (b/2, z)=0. (16) The zero value is of no matter since any constant can be added to the potential.Because the solution is a symmetrical one with respect to x=0 line and Eq. ( 16) we also have It is usual to use the boundary value problem with these additional conditions (Eqs.16 and 17) instead of the original one.Its solution is close to the solution of the original problem when b→∞.The conditions (Eqs.16 and 17) permit to continue the function (x, z) antisymmetrically (b−x, z)=− (x, z) and to get a periodical function with 2b period as in the previous formula (Eq.15).
The new function E 1 (x) equals to E 0 (x) (Eq.11) in the interval |x|<b−a and in contrast with E 0 (x) it can be presented as where The terms with even values 2n as well as all terms with sin (kx) are absent because the function E 1 (x) is antisymmetrical in respect of x=b/2 and symmetrical in respect of x=0.
The Fourier coefficients are equal Because of E 1 (x) symmetry and Eq. ( 16) The interval of integration may be decreased to 0<x<a since E 1 (x)=0 at a<x<b/2.Because of that E 1 (x)=E 0 (x) at 0<x<a the formula (Eq.11) can be used and the integral can be calculated analytically If k n a=π, that means 2n 0 −1=b/a, the denominator equals zero, but the numerator also equals zero and this f n =a/b.It can not occur if b/a is not integer.The coefficients f n are large for n close to this n 0 =(b/a+1)/2 and decrease as 1/n 3 for n→∞.
The general solution of equation (Eq.9) can be found due to separation of variables and is a superposition of exponential functions, depending on x and z separately +B n e n z , where The coefficients A n and B n can be derived from the boundary conditions given in Eqs. ( 10) and ( 14).
We use the presentation (Eq.15) for E 0 (x) since E 1 (x)=E 0 (x) for |x|<b−a and the parameter b is large enough and it does not disturb the solution.
With the constitutive relation, the coefficients are The components of the electric field in the atmosphere can be deduced from Eq. ( 6) where the potential (x, z) is calculated by the formula (Eq.21).The third component E y (x, z)=0 since there is no variation along the y coordinate.

Summary of the main results
The results of the electric field calculations are presented in Fig. 2. It presents the height distribution of the vertical component of the electric field at the line x=0.This graph is hardly distinguished from the line which corresponds to the fact that the vertical electric current density does not vary with height.The shape of horizontal cross-section of E z (x, z) is almost independent of the height as it can be seen in Fig. 3.The E z (x, z) distributions at the ground and at the upper boundary of the atmosphere are plotted in such scales that these two lines would be identical if Eq. ( 24) is exactly valid.For this purpose the dashed line presents the vertical electric field in the ionosphere E z (x, z up ) multiplied by exp (z up / h).So the maximal values are 100 V/m and 9µV/m, respectively.
Figures 2 and 3 show that the vertical component of the current density almost does not vary with height.The domain |x|<a of nonzero j z at the ground slightly increases up to the ionosphere.The total current per y=1 m does not depend on z in view of the charge conservation law.Here we regard 2a as a large distance, but less than b/2 to exclude the influence of the added periodically current sources at the ground.So this current can be calculated using the Ohm's law and given values of conductivity σ at the ground and E z (x, 0) (Eq.11) with values of the parameters Ē0 =100 V/m, a=200 km and σ =10 −13 S/m, which are chosen in Sects.3 and 4.1, respec- tively.This current J z goes to the ground through atmosphere from the ionosphere.In accordance with charge conservation law the same current must go along the ionosphere from its far regions, J z /2 from both infinities because of the symmetry.Horizontal electric field is necessary for such currents It is equal to ∓1 µV/m for x a and x<<−a if P =1 S. The typical values of P vary from 0.1 S in the night-time ionosphere till 10 S in the day-time ionosphere and in the auroral zone.
These limit values, i.e. ∓10 µV/m, in the night-time ionosphere are shown in Fig. 4 for |x|>200 km.The calculated E x (x) presented in this figure for the upper boundary of the atmosphere stays the same in the ionosphere in the frame of our model because of the large field-aligned conductivity in the ionosphere.
The electric field in the ionosphere is not more than 10 µV/m, since other values of P give decrease of E x .It can be seen in Fig. 4 where E x (x) for P =1 S is shown by dashed line.The day-time E x (x) distribution has almost the same shape, but it is invisible in this scale because its maximum equals 0.1 µV/m.
The electric fields of these magnitudes can not be observed in the ionosphere because much larger fields, definitely more than 1 µV/m are always present there.
Figure 5 gives general view of the electric potential in the atmosphere.Because of the simple properties of the electric field described above, it is possible to obtain analytical solution with a rather good precision.As the results of Eqs. ( 6), ( 8) and ( 24 if we approximately consider =0 in the ionosphere.Since the second term is small far from the ground, approximately (x, z)=|E 0 (x)|h exp (−z/ h).
The maximal value equals E 0 (x)h=600 kV.This value must be decreased if we take into account specific σ (z) behavior near the ground where h=3 km gives better approximation (Handbook of Geophysics, 1960).Then the maximal potential difference between ground and ionosphere becomes about 300 kV.Such an electric potential distribution is presented in Fig. 5.It almost does not depend on σ (z) distribution above z=5 km.
The described simple properties of E and space distributions are mainly due to large horizontal scale a h.If the region of the electric field generator near the analyzed earthquake becomes less, the ionospheric electric field is proportionally decreased in our model in accordance with the estimations (Eq.25).So 10 µV/m can be regarded as the upper limit of possible electric field penetrated into the ionosphere.

Discussion and conclusion
The details of E x (x, z) distribution are defined by the solution for the boundary value problem (Eqs.9, 10 and 14), but the scale of the penetrating field can be calculated by the formula (Eq.25) in a wide range of parameters which charac-terize the atmospheric and ionospheric conductivities while the horizontal scale of the process a is large enough.
This conclusion and estimations by the formula (Eq.25) contradict the former calculations (Pulinets et al., 1998), where the maximum of the horizontal electrostatic field penetration is in the mV/m range.
The electric field at the ground can not be much larger than 100 V/m, that is used both here and in former models, since it gives a few hundreds kV potential difference between ionosphere and ground.So, the only way to get large penetrated electric field may be due to atmospheric conductivity near the ground hundred times increased in comparison with usual conditions.
Because the conductivity along geomagnetic field lines is much greater than the conductivities perpendicular to the field lines, and because of the assumption, that geomagnetic field lines are parallel in the ionosphere, the electric field intensity will not change significantly in higher altitudes.This is taken into account in the effective upper boundary condition (Eq.14).Thus, the electrostatic field at the altitude z up =80 km can be used to calculate effects in higher regions.
The assumption of vertical geomagnetic field lines (I =90 • ) is almost fulfilled in polar regions.In the frame of satellite missions, data generally is recorded at latitudes below a specific region.In the case of DEMETER, observations are performed at latitudes less than 65 • on both hemispheres.This means, the model calculation obtained here must be expanded to oblique geomagnetic field lines to improve the results.
The designed model suppose vertical magnetic field, that is not valid in middle and low latitudes.It is of matter only for the value of the conductance of the ionosphere, since the atmospheric conductivity is a scalar that is independent of magnetic field.If we take the inclination into account, the conductance tensor with parameters P , H (Eq. 12) must be changed with tensor with coefficients xx , xy =− yx , yy (Hargreaves, 1979).Since these parameters are independent of x, y the terms with xy =− yx do not appear in the upper condition (Eq.14) after such a modification as well as H is not present in (Eq.14): If we exclude for simplicity a narrow region near the geomagnetic equator, then xx = P / cos 2χ, yy = P , where χ is the angle between vertical and magnetic field, that is in x, z plane (Hargreaves, 1979).Of cause xx = P if magnetic field is inclined in y, z plane and some average value would be in general case.Anyway, the conductance xx can only be increased in comparison with P .Hence the magnitude of the electric field in the ionosphere is smaller if we take the inclination into account.
From our investigations, it comes that the electric field penetration is approximately proportional to the atmospheric conductivity near the ground and inverse proportional to the integral ionospheric Pedersen conductivity.Also this penetrated electric field is a hundred times larger during night-time in comparison with day-time one, and its penetration of electrostatic fields depends on various characteristics of the conductivity distributions.However our estimations show that the field penetration is damped and no seismic precursor may be seen in the ionosphere.
Future investigation, the height distribution and the dynamic behavior of the atmospheric conductivity must be more better known, especially its increase near the ground during seismic events.

Fig. 1 .
Fig.1.Geometry of the model where the origin of the coordinate system is at the epicenter, and the x-and y-axis are perpendicular and along the fault, respectively.

Fig. 2 .
Fig. 2. The height distribution of the vertical electric field E z (0, z) above the point x=0.

Fig. 3 .
Fig. 3. Electric field at the ground E z (x, 0) -solid line, and at the bottom of the ionosphere E z (x, z up ) -dashed line.The maximal values are 100 V/m and 9 µV/m, respectively.

Fig. 4 .
Fig. 4. Horizontal electric field S E x (x, z up ), µV/m in the nighttime ionosphere with P =0.1 S -solid line, and in the ionosphere with P =1 S -dashed line.