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On the Electric Fields Produced by Dipolar Coulomb Charges of an Individual Thundercloud in the Ionosphere

In this paper we study the transmission of the electrostatic field due to coulomb charges of an individual thundercloud into the midlatitude ionosphere, taking into account the total geomagnetic field integrated Pedersen conductivity of the ionosphere. It is shown that at ionospheric altitudes, a typical thundercloud produces an insignificant electrostatic field whereas a giant thundercloud can drive the horizontal electrostatic field with a magnitude of ~270 μV/m for nighttime conditions.


INTRODUCTION
Thunderclouds are tropospheric sources of intense electrostatic fields and electromagnetic radiation. It is known that lightning-associated electric fields penetrate into the ionosphere; they have been observed in the E and F regions as transient electric fields with a typical duration of 10-20 ms and a magnitude of 1-50 mV/m (e.g., Kelley et al. 1985Kelley et al. , 1990Vlasov & Kelley 2009). According to the theoretical model of global atmospheric electricity developed by Hays & Roble (1979), the African array of multiple thunderclouds is responsible for the steady state electrostatic field of ~300 μV/m at ionospheric altitudes for nighttime conditions. The calculations by Park & Dejnakarintra (1973) showed that an isolated giant thundercloud could produce electrostatic fields of ~700 μV/m in the nighttime midlatitude ionosphere. However, Park & Dejnakarintra (1973) neglected the ionospheric Pedersen conductivity above 150 km. The purpose of this study is to theoretically examine the mapping of electrostatic fields of coulomb charges of an individual thundercloud into the midlatitude ionosphere, taking into account the height-integrated Pedersen conductivities of both hemispheres.

BASIC EQUATIONS
In the simplest thundercloud model, the electrical structure of a thundercloud is represented by two volume Coulomb charges of the same absolute value Q but opposite signs, with a positive charge in the upper part of the thundercloud and a negative charge in the lower part of the thundercloud (e.g., Chalmers 1967). Typical thunderclouds extend from 2-3 km to 8-12 km in altitude, and so-called giant thunderclouds extend above an altitude of 20 km (e.g., Uman 1969;Weisberg 1976). The magnitude of Q is estimated to range from 5 to 25 coulombs for the typical thunderclouds, whereas in giant thunderclouds, Q may exceed 50 coulombs (e.g., Malan 1963;Kasemir 1965).
We use a cylindrical coordinate system (r, φ, z), in which the origin is placed at the earth's surface and the z axis points upward and passes through the centers of thundercloud positive and negative volume charges.
The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : ∂ 2 /∂r 2 + (1/r)∂/∂r + (1/ P )∂( 0 ∂/∂z)/∂z = 0 , where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain (1) 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : ∂ 2 /∂r 2 + (1/r)∂/∂r + (1/ P )∂( 0 ∂/∂z)/∂z = 0 , where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain (2) 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain where 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain is the electric current density, 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain is the electrical conductivity tensor, and 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain and 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : ∂ 2 /∂r 2 + (1/r)∂/∂r + (1/ P )∂( 0 ∂/∂z)/∂z = 0 , where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain can be obtained from (1), (2), and (3) : 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain where 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain is the Pedersen conductivity and 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field. Equation (4) can be solved analytically if the conductivities 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain and 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain are exponential functions of z. In the case of isotropic conductivity (setting 4 4 undercloud electrostatic field into the ionosphere is t used by Park & Dejnakarintra (1973). In the steady state e the thundercloud is described by the following equations: (1) (3) e electrical conductivity tensor, and E and  are the . If we assume that the geomagnetic field B is vertical and ly on z, the following equation for the electrostatic , where b and h are constants), we obtain 4 4 negative volume charges. The mapping of thundercloud electrostatic field into the ionosphere is studied following a similar formalism to that used by Park & Dejnakarintra (1973). In the steady state case, the electrostatic field distribution above the thundercloud is described by the following equations: where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain where J 0 is the zero-order Bessel function of the first kind, A 1 and B 1 are coefficients, and 5 where J 0 is the zero-order Bessel function of the first kind, A 1 and B 1 are coefficients, and c 1 = -l/(2h) - where J ν and K ν are the ν-order modified Bessel functions of the first and the second kind, respectively, and A 2 and B 2 are coefficients, The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the boundary conditions.
The electrostatic field components are given by er modified Bessel functions of the first and the second kind, respectively, nd B 2 are determined from the boundary conditions. d components are given by (7) , 5 where J 0 is the zero-order Bessel function of the first kind, A 1 and B 1 are coefficients, and c 1 = -l/(2h) - where J ν and K ν are the ν-order modified Bessel functions of the first and the second kind, respectively, and A 2 and B 2 are coefficients, ν=h p /( h p -h 0 ), f=2νh 0 (b p / b 0 ) 1/2 exp[-z/(2 νh 0 )].
The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the boundary conditions.
The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the boundary conditions.
The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the boundary conditions.
The electrostatic field components are given by where J ν and K ν are the ν-order modified Bessel functions of the first and the second kind, respectively, and A 2 and B 2 are coefficients, 5 where J 0 is the zero-order Bessel function of the first kind, A 1 and B 1 are coefficients, and c 1 = -l/(2h) -[l/(4h 2 )+k 2 ] 1/2 , c 2 = -l/(2h) + [l/(4h 2 )+k 2 ] 1/2 . For the anisotropic region, where we let  0 = b 0 exp(z/h 0 ) and  p = b p exp(z/h p ), the solution to Equation (4) is where J ν and K ν are the ν-order modified Bessel functions of the first and the second kind, respectively, and A 2 and B 2 are coefficients, ν=h p /( h p -h 0 ), f=2νh 0 (b p / b 0 ) 1/2 exp[-z/(2 νh 0 )].
The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the boundary conditions.
The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the boundary conditions.
The electrostatic field components are given by . The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the boundary conditions.
The electrostatic field components are given by 5 and A 2 and B 2 are coefficients, ν=h p /( h p -h 0 ), f=2νh 0 (b p / b 0 The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the The electrostatic field components are given by E r = -∂Φ/∂r, Since the geomagnetic field B is assumed to be vertical, E r

B.
Above 90 km, the geomagnetic field lines are prac geomagnetic field aligned conductivity  0 is much higher us to consider the ionospheric region above 90 km as a thin line integrated Pedersen conductivity Ʃ p , and the continuit takes the following form: 5 The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the The electrostatic field components are given by E r = -∂Φ/∂r, Since the geomagnetic field B is assumed to be vertical, E r

B.
Above 90 km, the geomagnetic field lines are prac geomagnetic field aligned conductivity  0 is much higher us to consider the ionospheric region above 90 km as a thin line integrated Pedersen conductivity Ʃ p , and the continuit takes the following form: Since the geomagnetic field B is assumed to be vertical, E r is perpendicular to B, while E z is parallel to B.
Above 90 km, the geomagnetic field lines are practically equipotential because the geomagnetic field aligned conductivity where J is the electric current density,  is the electrical conductivity tensor, and E and  are the electrostatic field and potential, respectively. If we assume that the geomagnetic field B is vertical and the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain is much higher than the transverse conductivity where J is the electric current density,  is the electrical conductivity tensor, and E and  are th electrostatic field and potential, respectively. If we assume that the geomagnetic field B is verti the electrical conductivity tensor depends only on z, the following equation for the electrostatic potential  can be obtained from (1), (2), and (3) : where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h ar constants), we obtain . It allows us to consider the ionospheric region above 90 km as a thin conducting layer with a geomagnetic field line integrated Pedersen conductivity The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the boundary conditions.
The electrostatic field components are given by E r = -∂Φ/∂r, Since the geomagnetic field B is assumed to be vertical, E r is perpendicular to B, w

B.
Above 90 km, the geomagnetic field lines are practically equipotential beca geomagnetic field aligned conductivity  0 is much higher than the transverse condu us to consider the ionospheric region above 90 km as a thin conducting layer with a line integrated Pedersen conductivity Ʃ p , and the continuity equation of electric cur takes the following form: , and the continuity equation of electric current at z=90 km takes the following form:

5
The electrostatic field components are given by E r = -∂Φ/∂r, Since the geomagnetic field B is assumed to be vertical, E r

B.
Above 90 km, the geomagnetic field lines are prac geomagnetic field aligned conductivity  0 is much higher us to consider the ionospheric region above 90 km as a thin line integrated Pedersen conductivity Ʃ p , and the continuit takes the following form: where where ⊥ denotes the gradient operator in the two dimensions transverse to B accounts for a contribution of the Pedersen conductivity of the magnetically Equation (9) is explicitly expressed as  0 ∂/∂z = 2Ʃ p (∂ 2 /∂r 2 + 1/r ∂/∂r).
We use the conductivity model as shown in Fig. 1. Below 70 km, the and varies exponentially with z as  01 = p1 =b 1 exp(z/h 1 ) from 0 to 40 km, and Since the geomagnetic field B is assumed to be vertical, E

B.
Above 90 km, the geomagnetic field lines are prac geomagnetic field aligned conductivity  0 is much higher us to consider the ionospheric region above 90 km as a thi line integrated Pedersen conductivity Ʃ p , and the continuit takes the following form: accounts for a contribution of the Pedersen conductivity of the magnetically conjugate ionosphere. Equation (9) is explicitly expressed as where ⊥ denotes the gradient operator in the two dimensions transverse accounts for a contribution of the Pedersen conductivity of the magnetic Equation (9) is explicitly expressed as  0 ∂/∂z = 2Ʃ p (∂ 2 /∂r 2 + 1/r ∂/∂r).
We use the conductivity model as shown in Fig. 1 We use the conductivity model as shown in Fig. 1. Below 70 km, the conductivity is isotropic and varies exponentially with z as σ 01 = σ p1 = b 1 exp(z/h 1 ) from 0 to 40 km, and as σ 02 = σ p2 = b 2 [exp(z-z 1 )/h 2 ] from 40 to 70 km (where z 1 = 40 km) with the values of b 1,2 and h 1,2 to approximately fit the atmospheric conductivity models by Cole & Pierce (1965)  ∂ 2 /∂r 2 + (1/r)∂/∂r + (1/ P )∂( 0 ∂/∂z)/∂z = 0 , where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain , and 4 potential  can be obtained from (1), (2), and (3) : ∂ 2 /∂r 2 + (1/r)∂/∂r + (1/ P )∂( 0 ∂/∂z)/∂z = 0 , where  p is the Pedersen conductivity and  0 is the specific conductivity. The atmospheric conductivity below 70 km is isotropic since drifts of charged particles are not affected by the geomagnetic field.
Equation (4) can be solved analytically if the conductivities  0 and  p are exponential functions of z. In the case of isotropic conductivity (setting  0 = p =b exp(z/h), where b and h are constants), we obtain are exponentially extrapolated from 70 km to their equinoctial midday and midnight values at z=90 km. At z ≥ 90 km, the conductivities are found from 6 and varies exponentially with z as  01 = p1 =b 1 exp(z/h 1 ) from 0 to 40 km, and as  02 = p2 =b 2 [exp(zz 1 )/h 2 ] from 40 to 70 km (where z 1 =40 km) with the values of b 1,2 and h 1,2 to approximately fit the atmospheric conductivity models by Cole & Pierce (1965) below 40 km and by Swider (1988) where subscripts e and i denote electrons and the ith ion species, N e and N i are the electron and ion densities, e is the electron charge, m e and m i are the electron and ion masses, ν e and ν i are the electron and ion momentum transfer collision frequencies, and ω e and ω i are the electron and ion gyrofrequencies, respectively. The frequencies ν e and ν i are from Schunk (1988). The required input parameters are taken from the empirical ionospheric model IRI-2012 (http://omniweb.gsfc.nasa.gov/vitmo/iri2012_vitmo.html) and the neutral atmosphere model NRLMSIS-00 (http://ccmc.gsfc.nasa.gov/modelweb/models/nrlmsise00.php).
(11) 6 and varies exponentially with z as  01 = p1 =b 1 exp(z/h 1 ) from 0 to 40 km, and as  02 = p2 =b 2 [exp(zz 1 )/h 2 ] from 40 to 70 km (where z 1 =40 km) with the values of b 1,2 and h 1,2 to approximately fit the atmospheric conductivity models by Cole & Pierce (1965) below 40 km and by Swider (1988) where subscripts e and i denote electrons and the ith ion species, N e and N i are the electron and ion densities, e is the electron charge, m e and m i are the electron and ion masses, ν e and ν i are the electron and ion momentum transfer collision frequencies, and ω e and ω i are the electron and ion gyrofrequencies, respectively. The frequencies ν e and ν i are from Schunk (1988). The required input parameters are taken from the empirical ionospheric model IRI-2012 (http://omniweb.gsfc.nasa.gov/vitmo/iri2012_vitmo.html) and the neutral atmosphere model NRLMSIS-00 (http://ccmc.gsfc.nasa.gov/modelweb/models/nrlmsise00.php).
where subscripts e and i denote electrons and the ith ion species, N e and N i are the electron and ion densities, e is the electron charge, m e and m i are the electron and ion masses, ν e and ν i are the electron and ion momentum transfer collision frequencies, and ω e and ω i are the electron and ion gyrofrequencies, respectively. The frequencies ν e and ν i are from Schunk (1988). The required input parameters are taken from the empirical ionospheric model IRI-2012 (http://omniweb.gsfc.nasa.gov/vitmo/iri2012_vitmo.html) and the neutral atmosphere model NRLMSIS-00 (http:// ccmc.gsfc.nasa.gov/modelweb/models/nrlmsise00.php). Our calculations show that during solar minimum, in Equinox, the magnitude of 5 where J 0 is the zero-order Bessel function of the first kind, A 1 and B 1 are coefficients, and c 1 = -l/(2h) -[l/(4h 2 )+k 2 ] 1/2 , c 2 = -l/(2h) + [l/(4h 2 )+k 2 ] 1/2 . For the anisotropic region, where we let  0 = b 0 exp(z/h 0 ) and  p = b p exp(z/h p ), the solution to Equation (4) is where J ν and K ν are the ν-order modified Bessel functions of the first and the second kind, respectively, and A 2 and B 2 are coefficients, The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the boundary conditions.
The electrostatic field components are given by Since the geomagnetic field B is assumed to be vertical, E r is perpendicular to B, while E z is parallel to

B.
Above 90 km, the geomagnetic field lines are practically equipotential because the geomagnetic field aligned conductivity  0 is much higher than the transverse conductivity  p . It allows us to consider the ionospheric region above 90 km as a thin conducting layer with a geomagnetic field line integrated Pedersen conductivity Ʃ p , and the continuity equation of electric current at z=90 km takes the following form: at middle latitudes is commonly in the ranges of 5.0-8.0 S and 0.1-0.2 S for day and night, respectively. However, the nighttime 5 5 where J 0 is the zero-order Bessel function of the first kind, A 1 and B 1 are coefficients, and c 1 = -l/(2h) -[l/(4h 2 )+k 2 ] 1/2 , c 2 = -l/(2h) + [l/(4h 2 )+k 2 ] 1/2 . For the anisotropic region, where we let  0 = b 0 exp(z/h 0 ) and  p = b p exp(z/h p ), the solution to Equation (4) is where J ν and K ν are the ν-order modified Bessel functions of the first and the second kind, respectively, and A 2 and B 2 are coefficients, The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the boundary conditions.
The electrostatic field components are given by Since the geomagnetic field B is assumed to be vertical, E r is perpendicular to B, while E z is parallel to

B.
Above 90 km, the geomagnetic field lines are practically equipotential because the geomagnetic field aligned conductivity  0 is much higher than the transverse conductivity  p . It allows us to consider the ionospheric region above 90 km as a thin conducting layer with a geomagnetic field line integrated Pedersen conductivity Ʃ p , and the continuity equation of electric current at z=90 km takes the following form: can be as low as 0.05 S. Under solar maximum conditions, 5 5 where J 0 is the zero-order Bessel function of the first kind, A 1 and B 1 are coefficients, and c 1 = -l/(2h) -[l/(4h 2 )+k 2 ] 1/2 , c 2 = -l/(2h) + [l/(4h 2 )+k 2 ] 1/2 . For the anisotropic region, where we let  0 = b 0 exp(z/h 0 ) and  p = b p exp(z/h p ), the solution to Equation (4) is where J ν and K ν are the ν-order modified Bessel functions of the first and the second kind, respectively, and A 2 and B 2 are coefficients, The coefficients A 1 , B 1 , A 2 , and B 2 are determined from the boundary conditions.
The electrostatic field components are given by Since the geomagnetic field B is assumed to be vertical, E r is perpendicular to B, while E z is parallel to

B.
Above 90 km, the geomagnetic field lines are practically equipotential because the geomagnetic field aligned conductivity  0 is much higher than the transverse conductivity  p . It allows us to consider the ionospheric region above 90 km as a thin conducting layer with a geomagnetic field line integrated Pedersen conductivity Ʃ p , and the continuity equation of electric current at z=90 km takes the following form: is several times larger than in solar minimum.

RESULTS AND DISCUSSION
To compute the electrostatic potential above the thundercloud from (5) and (6), we impose the following boundary conditions: 7 Our calculations show that during solar minimum, in Equinox, the magnitude of Ʃ p at middle latitudes is commonly in the ranges of 5.0-8.0 S and 0.1-0.2 S for day and night, respectively. However, the nighttime Ʃ p can be as low as 0.05 S. Under solar maximum conditions, Ʃ p is several times larger than in solar minimum.

RESULTS AND DISCUSSION
To compute the electrostatic potential above the thundercloud from (5) and (6), we impose the following boundary conditions: km, h n =3 km, h p =8 km) and for the giant thundercloud (z b =20 km, h n =5 km, h p =17 km). Solar 7 Our calculations show that during solar minimum, in Equinox, the magnitude of Ʃ p at middle titudes is commonly in the ranges of 5.0-8.0 S and 0.1-0.2 S for day and night, respectively. owever, the nighttime Ʃ p can be as low as 0.05 S. Under solar maximum conditions, Ʃ p is several es larger than in solar minimum.

RESULTS AND DISCUSSION
To compute the electrostatic potential above the thundercloud from (5) and (6), we impose the llowing boundary conditions:  is continuous at z=40 km  0 ∂/∂z=2Ʃ p (∂ 2 /∂r 2 +1/r ∂/∂r) , at z=90 km here ε 0 is the vacuum permittivity, z b is the altitude of the plane setting directly above the undercloud top, and h p and h n are the altitudes of positive and negative charge centers of the undercloud, respectively. The first boundary condition follows from the accepted electrical model of e thundercloud. We assume that the thundercloud does not affect the atmospheric conductivity at z b . Fig. 2 shows the computed electrostatic field component E r normalized to Q as a function of r the nighttime and daytime midlatitude ionosphere at z≥90 km for the typical thundercloud (z b =10 , h n =3 km, h p =8 km) and for the giant thundercloud (z b =20 km, h n =5 km, h p =17 km). Solar 7 Our calculations show that during solar minimum, in Equinox, the magnitude of Ʃ p at middle latitudes is commonly in the ranges of 5.0-8.0 S and 0.1-0.2 S for day and night, respectively.
However, the nighttime Ʃ p can be as low as 0.05 S. Under solar maximum conditions, Ʃ p is several times larger than in solar minimum.

RESULTS AND DISCUSSION
To compute the electrostatic potential above the thundercloud from (5) and (6), we impose the following boundary conditions: at z=90 km where ε 0 is the vacuum permittivity, z b is the altitude of the plane setting directly above the thundercloud top, and h p and h n are the altitudes of positive and negative charge centers of the thundercloud, respectively. The first boundary condition follows from the accepted electrical model of the thundercloud. We assume that the thundercloud does not affect the atmospheric conductivity at

ULTS AND DISCUSSION
To compute the electrostatic potential above the thundercloud from (5) and (6), we impose the ing boundary conditions: is continuous at z=40 km ∂/∂z=2Ʃ p (∂ 2 /∂r 2 +1/r ∂/∂r) , at z=90 km ε 0 is the vacuum permittivity, z b is the altitude of the plane setting directly above the rcloud top, and h p and h n are the altitudes of positive and negative charge centers of the rcloud, respectively. The first boundary condition follows from the accepted electrical model of ndercloud. We assume that the thundercloud does not affect the atmospheric conductivity at Fig. 2 shows the computed electrostatic field component E r normalized to Q as a function of r ighttime and daytime midlatitude ionosphere at z≥90 km for the typical thundercloud (z b =10 =3 km, h p =8 km) and for the giant thundercloud (z b =20 km, h n =5 km, h p =17 km). Solar 7 Our calculations show that during solar minimum, in Equinox, the magnitude of Ʃ p at middle latitudes is commonly in the ranges of 5.0-8.0 S and 0.1-0.2 S for day and night, respectively.
However, the nighttime Ʃ p can be as low as 0.05 S. Under solar maximum conditions, Ʃ p is several times larger than in solar minimum.

RESULTS AND DISCUSSION
To compute the electrostatic potential above the thundercloud from (5) and (6), we impose the following boundary conditions: at z=90 km where ε 0 is the vacuum permittivity, z b is the altitude of the plane setting directly above the thundercloud top, and h p and h n are the altitudes of positive and negative charge centers of the thundercloud, respectively. The first boundary condition follows from the accepted electrical model of the thundercloud. We assume that the thundercloud does not affect the atmospheric conductivity at z≥z b . Fig. 2 shows the computed electrostatic field component E r normalized to Q as a function of r in the nighttime and daytime midlatitude ionosphere at z≥90 km for the typical thundercloud (z b =10 km, h n =3 km, h p =8 km) and for the giant thundercloud (z b =20 km, h n =5 km, h p =17 km). Solar 7 Our calculations show that during solar minimum, in Equinox, the magnitude of Ʃ p at middle s is commonly in the ranges of 5.0-8.0 S and 0.1-0.2 S for day and night, respectively. er, the nighttime Ʃ p can be as low as 0.05 S. Under solar maximum conditions, Ʃ p is several arger than in solar minimum.

ULTS AND DISCUSSION
To compute the electrostatic potential above the thundercloud from (5) and (6), we impose the ng boundary conditions: (Q/4πε 0 )[(r 2 +(z b -h p ) 2 ) -1 / 2 -(r 2 +(z b -h n ) 2 ) -1 / 2 ] at z=z b is continuous at z=40 km ∂/∂z=2Ʃ p (∂ 2 /∂r 2 +1/r ∂/∂r) , at z=90 km ε 0 is the vacuum permittivity, z b is the altitude of the plane setting directly above the rcloud top, and h p and h n are the altitudes of positive and negative charge centers of the rcloud, respectively. The first boundary condition follows from the accepted electrical model of ndercloud. We assume that the thundercloud does not affect the atmospheric conductivity at Fig. 2 shows the computed electrostatic field component E r normalized to Q as a function of r ighttime and daytime midlatitude ionosphere at z≥90 km for the typical thundercloud (z b =10 3 km, h p =8 km) and for the giant thundercloud (z b =20 km, h n =5 km, h p =17 km). Solar where ε 0 is the vacuum permittivity, z b is the altitude of the plane setting directly above the thundercloud top, and h p and h n are the altitudes of positive and negative charge centers of the thundercloud, respectively. The first boundary condition follows from the accepted electrical model of the thundercloud. We assume that the thundercloud does not affect the atmospheric conductivity at z≥z b . Fig. 2 shows the computed electrostatic field component E r normalized to Q as a function of r in the nighttime and daytime midlatitude ionosphere at z≥90 km for the typical thundercloud (z b =10 km, h n =3 km, h p =8 km) and for the giant thundercloud (z b =20 km, h n =5 km, h p =17 km). Solar minimum conditions are considered with 5 E z = -∂Φ/∂z.
Since the geomagnetic field B is assumed to be vertical, E r is perpendicular to B, while E z

B.
Above 90 km, the geomagnetic field lines are practically equipotential because th geomagnetic field aligned conductivity  0 is much higher than the transverse conductivity us to consider the ionospheric region above 90 km as a thin conducting layer with a geom line integrated Pedersen conductivity Ʃ p , and the continuity equation of electric current at takes the following form: =0.05 S at night and 5 E z = -∂Φ/∂z.
Since the geomagnetic field B is assumed to be vertical, E r is p

B.
Above 90 km, the geomagnetic field lines are practical geomagnetic field aligned conductivity  0 is much higher than us to consider the ionospheric region above 90 km as a thin con line integrated Pedersen conductivity Ʃ p , and the continuity equ takes the following form:  0 E z = ⊥·(2Ʃ p E⊥), =5.0 S by day. All curves show similar behavior, attaining first a maximum and then revealing a gradual lowering. At night, the thundercloud electrostatic field is transmitted into the ionosphere much better than during the daytime. For a typical thundercloud, E r reaches its nighttime maximum value of ~2.6 μV/m (for Q=25 coulombs) at r~35 km. In the case of the giant thundercloud, the nighttime maximum magnitude of E r is ~270 μV/m (for Q=50 coulombs) and r max~4 0 km. The daytime maximum values of E r are one order of magnitude less than their nighttime values. Thus, the steady state electrostatic fields associated with the individual typical thunderclouds have very small magnitudes at ionospheric altitudes. In the case of a giant thundercloud, E r is two orders of magnitude larger. Note that Park & Dejnakarintra (1973) discovered that the maximum magnitude of the transverse electrostatic field produced in the nighttime midlatitude ionosphere by a giant thundercloud with Q=50 coulombs can be as large as ~700 μV/m, which is about 2.6 times more than in our estimate. This difference can mainly be attributed to the fact that Park & Dejnakarintra (1973) ignored the Pedersen conductivity above 150 km.

Fig. 2.
Calculated magnitude of the thundercloud electrostatic field strength E r normalized to Q, as a function of r, at ionospheric altitudes z≥90 km for the typical and giant thundercloud at night and by day.