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.
1. INTRODUCTION
Thunderclouds are tropospheric sources of intense electrostatic fields and electromagnetic radiation. It is known that lightningassociated 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 1020 ms and a magnitude of 150 mV/m (e.g.,
Kelley et al. 1985
,
1990
;
Vlasov & 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 heightintegrated Pedersen conductivities of both hemispheres.
2. 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 23 km to 812 km in altitude, and socalled 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:
where
J
is the electric current density, σ is the electrical conductivity tensor, and
E
and Φ 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 Φ cna be obtained from (1), (2), and (3) :
where
σ
_{p}
is the Pedersen conductivity and
σ
_{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
σ
_{0}
and
σ
_{p}
are exponential functions of z. In the case of isotropic conductivity (setting
σ
_{0}
=
σ
_{p}
=
b
exp(
z/h
), where b and h are constants), we obtain
where
J
_{0}
is the zeroorder Bessel function of the first kind,
A
_{1}
and
B
_{1}
are coefficients, and
c
_{1}
= l/(2
h
) [l/(4
h
^{2}
)+
k
^{2}
]
^{1/2}
,
c
_{2}
= l/(2
h
) + [l/(4
h
^{2}
)+
k
^{2}
]
^{1/2}
. For the anisotropic region, where we let
σ
_{0}
=
b
_{0}
exp(
z/h
_{0}
) and
σ
_{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
=2
νh
_{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 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
σ
_{0}
is much higher than the transverse conductivity
σ
_{p}
. It allows us to consider the ionoshperic region above 90 km as a thin conducting layer with a geomagnetic field line integrated Pedersen conductivity Σ
_{p}
, and the continuity equation of electric current at
z
=90 km takes the following form:
where ▽
_{⊥}
denotes the gradient operator in the two dimensions transverse to
B
, and the factor 2 before Σ
_{p}
accounts for a contribution of the Pedersen conductivity of the magnetically conjugate ionosphere. Equation (9) is explicitly expressed as
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
Cold & Pierce (1965)
below 40 km and by
Swider (1988)
from 40 to 70 km. In the anisotropic region between 70 and 90 km,
σ
_{0}
, and
σ
_{p}
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
Model altitude profiles of specific (σ_{0}) and Pedersen (σ_{p}) conductivities. The numbers next to the curves indicate conductivity scale heights in kilometers within each altitude section.
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 IRI2012 (
http://omniweb.gsfc.nasa.gov/vitmo/iri2012_vitmo.html
) and the neutral atmosphere model NRLMSIS00 (
http://ccmc.gsfc.nasa.gov/modelweb/models/nrlmsise00.php
).
Our calculations show that during solar minimum, in Equinox, the magnitude of Ʃ
_{p}
at middle latitudes is commonly in the ranges of 5.08.0 S and 0.10.2 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.
3. RESULTS AND DISCUSSION
To compute the electrostatic potential above the thundercloud from (5) and (6), we impose the following boundary conditions:

1. Φ=(Q/4πε0)[(r2+(zbhp)2)1/2(r2+(zbhn)2)1/2] atz=zb

2. Φ is continuous atz=40 km

3.σ0∂Φ/∂z=2Ʃp(∂2Φ/∂r2+1/r∂Φ/∂r) atz=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 minimum conditions are considered with Ʃ
_{p}
=0.05 S at night and Ʃ
_{p}
=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, Er 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}
~40 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.
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.
4. CONCLUSION
Our computations show that the geomagnetic field line integrated Pedersen conductivity of the ionosphere plays an important role in troposphereionosphere electrostatic coupling. Even for nighttime conditions in solar minimum, when the values of Ʃ
_{p}
are minimal, the electrostatic charges of the individual thundercloud can drive only small electrostatic fields at ionospheric altitudes.
Acknowledgements
This work was supported by Russian Academy of Sciences through Atmospheric Electrodynamics program (project No. 12).
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