Journal of Atmospheric and Solar-Terrestrial Physics 62 (2000) 1311–1315
www.elsevier.nl/locate/jastp
On calculating the electric and magnetic elds produced in
technological systems at the Earth’s surface by a “wide”
electrojet
D.H. Boteler ∗ , R. Pirjola1 , L. Trichtchenko
Geomagnetic Laboratory, Geological Survey of Canada, 7 Observatory Crescent, Ottawa, Ont., Canada, K1A 0Y3
Received 7 July 1999; accepted 1 November 1999
Abstract
Forecasting the geomagnetic eects to technological systems on the ground requires rapid calculations of the electric and
magnetic elds produced by the auroral electrojet. The electrojet is often modelled as a line current at a height of 100 km, but
in reality it has a nite width and is typically spread over 5◦ or 6◦ of latitude. We show that the width of the electrojet can
easily be included in electric and magnetic eld calculations by assuming that the ionospheric current density has a Cauchy
distribution with a half-width a; j(x) = (I=)a=(x2 + a2 ), at a height h. It is shown that the electric and magnetic elds produced
at the surface of a layered earth by such a current are equivalent to the elds produced by a line current I at a height h + a. This
equivalence, combined with the complex image method, leads to simple formulas that provide a method for fast calculation
c 2000 Published by Elsevier
of the electric and magnetic elds that can aect ground-based systems. Crown Copyright Science Ltd. All rights reserved.
Keywords: Geo-electric eld; Electrojet
1. Introduction
Geomagnetic disturbances produce electric elds at the
Earth’s surface that can give rise to problems with technological systems, such as power systems, pipelines and phone
cables (see Boteler et al. (1998) for a review of geomagnetic
eects and further references). A major cause of geomagnetic disturbances is the auroral electrojet and an assessment
of the eects on technological systems requires that appropriate models of the electrojet are available. For time-critical
applications, such as forecasting the levels of geomagnetically induced currents in power systems, it is also important that the electrojet model allow rapid calculations of
the Earth-surface electric elds. The complete electrojet
∗ Corresponding author. Tel.: +1-613-837-2035; fax: +1-613824-9803.
E-mail address: boteler@geolab.nrcan.gc.ca (D.H. Boteler).
1 Permanent address: Finnish Meteorological Institute, P. O. Box
503, FIN-00101 Helsinki, Finland.
system involves ionospheric currents, spanning hundreds of
kilometres, connected to eld-aligned currents that couple to
currents in the magnetosphere. Techniques for calculating
the elds at the Earth’s surface produced by such a current
system have been presented by Hakkinen and Pirjola
(1986). However, these calculations are complicated and
are highly demanding of computer resources (Pirjola and
Hakkinen, 1991). Consequently, there has been interest in
developing simpler approximate expressions for the Earth
surface magnetic and electric elds.
The simplest approximation is to represent the ionospheric
current by an innitely long line current. Even for this case
the problem is complicated by the presence of induced currents in the Earth that create a magnetic eld which inuences the induction process. An approximate solution is
to replace the induced currents by an image current at a
complex depth derived from the surface impedance of the
Earth. Boteler and Pirjola (1998) recently presented a simple
derivation of the complex image formulas for an innitely
long line current and showed that they produce results very
c 2000 Published by Elsevier Science Ltd. All rights reserved.
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D.H. Boteler et al. / Journal of Atmospheric and Solar-Terrestrial Physics 62 (2000) 1311–1315
produced at the Earth’s surface by an ionospheric current
distribution of nite width. The accuracy of this approximate method is demonstrated by a comparison of the method
with results obtained by numerical integration.
2. The equivalence of current distributions
Fig. 1. (a) Schematic diagram of a current with a Cauchy distribution j(x)=(I=)a=(x2 +a2 ) where a=200 km at a height h=100 km
and the equivalent line current at a height, h + a = 300 km.
(b) Plot of the Cauchy distribution shown in (a).
close to the exact calculations for parameters typical of induction due to an auroral electrojet. Pirjola and Viljanen
(1998) have shown that the complex image method can also
be used for a current system with a nite-length electrojet
and eld-aligned currents.
The width of the electrojet is another aspect of the auroral current system that needs to be included in realistic calculations. McNish (1938) suggested that a “wide”
ionospheric current distribution is equivalent to a line current at a greater height. Kertz (1954) showed that the magnetic eld produced at the Earth’s surface by a current with a
Cauchy distribution is equivalent to that produced by a line
current at a greater distance, and Maurer and Theile (1978)
used this equivalence in studies of the auroral electrojet.
The same equivalence was also pointed out by Park (1973)
and used by Kannangara (1972) for analysing the magnetic
elds produced by the equatorial electrojet. These workers
were only concerned with the magnetic elds produced by
the current systems and either ignored or over-simplied
the eect of currents induced in the Earth. For calculating
the geomagnetic eects on technological systems one of the
most important parameters is the electric eld and this is
signicantly aected by the variation of conductivity with
depth within the Earth.
In this paper we present a derivation of the equivalence
of a current with a Cauchy distribution to a line current at a
greater height (see Fig. 1) in terms of both the electric and
magnetic elds and taking the inuence of induced Earth
currents exactly into account. These results are then combined with the complex image method to show how simple
calculations can be made for the electric and magnetic elds
We use the conventional geomagnetic coordinate system
with x northward, y eastward and z vertically down, and
consider an innitely long current owing parallel to the
y-axis at a height h. For a current distribution j(x) that is an
even function of x, the magnetic and electric elds produced
at the surface of the Earth can be expressed as integrals over
all wavenumbers :
Z ∞
1
Bx (x) =
0 J ()(1 + R)e−h cos x d;
(1)
2 0
Z ∞
1
Bz (x) = −
0 J ()(1 − R)e−h sin x d;
(2)
2 0
Z ∞
i!0
1
Ey (x) = −
(3)
J ()(1 − R)e−h cos x d;
2 0
where the current function is given by
Z ∞
J () =
j(x)e−ix d x:
−∞
(4)
For a line current of amplitude I the current distribution
jL (x) is
jL (x) = I(x);
(5)
where is the delta function. Then Eq. (4) gives
Z ∞
JL () = I
(x)e−ix d x = I;
(6)
−∞
which can be substituted into Eqs. (1) – (3) to obtain
I
Bx (x) =
2
Z
Bz (x) = −
I
2
Ey (x) = −
I
2
∞
0
Z
0 (1 + R)e−h cos x d;
∞
0
Z
∞
0
(7)
0 (1 − R)e−h sin x d;
(8)
i!0
(1 − R)e−h cos x d;
(9)
which are the familiar expressions for the elds due to a line
current above the Earth presented by Hermance and Peltier
(1970) and others.
Now consider a current with a Cauchy distribution (Korn
and Korn, 1961),
jC (x) =
a
I
:
x 2 + a2
(10)
D.H. Boteler et al. / Journal of Atmospheric and Solar-Terrestrial Physics 62 (2000) 1311–1315
Substituting into Eq. (4) and performing the integration
(Gradshteyn and Ryzhik, 1965, 3.354.5) gives
JC () = I e−||a :
(11)
Substituting into Eqs. (1) – (3) gives
Bx (x) =
I
2
Bz (x) = −
Z
0
I
2
Ey (x) = −
∞
I
2
Z
0
Z
0
∞
(12)
0 (1 − R)e−(h+a) sin x d;
(13)
i!0
(1 − R)e−(h+a) cos x d:
(14)
Comparing Eqs. (12) – (14) with Eqs. (7) – (9) shows that
the expressions for a Cauchy distribution are identical to
those for a line current except for the change in the exponential term. This is equivalent to an increase in the height
of the line current. Thus, the electric and magnetic elds
produced by a Cauchy distribution with half-width a at a
height h are exactly the same as the elds produced by a
line current at a height h+a. Cauchy distributions of current
at other heights would also produce the same magnetic and
electric elds on the ground if the sum of the height and the
half-width, h + a, was the same.
3. Complex image calculations
Making the approximation (see Boteler and Pirjola, 1998)
that the reection coecient can be expressed as
R = e−2p ;
(15)
where p is the complex skin depth of the Earth, allows
the magnetic eld expressions (Eqs. (12) and (13)) to be
written as
Z ∞
0 I
Bx =
[e−(h+a) + e−(h+a+2p) ] cos x d;
(16)
2 0
Z ∞
0 I
Bz = −
[e−(h+a) − e−(h+a+2p) ] sin x d:
(17)
2 0
R∞
−ax
2
0 I
2
Ey = −
i!0 I
2
Z
0
∞
e−(h+a) − e−(h+a+2p)
cos x d:
(20)
x
x
−
(h + a)2 + x2
(h + a + 2p)2 + x2
Z
0
∞
e−ax − e−bx
1 b 2 + m2
cos mx d x = ln 2
x
2 a + m2
(Gradshteyn and Ryzhik, 1965, 3.951.3) this becomes
#
"p
(h + a + 2p)2 + x2
i!0 I
p
:
(21)
Ey = −
ln
2
(h + a)2 + x2
These elds are equivalent to the elds produced by a line
current at height h + a and an image current at a complex
depth h + a + 2p.
As an example of the complex image calculations for a
Cauchy distribution we use Eqs. (18), (19), and (21) to calculate the elds at the surface of a multi-layer earth model
representing the conductivity structure of Quebec (see
Fig. 2 of Boteler and Pirjola, 1998). The model has layers
(from the surface down) with thicknesses of 15, 10, 125,
200, ∞ km and resistivities of 20 000, 200, 1000, 100,
3 m. Calculations are made for a period of 5 min. These
results are compared with calculations made for the exact
expressions (Eqs. (12) – (14)) using numerical integration.
Calculations were made for a current with a Cauchy distribution with a = 200 km at a height h = 100 km shown in
Fig. 1. The results are shown in Fig. 2. The close agreement
between the two sets of results illustrates the accuracy of
the complex image method. A similar good agreement was
obtained for dierent periods and with dierent Earth models. These results show that the complex image expressions
for a Cauchy current distribution provide an easy way of
including both the vertical conductivity structure of the
Earth and the horizontal extent of the ionospheric currents
in calculations of the magnetic and electric elds that aect
technological systems on the ground.
4. Conclusions
2
Using the relations 0 e cos bx d x = a=(a + b ) and
R ∞ −ax
e sin bx d x = b=(a2 + b2 ) (Gradshteyn and Ryzhik,
0
1965, 3.893.1, 3.893.2) these become
0 I
h + a + 2p
h+a
Bx =
;
+
2 (h + a)2 + x2
(h + a + 2p)2 + x2
(18)
Bz = −
Similarly, using the complex image approximation
(Eq. (15)) in Eq. (14) gives the electric eld expression
Using the integral relation
0 (1 + R)e−(h+a) cos x d;
∞
1313
:
(19)
The electric and magnetic elds produced at the Earth’s
surface by an ionospheric current at a height h with a current density having a Cauchy distribution with a half-width
a; j(x) = (I=)a=(x2 + a2 ); are identical to those produced
by a line current I at a height h + a. Combining this equivalence with the use of the complex skin depth, p, leads to
the simple expressions
0 I
h + a + 2p
h+a
Bx =
;
+
2 (h + a)2 + x2
(h + a + 2p)2 + x2
(22)
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D.H. Boteler et al. / Journal of Atmospheric and Solar-Terrestrial Physics 62 (2000) 1311–1315
Fig. 2. The horizontal (Bx ) and vertical (Bz ) magnetic elds and the horizontal (Ey ) electric eld produced by a current of 1 million amps
with the Cauchy distribution shown in Fig. 1. The calculations are made for a period of 5 min and an earth model representing Quebec.
Asterisks show the results of calculations made using the complex-image method; solid lines show results for the exact expressions obtained
by numerical integration.
Bz = −
0 I
2
x
x
−
(h + a)2 + x2
(h + a + 2p)2 + x2
i!0 I
Ey = −
ln
2
"p
(h + a + 2p)2 + x2
p
(h + a)2 + x2
;
(23)
#
(24)
for the magnetic and electric elds produced by an ionospheric current with a Cauchy distribution. Any assumed
shape for the current density can only be an approximation to the real, more complicated, structure of the electrojet. However, using the equivalence between a Cauchy
distribution and a higher line current greatly simplies the
magnetic eld and electric eld calculations. For forecasting
geomagnetic eects, where rapid calculations are required,
D.H. Boteler et al. / Journal of Atmospheric and Solar-Terrestrial Physics 62 (2000) 1311–1315
this approximation is especially useful and represents a signicant improvement over the use of a simple 100-km-high
line current model for the electrojet.
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1315
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