Calculation methods of the electric and magnetic fields at the Earth's surface produced by a line current

Authors


Abstract

[1] Space weather storms involve intense and rapidly varying electric currents in the ionosphere, which create electric and magnetic fields at the Earth's surface. The electric fields drive geomagnetically induced currents (GIC) in technological networks and may have serious impacts. For assessing the hazards it is necessary to estimate GIC magnitudes, and this requires calculations of the electric and magnetic fields produced at the Earth's surface by the ionospheric currents. The surface fields are also affected by currents induced within the ground and influenced by the conductivity of the Earth. This also has to be taken into account. The calculation methods should be fast enough that they can be applied to forecasting the fields and GIC, for example, by using satellite observations of the solar wind. In this paper, we consider an infinitely long horizontal line current, which is the basic model of an auroral electrojet and simple enough to give insight into the physics and calculation techniques. The Earth is assumed to be composed of horizontal layers. We consider the exact integral expressions of the fields at the Earth's surface. The applicability of a series expansion technique (SER) and the complex image method (CIM), both of which were originally developed for other disciplines, are reviewed and summarized by giving the expressions of the electric and magnetic fields at the Earth's surface and by considering the mathematical assumptions required. Numerical calculations and comparisons with exact solutions show that SER and CIM are very accurate.

1. Introduction

[2] The (geo)electric field occurring at the Earth's surface in connection with geomagnetic storms produces geomagnetically induced currents (GIC) in technological systems, such as electric power transmission grids, oil and gas pipelines, telecommunication cables, and railway equipment [Boteler et al., 1998]. GIC are the end of the space weather chain (Sun-solar wind-magnetosphere-ionosphere-Earth's surface) and thus constitute an essential part of today's active space weather research. GIC are a possible source of problems to the systems. Power system transformers may be saturated by GIC, leading to troubles in the operation of the system or even to permanent damage [Molinski et al., 2000]. The best known event is the blackout of electricity in Québec, Canada, for several hours during the large magnetic storm in March 1989 [Czech et al., 1992]. Pipelines may suffer from problems associated with corrosion and its control due to GIC and the accompanying pipe-to-soil voltages [Gummow, 2002]. Telecommunication and railway equipment can experience harmful overvoltages.

[3] The first GIC observations were already made in the early telegraph systems over 150 years ago; the history and several references are presented by Boteler et al. [1998]. Large GIC occur most frequently in the auroral regions that extend across North America and the Nordic countries. However, during intense space storms, high values of GIC may occur at lower latitudes, also depending on the network configuration and resistances. The problems GIC may cause to a power system vary with the electric power transmitted in the grid at the particular moment of a large GIC. The increasing number of technological systems and the growing dependence of society on their reliability make research on space weather, including GIC, very important today.

[4] It is usually convenient to make a theoretical calculation of GIC in a system in two separate steps: (1) The horizontal geoelectric field at the Earth's surface is determined. (2) This field is an “external” input to compute the currents in the particular technological system.

[5] In general, the second step, which is basically an application of electric circuit theory, is faster and easier to perform: A matrix formalism is appropriate for a power system [Lehtinen and Pirjola, 1985], and a pipeline network can be treated by using distributed-source transmission line theory [Boteler, 1997]. The first step requires models about ionospheric-magnetospheric currents and about the Earth's conductivity structure as the surface electric and magnetic fields are, besides being primarily produced by space currents, secondarily affected by currents induced in the conducting ground. At high latitudes, where the most significant geomagnetic effects occur, geoelectromagnetic disturbances are principally due to an auroral electrojet current system, which contains an intense east-west current (the electrojet) in the ionosphere. The electrojet is typically some hundreds of kilometers wide and on the order of thousands of kilometers long. The ionospheric current system is coupled, through field-aligned currents, to current systems farther out in the Earth's magnetosphere. The electrojet flows at a height of approximately 110 km and can reach a magnitude of millions of amperes.

[6] The simplest model for calculating the surface electric and magnetic fields due to an auroral electrojet is an infinitely long horizontal line current above a uniform or layered Earth [Price, 1962; Albertson and Van Baelen, 1970; Hermance and Peltier, 1970]. Examination of this problem provides a basic solution that can often be extended to include more realistic and sophisticated electrojet features. As a sheet current of a specified width can be constructed of adjacent line currents, generalizations to sheet currents are obtained by a trivial superposition from line current results. Furthermore, Boteler et al. [2000] demonstrate the equivalence of a sheet current with a line current at a greater height. Calculations of the electric and magnetic fields produced by a line current above the Earth are usable in other areas, too, such as studies of the coupling between power lines and other conductors and geophysical exploration of the ground structure.

[7] A straightforward calculation of the electric and magnetic fields at the surface of a layered Earth caused by an overhead infinitely long line current, which is based on solving Maxwell's equations and on using boundary conditions, leads to integral expressions of the fields [Price, 1962; Albertson and Van Baelen, 1970]. The fast Fourier (FFT) and fast Hankel (FHT) transforms provide efficient methods of computing the integrals. An alternative approach is to write the integrals as series expansions. This was used by Carson [1926] to provide an expression for the electric field at the surface of a uniform Earth. Recently, exact series expansions have been presented by Pirjola [1998] for both the electric and magnetic fields on a uniform Earth. Approximate series expansions may also be derived in the case of a layered Earth [Pirjola et al., 1999].

[8] In the complex image method (CIM) the secondary contribution from the ground to the surface fields is calculated by replacing the real ground by a perfect conductor located at a complex depth, which depends on the Earth's real conductivity structure and on the frequency [Wait and Spies, 1969; Thomson and Weaver, 1975; Bannister, 1986]. This greatly simplifies the problem and leads to fast and handy numerical computations. When the primary source is an infinitely long line current (as in this paper), CIM results in simple closed-form expressions for the surface electric and magnetic fields. While CIM was originally introduced and is widely used for engineering purposes, its applicability to space weather and GIC research as well has been emphasized and demonstrated recently [Boteler and Pirjola, 1998; Pirjola and Viljanen, 1998; Viljanen et al., 1999; Pirjola et al., 2000].

[9] Boteler and Pirjola [1999] provide a summary of and a comparison between the exact method (EXA), the series expansion method (SER), and the complex image method (CIM) of calculating the surface fields due to an infinitely long line current. In this paper, we consider the same geophysical model and the three methods, paying special attention to the mathematical approximations needed. This paper provides formulas readily usable for computing the surface fields with the three methods. The derivation of the equations is not included here, but pertinent references are given. Numerical computations are presented which show the good accuracy of SER and CIM. The great practical benefits of applying CIM do not become fully clear in connection with an infinitely long line current because EXA calculations can also be made fast then. The value of CIM would be much higher if a more complicated ionospheric-magnetospheric current system is considered. Thus this paper serves more as a summary of the three methods and as a proof of the accuracies of SER and CIM than as a plain demonstration of the efficiency of CIM.

2. Model

[10] We use the standard geomagnetic coordinate system, in which x is northward, y is eastward, and z is vertically down. The Earth's surface is the xy plane, and the line current is assumed to flow parallel to the y axis at height h (i.e., z = −h). The line current is assumed to have a sinusoidal time dependence with an angular frequency ω. The time factor eiωt would appear in the expressions but is not shown to avoid cluttering the equations. The Earth consists of N horizontal layers with thicknesses di, conductivities σi, permittivities εi, and permeabilities μi (i = 1, 2, …, N).

[11] Geoelectromagnetic frequencies are so slow that displacement currents do not play any role, so the permittivities may be ignored in practice. It is also a reasonable assumption that all permeabilities equal the vacuum value μ0. The Earth can be described by a surface impedance Z, which expresses the ratio between a horizontal electric field component and the perpendicular magnetic component (e.g., −Ey and Hx = Bx0) at the Earth's surface and depends on the frequency ω and on the horizontal wave number b associated with the x coordinate via a Fourier transform. The surface impedance can be obtained from a recursive formula, which is based on determining the impedance at the top of one layer in terms of the response at the top of the underlying layer [Albertson and Van Baelen, 1970; Boteler and Pirjola, 1999].

[12] Exact method (EXA) expressions for the electric and magnetic fields created at the Earth's surface can be derived in a straightforward manner based on Maxwell's equations and boundary conditions. The electric field only has a y component Ey, and the nonzero magnetic components are Bx and Bz. These can be written as integrals over b. We now give expressions that explicitly separate the primary and secondary contributions [Hermance and Peltier, 1970]:

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where J gives the magnitude (and the phase) of the line current and R = R(b) is the reflection coefficient at the Earth's surface given by

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Defining the complex skin depth p = p(b) by

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the reflection coefficient can be written as

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[13] It should be noted that p equals the “inductive response function” used in geoelectromagnetic induction studies of the Earth [Schmucker, 1970]. Substituting c into formulas (1)(3), the electric and magnetic fields also have the following EXA expressions [Pirjola et al., 1999]:

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[14] The EXA integrals can be solved by a variety of numerical techniques, for example, the Gauss integration formula [Abramowitz and Stegun, 1970] used by Pirjola and Häkkinen [1991] for a more elaborate ionospheric-magnetospheric current model, or Filon's technique [Tranter, 1956] used by Hermance and Peltier [1970]. Formulas (1)(3) and (7)(9) are Fourier cosine and sine transforms, so applying FFT is a possibility for computing the integrals numerically. Expressing sin bx and cos bx in terms of Bessel functions of the order of 1/2 and −1/2 leads to the use of FHT for numerical computations [Johansen and Sørensen, 1979; Boteler and Pirjola, 1999].

[15] Computations of exact formulas for the surface electric and magnetic fields tend to become time-consuming without powerful computing resources if more complicated (three-dimensional) ionospheric-magnetospheric current models are applied. Therefore it is necessary to search for approximate methods that permit faster calculations while still maintaining sufficient accuracy. It should be remembered that EXA solutions also always necessarily simplify the real space physical and geophysical situation, and so they are never really “exact.”

3. Series Expansions

[16] Series expansions for the surface electric field due to an overhead line current originally presented by Carson [1926] and subsequently rearranged for easier computation [Galloway et al., 1964] have been widely used in electrical engineering. However, Carson's expressions are only valid for a uniform Earth and do not contain any formulas for the magnetic field.

[17] Pirjola [1998] extended Carson's [1926] result by providing series expansions for both the magnetic and electric fields at the surface of a uniform Earth. The extension is based on equations (7)(9) and on the formulas

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where the propagation constant k of the Earth is defined by

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Substituting equations (11) and (12) into formula (7) permits the expression of Ey in terms of the Struve and Neumann functions, which have series expansions. The magnetic field is obtained by simple derivation operations from the electric field, and the final series expansions (SER) are the following:

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where

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The notation CEuler refers to Euler's constant (≈ 0.5772). The gamma function is denoted by Γ, and

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[18] It should be emphasized that the derivation of the series necessarily requires, as assumed above, that the permeability of the Earth equals μ0 and that the displacement currents are insignificant above the Earth. For consistency, displacement currents are also neglected in the Earth in equation (13).

[19] By looking at equations (7)(9), Pirjola et al. [1999] concluded that similar series expansions are also obtained for a layered Earth if p can be expressed as

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where the “equivalent propagation constant” ku is independent of b. A comparison of equations (5) and (24) defines the “equivalent surface impedance” Zu:

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Demanding that Zu and Z are equal when b = 0,

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where Zp = Z(b = 0) is the "plane wave surface impedance." When b goes to infinity, the correct layered-Earth surface impedance approaches iωμ0/b, and this is also the limit of Zu. Consequently, Z and Zu only differ for a range of small (nonzero) b values [Pirjola et al., 1999]. For a uniform Earth, Z and Zu, as well as k and ku, are equal (equations (10), (12), (25), and (26)).

[20] Summarizing, the SER expressions given by formulas (14)(23) are also valid for a layered Earth (with k replaced by ku) if p can be approximated as

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where p0 is the “plane wave complex skin depth”:

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[21] Substituting equation (27) into formula (6) yields the following expressions for the reflection coefficient, denoted by RSER now to indicate that this approximation is necessary for the series expansions to be valid:

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where s = p0b. Boteler and Pirjola [1999] give an approximate expression for RSER by assuming that |s| is small and using series expansions to the order s2 separately on the top and bottom of the formula for RSER. Their starting point is the latter form shown in equation (29), and so the formula is (1 − s + s2/2)/(1 + s + s2/2). However, if the expression were derived by expanding the top and bottom of the first form in equation (29), the result would be (1 − s + s3/2)/(1 + ss3/2). The different approximate expressions are, of course, due to mathematical inaccuracies involved in expanding the top and bottom separately. If |s| is small, permitting a series expansion of the reflection coefficient, RSER given by formulas (29) has to be considered a function of s and expanded as an ordinary Taylor series. We then obtain

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The next term in the series would be of the order s5 (i.e., not s4), so equation (30) represents the series of the fourth order. A numerical study (not shown here) indicates that the difference between expressions (29) and (30) is insignificant for values of |s| less than about 0.5–0.6, and the two incorrect expressions mentioned above are also close to the reflection coefficient RSER given by formula (29) then.

4. Complex Image Method

[22] Boteler and Pirjola [1998] present a simple derivation of the complex image method (CIM) for calculating the surface fields due to a line current [see also Boteler and Pirjola, 1999]: Assuming that |pb| is small, equation (6) can be expressed as a series:

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Up to the second order this is identical with e−2pb ≈ 1 − 2pb + 2(pb)2 − (4/3)(pb)3. Consequently, we replace R by e−2pb in equations (1)(3). If p can be assumed to be independent of b, the equations show that the secondary electromagnetic field at the Earth's surface due to currents within the Earth is simply obtained by replacing the Earth by an image current opposite to the primary line current that lies at the depth h + 2p; that is, a fictitious perfect conductor is located at the complex depth p. It is natural to assume that the value of p independent of b and necessary for CIM equals the plane wave complex skin depth p0 given by equation (28). Thus, with s = p0b, equation (6) gives

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A series expansion of RCIM, corresponding to formula (30) for RSER, is

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A numerical investigation shows that RCIM and its approximate expansion (equations (32) and (33)) do not practically differ for |s| less than about 0.5, and for |s| smaller than about 0.3 they are nearly equal to RSER given by equations (29) and (30).

[23] By using the CIM approximation R = exp (−2p0b), the integrals of equations (1)(3) can be calculated in closed forms [Boteler and Pirjola, 1999]:

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[24] Pirjola and Viljanen [1998] provide an important extension to the applicability of CIM as they show that for calculating the horizontal electric field and the magnetic field at the Earth's surface, a vertical primary source current can equivalently be replaced by a horizontal current distribution. This equivalence combined with the application of the CIM formulation that Thomson and Weaver [1975] presented for a horizontal current sheet makes it possible to use CIM in connection with three-dimensional ionospheric-magnetospheric current systems.

[25] CIM is an efficient mathematical concept that permits simple and fast computations of the electric and magnetic fields at the Earth's surface due to overhead currents. But the complex skin depth also has a physical interpretation as its real and imaginary parts correspond to the central depths of the in-phase and out-of-phase currents induced in the Earth at the particular frequency considered [Weidelt, 1972; Szarka and Fischer, 1989]. Practical computations of the surface electric and magnetic fields and of GIC often utilize data in the time domain and require the results as functions of the time. Thus, because CIM operates in the frequency domain, applications of (fast) Fourier transforms are needed.

5. Numerical Results

[26] To compare the three calculation methods (EXA, SER, and CIM) discussed in the paper, we compute the magnetic and electric fields that will be produced by an infinitely long line current of 1 MA at the height 110 km above the Earth's surface. The calculations are made for the period 600 s = 10 min. The Earth is assumed to have a five-layer structure with the following layer thicknesses and resistivities: [15, 10, 125, 200, ∞] km and [20000, 200, 1000, 100, 3] Ωm. This choice corresponds to the model for Québec, Canada, and is representative of a resistive area [Boteler and Pirjola, 1998]. Generally, horizontal electric field magnitudes tend to increase with the Earth's resistivity, emphasizing the possibility of GIC problems in resistive regions. The x range at the Earth's surface where the fields are calculated is from 0 to 200 km. The fields for negative x values are directly obtained by noting that Ey and Bx are even and Bz is odd with respect to x (equations (1)(3)). For |x| values larger than about 200 km, SER would experience numerical problems [Pirjola et al., 1999].

[27] The EXA calculations are performed by applying the FHT method to equations (1)(3) with equation (4). The SER computations are directly based on equations (14)(23) with k replaced by the equivalent propagation constant ku. In the CIM calculations we simply need to compute the electric and magnetic fields produced by two line currents (equations (34)(36)). For the magnetic field the real part is the largest, while for the electric field the imaginary part is the most significant. Therefore we only consider these quantities here. Figure 1 shows the results for the real part of the horizontal magnetic field Bx. Figure 2 is associated with the real part of the vertical magnetic field Bz. Figure 3 depicts the imaginary part of the electric field Ey. In each figure, EXA, SER, and CIM calculations are shown by solid lines, circles, and crosses, respectively.

Figure 1.

Real part of the perpendicular horizontal magnetic field component Bx at the surface of a layered Earth due to an infinitely long line current of 1 MA at the height 110 km. The period considered is 10 min, and the Earth has the “Québec” structure (see the text). The Earth's surface is the xy plane, and the line current is parallel to the y axis at x = 0. The solid line, the circles, and the crosses refer to exact calculations (section 2), to series expansions (section 3), and to the complex image method (section 4), respectively.

Figure 2.

. Similar to Figure 1, but refers to the real part of the vertical magnetic component Bz.

Figure 3.

Similar to Figure 1, but refers to the imaginary part of the parallel electric component Ey.

[28] All the figures show that there is very little difference between the results obtained by the different methods. When considering the less important quantities, i.e., Im (Bx), Im (Bz), and Re (Ey), slightly larger relative differences would be seen between the three methods. Anyway, the inaccuracies between EXA, SER, and CIM remain small compared to the uncertainties in the choice of a particular layered-Earth model to represent the conductivity structure of a region and in using a simplified model of the ionospheric-magnetospheric current distribution.

6. Concluding Remarks

[29] The electric and magnetic fields produced at the surface of the Earth by an infinitely long line current above the Earth are discussed in this paper. The fields are dependent on the amplitude, frequency, and height of the current. The fields are also affected by currents induced in the Earth, which are themselves influenced by the conductivity structure of the Earth. The line current model is the first approximation of the auroral electrojet and is simple enough to provide basic information about the geophysical processes and calculation techniques.

[30] Assuming that the conductivity of the Earth varies only with depth, the electric and magnetic fields at the Earth's surface due to the line current can be expressed as integrals over a horizontal wave number. Different techniques are available for performing numerical calculations. In practice, as the integrals are cosine and sine transforms, they can be solved using fast Fourier transform routines. The integrals may also be rewritten in terms of Bessel functions and solved using a fast Hankel transform.

[31] Another approach is to use series expansions for the integrals. This was originally done for the electric field at the surface of a uniform Earth by Carson [1926] and recently extended for the magnetic field, too, by Pirjola [1998]. The series expansions also give approximate results for a layered conductivity structure of the Earth [Pirjola et al., 1999].

[32] Boteler and Pirjola [1998] and Pirjola and Viljanen [1998] have demonstrated the applicability of the complex image method to geoelectromagnetic studies related to space weather research. Simple approximate solutions of the electric and magnetic fields at the Earth's surface can be found by representing the effect of induced currents in the Earth by an image current at a complex depth, whose value depends on the surface impedance for a layered conductivity model. The complex image method is particularly useful in connection with complicated three-dimensional ionospheric-magnetospheric current distributions since it permits very fast numerical computations.

[33] Although not exact, the errors in the results obtained by using series expansions or the complex image method are insignificant compared to typical uncertainties in the geophysical and space physical parameters usually involved in space weather and geoelectromagnetic studies.

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