On calculating the electric and magnetic elds produced in technological systems at the Earth s surface by a wide electrojet

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1 Journal of Atmospheric and Solar-Terrestrial Physics 6 ( On calculating the electric and magnetic elds produced in technological systems at the Earth s surface by a wide electrojet D.H. Boteler, R. Pirjola 1, L. Trichtchenko Geomagnetic Laboratory, Geological Survey of Canada, 7 Observatory Crescent, Ottawa, Ont., Canada, K1A Y3 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 1 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(=(i=a=( +a, 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 comple image method, leads to simple formulas that provide a method for fast calculation of the electric and magnetic elds that can aect ground-based systems. Crown Copyright c Published by Elsevier 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.: fa: address: boteler@geolab.nrcan.gc.ca (D.H. Boteler. 1 Permanent address: Finnish Meteorological Institute, P. O. Bo 53, FIN-11 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, Consequently, there has been interest in developing simpler approimate epressions for the Earth surface magnetic and electric elds. The simplest approimation 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 in- uences the induction process. An approimate solution is to replace the induced currents by an image current at a comple depth derived from the surface impedance of the Earth. Boteler and Pirjola (1998 recently presented a simple derivation of the comple image formulas for an innitely long line current and showed that they produce results very //$ - see front matter Crown Copyright c Published by Elsevier Science Ltd. All rights reserved. PII: S (71-7

2 131 D.H. Boteler et al. / Journal of Atmospheric and Solar-Terrestrial Physics 6 ( produced at the Earth s surface by an ionospheric current distribution of nite width. The accuracy of this approimate method is demonstrated by a comparison of the method with results obtained by numerical integration.. The equivalence of current distributions We use the conventional geomagnetic coordinate system with northward, y eastward and z vertically down, and consider an innitely long current owing parallel to the y-ais at a height h. For a current distribution j( that is an even function of, the magnetic and electric elds produced at the surface of the Earth can be epressed as integrals over all wavenumbers : Fig. 1. (a Schematic diagram of a current with a Cauchy distribution j(=(i=a=( +a where a= km at a height h=1 km and the equivalent line current at a height, h + a = 3 km. (b Plot of the Cauchy distribution shown in (a. close to the eact calculations for parameters typical of induction due to an auroral electrojet. Pirjola and Viljanen (1998 have shown that the comple 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 (197 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 eactly into account. These results are then combined with the comple image method to show how simple calculations can be made for the electric and magnetic elds B 1 (= J ((1 + Re h cos d (1 B 1 z(= J ((1 Re h sin d ( E 1 y(= where the current function is given by J (= i! J ((1 Re h cos d (3 j(e i d: (4 For a line current of amplitude I the current distribution j L( is j L(=I( (5 where is the delta function. Then Eq. (4 gives J L(=I (e i d = I (6 which can be substituted into Eqs. (1 (3 to obtain B I (= (1 + Re h cos d (7 B I z(= (1 Re h sin d (8 E I y(= i! (1 Re h cos d (9 which are the familiar epressions for the elds due to a line current above the Earth presented by Hermance and Peltier (197 and others. Now consider a current with a Cauchy distribution (Korn and Korn, 1961, j C(= I a + a : (1

3 D.H. Boteler et al. / Journal of Atmospheric and Solar-Terrestrial Physics 6 ( Substituting into Eq. (4 and performing the integration (Gradshteyn and Ryzhik, 1965, gives J C(=Ie a : (11 Substituting into Eqs. (1 (3 gives B I (= (1 + Re (h+a cos d (1 B I z(= (1 Re (h+a sin d (13 E I i! y(= (1 Re (h+a cos d: (14 Comparing Eqs. (1 (14 with Eqs. (7 (9 shows that the epressions for a Cauchy distribution are identical to those for a line current ecept for the change in the eponential 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 eactly 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. Comple image calculations Making the approimation (see Boteler and Pirjola, 1998 that the reection coecient can be epressed as R =e p (15 where p is the comple skin depth of the Earth, allows the magnetic eld epressions (Eqs. (1 and (13 to be written as B = I [e (h+a +e (h+a+p ] cos d (16 B z = I [e (h+a e (h+a+p ] sin d: (17 Using the relations e a cos b d = a=(a + b and e a sin b d = b=(a + b (Gradshteyn and Ryzhik, 1965, , these become B = I ( h + a (h + a + + h + a +p (h + a +p + (18 B z = I ( (h + a + (h + a +p + : (19 Similarly, using the comple image approimation (Eq. (15 in Eq. (14 gives the electric eld epression E y = i!i Using the integral relation e a e b e (h+a e (h+a+p cos d: ( cos m d = 1 ln b + m a + m (Gradshteyn and Ryzhik, 1965, this becomes [ E y = i!i ] (h + a ln +p + : (1 (h + a + These elds are equivalent to the elds produced by a line current at height h + a and an image current at a comple depth h + a +p. As an eample of the comple image calculations for a Cauchy distribution we use Eqs. (18, (19, and (1 to calculate the elds at the surface of a multi-layer earth model representing the conductivity structure of Quebec (see Fig. of Boteler and Pirjola, The model has layers (from the surface down with thicknesses of 15, 1, 15,, km and resistivities of,, 1, 1, 3 m. Calculations are made for a period of 5 min. These results are compared with calculations made for the eact epressions (Eqs. (1 (14 using numerical integration. Calculations were made for a current with a Cauchy distribution with a = km at a height h = 1 km shown in Fig. 1. The results are shown in Fig.. The close agreement between the two sets of results illustrates the accuracy of the comple image method. A similar good agreement was obtained for dierent periods and with dierent Earth models. These results show that the comple image epressions for a Cauchy current distribution provide an easy way of including both the vertical conductivity structure of the Earth and the horizontal etent of the ionospheric currents in calculations of the magnetic and electric elds that aect technological systems on the ground. 4. Conclusions 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(=(i=a=( + a are identical to those produced by a line current I at a height h + a. Combining this equivalence with the use of the comple skin depth, p, leads to the simple epressions B = I ( h + a (h + a + + h + a +p (h + a +p + (

4 1314 D.H. Boteler et al. / Journal of Atmospheric and Solar-Terrestrial Physics 6 ( Fig.. The horizontal (B and vertical (B z magnetic elds and the horizontal (E y 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 comple-image method solid lines show results for the eact epressions obtained by numerical integration. B z = I ( (h + a + (h + a +p + [ E y = i!i ] (h + a ln +p + (h + a + (3 (4 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 approimation 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,

5 D.H. Boteler et al. / Journal of Atmospheric and Solar-Terrestrial Physics 6 ( this approimation is especially useful and represents a signicant improvement over the use of a simple 1-km-high line current model for the electrojet. References Boteler, D.H., Pirjola, R.J., The comple image method for calculating the magnetic and electric elds produced at the surface of the Earth by the auroral electrojet. Geophysical Journal International 13, Boteler, D.H., Pirjola, R.J., Nevanlinna, H., The eects of geomagnetic disturbances on electrical systems at the Earth s surface. Advances in Space Research, Gradshteyn, I.S., Ryzhik, I.M., Table of Integrals, Series, and Products. Academic Press, New York. Hakkinen, L., Pirjola, R., Calculation of electric and magnetic elds due to an electrojet current system above a layered earth. Geophysica, Hermance, J.F., Peltier, W.R., 197. Magnetotelluric elds of a line current. Journal of Geophysical Research 75, Kannangara, M.L.T., 197. Some polarization characteristics of Pc 3-5 micropulsations observed at Colombo, Ceylon. Journal of Geophysical Research 77, Kertz, W., Modelle fur erdmagnetisch induzierte elektrische Strome im Untergrund. Nachr. Akad. Wiss. Gottingen, Math.-phys. Kl. Vol. IIa, pp Korn, G.A., Korn, T.M., Mathematical Handbook for Scientists and Engineers. McGraw-Hill, New York. Maurer, H., Theile, B., Parameters of the auroral electrojet from magnetic variations along a meridian. Journal of Geophysics 44, McNish, A.G., Heights of electric currents near the auroral zone. Terrestrial Magnetism 43, Park, D., Magnetic eld of a horizontal current above a conducting earth. Journal of Geophysical Research 78, Pirjola, R.J., Hakkinen, L.V.T., Electromagnetic eld caused by an auroral electrojet current system model. In: Kikuchi, H. (Ed., Environmental and Space Electromagnetics. Springer-Verlag, Tokyo, pp Pirjola, R., Viljanen, A., Comple image method for calculating electric and magnetic elds produced by an auroral electrojet of nite length. Annales Geophysicae 16,