The complex-image method for calculating the magnetic and electric fields produced at the surface of the Earth by the auroral electrojet

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1 Geophys. J. Int. (1998) 132, 31 4 The complex-image method for calculating the magnetic and electric fields produced at the surface of the Earth by the auroral electrojet D. H. Boteler and R. J. Pirjola* Geomagnetic L aboratory, Geological Survey of Canada, 1 Observatory Crescent, Ottawa, Ontario K1A Y3, Canada. boteler@geolab.nrcan.gc.ca Accepted 1997 July 7. Received 1997 June 23; in original form 1997 March 18 INTRODUCTION * Permanent address: Finnish Meteorological Institute, PO Box 53, FIN-11 Helsinki, Finland. SUMMARY For studying the auroral electrojet and for examining the effects it can produce in power systems on the ground, it is useful to be able to calculate the magnetic and electric fields that the electrojet produces at the surface of the Earth. Including the effects of currents induced in the Earth leads to a set of integral expressions, the numerical computation of which is complicated and demanding of computer resources. An approximate solution can be achieved by representing the induced currents by an image current at a complex depth. We present a simple derivation of the compleximage expressions and use them to calculate the fields produced by the auroral electrojet at the surface of an earth represented by layered conductivity models. Comparison of these results with ones obtained using the exact integral solution show that the errors introduced are insignificant compared to the uncertainties in the parameters used. The complex-image method thus provides a simple, fast and accurate means of calculating the magnetic and electric fields. Key words: auroral electrojet, electric field, induced currents, magnetic field. Knowledge of the electromagnetic fields produced by the auroral electrojet is required for work in a number of areas. Ground measurements of the magnetic field are used to study the morphology of the electrojet current system (e.g. Cramoysan, Bunting & Orr 1995) and the structured source fields of the electrojet complicate the interpretation of mag- netotelluric measurements of the conductivity structure of the Earth in auroral regions (e.g. Mareschal 1986). The auroral electrojet current system is also the principal cause of the magnetic disturbances that give rise to problems in power systems as a result of geomagnetically induced currents (e.g. Boteler, Pirjola & Nevanlinna 1997). This latter application in particular requires a fast method of calculation that can be used with the real-time forecasting schemes that are being planned for the next solar maximum. Calculations of the electromagnetic fields associated with the auroral electrojet are complicated by the effect of electric currents induced in the Earth. These induced currents them- selves create an electromagnetic field that adds to the field produced by the electrojet. The induction process is dependent on the frequency of the source variations and the conductivity structure of the Earth. The spatial extent of the source fields also has an effect and the complete solution at each frequency involves an integration over all spatial wavenumbers. Solutions of this integral have been obtained but involve a complicated computation that is demanding of computer resources and is unsuited to real-time calculations. Quick and easy solutions of the fields produced by a line current above the Earth are obtained in the electric power industry by using a technique that involves representing the currents induced in the Earth by an image current at a complex depth below the external line current. This complex-image technique was originally suggested by Wait & Spies (1969) and has been extensively used for examining the fields produced by conductors near the Earth s surface (e.g. Bannister 197, 1986). The technique has also been taken up by the power industry (Deri et al. 1981), where it is used extensively for calculating the mutual impedance between adjacent conduc- tors. At the frequencies and conductor heights found in the power-industry applications the complex-image method gives results that are indistinguishable from more complicated exact calculations. Complex-image calculations have been used by Thomson & Weaver (1975) for studying the effect of a linecurrent source on the magnetotelluric responses obtained on the ground. However, in general, the complex-image technique has not been applied to the problem of calculating the magnetic and electric fields produced by the auroral electrojet. In this paper we present a simple derivation of the complex- image expressions for the magnetic and electric fields produced at the Earth s surface by an infinite line current above the Earth. An infinite line current is a first approximation to the Downloaded from by guest on 3 November RAS 31

2 32 D. H. Boteler and R. J. Pirjola real auroral electrojet system which involves field-aligned currents as well as ionospheric currents. To test the accuracy of the complex image expressions, we use them to calculate the fields produced at the surface of an earth represented by two different layered earth models. One earth model represents the structure of Québec and is an example of a highly resistive region. The other earth model represents northern British Columbia and is an example of a more conducting region. For all cases, an exact integral solution based on the general electrojet model of Häkkinen & Pirjola (1986) is also used to calculate the fields. Comparison of the results shows that the complex-image method is nearly as accurate as the exact method and has the added benefits that it is much simpler and faster. the Fast Hankel Transform (Johansen & Sørensen 1979) and the Extended Simpson Rule (Abramowitz & Stegun 197), as done by Pirjola (1992). Independently of the numerical algorithm chosen, the computation is time-consuming and an efficient computer is required. COMPLEX IMAGE: MAGNETIC-FIELD EXPRESSIONS The incident parts in expressions (1) and (2) can be written (see Appendix) in the forms Binc x = m I h 2p h2+x2 (4) THE EXACT SOLUTION and An electrojet-system model applicable to the determination of Binc z = m I x 2p h2+x2, the geoelectromagnetic field at auroral latitudes was presented (5) by Häkkinen & Pirjola (1986). In the model the electrojet is which are simply the Biot Savart expressions for the external described by a horizontal sheet at a given height above the line current. Earth s surface, and its current may be any function of In the reflected part, the reflection coefficient, R, is given by the space coordinates and the time. The sheet current is accompanied by geomagnetic-field-aligned currents whose R= jvm /Z n magnitudes are fitted to make the total current non-divergent. jvm /Z+n, (6) Häkkinen & Pirjola (1986) presented accurate and rigorous formulae for the calculation of the geoelectromagnetic field where Z= E /H is the surface impedance of the Earth and y x owing to the general electrojet system in question at any point B =m H. To express the reflected part as the field of an image x x on the surface of the Earth, with the induction in a layered current flowing in the opposite direction to the external line earth also taken into account. Their solution is based on current we need to represent R in the form e 2pn. The reflection working in the 3-D Fourier domain (v, n, y) with respect to coefficient (eq. 6) can be rewritten as the time t and the horizontal space coordinates x and y. The case of a simple infinitely long line current discussed in the present paper is a special case where the wavenumber in R= 1/p n 1/p+n, (7) the y-direction, y, goes to zero. The fields at the Earth s surface where p is the complex skin depth, related to the surface are then given by impedance by B = m I x 2p P 2 (e hn+re hn) cos nx dn, (1) p= Z. jvm (8) B = m I [Note: sometimes Z is used to represent E /B, in which z 2p P 2 (e hn Re hn) sinnxdn, (2) y x case p=z/( jv).] Rewriting the expression for R in the form E = I y 2pP 2 jvm (1 R)e hn cos nx dn, (3) n R=1 2pn A 1+pnB 1 where h is the height of the line current, I, and x is the horizontal distance from the line current. R is the reflection and using the expansion 1/(1+x)=1 x+x2 x3+..., this coefficient and is dependent on the conductivity structure of becomes the earth and on the frequency v and the wavenumber n. These formulae have also been presented by Hermance & R=1 2pn+2(pn)2 2(pn)3+2(pn)4. (1) Peltier (197) and Wait (1992). A computer code has been developed to study the fields due Compare this to the expansion of the exponential function: to the general electrojet system model mentioned above (Pirjola &Häkkinen 1991). The numerical calculation of the electromagnetic field produced at the earth s surface at a single e 2pn=1 2pn+2(pn)2 4 3 (pn)3+ 2 ( pn)4. 3 (11) frequency v requires the computation of the inverse double- It will be seen that, for ( pn)3%1, these two expressions are Fourier integrals from the (ny) space to x and y. This is most identical. This condition is equivalent to that pointed out by often accomplished by using the the Gauss Integration Formula Wait & Spies (1969) and requires that the complex skin depth (Abramowitz & Stegun 197). Both in the n and in the y p is less than the horizontal wavelength of the source fields. integration, the interval is divided into subintervals, on each Thus, using the approximation of which the Gauss Integration Formula of the eighth order is used (Pirjola & Häkkinen 1991). Another approach is to use R=e 2pn, (12) Downloaded from by guest on 3 November RAS, GJI 132, 31 4

3 T he complex-image method 33 the reflected parts of the magnetic field can be written Brefl x = m I 2p P 2 e (h+2p)n cos nx dn, (13) Brefl= m I z 2p P 2 e (h+2p)n sin nx dn, (14) and using the relations from the Appendix these become Brefl x = m I h+2p 2p (h+2p)2+x2, (15) Brefl= m I x z 2p (h+2p)2+x2. (16) Thus the magnetic fields at the surface of the Earth due to a line current at height h are given by B x = m I 2p A h h2+x2 + B z = m I 2p A x h2+x2 h+2p Figure 1. Line current representation of the auroral electrojet and the (h+2p)2+x2 B, (17) position of an image current at a complex depth used to represent the effect of induced currents in the Earth. x (h+2p)2+x2b, (18) which is the field of the line current plus the field of an image current at complex depth h+2p. COMPLEX IMAGE: ELECTRIC FIELD EXPRESSIONS Using the approximation that R(n)=e 2pn in eq. (3) and expressing the cos nx in terms of exponential functions, we obtain E = jvm I y 4p CP 2 e (h+ix)n e (h+2p+ix)n dn n + P 2 e (h ix)n e (h+2p ix)n dn D n. (19) Using the integral relation P 2 e ax e bx dx=ln b x a this becomes, after minor algebra, E y = jvm I 2p ln C (h+2p)2+x2 h2+x2 D, (2) where the two square roots in the ln term are the distances to the line current and the image current as shown in Fig 1. COMPARISON OF EXACT AND COMPLEX-IMAGE RESULTS As part of an assessment of geomagnetic hazard to power systems in Canada, we have made calculations of the magnetic and electric fields produced at the surface of the Earth by an electrojet of 1 million amps. These calculations have been performed for layered earth models representing the conductivity structure of different parts of the country (Ferguson, private communication, 1996). Here we present results from the igneous rock area of Québec, which is part of the Canadian Shield, and from an area of younger rocks in northern British Columbia. The earth models for these two regions are shown in Fig. 2. These two models represent a highly resistive region Figure 2. Earth models of Québec and northern British Columbia. as well as a more conductive area and are representative of the range of earth structures for which these calculations are expected to be made. The complex image formulae (eqs 17, 18, 2) are used to calculate the horizontal and vertical magnetic fields and the horizontal electric field out to 1 km on either side of a line current at a height of 1 km. These results are compared with exact calculations made using the integral solution of Häkkinen & Pirjola (1986), as outlined previously. For this, the electrojet length is set to a large value which suitably represents an infinite line current, and the integration is performed using the method based on the Gauss integration formula. Calculations have been made for a number of periods from 2 min to 2 hr. Figs 3 and 4 show the results (* for complex-image method Downloaded from by guest on 3 November RAS, GJI 132, 31 4

4 34 D. H. Boteler and R. J. Pirjola Downloaded from by guest on 3 November 218 Figure 3. The horizontal (B x ) and vertical (B z ) magnetic fields and horizontal (E y ) electric field produced by a line current of 1 million amps 1 km above the Earth s surface. The calculations are made for a period of 5 min and the earth model representing Québec. Asterisks show the results of calculations made using the complex-image method; solid lines show the results of exact calculations made using the Häkkinen & Pirjola method RAS, GJI 132, 31 4

5 T he complex-image method 35 Downloaded from by guest on 3 November 218 Figure 4. Magnetic field and electric field calculations using the earth model for Québec as in Fig. 3, but for a period of 3 min RAS, GJI 132, 31 4

6 36 D. H. Boteler and R. J. Pirjola and solid lines for exact calculations) for the layered earth same as the exact calculations. For the electric fields the model of Québec at periods of 5 min and 3 min. Figs 5 and complex-image and exact calculations are virtually 6 show the results at the same periods but with the layered indistinguishable. earth model for British Columbia. The complex-image expressions are the same as the exact For the horizontal and vertical components of the magnetic solution when the complex skin depth is much less than the field, the contribution to the field from the electrojet itself and horizontal wavelength of the source field. However, even when the contribution from the induced currents are shown separ- this condition is not achieved, eqs (1) and (11) show that the ately. The total field, that is the sum of these external and complex-image results will differ little from the exact results. internal parts, is also shown. Separate plots depict the real For a source with a single wavelength and a uniform earth it and imaginary parts of the fields (taking fields in phase with is easy to evaluate how closely this condition is achieved. the electrojet as real). Thus the external part of the magnetic However, the fields produced by a line current are the sum of fields does not have an imaginary part. The complex-image multiple contributions with different wavelengths. Also, for a calculation of the external part involves no approximation and layered earth the complex skin depth has real and imaginary so always agrees with the exact solution. The external part is components whose relative sizes change with frequency. Thus shown in the figures to allow comparison of the relative it was felt necessary to make numerical calculations to compare contributions of the internal and external parts of the magnetic the complex-image method with the exact results. field. The approximation made in deriving the complex-image Examining results for all the periods between 2 min and 2 hr method involves the reflection coefficient and so only affects shows that there is a general trend for the error in the complex- the internal part. image results to be slightly greater at longer periods, as would The horizontal magnetic field has a maximum amplitude be expected. However, the difference between the complexdirectly underneath the electrojet. Here the external part of image and exact solutions is also a function of the conductivity the field is the major contributor. However, the contribution structure. For the Québec model the complex-image values for from the electrojet falls off faster than that due to the induced B directly under the line current are always less than the x currents and at a horizontal distance of 2 km from the exact solution. The British Columbia results, however, show electrojet the induced currents contribute 5 per cent of the that the complex-image values are above the exact values at total field and at larger distances they make the dominant periods of 2, 5 and 1 min (see Fig. 5) and decrease to below contribution. the exact results at longer periods (see Fig. 6). The errors in The vertical magnetic field is directed upwards on one side the real and imaginary parts of the complex-image results also of the electrojet and downwards on the other side, so it goes behave differently and complicate any comparison between through zero directly beneath the electrojet. Maximum amplitude results for different conductivity structures. For example, at a occurs at a horizontal distance of approximately 1 km period of 5 min the real and imaginary parts of the complex on either side of the electrojet. Here the contribution from the skin depth in Québec (132 j8.95 km) are considerably greater induced currents is again small compared to that from the than those in British Columbia (36.83 j48.54 km), because of electrojet itself. The vertical fields produced by the induced the higher resistivity of the Earth s crust in Québec. However, currents are directed opposite to those of the electrojet. At at a period of 3 min the fields penetrate deeper into the Earth increasing horizontal distances from the electrojet the contribution and the complex skin depths in Québec and British Columbia from the induced currents approaches the same ampli- are now ( j km) and ( j km), tude as the contribution from the electrojet and so the total respectively. The real part of the complex skin depth in British field goes to zero. How quickly this occurs depends on the Columbia is still less than half of that in Québec, but the density of the induced currents, which is influenced by the imaginary part for British Columbia is now greater than that conductivity of the ground and the period of the variations. for Québec. Electric fields produced at the Earth s surface are also shown in Figs 3 to 6 (where * shows the results from the compleximage method and solid lines are results from the exact DISCUSSION calculations). The electric field shows a maximum value directly The comparisons presented above show that the compleximage under the electrojet and a fall-off in amplitude on either side. method provides values for the magnetic and electric Examining individual plots, it can be seen that there is fields produced by a line current at a height of 1 km that generally very good agreement between the complex-image are in very good agreement with the exact results. A line calculations and the exact solution. (Remember that the com- current represents a highly structured source and is a severe plex-image calculation of the external part of the magnetic test of the complex-image technique. In reality, the auroral field is itself exact.) For the horizontal magnetic field the largest electrojet has considerable width and can be modelled by difference in the complex-image and exact calculations of the multiple line currents. The horizontal wavelengths of the fields internal part is about 1 per cent and occurs directly under- in this case will be greater than those produced by a line neath the electrojet. However, at this point the external part current, and the complex-image results are in even closer makes the dominant contribution to the magnetic field, so agreement with the exact results. The auroral electrojet current there is only a small difference between the complex-image system also includes field-aligned currents that contribute to and exact calculations of the total fields. For the vertical the magnetic fields observed at the Earth s surface. At present, magnetic field the largest difference (about 1 per cent) occurs to include field-aligned currents in the field calculations it is to the sides of the electrojet, near where the amplitude is at a necessary to use an exact solution such as that of Häkkinen maximum. However, again because the complex-image & Pirjola (1986). Further work is needed to examine whether approximation only affects the internal part, the total field the complex-image approach can be applied to modelling the results from the complex-image method are nearly the field-aligned currents. Downloaded from by guest on 3 November RAS, GJI 132, 31 4

7 T he complex-image method 37 Downloaded from by guest on 3 November 218 Figure 5. Magnetic field and electric field calculations as in Fig. 3 for a period of 5 min and an earth model representing northern British Columbia. The complex-image method allows the Earth to be modelled by multiple layers of different conductivities. This is an improvement on calculations which approximate the Earth by a uniform half-space, but still does not represent the real inhomogeneous conductivity structure of the Earth. The model of a line current above a layered conductivity structure is therefore only a rough approximation to the auroral electrojet current system above the Earth. Errors introduced by using 1998 RAS, GJI 132, 31 4

8 38 D. H. Boteler and R. J. Pirjola Downloaded from by guest on 3 November 218 Figure 6. Magnetic field and electric field calculations as in Fig. 3 for a period of 3 min and an earth model representing northern British Columbia. on the accuracy of the results, so the simplicity and greater speed of the complex-image method make this the obvious choice. Throughout this paper we have referred to the complex- the complex-image method are insignificant compared to the differences between the model and the real situation. Thus, for practical calculations of the fields produced by the auroral electrojet, the choice of computational method has no effect 1998 RAS, GJI 132, 31 4

9 T he complex-image method 39 image method, which is how it is known in other disciplines. method thus provides a quick and easy technique suitable However, it is important to note that the method actually for real-time calculations of the magnetic fields and electric involves a real image at a complex depth. Image currents have fields produced at the surface of the Earth by the auroral been used previously in electrojet calculations (e.g. Kisabeth electrojet. & Rostoker 1977; Pirjola & Viljanen 1989; Cramoysan et al. 1995), but these have always used an image at a real depth. In such studies the Earth is represented by a perfect conductor ACKNOWLEDGMENTS at a specified depth and the induced currents flow at the We are grateful to Peter Dick of Ontario Hydro Technologies surface of this conductor. The depth to the perfect conductor for suggesting the use of the complex-image method. Geological corresponds to a skin depth with only a real part, and the Survey of Canada contribution no currents at the surface of the conductor act to reflect the electromagnetic field of the electrojet and so can be represented by an image current at a real depth below the conductor. This perfect conductor technique is simple but cannot properly REFERENCES represent the effect of the Earth conductivity on the electric Abramowitz, M. & Stegun, I.A. (eds), 197. Handbook of Mathematical and magnetic fields. The complex-image method now provides Functions with Formulas, Graphs and Mathematical T ables. 9th printing, pp , Dover Publications, New York, NY. a technique of comparable simplicity that properly takes Bannister, P.R., 197. Utilization of image theory techniques in account of the induced currents in the Earth. determining the mutual coupling between elevated long horizontal The simplicity of the complex-image method makes it suit- line sources, Radio Sci., 5, able for time-critical applications such as calculating the Bannister, P.R., Applications of complex image theory, Radio geomagnetically induced currents (GIC) produced in power Sci., 21, systems by the auroral electrojet. Chains of magnetometers Boteler, D.H., Pirjola, R.J. & Nevanlinna, H., The effects of are being used to monitor the amplitude and position of the geomagnetic disturbances on electrical systems at the earth s surface, auroral electrojet. The complex-image method then provides Adv. space Res., in press. a convenient way of calculating the magnetic and electric fields Cramoysan, M., Bunting, R. & Orr, D., The use of a model experienced by power systems on the ground and this inforcurrent systems, Ann. Geophys., 13, current wedge in the determination of the position of substorm mation could be used to calculate the GIC flow throughout a Deri, A., Tevan, G., Semlyen, A. & Castanheira, A., The complex power system. A number of workers are also attempting to ground return plane: a simplified model for homogeneous and multidevelop algorithms to use real-time solar wind data from the layer earth return, IEEE T rans. Power App. & Sys., PAS-1, WIND satellite (and in the future from the ACE satellite) located at the L1 point between the Sun and the Earth to Häkkinen, L. & Pirjola, R., Calculation of electric and magnetic predict the position and amplitude of the auroral electrojet fields due to an electrojet current system above a layered earth, (Petschek & Feero, 1997). The complex-image technique could Geophysica, 22, be used with such information to provide a forecast of GIC Hermance, J.F. & Peltier, W.R., 197. Magnetotelluric fields of a line magnitudes expected in a power system. current, J. geophys. Res., 75, Hsu, H.P., 197. Fourier Analysis, revised edn, Simon and Schuster, New York, NY. CONCLUSIONS Johansen, H.K. & Sørensen, K., Fast hankel transforms, Geophys. Prospect., 27, The magnetic fields and electric fields produced at the sur- Kisabeth, J. & Rostoker, G., Modelling of three-dimensional face of the Earth by an auroral electrojet are, to a close current systems associated with magnetospheric substorms, Geophys. approximation, given by the expressions J. R. astr. Soc., 49, Mareschal, M., Modelling of natural sources of magnetospheric B = m I origin in the interpretation of regional induction studies: a review, x 2p A h h2+x2 + h+2p (h+2p)2+x2 B, (21) Surv. Geophys., 8, Petschek, H.E. & Feero, W.E., 1997, Workshop focuses on space B = m I weather s impact on electric power, EOS, T rans. Am. geophys. Un., z 2p A x h2+x2 x E y = jvm I 2p (h+2p)2+x2 B, (22) 78, Pirjola, R., On magnetotelluric source effects caused by an auroral electrojet system, Radio Sci., 27, ln C (h+2p)2+x2 h2+x2 D. (23) Pirjola, R.J. and Häkkinen, L.V.T., Electromagnetic field caused by an auroral electrojet current system model, in Environmental This corresponds to the fields produced by a line current at a and Space Electromagnetics, pp , ed. Kikuchi, H., Springerheight h and an image current at a complex depth h+2p, Verlag, Tokyo. where p is the complex skin depth, related to the surface Pirjola, R. & Viljanen, A., On geomagnetically induced currents impedance by in the Finnish 4 kv power system by an auroral electrojet current, IEEE T rans. Power Deliv., 4, p= Z. (24) Thomson, D.J. & Weaver, J.T., The complex image approxi- jvm mation for induction in a multilayered earth, J. geophys. Res., 8, Comparison of the results from these expressions with the Wait, J.R., Electromagnetic Wave T heory, corrected printing, results obtained using an exact solution show differences that J.R. Wait, Tucson, AZ. are small compared to the uncertainties in some parameters Wait, J.R. & Spies, K.P., On the representation of the quasiused in the calculations, such as the surface impedance of the static fields of a line current source above the ground, Can. J. Phys., Earth and the position of the electrojet. The complex-image 47, Downloaded from by guest on 3 November RAS, GJI 132, 31 4

10 4 D. H. Boteler and R. J. Pirjola APPENDIX A: FOURIER COSINE AND SINE TRANSFORMS The integrals in the expressions for the magnetic field are merely the Fourier cosine and sine transforms of an exponential function of the form e ax: F [e ax]= P 2 e ax cos bx dx, (A1) c F [e ax]= P 2 e ax sinbx dx, (A2) s These integrals can easily be evaluated as shown, for example by Hsu (197). First, integrate by parts: F c = C 1 a e ax cos bx D 2 F c = 1 a b a F s. b a P 2 e ax sinbx dx, (A3) (A4) Similarly, F s = C 1 a e ax sinbx D 2 F s = b a F c. + b a P 2 e ax cos bx dx, Then, combining eqs (A4) and (A6) gives F c [e ax]= and F s [e ax]= a a2+b2, b a2+b2. (A5) (A6) (A7) (A8) Downloaded from by guest on 3 November RAS, GJI 132, 31 4