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Australian Institute of Physics 17th National Congress 2006 - Brisbane, 3-8 December 2006 RiverPhys THE USE OF SPHERICAL CAP HARMONIC ANALYSIS IN PREDICTING GROUND MAGNETIC PERTURBATIONS FROM IONOSPHERIC ELECTRIC FIELD AND CONDUCTANCE MODELS D. L. GreenA† , C. L. WatersA and J. W. GjerloevB A School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan NSW, Australia B The Johns Hopkins University Applied Physics Laboratory, Laruel MD, USA † Email: david.lindsay.green@gmail.com Abstract Presently available models of the spatial distribution of ionospheric conductance are not functions of the interplanetary magnetic ﬁeld (IMF) conditions. Therefore, using model conductance data to predict the magnetic perturbations at the Earth’s surface caused by space weather events lacked accuracy. However, with new models of the ionospheric electric potential and new techniques for measuring ionospheric conductance becoming available, predicting the horizontal ionospheric current (J⊥ ) based on solar wind conditions may soon be possible. Surface magnetic perturbations are associated with only the divergence-free component of J⊥ in the ionosphere. This paper presents a technique for the reasonable extraction of the divergence-free component of J⊥ towards predicting the magnetic perturbations observed on the ground. 1 Introduction The interaction of the solar wind and interplanetary magnetic ﬁeld (IMF) with the Earth’s magnetic environ- ment produces a global system of electric currents that couples large amounts of energy to the high latitude ionosphere. Part of this system involves horizontal ionospheric currents that create time varying magnetic perturbations at the Earth’s surface. The perturbations are dependent on solar wind and IMF conditions. According to Faraday’s law, the time derivative of this magnetic perturbation creates a geoelectric ﬁeld which can induce electrical currents in man made conductors such as pipelines and power networks. These geomagnetically induced currents (GICs) can disrupt technological systems and methods for their prediction is an active area of research. In order to predict large-scale magnetic perturbations observed at the Earth’s surface, and hence the impact of GICs, the entire Sun-Earth system must be taken into account. Previous work in this area utilises empirically constructed models of electromagnetic parameters sorted according to solar wind conditions or geomagnetic activity levels. The appropriate models are combined such that the magnetic eﬀect at the Earth’s surface is predicted for a given set of solar wind conditions. A recent example is presented by Weimer (2005), where scalar potentials describing both the ionospheric electric ﬁeld (E⊥ ) and the curl- free component (Jcf ) of the horizontal ionospheric current (J⊥ ) are constructed based on observations from the Dynamics Explorer 2 (DE-2) spacecraft over 2645 orbits. Both models were sorted according to IMF and solar wind conditions. However, since magnetic perturbations at the Earth’s surface (bg ) are due solely to the divergence-free component (Jdf ) of J⊥ (Fukushima, 1971), assumptions concerning the height- integrated electrical conductivity (conductance) of the ionosphere are also required to predict bg . To avoid the requirement of an empirical conductance model, Weimer assumed a ﬁxed ratio between the Hall (ΣH ) and Pedersen (ΣP ) conductances. These conductances are deﬁned in terms of the height-integrated ionospheric current by ˆ J⊥ = ΣP E⊥ + ΣH B × E⊥ = Jcf + Jdf (1) Weimer (2005) further suggested that combining existing conductance models with the potentials describing E⊥ and Jcf does not yield good predictions for bg , perhaps since the models are constructed using diﬀerent data sources or sorting schemes. Presently available models of the auroral conductance enhancement (e.g., Hardy et al., 1987) are sorted according to the geomagnetic activity index Kp, not IMF. However, recent work by Coumans et al. (2004), Aksnes et al. (2005) and Green (2006) has made available new techniques for estimating ionospheric conduc- tance which provides the possibility of constructing new conductance models sorted according to solar wind Paper No. XXX 1 Australian Institute of Physics 17th National Congress 2006 - Brisbane, 3-8 December 2006 RiverPhys and IMF conditions. Such models could be combined with a model of the ionospheric electric ﬁeld, e.g., RG05 presented by Ruohoniemi and Greenwald (2005). In this way the requirement of an assumed conduc- tance ratio is unnecessary as J⊥ may be calculated directly from Eq. 1. The diﬃculty with this approach is that only Jdf creates a magnetic eﬀect at the Earth’s surface. Therefore, the direct use of a conductance model in predicting bg requires Jdf to be extracted from J⊥ . This paper describes a technique for the sep- aration of the large-scale ionospheric current (J⊥ ) into curl-free (Jcf ) and divergence-free (Jdf ) components using Spherical Cap Harmonic Analysis (SCHA). The mathematical framework is given in section 2. Since an IMF sorted conductance model is not yet available, example application of the technique is limited to a model horizontal ionospheric current constructed for a speciﬁc event using electric ﬁeld information calcu- lated from Super Dual Auroral Radar Network (SuperDARN) data and the statistical conductance models of Hardy et al. (1987) and Rasmussen et al. (1988). In section 3 the resulting separation is presented and the predicted magnetic perturbation at the Earth’s surface is qualitatively compared with observed values. 2 Method If model electric ﬁeld (Emod ) and conductance data (Σmod and Σmod ) are available then a model horizontal ⊥ P H ionospheric current may be constructed according to (Backus, 1986) µ0 Jmod = µ0 Σ . Emod = ⊥ ⊥ mod 1 p⊥ +r× q⊥ = µ0 Jmod + Jmod mod (2) cf df mod where pmod and q⊥ are the model poloidal and toroidal current scalars respectively and Σ is the model ⊥ conductance tensor. The divergence of Eq. 2 gives 2 pmod . Jmod = 1 ⊥ ⊥ (3) µ0 r which involves only the poloidal current scalar pmod , i.e., related solely to Jmod . Constructing pmod from a ⊥ ⊥ cf m linear sum of a set of spherical cap harmonic functions (Yn (θ, φ)) similar to (Haines, 1985) gives K M ≤k pmod = ⊥ m cm Ynk (θ, φ) k (4) k=1 m=0 Eq. 3 can be written 2 pmod K M ≤k 1 ⊥ 1 = cm k 2 m 1 Ynk (θ, φ) (5) µ0 r µ0 r k=1 m=0 Using the eigenvalues of the Laplacian on a sphere, 2 Y m (θ, φ) m = −n (n + 1) Yn (θ, φ), a simpliﬁed expres- 1 n sion is 2 pmod K M ≤k 1 ⊥ 1 = . Jmod = − nk (nk + 1) m cm Ynk (θ, φ) k (6) ⊥ µ0 r µ0 r k=1 m=0 Therefore, a least squares ﬁt to the numerical divergence of Jmod using a set of spherical harmonic basis ⊥ functions according to Eq. 6 gives the coeﬃcient set cm . Once the cm set is known, pmod and hence Jmod k k ⊥ cf may be generated from K M ≤k µ0 Jmod = mod 1 p⊥ = cm k m 1 Ynk (θ, φ) (7) cf k=1 m=0 Jmod is then removed from Jmod to give the desired model Jmod according to Eq. 2. A complication is that ⊥ cf df the process of separating Jmod into curl-free and divergence-free components is not unique. To solve this ⊥ problem a boundary condition is imposed at the lower latitude boundary (θc ) of the spherical cap. By using m only those spherical harmonics that satisfy Yn (θc , φ) = 0 in the expansion of Eq. 6 and excluding those that ∂ m (θ , φ) = 0 as suggested by Haines (1985), the azimuthal component of Jmod is constrained to satisfy ∂θ Yn c cf zero at the cap boundary. Paper No. XXX 2 Australian Institute of Physics 17th National Congress 2006 - Brisbane, 3-8 December 2006 RiverPhys Figure 1: Jmod and . Jmod Figure 2: Jcf and .J ⊥ ⊥ cf 3 Results and Discussion The technique described in section 2 was applied to a Jmod calculated from E⊥ data recorded by SuperDARN ⊥ on 10 January 2002 1100 UT and a model conductance. For this case K=10 and M=7 were used in the SCHA giving a latitude resolution of ∼5o . The cap size was set to θc =60o . However, in practice this would depend on magnetic activity. Figures 1 through 3 show Jmod , Jmod and Jmod including contours of their ⊥ df cf associated divergences. Examining the divergences of both Jmod (Fig. 1) and Jmod (Fig. 2) shows them to ⊥ cf be almost equal while that of Jmod (Fig. 3) is small indicating the separation of the current components has df been successful. Small diﬀerences are due to the smoothing eﬀect of the SCHA process. mod The predicted ground magnetic perturbation (bmod ) is calculated from Jmod by expanding q⊥ in terms g df of spherical harmonics as K M ≤k mod q⊥ = m hm Ynk (θ, φ) k (8) k=1 m=0 where the hm coeﬃcient set can be calculated from Jmod . By considering the boundary conditions across k df mod the ionospheric current sheet bmod can be shown to be related to q⊥ according to (Engels and Olsen, 1998; g Green, 2006) K nk + 1 n −1 M ≤k Re k bmod g = m hm Ynk (θ, φ) k (9) 2nk + 1 Ri k=1 m=0 Using this method gives the bmod shown in Fig. 4 (black). Figure 4 also shows bg observations (red) from g⊥ all available magnetometers for 10 January 2002, 1100 UT. Comparing the bmod vectors with observations g⊥ of bg⊥ shows reasonable agreement. It is expected that disagreements, particularly the underestimated magnitudes of bmod vectors, are associated with the use of a statistical conductance model. As suggested g⊥ by Aksnes et al. (2002), most statistical models of conductance underestimate the Hall conductance which has a large aﬀect on the magnitude of bg . 4 Conclusions This paper has presented a new technique to allow prediction of the large-scale ionospheric current systems, and hence magnetic perturbation at the Earth’s surface, utilising new models for the ionospheric electric Paper No. XXX 3 Australian Institute of Physics 17th National Congress 2006 - Brisbane, 3-8 December 2006 RiverPhys Figure 3: Jdf and .J df Figure 4: bmod and bg⊥ (red) rotated 90o clockwise. g⊥ ﬁeld and pending improved models of the auroral conductance distribution. Modiﬁcations to the standard application of SCHA has been shown to allow a convenient way for separating the complete horizontal iono- spheric current into reasonable curl-free and divergence-free components and therefore predict the ground magnetic perturbation. Acknowledgements For the ground magnetometer data we gratefully acknowledge the S-RAMP database, PI K. Yumoto and K. Shiokawa; the SPIDR database; Intermagnet; the institutes who maintain the IMAGE magnetometer array; AARI data, PI Oleg Troshichev; Danish Meteorological Institute, Ole Rasmussen and Project Scientist Jurgen Watermann; the CARISMA, PI Ian Mann; the MACCS program, PIs W. J. Hughes and M. Engebretson as well as the Geomagnetism Unit of the Geological Survey of Canada; GIMA, PI John Olson; MEASURE, UCLA IGPP and Florida Institute of Technology; USGS, Jeﬀrey J. Love; MAGIC, PI C. Robert Clauer; SAMBA, PI Eftyhia Zesta; 210 Chain, PI K. Yumoto; SAMNET, PI Farideh Honary; IMAGE, PI Ari Viljanen. References Aksnes, A., Amm, O., Stadsnes, J., Østgaard, N., Germany, G. A., Vondrak, R. R., Sillanpaa, I., 2005. Ionospheric conduc- tances derived from satellite measurements of auroral UV and X-ray emissions, and ground-based electromagnetic data: A comparison. Annales Geophysicae 23 (2), 343–358. Aksnes, A., Stadsnes, J., Bjordal, J., Østgaard, N., Vondrak, R. R., Detrick, D. L., Rosenberg, T. J., Germany, G. A., Chenette, D., 2002. Instantaneous ionospheric global conductance maps during an isolated substorm. Annales Geophysicae 20 (8), 1181–1191. Backus, G., 1986. Poloidal and toroidal ﬁelds in geomagnetic ﬁeld modeling. Reviews of Geophysics 24 (1), 75–109. e Coumans, V., G´rard, J.-C., Hubert, B., Meurant, M., Mende, S. B., 2004. Global auroral conductance distribution due to electron and proton precipitation from IMAGE-FUV observations. Annales Geophysicae 22, 1595–1611. Engels, U., Olsen, N., 1998. Computation of magnetic ﬁelds within source regions of ionospheric and magnetospheric currents. Journal of Atmospheric and Solar-Terrestrial Physics 60 (16), 1585–1592. Fukushima, N., 1971. Electric current systems for polar substorms and their magnetic eﬀect below and above the ionosphere. Radio Sci. 6, 269–275. Green, D. L., 2006. The mie and helmholtz representation of vector ﬁelds in the context of magnetosphere-ionosphere coupling. Ph.D. thesis, The University of Newcastle, University Drive, Callaghan, NSW 2308, Australia. Haines, G. V., 1985. Spherical Cap Harmonic Analysis. J. Geophys. Res. 90 (B3), 2583–2591. Hardy, D. A., Gussenhoven, M. S., Raistrick, R., McNeil, W. J., 1987. Statistical and functional representations of the pattern of auroral energy ﬂux, number ﬂux, and conductivity. J. Geophys. Res. 92 (A11), 12,275–12,294. Rasmussen, C. E., Schunk, R. W., Wickwar, V. B., 1988. A photochemical equilibrium model for ionospheric conductivity. J. Geophys. Res. 93 (A9), 9831–9840. Ruohoniemi, J. M., Greenwald, R. A., 2005. Depenencies of high-latitude plasma convection: Consideration of inter- planetary magnetic ﬁeld, seasonal, and universal time factors in statistical patterns. J. Geophys. Res. 110 (A09204), doi:10.1029/2004JA010815. Weimer, D. R., 2005. Predicting surface geomagnetic variations using ionospheric electrodynamic models. J. Geophys. Res. 110 (A12307), doi:10.1029/2005JA011270. Paper No. XXX 4

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THE USE OF SPHERICAL CAP HARMONIC ANALYSIS IN PREDICTING GROUND

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