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					Australian Institute of Physics 17th National Congress 2006 - Brisbane, 3-8 December 2006


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:

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
      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

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Australian Institute of Physics 17th National Congress 2006 - Brisbane, 3-8 December 2006

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 =
           ⊥             ⊥
                                       1 p⊥        +r×      q⊥ = µ0 Jmod + Jmod
                                                                              cf         df
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 ⊥
                       µ0 r
which involves only the poloidal current scalar pmod , i.e., related solely to Jmod . Constructing pmod from a
                                                 ⊥                                                  ⊥
linear sum of a set of spherical cap harmonic functions (Yn (θ, φ)) similar to (Haines, 1985) gives
                  K M ≤k
       pmod =
                            cm Ynk (θ, φ)
                             k                                                                                              (4)
                  k=1 m=0

Eq. 3 can be written
         2 pmod             K M ≤k
         1 ⊥         1
                  =                   cm
                                             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
                                                    1 Ynk (θ, φ)                                                            (7)
                                 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
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.

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Australian Institute of Physics 17th National Congress 2006 - Brisbane, 3-8 December 2006

             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
been successful. Small differences are due to the smoothing effect of the SCHA process.
    The predicted ground magnetic perturbation (bmod ) is calculated from Jmod by expanding q⊥ in terms
                                                    g                        df
of spherical harmonics as
                 K M ≤k
      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
the ionospheric current sheet bmod can be shown to be related to q⊥ according to (Engels and Olsen, 1998;
Green, 2006)
                       nk + 1              n −1 M ≤k
                                   Re       k
       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
all available magnetometers for 10 January 2002, 1100 UT. Comparing the bmod vectors with observations
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
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

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Australian Institute of Physics 17th National Congress 2006 - Brisbane, 3-8 December 2006

               Figure 3: Jdf and        .J
                                                                   Figure 4: bmod and bg⊥ (red) rotated 90o clockwise.

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.
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.

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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.
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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.
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Hardy, D. A., Gussenhoven, M. S., Raistrick, R., McNeil, W. J., 1987. Statistical and functional representations of the pattern
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