Computation of magnetic fields within source regions of ionospheric and magnetospheric currents

\ PERGAMON

Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 0474Ð0481

Computation of magnetic _elds within source regions of ionospheric and magnetospheric currents U[ Engelsa\\ N[ Olsenb b

a Institut fur Geophysik det Universitat Gottin`en\ D!26964 Gottin`en\ Germany Danish Space Research Institute\ Juliane Maries Vej 29\ DK!1099 Copenha`en\ Denmark

Received 13 October 0886^ received in revised form 07 August 0887^ accepted 14 August 0887

Abstract A general method of computing the magnetic e}ect caused by a predetermined three!dimensional external current density is presented[ It takes advantage of the representation of solenoidal vector _elds in terms of toroidal and poloidal modes expressed by two independent series of spherical harmonics[ In order to test the method\ it is applied to two ionosphericÐmagnetospheric model current densities[ The _rst example assumes a special large!scale current system which has an analytical solution for the magnetic _eld everywhere[ In the second example\ the results for a model current distribution that is of small!scale at high latitudes are compared with a direct integration using BiotÐSavart|s law[ It is demonstrated that the method provides a fast and numerically stable determination of the magnetic _eld both outside and inside the current density[ Þ 0887 Published by Elsevier Science Ltd[ All rights reserved[

0[ Introduction Magnetic _elds of external origin are of considerable interest for the analysis of geomagnetic _elds of internal origin using satellite observations[ A particular di.culty arises from the fact that satellites pass through the source regions of external _elds and thus\ accurate modelling requires the determination of the magnetic _eld B in the presence of a given ionosphericÐmagnetospheric current density J[ Most of the methods used for this purpose are based on BiotÐSavart|s law B"r# 

g

m9 J"r?#×"r−r?# dV?[ 3p =r−r?= 2

As the three!dimensional integral must be evaluated over the whole source region\ these methods are useful mainly for small!scale current systems[ Another disadvantage is the fact that numerical problems arise from the singu! larity of the integrand if B is to be determined within the

 Corresponding author[ Present address] GeoForschungs! Zentrum Potsdam\ Albert!Einstein!Strasse\ D!03362 Potsdam\ Germany[ Tel[] 9938 220 1770144^ fax] 9938 220 1770124^ e!mail] kassnerÝgfz!potsdam[de

source region\ where J is non!zero and a source!point r? may be identical with the _eld!point r[ In order to simplify the three!dimensional integration\ various approaches have been adopted[ They are based on a subdivision of the current system into a _nite number of small current elements[ In that way\ Olson and P_tzer "0863# represented the quiet!time magnetospheric ring and tail currents by in_nitesimally thin wire loops\ each of which is constructed from _nite length wire segments[ An extended technique was developed by Donovan "0882# to determine the magnetic e}ects of _eld!aligned currents[ He constructed a current distribution out of cylindrical\ volume!_lling elements of _nite length[ Using in_nitely long cylindrical current tubes\ Luehr et al[ "0885# evaluated several current models in order to test an algorithm for estimating the current density from single spacecraft magnetic _eld measurements[ Kisabeth "0868# introduced a matrix representation of BiotÐSavart|s law[ His model of _eld!aligned currents\ connected with the polar electrojets\ is made up of three!dimensional current loops with sheet currents[ Stern "0882# and Tsyganenko "0882# derived closed expressions for a model of the global Birkeland current systems\ whose magnetic _elds are calculated using BiotÐ Savart integration[ In the near!Earth region\ Stern|s model consists of _eld!aligned current sheets\ connected

S0253Ð5715:87:, ! see front matter Þ 0887 Published by Elsevier Science Ltd[ All rights reserved PII] S 0 2 5 3 Ð 5 7 1 5 " 8 7 # 9 9 9 8 3 Ð 6

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U[ En`els\ N[ Olsen:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 0474Ð0481

across a ~at polar cap sheet[ The magnetic _eld outside these current sheets is a potential _eld and Stern dem! onstrates how the magnetic _eld on each side of the sheet can be interpolated across the sheet in a way which guarantees that it is non!divergent[ However\ the prob! lem of calculating the magnetic _eld from a given current distribution remains and is solved using standard BiotÐ Savart integration[ Richmond "0863# used another approach[ He developed a method for computing the magnetic e}ect in the current!free region below the iono! sphere due to a three!dimensional continuous current system consisting of an ionospheric horizontal sheet cur! rent and magnetospheric _eld!aligned currents in dipole geometry[ In contrast to most of these methods\ the algorithm presented in the present paper is applicable and numeri! cally stable both outside and inside the source region and does not depend on a special current geometry[ Being restricted to the case of time!constant or slowly time! varying _elds\ the method is based on a decomposition of the current density and of the magnetic _eld into poloidal and toroidal modes\ which are expressed by a series of spherical harmonics[ Like the approach of Richmond "0863#\ the method is especially suitable for handling large!scale current due to the use of spherical harmonic expansions[ Applying a similar formulation\ Kosik "0873# derived a model of the magnetospheric magnetic _eld[ Olsen "0886# used a toroidalÐpoloidal decomposition to solve the inverse problem of estimating the ionospheric current density from averaged magnetic _eld measurements made by the MAGSAT satellite[ In the present paper\ an analogous algorithm is adopted to compute the magnetic e}ect due to a given current density[

9s1 

1 0 1 11 sin q ¦ 1 1 1q r sin q 1q r sin q 1l1

0

0

1

1

as the transverse part of the Laplacian[ F and C are the toroidal and poloidal scalars for B[ Following Backus "0875#\ they are determined uniquely by the additional requirement that their mean value over each sphere van! ishes\ i[e[ ÐF dV  ÐC dV  9 with dV  sin q dq dl[ From Ampere|s law\ it follows for the current density] m9 J  m9 Jpol ¦m9 Jtor  curl B  curl curl rF¦curl curl curl rC  curl curl rF−curl r91 C  curl curl rF¦curl rQ where Q and C are connected by the equation curl "rðQ¦91 CŁ#  9[

"2#

1

Then Q¦9 C  "0:r# dx:dr and C is the sum of two functions\ the general solution of Q  −91 C

"3# 1

and some function of r alone\ solving 9 C  "0:r# dx:dr[ However\ any part of C which depends on r alone does not contribute to "0# and may be omitted\ leaving only the solution of "3#[ The scalar _elds F\ C and Q are expanded into series of spherical harmonics F"r\ q\ l#  s fnm "r#Pnm "cos q# eiml

"4a#

n\m

C"r\ q\ l#  R s cnm "r#Pnm "cos q# eiml

1[ Mathematical framework

"1#

"4b#

n\m

The time and length scales of the current density and magnetic _elds considered are such that displacement currents can be neglected\ which is equivalent to a non! divergent current ~ow[ Because of div J  9 and div B  9 everywhere\ it is always possible to decompose the current density J and the associated magnetic _eld B into poloidal and toroidal parts "Stern\ 0865^ Backus\ 0875#[ In a spherical coordinate system "r\ q\ l# with radius r\ colatitude q and longitude l\ this decomposition reads as follows] B  Btor ¦Bpol

with "rC#?  d"rC#:dr and

0 s qm "r#Pnm "cos q# eiml R n\m n

"4c#

where Pnm "cos q# are the Schmidt normalized Legendre functions[ R is a reference radius\ e[g[ the Earth|s radius a  5260 km[ In this formulation the scalars fnm \ cnm and qnm have the units of B\ e[g[ in nanoteslas[ The components of the magnetic _eld and current den! sity follow from the expansion coe.cients fnm \ cnm and qnm as B  Btor ¦Bpol

 curl rF¦curl curl rC 9 J F −9s1 "rC# J F G G G G G 0 1 FG G 0 1 "rC#? G G  Gsin q 1l G¦G r 1q G G G G G − 1 F G G 0 1 "rC#?G j f 1q j fr sin q 1l

Q"r\ q\ l# 

"0#

9 F J F n"n¦0#cnm Pmn eiml J G G G G G im fnm Pmn eiml G G "rcm #? d Pm eiml G R n n  s G sin q sG G¦ G dq r n\m G n\m G G G G−fm d Pm eiml G G im "rcm #?Pm eiml G n n n f j fsin q j dq n

01

"5#

U[ En`els\ N[ Olsen:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 0474Ð0481

and

onm  −"n¦0#

m9 J  m9 Jpol ¦m9 Jtor 9 F n"n¦0#fnm Pnm eiml J F J G G G G im d 0 G "rfnm #? Pnm eiml G 0 G qnm Pnm eiml G  sG G¦ s G sin q G dq r n\m G G R n\m G G d G im "rfm #?Pm eiml G G−qm Pm eiml G n n n n fsin q j f j dq "6# with "rfnm #?  d"rfnm #:dr and a corresponding equation for cnm [ Note that the radial component of the current density is given by the expansion coe.cients fnm only and therefore\ these coe.cients can be found from a spherical harmonic expansion of Jr alone[ The toroidal magnetic _eld Btor  curl rF is directly related to the poloidal current density m9Jpol  curl curl rF[ However\ in order to obtain the poloidal magnetic _eld Bpol  curl curl rC from a given toroidal current density m9Jtor  curl rQ\ the Poisson equation 91C  −Q has to be solved[ Due to the spherical harmonic expansion and the fact that r1 9s1 "Pnm eiml #  −n"n¦0#Pnm eiml \ the three!dimensional equation 91C  −Q reduces to an ordinary di}erential equation in r] d 1 dcnm r r −n"n¦0#cnm  − dr dr R

0

1

1

01

qnm [

"7#

It can be solved by means of cnm "r# 

g



Gn "r\ s#

9

1

01 s R

qnm "s# ds

"8#

with the Green|s function "cf Arfken\ 0874\ chap[ 05[4# F rn G n¦0 \ 0 js Gn "r\ s#  1n¦0 J sn G \ frn¦0

r³s [

"09#

n\m

01

cnm\e "r#

"01#

and using eqn "8#\ this yields onm  −

n¦0 0 1n¦0 R

g

r1

qnm "s#

r0

n−0

01 R s

ds[

"02#

The magnetic _eld B  −grad Vi in the region r × r1 is a potential _eld\ too\ but representing internal sources only] Vi "r\ q\ l#  R s n\m

n¦0

01 R r

(nm Pnm eiml

"03#

with (nm  n



n¦0

01 r R

n 0 1n¦0 R

cnm\i "r#

g

r1

r0

qnm "s#

"04# n¦1

01 s R

ds[

"05#

One can choose any value of R^ however\ a suitable choice for practical applications is the Earth|s radius[ Note that onm and (nm do not depend on r ðsince the radial dependence is included in the expansion of the potential V\ cf eqns "00# and "03#Ł[ Therefore\ the magnetic _eld on any sphere of radius r ³ r0 or r × r1 is given by the same coe.cients onm or (nm \ respectively[ However\ in the current region r0 ³ r ³ r1\ eqn "8# has to be solved for every r for which the magnetic _eld is sought[ The integrals in eqns "02# and "05# give the height! integrated current function\ weighted by "R:s# n−0 and "s:R# n¦1\ respectively[ In the case of a sheet current at r  R\ this integral yields the expansion coe.cients of the equivalent current function and eqns "02# and "05# merge into the well!known relationship between the expansion coe.cients of the magnetic potential and of the equivalent current function "e[g[\ Chapman and Bart! els\ 0839\ p[ 529#[

2[ Method

n

01 r R

n

R r

r×s

A short discussion of eqn "8# might be helpful[ Let us assume that the currents only ~ow in the spherical shell r0 ³ r ³ r1[ The magnetic _eld in the region r ³ r0 is a Laplacian potential _eld B  −grad Ve  curl curl rCe\ where the potential Ve "r\ q\ l#  R s

0476

onm Pnm eiml

describes external sources only[ The coe.cients of Ve and Ce are connected as

"00# expansion

It is assumed that the current distribution is available on a dense grid in spherical coordinates covering the whole spherical source region r0 ³ r ³ r1 "r1 : #[ The method of determining the magnetic e}ect due to a given current density consists of the following steps] "0# For a sequence of spheres of radius rk "k  0\ [ [ [ \ kmax# a harmonic analysis in longitude of the current density J is performed for a _xed latitude and the expansion coe.cients fnm "rk # and qnm "rk # are estimated using a least!squares technique[ Jr "rk# alone

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yields the coe.cients fnm "rk #\ whereas an analysis of the horizontal components Jq "rk# and Jl "rk# yields qnm "rk #[ The choice of the maximum degree nmax and order mmax of the spherical harmonic expansion depends on the shape of the current distribution\ of course[ "1# The radial dependence qnm is interpolated between the grid points rk using splines and the expansion coe.cients cnm are obtained from these splinesÐ interpolated values by means of numerical Romberg integration over the current region r0 ³ r ³ r1 according to eqn "8#[ "2# Finally\ the magnetic _eld B is obtained from the toroidal and poloidal coe.cients fnm and cnm by using eqns "4a#\ "4b# and "5#[ Limitations of the method arise from the need of a spheri! cal harmonic expansion and from the close association between the toroidalÐpoloidal formalism of eqns "0#Ð"4# and spherical geometries[ If the geometries encountered are quite non!spherical and the current is con_ned to some thin layers\ in these regions the spherical harmonic expansions become unwieldly if carried to full accuracy and inaccurate when cut too short[

F J 9 G G b 2 1il G G K"q\ l#  G−J9 sin q e G\ r  b 3 G G 9 f j

"06#

which allows a solution in closed form^ J9 "dimension A:m1# is an amplitude factor[ This sheet current has non! vanishing divergence and hence the radial current density Jr "b¦\ q\ l# just above r  b can be determined as Jr  −div Kd"r−b#  J9 sin1 q cos q e1il[ Assuming that the currents in the region r × b ~ow along the _eld lines of a dipole\ the current density is given by 3 FJ b sin1 q cos q e1il J 9 G G r G G J> "r\ q\ l#  G 0 b 3 2 1il G\ r × b[ sin q e G G J9 r G G 1 j f 9

01 01

"07#

There are no currents in the region r ³ b[ According to Siebert\ this current distribution gen! erates a magnetic _eld B\ which can be derived in a somewhat tricky direct way] B  B0 −grad V0

3[ Examples The algorithm is tested with two models of iono! sphericÐmagnetospheric currents[ The _rst example has the purpose of demonstrating how the numerical results for a special spatially smooth current system can be veri! _ed by an analytical solution[ Although the assumed current distribution is not very realistic\ it is chosen here for test purposes because it is one of the few current systems with an elementary analytical solution for the magnetic _eld[ For the second example\ in which the current density varies rapidly as a function of latitude and height\ the method is compared with a direct inte! gration of BiotÐSavart|s formula[ 3[0[ Comparison with an analytical solution The details of the _rst example are quite complicated so only an outline is presented here[ The full treatment of the _rst analytic part is due to Siebert "0855\ private communication#[ The basis of this model is a current distribution J  J>¦K = d"r−b# consisting of a _eld!alig! ned current density J> "dimension A:m1# in the region r × b and a horizontal sheet current density K "A:m# in the in_nitesimally thin shell at r  b[ The current J ~ows between the two hemispheres along all dipole _eld lines outside a sphere r  b and its strength on any such line is determined by the divergence of the sheet current K which closes the circuit on r  b^ K has only a q!com! ponent and is given by

F J F b 2 5 b 3 1 1il G − P1 e −1 G G G r 4 r G G G G Gim J bG 2 3 1 3 b dP1 1il b G\ r − b G 9 9 G e − G r 4 r dq G 3z2 G G G G G G 2 3 G 3 b b 1i G P11 e1il G cos1 q− G 4 r sin q j f r G j F r 1 1il J J G G 1 b P1 e G G G G G Gim J bG r dP1 1 1il G G 9 9 G e \ r³b G G 19z2 G b dq G G G G G r 1i G P11 e1il G G G b sin q j f f "08#

$0 1 0 1 % $0 1 0 1 % $0 1 0 1% 01 01 01

where B0 is a particular solution of curl B0  m9J and V0 is chosen so as to guarantee the boundary conditions at r  b "continuity of Br and Bq\ jump of Bl as given by K#[ Equation "08# can be veri_ed by decomposing the cur! rent density of eqns "07# and "06# into toroidal and polo! idal parts according to eqns "4a# and "4c# with reference radius R  b[ It turns out that the only non!vanishing expansion coe.cients are f21 and q11 ] m9 J9 b b 2 \ f21 "r#  5z04 r

r−b

9\

r³b

8

01

"19#

U[ En`els\ N[ Olsen:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 0474Ð0481

1 1

q "r# 

8

−im9 J9 b 2z2

b 3 b − d"r−b# \ r 1

$0 1

%

9\

r−b

[

0478

"10#

r³b

The poloidal scalar F changes discontinuously at r  b\ which leads to a horizontal sheet current density "Kq\ Kl# at r  b\ while the l!component of the total sheet current vanishes due to the contribution of the toroidal part\ cf the delta function in eqn "10#[ Integrating eqn "10# according to eqn "8# yields F−im9 J9 b b 1 5 b − G r 4 r 01z2 j 1 c1 "r#  J 1 G¦im9 J9 b r \ f 59z2 b

2

$0 1 0 1 % 01

\

r−b [

"11#

r³b

The resulting magnetic _eld\ determined via eqns "5#\ "19# and "11#\ is identical to that of eqn "08#\ as expected[ In the current!free region r ³ b\ the magnetic _eld is a Laplacian potential _eld B  −grad V\ with V  bo11 "r:b# 1 P11 exp "1il# and o11  −im9 J9 b:"19z2# according to eqn "02#[ In order to apply the numerical algorithm\ the iono! spheric sheet current K at r  b  5370 km is assumed to ~ow in a shell of _nite thickness D  39 km between b−D:1 and b¦D:1[ In this shell\ the radial dependence of the _eld!aligned current density is chosen according to

0

J> "r\ q\ l#  cos1 p

1

r−b 2p J> "b\ q\ l# ¦ 1D 3

"12#

with J> "b\ q\ l# given by eqn "07#[ An analytical expression for the ionospheric horizontal currents Jq "rep! resenting the q!component of the sheet current density K# follows from div J  9[ The so de_ned current dis! tribution is calculated on a grid with radial grid sep! aration Dr  1 km for a _rst interval of 099 km "starting at r  b−D:1#\ Dr  19 km for the following 0999 km and Dr  1999 km for a last interval of 199\999 km[ The horizontal resolution on each sphere is Dq  0> and Dl  04>[ Application of the method described in Section 2 with a maximal degree nmax  2 and order mmax  1 yields "outside the shell b2D:1# expansion coe.cients and magnetic _eld variations which agree very well with the analytic results given by eqns "08#\ "19# and "11#^ the di}erences are less than 9[90)[ 3[1[ Application to a model of _eld!ali`ned currents at hi`h latitudes The second example assumes a high!latitude current system with currents ~owing along magnetic dipole _eld lines to and from the ionosphere and closing along rela! tively narrow slabs in the polar ionosphere and in the magnetospheric equatorial plane\ respectively "Fig[ 0#[

Fig[ 0[ A schematic diagram illustrating the ~ow directions of the _eld!aligned and ionospheric model currents[ Unlike in this drawing\ the model currents are not line currents but a three! dimensional current density[ The detailed pattern of the high! latitude currents is shown in Fig[ 1[

This current distribution is chosen as an example of the typical scale!lengths and intensities of high!latitude cur! rents[ The ionosphere is modelled by a shell extending from b  a¦89 km to c  a¦129 km\ with a  5260 km as the Earth|s radius[ Above that shell\ that means above 129 km altitude\ the current density is entirely aligned with the dipole\ up to the equatorial plane\ where the circuit is closed[ By using the following analytical expression for the radial component of the model current density at the top of the ionosphere\

0

Jr "c\ q\ l#  J9 = sin2 1p

1

q−qN = sin "l−l9 # qS −qN

"13#

with l corresponding to magnetic local time "MLT# and l9  9>\ the spatial distribution of the currents is chosen to resemble a model proposed by Iijima and Potemra "0867#[ There are no ionospheric currents outside the colatitude band qN  04> ³ q ³ qS  14>[ Choosing the amplitude to J9  9[83 mA:m1 yields a maximum _eld! FAC aligned current density of =Jmax =  0 mA:m1[ The upper part of Fig[ 1 shows the statistical directions of the _eld! aligned currents as deduced from TRIAD data during weakly disturbed conditions as found by Iijima and Pot! emra "0867#^ the lower part shows isolines of the radial component of the model current density[ According to a typical height pro_le of ionospheric currents\ the radial dependence of the radial current den! sity in the ionosphere b ³ r ³ c is chosen according to

$

0

Jr "r\ q\ l#  0−sin1k p

r¦c−1b 1"c−b#

1%

= `"r\ q# = Jr "c\ q\ l# "14#

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Fig[ 2[ Vertical "Jr# and horizontal "Jq# components of the model current density as functions of altitude at q  06[4> and l  89>[

Fig[ 1[ A polar projection showing the spatial distribution and directions of large!scale _eld!aligned currents as determined from TRIAD data during weakly disturbed conditions "=AL= ³ 099 nT# ðtop\ redrawn after Iijima and Potemra "0867#Ł and as assumed for the model current distribution "bottom#\ respectively[

with k  4^ `"r\ q# follows from div J  9 to `"r\ q#  "c:r# 2 "cos q:cos qc#\ where qc is given by r sin1 qc  c sin1 q[ Assuming vanishing EastÐWest cur! rents "Jl  9#\ the horizontal current density follows from div J  9[ Figure 2 shows the height dependence of Jr and Jq at q  06[4> and l  89> where the radial current density is maximal[ As the currents vary rapidly with colatitude in the polar ionosphere\ a maximum degree of nmax  064 and the order m  9 and m  0 of the spherical harmonic expan! sions are used[ This requires a large number of grid points] the horizontal grid resolution is chosen to Dq  9[1> for q $ ð9>\ 59>Ł and ð019>\ 079>Ł\ Dq  9[4> for

q $ ð59>\ 019>Ł and Dl  04>[ Hence\ each shell of a _xed radius rk contains 698×13  06\905 grid points[ The rad! ial grid separation amounts to Dr  0 km for a _rst inter! val of 49 km "starting at r0  b#\ Dr  09 km for a second interval of 499 km\ Dr  099 km for the following 09\999 km and Dr  0999 km for a last interval of 099\999 km\ ending at r1  07[3a[ A dipole _eld line with this apex cuts the Earth|s surface at a colatitude q  02[4> and since the model currents in the northern hemisphere are restricted to q × 04>\ there are no currents beyond r1[ Figure 3 shows the resulting magnetic e}ects within the source region at 349 and 019 km altitude on two pro_les at 5 and 04 MLT\ respectively[ For comparison\ the symbols represent the magnetic _eld obtained by means of BiotÐSavart|s formula using the algorithm of Kisabeth "0868#[ Performing the BiotÐSavart integration numerically\ an extremely high grid resolution in the immediate vicinity of the singularity is used\ while the singularity itself is excluded[ It is obvious that the results of the di}erent methods correspond to each other to high accuracy^ the di}erences are less than 9[0)[ However\ the new method is considerably faster in computing the magnetic _eld on an entire pro_le than the numerically problematic evaluation of BiotÐSavart|s integral\ although the chosen current distribution is not par! ticularly designed for being represented by spherical har! monic functions\ since the small!scale ionospheric current closure requires high!degree spherical harmonics[

4[ Discussion The method described enables the calculation of the magnetic e}ect of a given solenoidal current density out! side as well as inside the source region[ Since the poloidal and toroidal parts of the current density and the associ! ated magnetic _eld are expanded in terms of spherical harmonic functions\ the method is not useful for currents which vary drastically with longitude or latitude[

U[ En`els\ N[ Olsen:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 0474Ð0481

0480

Fig[ 3[ Magnetic _eld due to the model of polar _eld!aligned currents on two latitude pro_les at 04 MLT "magnetic local time# and 5 MLT\ respectively\ at altitudes of 349 km "top# and 019 km "bottom#[ Additionally for some data points\ the results obtained by using BiotÐSavart|s law are marked by circles\ diamonds and triangles[

However\ it has been applied successfully to a model consisting of _eld!aligned currents and small!scale "hori! zontal dimension of a few hundred kilometres# closure currents in the polar ionosphere[ These two examples demonstrate that the method pro! vides a useful tool for modelling the near!Earth magnetic _eld[ A model study using this algorithm to examine the in~uence of _eld!aligned and meridional currents on main!_eld models estimated from satellite data is in pro! gress[

Acknowledgements We wish to thank Manfred Siebert and Ulrich Schmucker for their continuous support and encourage! ment[ The constructive comments by the referee are greatly acknowledged[ One of us "U[E[# is grateful for a scholar! ship from the Cusanuswerk and would like to thank the Danish Interdisciplinary Inversion Group for support[

References Arfken\ G[\ 0874[ Mathematical Methods for Physicists\ 2rd ed[ Academic Press[ Backus\ G[\ 0875[ Poloidal and toroidal _elds in geomagnetic _eld modeling[ Reviews of Geophysics 13\ 64Ð098[ Chapman\ S[\ Bartels\ J[\ 0839[ Geomagnetism\ vols I and II[ Clarendon Press\ Oxford[ Donovan\ E[F[\ 0882[ Modeling the magnetic e}ects of _eld! aligned currents[ Journal of Geophysical Research 87\ 02\418Ð 02\432[ Iijima\ T[\ Potemra\ T[A[\ 0867[ Large!scale characteristics of _eld!aligned currents associated with substorms[ Journal of Geophysical Research 72\ 488Ð504[ Kisabeth\ J[L[\ 0868[ On calculating magnetic and vector poten! tial _eld due to large!scale magnetospheric current systems and induced currents in an in_nitely conducting earth[ In] Olson\ W[P[ "Ed[#\ Quantitative Modelling Magnetospheric Processes[ American Geophysical Union[ Kosik\ J[C[\ 0873[ Quantitative magnetospheric _eld modelling with toroidal and poloidal vector _elds[ Planetary Space Sci! ence 21\ 854Ð863[

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U[ En`els\ N[ Olsen:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 0474Ð0481

Luehr\ H[\ Warnecke\ J[\ Rother\ M[\ 0885[ An algorithm for estimating _eld!aligned currents from single spacecraft mag! netic _eld measurements] a diagnostic tool applied to freja data[ IEEE Transactions on Geoscience and Remote Sensing 23\ 0258Ð0265[ Olsen\ N[\ 0886[ Ionospheric F region currents at middle and low latitudes estimated from Magsat data[ Journal of Geophysical Research 091\ 3452Ð3465[ Olson\ W[P[\ P_tzer\ K[A[\ 0863[ A quantitative model of the magnetospheric magnetic _eld[ Journal of Geophysical Research 68\ 2628Ð2637[

Richmond\ A[D[\ 0863[ The computation of magnetic e}ects of _eld!aligned magnetospheric currents[ Journal of Atmo! spheric and Terrestrial Physics 25\ 134Ð141[ Stern\ D[P[\ 0865[ Representation of magnetic _elds in space[ Reviews of Geophysics 03\ 088Ð103[ Stern\ D[P[\ 0882[ A simple model of Birkeland currents[ J[ Geophys[ Res[ 87\ 4580Ð4695[ Tsyganenko\ N[A[\ 0882[ A global analytical representation of the magnetic _eld produced by the region 1 Birkeland currents and the partial ring current[ J[ Geophys[ Res[ 87\ 4566Ð 4589[