Magnetic multipole moments (Gauss coefficients) and vector potential given by an arbitrary current distribution

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1 RESEARCH NEWS Earth Planets Space, 63, i vi, 0 Magnetic ultipole oents (Gauss coefficients) and vector potential given by an arbitrary current distribution F. J. Lowes and B. Duka School of Cheistry, Newcastle University, Newcastle Upon Tyne, NE 7RU, U.K. Departent of Physics, Faculty of Natural Sciences, Tirana University, Albania (Received Noveber 5, 009; Revised Deceber 7, 00; Accepted August 4, 0; Online published February, 0) Until recently there has been nothing in the geoagnetic literature giving the Gauss coefficients (equivalent to agnetic ultipole oents) for the agnetic scalar potential produced outside a finite-sized region of electric current. Nor has there been an expression for the corresponding agnetic vector potential. This paper presents a siple expression for the Gauss coefficients in ters of a volue integral over the current, and also a series expansion of the vector potential in ters of these coefficients. We show how our result is related to the classical expressions for the scalar potential given by a spherical current sheet, and to the results of the recent papers by Engels and Olsen (998), Stup and Pollack (998) and Kazantsev (999). Key words: Gauss coefficients, agnetic field, agnetic scalar potential, ultipole oents, spherical haronics, vector potential, vector spherical haronics.. Introduction In considering the ain geoagnetic field outside the Earth, ost workers specify the field by its scalar potential expanded in ters of spherical haronics, and the corresponding Gauss coefficients, which are scaled versions of the classical ultipole oents. There is a siilar approach in electrostatics, and any classical texts show how to calculate these oents by integrating over the electric charges (or the equivalent agnetic onopoles) that are the source of the field. However the source of the ain geoagnetic field is not onopoles but electric currents, and there did not appear to be in the literature general expressions for calculating the oents (higher than the dipole) by integrating over the current syste. Nor were there readily available expressions for the equivalent vector potential distribution. The present paper derives explicit expressions for the Gauss coefficients as integrals over an arbitrary current distribution. It also presents the vector potential analogue of the scalar spherical haronics, and relates the oents in the two approaches. It copares our results with those of previous workers in a consistent notation. As is usual, we assue that the source current density J is varying so slowly in tie that we can assue that div J = 0. We are concerned only with real current density, so ignore the effect of any agnetisation. The rest of this Introduction presents the basic ideas of agnetic scalar and vector potentials, and of toroidal and poloidal fields.. Magnetic scalar potential, poles and ultipoles If the sources of agnetic field are within a liited region V, then outside this source region we have curl H = 0, Copyright c The Society of Geoagnetis and Earth, Planetary and Space Sciences (SGEPSS); The Seisological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. doi:0.5047/eps so the resulting agnetic field at the field point r can be expressed as the gradient of a scalar potential, H(r) = grad φ(r). () In the external region we also have div B = div(µh) = 0, so in a region of constant pereability µ the potential φ is a solution of the Laplace equation φ = 0. If the source is a single agnetic pole of strength p (a fictitious, but useful, analogue of an electric charge) at the origin, then we have φ = p/r = oent (0) /r = M (0) /r, () where the choice of notation, degree l = 0, will be explained later. If we have two poles of strength +p and p, centred at the origin but displaced fro each other along the z-axis by a distance δz, the resulting potential is approxiately φ = ( pδz/) (/r)/ z = (pδz/)(z/r 3 ). (3) If we keep the product pδz constant at the value M z () as we go to the liit δz 0, we have a (fictitious) degree point-dipole source for which the external potential is φ = M () z (z/r 3 ) = M () z (cos θ/r ). (4) Siilarly, taking two z-axis dipoles of opposite sign displaced along the x-axis would give a zx-quadrupole. In general, a degree l ultipole has a oent M (l) involving l factors of displaceent, and potential falling off with distance as /r l+ ; an arbitrary degree l ultipole can be specified in ters of (l + ) independent oents.. Scalar potential of a distributed source If the source is a distribution having agnetic pole density ρ(s) at source point s then at the field point r Eq. () i

2 ii F. J. LOWES AND B. DUKA: MAGNETIC MULTIPOLE MOMENTS generalises to φ(r) = ρ(s) dv(s), (5) r s where the integration is over the source region. It is convenient if (5) can be expressed in the for φ(r) = M i R i (r) (6) where the M i are oents that depend only on the source, i.e. on the distribution of ρ(s), and the R i (r) are functions that depend only on the position r of the field point; we will see that this corresponds to replacing the actual source distribution ρ(s) by a series of (fictitious) point ultipole sources, all at the origin. (Note that if we choose to easure r fro a different origin, then in general the oents M i will change; only the first non-zero oent is independent of the choice of origin see e.g. Raab and De Lange, 005, section.7.) Provided we are content to restrict the field point to being outside a sphere that contains all the sources (so that r > s ax ), we can do this by expanding / r s as a power series in s. The successive degree l = 0,,,... ters in this series lead to oents that are integrals over the source in which the source density ρ(s) is (in effect) weighted by s l, and the spatial function R i (r) decays with distance as /r (l+). However there are several possible choices of power series, and different choices lead to different selections and nubers of oents. We will use the spherical haronic addition theore for s < r, r s = r i ) l l Sl (θ r,λ r )Sl (θ s,λ s ). (7) r l=0 = l (Throughout our paper we use real Sl, and Schidt seinoralised Pl (cos θ), with Sl (θ, λ) = Pl (cos θ)cos λ for positive, and Sl (θ, λ) = P l (cos θ)sin λ for negative. These Sl (θ, λ) have ean-square value over the sphere of /(l + ).) This gives φ(r) = r l, l+ S l (θ r,λ r ) s l Sl (θ s,λ s )ρ(s)dv = Rl (r)ml (8) l, where the spatial functions Rl (r) are the haronic functions (orthogonal over the sphere) Rl (r,θ r,λ r ) = r l+ S l (θ r,λ r ), (9) and the oents Ml are given by Ml = s l Sl (θ s,λ s )ρ(s)dv. (0) The volue integral is to be taken over the whole source region, and (8) is convergent outside the sphere r = s ax that circuscribes the source region. Successive ters of (8) correspond to the potential of a pole (l = 0), three dipoles (l = ), five quadrupoles (l = ), etc. at the origin. (In our agnetic case there are no real onopoles, and the series starts with the dipole, l =, ters.) In the integral of (0), if the volue eleent dv is put in the for ds da, the volue integral can be organised into a succession of surface integrals over thin spherical shells of thickness ds, followed by integration over radius. This gives Ml = s l ds ρ(s)sl (θ, λ)da s s A = s l+ ds ρ(s,θ,λ)s l (θ, λ)d. () (Integrations are always only over the source region, and potentials are expressed only in the external region, so fro now on the subscripts s and r will be oitted.) The integration over is just a surface haronic analysis giving the (l, ) contribution fro that radius. So if at radius s the source density variation is we have and hence ρ(s,θ,λ)= l, ρ l (s) = l + Ml = l + ρ l (s)s l (θ, λ) () ρ(s,θ,λ)s l (θ, λ)d (3) ρ l (s)s l+ ds. (4) Algebraically (0) and (4) are the sae whatever the shape of the real source volue; if this is not spherical ρ(s) is siply put to zero as necessary when aking the surface haronic analysis. (There ight of course be arithetic probles in the surface haronic analysis if ρ(s) is large next to a boundary.) Using this spherical haronic approach has the advantage that for a given degree l the series autoatically produces the correct nuber (l + ) of independent functions R l (r) (orthogonal over spherical surfaces) and corresponding oents. Alternatively, if / r s is expanded using a Cartesian approach, this leads to ters ost easily expressed using tensor notation (see e.g. Raab and De Lange, 005). However, forally this leads to ore than (l + ) oents/spatial functions, but not all of these are independent (orthogonal); this redundancy is discussed in physical ters in Wikswo and Swinney (984), and in ters of traceless tensors by Raab and De Lange (005, section.6). In the context of geoagnetis, it is conventional to assue that µ = µ 0 everywhere, and to include a factor of µ 0 in the definition of a different potential ψ, giving B = grad ψ and ψ = 0. (5) It is also conventional to introduce a reference radius a,and to scale the oents M l and spatial functions R l so that each now has the sae diensions for all degrees l. In our notation the solution of (5) used in geoagnetis is ψ(r) = a l gl l= = l ( a ) l+ S r l (θ, λ); (6)

3 F. J. LOWES AND B. DUKA: MAGNETIC MULTIPOLE MOMENTS iii the gl l for negative ) are the so-called Gauss coefficients, appropriate to the reference radius a. By coparison with (8) we see that (h g l = (µ 0 /a l+ )M l. (7).3 Magnetic vector potential Of course agnetic fields are produced not by poles, but by currents, having current density say J(s). The agnetic field is then given by the Biot-Savart Law B(r) = µ 0 J(s) (r s) dv. (8) r s 3 Inside the source region the concept of scalar potential is no longer useful. We can however use the vector potential A, with B = curl A, everywhere, and the equivalent equation to (5) is then A(r) = µ 0 J(s) dv. (9) r s (When putting B = curl A, the gauge of A is not unique; we have selected the Coulob gauge (see e.g. Jackson, 975, section 6.4). However this does not affect the use of A here; see the discussion at the end of Section 4.) It is a standard result (see e.g. Gray (978, equation (6)), or Backus et al. (996, pp. 3 3) that (9) leads to (in our notation) rb r (r) = µ 0 div(s J(s))dV r s = µ 0 r s [s curl h J(s)]dV, (0) where the subscript h refers to the surface operator. Outside the source region the sae technique of expanding / r s as a series can be applied, though of course we now have to use vector algebra. For exaple, Backus et al. (996) expanded / r s using the spherical haronic approach, and by coparing the expressions for the resulting radial field ters with those given by the scalar potential (6), derived a general expression for the g l in ters of integrals involving derivatives of J(s) gl = µ 0 (l + )a l+ s l S l (θ, λ)s curl h J(s)dV; () in effect they ade a radially weighted spherical haronic analysis of s curl h J(s). However we are looking for a ethod that avoids taking derivatives of J, as this could lead to errors if J was specified nuerically. For each gl it is possible to use vector algebra, and the constraint div J = 0, to convert the integrand of () into a for using J directly. Unfortunately, working in Cartesian coponents often leads to integrands involving all three coponents J x, J y and J z ; further anipulations using div J = 0 have to be used to reduce the integrands to two coponents. For exaple for l =, we get g 0 = µ 0 z(xj a 4 y yj x )dv () g = µ 0 3 (x z )J a 4 y dv (3) g = µ 0 3 (z y )J a 4 x dv (4) g = µ 0 3 (y x )J a 4 z dv (5) g = µ 0 3 xyj a 4 z dv. (6) Another possible approach is to expand the inverse radius in (9) using a Cartesian Taylor series expansion. Converting to vector notation, we get for the quadrupole l = ter A () = µ 0 3(ˆr s) s J(s)dV. (7) r 3 Taking the curl, using considerable vector algebra, applying the constraint div A = 0 (equivalent to div J = 0), and coparing the ters with those given by the scalar potential approach, we can again obtain the expressions () to (6) above. However these are ad hoc process, only feasible for low degree ters. In Section we present a systeatic ethod for obtaining the Gauss coefficients in ters of integrals over J(s) itself, and the corresponding series expansion for the vector potential. In Section 3 we briefly discuss other approaches and recent work, using a consistent notation..4 Toroidal and poloidal electric currents and agnetic fields When considering the effects of a current distribution in a sphere, it is coon practice to separate the toroidal current systes, which have current purely in concentric spherical surfaces and have no radial coponent, fro the poloidal current systes, which do have radial (as well as tangential) coponents); see e.g. Backus et al. (996, section 5.3). It is a standard result that toroidal currents produce only poloidal agnetic fields, both inside and outside the region of current, while poloidal current systes produce only toroidal agnetic fields, and these only inside the current region. So we know that the whole of a poloidal current syste (the su of its radial and associated tangential parts) produces no agnetic field outside the source region. But the iddle ter of (0) shows that the radial part of such a poloidal current syste gives no external field, so it follows that the tangential part ust also give no external field. Therefore it does not atter if the tangential part of any poloidal current syste is included or not in our calculation; we do not need to ake the toroidal/poloidal separation before integration. Following Engels and Olsen (998), in the source region we can put B = B tor + B pol = curl st + curl curl sp, (8) where T and P are the defining toroidal and poloidal scalar fields. This leads to the corresponding poloidal and toroidal

4 iv F. J. LOWES AND B. DUKA: MAGNETIC MULTIPOLE MOMENTS currents µ 0 J = µ 0 (J pol + J tor ) = curl B = curl(b tor + B pol ) = curl curl st + curl curl curl sp, (9) (note that it is J tor that gives B pol ) which can be put in the for µ 0 (J pol + J tor ) = curl curl st + curl sq, (30) where Q is another scalar field.. Series Expansion of the Vector Potential In geoagnetis, until recently the vector potential has been used only indirectly, as a eans of obtaining expressions for the scalar potential Gauss coefficients. However Kazantsev (999) produced expressions for the vector potential produced outside an arbitrary current distribution, using vector potential oents. His paper is difficult to follow, so we now give an equivalent approach for the deterination of these oents; we also show how they are related to the scalar potential Gauss coefficients. We saw in Section. that a particular external scalar potential Sl (θ, λ)/r l+ cae fro that part of the charge/pole source distribution in a thin spherical shell that was proportional to Sl (θ, λ). Because of the orthogonality of the Sl (θ, λ), the total (l, ) oent could be obtained by (in effect) perforing a spherical haronic analysis of the charge/pole source distribution within each shell, then weighting by s l+ when integrating over radius to give the total oent. What is now described is the equivalent for an arbitrary current distribution. The iddle ter of (0) shows that (when integrated over the source region) any radial coponent of current produces no external field, so in our spherical shell we need consider only the tangential coponents of current. As our basis functions we use the diensionless surface vector haronics Kazantsev (999) called αl (θ, λ), α l (θ, λ) = curl(rs l (θ, λ)) = r grad h S l (θ, λ). (3) This r grad h operator is essentially the sae as the angular oentu operator L of quantu echanics. These αl (θ, λ) have spherical polar coponents α l (θ, λ) = ( 0, sin θ Sl (θ, λ), S l (θ, λ) λ θ ) (3) and are essentially the vector spherical haronics used in electroagnetic wave theory (see e.g. Jackson, 975, section 6.), and in the separation of toroidal and poloidal current systes; in the notation of Section.4 they are toroidal fields. Just as the individual scalar functions S l (θ, λ) are orthogonal over the sphere (having ean-square value of /(l + )), so also are the corresponding surface vector functions α l (θ, λ) (see e.g. Jackson, 975), which have a ean-square value of l(l + )/(l + ). (Although this orthogonality is not usually stated explicitly in the geoagnetic context, it is a standard result (see e.g. Lowes, 975) that the tangential vector fields grad h S l (θ, λ) are orthogonal for different (l, ); the r operator essentially just interchanges the θ and λ coponents.) By analogy with the expansion (8) for the scalar potential, we can expand the external vector potential given by a current distribution in the for A(r) = µ 0 r l, l+ α l (θ, λ)l. (33) (Note that, by analogy with the Ml oents we used for the agnetic scalar potential, we use l for the corresponding oents for the vector potential; there should not be any confusion with the superscript used to denote spherical haronic order.) For a given (l, ) the field curl[αl (θ, λ)/r l+ ] given by this vector potential approach, ust have the sae field geoetry as the field grad[sl (θ, λ)/r l+ ] given by a scalar potential approach; the two fields can differ only by a constant factor. It is straightforward to show that curl ( αl (θ, λ)/r l+) = curl ( r grad ( Sl (θ, λ)/r l+)) = lgrad ( Sl (θ,λ)/r l+) (34) (a proof is given by Stup and Pollack, 998, p. 806), so for a given source the vector potential oents l are a factor l saller than the corresponding scalar potential oents Ml. Using the orthogonality of the αl (θ, λ), we can expand a general volue tangential current distribution J h (s,θ,λ) as the series J h (s,θ,λ)= l, J l (s)α l (θ, λ) (35) where Jl (s) = l + l(l + ) J(s,θ,λ) αl (θ, λ)d. (36) (Strictly, these Jl give only the toroidal part of the surface current J h ; but this is exactly the part that produces external poloidal agnetic field.) It follows fro (34) and (46) below that the oents l are given by l = l + s l+ J l (s)ds (37) analogous to (4) for the scalar potential oent. Reversing the arguent used to go fro (0) to (4),we get l = J(s) αl (θ, λ)s l dv (38) l(l + ) for an arbitrary source volue, analogous to (0); the extra factor of /l(l + ) occurs because the Sl and αl have different ean-square values. (In (38) J(s) can be the full (toroidal + poloidal) current density; the scalar product with αl (θ, λ) picks out only the toroidal part.) So fro (7) and (37) we get gl = µ ( 0l s J (l + ) a l (s)ds. (39) Fro (7) and (38), g l = µ 0 a (l + ) ) l J(s) αl (θ, λ)dv. (40) a

5 F. J. LOWES AND B. DUKA: MAGNETIC MULTIPOLE MOMENTS v We can then write (33) in the for ( a ) l+ =l A(r) = a gl αl (θ, λ), (4) l r l= = l giving the vector potential directly in ters of the Gauss coefficients; this is the equivalent of the scalar potential expression (6). A result equivalent to (40) was obtained by Gray (978) by applying unspecified vector algebra to (). He also in effect gave (4). The anonyous referee has shown that () can be put in the alternative fors gl = µ 0 (l + )a l+ ( grad h s l Sl (θ, λ) ) (s J(s))dV (4) and gl = µ 0 (l + )a l+ ( curl h s l Sl (θ, λ)s ) J(s)dV. (43) Equation (43) is equivalent to gl = µ 0 s l ( curl (l + )a l+ h S l (θ, λ)s ) J(s)dV, (44) which is the sae as our (40). 3. Other Approaches Using the notation of this paper, we now suarise the relevant parts of an old classical approach, and of three recent papers. 3. Magnetic field produced outside a spherical surface current distribution Chapan and Bartels (940) Workers on ionospheric phenoena needed to calculate the agnetic field produced by the ionospheric current syste, which is often approxiated by a surface current distribution. The standard approach (see e.g. Chapan and Bartels, 940, p. 630, but note that their factor of /0 is not needed when working in SI units) is not to work in ters of the current itself, but to specify the current flowing in a spherical surface by the equivalent scalar current function F(θ, λ) (analogous to the strea function of hydrodynaics), that can be expressed as contours on the sphere. This current function is such that between the two contours F and F + F there is a current flow of F, parallel to the contours, clockwise round a iniu. If a surface haronic analysis is perfored on the F(θ, λ) at radius s to give F(θ, λ) = Il Sl (θ, λ) (45) l, then the external field produced by the Il coponent has the scalar potential φl (r,θ,λ)= l ) l+ l + I l S r l (θ, λ), (46) corresponding to a Gauss coefficient gl l = µ 0 I l + a l. (47) In fact for a given current function F(θ, λ) the actual surface current density is i(θ, λ) = curl[ˆrf(θ, λ)] = ˆr grad h F(θ, λ). (48) Coparing this with the definition of αl (θ, λ) in (3) we see that the surface current distribution corresponding to the current function Il Sl (θ, λ) at radius s is just i l (θ, λ) = i l α l (θ, λ) with i l = (I l /s). (49) Expressing (47) in ters of the surface current coefficient il gives gl l = µ 0 si l + a l. (50) The concept of a current function used here is forally only applicable to a surface current (having no coponent noral to the surface), or to a current sheet thin enough that it is sufficiently accurate to use the thickness-integrated current density as an equivalent surface current; it cannot be applied to a general 3-diensional current syste. However no external agnetic field is produced by any radial coponent of current (Eq. (0)), so there is no reason why equations such as (46) or (47) should not be integrated over radius, to give the integrated effect of the tangential coponent of a 3-diensional current as in (39). But workers using this Chapan and Bartels approach do not appear to have seen this possibility. 3. Toroidal currents and poloidal agnetic fields Engels and Olsen (998) The approach of these authors starts with Eq. (30): µ 0 (J pol + J tor ) = curl curl st + curl sq. (5) Coparing the curl sq toroidal part of this with (48) we see that Q(s,θ,λ), the toroidal current scalar at radius s, is siply a scaled version of F(s,θ,λ), the current function at radius s. The current function approach used to calculate ionospheric and siilar agnetic fields is essentially the sae as the toroidal current/poloidal field relationship used in geodynao theory and elsewhere, and Engels and Olsen explicitly note the identity for the case of a current sheet. In each case, the current function F (or the toroidal scalar Q) is subject to surface haronic analysis at a given radius, and the resulting partial coefficient is then weighted by s l+ and integrated over radius (though this last step does not appear to have been done previously by workers using the current function approach). Using the notation of the present paper, for a given radius s, Engels and Olsen analysed J tor in the for µ 0 J tor (s,θ,λ)= ql (s)αl (θ, λ). (5) a They worked in ters of the scalar potential, and used a Green s function approach to show that if the coefficients l,

6 vi F. J. LOWES AND B. DUKA: MAGNETIC MULTIPOLE MOMENTS ql (s) are known as a function of radius s, the resulting external field corresponds to a Gauss coefficient gl = l q l + a s a l (s)ds (53) equivalent to our (39). In their exaples they used standard nuerical ethods to estiate the coefficients at a succession of discrete radii, and then to estiate the gl. By going to sufficiently high order (and using also the internal fields that we have not considered) they had no proble in applying the ethod to give the field fro thin field-aligned currents joining the north and south auroral zones. 3.3 Series expansion for vector potential Kazantsev (999) Kazantsev introduced the approach of analysing the tangential current density in ters of the αl (θ, λ) and then integrating over radius to give our oents l. (Note that his oents jl are nuerically l ties larger than our l ). Unfortunately his notation and atheatical approach are difficult to follow for readers with a geophysical background. Kazantsev did not relate his oents to the Gauss coefficients, and although he gives explicit expressions for the αl, he does so using Cartesian coponents, which adds further coplication. 3.4 The paper by Stup and Pollack (998) Unlike Engels and Olsen (998) and Kazantsev (999), the paper by Stup and Pollack (998), was restricted to the calculation of the field produced outside the current distribution on a single surface, and they apparently did not know of the Chapan and Bartels (940) current function approach. As in the other two papers they expanded the current density at radius s in the for (their vector spherical haronics had the opposite sign) i(θ, λ) = kl αl (θ, λ), (54) l, used the spherical haronic expansion of / r s to give a series representation of the vector potential in the for A(r) = µ 0 a k ) l+ l α l (θ, λ), (55) l + r l, and used the result (34) to give gl l = µ 0 k l + a l, (56) equivalent to (47) above. 4. Conclusion In geoagnetis, once the Gauss coefficients are known, it has been usual to calculate the external field using the corresponding scalar potential ψ(r) = a ( a ) l+ gl S r l (θ, λ), (6) l, but there does not see to have been a siple expression enabling the Gauss coefficients to be derived by the volue integration over an arbitrary current distribution. We show that g l = µ 0 a (l + ) ) l J(s) αl (θ, λ)dv, (40) a where the vector haronics α l (θ, λ) are defined in (3) or (3). If the horizontal part of the current density at radius s is expanded in the for we have J h (s,θ,λ)= l, g l = µ 0l (l + ) J l (s)α l (θ, λ), (35) J a l (s)ds. (39) There ight be situations where it is ore convenient to work in ters of the vector potential, and the present paper leads to the siple expression A(r) = a l, g l ( a ) l+ α l (θ, λ), (4) l r giving the external vector potential in ters of the Gauss coefficients and the surface vector haronics α l (θ, λ). (Of course when using vector potential, possible gauge transforations ean that in a particular situation different approaches ight give different A(r); however the difference will siply be the gradient of a scalar function, and will not affect the resultant field.) Acknowledgents. We thank Angelo De Santis and an anonyous referee for helpful suggestions. References Backus, G., R. Parker, and C. Constable, Foundations of Geoagnetis, Cabridge University Press, 996. Chapan, S. and J. Bartels, Geoagnetis, Oxford University Press, 940. Engels, U. and N. Olsen, Coputation of agnetic fields within source regions of ionospheric and agnetospheric currents, J. Atos. Sol.-Terr. Phys., 60, , 998. Gray, C. G., Multipole expansion of electroagnetic fields using Debye potentials, A. J. Phys., 46(), 69 79, 978. Jackson, J. D., Classical Electrodynaics, nd Ed., Wiley, New York, 975. Kazantsev, V. P., Spherical agnetic ultipole oents systes of currents, Russ. Phys. J., 4(0), 96 9, 999. (Translated fro Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 0, pp. 66 7, Oct. 999.) Lowes, F. J., Vector errors in spherical haronic analysis of scalar data, Geophys. J. R. Astron. Soc., 4(), , 975. Raab, R. E. and O. L. De Lange, Multipole Theory in Electroagnetis, Clarendon Press, Oxford, 005. Stup, D. R. and G. L. Pollack, A current sheet odel for the Earth s agnetic field, A. J. Phys., 66(9), 80 80, 998. Wickswo, J. P. and K. R. Swinney, A coparison of scalar ultipole expansions, J. Appl. Phys., 56(), , 984. F. J. Lowes (e-ail: f.j.lowes@ncl.ac.uk) and B. Duka