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Magnetic multipole moments (Gauss coefficients) and vector potential given by an arbitrary current distribution
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2011
- Received: 25 November 2009
- Accepted: 4 August 2011
- Published: 21 February 2012
Until recently there has been nothing in the geomagnetic literature giving the Gauss coefficients (equivalent to magnetic multipole moments) for the magnetic scalar potential produced outside a finite-sized region of electric current. Nor has there been an expression for the corresponding magnetic vector potential. This paper presents a simple expression for the Gauss coefficients in terms of a volume integral over the current, and also a series expansion of the vector potential in terms 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 (1998), Stump and Pollack (1998) and Kazantsev (1999).
- Gauss coefficients
- magnetic field
- magnetic scalar potential
- multipole moments
- spherical harmonics
- vector potential
- vector spherical harmonics
In considering the main geomagnetic field outside the Earth, most workers specify the field by its scalar potential expanded in terms of spherical harmonics, and the corresponding Gauss coefficients, which are scaled versions of the classical multipole moments. There is a similar approach in electrostatics, and many classical texts show how to calculate these moments by integrating over the electric charges (or the equivalent magnetic monopoles) that are the source of the field. However the source of the main geomagnetic field is not monopoles but electric currents, and there did not appear to be in the literature general expressions for calculating the moments (higher than the dipole) by integrating over the current system. 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 harmonics, and relates the moments in the two approaches. It compares our results with those of previous workers in a consistent notation.
As is usual, we assume that the source current density J is varying so slowly in time that we can assume that div J = 0. We are concerned only with real current density, so ignore the effect of any magnetisation. The rest of this Introduction presents the basic ideas of magnetic scalar and vector potentials, and of toroidal and poloidal fields.
1.1 Magnetic scalar potential, poles and multipoles
Similarly, taking two z-axis dipoles of opposite sign displaced along the x -axis would give a zx-quadrupole. In general, a degree l multipole has a moment M(l) involving l factors of displacement, and potential falling off with distance as an arbitrary degree l multipole can be specified in terms of (2l + 1) independent moments.
1.2 Scalar potential of a distributed source
The volume integral is to be taken over the whole source region, and (8) is convergent outside the sphere that circumscribes the source region. Successive terms of (8) correspond to the potential of a pole , three dipoles , five quadrupoles , etc. at the origin. (In our magnetic case there are no real monopoles, and the series starts with the dipole, l = 1, terms.)
Algebraically (10) and (14) are the same whatever the shape of the real source volume; if this is not spherical is simply put to zero as necessary when making the surface harmonic analysis. (There might of course be arithmetic problems in the surface harmonic analysis if is large next to a boundary.)
Using this spherical harmonic approach has the advantage that for a given degree l the series automatically produces the correct number of independent functions (orthogonal over spherical surfaces) and corresponding moments. Alternatively, if is expanded using a Cartesian approach, this leads to terms most easily expressed using tensor notation (see e.g. Raab and De Lange, 2005). However, formally this leads to more than moments/spatial functions, but not all of these are independent (orthogonal); this redundancy is discussed in physical terms in Wikswo and Swinney (1984), and in terms of trace-less tensors by Raab and De Lange (2005, section 1.6).
1.3 Magnetic vector potential
However these are ad hoc process, only feasible for low degree terms. In Section 2 we present a systematic method for obtaining the Gauss coefficients in terms 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.
1.4 Toroidal and poloidal electric currents and magnetic fields
When considering the effects of a current distribution in a sphere, it is common practice to separate the ‘toroidal’ current systems, which have current purely in concentric spherical surfaces and have no radial component, from the ‘poloidal’ current systems, which do have radial (as well as tangential) components); see e.g. Backus et al. (1996, section 5.3). It is a standard result that toroidal currents produce only poloidal magnetic fields, both inside and outside the region of current, while poloidal current systems produce only toroidal magnetic fields, and these only inside the current region. So we know that the whole of a poloidal current system (the sum of its radial and associated tangential parts) produces no magnetic field outside the source region.
But the middle term of (20) shows that the radial part of such a poloidal current system gives no external field, so it follows that the tangential part must also give no external field. Therefore it does not matter if the tangential part of any poloidal current system is included or not in our calculation; we do not need to make the toroidal/poloidal separation before integration.
In geomagnetism, until recently the vector potential has been used only indirectly, as a means of obtaining expressions for the scalar potential Gauss coefficients. However Kazantsev (1999) produced expressions for the vector potential produced outside an arbitrary current distribution, using vector potential ‘moments’. His paper is difficult to follow, so we now give an equivalent approach for the determination of these moments; we also show how they are related to the scalar potential Gauss coefficients.
Using the notation of this paper, we now summarise the relevant parts of an old classical approach, and of three recent papers.
3.1 Magnetic field produced outside a spherical surface current distribution—Chapman and Bartels (1940)
The concept of a current function used here is formally only applicable to a surface current (having no component normal 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-dimensional current system. However no external magnetic field is produced by any radial component of current (Eq. (20)), 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 component of a 3-dimensional current as in (39). But workers using this Chapman and Bartels’ approach do not appear to have seen this possibility.
3.2 Toroidal currents and poloidal magnetic fields— Engels and Olsen (1998)
3.3 Series expansion for vector potential—Kazantsev (1999)
Kazantsev introduced the approach of analysing the tangential current density in terms of the and then integrating over radius to give our moments . (Note that his moments are numerically times larger than our ). Unfortunately his notation and mathematical approach are difficult to follow for readers with a geophysical background. Kazantsev did not relate his moments to the Gauss coefficients, and although he gives explicit expressions for the , he does so using Cartesian components, which adds further complication.
3.4 The paper by Stump and Pollack (1998)
We thank Angelo De Santis and an anonymous referee for helpful suggestions.
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