Exact solutions for internally induced magnetization in a shell

Summary

We analyse the external magnetic field generated by a spheroidal shell of constant susceptibility when it is magnetized by an internal magnetic field of arbitrary complexity. The analysis is relevant to the generation of the Earth’s crustal magnetic field by the internal core field. We find an analytical expression for such a crustal field that is expressed in an oblate spheroidal coordinate system, valid for arbitrary flattening, shell thickness and susceptibility. Our exact calculation takes into account magnetization due to the magnetic field of the crust itself, generating an external field that depends on terms both linear and non-linear in the susceptibility. Each spheroidal harmonic coefficient of the external field is generated by and proportional to the same coefficient in the expansion of the inducing field. The terms linear in the susceptibility are generated by the flattening of the Earth, and would otherwise vanish (by Runcorn’s theorem) in the spherical case. For a geophysically relevant model, the crustal field is weak, generated primarily by the non-linear terms, and dominated by long wavelengths.

Earth’s magnetic field, geomagnetic field, induced magnetization

1 Introduction

At long wavelengths it is generally agreed that the Earth’s magnetic field originates principally from dynamo action in the Earth’s outer core. Nevertheless, some magnetic fields originate close to the Earth’s surface due to the magnetization of minerals that are below their Curie temperature; we refer to the magnetic field generated by this magnetization as the crustal field. As a result of Gauss’s theorem, it is impossible to separate the core and crustal fields at a given wavelength. Consequently, the (weak) contribution of the crustal magnetic field for long wavelengths is essentially unknown. For a better understanding of both sources of magnetic fields it may be appropriate to study systematically all the possible sources that can generate a very long-wavelength crustal magnetic field.

It is well known since the work of Runcorn (1975) that much of the induced crustal magnetization does not generate a crustal magnetic field since, to first order in the susceptibility, a spherical shell of constant thickness and susceptibility magnetized by internal sources does not produce any external field. This remarkable theorem was applied successfully by Runcorn to the case of thermoremanent magnetization to explain the very small external magnetic field of the moon.

The problem of the magnetic field induced by a spheroidal shell of constant susceptibility has been addressed by Jackson (1999). The result, which is obtained by a more general approach in the present work, is that this induced field is very small (≃ 10^{− 2} nT), even compared to the amplitude of the crustal field (a few nanoteslas) in the highest spherical harmonic degrees. Other estimates of the magnitude of the crustal magnetic field have been obtained in earlier work from statistical or geological considerations, e.g. Hahn (1984), Jackson (1990, 1994, 1996), Langel (1989) and Cohen & Achache (1990, 1994). A comparison of the power spectrum of the observed magnetic field with the predictions of two of these geological models is given in Fig. 1, showing their ability to reproduce, to a large extent, the observed short-wavelength power spectrum and to generate a small field at the longest wavelengths (l<10).

Figure 1

Power spectrum of the magnetic field as a function of the spherical harmonic degree from the model M07AV6 of Cain (1989) (solid line). Also plotted are the induced power spectrum generated by the continent function of Counil (1991) with an internal source (squares) when χd=1.4 km and the spectrum for the model of the induced field CRST-70-F-22-22 of Hahn (1984) (crosses).

In those previous works that treated induced magnetization, the external field is calculated in the linear approximation (i.e. to first order in susceptibility), and in doing so, the field induced in the crust by itself is neglected. This approximation may be reasonable since the susceptibility is usually very small. However, because of the weakness of the field generated by a spheroidal shell in the linear approximation, it is worth looking at the non-linear terms. This has already been done in the simplified case of a spherical geometry for spherical harmonics of degree l=1 only, by Stephenson (1975, 1976), Srnka (1976) and Srnka & Meldenhall (1979). In Section 2 we compute the exact external magnetic field due to a spheroidal shell of constant susceptibility magnetized by an internal potential. Starting from the result that the magnetic potential must satisfy Laplace’s equation in a medium of constant susceptibility, the technique we use is to solve this equation in each of the three distinct regions of 3-D space: in the interior of the shell, in the shell itself and in the exterior of the shell. The general solution is subsequently found by imposing the usual continuity conditions on the boundaries of the shell. The same calculus is presented for an exterior inducing potential magnetizing a spherical shell in Jackson (1975) or a spheroidal shell in Rikitake (1995). Obviously, this technique may be applied to a series of layers of constant susceptibility and therefore is applicable to the problem of susceptibility varying with depth.

Such an exact representation of the external potential field leads to an unwieldy result since the spheroidal shape of the Earth introduces Legendre functions of both the first and second kinds with complex arguments into the solution. Thus, in Section 3 an approximate result is developed in which powers higher than three in susceptibility and higher than two in the flattening of the Earth are neglected in a power series expansion of the exact solution. This turns out to be an excellent approximation. The last section discusses the results.

2 Theory

In a region of constant susceptibility and free of electrical currents, the magnetic field is B=−∇Φ, where the magnetic potential Φ satisfies Laplace’s equation,

(e.g Jackson 1975). The most general solution of this equation in spherical coordinates (r, θ, φ) is the well-known expression

where the Y_l^m(θ, φ) are spherical harmonics and s_{l, m} and k_{l, m} are Gauss coefficients. For the case of an oblate spheroidal coordinate system, we use the notation (r, θ, φ) to denote the position; there should be no confusion with the similar use for the spherical case. The spherical case is the limit of the spheroidal coordinate system, in which the focal distance is zero. For an oblate spheroidal coordinate system (r, θ, φ), a surface of constant r is an oblate spheroidal surface. A point on this surface is given in Cartesian coordinates by

where in terms of a standard ellipse with semi-major axis a, semi-minor axis b, c is the focal distance and a=√b²+c². In this spheroidal coordinate system expression (2) becomes

where F₁(r) and F₂(r) are functions of r, l and m such that Laplace’s equation is satisfied. These functions are the Legendre functions of the first and second kinds, P_l^m(ir/c) and Q_l^m(ir/c), respectively (Schmidt 1889; Winch 1967).

The geomagnetic problem in which we are interested comprises a spheroidal shell of constant susceptibility χ with inner and outer boundaries that are given in the spheroidal coordinate system by r=r₁ and r=r₂, with r₂>r₁, magnetized by an internal (inducing) potential whose sources lie in a region r<r₀, where r₀<r₁:

where there is no term in F₁(r) since this potential must vanish for very large r. The 3-D space can be divided into three regions in which the potential can be expressed in terms of spherical harmonics:

The expression for the potential in regions 1 and 3 does not contain the functions F₂(r) and F₁(r)respectively in order to describe a potential field that is finite at r=0 and vanishes at r=∞. For each l, m, four geomagnetic coefficients must be determined for a complete description of the potential field.

The magnetic potential can be calculated everywhere in 3-D space by demanding the continuity of the potential and of μ∂_ηΦ(r) at a surface of discontinuity of the susceptibility; here μ is the magnetic permeability and is related to the susceptibility by μ=μ₀(1 +χ), and ∂_η is the gradient normal to the surface. Imposing the two continuity constraints at the two spheroidal surfaces r=r₁ and r=r₂ leads to the four equations

where r₁^±, r₂^± are points on the positive and negative sides of the boundaries r₁, r₂ oriented in the direction of increasing r. When the boundary conditions (7) are applied to (6), by using the orthogonality of the spherical harmonics (Jackson 1975) we find

where F_ij=F_i(r_j) and F′_ij=∂_rF_i(r_j) for i=1, 2 and j=1, 2. Solving the system of linear equations (8) for κ³_{l, m} is tedious but straightforward. The result is

where

The coefficients κ³_{l, m} give the total potential field outside the spheroidal shell for r>r₂. We may be interested in that part of the field that is induced by the spheroidal shell. Written in terms of spheroidal harmonics, this field can be defined as

with

The expressions for A₁ and A₂ are given in Appendix A. This gives the exact solution to our problem. The amount by which the inducing field is amplified can be described in terms of an amplification factor:

3 Approximation

With the dependence of the results on the Legendre functions in the above section, it is difficult to estimate the relative importance of the different terms in (13). In the following we first write (13) separating linear and non-linear terms in χ, These terms are then approximated to clarify their dependence on the focal distance c.

Let us rewrite (13) as

or, using the notation ~ α _l^m to denote the approximate value of α_l^m,

where in this last equation terms in χⁿ with n > 3 are neglected.

Now, neglecting terms in (c/r)ⁿ with n>2, the F₁(r) and F₂(r) functions can be approximated (Schmidt 1889; Winch 1967) by

which lead to the relations

and

where the second line in each of (17) and (18) holds when l is small (say l≤ 10) since both r₁ and r₂ are very large compared to c and the thickness of the crust d≃ (r₂−r₁) is small compared to r₂. Rewriting A₁ such that it exhibits the above ratio gives

or using eqs (17) and (18),

where the approximations are the same as those in (17) and (18) and τ_l^m is defined by

Therefore, taking into account only terms linear in χ, the approximate amplification factor ~ α _l^m in (15) is

where d≃r₂−r₁ is the thickness of the crust and b is a mean radius, i.e. b≃r₁≃r₂. This is in agreement with the amplification factor given for l=1 in Jackson (1999). In the same way we find for A₂

The term A_2o is independent of c and therefore contributes to the induced crustal field even for a spherical earth.

The result (20) allows further simplifications to be applied to (15). Neglecting terms in (c/r)ⁿ with n>2, the approximated amplification factor ~ α _l^m is now given by

This relation holds for small lif the susceptibility, the shell thickness and the ratio c/r are small.

For a spherical Earth, we have c=0, so the terms (1 −A₁) and the A_2c part of A₂ vanish and we obtain

Further simplification can be applied, and the amplification factor becomes

where d≃r₂−r₁ is the thickness of the crust and b is a mean radius, i.e. b≃r₁≃r₂. This approximation may be compared with (22). It follows that the amplification factor for the term dominant in the spectrum (l=1, m=0) will behave like χ if χ≪ (c/b)² and like χ² if 1 ≫χ≫ (c/b)².

4 Discussion

We begin by considering the following geophysically relevant but simple model used previously in Jackson (1999): χ=0.1, c=522 km, r₂=6356 km and r₁=6346 km. Fig. 3 along with the value from the approximation (20). The exact results have already been published in Jackson (1999). These terms are dependent on c (see eq. 20) and, as stated in Runcorn (1975), drop to zero if the flattening of the earth is neglected.

Figure 3

Amplification factor α_l^m using only terms linear in χ and computed for each coefficient with degree l and order m running from degree 1 up to 10. The coefficients are ordered with increasing land mincreasing within each degree l; χ=0.1 and d=10 km. The diamonds give the exact values, using the exact A₁ from eq. (A3) while the crosses use the approximations of eq. (20).

As noted in Section 3, the term A_2o in (23) is independent of c, therefore contributing to the crustal induced field even for a spherical shell. The product − (χ²−χ³)A_2o is, in the spherical case, an approximation of the exact amplification factor α_l^m. This exact factor, which can be obtained in the same way as in Section 2 by using r^l and r^{− (l+ 1)} for F₁(r) and F₂(r), is

This amplification factor is independent of m and obviously negative since r₂ > r₁. The sign indicates that the induced external magnetic field in the spherical case is related to the ‘demagnetization’ field in the shell. [For details about the ‘demagnetization’ field, see e.g. Parasnis (1997) or Dunlop & Özdemir (1997).]

With (20) and (23) and the simple model described above, we see from Figs 2 and 3 that terms in A_2oχ² are more than one order of magnitude larger than terms in (1 −A₁)χ, Therefore, the external induced magnetic field of an Earth-like spheroidal shell is dominated by the non-linear terms that are almost independent of the flattening of the Earth (A_2c≪A_2o). This is illustrated in Fig. 4, where we plot the power at degree l=1 [i.e. 2 Σ_m (κ_{1, m}^I)²] of the induced magnetic field as a function of the susceptibility and the crustal thickness (d≃r₂−r₁). The Gauss coefficients used for the inducing field are those of the core field model from Bloxham & Jackson (1992) for the year 1985. For susceptibility between 0.01 and 0.1, terms in χ² dominate.

Figure 2

Amplification factor α_l^m for each coefficient with degree l and order m running from degree 1 up to 10. The coefficients are ordered with increasing l and m increasing within each degree l; χ=0.1 and d=10 km. The diamonds give the exact values, using eq. (13) and the exact A₁, A₂ from eqs (A3) and (A4), while the crosses use the approximations of eqs (20), (23) and (24).

Figure 4

Power computed at the Earth’s surface at degree l=1 of the induced crustal magnetic field as a function of susceptibility for different thicknesses. The inducing field used is described in Bloxham & Jackson (1992). At low susceptibility the behaviour of the field is dominated by terms linear in χ whereas at large susceptibility the behaviour is dominated by terms in χ². In all the cases, the field is roughly linear in the thickness d of the layer.

In Fig. 5 we plot the power spectra of the Earth’s magnetic field and of the induced magnetic fields due to a spheroidal shell for several values of the thickness and susceptibility. For reasonable values of susceptibilities and thicknesses, the crustal magnetic field induced by a spherical or spheroidal shell is weak. It reaches the same order of magnitude as the crustal field models in Fig. 1 only for the very longest wavelengths (i.e. degree l≃ 1). These models are computed in the linear approximation, whereas our result is mainly due to the non-linear terms.

Figure 5

Power spectrum of the induced crustal magnetic field for different susceptibilities and thicknesses compared to a core field power spectrum, all taken at the Earth’s surface. The inducing core field used is described in Bloxham & Jackson (1992).

Acknowledgements

This work was supported by NERC grant GR3/R9510 and the Royal Society. We are indebted to Frank Lowes for his extensive review of the paper, which improved the manuscript.

Appendix

Appendix A: Expressions for A₁ and A₂

In this Appendix we present expressions for the functions A₁ and A₂ in eq. (10), where F_ij=F_i(r_j) for i=1, 2 and j=1, 2. The two functions F₁(r) and F₂(r) are

Since A₁ and A₂ are ratios of these functions or their derivatives, they are independent of the P_l^m(fracirc) and Q_l^m(fracirc) normalization. We use the normalization given in Abramowitz & Stegun (1965) and their expression of the Wronskian (formula 8.1.8):

Therefore, we have

and