Possible effects of lateral heterogeneity in the D layer on electromagnetic variations of core origin

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1 Physics of the Earth and Planetary Interiors 129 (2002) Possible effects of lateral heterogeneity in the D layer on electromagnetic variations of core origin Takao Koyama, Hisayoshi Shimizu, Hisashi Utada Earthquake Research Institute, University of Tokyo, Yayoi 1-1-1, Bunkyo-ku, Tokyo , Japan Received 3 May 2000; received in revised form 10 June 2001; accepted 26 July 2001 Abstract This paper examines the consistency of a model for typical spatial features of rapid geomagnetic secular variations in which electromagnetic (EM) scattering due to the lateral heterogeneity of mantle conductivity plays an important role. When geomagnetic field variations of a core origin were separated into poloidal and toroidal modes, previous studies suggested that the scattering effect on the poloidal mode is probably weak. However, the toroidal field originating in the core may extend to the (very) deep mantle and can be converted into a poloidal field of observable intensity by scattering. In this paper, scattering of the EM field due to the D layer, where the lateral heterogeneity is supposed to be most significant in the mantle, is studied using three-dimensional numerical calculations. The results show that the spatial features of 60-year variations at the surface can be explained by scattering, although it is possible to interpret that they are directly reflecting those of poloidal field variations originating in the core. The same explanation may be applicable to geomagnetic jerks, assuming that they are represented by a shorter time-scale (1 year) variation. Although a magnetic field observation itself does not enable us to distinguish whether or not observed variations originated from poloidal field variations in the core or by a scattering of the toroidal field in the D layer, our calculation results indicate that detection of electric field and LOD variations of corresponding periods provide strong constraints on the mechanisms of these phenomena, especially the spatial pattern of decadal variations Elsevier Science B.V. All rights reserved. Keywords: D layer; Lateral heterogeneity; Toroidal magnetic field; Mode conversion 1. Introduction It is well known that the geomagnetic field varies over wide range of time scales. Some of the variations, especially at higher frequencies, are of external origin generated in the ionosphere and magnetosphere (e.g. Merrill et al., 1998). In contrast, variations that originated inside the Earth and observed at the surface are believed to have time scales longer than a few years. Studies on the characteristics of the internal Corresponding author. Tel.: ; fax: address: tkoyama@eri.u-tokyo.ac.jp (T. Koyama). field variations have provided valuable information on the physical processes in the Earth s core and mantle (Merrill et al., 1998). The internal field itself has wide range of temporal variations, from a few years to millions of years. Among them, variations having shorter time scales, of a few years to decades, may be influenced by the electrical conductivity of the mantle; electromagnetic (EM) induction in the mantle filters out short time and small spatial scale fields that may exist in the core. Moreover, heterogeneous electrical conductivity of the mantle might cause a distortion of the electric current and corresponding magnetic field, so that spatial distribution of the magnetic field at the surface may be /02/$ see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S (01)

2 100 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) very different from that at the core mantle boundary (CMB). A quantitative examination made by Poirier and Le Mouel (1992) showed that heterogeneity in the mantle does not have a significant influence on altering the distribution of the magnetic field at the surface if the source field is poloidal at the CMB, even for quick geomagnetic variations. Some works concerned the effect of the heterogeneous lowermost mantle on VGP reversals (Brito et al., 1999; Holme, 2000) and concluded that the heterogeneity of the mantle has little effects on VGP reversals. However, the toroidal magnetic field at the CMB, which is not necessarily zero due to the slightly conducting mantle, was not examined as a source field in their analysis, perhaps because the toroidal magnetic field is not observable. Recently, Holme (2000) estimated EM torque due to the toroidal magnetic field with an assumption for core surface flow, and concluded that torque fluctuation was not affected by mantle heterogeneity. Nevertheless, it is worth examining the scattering effect in the mantle on the toroidal magnetic source field; i.e. testing the distribution and the strength of the poloidal field, which is converted from a toroidal field due to the heterogeneity of the mantle. This paper aims to study the EM induction problem in the mantle on a time scale from a few years to decades quantitatively to examine if such a distortion effect in the mantle could exist and if the resulting poloidal field is observable at the surface of the Earth. In particular, we apply this to explain some features of geomagnetic jerk and decadal variations. Geomagnetic jerk is a peculiar variation of geomagnetic field detected by global observatories, which is believed to be a variation of a few years of an internal origin (Malin and Hodder, 1982). Several geomagnetic jerks have been observed in the last century (Macmillan, 1996). The largest event among them occurred in the late 1960s and has been studied from various aspects (e.g. Malin and Hodder, 1982; Alldredge, 1984; McLeod, 1985). Using this phenomena, some works estimated the upper bound of the electrical conductivity of the lowermost mantle (Ducruix et al., 1980; Alexandrescu et al., 1999). It is important and curious, however, why the geomagnetic jerks most often appear clearly in the eastward component of the magnetic field in the European region (Alexandrescu et al., 1995). One possible interpretation is that the jerk field is generated only under Europe (Alldredge, 1987); that is, the spatial pattern of the geomagnetic variation on the surface is merely reflecting the field distribution at the CMB. Another possibility may be ascribed to the scattering effect or the mode conversion of the EM field due to the lateral heterogeneity of electrical conductivity in the mantle; that is, the mantle is acting as a filter having a window of short time-scale variations only beneath Europe. This idea is examined by a numerical simulation of the induction problem. Also, suggestions for other observations to test the consistency of the model are provided. Geomagnetic field variations with decadal time scales, typically 30 or 60 years, have also been reported (Papitashvili et al., 1982; Yokoyama and Yukutake, 1991; Yokoyama and Yukutake, 1993). Yokoyama and Yukutake (1991), Yokoyama and Yukutake (1993) studied the spatial distribution of a 60-year variation using a spherical harmonic analysis, and concluded that the most manifest mode is axial dipole (namely Y1 0 (θ, φ) of spherical harmonics), of which amplitude is about 100 nt. Among non-dipole components, Y3 2 (θ, φ) component is also conspicuous, having an amplitude of about 50 nt (Yokoyama and Yukutake, 1993). If strong lateral variations exist in mantle conductivity, the distribution of EM variations at the surface as reported in Yokoyama and Yukutake (1993) may be different from one at the CMB, due to the scattering effect of conductivity anomalies. In this paper, it is found that the Y3 2 field can be easily generated by the toroidal field at the CMB. The physical consistency of this model is tested with reference to the EM torque generated by the toroidal field, the Joule heat induced in the mantle, and the electric field at the surface. In this paper, the model of the conductivity structure and the EM source field are set-up in Section 2. The method to solve the induction equation when the mantle has a (three-dimensional) 3-D conductivity structure is explained in Section 3. The results of a calculation to examine if the EM variation is distorted by the heterogeneous conductivity structure in the mantle are shown in Section 4. A discussion is given in Section 5, and conclusions are given in Section 6.

3 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Physical set-up We perform forward modeling of the EM field to examine the screening and scattering effects caused by a 3-D electrical conductivity distribution in the mantle. During modeling, source magnetic field variations at the CMB are given and induction equations in the heterogeneous mantle are solved to obtain EM field variations at the Earth s surface. It is necessary to assume an appropriate electrical conductivity model of the mantle and source field configuration at the CMB. These are described in this section Model of the conductivity structure in the mantle The upper mantle is known to be laterally heterogeneous due to, for example, the presence of subducting slabs and mantle hotspots (Schultz and Larsen, 1990). Even if such heterogeneities exist, however, the structure at this depth range will have only a little influence on EM variations with periods longer than a few years. This is because the upper mantle is rather resistive with a mean conductivity between 10 3 and 10 1 S/m (e.g. Banks, 1969), and the induction scale length corresponding to 1-year variation is of the order of 10, ,000 km, which is much larger than the thickness of the upper mantle. Therefore, we neglect the effects of heterogeneity in the upper mantle on EM screening and scattering, and assume a laterally uniform upper mantle in this study. Very limited information about the heterogeneity of the lower mantle is available from both induction studies and laboratory experiments. According to the results of seismic tomography, the lower mantle is supposed to be more homogeneous than the upper mantle (Su et al., 1994). Shankland et al. (1993) showed, from a laboratory experiment, that the lateral heterogeneity of the electrical conductivity in the lower mantle is weak if temperature change is its major cause. Note that the mean value of lower mantle conductivity is found to be in the order of 1 S/m (Shankland et al., 1993). The screening of EM field variations of periods of years will be small if the conductivity of the lower mantle is of the order of 1 S/m. However, there is a possibility that conductivity may be exclusively high in the D layer, which is the layer just above the CMB. Possible presence of iron infiltrating from the outer Fig D conductivity model used in this study. core, partial melting of the mantle material or products of reactions of iron and silicates (Knittle and Jeanloz, 1991) are considered to enhance electrical conductivity by as much as several orders of magnitude. Therefore, the screening effect in the D layer may not be negligible for the time scale of variations. Also, it is highly probable that enhanced conductivity is laterally heterogeneous (Buffett et al., 2000). Such heterogeneity in the D layer will cause significant scattering of the EM fields through EM induction. We employ a mantle conductivity model mainly containing two regions: (1) the D layer and (2) the region above the layer (Fig. 1). The mantle above the D layer is assumed to be laterally homogeneous. Recent results of a radially symmetric conductivity model are employed (e.g. McLeod, 1994; Honkura and Matsushima, 1997). The 1-D model used in this paper is simplified from recent inversion results of global EM responses (Constable, 1993; McLeod, 1994; Honkura and Matsushima, 1997) (see Fig. 2). There is no direct information on lateral variations of conductivity. We refer to the results of seismic tomography to construct a laterally heterogeneous conductivity model in the D layer. According to recent tomographic works, the D layer has the largest

4 102 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Fig. 2. The profile of the conductivity model of the Earth. 1-D and 3-D represent one-dimensional and three-dimensional, respectively, in the regions. perturbation of seismic velocity in the mantle (e.g. Dziewonski, 1984; Su et al., 1994; Masters et al., 1996; Obayashi and Fukao, 1997; van der Hilst and Karason, 1999). Su et al. (1994), Masters et al. (1996), and Obayashi and Fukao (1997) expanded heterogeneity into spherical harmonics and all of them found that the degree 2 sectoral component (Y 2 2 ) is the most dominant. For simplicity, we adopted this distribution for the conductivity distribution in the D layer. Although the thickness of the D layer is also not well-known and may be uneven, we assumed it to be 300 km according to Obayashi and Fukao (1997). The presence of iron infiltrating from the outer core is a possible cause of the lateral heterogeneity in the Fig. 3. The profile and the plane view of the conductivity model of the D layer. σ max is variable, while σ min is set at 5 S/m in all cases.

5 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) D layer, and this is most effective for enhancing conductivity (Knittle and Jeanloz, 1991; Poirier and Le Mouel, 1992; Buffett et al., 2000). In this case, the maximum value of the conductivity anomaly in the D layer is supposed to be between 20 and 4000 S/m, depending on the iron content (Poirier and Le Mouel, 1992). We examine the effects of heterogeneity by changing the maximum value of the enhanced conductivity of the D layer (hereafter we call this parameter σ max ) from 20 to 4000 S/m. The minimum value of conductivity in the layer, σ min, is set at 5 S/m, which is the value of the background 1-D model (Fig. 3). Although it is supposed that upper mantle heterogeneity does not cause EM scattering on yearly time scales, the presence of oceans at the Earth s surface might cause some distortion. Surely the ocean causes distortion of the electric field due to accumulated effects at the ocean land interface, known as the coast effect. This is known to affect rapid geomagnetic field variations of ionospheric and magnetospheric origins (Parkinson, 1964). The coast effect might be taken into account, even when we deal with variations of longer time scales. This effect is examined separately in Section 5. For the moment, calculations are made for an Earth model without lateral conductivity variations at the surface Assumed source field configuration at the CMB Source EM field variations at the CMB are given to solve the induction problem. As is usually done, the EM field is decomposed into poloidal and toroidal part (e.g. Gubbins and Roberts, 1987) in numerical calculations, as is the source field at the CMB. Although we do not know the actual field configuration at the CMB, we assumed that the most dominant part of the variation of the poloidal field is the axial dipole (S1 0 ), which is a fluctuating part of the main field. Variation of the toroidal field at the CMB is not known, but it may be natural to assume that the toroidal field variation at the CMB has the form T2 0, because this component is generated by axial dipole and zonal shear flow in the fluid core by the ω-effect (e.g. Gubbins and Roberts, 1987). Note that the actual magnetohydrodynamic interaction is not very simple and the dominant toroidal mode at the CMB might be different. Periods of the variations of the source field S1 0 and T 2 0 are assumed to be 1 and 60 years. One year is chosen to consider an EM field variation that has a similar time scale to the geomagnetic jerk. 3. Formulation and test of the code During recent years, significant progress has been made in numerical calculations of EM induction in a medium having a 3-D conductivity structure. Methods based on an integral equation (IEM, e.g. Wannamaker, 1991), finite element (FEM, e.g. Everett and Schultz, 1996) and finite difference (FDM, e.g. Mackie et al., 1993) are often employed for modeling the EM field. IEM has an advantage in terms of computational memory over the other two methods if the conductivity structure is relatively simple. On the other hand, FEM and FDM are superior to IEM when the conductivity structure is complicated, although they require a large memory space for calculators. In this paper we employ one of the IEMs for the calculation, because only relatively simple large-scale structures are essential for the large-scale EM induction, upon which we are going to focus. The actual method employed in our calculation is an IEM using the modified iterative dissipative method (MIDM) developed by Singer (1995) Formulation based on the MIDM The equations governing the EM field are Maxwell equations: ( H = σ 3-D + ɛ ) E+j t s and ( σ 0 + ɛ ) E + (j t s + δσe) (1) E = µ t H (2) where E and H are the vectors of the electric field and the magnetic field, respectively, j s the source electric current density, ɛ and µ the electric permittivity and the magnetic permeability, σ 3-D the electrical conductivity having 3-D heterogeneities, σ 0 a simple background structure, and δσ ( σ 3-D σ 0 )isthe deviation of the conductivity from the background structure. In the case of a spherical Earth, the EM

6 104 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) field must be bounded at the center. A radiation condition, which states that the EM energy flux must propagate outward, must be satisfied if r approaches infinity. Also, at the interfaces of different conductivities, magnetic flux density B, electric field horizontal to the interface, and electric current normal to the surface that includes the displacement current must be continuous. The Maxwell equations are solved in the frequency domain. In this approach we need to find the 3 3 electric and magnetic Green s function tensors, q E 0(r; r ) and q H 0(r; r ), for a radially symmetric background structure σ 0 (r). The Green s function tensors satisfy q H 0 (r; r ) = (σ 0 (r) + iωɛ)q E 0 (r; r ) + δ(r )I (3) q E 0 (r; r ) = iωµq H 0 (r; r ) (4) where ω is angular frequency and I is the 3 3 identity matrix. Using Green s function, the integral equation for the electric field can be written as E = dvq E 0 (j s + δσe) (5) V where V stands for a whole space. The IDM represents a solution of the electric field as a series of volume integrals ( ) 2 E = E 0 + E 0 dvq E 0 δσe 0 + dvq E 0 δσ V het V het ) 3 E 0 + (6) ( + dvq E 0 δσ V het B = B 0 + µ dvq H 0 δσe V het (7) where E 0 = dvq E 0 j s V (8) B 0 = µ dvq H 0 j s V (9) and ( ) n [ { ( ) }] n 1 dvq E 0 δσ E 0 dr q E 0 (r; r )δσ (r ) dvq E 0 δσ E 0 V het V het V het with V het representing the region of anomalous conductivity where δσ is non-zero. E 0 and B 0 are the electric and magnetic fields for the background structure σ 0. Therefore, for a simple structure with a small heterogeneous region, the time for calculating these integrals, which occupies a major part of the calculation time, becomes relatively short for one iteration, because the integrated volume is small. At this stage, the convergence of the Neumann series (Eq. (6)) is not guaranteed for general conductivity structures. This problem is overcome by Singer (1995) by introducing MIDM or Pankratov et al. (1995) by introducing IDM with the modified Neumann series by transforming from the EM field vector to another vector where the Neumann series can converge. In a later part of this paper, the source current j s is given in the core to discuss the EM field of core origin. We give source field variations by, for example, unit current density in the region 3400 r 3500 km, which is beneath the CMB. E 0 and B 0 in the mantle are simply calculated by an upward continuation. After the 3-D calculation, the values of EM fields are rescaled so that calculated magnitude of magnetic field variation is the same as that of the observed values at the surface Testing the code A code based on MIDM was developed and its numerical accuracy was tested by comparing the numerical results obtained by other codes based on different algorithms. We will show, in particular, the EM response calculated by our code and that by the staggered-grid finite difference (SGFD) method of Uyeshima and Schultz (2000). The conductivity structure employed is the so-called double hemisphere model, which contains a non-homogeneous layer at a depth of 400 km. An external axial dipole field having a period of one day is given as the source field with an intensity of 100 nt at the surface (see Uyeshima and Schultz, 2000). The memory size used was about 10 Mbytes. The job execution time was about 5 s on a single 300 MHz MIPS RISC R12000 CPU. Fig. 4 (n 1) (10)

7 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Fig. 4. The EM fields at the surface for the test model. SGFD is calculation results provided by the staggered-grid finite difference method of Uyeshima and Schultz (2000).

8 106 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) shows the numerical results. The rms misfit between the numerical results of SGFD and MIDM is 1.5% or less for every component of the EM fields, which indicates that our code is as accurate and usable as that of Uyeshima and Schultz (2000). 4. Numerical results Using the numerical code, we solved induction equations for periodic fields and compared the numerical results with observed data of jerks and decadal variations to examine how scattering effects modify the EM fields. We estimated the induced field amplitudes at the Earth s surface, which were originally S1 0 or T2 0 as assumed in Section 2, and then estimated the EM torque, the Joule heat, and the electric field at the surface Mode conversion of EM fields Here, we take the geomagnetic jerk into consideration, for example, to see how scattering due to heterogeneity occurs. One of the most important features of jerks is that they often appear most clearly in the declination of observatories in Europe (Malin and Hodder, 1982). The intensity of the late 1960s jerk is estimated to be about 5 nt year 2 (Ducruix et al., 1980). Letting the typical time scale of this change be 1 year, the maximum value of the amplitude of B φ was simply assumed to be 5 nt (Figs. 5 and 6, Tables 1 and 2). Here, we define the intensity I of the magnetic field as the square root of the power P, i.e. I S P B 2 ds S ds (11) where SdS denotes the integral over a surface S. Table 1 shows the intensity of each mode at the surface for the S1 0 source. Coupling the S0 1 field and the P2 2 type heterogeneity causes only odd degree and even order harmonics. S2n+1 2m (n = 0, 1,... ; m = 0, 1,...,n) are known to be non-zero at the surface from the selection rules for the Gaunt and Elsasser integrals (e.g. Holme, 1997). Fig. 5 shows the distribution of EM field components calculated at the surface. This result indicates that, if the geomagnetic source is an axial dipole S1 0, the magnetic field at the surface should appear most clearly in the radial and northward components, but the eastward component will be negligibly small. On the other hand, Table 2 and Fig. 6 show the response for the T2 0 source. The selection rules indicate that S2n+1 2m (n = 1, 2,... ; m = 1, 2,...,n) are non-zero at the surface in this case. As shown in Fig. 6, the electric field at the surface seems to conserve the original field pattern; that is, zonal degree 2 order 0 harmonics. The magnetic field, however, has a completely different spatial distribution. As in the case of the S1 0 source, the results show that the poloidal magnetic field produced from the T2 0 source by mode conversion of the P2 2 distributed D layer consists of only odd degree and even order harmonics, where S3 2 is especially dominant Numerical calculations Numerical calculations were carried out by giving source field variations of three types at the CMB; S1 0, T2 0, and also S2 3. Yokoyama and Yukutake (1993) found that there exist variations with a 60-year period having amplitudes of 100 nt in S1 0 and 50 nt in S2 3. We take them as the amplitudes of observed variations, and try to estimate the intensity of core fields for various σ max. For EM source models of the core origin with a period of 60 years, we took into consideration three cases named: (1) S1 0 model; (2) T 2 0 model; and (3) S2 3 model. The first two models are described in the previous section. For the S1 0 model, values of the EM field are normalized so as that an intensity of S1 0 at the surface can be 100 nt. For the T2 0 model, the values of the EM field are normalized so as that an intensity of S3 2 at the surface can be 50 nt, which is produced by the mode conversion of T2 0 source variation. We also apply the S3 2 model, because the S2 3 source variation in the core can produce the S3 2 distribution of the magnetic field at the surface. For the S3 2 model, values of the EM field are normalized so as that an intensity of S3 2 at the surface can be 50 nt. Table 3 shows the intensity of the source field on the CMB. For the S1 0 model, we can see that S0 1 variation at the CMB is of the order of 1000 nt and the intensity tends to be larger for larger σ max, because both the effects of mode conversion and screening become stronger. For the T2 0 model, the intensity of T 0 2

9 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Fig. 5. The characteristic distribution of the EM variation in the case of a poloidal magnetic source (S1 0 ) with a period of 1 year. Each snapshot of EM variations at the moment when the peak intensity of each component is largest is shown (σ max = 50 S/m). The characteristic distribution of the EM variation in the case of a poloidal magnetic source (S1 0 ) with a period of 1 year. Each snapshot of EM variations at the moment when the peak intensity of each component is largest is shown (σ max = 50 S/m).

10 108 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Fig. 6. The characteristic distribution of the EM variation in the case of a toroidal magnetic source (T2 0 ) with a period of 1 year and σ max = 50 S/m.

11 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Table 1 The intensity of modes at the surface for magnetic S 0 1 variations at the CMB with a period of 1 year a Degree n Order m Ratio Intensity (nt) a Ratio is Sn m/s2 3 ; intensity is the magnetic field in case the maximum of B φ is 5 nt (a gauss coefficient of S3 2 is 2.5 nt). The period is 1 year and σ max is 50 S/m. Table 2 The intensity of modes at the surface for magnetic T2 0 variations at the CMB with a period of 1 year a Degree n Order m Ratio Intensity (nt) a Ratio is Sn m/s2 3 ; intensity is the magnetic field in case the maximum of B φ is 5 nt (a gauss coefficient of S3 2 is 2.5 nt). The period is 1 year and σ max is 50 S/m. variation at the CMB is not proportional to σ max,but reaches the minimum when σ max is 1000 S/m. When σ max is smaller than 1000 S/m, i.e. the D layer is less heterogeneous, the effects of mode conversion are weaker. Therefore, the presence of a stronger source field at the CMB is needed to provide S3 2 of 50 nt at the surface. When σ max is larger than 1000 S/m, the T2 0 source intensity at the CMB is roughly constant at about 50,000 nt. For the S3 2 model, the intensity is almost constant, although a source with a larger intensity is required to compensate for the scattering effect of a larger σ max. 5. Discussion In this section, other physical parameters are calculated such as EM torque, Joule heat, and electric field on the surface. Then the calculated results are compared with observed geomagnetic field variations to examine possible mechanisms producing particular features of geomagnetic 60-year variations. We also examine the geomagnetic jerk with 1-year source variations. First, EM torques for all the cases are calculated to examine the consistency of the models. EM torque is calculated, using values of magnetic fields at the CMB, i.e. (e.g. Gubbins and Roberts, 1987) Γ = c3 (B r B φ ) sin 2 θ dθ dφ (12) µ 0 CMB where c, θ, and φ are the radius of core, colatitude, and longitude, respectively. If either of B r or B φ is periodic, then the torque fluctuates at the same periodicity, that is, the fluctuating torque Γ is Γ = c3 µ 0 CMB ( B r B 0 φ + B0 r B φ ) sin 2 θ dθ dφ (13) where B 0 r and B0 φ denote the steady part and B r and B φ denote the fluctuating part. Here, the steady and fluctuating fields are assumed to be expressed by DGRF Table 3 The calculated intensity of the magnetic field on the CMB a σ max (S/m) (conductance (S)) S model (nt) T2 model (nt) S2 3 model (nt) 50 ( ) ( ) ( ) ( ) ( ) ( ) ( ) a The period of the EM variation is 60 years. The S1 0 model is assumed to produce S0 1 with an amplitude of 100 nt on the surface. The T2 0 and S2 3 models are assumed to produce S2 3 with an amplitude of 50 nt on the surface.

12 110 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Table 4 The calculated intensity of EM torque a σ max (S/m) S 0 1 model (Nm) T 0 2 model (Nm) S 2 3 model (Nm) a The period of the EM variation is 60 years. The assumptions for each model are the same as shown in Table 3. Table 6 The calculated voltage between Guam and Japan a σ max (S/m) S 0 1 model (mv) T 0 2 model (mv) S 2 3 model (mv) a The period of the EM variation is 60 years. The assumptions for each model are the same as shown in Table and the calculated fields, respectively. We did not consider the steady toroidal field, which is unknow. The resulting amplitudes of EM torque variations are given in Table 4. For the S1 0 model, the EM torque is of the order of Nm and is roughly proportional to σ max. For the T2 0 model, the torque is of the order of Nm, minimum when σ max is 1000 S/m and proportional to the intensity of source fields, which we can understand from Eq. (12). For the S3 2 model, the torque is almost constant as Nm. Joule heat produced by the EM field variation in the whole mantle is also calculated. Total flux is calculated using values of EM fields the CMB, i.e. J = c2 (E θ B φ E φ B θ ) sin θ dθ dφ (14) µ 0 CMB Resulting values of Joule heat generated in the mantle are shown in Table 5. For the S1 0 and S2 3 models, the total flux of Joule heat is of the order of 10 5 W and is roughly proportional to σ max. For the T2 0 model, the Table 5 The calculated total flux of Joule heat a σ max (S/m) S 0 1 model (W) T 0 2 model (W) S 2 3 model (W) a The period of the EM variation is 60 years. The assumptions for each model are the same as shown in Table 3. total flux of Joule heat is of the order of 10 9 W and is inversely proportional to σ max. The last is the electric field. Here, as a demonstration, we calculated the voltage change between Guam and Japan expected by each model. The voltage difference between the two points has been measured using the submarine cable formerly named TPC-1. We showed the calculated values in Table 6. For the S1 0 and S3 2 models, the voltage is of the order of 0.1 mv, while the voltage is of the order of 1000 mv for the T2 0 model. The last column of Table 7 shows the intensity of S3 2 at the surface expected for the S0 1 model whose source intensities were shown in Table 3. The intensity of S3 2 is smaller by one or two orders of magnitude than the observed S3 2 amplitude of 50 nt. Therefore, we can conclude that the S3 2 variation cannot be produced by the effects of mode conversion from the S1 0 source of an internal origin. This is consistent with Table 7 The calculated intensity of S1 0 on the CMB and S2 3 on the surface, which is produced by mode conversion due to the heterogeneity of the D layer a σ max (S/m) S1 0 on CMB (nt) S2 3 on surface (nt) a The amplitude of S1 0 variations on the surface is assumed to be 100 nt. The period of the EM variation is 60 years.

13 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Fig. 7. Mechanism of mode conversion from toroidal to poloidal magnetic field. The part of the D layer of a lighter color is more conductive and the magnetic field variation hardly penetrates upward. the conclusion of Poirier and Le Mouel (1992), i.e. mode conversion due to mantle heterogeneity from a poloidal mode to other poloidal modes is not effective. On the other hand, in the case of the T2 0 model, as shown in Fig. 7, the presence of a strong lateral heterogeneity in the D layer causes a conversion from toroidal to poloidal modes. Comparing the values in Table 3, the toroidal poloidal amplitude ratio is estimated to be between 10 and 100 for various σ max. Because direct observation of the toroidal magnetic field is not possible, it seems to be difficult to confirm the consistency of this model. Here, we examine whether or not the obtained T2 0 amplitude is consistent with other geophysical information such as LOD variation and surface heat flow. The intensity of the EM torque variation is of the order of Nm (Table 4), which is of the same order of magnitude as the estimated torque from the LOD variation observed by Morrison (1979). Therefore, the intensity of the calculated T2 0 variation is not inconsistent with the LOD observation. The presence of the leakage current corresponding to the toroidal field in the conducting mantle will produce additional Joule heat. The present model would not be acceptable if the heat generated affects the global heat flow significantly. In Table 5, Joule heat in the T2 0 case is smaller by at least three orders of magnitude than the observed surface total heat flow, of the order of W (Pollack et al., 1993). Thus it has been shown that conversion of T2 0 mode by mantle heterogeneity will be able to produce 50 nt S3 2 variation at the surface, and does not contradict LOD variation and the surface heat flow. Although the T2 0 model seems to be successful, there remains the possibility that the surface feature simply reflects the presence of the S3 2 mode at the CMB. These two models cannot be distinguished from each other only by geomagnetic observations. The presence of the correlating LOD variation described above will work for this purpose. In addition, we emphasize that observations of electric fields will provide a significant constraint on selecting the preferred one between the T2 0 and the S2 3 models. In recent years, continuous observations of electric fields in the oceans have become possible using a network of retired submarine telecommunication cables. Shimizu et al. (1998) have shown that a signal of the core origin may be observable. From Table 6, in the case of the S3 2 model the calculated voltage is too small, typically 0.1 mv, to observe, while in the case of the T2 0 model the voltage ranges from 40 mv to 4 V. If we detect a significant voltage variation of decadal time scales correlating with main geomagnetic field and LOD variations, our result suggests it will strongly support the T2 0 model. Furthermore, it may provide information on the heterogeneity in the D layer, because the amplitude of this variation is simply inversely proportional to σ max as shown in Table 6. Other studies on EM coupling phenomena such as LOD variation and nutation phase prefer a thin layer with conductance of 10 8 S or more (Holme, 1998; Buffett et al., 2000). In our study, the efficiency of producing the S3 2 mode from the T 2 0 model was shown to be saturated at a conductance value of 10 8 S(Table 3). Although this study cannot further discuss this coincidence of conductance values suggested by independent studies, it might contain some implication about the conductance of the D layer. For the geomagnetic jerk, we also calculated EM variations in the case of both S1 0 and T 2 0 sources at the

14 112 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Table 8 The calculated intensity of S1 0 at the CMB and at the surfacea σ max (S/m) S1 0 at CMB (nt) S0 1 at surface (nt) a The amplitude of B φ variations on the surface, which are produced by the mode conversion due to heterogeneous D layer and S1 0 magnetic source, is assumed to be 5 nt. The period of the EM variation is 1 year. CMB (Tables 8 and 9). Our calculation predicts that in the case of S1 0 source it is impossible to explain the remarkable variation of the eastward component of the geomagnetic field. On the other hand, in the case of T2 0 source, it was found to be plausible for explaining the eastward component of the magnetic field, especially in Europe (Fig. 6). The calculated values of variations of EM torque and Joule heat are both far below observable levels. This implies that the model is not inconsistent with observations of the LOD variation and surface heat flow, although proof of consistency of the model may not be as straightforward as in the case of the 60-year variation. The electric field observation cannot provide as strong a constraint as in the case of the 60-year variation, because the voltage between Guam and Japan is estimated to be of the order of 10 mv, which is again difficult to observe. The results above suggest that the mode conversion of the T2 0 field at the CMB due to strong lateral heterogeneity only in the D layer may cause the typical spatial patterns observed in short-period geomagnetic variations. Considering the high conductivity of sea water, however, it may be necessary to take not only the D layer heterogeneity but also the effects of land ocean contrast into account. We can predict that the ocean effect will hardly influence the magnetic field, because the depth of the ocean is only several kilometers, which is much smaller than the induction scale length. On the other hand, because the charges at the boundary of the conductivity contrast distorts the electric field, the electric field will be possibly modified at the ocean land interface. Using the distribution of the ocean taken from ETOPO5 and assuming a value for the conductivity of the ocean of 3 S/m, we calculated the ocean effect with the T2 0 source variation field at the CMB for a period of 60 years. Comparing the results shown in Figs. 8 and 9 with and without land ocean contrast, a significant difference can be seen only in the resulting electric field, while little difference can be noticed in the magnetic field components. Tables 10 and 11 show the voltage between Guam and Japan calculated using models without and with ocean effect, respectively. Regardless of σ max, the values of cable voltage with the ocean effect are smaller by a factor of 2.8 than those without the ocean effect. This means we can consider only the heterogeneity of D layer, but including surface heterogeneity is not essential. An ocean effect appears only in the electric field, which can be evaluated simply by multiplying the observed voltage by this constant value. Table 9 The amplitude of B φ variations on the surface, which are produced by the mode conversion due to the heterogeneous D and T2 0 magnetic source, is assumed to be 5 nt a σ max (S/m) T 0 2 at CMB (nt) EM torque (Nm) Joule heat (W) Voltage (mv) a The period of the EM variation is 1 year.

15 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Fig. 8. The characteristic distribution of EM variations in the case of a toroidal magnetic source (T2 0 ) with a period of 60 years when land ocean contrast is included.

16 114 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Fig. 9. The characteristic distribution of EM variations in the case of a toroidal magnetic source (T2 0 ) with a period of 60 years when land ocean contrast is ignored.

17 T. Koyama et al. / Physics of the Earth and Planetary Interiors 129 (2002) Table 10 The calculated voltages between Guam and Japan with and without land ocean discontinuity a σ max (S/m) Amplitude without ocean (mv) Amplitude with ocean (mv) Ratio (without/with) a The source field is T2 0 on the CMB. The period of the EM variation is 60 years. The amplitude of S3 2 magnetic variations on the surface is assumed to be 50 nt. Table 11 The calculated phase difference between the voltage and the S3 2 magnetic field at the surface with and without land ocean discontinuity a σ max (S/m) Phase without ocean ( ) Phase with ocean ( ) a The source field is the same as shown in Table Conclusions In this paper, we examine the possible effects of lateral conductivity heterogeneity in the D layer on EM field fluctuations of a core origin to account for typical spatial features of rapid geomagnetic variations such as the geomagnetic jerk and geomagnetic 60-year variations. For that purpose, a 3-D EM induction equation was solved in the spherical coordinates by an integral equation approach. During modeling, we assumed a 3-D heterogeneous conductivity structure only in the D layer, with its spatial distribution being of the P2 2 type as inferred from seismic tomographic work. It was found that conversion from the originally toroidal magnetic mode to poloidal modes may be effective for generating a typical spatial distribution of geomagnetic field variations on the Earth s surface, although mode conversion of the originally poloidal mode into poloidal modes of higher degrees will not make a significant contribution. For the geomagnetic 60-year variation, for example, our calculation indicated that the T2 0 variation at the CMB with an amplitude of about 50,000 nt can produce a S3 2 variation at the surface with an amplitude of 50 nt. Furthermore, it was shown that the model can predict other geophysical observables such as length-of-day fluctuations and trans-oceanic cable voltage variations of decadal time scales. The result indicates that the LOD amplitude is of the same order of magnitude as the observation, and the cable voltages are expected to be detectable. Consequently, we can conclude that the consistency of the model can be tested using these observations. Similar calculations were carried out for the geomagnetic jerk. The results show that, although the consistency of the model cannot be tested because parameters from other geophysical observations are predicted to be too small to detect, the same T2 0 model may be able to explain a particular feature of the jerk, which is often remarkable in its east west components in Europe. Acknowledgements We thank A.V. Kuvshinov, O.V. Pankratov and D.B. Avdeev for explaining MIDM. M. Uyeshima has kindly provided us with the numerical results of SGFD in fourth figure. We also thank R.N. Edwards for giving us useful comments. We are very grateful to R. Holme and one anonymous reviewer for their constructive reviews. References Alexandrescu, M.M., Gibert, D., Hulot, G., Le Mouël, J.-L., Saracco, G., Detection of geomagnetic jerks using wavelet analysis. J. Geophys. Res. 100 (7), Alexandrescu, M.M., Gibert, D., Le Mouël, J.-L., Hulot, G., Saracco, G., An estimate of average lower mantle conductivity by wavelet analysis of geomagnetic jerks. J. Geophys. Res. 104 (8), Alldredge, L.R., A discussion of impulses and jerks in the geomagnetic field. J. Geophys. Res. 89 (6), Alldredge, L.R., Localized sudden changes in the geomagnetic secular variation. J. Geomag. Geoelectr. 39,

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