FINITE ELEMENT MODELING FOR SWARM ELECTROMAGNETIC INDUCTION STUDIES Joseph Ribaudo, Catherine Constable, Robert Parker Institute for Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California San Diego 9500 Gilman Dr., La Jolla, CA 92093-0225
[email protected] ABSTRACT Magnetic field data from the Magsat, Ørsted, and CHAMP satellites have been successfully used for global electromagnetic induction studies to recover information about one-dimensional electrical conductivity variations in Earth’s mantle. The induction field may be isolated in satellite magnetic data by subtracting the influence of other field components with the aid of a comprehensive global geomagnetic model like CM4 [1] or CHAOS [2]. Once isolated, the induction field data is usually separated geometrically into its primary and secondary (or external and internal, respectively) components using simplifying assumptions about the spatial structure of both external and internal fields. In the frequency domain, this separation is used to define electromagnetic response functions that can be inverted for radial electrical conductivity estimates. We present recent progress in forward modeling of the global electromagnetic induction problem with non-uniform primary field structure, 3D Earth conductivity, and rotation. The modeling is performed in either the time or frequency domain via a commercially-available, general-purpose, finite element modeling package called FlexPDE, and has been validated against known solutions to 3D steady state and time-dependent problems. The induction problem is formulated in terms of the magnetic vector potential and electric scalar potential, and mesh density is managed both explicitly and through adaptive mesh refinement. The modeling routine allows for arbitrary internal conductivity and time-varying primary field structure. Simple asymmetric time-varying external field structures have been tested, and show an inductive response that depends on local-time for a rotating Earth. Upon the collection of a long time-series of synthetic magnetic field data, the anticipated effects on Weidelt’s c-response [3] will be evaluated, allowing for the characterization of and possible correction for local-time bias in satellite estimates of c that results from Earth’s rotation through an asymmetric primary field. Earth’s most prominent 3D electrical conductivity structure is the global ocean, but its parameterization within the finite element mesh is too computationally intensive, so we adopt a discontinuous spatially varying magnetic boundary condition to model its conductance. This can be validated by comparison to solutions from existing integral equation methods such as those in [4]. INTRODUCTION As charged particles from the solar wind flow around Earth, they compose the magnetospheric ring current and create an externally-generated primary component, Bp , of Earth’s magnetic
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Figure 1: The 3D solution domain. Earth is modeled as a conductive sphere, while infinite space is modeled as a surrounding sphere with zero conductivity and a radius 10 times that of Earth. The externally-sourced primary magnetic field Bp is zdirected throughout the solution domain. The internally-sourced secondary field Bs is generated by induced electrical currents in the conducting Earth, and is fully attenuated at the boundary of modeled space. field. Variations in the magnitude of this primary field induce currents in the interior of Earth, thus producing Bs , an internally-generated, secondary contribution to the geomagnetic field. The vector sum of these two fields is known as the geomagnetic induction field B. If both primary and secondary fields are known, the electrical conductivity of Earth’s mantle may be investigated. Our work focuses on developing numerical tools for the accurate analysis of magnetic field data obtained from satellite instruments, and applying these tools to the study of the electrical properties of Earth’s mantle. The induction field may be isolated in satellite magnetic data by subtracting the influence of other field components with the aid of a comprehensive global geomagnetic model (e.g. CM4 [1], which is based on a combination of satellite and observatory data). Once isolated, the induction field data may be separated geometrically into its primary and secondary components. In the frequency domain, this separation can be used to define electromagnetic response functions that can be inverted for radial electrical conductivity estimates. Several estimates of the so-called c-response, like those of [5] and [6], have been produced in this way. However, it has been shown that global response function estimates from these data are systematically dependent on the local time of the longitude where the satellite measurement was performed [7]. This bias makes interpretation of empirical c-responses very difficult, and is
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Figure 2: A snapshot of the magnetic induction field in Earth’s vicinity as calculated by FlexPDE. In this simulation, Earth is modeled with a 0.01 S/m upper mantle, a 0.1 S/m transition zone, a 1 S/m lower mantle, and a 5 × 105 S/m core. The external field representing the field from the ring current is spatially uniform and sinusoidal in time with a frequency of ω = 3 × 10−8 rad/s, and is shown on the left. In the middle is the internal field, which represents Earth’s inductive response to changes in the external field. On the right is the their sum, the total induction field. White lines represent magnetic field lines, and colors represent vertical-field magnitudes, with blue indicating a strong upwardly-directed field, red indicates a strong downward field, and green indicates a field that is either small in magnitude or horizontally-directed. thought to be caused by asymmetry in Earth’s primary field. New techniques must be invented to account for this asymmetry in the data inversion, but before this can be done, the response of Earth to a non-axisymmetric driving field must be modeled and understood. That is, we must be able to reproduce with numerical simulations the local time bias demonstrated by [7]. The numerical simulations are carried out via a commercially available, general-purpose FEM software program called FlexPDE, which allows the user to design and implement detailed simulations in a scripting environment, leaving the mechanics of the modeling routine to be managed by the program, and allowing for great flexibility in modeling. By simply adapting the script, we can convert an existing problem into any number of others, without extensive and time-consuming code modification. FlexPDE also features adaptive mesh refinesment (AMR), and time-stepping capabilities. The latter allows induction simulations to be carried out in the time domain as well as in the frequency domain, which is especially important for this project because the movements of satellites and the rotation of the Earth make operating in the frequency domain less practical. GOVERNING EQUATIONS The geomagnetic induction field is governed by the Pre-Maxwell equations and Ohm’s law. These can be combined to give the pair of linked equations ∇ × ∇ × As (t) − µ0 σ∂t A(t) − µ0 σ∇V (t) = 0,
(1)
and ∇ · (σ∂t A + σ∇V ) = 0,
(2)
where A = Ap + As is the magnetic vector potential, V is the electric scalar potential, µ is the magnetic permeability, which we assume to equal that of vacuum, and σ is electrical conductivity. These two equations uniquely determine the potentials in Earth and space when appropriate boundary conditions are specified. The use of potentials for modeling has two main advantages over direct modeling of the fields. First, since B = ∇ × A we can easily ensure that ∇ · B = 0, as required by the pre-Maxwell equations. Furthermore, unlike the fields themselves, all components of the potentials are continuous across conductivity interfaces. The usual approach to geomagnetic induction studies [8] [9] is to separate the induction field into Bp (t) and Bs (t), Fourier transform them, and then estimate the complex frequency-domain electromagnetic response functions e0 (ω) Q(ω) = 01 , (3) i1 (ω) and c(ω) =
1 − 2Q(ω) a, 2(1 + Q(ω))
(4)
where a is the radius of Earth, and i01 and e01 are internal and external dipole coefficients in a spherical harmonic expansion of the fields. The c-response can provide information about the depth of penetration of the external field into the crust or mantle, and knowledge of it over all frequencies is sufficient to recover the 1D conductivity structure of Earth [3]. It is known, however, that Earth’s ring current and primary field are asymmetric [10]. The enhanced magnitude on the night side is believed to be caused by magnetic reconnection effects in Earth’s magnetotail, and is represented mathematically by a significant quadrupole component in the spherical harmonic expansion of the fields. This quadrupole component is neglected in the above equations, and it is suspected that Earth’s twenty-four hour rotation through this nondipolar primary field has a confounding effect on the accurate estimation of c-responses from satellite data. At present there is no known correction for this effect. Providing such a correction is one goal of this project. PRIMARY FIELD MODELS In Earth’s vicinity, the external magnetic dipole caused by the axisymmetric component of the magnetospheric ring current can be approximated by a uniform, z-directed primary field, Bp = B0 (t)ˆ z,
(5)
which can be generated with the vector potential Ap =
B0 (t) (−yˆ x + xˆ y) , 2
(6)
where B0 (t) is a long time-series of primary field magnitudes, which will be derived from some variant of Dst or its external coefficient. The resulting time-series of magnetic fields can be analyzed with existing techniques in order to generate synthetic c-responses.
The addition of a quadrupole component, representing the enhanced magnitude of the external field on the nightside resulting from the non-axisymmetric component of the ring current, can be approximated by a linear increase in primary field strength along the x-axis (defined as perpendicular to z and parallel to the 10 am - 10 pm local time plane): Bp = (B0 + mx)ˆ z,
(7)
where m is the empirically estimated x-derivative of the primary field magnitude. This nonaxisymmetric field is produced by adding a quadratic term in x to the y-component of Ap in (6): 1 Ap = −B0 (t)yˆ x + B0 (t)x + mx2 y ˆ . (8) 2 Note that in this primary field model, while the magnitude of the field itself is not spatially uniform, the time-derivatives of the fields are spatially uniform. In other words, although the primary field is stronger on the nightside, the change in primary field magnitude from one time step to the next is the same everywhere in the model. Since electromagnetic induction is driven by time variation of the magnetic field, we should see no effect of this asymmetric primary field structure in the secondary field for a non-rotating Earth. However, Earth’s rotation through this asymmetric primary field constitutes a time-varying signal in the primary field when measured in a reference frame that rotates with Earth, even if the primary field is not driven by a time-series. Indeed, we have simulated the response of Earth to rotation through just such a field. This z-directed field increased linearly in magnitude from -4 nT on the dayside to +4 nT on the nightside, and did not vary with time, except for rotating with a daily period. The rotation of our simulated Earth through this constant field produced a secondary field of 1 nT at satellite altitude. For computational reasons, it is desirable to rotate the primary field and hold the geographic reference frame still. Thus we simulate Earth rotation by defining the primary field with reference to a reference frame that rotates around the existing z-axis. In future models we will account for the tilt of Earth’s magnetic dipole by aligning the rotating external field with the dipole axis. Additionally, with simple modifications to (8), it is possible to generate primary fields with both magnitudes and time derivatives that are stronger on the nightside. CONDUCTIVITY MODEL Using FlexPDE, we model this system in a 3D cartesian coordinate system with the origin at the center of the Earth, as shown in Fig. 1. The z-axis coincides with Earth’s geomagnetic pole, and the x- and y-axes define the plane of the geomagnetic equator. Earth is modeled as a conductive sphere of radius a, while infinite space is modeled as a sphere of vanishing conductivity of radius 10a, also centered on the origin. The size of the solution domain was chosen to allow the secondary field to fully attenuate at the surface of the model. This boundary condition is imposed by forcing the secondary potentials to vanish at the boundary as well. The conductivity profile of Earth is motivated by [11], and is given by concentric, discontinuous shells of 5,000 S/m (core), 1 S/m (mantle below 800 km depth), 0.1 S/m (400 - 800 km), and 0.01 S/m (0 - 400 km). Explicitly modeling the 3D conductivity of the crust and oceans within the finite element mesh would be computationally expensive. Instead, we impose a novel boundary condition at the interface between Earth and space. Ampere’s Law tells us that an infinitely thin sheet of current will introduce a discontinuity in the surface-parallel component of the magnetic field that can be written
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Figure 3: Real (positive) and imaginary (negative) simulated and observed c-responses. The open black circles refer to the simulated response of Earth’s mantle, with no crust or oceans. The orange lines refer to a surface conductance of 10 S, which reflects the average vertically integrated conductance of continental interiors. The light blue lines refer to a surface conductance of 1000 S, representing the average for coastal plains. The green lines refer to a surface conductance of 8,000 S, representing the average ocean. Also plotted are empirical estimates from satellite data from [5], [6], and [13].
∆B|| = ∆(∂n A) = µ0 τ n ˆ × E,
(9)
where n ˆ , is the unit vector normal to the current sheet, and τ is the conductance of the surface. By imposing this boundary condition at the Earth-space interface and supplying a map of conductance as a function of latitude and longitude (identical to the one used by [12], we can incorporate the effect of the oceans and continents into the solution process. This technique will produce accurate results for primary fields in the period range of interest (roughly 104 - 107 seconds). The c-responses produced by applying this boundary condition with different uniform surface maps are seen in Fig. 3 and demonstrate that a 1D global ocean produces a visible inductive signal for periods up to and exceeding one day. REFERENCES [1] T.J. Sabaka, N. Olsen, M.E. Purucker, “Extending comprehensive models of the Earth’s
magnetic field with Ørsted and CHAMP data.” Geophys. J. Int., 159, 521–547, 2004. [2] N. Olsen, H. Lhr, T.J. Sabaka, M. Mandea, M. Rother, L. Tffner-Clausen, S. Choi, “CHAOS a model of Earth’s magnetic field derived from CHAMP, rsted, and SAC-C magnetic satellite data.” Geophys. J. Int., 166 6775, 2006. [3] P. Weidelt, “The inverse problem of geomagnetic induction.” Zeit. f¨ ur Geophys., 38, 257–289, 1972. [4] A.V. Kuvshinov, “3-D global induction in the oceans and solid Earth: Recent progress in modeling magnetic and electric fields from sources of magnetospheric, ionospheric and oceanic origin.” Surv. Geophys., 29, 139-186, 2008. [5] S.C. Constable, C.G. Constable, “Observing geomagnetic induction in magnetic satellite measurements and associated implications for mantle conductivity.” Geophys. Geochem. Geosyst., 5, Q01006, 2004. [6] A.V. Kuvshinov, N. Olsen, “A global model of mantle conductivity derived from 5 years of CHAMP, Ørsted, and SAC-C magnetic data.” Geophys. Res. Let., 33, L18301, 2006. [7] G. Balasis, G.D. Egbert, S. Maus, “Local time effects in satellite estimates of electromagnetic induction transfer functions.” Geophys. Res. Lett., 31, L16610, 2004. [8] B. Lahiri, A. Price, “Electromagnetic induction in non-uniform conductors, and the determination of the conductivity of the Earth from terrestrial magnetic variations.” Phil. Trans. R. Soc. Lond., A 237, 509–540, 1939. [9] R.J. Banks, “Geomagnetic variations and the conductivity of the upper mantle.” Geophys. J. R. Astron. Soc., 17, 457–487, 1969. [10] G. Balasis, G.D. Egbert, “Empirical orthogonal function analysis of magnetic observatory data: Further evidence for non-axisymmetric magnetospheric sources for satellite induction studies.” Geophys. Res. Lett., 33, L11311, 2006. [11] A. Medin, R. Parker, S. Constable, “Making sound inferences from geomagnetic sounding.” Phys. Earth Planet. Inter, 160, 51–59, 2007. [12] M.E. Everett, S.C. Constable, C.G. Constable, “Effects of near-surface conductance on global satellite induction responses.” Geophys. J. Int, 153, 277–286, 2003. [13] N. Olsen, “Induction studies with satellite data.” Surv. Geophys., 20, 309–340, 1999.
