Technical Brief Volume 11, Number 11 20 November 2010 Q11010, doi:10.1029/2010GC003283 ISSN: 1525‐2027
Triangulated finite difference methods for global‐scale electromagnetic induction simulations of whole mantle electrical heterogeneity C. J. Weiss Department of Geosciences, Virginia Polytechnic Institute and State University, 4044 Derring Hall (0420), Blacksburg, Virginia 24061, USA (
[email protected]) [1] Long‐period, global‐scale electromagnetic induction data have long been recognized as containing potentially useful information for constraining our understanding of the dynamics and physiochemical state of Earth’s deep interior. A key component to realizing the potential of this data set is the ability to compute the induction response of a fully 3‐D Earth mantle in global spherical geometries. In this contribution a novel finite difference method is described which preserves many algorithmic advantages of staggered grid discretizations in Cartesian computational electromagnetics but is generalized for the sphere by introducing a hybrid, tensor product mesh consisting of concentrically nested, triangulated, spherical shells topologically connected node‐for‐node in the radial direction. Such a discretization allows for a flexible distribution of mesh nodes in the lateral sense and avoids the problems associated with excessive node density near the poles which results from conformally mapping the standard Cartesian mesh. Modified finite difference templates for spherical geometries are derived and presented, as is a comparison with integral equation solutions for a simple double‐hemisphere test model. As a first step toward applying the algorithm for mantle induction studies, results of a conservative, thermally activated mantle heterogeneity model confirm that mantle structure remains observable in the presence of the strongly conductive ocean and may manifest important, diagnostic signatures in the orientation of the horizontal component of induced magnetic field. Components: 8100 words, 9 figures, 2 tables. Keywords: geomagnetic induction; numerical modeling; mantle conductivity. Index Terms: 1515 Geomagnetism and Paleomagnetism: Geomagnetic induction; 0545 Computational Geophysics: Modeling (1952, 4255); 1025 Geochemistry: Composition of the mantle. Received 1 July 2010; Revised 13 September 2010; Accepted 17 September 2010; Published 20 November 2010. Weiss, C. J. (2010), Triangulated finite difference methods for global‐scale electromagnetic induction simulations of whole mantle electrical heterogeneity, Geochem. Geophys. Geosyst., 11, Q11010, doi:10.1029/2010GC003283.
1. Introduction [2] Constraining our understanding of the physio-
chemical state of Earth’s deep interior has been and continues to be an area of active geoscience research that impacts a broad spectrum of fundamental questions concerning the nature of planetary evolution
Copyright 2010 by the American Geophysical Union
and the interaction between mantle and surficial processes. For example, the presence of hydrous minerals in the transition zone has recently been inferred from elevated electrical conductivity values recovered through inversion of long‐period geomagnetic transfer functions [Koyama, 2001; Kelbert et al., 2009; Utada et al., 2009; Shimizu et al.,
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2010], a finding that, if true, provides a valuable insight into the role of water in the global geodynamic cycle. Alternatively, the anticorrelation of seismic p and s mantle tomograms provides evidence for strong thermochemical heterogeneity and supports the hypotheses of “thermochemical piles” [Tackley, 2002; Jellinek and Manga, 2004; McNamara and Zhong, 2004, 2005], metastable superplumes [Tan and Gurnis, 2005] and thermochemical superplumes [Davaille, 1999] at the base of the lower mantle. In these and similar less recent examples in the global geophysics literature, the geodynamic implications of such inferences are bolstered by corroborating geophysical evidence from methods sensitive to independent, but complementary, physical properties of mantle materials (e.g., electrical conductivity, seismic wave speed, attenuation, density, and anisotropic fabric). Hence, because of the sensitivity of electrical conductivity to compositional variability, thermal structure, and the presence of hydrous minerals, global‐scale electromagnetic induction studies of the mantle remain an important tool for resolving mantle structure. Key to the successful application of electromagnetic methods on the global scale is a scheme for solving Maxwell’s equations in spherical geometries which strikes an acceptable compromise of usability, efficiency, speed, and accuracy. In this manuscript, a new electromagnetic induction simulator is presented whose underlying discretization addresses a principal shortcoming of previously published finite difference methods in spherical geometries. [3] Four main approaches for solving the time harmonic 3‐D Maxwell equations on a sphere appear in the geophysical literature with respect to global geomagnetic sounding: integral equation, spectral, finite element, and finite difference. Integral equation methods [Kuvshinov and Pankratov, 1994; Koyama, 2001; Koyama et al., 2006; Kuvshinov, 2008] are well known to be fast, and for simple geometries, result in a small, dense system of linear equations to be solved. Increasing model complexity increases the size of the resulting linear system. Spectral methods have the advantage of an economy of global basis function for representing smooth model responses [Tarits, 1994; Martinec, 1999; Grammatica and Tarits, 2002; Plotkin, 2004], however sharp jumps (such as at the ocean/continent interface) can be problematic. Finite element methods [Everett and Schultz, 1995, 1996; Yoshimura and Oshiman, 2002] are attractive because of their natural geometric adaptivity to the spherical domain.
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However, the inherent discontinuity of electric fields normal to conductivity boundaries in the model requires finite element parametrization in terms of either continuous potentials or edge elements [Whitney, 1957; Nedelec, 1980; Bossavit, 1988; Bíró and Preis, 1990; Bíró, 1999]. Either way, the apparent advantage these two strategies may have for finite elements is immediately compromised by loss of numerical accuracy in differentiating potentials in order to compute the fields, or in the added algorithmic complexity of the edge elements themselves. Hence, there appears to be a conservation principle at work, the conservation of evil, in which advantages in one part of the algorithm result in disadvantages elsewhere. Lastly, it is worth noting that these advances in modeling the spatial distribution of monochromatic fields has led to the recent development of a handful of time domain methods for the global problem, particularly well suited for spatiotemporal modeling of magnetic fields at satellite altitudes. These includes both semi‐ implicit [Hamano, 2002; Velímský and Martinec, 2005] and frequency transform [Kuvshinov et al., 2006] methods. [4] Finite difference methods that discretize Maxwell’s equations on a 3‐D staggered Cartesian grid have been a popular and powerful approach since their introduction over 40 years ago [Yee, 1966]. In part, their popularity is rooted in the simple 13‐point finite difference templates which arise from the staggered grid discretization where the degrees of freedom are isolated components of electric or magnetic field centered on, and parallel to, the edges connecting nodes of the Cartesian grid. On the dual (or Voronoi) grid lie the components of the magnetic or electric field. Components of the two fields are therefore not collocated and, hence, “staggered” in space. For time domain calculations, each of these field quantities are computed at alternating time steps, but in the case of frequency domain (or time harmonic) sources, the ultimate finite difference system of equations is given entirely in terms of a single field quantity: either electric E or magnetic H field. For the global geomagnetic induction problem considered here, we concern ourselves with the time harmonic sources of magnetospheric origin and formulate our problem entirely in terms of the observable of interest, H, for eventual comparison with magnetic field measurements at the global observatory network as well as those from satellite platforms. [5] However, translation of a Cartesian grid to a conformal, fully spherical geometry, where x y and
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z coordinates are mapped onto latitude, longitude, and depth coordinates [Uyeshima and Schultz, 2000], introduces at least two unattractive features to the spherical grid, in spite of the relative straightforwardness of the coordinate transformation. First of these is the strong geographical bias to the node distribution with the greatest concentration of grid nodes poleward of ±50° latitude where there are few observatories. Sufficiently fine node spacing at mid latitudes (where the observatories are) requires excessively fine discretization at high latitudes. Second of these two features is the special position, algorithmically speaking, held by the pole nodes since they are topologically connected to many nodes (those which immediately circumscribe the poles) when compared to other nodes on the mesh which are connected only to their immediate north/ south/east/west/up/down neighbors. Hence, the finite difference equation for the pole nodes is a complex, dense expression conveying the coupling relation for all nodes immediately adjacent to the poles in comparison to the efficient and sparse equation that arises for nodes elsewhere in the mesh.
netic field h in and around a conducting Earth excited by time harmonic magnetic source field of the form rY. Values of the DoFs are the discrete solution to the partial differential equation
[6] The present work addresses these deficiencies
ð2Þ
by introducing a finite difference discretization in which the lateral node distribution over a given radius in the spherical grid is more or less uniform, thus eliminating the spatial aliasing of the warped Cartesian grid, as well as eliminating any topological significance to the pole nodes. In summary, the grid is composed of nested spherical shells which themselves are triangulated (in this case) to produce a regular node spacing. Shells are connected topologically, node for node, in the radial direction to build full 3‐D spherical mesh. With this discretization, it is shown how the integral representation of the Maxwell equations, from which the Cartesian finite difference system of linear equations is derived [Mackie et al., 1994], also can be applied to this new mesh topology, resulting in a linear system of spherically conformal finite differences. Numerical examples are given which validate the results against those obtained from an 3‐D integral equation code [Kuvshinov and Pankratov, 1994; Kuvshinov, 2008]. Lastly, results of a 3‐D simulation are presented showing the effect of thermally induced conductivity heterogeneity on the global electromagnetic response.
2. Methodology [7] Specifically, we choose as the degrees of freedom (DoF) the scattered, frequency domain mag-
1 r r h þ i!0 h ¼ i!rY
ð1Þ
iwt where p ffiffiffiffiffiffiffi an e time dependence is imposed with i = 1, spatially variable electrical conductivity is denoted by s, and magnetic permeability is assumed constant, taking the value of free space m0 = 4p × 10−7 H/m. The spatial domain on which a solution is sought is defined as extending from just inside the core‐mantle boundary to several Earth radii, a distance sufficiently far such that fields due to induction in the Earth have decayed approximately to zero. The source field is defined in terms of m a finite set of Gauss coefficients {gm n , hn } in the generalized spherical harmonic expansion:
Yðr; ; Þ ¼
1 X n n X m r gnm cos m þ hm n sin m Pn ðcosÞ; a n¼1 m¼0
with a = 6371 km as the normalization constant, independent variables r, and the coordinates in spherical polar coordinates, and Pnm the associated Legendre function of the first kind. Because h is the scattered magnetic field due to induction in the electrically conducting Earth, the total magnetic induction vector is given by the sum m0h + rY. [8] For simplicity, the numerical mesh used for the calculations presented here is based on successive sub‐4 triangular subrefinements of the facets on a base octahedron (Figure 1). Such a mesh generation scheme is not a requirement, and there is nothing inherent in the finite difference algorithm that uses this mesh which prohibits localized mesh refinement. Rather, the octahedral‐based mesh is simple to implement and has the attractive property that each of the nodes in the mesh is topologically connected to an even number of lateral neighbor nodes. The consequence of this mesh topology is that the orientations of magnetic field DoFs along triangle edges are geometrically compatible, representing a consistent circulation around the perimeter of each triangle. To complete the construction of the fully 3‐D mesh, nodes of the triangular shell are projected radially onto a set of discrete radii r H i (Figure 2), with the innermost shell located just below the core‐mantle boundary, and the outermost at a distance of 3–10 Earth radii.
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of the primary mesh resulting from these radially stitched spherical shells are thus triangular prisms, the end caps of which are taken to be spherical polygons to minimize faceting errors. Given this definition of the primary mesh with nodes located at radii r Ni , a dual Voronoi mesh follows, consisting (again) of spherical right prisms with vertices at radii r Vi . For the case of the Voronoi mesh, the end caps of these prisms are arbitrary spherical polygons that depend on the lateral connectivity of those nodes in the primary mesh. Choosing a primary mesh based on the successive sub‐4 refinement of the octahedron, as was done here, results in each Voronoi cells being capped by spherical hexagons, except for those cells centered directly on the 8 cardinal directions, which are capped by the seemingly paradoxical “spherical square.” Figure 1. Illustration of the sub‐4 refinement procedure for the lateral triangulation of grid nodes. Shown in red are the node positions prior to the spacing equilibration procedure described in the text.
Along radial edges connecting corresponding nodes of adjacent spherical shells are the final DoFs in the discretization, the radial component of magnetic field, which is taken as positive outward and located at the midpoint rVi between shells (Figure 2). Cells
[9] Choosing the prisms of Voronoi as cells of constant electrical conductivity, the discretization described above yields magnetic field DoFs that are normal to the facets of the cell (Figure 3). Furthermore, circulation of magnetic field between neighboring cells yields the component of electric field along the edges of the Voronoi cell, which themselves are tangent to any conductivity jumps. Thus, the mesh avoids the implicit use of electric field components orthogonal to and on the boundary between conductivity jumps, such as the interface
Figure 2. Schematic diagram showing the relationship among radii at which lateral triangulations of the primary V grid are defined, r H i ; radii at which nodes on the dual grid are defined, ri ; and major conductivity boundaries such as the air‐Earth interface (AEI) and core‐mantle boundary (CMB). 4 of 15
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Figure 3. Sketch of a dual‐grid (Voronoi) cell and nodes of primary grid between which are shown the orientation of the edge‐based DoFs in finite difference discretization. Half integer indices on radii r correspond to dual‐grid radii rV in Figure 2.
between the resistive air region and the conducting Earth, locations precisely where that component of electric field is undefined.
defining edge length and area by ‘ and D, respectively, integration of Ampère’s law for Ohmic conduction results in the following expressions:
[10] Use of this grid for discretization of the gov-
erning Maxwell equations follows analogously from earlier work on orthogonal Cartesian grids in computational electromagnetics [Yee, 1966] and its application to electromagnetic geophysics [Mackie et al., 1994; Newman and Alumbaugh, 1995; Weiss and Newman, 2002; Weiss and Constable, 2006]. In each of these earlier efforts, successive application of Ampère’s and Faraday’s law around the facets of the primary mesh were used to derive a set of difference equations corresponding to each DoF. Because of the topological symmetry of the mesh, once this procedure was completed for a DoF in one Cartesian direction, the resulting template could be rotated and applied to the remaining two. In contrast, use of the spherical triangular mesh described here requires two distinct templates: one for the radial DoFs; and, another for the tangential DoFs. [11] Consider first the facets of the primary grid
emanating from a given tangential DoF (Figure 4). Using a local enumeration scheme for the DoFs hi {i = 0, 1,…, 11} along the edges of the facet and
h1 ‘1 þ h2 ‘2 þ h3 ‘3 a ea Da ;
ð3Þ
h1 ‘1 þ h6 ‘6 h7 ‘7 h8 ‘8 b eb Db ;
ð4Þ
h1 ‘1 þ h4 ‘4 þ h5 ‘5 c ec Dc
ð5Þ
h1 ‘1 h9 ‘9 h10 ‘10 þ h11 ‘11 d ed Dd ;
ð6Þ
and
where s is the electrical conductivity at the facet centroid and e is the electric field, a temporary variable that does not appear in the final system of linear equations coupling the various DoFs on the mesh. Likewise, integration of Faraday’s law along the path a‐b‐c‐d gives ea ‘a þ eb ‘b ec ‘c ed ‘e i!ð0 h1 D1 þ ^t 1 rYÞ;
ð7Þ
where ^t 1 is a unit vector in the “1” direction at the center of the edge. Note that along each segment of this path a–d, the electrical conductivity is constant 5 of 15
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Figure 4. Local enumeration and orientation of DoFs used to construct the discrete form, equation (8), for magnetic field components centered along the lateral edges of primary grid. Path a‐b‐c‐d is the circulation of electric field which results in the 11‐point finite difference stencil for primary edge 1. Note from Figure 3 that the Voronoi cells of constant conductivity are centered on the endpoints of edges where H DoFs are defined.
and takes the value of the volume weighted average of the 3 (for triangular facets) or 4 (for quadrilateral) Voronoi cells which share these common edges (see Figure 3). [12] Taking these approximations to be exact, sub-
stitution of equations (3)–(6) into equation (7) results in the generalized 11‐point template for all tangential edges, orthogonal to the radial direction ^r, ! X ‘1 ‘i ‘a ‘2 ‘a ‘3 ‘c ‘4 þ i!0 D1 h1 þ h2 þ h3 þ h4 D D D i i a a a a c Dc i¼a;...;d ‘c ‘5 ‘b ‘6 ‘b ‘7 ‘b ‘8 ‘d ‘9 h5 þ h6 h7 h8 h9 c Dc b Db b D b b Db d Dd ‘d ‘10 ‘d ‘11 h10 þ h11 ¼ i!D1^t 1 rY: ð8Þ d Dd d Dd þ
and Faraday’s law around the facets connected to a given radial DoF (Figure 5) as was done for equations (3)–(7), the corresponding 19‐point stencil for radial DoFs is
‘a ‘2 ‘b ‘3 ‘c ‘4 ‘d ‘5 ‘e ‘6 h2 þ h3 h4 þ h5 h6 a Da b D b c Dc d D d e De ‘f ‘7 ‘a ‘8 ‘b ‘9 ‘c ‘10 ‘d ‘11 þ h7 h8 h9 h10 h11 f Df a Da b Db c Dc d Dd ‘f ‘13 ‘e ‘12 ‘a ‘14 ‘b ‘15 h12 h13 þ h14 h15 e De f D f a Da b Db ‘f ‘19 ‘c ‘16 ‘d ‘17 ‘e ‘18 þ h16 h17 þ h18 h19 c Dc d Dd e De f Df ! X ‘1 ‘i þ þ i!0 D1 h1 ¼ i!D1^t 1 rY: ð9Þ D i¼a;...;f i i
[14] Multiplication of these equations by their
[13] With the basic procedure for generating the
finite difference templates now established, treatment of the radially directed DoFs is straightforward. Following the same intermediate steps of approximating the integral form of Ampère’s
respective values of ‘1 and mapping the local enumeration to a global one yields the complete finite difference system of linear equations in terms of the unknown DoFs, h, where a homogeneous Dirichlet boundary condition is applied to DoFs on inner and 6 of 15
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Figure 5. Local enumeration and orientation of DoFs used to construct the discrete form, equation (9), for magnetic field components centered along the radial edges of primary grid. Path a‐b‐c‐d‐e‐f is the circulation of electric field which results in the 19‐point finite difference stencil for primary edge 1. Note from Figure 3 that the Voronoi cells of constant conductivity are centered on the endpoints of edges where H DoFs are defined.
outer boundaries of the mesh. Observe that the tangential stencil in equation (8) holds for all tangential degrees of freedom in the mesh. However, in the implementation exercised here, where the 3‐D grid is based on a primal octahedron, radial nodes along each of the 8 cardinal directions are topologically connected to only 4 lateral nodes, not 6 like all the others. Hence, the stencil has only 4 “flaps” defined over 13 DoFs. For these special nodes in this particular mesh, the sum in equation (9) only extends over hi on i = 1, …, 13 and ‘x,sx,Dx on x = a, …, d. Note that equations (8) and (9) require a finite conductivity value in the AIR region for the quotients to remain bounded. For the calculations presented here, this value is fixed at 10−6 S/m. The system of linear equations is solved iteratively with Jacobi preconditioning using the quasi‐minimal residual method (QMR) [Freund and Nachtigal, 1994], a strategy that has previously been shown to be reasonably efficient in solving large, sparse, complex‐symmetric linear systems arising from finite difference approximations to Maxwell’s equations
in geophysics [Mackie et al., 1994; Newman and Alumbaugh, 1995; Weiss and Constable, 2006].
3. Validation [15] To assess whether the finite difference system
of linear equations described above, modified for spherical geometries consisting of concentrically nested and triangulated spherical shells, obeys the requisite physics, results from an azimuthally symmetric test problem are compared here against those derived independently by integral equation methods [Kuvshinov and Pankratov, 1994; Kuvshinov, 2008]. The model consists of a thin spherical sheet in the upper mantle whose total conductance in the Northern Hemisphere is 40000 S, and in the Southern Hemisphere is 2000 S (Figure 6). Upper and lower mantle conductivities 0.0025 S/m and 2.5 S/m, respectively, are within the range of values inferred from long‐period geomagnetic depth sounding experiments [Schultz and Larsen, 1987; Kuvshinov
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introduced by the inherently nonuniform node spacing from triangular subrefinement on a sphere, an ad hoc node redistribution scheme was used that shifted nodes laterally toward the center of their corresponding octahedral facet based on the inverse of their distance from its center (Figure 1). The effect of this node equilibration scheme is to effectively decrease by a factor of 2 (Figure 6) the range of lateral node spacings in the mesh and hence, their corresponding effect on the spatial distribution numerical error throughout the mesh. [17] Numerical results from the finite difference
Figure 6. (left) Sketch of the double‐hemisphere model used to validate the finite difference code. Radii of the triangulated shells are shown by the dots with (bottom right) a zoomed view. (top right) Histogram of node spacings demonstrates the effect of equilibration procedure on the model used for these tests.
and Olsen, 2006; Kelbert et al., 2009] A zonal source term g01 = 1 nT at excitation period T = 1/2p w = 1 d is used to simulate the effects of a long‐period magnetospheric ring current excitation. Although no comprehensive suite of test models has yet been proposed or explored for intercomparison of global induction forward solvers, the double hemisphere model analyzed here has emerged as the most consistent measure of intercode compatibility [Everett and Schultz, 1996; Yoshimura and Oshiman, 2002; Uyeshima and Schultz, 2000]. [16] The computational mesh for these numerical
results consists of 20 layers in the air region and 30 layers in the Earth region (Figure 6). To ensure that node spacing is small where the fields change most rapidly but not overly so where they are not, nodes in the air region are distributed geometrically with a 10% increase in radial node spacing as a function of distance from the air/Earth interface, terminating at a distance of 2.5 Earth radii. Radial node spacing in the uppermost 800 km of the mantle is ∼35.3 km, but then also increases with depth at a 10% growth rate, terminating at roughly 300 km at the core‐mantle boundary. Seven levels of sub‐4 triangular refinement were used to generate a mesh with a mean lateral node spacing of approximately 0.85° (95 km) and approximately 13.8 million DoFs. To minimize numerical errors
solution on air/Earth interface (AEI) show generally favorable agreement with those obtained via integral equation methods (Figure 7). To verify the azimuthal symmetry of the finite difference solution, field components from all nodes on the AEI are plotted as a function of colatitude. Hence, any azimuthal variation due to numerical error would manifest as scatter for a given colatitude. Note that because of the staggered grid used for the finite difference formulation (Figures 2 and 3), the radial component of magnetic induction (Z) is located directly on AEI and shows the best agreement with the integral equation solution. Estimation of the northward field component (X) at the AEI is done by simple linear interpolation between DoFs computed on the two spherical shells that straddle the AEI.
4. Example Application: Global Induction Response to Thermally Activated Lateral Heterogeneity [18] To put lower bounds on the magnitude of the
induced magnetic field observable at Earth’s surface due to mantle heterogeneity, a reasonable model of mantle heterogeneity must first be proposed. Although temperature clearly plays a role in the electrical conductivity of mantle minerals, the most striking electrical heterogeneity arises from chemical heterogeneity through, for example, the presence of hydrogen in the transition zone [Huang et al., 2005; Yoshino et al., 2008; Kelbert et al., 2009; Karato and Dai, 2009], iron enrichment in ringwoodite [Yoshino and Katsura, 2009], and pressure‐induced iron spin state transition that results in metallic conduction [Shim et al., 2009]. In the lowermost mantle, chemical heterogeneity associated with postperovskite and/or accumulated basaltic materials is expected to be the dominant source of lateral chemical heterogeneity [Shim et al., 2009]. 8 of 15
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Figure 7. Comparison of magnetic field components from finite difference (symbols) and integral equation (lines) solutions for the double hemisphere model in Figure 6.
4.1. Model Construction [19] As a conservative initial step toward predic-
tive modeling and hypothesis‐driven numerical experiments on the geophysical signature of candidate mantle processes, a 3‐D model is constructed based on spatially correlated electrical and seismic wave speed structure. Clearly, creating a set of self‐consistent geophysical and geodynamic model parameters is an important step and deserves careful consideration. A number of studies have already addressed the issue of mapping of seismic wave speed perturbations to temperature perturbations [cf. Kellogg et al., 1999; Goes et al., 2004; Samuel et al., 2005; Wüllner et al., 2006]. Following King and Masters [1992] we start by noting variations in shear wave speed vs are related to changes in bulk modulus G and density r by the expression, @ ln vs 1 ¼ @ ln 2
@ ln G 1 : @ ln
ð10Þ
Variations in @@ lnln G from 5.5 for garnets to 7.9 for SrTiO3 [Anderson, 1988] yield a range of values for @@ lnlnvs from 2.25 to 3.4, with a value of 2.5 chosen here for simplicity. Density perturbations dr are scaled to temperature dT following dr = roadT, where a is assumed to be 3.0 × 105 and r o is 4.0 × 103 kg/m3 [e.g., Turcotte and Schubert, 1982]. In turn, the temperature variations at a given depth d are mapped to conductivity by the low‐order expansion ðT þ T Þ ðT Þ þ T
@ : @T
ð11Þ
Assuming an Arrhenius equation as the functional form between conductivity, temperature and pressure P U ðd Þ þ Pðd ÞV ðd Þ ðd Þ ¼ exp s0 ðd Þ kB T ðd Þ
ð12Þ
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Table 1. Effective Arrhenius Parameters Estimated From Xu et al.’s [2000] 1‐D Conductivity Profile Based on Effective Medium Theory Applied to Laboratory‐Derived Conductivity Dataa Depth, d (km)
s0 (log10 S/m)
dU (eV)
dV (cm3/mol)
