Mantle convection modeling of the supercontinent cycle: Introversion

Geoscience Frontiers 5 (2014) 77e81

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Research paper

Mantle convection modeling of the supercontinent cycle: Introversion, extroversion, or a combination? Masaki Yoshida a, *, M. Santosh b, c a Institute for Research on Earth Evolution (IFREE), Japan Agency for MarineeEarth Science and Technology (JAMSTEC), 2-15 Natsushima-cho, Yokosuka, Kanagawa 237-0061, Japan b Division of Interdisciplinary Science, Faculty of Science, Kochi University, Akebono-cho 2-5-1, Kochi 780-8520, Japan c School of Earth Sciences and Resources, China University of Geosciences (Beijing), 29 Xueyuan Road, Beijing 100083, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 22 April 2013 Received in revised form 4 June 2013 Accepted 18 June 2013 Available online 29 June 2013

The periodic assembly and dispersal of continental fragments, referred to as the supercontinent cycle, bear close relation to the evolution of mantle convection and plate tectonics. Supercontinent formation involves complex processes of “introversion” (closure of interior oceans), “extroversion” (closure of exterior oceans), or a combination of these processes in uniting dispersed continental fragments. Recent developments in numerical modeling and advancements in computation techniques enable us to simulate Earth’s mantle convection with drifting continents under realistic convection vigor and rheology in Earth-like geometry (i.e., 3D spherical-shell). We report a numerical simulation of 3D mantle convection, incorporating drifting deformable continents, to evaluate supercontinent processes in a realistic mantle convection regime. Our results show that supercontinents are assembled by a combination of introversion and extroversion processes. Small-scale thermal heterogeneity dominates deep mantle convection during the supercontinent cycle, although large-scale upwelling plumes intermittently originate under the drifting continents and/or the supercontinent. Ó 2013, China University of Geosciences (Beijing) and Peking University. Production and hosting by Elsevier B.V. All rights reserved.

Keywords: Mantle convection Numerical simulation Tectonics Supercontinent cycle Introversion Extroversion

1. Introduction Recognition of the repeated assembly of supercontinents throughout much of Earth history and the profound effects of this cyclic process on Earth’s geologic, climatic and biogeochemical evolution demonstrates the fundamental importance of the socalled “supercontinent cycle” (e.g., Condie, 2011). However, the processes by which supercontinents assemble remains poorly understood. Unlike Wilson’s (1966) proposal involving the repeated opening and closure of individual ocean basins known as “Wilson cycles”, the episodic assembly and disruption of supercontinents involve more complex processes that induce major changes in the

* Corresponding author. Tel.: þ81 46 867 9814; fax: þ81 46 867 9315. E-mail address: [email protected] (M. Yoshida). Peer-review under responsibility of China University of Geosciences (Beijing)

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pattern of mantle convection (e.g., Gurnis, 1988; Phillips and Bunge, 2007). Early studies recognized that the assembly of supercontinents involves either the closure of “interior” oceans created by the breakup of the previous supercontinent (e.g., Worsley et al., 1984), or the closure of the “exterior” ocean that surrounded the supercontinent following its assembly (e.g., Gurnis, 1988). Murphy and Nance (2003) introduced the terms “extroversion” (closure of the exterior ocean) and “introversion” (closure of the interior oceans) to distinguish these modes of supercontinent assembly. Whereas most geodynamic models favor extroversion (e.g., Gurnis, 1988; Hoffman, 1991), the well-studied example of the Pangea supercontinent exemplifies introversion through the closure of the interior Iapetus and Rheic oceans (e.g., Murphy and Nance, 2008; Murphy et al., 2009; Nance et al., 2013). Previous numerical simulations of mantle convection incorporating a rigid, nondeformable supercontinent in 3D spherical geometry suggest that supercontinents trigger large-scale horizontal mantle flow, thereby reorganizing the thermal structure of the mantle interior (Yoshida et al., 1999; Phillips and Bunge, 2005; Yoshida, 2010b). These models also show large-scale upwelling (plumes) originating from the base of the mantle beneath the supercontinent and a thermal structure in the mantle dominated by a

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M. Yoshida, M. Santosh / Geoscience Frontiers 5 (2014) 77e81

spherical harmonic degree of one (i.e., degree-one convection), in which upwelling in one hemisphere is balanced by an antipodal downwelling. The formation of degree-one convection may be closely related to the periodicity of supercontinent cycles (Phillips and Bunge, 2005, 2007). So far, regular periodic supercontinent cycles with continents assembled through extroversion has been observed in most previous 2D and 3D simulation models using rigid, nondeformable continents (Gurnis, 1988; Phillips and Bunge, 2005, 2007). On the other hand, recent numerical studies presented 3D spherical models with deformable continents and self-consistent plate tectonics (Yoshida, 2010a, 2012, 2013; Rolf and Tackley, 2011; Coltice et al., 2012; Rolf et al., 2012). In particular, Rolf et al. (2012) performed a systematic numerical simulation of mantle convection with relatively weakly deformable continents and selfconsistent plate tectonics to investigate the quantitative dynamic feedback between continental drift and the thermal state of subcontinental mantle. More recently, Yoshida (2013) performed a numerical simulation of mantle convection with relatively strongly deformable continents with different types of rheological heterogeneity to evaluate the average subcontinental mantle temperature during a supercontinent cycle. In this study, using the model of Yoshida (2013), we describe a numerical simulation of 3D spherical-shell mantle convection, incorporating drifting deformable continents, to evaluate the process of supercontinent assembly in mantle convection under Earthlike conditions. 2. Model description Mantle convection with drifting continents is computed numerically using a staggered grid-based, particle-in-cell, finitevolume code, ConvGS (e.g., Yoshida, 2008a,b, 2013). The mantle is modeled as a Boussinesq fluid with an infinite Prandtl number confined in 3D spherical-shell geometry with a thickness of 2867 km, and is assumed to be heated from the bottom and from within by the decay of radioactive elements. Impermeable, freeslip, and isothermal conditions are imposed on both the top and bottom surface boundaries of the spherical-shell. The number of computational grids is taken to be 64 (r) 64 (q) 192 (4) 2 (for Yin and Yang grids) (e.g., Yoshida and Kageyama, 2004). Following a preliminary model of Yoshida (2013), the thermal Rayleigh number is fixed at 5.72 106 (corresponding to a reference viscosity of href ¼ 1022 Pa s and a temperature difference across the mantle of DT ¼ 2500 K), and the internal heat production rate is fixed at 3.52 1012 W kg1. The resultant basal heating ratio (i.e., the ratio of the total heat flow at the bottom and the top surfaces of the mantle) is about 50e60%, which is comparable to the real Earth’s mantle (Lay et al., 2008), for two models presented here. One of the major differences between the models of Rolf et al. (2012) and Yoshida (2013) is the heating mode. In the model of Rolf et al. (2012), the bottom boundary is the insulating (i.e., adiabatic) condition. The viscosity of the mantle, h, depends on temperature according to the formula:

h

h ¼ exp½C$lnðDhÞ$max href expfEðT=DT 0:5Þg; hmin

i

(1)

where C is the dimensionless composition field (0 C 1); Dh, the viscosity ratio between continental material and the surrounding mantle (see below); T, the temperature; E, the dimensionless activation energy that controls the viscosity contrast across the mantle; and hmin, the minimum viscosity of the mantle. In the present model, E and hmin are set as ln(106) ¼ 13.82 and 1021 Pa s, respectively. A yield stress (i.e., maximum limit of lithosphere strength),

sy, is applied to the top part of the model to realize plate-like behavior in the highly viscous oceanic lithosphere. For simplicity, a low-viscosity asthenosphere generated by the implementation of solidus temperature (e.g., Tackley, 2000; Rolf et al., 2012) is not considered. Following the preliminary model (Yoshida, 2013), the “initial supercontinent” in the present model is composed of three types of continental material: four pieces of Archean cratonic lithosphere (CL); weak continental margins (WCM) along the sides of the CLs to mimic rheologically weak, Proterozoic mobile belts that surround Archean cratonic lithosphere (e.g., Ring, 1994; Vauchez et al., 1997); and weak breakup zones (WBZ), which mimics preexisting suture zones (e.g., Burke et al., 1977; Dewey, 1977). The supercontinent occupies part of a spherical cap with a uniform initial thickness of 250 km, covering about 39% of the total surface area. The widths of the WCM and WBZ are about 625 km (see Fig. 1 in Yoshida (2013) for details of the initial state of the simulation). The maximum viscosity ratio between CL and the surrounding mantle, DhCL, is fixed at 103 (Yoshida, 2012), and the maximum viscosity ratios between WCM and WBZ, DhWCM and DhWBZ, are free parameters in the present study. The continental materials are more buoyant (i.e., lighter) than the reference mantle and are assumed to exhibit a density contrast of 100 kg m3. The yield stress in the continental materials is set to be infinite, instead of an implementation of WBZs (Yoshida, 2013). In the present simulation, the initial supercontinent is fixed at a location above a region of mantle convection until the thermal/mechanical conditions of this convection reach a statistically steady state. Considering the average viscosity of the Earth’s upper mantle (including a low-viscosity asthenosphere) to be hum ¼ 5 1020 Pa s (e.g., Lambeck and Johnston, 1998) and assuming a relationship where (Rayleigh number) w (fluid velocity)2/3 based on the boundary layer theory (Turcotte and Schubert, 2002), the elapsed time is defined by te h tth∙(href/hum)2/3, where tth is the thermal diffusion time. 3. Results and discussion The preliminary simulation runs revealed that the realization of the supercontinent cycle depends largely on the choice of values for DhWCM and DhWBZ (Yoshida, 2012, 2013). In other words, the actual continental breakup and dispersal for the real Earth may be realized only for a limited set of geophysical conditions, particularly with regard to the rheology of the continental interior. When the yield stress, sy, is set at 200 or 300 MPa, the oceanic lithosphere breaks and the resultant oceanic plates subduct along a large part of the supercontinent margin. When the moderate temperaturedependent viscosity is considered, degree-one convection originates (e.g., Yoshida and Kageyama, 2006), whereas a supercontinent is imposed under this situation, mantle downwellings (i.e., subduction of the oceanic plates) concentrates along half of the supercontinent margin (Yoshida, 2013). In this paper, only the models with sy ¼ 200 MPa are presented because similar phenomena are observed throughout the range sy ¼ 200e300 MPa. The continent starts to drift at an elapsed time of 0 Myr. When DhWCM is relatively high (102) and DhWBZ is relatively low (101, 1, or 10), the four CLs drift freely on the Earth’s surface after the breakup of the initial supercontinent. As a result, stable supercontinent cycles are realized. Fig. 1 illustrates the temporal evolution of drifting continents and mantle convection over 750 Myr for Model Y200A with DhWBZ ¼ 101 and Model Y200B with DhWBZ ¼ 1. In both models presented here, the four CLs gather and form a new supercontinent-like assembly (Fig. 1e and k). The motions of each CL during the supercontinent cycle are quite


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Figure 1. Time sequence of the drifting continents and mantle convection at every 125 Myr for (aef) Model Y200A and (gel) Model Y200B. The left panel in each model shows the time sequence of the positions of four cratonic lithospheres, labeled “CL-A” to “CL-D”. The gray and light gray regions show weak continental margins (WCM) and weak breakup zones (WBZ). The blue region indicates the region of oceanic lithosphere. The white line contour shows the temperature anomaly (i.e., the deviation from the horizontally averaged temperature) of the mantle at a depth of 493 km. The contour intervals are 250 C. The solid and dashed lines represent positive and negative temperature anomalies, respectively. “Initial SC” (a, g) and “Next SC” (e, k) indicate the initial and next supercontinents. The white and blue arrows indicate continental collisions accompanied by introversion and extroversion, respectively. Each right panel shows the time sequence of the 3D view of mantle convection, corresponding to the left panel. The blue and yellow isosurfaces of the temperature anomaly are 100 C and þ100 C, respectively. The positions of the CLs are represented by transparent orange caps. The bottom boundary of the mantle (i.e., the core-mantle boundary) is represented as a red spherical surface. “P” and “S” arrows show the upwelling plumes from the core-mantle boundary under the supercontinents and the subducting plates along the supercontinent margins, respectively.

irregular because the cold subducting plates along the margins of the separated CLs and hot upwelling plumes originating in a random manner from the base of the mantle would enhance the toroidal component of mantle flow. Fig. 2 shows a time tracking of the centers of the four CLs (CL-A to CL-D) for 800 Myr in Models Y200A and Y200B. It is apparent that the motions of each continent show complex traces with time

during the supercontinent cycle. In both models, CL-A (red lines) and CL-D (light green lines) move longer distances, whereas CL-B (brown lines) and CL-C (yellow lines) move around in relatively narrow areas of the Earth’s surface for 800 Myr. For instance, in Model Y200A, CL-B and CL-C move around in narrow areas from the breakup of the initial supercontinent (Fig. 1b) to the formation of the next supercontinent at ca. 625 Myr, combining with CL-B and

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125

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Figure 2. Time tracking of the center of four cratonic lithospheres (CLs; CL-A to CL-D) for 800 Myr in Models Y200A (thick lines) and Y200B (thin lines). The solid circles indicate the initial positions of each CL. The open circles represent the positions of each CL at the elapsed times of 125, 250, 375, 500, 625, and 750 Myrs, which correspond to the time-step shown in Fig. 1. The numbers showing the elapsed time are added only for Model Y200A.

CL-C (Fig. 1e). This continental collision is accompanied by introversion (see white arrows in Fig. 1bed). On the other hand, CL-D first separates from the initial supercontinent, then circles the Earth for ca. 500 Myr, before eventually combining with CL-B and CL-D to form the next supercontinent. This continental collision is typical of extroversion (see light blue arrows in Fig. 1bed). Thus, the assembly of the supercontinent is accompanied by a combination of introversion and extroversion. This tendency is also apparent in Model Y200B (Fig. 1hej). Hence, as implicitly suggested by the previous mantle convection model with weakly deformable continents (Coltice et al., 2012; Rolf et al., 2012), the present model with relatively strongly deformable continents do not support the regular periodic supercontinent cycles involving only extroversion observed in previous 2D and 3D simulation models with rigid, nondeformable continental lids (Gurnis, 1988; Phillips and Bunge, 2007). The temperature anomaly immediately beneath the supercontinent is generally positive owing to the thermal insulation effect and the active upwelling plumes from the core-mantle boundary (“P” arrows in Fig. 1a, f, g, and l). This temperature increase would produce sufficient tensional force to breakup the supercontinent. When the next supercontinent forms, the subduction of oceanic plates along the supercontinent is enhanced (“S” arrows in Fig. 1f and l). In the real Earth, it has been speculated that circumsupercontinent subduction leads to the formation of new largescale upwelling plumes beneath the supercontinents, and eventually leads to the breakup of the supercontinents (e.g., Honda et al., 2000; Zhong et al., 2007; Li and Zhong, 2009). Supercontinent cycles with complex continental motions like those shown in Fig. 1 are closely related to the behavior of mantle convection. The heterogeneity spectrogram of the mantle temperature field (Fig. 3) shows that small-scale thermal heterogeneity (i.e., spherical harmonic degrees 10) dominates to considerable depths in the mantle under drifting continents during the supercontinent cycle, although several active upwelling plumes intermittently originate beneath the drifting continents and/or the supercontinent. This result suggests that, during a supercontinent cycle, actively subducting cold plates along convergent margins generate thermal heterogeneity with short-wavelength structure. However, long-wavelength thermal heterogeneity (i.e., degree-one or -two) sometimes coexists with the short-wavelength thermal heterogeneity as a result of complex relationships among the distribution of continents, the upwelling plumes, and the subducting plates.

Figure 3. Heterogeneity spectrogram of the mantle temperature field under a depth of 500 km for 800 Myr in Models (a) Y200A and (b) Y200B. Red to blue range shows strong to weak thermal heterogeneity at spherical harmonic degrees of 1e32. The logarithmic power spectra are normalized by the maximum values at each elapsed time.

The results from the numerical studies presented in this study confirm that supercontinent cycles involve a combination of introversion and extroversion. Hoffman (1991) proposed an “inside-out” (extroversion) mechanism of assembling continental fragments after the breakup of the Rodinia supercontinent, exemplified today by the Pacific region (Santosh et al., 2009). Conversely, as stated in Section 1, the Tethyan region exemplifies an “inside-in” (introversion) configuration after the breakup of the Pangea supercontinent (e.g., Murphy and Nance, 2008; Murphy et al., 2009; Nance et al., 2013). Hence, we conclude that both processes (i.e., introversion and extroversion) operate in the Earth’s history. Our results obtained from the model with basal and internal heating is essentially consistent with the previous work of Phillips and Bunge (2007) in which the active upwelling plumes from the core-mantle boundary disrupt the regular periodicity of the supercontinent cycle. The oceanic-continental margins with active subduction zones during the dispersal of the continental fragments decrease the subcontinental mantle temperature, which would be common in our model and Rolf et al.’s model (see Rolf et al., 2012; Yoshida, 2013 for details). Acknowledgments We thank Taras Gerya, an anonymous reviewer, and the Associate Editor, Guochun Zhao for their careful reading and valuable comments, which helped to improve the manuscript. Some of the figures were produced using the Generic Mapping Tools (GMT) software (Wessel and Smith, 1998). The calculations presented herein were performed on the supercomputer facilities (SGI ICE X) at JAMSTEC. The


present study M.Y. was supported partly by a Grant-in-Aid for Scientific Research (B) (No. 23340132) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. This paper also contributes to the 1000 Talent Award from the Chinese Government to M.S. References Burke, K., Dewey, J.F., Kidd, W.S.F., 1977. World distribution of sutures; the sites of former oceans. Tectonophysics 40 (1e2), 69e99. http://dx.doi.org/10.1016/ 0040-1951(77)90030-0. Coltice, N., Rolf, T., Tackley, P.J., Labrosse, S., 2012. Dynamic causes of the relation between area and age of the ocean floor. Science 336 (6079), 335e338. http:// dx.doi.org/10.1126/science.1219120. Condie, K.C., 2011. Earth as an Evolving Planetary System, second ed. Academic Press, 574 pp. Dewey, J.F., 1977. Suture zone complexities: a review. Tectonophysics 40 (1e2), 53e67. http://dx.doi.org/10.1016/0040-1951(77)90029-4. Gurnis, M., 1988. Large-scale mantle convection and the aggregation and dispersal of supercontinents. Nature 332, 695e699. http://dx.doi.org/10.1038/332695a0. Hoffman, P.F., 1991. Did the breakout of Laurentia turn Gondwanaland inside-out? Science 252 (5011), 1409e1412. http://dx.doi.org/10.1126/science.252.5011.1409. Honda, S., Yoshida, M., Ootorii, S., Iwase, Y., 2000. The timescales of plume generation caused by continental aggregation. Earth and Planetary Science Letters 176 (1), 31e43. http://dx.doi.org/10.1016/S0012-821X(99)00319-2. Lambeck, K., Johnston, P.,1998. The viscosity of the mantle: evidence from the analysis of glacial rebound phenomena. In: Jackson, I. (Ed.), The Earth’s Mantle, Composition, Structure and Evolution. Cambridge University Press, U.K., pp. 461e502. Lay, T., Hernlund, J., Buffett, B.A., 2008. Core-mantle heat flow. Nature Geoscience 1, 25e32. http://dx.doi.org/10.1038/ngeo.2007.44. Li, Z.X., Zhong, S.J., 2009. Supercontinent-superplume coupling, true polar wander and plume mobility: plate dominance in whole-mantle tectonics? Physics of the Earth and Planetary Interiors 176, 143e156. http://dx.doi.org/10.1016/jpepi.2009.05.004. Murphy, J.B., Nance, R.D., 2003. Do supercontinents introvert or extrovert? Sm-Nd isotope evidence. Geology 31 (10), 873e876. http://dx.doi.org/10.1130/G19668.1. Murphy, J.B., Nance, R.D., 2008. The Pangea conundrum. Geology 36 (9), 703e706. http://dx.doi.org/10.1130/G24966A.1. Murphy, J.B., Nance, R.D., Cawood, P.A., 2009. Contrasting modes of supercontinent formation and the conundrum of Pangea. Gondwana Research 15 (3e4), 408e420. http://dx.doi.org/10.1016/j.gr.2008.09.005. Nance, R.D., Murphy, J.B., Santosh, M., 2013. The supercontinent cycle: a retrospective essay. Gondwana Research. http://dx.doi.org/10.1016/j.gr.2012.12.026. in press. Phillips, B.R., Bunge, H.-P., 2005. Heterogeneity and time dependence in 3D spherical mantle convection models with continental drift. Earth and Planetary Science Letters 233 (1e2), 121e135. http://dx.doi.org/10.1016/j.epsl.2005.01.041. Phillips, B.R., Bunge, H.-P., 2007. Supercontinent cycles disrupted by strong mantle plumes. Geology 35 (9), 847e850. http://dx.doi.org/10.1130/G23686A.1. Ring, U., 1994. The influence of preexisting structure on the evolution of the Cenozoic Malawi rift (East African rift system). Tectonics 13 (2), 313e326. http://dx.doi.org/10.1029/93TC03188. Rolf, T., Coltice, N., Tackley, P.J., 2012. Linking continental drift, plate tectonics and the thermal state of the Earth’s mantle. Earth and Planetary Science Letters 351e352, 134e146. http://dx.doi.org/10.1016/j.epsl.2012.07.011.

81

Rolf, T., Tackley, P.J., 2011. Focussing of stress by continents in 3D spherical mantle convection with self-consistent plate tectonics. Geophysical Research Letters 38, L18301. http://dx.doi.org/10.1029/2011GL048677. Santosh, M., Maruyama, S., Yamamoto, S., 2009. The making and breaking of supercontinents: some speculations based on superplumes, superdownwelling and the role of tectosphere. Gondwana Research 15 (3e4), 324e341. http:// dx.doi.org/10.1016/j.gr.2008.11.004. Tackley, P.J., 2000. Self-consistent generation of tectonic plates in time-dependent, three-dimensional mantle convection simulations 2. Strain weakening and asthenosphere. Geochemistry, Geophysics, Geosystems 1 (8), 2000GC000043. http://dx.doi.org/10.1029/2000GC000043. Turcotte, D.L., Schubert, G., 2002. Geodynamics, second ed. Cambridge University Press, U.K., 456 pp. Vauchez, A., Barruol, G., Tommasi, A., 1997. Why do continents break-up parallel to ancient orogenic belts? Terra Nova 9 (2), 62e66. http://dx.doi.org/10.1111/ j.1365-3121.1997.tb00003.x. Wessel, P., Smith, W.H.F., 1998. New, improved version of the Generic Mapping Tools released. Eos, Transactions American Geophysical Union 79 (47), 579. http://dx.doi.org/10.1029/98EO00426. Wilson, J.T., 1966. Did the Atlantic close and then re-open? Nature 211, 676e681. http://dx.doi.org/10.1038/211676a0. Worsley, T.R., Nance, R.D., Moody, J.B., 1984. Global tectonics and eustasy for the past 2 billion years. Marine Geology 58 (3e4), 373e400. http://dx.doi.org/ 10.1016/0025-3227(84)90209-3. Yoshida, M., 2008a. Core-mantle boundary topography estimated from numerical simulations of instantaneous mantle flow. Geochemistry, Geophysics, Geosystems 9 (7), Q07002. http://dx.doi.org/10.1029/2008GC002008. Yoshida, M., 2008b. Mantle convection with longest-wavelength thermal heterogeneity in a 3-D spherical model: degree one or two? Geophysical Reseach Letters 35, L23302. http://dx.doi.org/10.1029/2008GL036059. Yoshida, M., 2010a. Preliminary three-dimensional model of mantle convection with deformable, mobile continental lithosphere. Earth and Planetary Science Letters 295 (1e2), 205e218. http://dx.doi.org/10.1016/j.epsl.2010.04.001. Yoshida, M., 2010b. Temporal evolution of stress state in a supercontinent during mantle reorganization. Geophysical Journal International 180 (1), 1e22. http:// dx.doi.org/10.1111/j.1365-246X.2009.04399.x. Yoshida, M., 2012. Dynamic role of the rheological contrast between cratonic and oceanic lithospheres in the longevity of cratonic lithosphere: a threedimensional numerical study. Tectonophysics 532-535, 156e166. http:// dx.doi.org/10.1016/j.tecto.2012.01.029. Yoshida, M., 2013. Mantle temperature under drifting continents during the supercontinent cycle. Geophysical Reseach Letters 40 (4), 681e686. http:// dx.doi.org/10.1002/grl.50151. Yoshida, M., Iwase, Y., Honda, S., 1999. Generation of plumes under a localized high viscosity lid on 3-D spherical shell convection. Geophysical Reseach Letters 26 (7), 947e950. http://dx.doi.org/10.1029/1999GL900147. Yoshida, M., Kageyama, A., 2004. Application of the Yin-Yang grid to a thermal convection of a Boussinesq fluid with infinite Prandtl number in a threedimensional spherical shell. Geophysical Research Letters 31 (12), L12609,. http://dx.doi.org/10.1029/2004GL019970. Yoshida, M., Kageyama, A., 2006. Low-degree mantle convection with strongly temperature- and depth-dependent viscosity in a three-dimensional spherical shell. Journal of Geophysical Research 111 (B3), B03412. http://dx.doi.org/ 10.1029/2005JB003905. Zhong, S., Zhang, N., Li, Z.-X., Roberts, J.H., 2007. Supercontinent cycles, true polar wander, and very long-wavelength mantle convection. Earth and Planetary Science Letters 261 (3e4), 551e564. http://dx.doi.org/10.1016/j.epsl.2007.07.049.