Two-dimensional (2D) and three-dimensional (3D) simulations are used to explore the effects of continental distribution on mantle convection. In both 2D and 3D, at total surface areas < 50%, internal temperature is weakly sensitive to continental configuration. Mantle heat flux values show mild variations with changing configuration. In 3D, at total continental area > 50%, the dependence of mantle heat loss on continental configuration becomes stronger. Mantle temperature continues to increase with total continental area but now varies by 5%–7% with changing configurations. When distributed, continents can cause flow patterns to become locked. This leads to significant variations in mantle temperature below continental and oceanic regions. This differs from the expectation that supercontinents would preferentially lead to large lateral variations in mantle temperatures and suggests that insulation-induced thermal anomalies could exist below continents today if they have remained fixed relative to mantle flow (e.g., Africa).
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 Continents have covered a portion of the Earth's surface for ~90% of the planet's history and have influenced the Earth's thermal evolution [e.g., Bowring and Williams, 1999]. As continents do not participate in convective overturn, heat transfer through them is dominantly via conduction (with episodes of magmatic heat transfer). For this reason, continents are considered to act as thermal insulators above the convecting mantle [e.g., Pekeris, 1935; Busse, 1978; Gurnis, 1988; Zhong and Gurnis, 1994]. The oceanic lithosphere does actively participate in convective overturn of the mantle. This duality leads to a conjugate heat transfer system with two distinct heat transfer modes affecting mantle heat loss: conduction through the stable, continental lithosphere, and a more efficient balance of horizontal advection and vertical conduction in the actively overturning oceanic thermal boundary layer. If the subcontinental and suboceanic mantle are well mixed, then the network can buffer global heat loss against the insulating effect of continental regions [Lenardic and Moresi, 2003; Lenardic et al., 2005; Cooper et al., 2006; Lenardic et al., 2011a]. The diagnostic for this condition is a negligible difference in temperature between subcontinental and suboceanic mantle. If, however, these two regions remain thermally isolated from each other, then the local temperature beneath the continental lithosphere can increase with respect to the suboceanic temperature [Phillips and Coltice, 2010; Lenardic et al., 2011a]. Thermal isolation can be induced by subduction zones surrounding a continent [e.g., Lenardic et al., 2011a; Heron and Lowman, 2011; Yoshida, 2013] or if convection cells within the mantle become locked to the continents [e.g., Phillips and Coltice, 2010]. Each of these scenarios could be triggered by the spatial distribution of continents.
 We present convection simulations, with multiple continental blocks in assembled versus dispersed configurations, designed to elucidate the tradeoffs between total continental area and continental distribution on the heat transfer properties of the system. We compare our simulation results against the thermal network theory of Lenardic, et al. , which predicts the relationship between continental surface area, convective heat loss, and average mantle internal temperature. This theory does not explicitly account for the effects of wavelength or convective patterns that continental material can invoke. Thus, deviations away from the theoretical values can help quantify the additional influence of geometric effects.
 The model system is simplified relative to the Earth in order to isolate the effects of configuration versus total continental area free of added complexities. For example, assuming an isoviscous, bottom-heated mantle allows us to isolate the wavelength effects due solely to the emplacement of continents. Convective wavelength dependencies can also be introduced by depth- and temperature-dependent mantle viscosity as well as heating mode [e.g., Bunge et al., 1996; Lowman et al., 2011; O'Farrell et al., 2013]. Addressing these simplifying assumptions will be left for future work.
 All three-dimensional (3D) simulations were run in a 3 × 3 × 1 model domain with a mantle Rayleigh number of 107 and a grid resolution of 128 × 128 × 48 elements. These simulations were run using the numerical code Underworld [Moresi et al., 2007]. Continents were emplaced in the top thermal boundary layer as rigid rectangular blocks, which extend to a depth of 5% of the total domain thickness. The continents were fixed to the grid, while the mantle was free to move through the domain using periodic boundary conditions. This allowed continents to “drift” with respect to changing mantle flow patterns while still maintaining a fixed location with respect to each other. The continents were assigned a stabilizing viscosity contrast 10,000 times that of the interior fluid. The continental material was assigned a nondimensional density value equal to that of the hot mantle, ensuring that it is always positively buoyant. The surface area covered by continents and continental configuration varied. The three layouts were: (1) a single block centered in the model domain, (2) two blocks of equal size centered in the model domain, or (3) four blocks of equal size centered in the model domain (Figure 1). The distance between the blocks was chosen such that the distance between each block was equivalent in each direction. As such, block separation depended on the total continental coverage. The two-dimensional (2D) simulations followed the same approach as above, but a 4 × 1 box with a resolution of 256 × 64 elements (and additional 192 × 48 and 512 × 128 elements for resolution tests). The continental block depth was set to 5.5% of the total domain thickness to align with the mesh and to allow for the use of enhanced resolution near the surface of the model domain [e.g., Lenardic et al., 2005]. For all simulations, we tracked the internal temperature of the mantle and the Nusselt number, i.e., the nondimensional heat flux of the system. The mantle internal temperature was defined as the average of the temperature in the region between the top and bottom boundary layers. As the upper boundary layer is partially stabilized by buoyant material, we defined an apparent boundary layer thickness by using the near-surface geotherm to predict the depth to the average internal temperature (assuming that the system was in a statistically steady state). We did the same for the lower boundary and the values of layer thicknesses and internal temperature were obtained through an iterative process. Average values for the bulk internal temperature and the Nusselt number represent a space and time average over the duration of the simulations, all of which were run to statistically steady state (Figure 2, insert).
 We compared our numerical results to the predicted values from the isoviscous version of a thermal network theory [Lenardic et al., 2005]. The network theory contains a free parameter related to the critical boundary layer Rayleigh number which was determined from a robust fitting to 2D data [Lenardic et al., 2005]. The same value was used for comparison to 3D results to highlight differences between the two suites. For the reference cases (no continents), the effect of different aspect ratio between the 2D and 3D simulations was accounted for using boundary layer theory predictions for the effects of wavelength on isoviscous convection free of continents [Turcotte and Schubert, 1982].
 Increasing the total continental surface area always decreases the global heat flux and increases the average mantle interior temperature, independent of block configuration. In 2D, the simulation results are also independent of configuration and follow the theoretical scaling trends (Figures 2 and 3). In 3D, for total continental area <50%, internal temperature varies by less than 1% with continental configuration and agrees well with the scaling theory (Figure 3). In addition, the value of the Nusselt number is lower for the assembled continent (Figure 2). This suggests that changes in convective wavelength are causing a different structure of the thermal boundary layers. This is consistent with the thermal fields (e.g., Figure 1), which show that multiple continents resulted in convective cell wavelengths near unity, while the single continent cases resulted in longer characteristic mantle flow wavelengths. For isoviscous convection, longer aspect ratio cells are less efficient at global heat transfer [e.g., Turcotte and Schubert, 1982].
 The numerical trends change for the 3D simulations as the total coverage increases beyond 50%. Different configurations now have an increased effect on the global heat flux, and the internal temperature also becomes dependent on configuration. Variations in the internal temperature of up to 7% are observed for the largest total continental area. Lower heat flux in the greater surface coverage cases enhances the tendency for convective flow patterns to become spatially fixed [e.g., Jellinek and Lenardic, 2009]. Our 3D simulations results are consistent with this expectation as we observed in some scenarios that upwelling plumes lock into the spaces between continental blocks. This change in trend is not as apparent within the 2D simulations. This suggests that the variation is due primarily to the changes in planform rather than wavelength as 2D simulations can only accommodate wavelength variations. The enhanced potential for locking mantle flow patterns increases the likelihood of regime shifts from a thermally well-mixed mantle to a situation in which lateral advection from the subcontinental to the suboceanic mantle is inhibited [Lenardic et al., 2011a].
 Figure 4 shows the temperature profiles sampled in various locations both on and off the continental blocks. Each geotherm represents a temperature profile from a single time step of a simulation (as opposed to the average internal temperature, shown as a vertical solid, thick gray line, which is averaged over the entire duration of the simulation). All results plotted are from the times when the system was in a statistical steady state. Continental geotherms are from the center of each block. We strived to avoid transient upwellings and downwellings; however, sampling upwellings was inescapable if they were locked beneath the continental blocks (for example, Figures 4c and 4f). Figure 4 shows that thermal isolation occurs for simulations with larger total continental coverage or when the continental material is dispersed into four blocks (or both). The grouping in temperature variations, continental versus oceanic geotherms, for simulations with same total coverage, but differing configurations, is tighter for smaller total continental coverage, indicating thermal mixing except in the case of the four-block configuration. At larger surface areas, more variation is apparent between the average internal temperatures and continental versus oceanic geotherms, which suggests thermal isolation can occur for dispersed or aggregated continents. This is consistent with the idea that the reduced convective vigor of the higher total surface area suites allows for mantle flow patterns to become fixed in space and in time relative to the position of continents, which favors thermal isolation.
4 Discussion and Conclusions
 Our results are consistent with previous studies for total continental surface areas <30%–40% [Phillips and Coltice, 2010; Rolf et al., 2012] which suggest that the effects of convective wavelength and planform have less influence on total system heat loss when the continental coverage is lower. Smaller overall continental coverage promotes thermal mixing which works in favor of a homogenized interior temperature (an exception to this is the case where the continental material is dispersed into four blocks). Below 50% coverage within our models, the deep mantle temperature beneath the supercontinents is not significantly changed from that beneath two smaller continents and the shallow, sublithospheric mantle is somewhat warmer (Figure 4). This suggests that for the Earth's continental coverage (~36%), the average mantle temperature is insensitive to single versus multiple block configurations. As such, any anomalous mantle temperatures associated with supercontinents must be caused by something other than the assemblage of continental material. Evidence for increased mantle temperatures prior or during supercontinental breakup [e.g., Kelemen and Holbrook, 1995; Brandl et al., 2013] suggests then that thermal isolation was induced by other means, such as subduction zones surrounding the continent [e.g., Lenardic et al., 2011a; Heron and Lowman, 2011; Yoshida, 2013].
 When the continental material is dispersed into four blocks, the configuration imposes a flow pattern on the system that can also force thermal isolation of the subcontinental mantle and result in significantly higher relative temperatures beneath continents and lower relative temperatures beneath oceanic material with minimal impact on the average internal temperature. This result demonstrates that sublithospheric topography can have a strong influence on convective planform and that this effect makes it possible to observe the transition between isolated and mixed thermal regimes at a lower continental coverage. We note that, in this four-block case, the continental blocks are still closer together than the typical scale of the horizontal flow and conclude that further work in higher aspect ratio domains is needed.
 The effect discussed above could have implications for stationary continents, such as Africa, which has remained relatively fixed for at least 30 million years [Burke and Wilson, 1972; O'Connor et al., 1999]. Africa experiences anomalously high topography and evidence of thermal perturbations, which have been attributed to interaction with a deep-seated plume [e.g., Nyblade and Robinson, 1994; Lithgow-Bertelloni and Silver, 1998]. Our simulations predict that at least a 100°C anomaly can occur below a fixed continent (as calculated from the geotherms in Figure 4 assuming an average mantle temperature of 1400°C) without requiring the presence of a plume. This temperature increase could cause an increase in topography. This suggests that such anomalous conditions do not require preexisting deep-seated mantle plume; rather, a continent can impose a long-term deep thermal structure by becoming fixed to a convective pattern and introducing thermal isolation from the surrounding mantle. This may also provide a mechanism for continental breakup. If allowed to drift, continents tend to move away from geoid highs induced by thermal upwellings [Gurnis, 1988]. However, if the continent is fixed, with respect to the mantle by the geometry of the plates, the increased mantle temperatures and topographic swells could lead to the rifting and ultimate demise of the continent as Africa is currently experiencing [Girdler et al., 1969].
 At total continental coverage >50%, there is a change in the behavior of the system for the 3D simulations. The slope of variations in Nusselt number with increasing continental coverage changes for single versus multiple block scenarios (Figure 2). The separated blocks have a noticeably higher internal temperature than the single block for larger total coverage (Figure 3). This is consistent with a transition from a thermally well-mixed to an isolated regime. While this transition does occur at continental surface area coverage larger than Earth's present today values, it does raise the question of what if the Earth or other planets possessed greater continental coverage? This change in system behavior at 36%–50% continental surface area is consistent with several prior studies, which also show a systematic shift in behavior of the convective system at this range [Lenardic and Moresi, 2003; Lenardic et al., 2005; Cooper et al., 2006; O'Neill et al., 2005; Lenardic et al., 2011b].
 This consensus supporting a shift in global dynamics driven by continental coverage suggests that continental growth past a critical threshold could fundamentally alter a planet's thermal history. This begs the question as to whether mechanisms are in place to buffer continental growth below a threshold value. Finally, this work demonstrates that while 2D simulations and theory can provide valuable insight into the influence of continental material on convective system, caution is necessary when extending such work to scenarios where the influence of imposed convective patterns becomes important.
 This research was supported in part by the National Science Foundation through TeraGrid resources provided by TACC [Catlett et al., 2007]. This material is based upon the work supported by the National Science Foundation under grant EAR-1112820.
 The Editor thanks Julian Lowman and an anonymous reviewer for their assistance in evaluating this paper.