Supercontinent Cycle and Thermochemical Structure in the Mantle: Inference from Two-Dimensional Numerical Simulations of Mantle Convection

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1 geosienes Artile Superontinent Cyle and Thermohemial Struture in the Mantle: Inferene from Two-Dimensional Numerial Simulations of Mantle Convetion Masanori Kameyama * ID and Akari Harada Geodynamis Researh Center, Ehime University, 2 5 Bunkyo-ho, Matsuyama , Ehime, Japan; akari@si.ehime-u.a.jp * Correspondene: kameyama@si.ehime-u.a.jp Reeived: 27 September 217; Aepted: 3 November 217; Published: 5 Deember 217 Abstrat: In this study, we ondut numerial simulations of thermohemial mantle onvetion in a 2D spherial annulus with a highly visous lid drifting along the top surfae, in order to investigate the interrelation between the motion of the surfae (super)ontinent and the behavior of hemial heterogeneities imposed in the lowermost mantle. Our alulations show that assembly and dispersal of superontinents our in a yli manner when a suffiient amount of hemially-distint dense material resides in the base of the mantle against the onvetive mixing. The motion of surfae ontinents is signifiantly driven by strong asending plumes originating around the dense materials in the lowermost mantle. The hot dense materials horizontally move in response to the motion of ontinents at the top surfae, whih in turn horizontally move the asending plumes leading to the breakup of newly-formed superontinents. We also found that the motion of dense materials in the base of the mantle is driven toward the region beneath a newly-formed superontinent largely by the horizontal flow indued by old desending flows from the top surfae ourring away from the (super)ontinent. Our findings imply that the dynami behavior of old desending plumes is the key to the understanding of the relationship between the superontinent yle on the Earth s surfae and the thermohemial strutures in the lowermost mantle, through modulating not only the positions of hemially-dense materials, but also the ourrene of asending plumes around them. Keywords: mantle onvetion; superontinent yles; hemial heterogeneity in the lowermost mantle 1. Introdution It is often onjetured that the assembly and dispersal of superontinents ourred several times during the Earth s history (e.g., [1]). This yli behavior, ommonly termed superontinent yle or Wilson yle [2], inludes the formation of Pangea (about 33 Ma) and its breakup (starting about 175 Ma) [3 5], the formation of Rodinia (about 9 Ma) and its breakup (during 825 and 75 Ma) [6] and those of far earlier ones. Among the events of breakup, some are aused by the impingement of asending plumes beneath the superontinent [7], even though some do not have any sign of asending plumes [8]. This fat suggests the signifiane of the interrelations between the superontinents and asending plumes from the deep mantle in driving the superontinent yle. In partiular, it is of ruial importane to understand the mehanism that generates asending plumes in a yli manner in response to the assembly and dispersal of superontinents. On the other hand, it has been also well aknowledged that the generation of asending plumes from the deep mantle is signifiantly affeted by the presene of hemial heterogeneities in the mantle (e.g., [9]). One of the most relevant examples is the large low-shear veloity provines (LLSVPs) in the lowermost mantle beneath the Paifi Oean and Afria (for a review, see [1]). From several lines of seismologial evidene, suh as their sharp-sided struture, low V s /V p ratio and inreased Geosienes 217, 7, 126; doi:1.339/geosienes

2 Geosienes 217, 7, of 25 density, it is generally onsidered that the LLSVPs are hemially distint from the surrounding, although some reent studies suggest their thermal origin [1,11,12]. Moreover, Burke and oworkers demonstrated the spatial orrelations between the eruption sites of large igneous provines (LIPs) [13] within reent 3 Ma and the distributions of S-wave perturbation in the lowermost mantle [14 16]: the hot asending plumes originate from the margins of the LLSVPs, not from elsewhere or not from within the LLSVPs. These observations suggest that the presene of hemial heterogeneity in the lowermost mantle restrits the regions of origin of asending plumes at least for the latest 3 Ma, although it is still ontroversial how strongly a hemial heterogeneity tends to fix the roots of asending plumes [17]. This further implies an interrelation between the superontinent yle at the Earth s surfae and the distributions of hemial heterogeneity in the lowermost mantle, through the generation of asending plumes. In this study, we will ondut numerial experiments of thermohemial mantle onvetion, whih inorporates the effets of the presene and drifting motion of the surfae ontinent. The aim of this study is to investigate the interrelation between the motion of the surfae (super)ontinent and the behavior of hemial heterogeneities in the lowermost mantle, both of whih are thought to be related to the asending plumes in the mantle. As an be inferred from the interrelations with the generation of asending plumes, the operation of the superontinent yle at the Earth s surfae is aided by a yli hange in the lateral distributions of hemial heterogeneity in the deep mantle. We will in partiular fous on the onditions under whih the superontinent yle, i.e., the yli assembly and dispersal, an take plae in onert with the presene of hemial heterogeneity in the lowermost mantle. As will be desribed later, our study relies on an extremely simplified numerial model of thermohemial onvetion in the mantle and drifting motion of the ontinental lid along the top surfae. Despite the simpliity, we do believe that our model aptures the fundamental nature of the interrelation between the motion of the surfae (super)ontinent and the behavior of hemial heterogeneities in the lowermost mantle. 2. Model Desription We onsider a time-dependent basally-heated onvetion of a two-omponent fluid in a model of a two-dimensional spherial annulus spanning an angle of θ max θ min = π radians (18 ) whose outer and inner radii are r max and r min, respetively. The values of r max and r min are taken to satisfy r min /r max =.55 and d r max r min = 29 km, in order to mimi the values of the Earth s mantle. Both the top (r = r max ) and bottom (r = r min ) surfaes are taken to be impermeable (i.e., zero normal veloity) and free-slip (i.e., zero tration). On the other hand, we assumed mirror symmetry at the vertial side walls (θ = θ min and θ max ), in order to reprodue onvetion ells with the largest horizontal wavelength. This allows us to virtually onsider the onvetion in a full spherial annulus spanning an angle of 2π radians (36 ) with low omputational osts. Temperature is fixed to be T top (= C) and T bot (= 35 C) at the top and bottom boundaries, respetively. We take into aount the effet of the endothermi phase transition between ringwoodite and bridgmanite plus ferroperilase at around a 66-km depth from the top surfae. In addition, we impose a lid of highly visous fluid along the top surfae (e.g., [18 2]) as a model of the (super)ontinent, whose vertial and horizontal dimensions are fixed to be d lid = d/16 = km and w lid = π/3 radians, respetively. That is, the ontinental lid is taken to oupy a third of the top surfae (w lid = (θ max θ min )/3). We also take into aount the assembly and breakup of a superontinent, by allowing a oherent horizontal motion of ontinental lid driven by the overall onvetion in the mantle. In this study, we mean by the presene of a superontinent the state where the ontinental lid is in ontat with vertial side walls: owing to the mirror symmetry at θ = θ min or θ max, the lid is ombined with its mirror image to form a superontinent whose horizontal dimension is 2w lid. Before going into the detailed desription of our numerial model, we admit that intensive simplifiations were made in developing our model. For example, we assumed that thermal onvetion in the mantle takes plae solely by the heating from the bottom and, in other words, ignored the

3 Geosienes 217, 7, of 25 effet of internal heating. This means that our model is unable to orretly reprodue not only the thermal evolution of the mantle owing to the lak of dominant heat soures, but also the styles of mantle onvetion suh as horizontal wavelength (e.g., [21,22]), the relative strength of mantle plumes (e.g., [23]) and the dynami auses for the breakup of superontinents [8,2]. We assumed that in most regions, the visosity is smoothly dependent on temperature and depth, by ignoring the loalized redution in visosity in the uppermost old regions due to, for example, plasti yielding (e.g., [24]) or damage (e.g., [25]), whih may play a ruial role in reproduing the plate-like behavior of otherwise stagnant lids. In addition, we onsider the thermohemial onvetion only in a half of two-dimensional spherial annulus, not in a full annulus. We also employed a single ontinental lid, whih is a rude assumption ompared with sophistiated models of ontinental drift above the onveting mantle, whih inlude multiple lids (e.g., [26 28]). Among the simpliities desribed above, however, using half of the spherial annulus is not likely to do any harm to the horizontal sales of onvetion ells, beause we an reprodue onvetion ells even with the largest horizontal wavelength virtually spanning a full annulus by the help of mirror symmetry assumed in the horizontal diretion (for example, onvetion ells with a pair of single asending and desending flows at θ = θ max and θ = θ min ). We therefore believe that our numerial model suffiiently aptures the fundamental nature of thermohemial onvetion in the Earth s mantle, whih interats with the surfae ontinents. In partiular, we an trak the drifting motions of surfae ontinents from the end of one hemisphere to the end of another one [2] at whih the lid is ombined with its mirror image to form a superontinent. In the following, we aordingly define the term superontinent yle by a yli formation of superontinents at one hemisphere entered at θ = θ min and another entered at θ = θ max Physial Properties of Modeled Fluids We assume that the modeled fluid onsists of two omponents (denoted by Components A and B) whose densities are different. The densities of Components A ( normal mantle) ρ A and B (dense omponent) ρ B are given by, ρ A = ρ A (1 αt + βγ), (1) ( ρ B = ρ B (1 αt + βγ) ρ A 1 αt + βγ + ρ ) B ρ A. (2) ρ A Here, ρ A and ρ B are the densities of a referene state (T = Γ = ), α is the thermal expansivity, β is the rate of density jump by the phase transition at around a 66-km depth and Γ is the phase funtion representing the relative fration of the high-pressure phase of the transition ( Γ 1). For simpliity, α and β are taken to be ommon for Components A and B. As in many earlier studies (e.g., [29,3]), the funtional form of phase funtion Γ is taken to be dependent on temperature T and depth from the top surfae by way of an exess pressure Π as, Γ 1 2 [ ( )] Π 1 + tanh, (3) p Π ρ A g(r max r) (p t γt), (4) where p t is the pressure of the phase transition at T =, γ is the Clapeyron slope and p is the range of pressure over whih the phase hange takes plae. In the right-hand side of (4), the first term represents the hydrostati pressure at the depth r max r from the top surfae, while the seond term represents the pressure of the phase transition for the temperature T. The thermal expansivity α depends on depth from the top surfae as: ( ) r max r α = α top exp E α. (5) d

4 Geosienes 217, 7, of 25 Here, α top is the thermal expansivity at the top surfae (r = r max ) and E α is a dimensionless parameter desribing the ratio of α top to the thermal expansivity α bot at the bottom surfae (r = r min ). In this study, we hoose E α = ln 3, implying that the value of α dereases with depth by a fator of three (α top /α bot = 3). The visosity η of the modeled fluid depends on temperature T and depth from the top surfae as: ( η = η top exp E T ) T T top r max r + E p. (6) T bot T top d Here, η top (= Pa s) is the visosity at the top surfae (T = T top and r = r max ) and E T and E p are onstants, whih desribe the dependene of η on the T and r, respetively. We fixed E T = ln(1 4.5 ) and E p = 1 2 throughout this study. In Figure 1, we show typial profiles of visosity in the radial diretion. Note here that the smooth profiles in visosity ome from the absene of various omplexities, whih may affet the values of visosity. In addition to the lak of loalized redution in visosity in the uppermost old regions, we did not inlude the visosity jump at the 66-km disontinuity, whih may enlarge the horizontal sale of onvetion ells (e.g., [31,32]). The absene of a visosity jump at the disontinuity in this study results in a more visous upper mantle and less visous lower mantle than those of earlier numerial studies on superontinent yles (e.g., [28]). 5 1 depth [km] visosity [Pa s] Figure 1. Plots of the typial profiles of visosity in the radial diretion. The profiles are alulated from Equation (6) together with the radial profiles of horizontally-averaged temperature for the snapshots shown in Figure 6. The thermal diffusivity κ is assumed to depend on temperature T as, ( ) ( ) κbot T 2 ( Ttop κ = κ top [ T T ) ] top, (7) κ top T bot T top T bot T top where κ top and κ bot are the values of thermal diffusivity at the top and bottom surfaes, respetively. In this study, we take κ bot /κ top = 3, in order to mimi the inrease in thermal ondutivity throughout the mantle [33,34] Fundamental Equations and Dimensionless Parameters In order to ompute the temporal evolution of the distributions of thermal and ompositional field in the mantle, we numerially solve the fundamental equations of mantle onvetion in the two-dimensional spherial polar oordinates (r and θ). In this study, we employed the extended

5 Geosienes 217, 7, of 25 Boussinesq approximation (e.g., [3]), where the effets of adiabati heating, visous dissipation and latent heat exhange are expliitly inluded in the energy transport. The values of parameters used in the present alulations are summarized in Table 1. Table 1. Adopted values of parameters. Symbols Meanings Values ρ A density of Component A ( normal mantle) kg/m 3 T = T bot T top temperature differene aross the mantle 35 K d = r max r min thikness of the mantle m α top thermal expansivity K 1 C p speifi heat J/kg K η top visosity Pa s κ top thermal diffusivity m 2 /s g gravitational aeleration 9.8 m/s 2 γ Clapeyron slope of phase hange at around 66 km depth 3 MPa/K The basi equations are written in dimensionless forms as, v =, (8) [ α(r) p + [η ( v + v )] + Ra T Rb α top Ra Γ R ] Ra e r =, (9) T t = ( vt κ T) Di α(r) α top Tv r + Φ + L, (1) + v =. (11) t Here, v is the vetor of flow veloity, p is pressure, is the fration of dense material (Component B), e r is the unit vetor in the radial (r-) diretion, v r is the flow veloity in the r-diretion positive outward and stands for the tensor produt of vetors. In (9), the first and seond terms of the left-hand side represent the effets of the pressure gradient and visous resistane, respetively. The third term of the left-hand side of (9) represents the effet of buoyany fore, whose first, seond and third terms in the square brakets ome from the hanges in density due to the hanges in temperature T, phase Γ and hemial omposition, respetively. In (1), the first and seond terms in the right-hand side represent the effets of the apparent heat transport (advetion plus ondution) and the adiabati temperature hange, respetively. The third and fourth terms of the right-hand side of (1) stand for the effets of visous dissipation (fritional heating) and the latent heat exhange due to the phase transition, respetively, whose funtional forms are expliitly given by, Φ Di Ra η ( ε II) 2, (12) L DiTv r γ Rb dγ Ra dπ, (13) where ε II is the seond invariant of the strain rate tensor. In this study, we ignored the effet of internal heating for simpliity. In the present formulation desribed above, we have four dimensionless parameters. The first one is the dissipation number Di defined by: Di α topgd C p, (14)

6 Geosienes 217, 7, of 25 where g is the gravitational aeleration and C p is the speifi heat. Based on the physial properties shown in Table 1, Di =.47 is used throughout this study. The seond one is the Rayleigh number Ra of thermal onvetion defined by, Ra ρ Aα top Tgd 3 η top κ top, (15) where T T bot T top is the temperature differene aross the layer. Throughout this study, we fixed Ra = The third and fourth ones are the phase boundary Rayleigh number Rb and hemial boundary Rayleigh number R given by: Rb ρ Aβgd 3 η top κ top = R (ρ B ρ A )gd 3 η top κ top β Ra, (16) α top T = ρ B ρ A Ra, (17) ρ A α top T respetively. These parameters represent the effets of the buoyany due to the hanges in phase Γ and hemial omposition, respetively. In this study, we fixed Rb/Ra =.7143, indiating that the value of β ( 5%) is smaller than the density jump at at the 66-km disontinuity ( 7.5%) estimated by the PREM[35]. On the other hand, the values of R are varied in order to study how the density differene between the two omponent fluids affets the motion of the ontinental lid, as well as the thermohemial state in the mantle. Note here that, under the extended Boussinesq approximation, the effet of the adiabati hange in temperature is taken into aount. In order to see how strong its effet is, let us estimate the potential temperature T pot at given (r = r, θ) alulated by: ln [ Tpot (r = r ], θ) T(r = r = Di 1 r=rmax α(r) dr. (18), θ) d r=r α top Taken together with Equation (6), the differene T pot in the potential temperature aross the layer is about 2563 K, whih is 73% of the differene T in the raw temperature. In the following, the density differene between the two omponent fluids is represented by the buoyany ratio B instead of R. Following [36], B is defined as, B ρ B ρ A ρ A α bot T = α top α bot R Ra. (19) By definition, B represents the ratio of the density differene between the two omponent fluids (ρ B ρ A ) to the density hange by thermal expansion aross the layer measured by the thermal expansivity at the bottom surfae (ρ A α bot T). In this study, we varied B in the range of B 1.5. The range of B employed in this study inludes the values of B hosen in [36] for the materials forming LLSVPs (B =.8). In addition, the value of the density differene between MORB (mid-oean ridge basalt) and the surrounding materials in the lowermost mantle is estimated to be about 1 kg/m 3 [37], whih roughly orresponds to the values of B in the range of.9 B 1.2 [38] Model of Continental Drift We alulate the temporal hange in the position of the ontinental lid in a similar manner as that employed in [2]. The key to our tehnique lies in imposing different onstraints on the horizontal veloity of the lid depending on whether or not the lid is in ontat with the side boundaries (θ = θ max or θ min ).

7 Geosienes 217, 7, of 25 Let θ lid be the horizontal oordinate of the enter of the ontinental lid at the time instane t. Considering the horizontal dimension w lid of the lid, the value of θ lid varies only in the range of θ lid,min θ lid θ lid,max where: θ lid,min θ min + w lid /2, θ lid,max θ max w lid /2. (2) The lid is in ontat with either side boundary when θ lid = θ lid,min or θ lid = θ lid,max, while it is away from both boundaries when θ lid,min < θ lid < θ lid,max. When (i) the lid is not in ontat with either side boundary (θ lid,min < θ lid < θ lid,max ), we solve the flow field both in the ontinental lid and that in the mantle as a whole. We next alulate an average angular veloity Ω lid within the ontinental lid as: Ω lid = (vθ /r)w lid (r, θ)rdrdθ, (21) Wlid (r, θ)rdrdθ where v θ is the flow veloity in the θ-diretion and W lid (r, θ) is a funtion given by: W lid (r, θ) = { 1 inside of the lid ( r max r d lid and θ θ lid w lid /2) outside of the lid (otherwise). (22) We then alulate the value of θ lid at a new time instane t = t + t by: dθ lid dt = Ω lid. (23) Note the impliit assumption in the above tehnique that the ontinental lid behaves as a rigid body. In this study, we inreased the visosity of the lid in order to keep the deformation in the interior of the lid as small as possible. We onfirmed that the ontinental lid an be regarded to be suffiiently rigid by taking the visosity in the ontinental lid as 1 3 -times larger than that of the surrounding fluid. When (ii) the lid is in ontat with either side boundary (θ lid = θ lid,min or θ lid,max ), on the other hand, we seek for a valid flow pattern where the ontinental lid does not penetrate the side boundaries in the following manner. First, we tentatively alulate an overall flow pattern and an angular veloity Ω lid of the ontinental lid by the same manner as in the ase (i) where the lid is not in ontat with either side boundary. We then examine whether or not the tentative flow pattern is valid, and in other words, the sign of Ω lid is onsistent with the horizontal position θ lid of the lid. That is, we regard the tentative flow field to be valid if it does not fore the lid to penetrate the side boundaries (i.e., Ω lid for θ lid = θ lid,min or Ω lid for θ lid = θ lid,max ) and alulate the position of lid θ lid at the new time instane by the same manner as in (i). If it fores the lid to penetrate the side wall, in ontrast, we disard the tentative flow pattern and solve for a new one together with the onstraint of no horizontal motion (v θ = ) in the region of the highly visous lid (in short, we need to ompute the flow field of one time instane at most twie in the ases where the ontinental lid is in ontat with the side walls). Note that the above proedure allows the motion of the ontinental lid away from the side boundaries. In addition, we redued the visosity by a fator of 1 2 in the region of ontinental margins whose thikness and width are d/16 = km and π/128 radians, respetively [19,2]. These weak regions help failitate the motion of the lid away from the side walls, as well as indue subduting flows at the ontinental margins Initial Conditions The initial distribution of temperature T is given by an adiabati profile with potential temperature of 135 C exept in thin regions near the top and bottom boundaries. In order to drive thermal onvetion, a small perturbation is added to the initial profile of T.

8 Geosienes 217, 7, of 25 The initial distribution of the fration of dense material (Component B) is given by: = { 1 for r min r r min + d for r min + d r r max. (24) This equation means that a layer of dense material is initially imposed above the bottom boundary whose thikness is d. In this study, we varied d in the range of 1/32 d /d 3/32. By assuming d = d/32, the three-dimensional volume of the layer is about 1.6%, whih is lose to the ratio of the volume of the LLSVPs to that of the entire mantle [16]. The initial position of the ontinental lid is taken to be θ lid = θ lid,min (although in some ases, we onduted additional alulations by initially hoosing θ lid = (θ min + θ max )/2). This means that the ontinental lid is initially in ontat with the side wall of θ = θ min, and in other words, a superontinent of horizontal dimension of 2w lid is initially present at the side wall by the symmetry at θ = θ min. Note also that, from the symmetry with respet to θ = (θ min + θ max )/2, this is equivalent with the ases where the lid is initially in ontat with the other side wall of θ = θ max. In short, all of the alulations in this study were started from the initial ondition where (i) a signifiant portion of the top surfae is overed by a ontinental lithosphere and (ii) a layer of hemially-dense materials is formed above the bottom surfae of the Earth s mantle. Although suh a state is not likely in the very early Earth where the amount of ontinental rust is muh smaller than the present one, it is most likely to take plae through the Earth s thermohemial evolution. Indeed, in the hot mantle in the very early Earth, a vigorous magmatism is expeted to signifiantly differentiate the mantle, forming a growing amount of ontinental rust at the Earth s surfae (e.g., [39]). At the same time, a substantial amount of basalti materials (e.g., [4 42]) an be generated, whih is later taken into the mantle through old desending flows and deposited in the lowermost mantle owing to its higher density than surrounding materials (e.g., [37]). Taken together, at a ertain time instane in the Earth s evolution, there might our a state where a ontinental lithosphere overs a third of the Earth s surfae and a layer of hemially-dense material is formed at the base of the mantle. In other words, our present alulations onsider the temporal evolutions of the mantle and (super)ontinents only in a later stage of Earth s history where the vigor of magmatism subsides (e.g., [38]). We admit that our alulations were started from arbitrarily-hosen initial distributions of temperature T, omposition and the surfae lid, in ontrast to earlier works (e.g., [43,44]), where preparatory alulations were onduted to make the steady-state distributions. Sine the employed initial state is far from the statistially-steady one, our models needed some length of time just after the start of eah alulation in order to rearrange the flows for the initial onditions. After the very early stage for the initial rearrangement, however, the temporal evolutions are expeted to be qualitatively similar regardless of the initial onditions. As will be shown later (see Setion 3.2), we onfirmed that the hange in the initial positions of the ontinental lid does not signifiantly affet the motion of the ontinental lid, nor the temporal evolution of thermohemial states Numerial Tehniques The omputational domain is divided uniformly into 1536 (horizontal) and 256 (radial) meshes based on a finite-volume sheme. The energy Equation (1) is disretized by the Crank Niholson method in time. An upwind sheme, alled the power-law sheme [45], is used to evaluate the advetion and ondution terms in the energy equation. Equations (8) and (9) for the flow field are solved for a stream funtion ψ defined by, v r = 1 r ψ θ, v θ = ψ r, (25)

9 Geosienes 217, 7, of 25 instead of diretly solving for primitive variables (veloity v and pressure p). The veloity field v defined thus automatially satisfies the equation of ontinuity (8). The disretized equation for ψ is solved by the Gaussian elimination for a non-symmetri band matrix. The reliability of the numerial ode has been verified by the omparison with the results of [18]. The advetion Equation (11) for hemial omposition, on the other hand, is solved by the Flux-Correted Transport (FCT) method [46] together with the Zalesak s fully-two-dimensional flux-limiter [47] and Euler-leap-frog time stepping. A third-order interpolation sheme and donor ell sheme with a zeroth order diffusion are used to evaluate the higher and lower order flux in the FCT method, respetively. As had been already demonstrated in our earlier work [4], this numerial tehnique allows us to transport in a perfetly onservative and less diffusive manner. Although the atual alulations are onduted in a non-dimensional form, the results are presented in a dimensional form. The onversion to dimensional quantities is arried out with a length sale of d = 29 km, a time sale of d 2 /κ top years, a veloity sale of κ top /d m/y, and a temperature sale of T = 35 K. 3. Results We arried out numerial experiments at various values of the two parameters, the buoyany ratio B and initial thikness d of the dense material (Component B). The adopted values of parameters are shown in Table 2. In addition, we will arbitrarily employ θ min = and θ max = π radians in some figures presented in this setion. Table 2. The values of parameters B and d adopted for the alulations in this study. Cases B d /d 1 1/ / / / / / / / / Effets of the Density Differene of Chemial Heterogeneity in the Lowermost Mantle on the Superontinent Cyle We first onduted the alulations of Cases 1 5, in order to see how the motion of the ontinental lid at the top surfae is affeted by the buoyany ratio B of dense material initially imposed in the lowermost mantle. In these alulations, we varied B in the range of B 1.5, while the value of d /d = 1/16 is kept unhanged (see Table 2). We show in Figure 2 the plots against the elapsed time of the horizontal position θ lid of the enter of the ontinental lid obtained for these ases. In the figure, the presene of a superontinent is meant for the state with θ lid = θ lid,min or θ lid,max where the ontinental lid is in ontat with vertial side walls. The motions of the lid in ontat with and away from the side walls (θ lid = θ lid,min or θ lid,max ) aordingly mean the assembly and dispersal of the superontinent, respetively. In addition, the formation of a superontinent alternately at one side wall and another is termed an ourrene of the superontinent yle. The plots in Figure 2 learly show that for B.75 (Cases 3 5), the superontinent yle is suessfully reprodued, while for B.375 (Cases 1 and 2), the motion of the ontinental lid is not appropriate for the reprodution of a superontinent yle. This means that the presene of suffiiently dense and hemially-heterogeneous material in the deep mantle is of

10 Geosienes 217, 7, of 25 ruial importane for the operation of the superontinent yle at the Earth s top surfae assoiated with the yli assembly and dispersal of the superontinent. We will next study how the density differene of hemial heterogeneity in the lowermost mantle affets the mobility of the ontinental lid at the top surfae by omparing the temporal evolutions of thermal and hemial states in these ases with different B. In Figures 3 6, we present the flow patterns and thermohemial states for Cases 2 (B =.375) and 3 (B =.75). Figures 3 and 5 show the plots against elapsed times of (a) root-mean-squared veloities in the entire domain v rms (red), along top surfae v top (green), and absolute drifting veloity of the ontinental lid v lid and (b) the horizontal oordinate θ lid of the enter of the ontinental lid for the ases. Here, the value of v lid is alulated by v lid = r max Ω lid where Ω lid is the angular veloity of the ontinental lid (see (21)). Figures 4 and 6 show, on the other hand, the snapshots of the distributions of fration of dense Component B (left panels) and temperature T (right panels) at the elapsed times indiated in the figure, together with the positions of the ontinental lid. Note that the mirror symmetry at the side walls (θ = θ min and θ max ) is assumed in Figures 4 and 6: In eah snapshot, θ = θ max is loated at its top while θ = θ min is at its bottom, and θ inreases from θ min to θ max in lokwise and ounterlokwise diretions in the snapshots of omposition and temperature T, respetively. θ lid,max 5π/6 horizontal position of lid θ lid θ lid,min elapsed time [Gy] π/6 Case 1 Case 2 Case 3 Case 4 Case 5 Figure 2. Plots against elapsed time of the horizontal position of the ontinental lid θ lid for Cases 1 5. In the figure, θ lid,min θ min + w lid /2 (= π/6) and θ lid,max θ max w lid /2 (= 5π/6) are the minimum and maximum values of θ lid, respetively. veloity [m/yr] (a) 8 v rms v top v lid (b) θ max 1 π θ lid θ min elapsed time [Gy] Figure 3. Plots against elapsed times of (a) root-mean-squared veloities in the entire domain v rms (red), along the top surfae v top (green), and the absolute drifting veloity of the ontinental lid v lid and (b) the horizontal oordinate θ lid of the enter of the ontinental lid for Case 2. For the plot of v lid in (a), the solid line indiates v lid > (namely, θ lid inreases with time), while the dashed line indiates v lid < (namely, θ lid dereases with time). In (b), the shaded region indiates the range of the horizontal oordinate θ where the ontinental lid overs (θ lid w lid /2 θ θ lid + w lid /2).

11 Geosienes 217, 7, of 25 temperature T omposition temperature T 2 omposition T [ C] Gy Gy Gy Gy Gy Gy Figure 4. Snapshots of the thermohemial states of Case 2 at the elapsed times indiated in the figure. Shown by olor and ontour lines in eah row are the distributions of fration of the dense omponent (left) and temperature T (right). For the olor sale for the distributions of and T, see the sale bars in the figure. In eah figure, the thik white lines indiate the positions of the ontinental lid at the top surfae. Note that the mirror symmetry at the side walls (θ = θmin and θmax ) is assumed in displaying the distributions of and T: in eah snapshot θ = θmax is loated at its top, while θ = θmin is at its bottom, and θ inreases from θmin to θmax in lokwise and ounterlokwise diretions in the snapshots of omposition and temperature T, respetively. (a) 8 veloity [m/yr] vrms vtop vlid (b) 1 θmax π θlid θmin elapsed time [Gy] 5 Figure 5. Same as Figure 3, but for Case 3. 6

12 Geosienes 217, 7, of omposition temperature T temperature T 2 omposition Gy Gy Gy Gy Gy Gy Figure 6. Same as Figure 4, but for Case 3. The omparison of Figures 3a and 5a shows that for both ases, the veloity vlid of the ontinental lid of the top surfae beomes large when the flow veloities in the entire region vrms and at the top surfae vtop are large. This implies that, regardless of the values of B, the motion of the ontinental lid at the top surfae is driven by the overall onvetion motion in the mantle. However, as an be learly seen from the omparison between Figures 3b and 5b, we note the differene in the motion of the ontinental lid with time between the ases with B =.375 and B =.75. For B =.75 (Case 3; Figure 5b), the lid moves with time in a quite smooth and intermittent manner, resulting in a superontinent yle assoiated with an alternate formation of the superontinent at θ = θmin and θmax. For B =.375 (Case 2; Figure 3b), in ontrast, the lid moves in a rather haoti manner during the period of alulation without forming a superontinent at either side boundary θ = θmin or θmax. From the omparison of Figures 4 and 6, we an learly see the differene in the thermohemial strutures in the mantle depending on the values of B. For Case 3 when B =.75 (Figure 6), a substantial amount of dense material resides in the lowermost mantle up to t = 3 Gy, even though it is slightly entrained into the overall onvetion of the entire mantle. There our several hemially-distint piles with a high fration of dense Component B just above the bottom surfae, whose shapes and horizontal positions hange with time in the lowermost mantle. These piles of high are signifiantly hotter than their surroundings with low, beause the effet of inreasing density due to large dominates that of dereasing density due to high temperature T. In addition, the roots of hot asending plumes signifiantly oinide with the hemially-dense piles just above the bottom surfae, refleting the heating from the hot and dense materials in the deep mantle. In other words, the hot hemially-distint piles tend to anhor the roots of hot asending plumes. Indeed, some asending plumes originate from near the margins of the hot hemial piles. The asent of

13 Geosienes 217, 7, of 25 plumes originating around the hemially-distint piles of dense materials therefore results in a smooth and intermittent motion of the ontinental lid (see Figure 5). For Case 2 when B is as small as.375 (Figure 4), in ontrast, the dense material, whih is initially imposed within a layer at the bottom surfae, is rapidly destroyed by the overall onvetion and mixed up into the entire mantle, leaving almost no trae of an initial layer of hemial heterogeneity for t > 1 Gy. The hot asending plumes take plae at the bottom surfae in a haoti manner owing to the absene of anhoring effets oming from the hemially-distint dense materials, whih in turn results in a disordered motion of the ontinental lid and, hene, the lak of superontinent yles (see also Figure 3). One may think that the differene in the dynamis of ontinental lids by the hange in B is due to the differene in the strength of hot asending plumes, rather than the anhoring of plumes by dense materials. In order to ompare the strength of vertial flows, we show in Figure 7 the two-dimensional distributions of dimensionless buoyany b and its radial distributions below a 1-km depth from the top surfae, together with the distributions of and T, for Cases 2 and 3 at the time instanes where the flow veloities are large (see also Figures 3a and 5a). Here, the dimensionless buoyany b is alulated by: b [ 1 ρ t ρ A α top T 1 θ max θ min θmax θ min ] ρ t dθ, (26) where ρ t ρ A αt (ρ B ρ A ) represents the hange in density due to the hanges in temperature T and omposition (in alulating the buoyany b, we ignored the effet of phase transition for simpliity). Note that b is similar to the residual buoyany used in [48] and is a measure of the onvetive vigor of asending and desending plumes at given depths. The snapshot of Case 2 learly shows that the distribution of b is quite similar to that of temperature T, beause the distribution of is almost uniform in the entire mantle. The snapshot of Case 3 shows, in ontrast, that a broad region of negative b develops just above the bottom surfae owing to the high fration of suffiiently dense omponent (B =.75) despite the high temperature T. We an also learly see that the broad region of b < is surrounded by a thin region of positive b from whih hot asending plumes originate. In addition, the omparison between Cases 2 and 3 of the radial profiles of b shows that neither the magnitude of positive b nor the standard deviation of b are very different in the range of depth exept near the bottom boundaries. This means that the vigor of hot asending plumes is almost unhanged with inreasing B from.375 (Case 2) to.75 (Case 3). In other words, the ourrene of the superontinent yle for suffiiently large B is not due to the hange in the vigor of asending plumes, but due to the anhoring of plumes by dense materials. In order to study in muh more detail how the hemial heterogeneity of dense materials behaves in the lowermost mantle during the superontinent yle, we show in Figures 8 and 9 the temporal evolutions of flow patterns and thermohemial states for Case 4 (B = 1.125) where the density differene between two omponents is omparable with that between MORB and the surrounding materials in the lowermost mantle [36 38]. In Figure 8, we show by olor the temporal evolution as a funtion of the horizontal oordinate θ of the fration of dense Component B averaged over a d/16 (= km) depth just above the bottom surfae. In order to learly show the spatial and temporal orrelation with the flows near the top surfae, we also plot by the blak lines and purple dots in the figure the positions of the margins of the ontinental lid (θ = θ lid ± w lid /2) and those of old desending plumes at the top surfae as a funtion of elapsed time, respetively. In Figure 9, we show, on the other hand, the snapshots of the distributions of fration of dense Component B (left panels) and temperature T (right panels) at the elapsed times indiated in the figure, together with the positions of the ontinental lid and the vetors of flow veloity v. Figure 9a shows the snapshots just before and after the breakup of the superontinent at around t 1.75 Gy, while Figure 9b shows those for the time instanes prior to the breakup. Note that in Figure 9, the snapshots of omposition are superimposed by the iso-lines of temperature T, in order to learly show how the distributions of are affeted by the overall flow strutures.

14 Geosienes 217, 7, of Case 3 (B =.75, Gy) omposition temperature T 5 3 Case 2 (B =.375, Gy) omposition temperature T 1 1 buoyany b buoyany b radial profiles distribution radial profiles distribution depth [km] depth [km] b b b b Figure 7. Comparison of the snapshots of the thermohemial states of Cases 2 and 3 at the elapsed times indiated in the figure. Shown by olor and ontour lines in the first row of eah olumn are the distributions of fration of the dense omponent (top left) and temperature T (top right). The meanings of the olors and thik white lines are the same as in Figure 4. Shown in the seond row of eah olumn are the distributions of dimensionless buoyany b oming from the differenes in T and at given depth (bottom right) and its radial profiles in the depth ranges below 1 km from the top surfae (bottom left). For the olor sales for the distributions of b, see the sale bars in the figure. In the plots of the bottom left of eah row, the red and blue lines indiate the maximum and minimum values of b at given depth, respetively, while the shaded regions delineate the standard deviation of b at the depth. θmax π P1 θ d3 d2 p1 d1 θmin P3 p3 p2 P elapsed time [Gy] Figure 8. Temporal evolution as a funtion of the horizontal oordinate θ of omposition averaged over a d/16 (= km) depth just above the bottom surfae for Case 4. For the olor sale for the distributions of, see the sale bar in the right of the figure. The two blak lines indiate the positions of the margins of the ontinental lid as a funtion of the elapsed time (i.e., θ = θlid ± wlid /2), while purple dots indiate the positions of old desending flows at the top surfae for given elapsed times. See the text for the meanings of the labels P1 P3, p1 p3 and d1 d3.

15 Geosienes 217, 7, of 25 (a) just before and after the breakup of superontinent omposition temperature T omposition P1 P temperature T Gy 1 5 m/yr Gy (b) prior to the breakup of superontinent omposition temperature T omposition d1 d1 5 m/yr p Gy p temperature T m/yr 5 3 d1 5 m/yr d Gy Figure 9. Snapshots of the thermohemial states of Case 4 at the elapsed times indiated in the figure. In the snapshots both for the distributions of omposition and temperature T, the purple arrows indiate the veloity vetors ~v and thin blak lines are the iso-lines of temperature T. The meanings of the olors and thik white lines are the same as in Figure 4. See the text for the meanings of the regions labeled P1 P3, p1 p3 and d1 d3. We an easily note from these figures that, in Case 4 where B = 1.125, a suffiient amount of dense material resides against the mixing by the overall onvetion in the lowermost mantle throughout the period of alulation. In addition, the distribution of dense material takes the form of an almost ontiguous layer aompanied by several usp-like strutures in Case 4, rather than non-ontiguous piles in Case 3 where B =.75 (see also Figure 6). This simply reflets the inrease in density differenes between the hemially-distint materials and surrounding ones by inreasing B, whose effet is further onfirmed by the alulation of Case 5 with larger B = 1.5. However, what is most striking in Figures 8 and 9 is the spatial and temporal interrelations between the superontinent yle at the top surfae and the distributions of dense materials in the lowermost mantle. In the following, we will see the interrelations between them in detail. From Figures 8 and 9a, we an first learly see that the breakup of the superontinent is losely related to the onentration of dense materials in the lowermost mantle beneath the lid. As indiated by the letter P1 in Figure 8, a broad region with a high fration of dense materials is formed in the lowermost mantle beneath the superontinent prior to the breakup of the superontinent away from θ = θmax at around t 1.75 Gy. As indiated again by the letter P1 in the snapshots of Figure 9a, a broad region of dense material is indeed formed above the bottom surfae beneath the superontinent just before and after the breakup of the superontinent at t 1.75 Gy. In addition, we note from Figure 9a the ourrene of hot plumes around the usp-like strutures of dense materials developed on the region P1. The asent of these plumes exerts an extensional fore on the superontinent from its bottom, eventually leading to its breakup at t 1.78 Gy. Similarly, prior to the breakup away from the side walls at around t 2.8 Gy and t 4.15 Gy, regions with high are formed at depth beneath the superontinent as indiated by P2 and P3 in Figure 8. This means that the

16 Geosienes 217, 7, of 25 key to the operation of the superontinent yle lies in the formation of the basal regions of the high fration of dense materials in the lowermost mantle beneath the lid alternately at eah side wall (θ = θ min, θ max ). In addition, from a loser look at Figure 8, we an note that the breakup of the superontinent (exept the very initial one) preedes the arrival of hemially-dense materials to the enter of the superontinent (namely, θ = θ min or θ max ). This is onsistent with the earlier finding [43] that asending plumes are not neessarily formed under the enter of a suffiiently wide superontinent. This fat also indiates that the onentration of dense materials under the superontinent is finished by sution due to the flows indued by the breakup of the superontinent. On the other hand, from Figures 8 and 9b, we an learly see that the formation of the basal regions of high beneath the lid is largely due to the horizontal motion of dense materials indued by the flows in the lowermost mantle. As indiated by the letter p 1 in Figure 8, a small region with high moves toward θ = θ max and eventually merges into the existing broad region P 1 of high. The horizontal motion of the region p 1 an be also seen from the omparison of the two snapshots in Figure 9b. As indiated by the arrows of flow veloity v in Figure 9b, a usp-like struture of dense material above p 1 is driven toward θ = θ max by the horizontal flow in the lowermost mantle. In addition, we an see from the iso-lines of temperature T that the horizontal flow in the lowermost mantle is largely indued by the old desending flow from the top surfae far from the lid indiated by the letter d 1. In other words, the desent of old flows ourring outside of the ontinental lid is the key to the horizontal motion of dense materials in the lowermost mantle toward the superontinent. Similarly, as an be seen from Figure 8, the motions of regions p 2 and p 3 toward the superontinent are spatially and temporarily related to the desending flows indiated by d 2 and d 3, respetively. In summary, our alulations of Cases 1 5 demonstrated that the assembly and dispersal of superontinents our in a yli manner when the hemially-distint material is suffiiently denser than the surrounding ones so that regions of dense materials an be formed at the base of the mantle. The motion of surfae ontinents is signifiantly driven by asending plumes originating around the hot piles or usp-like strutures of hemially-distint dense materials in the lowermost mantle. The hot dense materials, on the other hand, horizontally move in the lowermost mantle in response to the motion of ontinents at the top surfae, whih in turn horizontally move the asending plumes leading to the breakup newly-formed superontinents. We also found that the motion of dense materials in the base of the mantle is driven toward the region beneath a newly-formed superontinent largely by the horizontal flow indued by old desending flows from the top surfae ourring away from the (super)ontinent Effets of the Initial Positions of the Continental Lid In order to study how the initial positions of the ontinental lid affet the temporal evolution of its motion, we onduted additional alulations of Cases 3C to 5C where the horizontal position θ lid is initially taken to be θ lid = (θ min + θ max )/2 while the values of B and d are kept unhanged from those employed in Cases 3 5, respetively. In Figure 1, we show the plots of θ lid against the elapsed time obtained for Cases 3C 5C. Compared with the ases where the ontinental lid is initially loated at the vertial side walls (Cases 3 5; see also Figure 2), it takes a longer time for the onset of the motion of the ontinental lid when the lid is initially loated at the enter of the model domain. The delayed onset of ontinental motions reflets the longer duration of the very early stage where the overall flow pattern is rearranged for the initial onditions (namely, the positions of ontinental lid and small perturbation in temperature). However, the omparison also shows that, one the motion of the ontinental lid beomes signifiant, the temporal hanges in θ lid are qualitatively the same regardless of the initial position of the ontinental lid, although there is a slight differene in the threshold values of B above whih the superontinent yle an take plae. Indeed, as an be seen from the plots in Figure 1, the superontinent yle is suessfully reprodued for B (Cases 4C and 5C), while for B =.75 (Case 3C), the motion of the ontinental lid is not appropriate for the reprodution of a superontinent yle. This means

17 Geosienes 217, 7, of 25 that, one properly settled in the lowermost mantle, the distributions and shapes of dense materials strongly affet the overall flow patterns, as well as the motion of the ontinental lid. θ lid,max 5π/6 horizontal position of lid θ lid θ lid,min elapsed time [Gy] π/6 Case 3C Case 4C Case 5C Figure 1. Same as Figure 2, but for Cases 3C to 5C where the ontinental lid is initially positioned at the enter of the top surfae (θ lid = (θ min + θ max )/2 = π/2). We an therefore onlude that, rather than the hoie of the initial position of the ontinental lid, the presene of suffiiently dense and hemially-heterogeneous material in the deep mantle is of ruial importane for the operation of the superontinent yle at the Earth s top surfae assoiated with the yli assembly and dispersal of the superontinent. In the following, we will only onsider the ases where the ontinental lid is initially loated at the vertial side wall Effets of the Amount of Chemial Heterogeneity in the Lowermost Mantle on the Superontinent Cyle In this subsetion, we arried out alulations of Cases 6 9 where the thikness d of the initial layer of dense materials is above the bottom surfae, in order to study the influenes of the amount of dense materials on the superontinent yle. In these alulations, we employed B.75 (see Table 2) sine, as had been found in the previous subsetions, a suffiiently dense hemial heterogeneity is important for the operation of the superontinent yle. We first present in Figure 11 the result of Case 6 where the initial thikness of the layer of dense materials is redued by a fator of 2 (d /d = 1/32) while keeping B =.75 the same as that in Case 3. As an be seen from the figure, the initial superontinent loated at θ = θ min had been broken up, and the ontinental lid is horizontally driven to the other side walls, eventually forming another superontinent at the other side wall θ = θ max at around t 2 Gy. Compared with the results of Case 3 where the same B =.75 is employed, however, it took a longer time for the onset of the ontinental drift in Case 6. In addition, the initially imposed dense materials in the base of the mantle are rapidly entrained into the entire mantle by the overall onvetion motion. Indeed, the dense materials had been almost ompletely mixed up into the entire mantle by t 2 Gy, leaving almost no trae of the initial layer. Although we ontinued the alulations up to t 3 Gy, we do not observe the dispersal of the superontinent away from θ = θ max. We next present the results of Cases 7 9 where the initial thikness of the layer of dense materials is inreased (see Table 2). In Figure 12, we show the plots against the elapsed time of the horizontal position θ lid of the enter of the ontinental lid obtained for these ases. Figure 12 learly shows that the superontinent yle suessfully takes plae for all the ases in this figure. We also show in Figures 13 and 14 the results of the alulation of Case 7. Comparing with the results of Cases 7 and 3 (see Figures 5 and 6) where B =.75 is the same, we an intuitively see that a larger amount of dense materials resides in the lowermost mantle in Case 7 than in Case 3, whih results in larger piles of dense materials formed in Case 7. We also note, however, hat the temporal evolutions are qualitatively

18 Geosienes 217, 7, of 25 the same for these two ases. Indeed, there our hot asending plumes around the piles of dense materials above the bottom surfae, whih help the drifting motion of the ontinental lid at the top surfae. The piles of dense materials move in the horizontal diretion in the lowermost mantle along with the yli motion of the ontinental lid. temperature T omposition temperature T 1 omposition Gy Gy T 2 1 [ C] Gy Gy Gy Gy Figure 11. Same as Figure 4, but for Case 6. θlid,max horizontal position of lid θlid 5π/6 Case 7 Case 8 Case 9 θlid,min π/ elapsed time [Gy] 6 Figure 12. Same as Figure 2, but for Cases 7 9.

19 Geosienes 217, 7, of 25 (a) 5 veloity [m/yr] vrms vtop vlid (b) 1 θmax π θlid θmin elapsed time [Gy] 5 6 Figure 13. Same as Figure 3, but for Case T [ C] temperature T omposition temperature T 3 omposition 1. Gy Gy Gy T [ C] Gy T [ C] Gy Gy Figure 14. Same as Figure 4, but for Case 7. In summary, from the series of alulations by varying the amount of dense materials, whih is imposed as an initial ondition, we found that a suffiient amount of dense material must be present in the lowermost mantle, in order to operate the superontinent yle. By initially imposing a great

20 Geosienes 217, 7, of 25 amount of dense materials in the lowermost mantle, a suffiient amount of dense materials an survive against the onvetive mixing, resulting in the formation of either non-ontiguous piles or a ontiguous layer of dense materials above the bottom surfae depending mainly on the values of B. The dense materials in the lowermost mantle indue the hot asending plumes originating around them, whih in turn leads to the superontinent yles and horizontal motion in the dense materials at depth. 4. Disussion and Conluding Remarks In this study, we arried out numerial experiments of thermohemial mantle onvetion, whih inorporate the effet of the presene and drifting motion of the surfae ontinent together with initially imposed hemially-dense materials, aiming at understanding the interrelation between the motion of the surfae (super)ontinent and the behavior of hemially-dense heterogeneous materials in the lowermost mantle. We found that the presene of dense hemial heterogeneities in the lowermost mantle is of ruial importane for the ourrene of the superontinent yle. Indeed, the assembly and dispersal of the superontinent at the top surfae takes plae in a yli manner only when a suffiient amount of dense material resides in the deep mantle against onvetive mixing. The hemially distint dense materials, whih take the form of either non-ontiguous piles or a ontiguous layer depending on the density differene of the surrounding materials, indue asending plumes around them. In partiular, the asent of hot plumes originating from the dense materials beneath the superontinent exerts an extensional fore on the superontinent from its bottom, eventually leading to its breakup and dispersal followed by an assembly of the next superontinent at a new position. We also found that old desending flows ourring outside of the (super)ontinent play an important role in the repetitive breakup of the superontinent by suessively onentrating dense materials beneath a newly-formed superontinent. The desent of old flows ourring outside of the newly-formed superontinent indues the horizontal flows in the lowermost mantle partiularly toward the new superontinent. Owing to the basal flows in the diretion of the new superontinent, the dense materials loated beneath the former superontinent are swept and onentrated in the lowermost mantle beneath the new superontinent, whih further enhanes its breakup. This result is onsistent with the earlier findings (e.g., [49]), whih demonstrated the strong ontrols of old desending flows on the loations of deep hemial heterogeneity. In addition, our findings imply that the dynami behavior of old desending plumes is the key to the understanding of the relationship between the Wilson yle on Earth s surfae and the thermohemial strutures in the lowermost mantle, through modulating not only the positions of hemially-dense materials, but also the ourrene of asending plumes there. Our present finding on the spatial orrelation between the surfae motion and hemial heterogeneity in the lowermost mantle is not in good agreement with that of the series of papers by Trim and oworkers [44,48], who indiated no lear orrelation between the two. The differene of our finding from theirs is probably due to the differene in the assumed dependene of visosity on temperature and that in the geometries of onveting vessels. In the works by Trim and oworkers [44,48], the visosity is assumed to be dependent on the geotherm (horizontally-averaged temperature) and, hene, is taken to be laterally unhanged regardless of variations in loal temperature. That is, the old desending flows in their works are not likely to be so stiff ompared to ours where the visosity loally depends on temperature. In addition, in ontrast to our use of (2D) spherial geometry, their use of 2D Cartesian geometry does not allow one to inorporate the effet of the derease with depth in the areas of horizontal planes. Considering these two influenes, the models by Trim and oworkers [44,48] are most likely underestimating the roles of old desending flows in deforming and/or sweeping of the hemially-dense materials in the lowermost mantle. In other words, the differene between our results and those of earlier works may indiate the importane of fully-temperature-dependent visosity and model geometries in the studies of the relation between the surfae motions and the thermohemial state in the lowermost mantle.

21 Geosienes 217, 7, of 25 Among the intensive assumptions employed in this study, the lak of internal heating is not likely to spoil the signifiane of the present numerial experiments. Certainly, many earlier studies showed that a uniformly-distributed internal heating strongly affets the styles of overall onvetion suh as horizontal wavelength (e.g., [21,22]) and the relative strength of mantle plumes (e.g., [23]). However, onsidering their inompatible nature, the heat-produing elements are most likely to be distributed non-uniformly in the real mantle, by their frationation into magmati produts and reyling into the mantle through subdution. In partiular, assuming that the origin of hemially-dense materials in the lowermost mantle is anient magmati produts suh as subduted basalts (e.g., [38]), the heat-produing elements must be onentrated muh more highly in the dense materials than in the ambient (depleted) mantle. That is, if distributed highly non-uniformly in the mantle, the internal heating is expeted to loally heat up the hemially-dense materials, as well as to alter the style of overall onvetion. On the other hand, sine the internal heating the temperature of the hemially-dense materials in partiular, it an in turn heat up the surrounding mantle loally, but more strongly than in the ases without internal heating, and an aordingly enhane asending flows that originate preferentially around the dense materials. We therefore expet that the superontinent yles an our more readily by adding the effet of the heterogeneous distribution of internal heating to the present model. One may think that our highly simplified model an be hardly applied to the dynamis of the Earth s mantle or the motion of the surfae ontinent ourring in a three-dimensional spherial shell. However, as had been disussed in Setion 2, we do believe that the geometry of our numerial model (a half of a two-dimensional spherial annulus) does no harm to the motion of the surfae ontinent indued by the overall onvetion in the mantle, sine we an reprodue onvetion ells even with the largest horizontal wavelength virtually spanning a full annulus. On the other hand, we admit that the use of a two-dimensional model disallows us to preisely predit the dynamis of asending plumes or the extent of onvetive mixing of hemially-dense materials, both of whih are in essene three-dimensional phenomena. We nevertheless believe that our numerial results an orretly highlight, at least, the importane of old desending flows. This is beause old desending flows take the form of two-dimensional slabs in the three-dimensional Earth s mantle. In other words, the dynami behavior of old desending flows an be suffiiently well represented even in our two-dimensional model, although this is not likely the ase for hot asending plumes, whih an be best illustrated in three-dimensional models. In addition, onsidering that the effet of the derease with depth in the areas of horizontal planes is more signifiant in three-dimensional models than in two-dimensional ones, old desending plumes are most likely to exert greater ontrols on the hemial heterogeneity in the lowermost mantle than what is shown in the present two-dimensional models. Although our result highlights a ruial role of hemial heterogeneity in the deep mantle in the superontinent yle, several earlier studies obtained its ourrene in the numerial models without the effet of thermohemial onvetion (e.g., [26 28,5]). Among them, Zhong and oworkers proposed an idea alled the model [1,32], whih delineates the relations between the superontinent yle and the assoiated hanges in the onvetive patterns in the mantle. The heart of their model lies in an observation from their numerial experiments where the patterns of thermal onvetion in a three-dimensional spherial shell vary depending on whether a superontinent is absent or present at the Earth s surfae. When a superontinent is absent, the onvetive planform is haraterized by a pattern of spherial harmoni of degree-one, with a major asending flow in one hemisphere and a major desending flow in the other. When a superontinent is present, in ontrast, the onvetion takes a planform of degree-two pattern, with two major asending flows in an antipodal manner. These two modes of mantle onvetion alternately our depending on the modulation of the superontinents, ausing in turn the yli proesses of assembly and breakup of superontinents. This model properly explains not only the superontinent yle, but also the dominane of the degree-two struture for the present-day mantle despite the absene of superontinents, as a transient state bak to the degree-one struture. However, their model does not

22 Geosienes 217, 7, of 25 fully demonstrate any yli assembly or dispersal of the superontinent, sine they solely predit the hange in flow patterns of purely thermal onvetion and the ourrene of asending plumes beneath a new superontinent, whih is instantaneously formed above the major desending flow. In addition, based on our results of Cases 2 6 where the effet of dense hemial heterogeneity is minor (see Figures 4 and 11), the asending plumes are not likely to indue superontinent yles if they are laking in deep hemial roots, whih is firm enough to anhor their positions. We thus believe in the ultimate importane of hemial heterogeneities in the deep mantle on the superontinent yle. The importane of hemial heterogeneities in the deep mantle on the superontinent yle also enables us to understand the reason why several earlier studies obtained its ourrene in the numerial models without the effet of thermohemial onvetion (e.g., [26 28,5]). These earlier models employed an abrupt hange in visosity at the 66-km disontinuity between the upper and lower mantle. The jump in visosity at this depth is known to inrease the horizontal wavelength of onvetion ells (e.g., [31,32]) and, hene, to stably maintain the large-sale struture of onvetion. Taken together with the finding from our thermohemial model that the breakup of the superontinent is enhaned when the roots of asending plumes are anhored in the deep mantle, the visosity jump at the disontinuity an promote the breakup of the superontinent even in the ases with purely thermal onvetion, by maintaining the overall flow struture for a suffiiently long time. In ontrast to the model desribed above, several researhers argue that a degree-two struture of hemially-dense materials persists over a geologially long time in the Earth s mantle, based on the observational evidene for the spatial orrelation between the reonstruted eruption sites of LIPs (large igneous provines) and the margins of LLSVPs (e.g., [15,51]). However, the survival of the degree-two struture over a very long time is apparently inonsistent with our numerial results. Based on our experiments, the thermohemial struture in the mantle hanges with time along with the superontinent yle owing to the dynami oupling between them. In addition, the dispersal and assembly of superontinents are not likely to our in a yli manner above the stable degree-two struture assoiated with two major asending flows at antipodal positions. That is, if the degree-two hemial struture is indeed stable by the help of, for example, the presene of high visosity regions in the mid-mantle [52], the dynamis of surfae ontinents must be independent of that in the underlying mantle in order for the superontinent yle to surely our; or there must be some other mehanisms that an drive the yli motions of the (super)ontinent above the stable degree-two struture in the mantle. One possible andidate is the ontribution from flows with shorter horizontal wavelengths, whih disturb the stable long-wavelength struture (e.g., [28]), although this idea may not fully exlude the yli hanges in onvetion strutures suh as ours or the model. It still remains a hallenging task left to numerial modelers of mantle dynamis to fully reveal the manner of interrelation between the dynamis of surfae motions and the thermohemial evolution in the mantle. In order to fully understand the origin of the superontinent yle from the interations with the thermohemial onvetion in the Earth s mantle, it is very important to ondut numerial modelings by inorporating various omplexities ignored in the present study. These omplexities inlude three-dimensionality, onvetions under more vigorous onditions, the transition into the post-perovskite phase and the rheology appropriate for surfae plates. For example, the signifiane of the present finding should be verified by three-dimensional models with a full spherial shell geometry, although we do believe that the fundamental nature of superontinent yles an be appropriately aptured by our numerial model of half of a two-dimensional spherial annulus. In addition, in order to improve the present model, the effet of omplex rheology should be inluded, whih an indue the plate-like behavior of the top old fluid. In partiular, it is quite important to verify if the subdution of old fluids an frequently our not only at the ontinental margins, but also away from them and, in turn, effetively sweep the dense materials in the lowermost mantle under the newly-formed superontinent. On the other hand, the disrepany in the time sales of the superontinent yles obtained in the present study ( Gy) from those for the atual ones ( 4 My) [2,5] is partly due to

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