Chemical stratification in a two-dimensional convecting mantle with magmatism and moving plates

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B12, 2561, doi: /2002jb002205, 2003 Chemical stratification in a two-dimensional convecting mantle with magmatism and moving plates Masaki Ogawa Department of Earth Sciences and Astronomy, University of Tokyo at Komaba, Tokyo, Japan Received 17 September 2002; revised 27 August 2003; accepted 19 September 2003; published 19 December [1] A self-consistent numerical model is presented for magmatism in a convecting mantle with moving plates. Mantle convection is modeled as a thermal-chemical convection of binary eutectic material with Newtonian rheology in a two-dimensional rectangular box internally heated by radioactive elements. Viscosity is assumed to depend on stress history as well as temperature and pressure to self-consistently reproduce moving plates. The barrier effect of the solid-solid phase transitions at depths around 660 km to convective flow across the 660 km phase boundary is also taken into account. Magmatism is modeled as a permeable flow of basaltic melt produced upon decompression melting of the convecting material through the coexisting matrix. Magmatism makes mantle chemically stratified with the shallower part occupied by chemically buoyant residue of magma and the deeper part occupied by hot but chemically dense materials enriched in basaltic component even under the influence of moving plates and subducting slabs. The magmatism is induced by moving plates (ridge volcanism), mantle overturn, and hot plumes uprising from the 660 km phase boundary or the lower mantle, depending upon the internal heating rate and the strength of the 660 km barrier. A weaker 660 km barrier leads to more continuous plate tectonics. When plates move, subducting slabs penetrate deep into the lower mantle and induce broad thermal and chemical heterogeneity at depth in the lower mantle. The chemically stratified mantle with moving plates is compared to the Earth s mantle. INDEX TERMS: 8120 Tectonophysics: Dynamics of lithosphere and mantle general; 8121 Tectonophysics: Dynamics, convection currents and mantle plumes; 8124 Tectonophysics: Earth s interior composition and state; 8125 Tectonophysics: Evolution of the Earth; 8434 Volcanology: Magma migration; KEYWORDS: plate motion, magmatism, mantle stratification, numerical models Citation: Ogawa, M., Chemical stratification in a two-dimensional convecting mantle with magmatism and moving plates, J. Geophys. Res., 108(B12), 2561, doi: /2002jb002205, Introduction [2] Geochemical observations indicate that the Earth s mantle contains plural chemical reservoirs including the source of mid oceanic ridge basalt (MORB), which is depleted in incompatible elements and occupies the uppermost mantle, and several sources of oceanic island basalt (OIB), which are more enriched in incompatible elements than the MORB source and are somewhere at depth in the mantle [Hofmann, 1997]. Seismic observations also suggest that the deeper part of the lower mantle is chemically heterogeneous on the spatial scale of 1000 km or more and the relationship between the seismic heterogeneity and the geochemical reservoirs has been discussed [e.g., Yuen et al., 1993; Su and Dziewonski, 1997; Kennett et al., 1998; Kellogg et al., 1999; van der Hilst and Karason, 1999; Ishii and Tromp, 1999; Tackley, 2002]. Understanding how chemical heterogeneity has developed in the Earth s convecting mantle is one of the major issues in the study of Copyright 2003 by the American Geophysical Union /03/2002JB002205$09.00 mantle dynamics and evolution (for reviews, see Tackley [2000a] and van Keken et al. [2002]). To get insight into this issue, here, I present a self-consistent numerical model of magmatism in a convecting mantle with moving plates. [3] The convective flow induced by moving plates and subducting slabs extends from the surface to, at least, mid lower mantle [van der Hilst et al., 1997]. This convective flow would have well stirred the entire mantle and large scale chemical heterogeneity would not have survived convective stirring for billions of years in the mantle even under the influence of the phase boundary at 660 km depth and viscosity stratification if the chemical buoyancy that accompanies the heterogeneity is negligibly small [van Keken and Ballentine, 1999]. The viscosity variation that possibly accompanies the chemical heterogeneity [Manga, 1996; Becker et al., 1999] is probably not important in preserving the heterogeneity [Tackley, 2000a; van Keken et al., 2002]. The observations of chemical heterogeneity, therefore, suggest that the heterogeneity occurs as some forms of chemical stratification stabilized by chemical buoyancy [Tackley, 2000a]. In particular, the deeper part of lower mantle including the D 00 layer has been suggested ETG 5-1

2 ETG 5-2 OGAWA:MAGMATISM AND MOVING PLATES to be chemically denser than the overlying mantle [Davies and Gurnis, 1986; Hansen and Yuen, 1988; Olson and Kincaid, 1991; Christensen and Hofmann, 1994; Sidorin and Gurnis, 1998; Kellogg et al., 1999; Montague and Kellogg, 2000; Tackley, 2002; Davies, 2002]. [4] Even when chemical buoyancy is taken into account, however, the development of such a large scale chemical stratification in deep mantle still remains an open issue. The most likely candidate for the agent that induces the chemical stratification is magmatism, in particular, ridge volcanism [Davies, 1984; Olson and Kincaid, 1991; Coltice and Ricard, 1999; van Keken et al., 2002]. Yet it is not clear, from the earlier numerical simulations of mantle convection with simplified models of ridge volcanism, if ridge volcanism has been vigorous enough to generate oceanic crusts that segregate from convecting mantle and accumulate at the base of the mantle to form a layer as thick as the D 00 layer in the Earth s history [Gurnis, 1986; Christensen and Hofmann, 1994; Sidorin and Gurnis, 1998; Davies, 2002]. Other agents (e.g., reaction of mantle materials with the core [Knittle and Jeanloz, 1991]) are less likely to produce sufficient amount of chemically dense materials at the base of the mantle [Sidorin and Gurnis, 1998]. Furthermore, once formed at the base of the mantle, a chemically dense layer, which should be enriched in heat producing elements [Davies, 1984; Kellogg et al., 1999; van Keken et al., 2002], may eventually become unstable depending on the magnitude of chemical buoyancy of the layer, because of the excess heat and hence thermal buoyancy that build up in the layer [e.g., van Keken et al., 2002]. More comprehensive and selfconsistent models of magmatism and mantle convection with moving plates are necessary to fully understand the structure and evolution of the Earth s mantle. [5] Models of magmatism as magma generation by decompression melting and resultant magma migration that occurs as a permeable flow are formulated by McKenzie [1984] and Bercovici et al. [2001a, 2001b]. On the basis of the formulation, numerical models of magmatism in convecting mantle have been developed [Ogawa and Nakamura, 1998; Ogawa, 2000] (hereinafter called papers 1 and 2, respectively) to study how a convecting mantle can be chemically stratified. It is found in papers 1 and 2 that magmatism and mantle convection strongly influence each other to form a coupled system and that chemical stratification develops in convecting mantle under the control of the coupled system when internal heating is sufficiently strong. Mantle convection is, however, modeled as a convection of Newtonian fluid with constant viscosity (paper 1) or temperature-dependent viscosity (paper 2) and there are no moving plates in these models. Plate motion does not self-consistently occur in the simple numerical models of magmatism in convecting mantle of Dupeyrat et al. [1995] and de Smet et al. [1999], too. Since the efficiency of convective stirring in the mantle depends on the vigor and complexity of plate motion [Gurnis and Davies, 1986; Gurnis, 1986; Christensen, 1989; Ferrachat and Ricard, 2001], it is crucial to self-consistently reproduce moving plates in numerical studies of structure and evolution of the mantle [Ferrachat and Ricard, 2001]. After the efforts made by earlier workers (see the reviews by Bercovici et al. [2000] and Tackley [2000a]), Ogawa [2003] (hereinafter called paper 3) reached a regime where moving plates selfconsistently develop in his numerical model of mantle convection. In the present numerical model, I build the model of moving plates developed in paper 3 in the earlier model of coupled magmatism-mantle convection system developed in papers 1 and 2 to clarify the qualitative features of thermal and chemical state of the mantle controlled by magmatism and mantle convection with moving plates. Though I will use Earth-like values for the parameters that arise in the model, a realistic simulation of the Earth s mantle evolution is still beyond the scope of the present numerical experiment. 2. Model Description [6] Mantle convection is modeled as a thermal-chemical convection of a binary eutectic material with Newtonian rheology in a two-dimensional rectangular box internally heated by heat producing elements. Boussinesq approximation is employed. The height of the box is 2000 km, which is the ratio of volume to surface area of the Earth s mantle, and the aspect ratio of the box is shown in Table 1. The composition of convecting material is denoted as A x B 1-x, where the end-member A is a model of harzburgite and the end-member B is a model of a mixture of clinopyroxene and garnet in the uppermost mantle. The eutectic composition in the upper mantle, which is A 0.1 B 0.9, corresponds to the basaltic composition. The phase diagram of the convecting material is schematically illustrated in Figure 1 and its chemical potential is defined in paper 1. The assumed solidus is close to the solidus of a mantle material in the upper mantle [Herzberg and Zhang, 1996]. The end-member A is assumed to transform into its high pressure phase at a depth around 660 km as is the case for olivine [Ito and Takahashi, 1989], while the end-member B is assumed to gradually transform into its high pressure phase in the depth range from 620 km to 740 km to model the garnetperovskite transition [Irifune and Ringwood, 1993; Hirose et al., 1999]. (Hereinafter the phase boundary for the endmember A will be called the 660 km phase boundary.) The adopted Clausius-Clapeyron slope of the 660 km phase boundary is shown in Table 1; the slope of 2 MPa K 1 ( 0.2 MPa K 1 ) in Table 1 is reasonable for pyrolite at temperature lower (higher) than about 2000 K [Bina and Helffrich, 1994; Hirose, 2002]. The adopted Clausius-Clapeyron slope is 0 MPa K 1 for the end-member B, close to the slope of garnet-perovskite transition. The end-member B is also assumed to transform into its crustal phase above the depth of 40 km to model the basalt-eclogite transition. The density of convecting material depends on temperature, composition, melt content, and the phase of the matrix (see section 2.1 below), while the viscosity of the material depends on temperature, pressure, and degree of damage defined in section 2.2 below. [7] Magmatism is modeled as decompression melting of the convecting materials and gravitationally driven permeable flow of the produced melt through the coexisting matrix [McKenzie, 1984; Bercovici et al., 2001a, 2001b]. Chemical equilibrium is assumed to always hold. Since melting occurs only in the uppermost mantle in the present numerical experiment as will be shown below, the composition of melt is almost always basaltic. The melt composition is different from the bulk composition of convecting mantle materials in

3 OGAWA:MAGMATISM AND MOVING PLATES ETG 5-3 Table 1. Adopted Parameter Values a Case T init,k q 1,10 8 Wm 3 general and magmatism makes the mantle chemically heterogeneous. Magmatism also induces heterogeneity in the distribution of heat producing elements when the heat producing elements are incompatible (see Table 1) Density [8] The density of convecting material r depends on temperature T, bulk composition x, melt content j, and the proportion of high pressure phase in the end-member i in matrix z i (i = A, B) as r ¼ r 0 f1 at þ bð1 xþ jðv m =V 0 Þ½1 þ bð1 xþš þ ð1 þ jþðv s =V 0 Þ½x s z A þ ð1 x s Þz B Šg: ð1þ Here, the bulk composition x is related to the composition of melt x m and that of matrix x s as x ¼ q, 10 8 Wm 3 CC, MPa K 1, 10 6 Pa 1 ð1 jþx s þ jx m : ð2þ V 0 is reference molar volume, r 0 is reference density 3300 kg m 3, V is change in molar volume due to melting (subscript m ) and due to the solid-solid phase transitions at the depths around 660 km (subscript s ) and a is thermal expansivity deg 1. The value of b is shown in Table 1; b is assumed to be negative at depths less than 40 km to take the effect of basalt-eclogite transition into account. V m /V 0 is calculated from the assumed solidus curve by the Clausius-Clapeyron equation. V m /V 0 gradually decreases with depth and is 0.21 and 0.03 at depths z = 0 and 660 km, respectively. V s /V 0 is taken to be 0.1 to realize penetrative convection [Silver et al., 1988] (see below). [9] The dependence of density on composition and the phase of matrix is illustrated in Figure 1c. The basaltic material (A 0.1 B 0.9 ) is chemically denser than the primitive mantle material (A 0.64 B 0.36, see section 2.5 below) except at depths less than 40 km and just beneath the 660 km phase boundary. This density relationship is consistent with earlier estimates of densities of basaltic materials and pyrolite from the upper mantle to middle lower mantle [Irifune and Ringwood, 1993; Hirose et al., 1999; see also S. Ono et al., b >40 km <40 km Aspect Ratio = 3, PC = 0.01, Heat Capacity of Heat Bath = 0.25 HCM WK ST Rev Aspect Ratio = 2, PC = 1, Heat Capacity of Heat Bath = 0.05 HCM LIHWK LIHST WKCH WKCH WKPL WKPL a T init, PC, CC, and HCM stand for the initial temperature at depth in the mantle, partition coefficient of heat-producing elements between solid and melt, the Clausius-Clapeyron slope of the 660 km phase boundary, and the heat capacity of the mantle, respectively. See equation (7) for q 1 and q, equation (1) for b, equation (4) for, and section for the initial temperature of case Rev. The value of b is shown for both depths less than and greater than 40 km. manuscript submitted to Physics of the Earth and Planetary Interiors, 2003]. Whether or not basaltic materials are denser than pyrolite at greater depths is still an open issue [e.g., Ono et al., 2001]. Here, basaltic materials are assumed to be denser as has been done in the earlier models of mantle convection [e.g., Gurnis, 1986; Christensen and Hofmann, 1994]. The density contrast between basaltic materials and primitive mantle materials is 120 kg m 3 and the corresponding chemical buoyancy parameter r B [e.g., Davies and Gurnis, 1986] is 0.6 except just beneath the 660 km phase boundary when typical temperature contrast T = 2000 K and b = (see Table 1). A thermal and chemical structure similar to the one suggested for the Earth [Su and Dziewonski, 1997; Kennett et al., 1998] has been found to arise in deep mantle at this value of r B in the earlier numerical models of mantle convection [Davies and Gurnis, 1986; Sidorin and Gurnis, 1998; Montague and Kellogg, 2000]. The chemically induced positive buoyancy of basaltic materials just beneath the 660 km phase boundary (Figure 1c) implies that the barrier effect of the garnet-perovskite transition to convective flow of basaltic materials across the 660 km phase boundary [Irifune and Ringwood, 1993] is included in the present model. (This barrier will be called basalt barrier, hereinafter.) [10] The density jump at the 660 km phase boundary implies that the barrier effect of the post spinel transition of olivine (see review by Schubert et al. [2002]) is also included in the cases with Clausius-Clapeyron slope of 2 MPa K 1 for the boundary (Table 1). (This barrier will be called olivine barrier, hereinafter.) The phase buoyancy parameter [Christensen and Yuen, 1985] calculated from the parameter values described above is 0.1x or for primitive mantle materials with x = 0.64 for these cases. At this value of phase buoyancy parameter, the 660 km phase boundary is expected to stop narrow plumes with thickness less than about 200 km crossing the boundary when the plumes are driven only by thermal buoyancy and viscosity is constant in and around the plumes according to an earlier estimate [Tackley, 1995]. It would be, however, difficult for the phase boundary to stop broad convective flow induced by, say, flushing [e.g., Christensen and Yuen, 1985] and also to stop subducting slabs, which are much more viscous than the surrounding mantle [Christensen, 1996]. This assumption of penetrative convection [Silver et al., 1988] is consistent with a tomographic image of the mantle [Gu et al., 2001]. It is, however, important to carry out numerical experiments with weaker olivine barrier, too. The Clausius-Clapeyron slope becomes almost 0 or even positive and the olivine barrier does not work when temperature is higher than about 2000 K [Hirose, 2002]. Furthermore, the olivine-spinel transition at the depth of 400 km, which is not included in the present model, partly cancels out the effect of olivine barrier (for a review, see Schubert et al. [2002]). I calculated two cases with the slope of 0.2 MPa K 1 or the phase buoyancy parameter of 0.01x for the 660 km phase boundary (cases WK660 and LIHWK in Table 1) Viscosity and Plate-Like Behavior [11] The viscosity of convecting material depends on temperature T, depth z, and degree of damage w as h ¼ h 0 exp½et ð ref TÞþVz Fw= ð1 þ wþš; ð3þ

4 ETG 5-4 OGAWA:MAGMATISM AND MOVING PLATES where the constants h 0, E, T ref, V, and F are taken as Pa s, K 1, 1573 K, m 1, and ln ( )= 8.06, respectively. The parameter values imply that a viscosity contrast of 10 6 arises for temperature contrast of 1300 K, a typical temperature contrast between the surface and the asthenosphere of the Earth, and a viscosity contrast of 100 arises between the top and bottom boundaries of convecting box due to the depth dependence. [12] The degree of damage w changes with time as [Bercovici, 1998; Tackley, 2000b; Ogawa, 2003] dw dt ¼ s ij _e ij lw exp½et ð T srfc ÞŠ; ð4þ where d/dt is used for Lagrangean time derivative, _e ij is strain rate, s ij is stress, the constant l is s 1,

5 OGAWA:MAGMATISM AND MOVING PLATES ETG 5-5 and the surface temperature T srfc is 273 K; the value of is shown in Table 1 (see below for the choice of parameter values). Namely, the convecting material is damaged (w increases) and its viscosity drops when strong viscous dissipation occurs but recovers from the damage with a characteristic time of 1/{l exp [E(T T srfc )]} < 2 Myr, much shorter than the turnover time of mantle convection. The short recovery time implies that quasi-steady state approximation holds well in equation (4) and w, and hence viscosity h, can be well calculated as a function of stress p s ¼ ffiffiffiffiffiffiffiffiffiffi s ij s ij from equation (4) with the approximation of dw/dt = 0. Viscosity calculated in this way is schematically illustrated in Figure 1d for F > 4. There are two branches, the intact branch characterized by w 1 and high viscosity and the damaged branch characterized by w 1 and low viscosity, with a hysteresis between the two branches in the stress range of (s D, s I ), where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi lh s I;D ¼ 0 exp½et ð ref T srfc ÞþVzŠ F 2 1 F 2 4 F " exp F r ffiffiffiffiffiffiffiffiffiffiffiffi! # : ð5þ 4 F Here, the plus/minus takes plus (minus) for s D (s I ) (see paper 3). The hysteresis induces a stress history dependence of viscosity. At a stress in the range, a convecting material chooses the damaged branch (intact branch) if the material has experienced a stress higher than s I (lower than s D )in the past. [13] The stress history dependence of viscosity due to hysteresis induces a regime of thermal convection where the numerically modeled lithosphere shows a plate-like behavior as shown in paper 3. The lithosphere is divided into several rigid pieces or plates, which are on the intact branch, separated by soft and narrow plate margins, which are on the damaged branch, and each plate rather steadily moves on this regime. Paper 3 also shows that the criterion for the plate-like regime to arise is s D < s R < s I ; where s R is the stress induced in the lithosphere by ridge push force, while the rupture strength of plates s I and that of plate margins s D are evaluated at z = 0. The plates become stagnant when s D > s R, while the plates frequently fragment into smaller pieces owing to ridge push force when s R > s I (the weak plate regime of paper 3). Both are not the case for the Earth. (For the comparison of s I and s R, see Mueller and Phillips [1991] and Erickson and Arkani-Hamed [1993].) I ð6þ chose the values of and other parameters equal to that of case Had1 of paper 3, where the lithosphere falls on the plate-like regime, in most of the cases shown in Table 1. In cases WKPL2 andwkpl3, however, I calculated at higher to see the effect of lower rupture strength of plates on numerical results. [14] Criterion (6) implies that plate motion would be maintained for billions of years in the present model only when there are agents, outside the lithosphere, that break the lithosphere to newly form plate margins (paper 3). Otherwise, the plate motion, if occurs, would stop and would never resume when all of plate margins happen to disappear by some processes like ridge subduction. This is because ridge push force, i.e., the weight of lithosphere itself, is not sufficient for causing lithospheric rupture (s R < s I in equation (6)). An example of the agents that break the lithosphere found in paper 3 is hot uprising plumes; the lithosphere is broken when tapped by the heads of sufficiently buoyant hot plumes uprising from deep mantle Magmatism [15] Magmatism is modeled as a decompression melting of hot upwelling mantle materials and subsequent gravitational separation of the generated magma from the upwelling matrix, which occurs as a permeable flow of magma through matrix [McKenzie, 1984; Bercovici et al., 2001a, 2001b]. The relative velocity of magma with respect to matrix is proportional to the density difference between magma and matrix. Magma soliton [Scott and Stevenson, 1986; Olson and Christensen, 1986], therefore, does not occur in the present model. The approximation is valid on a spatial scale much larger than the compaction length, on the order of 1 km [McKenzie, 1984]. [16] The rate of magma generation modeled in this way rapidly increases with increasing temperature of the upwelling mantle materials [McKenzie, 1984] and hence with increasing internal heating rate in the mantle. The numerical studies of coupled magmatism-mantle convection system with constant viscosity in paper 1 show that a bifurcation occurs owing to this sensitive dependence of magma generation rate on internal heating rate. When the internal heating rate is lower than the bifurcation value (called the thermal convection regime hereinafter), magmatism does not occur at all or is negligibly weak, mantle convection occurs as a thermal convection, and the mantle is kept chemically homogeneous as a whole by convective stirring even when magmatism takes place. As the internal heating rate increases, however, magma generation rate increases, the mantle becomes chemically more heterogeneous, and Figure 1. (opposite) (a) Phase diagrams of the convecting binary eutectic material illustrated on the plane of composition x versus temperature T at the depths indicated in Figure 1b. The end-members A and B are models of harzburgite and a mixture of clinopyroxene and garnet, respectively. Ah ( Al ) implies that the end-member A in matrix is in its high- (low-) pressure phase, while L means liquid phase. The initial composition of convecting material x init = 0.64 and basaltic composition x b = 0.1 are indicated by dashed lines. (b) Phase diagram of the convecting material with x = x init illustrated on the plane of T versus depth z. The thick dashed line at z = 40 km indicates basalt-eclogite boundary while the thick dashed lines at z = 620 and 740 km show the depth range for the garnet-perovskite transition. (c) Depth dependence of density for the convecting materials with x = x init and x = x b at the temperature indicated by the dash-dotted line in Figure 1b. (d) Illustration of dependence of viscosity on stress calculated from equation (4) with dw/dt = 0. The solid (dashed) lines imply that the branches are stable (unstable) and physically realizable (unrealizable). Figures 1a to 1c are taken from paper 1 with modification.

6 ETG 5-6 OGAWA:MAGMATISM AND MOVING PLATES the chemical buoyancy that accompanies the induced heterogeneity becomes stronger. When the internal heating rate exceeds the bifurcation value, the chemical buoyancy becomes so strong that (1) the mantle suddenly becomes chemically stratified, (2) the vigor of mantle convection suddenly drops, and (3) heat builds up at depth in the mantle to make magmatism even stronger. The chemical stratification develops as a result of dynamic balance between convective stirring and chemical differentiation due to magmatism. The regime above the bifurcation point, which will be called the chemically stratified regime hereinafter, occurs when the viscosity of convecting materials strongly depends on temperature too (paper 2). The chemically stratified regime is the key to understanding the numerical result presented below. [17] The chemically stratified regime does not occur in the models of magmatism and mantle convection of Gurnis [1986], Christensen and Hofmann [1994] and Davies [2002], because the temperature dependence of magma generation rate is neglected in these models. In contrast, the thermal and chemical state of the chemically stratified regime seems to temporarily occur in the numerical models of magmatism and mantle convection of Dupeyrat et al. [1995] and de Smet et al. [1999], where magma generation rate sensitively depends on temperature. The thermal and chemical state, however, does not continue long in these models since the recycling of basaltic crust into the mantle is not fully taken into account Heat-Producing Elements [18] The heat-producing elements are highly incompatible in some of the calculations described below and the adopted values for the partition coefficient (solid/melt) are shown in Table 1. Owing to the decay of heat producing elements, the internal heating rate changes with time as q ¼ q 1 þ q expð t=tþ; ð7þ when melting does not occur at all and the heat producing elements do not segregate from the convecting matrix. The first term on the right hand side of equation (7) is a model of heating due to heat producing elements with long life time ( 238 U and 232 Th and) while the second term is a model of heating due to heat producing elements with short life time ( 235 U and 40 K). (When magmatism occurs, I generalized equation (7) to include the contribution of transport of heat producing elements by migrating magma as described in paper 2. In the generalized equation, the spatial average of the internal heating rate still follows equation (7).) I fixed t at 1.5 Gyr and adopted the values of q 1 and q shown in Table 1. When q 1 = Wm 3, and q 1 + q = Wm 3 (cases WK660 and ST660 in Table 1), the range of internal heating rate during the period of t =0 2 Gyr ( to Wm 3 ) covers a range suggested for the Earth s mantle during the period from the early Archean to the present [Schubert et al., 2002] and I calculated for a period longer than 2 Gyr in these cases Basic Equations, Boundary Conditions, and Initial Conditions [19] The basic equations are already shown in the earlier papers (chemical potential in paper 1 and the momentum, continuity, mass transport, and energy equations as well as the equation for transport of heat producing elements in paper 2), and only a brief description of the equations are presented here. The volume changes from phase transitions are neglected in the continuity equation. The energy equation is written in terms of enthalpy, which is calculated from the assumed chemical potential and includes the contributions from sensible heat, latent heat of melting, and that of solid-solid phase transition for the end-member A. Enthalpy distribution changes with time owing to convective transports by magma and matrix, thermal diffusion, internal heating, and the work done by volume changes from phase transitions. Magma migration is calculated as the transport of latent heat of melting by migrating magma in the energy equation. Mass transport equation and the equation for the transport of heat producing elements include the contribution of convective transports by magma and matrix but do not include the contribution of chemical diffusion. The content of heat producing elements changes owing to their decay, too. Chemical equilibrium is assumed to always hold and the phase relationship is calculated from the assumed chemical potential. [20] The employed mechanical boundary condition is impermeable and shear stress free for all of the boundaries. The employed thermal boundary condition is fixed temperature for the surface boundary (T = 273 K) and adiabatic for the vertical sidewalls. As for the bottom boundary, I placed a heat bath of uniform temperature beneath the bottom boundary as a model of the core. Heat capacity of the heat bath is shown in Table 1. The heat bath does not contain internal heat source and the effect of latent heat released from growing inner core [e.g., Labrosse et al., 1997] is neglected; the thermal state of the mantle is not affected much by the detailed value of heat flow from the core [Davies and Richards, 1992]. [21] The initial condition for chemical distribution is uniform with a primitive content of the end-member A x init = 0.64 and primitive internal heating rate of q = q 1 + q except for case Rev in Table 1. The initial temperature distribution is the minimum of T = T init (see Table 1) and the solidus temperature except in a layer of thickness 100 km along the surface boundary, where temperature linearly increases with depth starting from 273 K on the surface, except for case Rev; the initial temperature of the heat bath is also T init. I assumed a high T init in cases ST660 and WK660 because of the hot origin of the Earth. The initial condition for equation (4) is w = 0 everywhere in the mantle except for case Rev. To start the convection, I added a sinusoidal perturbation to the initial temperature distribution. As for the initial condition for case Rev, see section [22] The basic equations are numerically solved by finite difference method explained in papers 1 to 3 and [Ogawa, 1993]. The employed numerical mesh is uniform in horizontal direction but nonuniform in vertical direction with higher resolution around the boundaries. The number of mesh points is 360 (horizontal) times 150 (vertical) in the calculation of continuity and momentum equations, and twice that resolution is employed in the calculation of equation (4), energy equation, and mass transport equations. The numerical integration of the basic equations for the period of 2 Gyr takes 2400 to 3500 hours on a vector

7 parallel computer (VPP-5000, 8CPU) when the aspect ratio of the convecting vessel is Results 3.1. Cases With Weak Olivine Barrier Case WK660 [23] First, I present the result of case WK660, where the internal heat source decays with time and the Clausius- Clapeyron slope of the 660 km phase boundary is 0.2 MPa K 1 (Table 1). The olivine barrier is negligible but the basalt barrier is still effective in this case. Such a negligible olivine barrier is appropriate especially for a hot mantle as discussed in section 2.1 and I calculated case WK660 for the first 2.3 Gyr, i.e., when the mantle is particularly hot owing to strong internal heating. The model of case WK660 is presented in Figures 2 to 4. In Figure 2, I add the scale of average internal heating rate calculated from equation (7) besides the scale of time in the abscissa for convenience. In Figure 3b, the distribution of degree of damage w is shown only along the surface boundary because w rapidly decreases with increasing depth from its surface value w s and becomes negligibly small beneath the lithosphere (see paper 3). The normalized internal heating rate q norm in Figure 3c is the ratio of internal heating rate to its spatial average calculated from equation (7) with the elapsed time t for each snap shot; q norm <1(q norm 1) implies that the material is depleted (undepleted) in heat producing elements. The reduced density deviation dr in Figure 4 is the deviation of density from the reference density r 0 due to thermal expansion, compositional change, melting, and the basalt-eclogite transition and is defined from equation (1) as dr ¼ rðt; x; j; z A ¼ 0; z B ¼ 0Þ r 0 : ð8þ OGAWA:MAGMATISM AND MOVING PLATES ETG 5-7 Figure 2. The plots of (a) average temperature in the upper mantle T UM and in the entire mantle T WM as well as the temperature of heat bath beneath the convecting mantle T hb, (b) horizontally averaged surface heat flow, (c) rootmean-square average of horizontal velocity on the surface boundary, (d) root-mean-square average of velocity in the entire convecting mantle, and (e) horizontally averaged magma eruption rate on the surface boundary, all against time for case WK660. Besides the scale of time in abscissa, the scale of average internal heating rate calculated from equation (7) is added. For the horizontal arrows stage I and stage II at the top, see the text. The effects of the phase transitions at depths around 660 km are not included in dr. [24] The plots of heat flow and other average quantities shown in Figures 2b to 2e indicate that the mantle of case WK660 evolves in two stages. On the earlier stage at elapsed time t less than 1.4 Gyr (stage I in Figure 2), there are spikes in the plots of these average quantities. The spikes imply that episodes of vigorous magmatism (Figure 2e) and mantle convection (Figures 2c and 2d) repeatedly occur on stage I when the mantle is strongly heated by heat producing elements. As the heat producing elements decay, however, such spikes disappear and magmatism and mantle convection become more steady (stage II in Figure 2). [25] Figure 3 shows (1) how the thermal and chemical state of the mantle evolves in case WK660 and (2) how the episodes of vigorous magmatism and mantle convection occur on stage I: [26] The lithosphere as a cold and highly viscous thermal boundary layer develops along the surface boundary as indicated by the thin layer of blue color in Figure 3b. At t = 0.48 Gyr, a part of the lithosphere moves to the right as a rigid plate as expected from the model setting; w s 1 and the surface velocity u s is positive and constant on the right side of the partially molten region indicated by arrow a in Figure 3b except in the narrow plate margin at the top right corner of the convecting vessel. The plate motion starts at t around 0.2 Gyr when the initial transient behavior disappears (Figure 2c). The plate motion induces partial melting and the resulting ridge volcanism beneath the spreading center as indicated by arrow a. The ridge volcanism generates a basaltic crust, 15 to 30 km in thickness (see the thin blue layer indicated by arrow b in Figure 3a), and a layer made of residue of magma, characterized by low x and low internal heating rate, just beneath the basaltic crust (see the thin orange layer beneath the crust indicated by arrow c). The veneer of basaltic crust and the residual layer

8 ETG 5-8 OGAWA:MAGMATISM AND MOVING PLATES

9 OGAWA:MAGMATISM AND MOVING PLATES ETG 5-9 Figure 4. Distribution of reduced density deviation dr defined by equation (8), i.e., the deviation of density from reference density due to thermal expansion, compositional change, melting, and basalt-eclogite transition, at two elapsed times shown in the figure for case WK660. The effects of solid-solid phase transitions of both the endmembers A and B at the depths around 660 km are not included in dr. sinks to the bottom boundary as a part of the subducting slab along the right sidewall (Figure 3a for t = 0.81 Gyr). At the base of the mantle, the chemically buoyant residual materials separate from the chemically dense basaltic crust as the subducted slab is warmed up by thermal diffusion from the surrounding hot mantle as indicated by the circle in Figure 3a. The ridge volcanism chemically differentiates the mantle. The plate motion and the chemical differentiation due to ridge volcanism continue throughout the period of time calculated in case WK660. [27] Besides the ridge volcanism, hot spot magmatism occurs at t = 0.81 Gyr (see arrow p in Figure 3b) owing to the hot plume that rises up from the broad hot region in the lower mantle indicated by arrow d. The hot spot magmatism breaks the lithosphere to induce a new plate margin. Similar hot spot magmatism repeatedly occurs on stage I and induces the spikes in the plots of Figures 2b to 2e (see Figure 3b for t = 1.11 Gyr, too). (Hot plumes from broad hot regions in deep mantle have been observed in earlier laboratory experiments [Davaille, 1999] and numerical experiments [e.g., Hansen and Yuen, 2000; Montague and Kellogg, 2000; Tackley, 2002] of thermal-chemical convection, too.) The hot spot magmatism also chemically differentiates the mantle. A large portion of the chemically dense basaltic materials formed by the hot spot magmatism sinks deep into the lower mantle as exemplified by arrows e in Figure 3a for t = 0.81 and 1.11 Gyr while a large portion of the chemically buoyant residue of magma remains in the uppermost mantle. [28] The chemical differentiation due to ridge and hot spot volcanism makes the mantle chemically stratified in spite of the mantle-wide stirring induced by moving plates and subducting slabs that sink to the bottom boundary. The upper mantle and the top of the lower mantle are mostly occupied by chemically buoyant residue of magma depleted in heat producing elements while the deeper part of the lower mantle, especially the region along the bottom boundary, is mostly occupied by materials enriched in chemically dense basaltic components and heat producing elements (see Figures 3a and 3c). The effect of basalt barrier on the chemical structure of the mantle is also visible in the snap shots of t = 1.5 to 2.25 Gyr in Figure 3. A thin layer of residual materials highly depleted in heat producing elements develops just beneath the 660 km phase boundary in these snap shots (Figures 3a and 3c). [29] Superposed to this chemical stratification, broad lateral heterogeneity develops in temperature and compositional distributions in the lower mantle owing to subducting slabs. The subducted slabs accumulate to form broad cold regions at depth in the lower mantle around the vertical sidewalls. Away from these broad cold regions, a broad hot region develops (see arrow d in Figure 3b for t = 0.81 Gyr). The hot region is occupied by materials enriched in basaltic component and heat producing elements (Figures 3a and 3c). The hot region does not rise up as a whole in spite of its thermal buoyancy because the region is also enriched in chemically dense basaltic materials. Indeed, Figure 4 shows that the density of the materials in the hot region is almost the same as the density of the surrounding cold materials at the same depth level and hence that the hot region is almost neutrally buoyant. [30] The chemical stratification observed in Figure 3a is the reason why the transition from stage I to stage II occurs. On stage I, a significant volume of the materials in the hot region in the lower mantle is as buoyant as the upper mantle materials (arrow a in Figure 4 for t = 0.81 Gyr) though the materials in the hot region is more enriched in chemically Figure 3. (opposite) Distributions of (a) bulk composition x, (b) temperature T, and (c) normalized internal heating rate q norm, all indicated by color, at the elapsed time shown for case WK660. Also shown are the location of 660 km phase boundary (solid line in Figure 3a), the distribution of magma j (contour lines in Figure 3b), and the plots of horizontal velocity u s (black curve and black scale in Figure 3b) and degree of damage w s (red curve and red scale in Figure 3b) both along the surface boundary. In Figure 3a, basaltic materials (x = 0.1) are indicated by blue, while residue of magma (x > x init = 0.64) is indicated by yellowish green to orange. In Figure 3c, depleted materials (q norm < 1) are indicated by blue to green, while undepleted materials (q norm > 1) are indicated by green to orange. The contour interval for magma content j in Figure 3b is 5%. The scales for the plots of u s and w s are in cm yr 1 and nondimensional, respectively. See color version of this figure at back of this issue.

10 ETG 5-10 OGAWA:MAGMATISM AND MOVING PLATES dense basaltic component; the materials are strongly heated by abundant heat producing elements and are thermally buoyant. Only small amount of additional heating is sufficient for letting the materials in the hot region rise up to the surface as hot plumes. As the heat producing elements decay (stage II), however, the temperature becomes lower in the hot region (Figure 3b for t = 1.50 to 2.25 Gyr) and the reduced density in the hot region becomes significantly higher than the reduced density in the uppermost mantle (see Figure 4 for t = 1.64). The higher density inhibits hot plumes to rise up directly from the lower mantle to the surface by their own buoyancy on stage II. [31] The hot materials in the lower mantle, though cannot rise up as plumes by their own buoyancy, still rise up to the base of the upper mantle as a part of the return flow of subducting slabs (see arrow f in Figure 3b for t = 1.64 Gyr) and accumulate there to form a pool on stage II. (See the region slightly enriched in basaltic component and heat producing elements indicated by arrow g.) They do not return back to the lower mantle owing to the basalt barrier. Occasionally, the hot materials in the pool rise up to the surface to induce mild hot spot magmatism (see arrow h in Figure 3b). This hot spot magmatism often breaks the lithosphere to form a new plate margin as observed in Figure 3b for t = 1.64 Gyr and hence maintains plate motion on stage II. [32] In summary, case WK660 suggests that chemical stratification well develops even under the influence of convective stirring induced by moving plates and subducting slabs provided that the mantle is kept hot enough to induce vigorous ridge and hot spot magmatism Case LIHWK [33] To see if the qualitative features of case WK660, i.e., plate motion that continues for billions of years and the development of chemical stratification beneath moving plates, are the robust features of the coupled magmatismmantle convection system when the olivine barrier is negligible, I carried out the numerical experiment of case LIHWK, where both the internal heating rate and the initial temperature are lowered (see Table 1). I assumed that the internal heating rate is constant in time and is uniformly distributed for simplicity. The lower internal heating rate and initial temperature imply that the magmatism is milder in case LIHWK than in case WK660. The main purpose of case LIHWK is to see if chemical stratification well develops in the mantle even under this mild magmatism. [34] The result is presented in Figure 5. Plate motion continues throughout the calculated period of time, about 5 Gyr (Figure 5c). Ridge volcanism (see arrow a in Figure 5g for t = 1.17 Gyr) actively occurs at t < 1 Gyr but declines during the period of 1 Gyr < t < 2.9 Gyr and stops at t = 2.9 Gyr (Figure 5e) because of cooling mantle (Figure 5a). Though the ridge volcanism in case LIHWK is considerably milder than the ridge and hot spot volcanism in case WK660 (see Figures 2e and 5e), chemical stratification still develops well at t < 2 Gyr (Figure 5f for t = 1.17 and 2.35 Gyr); a layer of materials enriched in basaltic component develops well along the bottom boundary, while the residual materials separate from the subducted basaltic crust in the lower mantle and rise up to the upper mantle. During the period of t > 2.5 Gyr, however, ridge volcanism becomes negligible and the mantle becomes more chemically homogeneous (Figure 5f for t = 3.47 and 5.13 Gyr) owing to convective stirring by moving plates and subducting slabs. Even at t = 5.13 Gyr, more than 2 Gyr after the ridge volcanism stops (Figure 5e), however, the chemical heterogeneity in the lower mantle still survives convective stirring (Figure 5f ). Once formed, chemical heterogeneity can survive convective stirring for billions of years. This result is consistent with the results of earlier numerical experiments of thermal and chemical convection [Davies and Gurnis, 1986; Sidorin and Gurnis, 1998; Hansen and Yuen, 2000; Montague and Kellogg, 2000; Tackley, 2002]. [35] In summary, case LIHWK suggests that chemical stratification and continuous plate motion observed in case WK660 are robust features of the coupled magmatismmantle convection system with moving plates modeled here when the olivine barrier is negligible Cases With Stronger Olivine Barrier [36] The olivine barrier is stronger than assumed in section 3.1 in the Earth when temperature is less than about 2000 K [Hirose, 2002]. To see how the stronger olivine barrier affects the development of chemical stratification and continuous plate motion observed in section 3.1, I carried out several numerical experiments with 2 MPa K 1 for the Clausius-Clapeyron slope of the 660 km phase boundary (Table 1). Since the effects of positive Clausius-Clapeyron slopes of the olivine-spinel transition at 400 km depth [Schubert et al., 2002] and the garnet-perovskite transition [Hirose et al., 1999] are neglected, the barrier effect of the phase transitions in the Earth s transitional layer to the mass exchange between the upper and lower mantle is overestimated in these numerical models. The Earth s mantle is somewhere between these models and cases WK660 and LIHWK in section 3.1 when the temperature at 660 km depth is less than about 2000 K Case ST660 [37] Figures 6 to 8 show the numerical model of case ST660 where the internal heat source decays with time (Table 1). The reduced density contrast between the mid upper mantle and the uppermost part of the lower mantle shown in Figure 8 is defined from equation (8) as ½drŠ ¼ Z w 0 dx Z 220þz 220 dzdr Z 710þz 710 dzdr = ðwzþ; ð9þ where w is the width of convecting box, z is depth in km, and z is taken to be 150 km. The depth ranges for integration in equation (9) are chosen to avoid the thermal and chemical boundary layers that develop along the surface and the 660 km phase boundary (see below). [38] Figure 6 shows that the mantle of case ST660 evolves in three stages instead of two stages in case WK660. Plates continuously move only on stage I (Figure 6c), and surface velocity is zero and plates are stagnant except when there are spikes in the plots of Figure 6b to 6e on stages 0 and II. I will describe the three stages on the order of their occurrences Stage 0, Mantle Overturn [39] On stage 0 (t < 0.9 Gyr), spikes arise in the plots of Figures 6b to 6e at t = 0.43 and 0.62 Gyr, while average temperature in the entire mantle T WM drops and the average temperature in the upper mantle T UM jumps up at the

11 OGAWA:MAGMATISM AND MOVING PLATES ETG 5-11 Figure 5. Model of case LIHWK. (a) to (e) Same as Figures 2a to 2e, respectively. Also shown are (f ) the distributions of bulk composition in the mantle and degree of damage along the surface boundary w s and (g) the distributions of temperature and magma in the mantle and surface velocity u s at the elapsed time shown. The contour interval for magma distribution is 5% and u s is in cm yr 1. See color version of this figure at back of this issue.

12 ETG 5-12 OGAWA:MAGMATISM AND MOVING PLATES Figure 6. Same as Figure 2 but for case ST660. moment of the spikes (Figure 6a). Figure 7 for t = 0.01 to 0.92 Gyr shows that the spikes and the sudden changes in T UM and T WM are the results of mantle-wide overturn that takes place at t = 0.43 and 0.62 Gyr: [40] After the vigorous magmatism induced by the high initial temperature (see arrow a in Figure 7b for t = 0.01 Gyr), both the upper mantle and the lower mantle become chemically stratified. A basaltic crust develops along the surface boundary (arrow b in Figure 7a for t = 0.15 Gyr), a layer of residual materials depleted in the basaltic component and heat producing elements develops just beneath the basaltic crust (arrow d), and a layer of materials enriched in the basaltic component and heat producing elements develops along the bottom boundary (arrow e in Figure 7a). The chemical buoyancy induced by the stratification makes the entire mantle stagnant at t = 0.15 Gyr. At this moment, I found no difference between case ST660 and case WK660 described in section 3.1. [41] The effect of stronger olivine barrier becomes, however, important during the subsequent period of time on stage 0. As the upper mantle is cooled from the surface boundary, secondary cold plumes start to grow at the base of the stagnant lithosphere (see, for example, the cold plume indicated by arrow f in Figure 7b for t = 0.42 Gyr). The secondary plumes are opposed at the 660 km phase boundary by the olivine barrier and stirs only the already cooled upper mantle; a secondary convection occurs within the upper mantle and the lithosphere remains stagnant till t = 0.4 Gyr. (In case WK660 of section 3.1, in contrast, the secondary plumes penetrate into the still hot lower mantle to let the hot lower mantle materials rise up to the surface as a return flow. The return flow induces magmatism that breaks the lithosphere to form plate margins and hence induces plate motion. This is the reason why continuous plate motion starts at t = 0.2 Gyr in case WK660.) [42] At t = 0.42 Gyr, a hot plume rises up to induce magmatism and to break the lithosphere as indicated by arrow g in Figure 7b. In contrast to case WK660, however, the resulting plate motion does not continue long. When the subducted slab (see arrow t in Figure 7b for t = 0.42 Gyr) hits the lower mantle, mantle overturn starts (Figure 7 for t = 0.43 Gyr). The hot material indicated by arrow h (Figure 7b) rises up from the lower mantle as a huge plume to induce a large partially molten region in the upper mantle. This magmatism induced by mantle overturn is the cause of the pulses in the plots of surface heat flow etc. in Figures 6b to 6e and the sudden changes in T WM and T UM in Figure 6a at t = 0.43 Gyr. The mantle overturn at t = 0.43 Gyr occurs because, by that time, (1) the secondary convection has cooled the upper mantle to make it sufficiently dense (see the decrease in T UM with time in Figure 6a) and (2) the lower mantle, which is mechanically isolated from the surface boundary by the olivine barrier, has been heated up by the strong internal heating of stage 0 (Figure 7b for t = 0.43 Gyr) and has become thermally buoyant. Indeed, the reduced density contrast [dr] in Figure 8 is as large as 15 kg m 3 at t = 0.42 Gyr. After the magmatism and convective stirring due to mantle overturn subside, the mantle becomes chemically stratified and the entire mantle including the lithosphere becomes stagnant owing to the chemical buoyancy, again. Then mantle overturn again occurs at t = 0.62 Gyr. By the end of stage 0 (t 0.9 Gyr), the repeated magmatism due to mantle overturn makes the entire mantle chemically stratified (Figure 7a and 7c) Stage I, Plate-Dominated Mantle [43] As the internal heat source decays, mantle overturn stops and, instead, I observed the dynamic behavior of the mantle essentially the same as that of stage I of case WK660 during the period of 0.92 Gyr < t < 2 Gyr. A continuous plate motion starts at t = 0.92 Gyr when the lithosphere is hit by a partially molten plume (arrow i in Figure 7b) and is broken (see w s in Figure 7b). This plate motion continues during the subsequent period of t = 1 to 2 Gyr (stage I of Figure 6) and induces ridge volcanism as indicated by arrows r in Figure 7b for t = 1.03 and 1.17 Gyr. The plate motion does not induce mantle overturn on stage I in contrast to stage 0 since the lower mantle is not sufficiently hot and buoyant any more because of the decayed internal heating. The basaltic crust and residual materials generated by the ridge volcanism sink to the bottom boundary as a part of subducting slab (see arrows j in Figure 7a for t = 1.03 to 1.81 Gyr) and separate from each other at depth in the lower

13 OGAWA:MAGMATISM AND MOVING PLATES ETG 5-13 Figure 7. Same as Figure 3 but for case ST660. The contour interval for magma distribution is 10%. See color version of this figure at back of this issue.

14 ETG 5-14 OGAWA:MAGMATISM AND MOVING PLATES Figure 7. (continued)

15 OGAWA:MAGMATISM AND MOVING PLATES ETG 5-15 Figure 8. Reduced density contrast, i.e., difference in average reduced density deviation, between the mid upper mantle and the uppermost part of the lower mantle defined by equation (9) plotted against time for case ST660. The averaging is spatially carried out in the depth range of 220 to 370 km for the mid upper mantle and 710 to 860 km for the uppermost part of the lower mantle. In a crude sense, positive value of the contrast means top heavy density distribution. mantle (see the circle in Figure 7a for t = 1.81 Gyr). Namely, the ridge volcanism induces chemical differentiation of the mantle to maintain the chemical stratification originally formed on stage 0. Because of the deep slab penetration into the lower mantle, a broad lateral heterogeneity develops in thermal and compositional distributions at depth in the lower mantle (Figures 7a and 7b for t = 1.04 to 1.81 Gyr). Occasionally, hot plumes rise up from the lower mantle to induce hot spot magmatism as indicated by arrows p and q in Figure 7b for t = 1.04 to 1.81 Gyr. The hot plumes are more enriched in heat producing elements than the surrounding upper mantle (Figure 7c). Some of the hot plumes break the lithosphere to induce new plate margins. Owing to these plumes, spikes arise in the plots of surface heat flow etc. in Figures 6b to 6e. [44] Though the effect of the stronger olivine barrier on the overall convective flow pattern is minor on stage I, the barrier does affect the way hot plumes grow. In case WK660, hot plumes always start from the broad hot regions in the lower mantle (Figure 3 for t = 0.81 Gyr). This type of hot plumes occurs in case ST660, too (arrows p in Figure 7b for t = 1.04 and 1.81 Gyr). The hot plumes indicated by arrows q in Figure 7b for t = 1.04 to 1.61 Gyr, however, start from the 660 km phase boundary away from the broad hot region at depth in the lower mantle (arrow k in Figure 7b for t = 1.81 Gyr). These hot plumes occur as a return flow of subducting slabs and are hotter than the surrounding upper mantle because the temperature in the return flow is raised by the latent heat release at the endothermic 660 km phase boundary; the return flow takes the form of narrow jets rather than a diffuse back ground flow when the flow crosses the 660 km phase boundary. Similar plumes are observed in earlier numerical models of thermal convection with the 660 km phase boundary [see, e.g., Steinbach and Yuen, 1998, Figure 1a], too Stage II, Mantle With Weak Internal Heating [45] As the internal heat source further decays, the activity of hot uprising plumes declines and hot plumes stop inducing new plate margins as in case WK660 (see the low spikes at t = 1.7 to 1.95 Gyr in Figures 6b to 6e). On account of the decay of plume activity, the plate motion stops at t 2 Gyr, when the only ridge at the moment merges with a subduction zone and disappears, and does not resume on stage II in contrast to case WK660 (see Figure 6). This occurs because the olivine barrier strongly opposes the upwelling flow of hot lower mantle materials into the upper mantle that occurs as a part of the return flow of subducting slabs in case ST660; the hot materials are the agent that induces new plate margins and maintains plate motion on stage II in case WK660 (see arrows f to h in Figure 3 for t = 1.64 Gyr). As a result, mantle convection occurs separately in the shallower part of the lower mantle and the upper mantle beneath the stagnant lithosphere, and the deeper part of lower mantle, which used to be laterally heterogeneous on stage I, gradually becomes laterally homogeneous during the period of 2 to 3.3 Gyr (Figure 7). Because of the lack of mantle wide convective circulation, internally generated heat builds up in the lower mantle while the upper mantle is cooled by secondary convection (Figure 7b for t = 2.04 and 3.31 Gyr) and the lower mantle becomes more buoyant than the upper mantle by t = 3.5 Gyr (Figure 8). [46] The top heavy density distribution induces mantle overturn at t around 3.5 Gyr (Figure 7). When the turmoil by mantle overturn subsides, however, the lithosphere becomes again stagnant; the chemical stratification induced by the vigorous magmatism due to the overturn makes the lithosphere too chemically buoyant to subduct as is the case for the mantle overturn at t = 0.43 Gyr (see Figure 7). Mantle convection occurs as a layered convection beneath a stagnant lithosphere and the lower mantle becomes laterally homogeneous again after the mantle overturn (see Figure 7 for t = 4.91 Gyr) Reverse Experiment for Stage I [47] Stage I characterized by moving plates continues only for 1Gyr in casest660. Since the basic equations aresolved as an initial value problem starting from an arbitrarily set initial condition, a possibility still remains that stage I is a transient phenomenon induced by the assumed initial condition. To confirm that this is not the case, I carried out a reverse experiment (case Rev in Table 1). First, I picked up the snap shot of internal heating rate distribution at t = 2.78 Gyr, when the thermal and chemical state of mantle is on stage II. Then, I uniformly multiplied the internal heating rate by a factor so that the spatially averaged internal heating rate becomes Wm 3.This value ofaverage internal heating rate occurs at t = 1.47 Gyr, well within stage I, in case ST660 as shown in Figure 6. Starting from this enhanced internal heat source together with the temperature, composition, and degree of damage distributions obtained at t = 2.78 Gyr in case ST660, I numerically integrated the basic equations till the initial transient behavior disappears. I assumed q = 0 in equation (7) to keep the average internal heating rate constant (see Table 1). I confirmed that plate motion resumed and hence that stage I resumed in case Rev. Stage I is, therefore, not a transient phenomenon induced by the assumed initial condition Chemical Stratification and Plate Motion Under Strong Olivine Barrier [48] In case ST660, chemical stratification well develops as in case WK660 but plates continuously move only on stage I and are stagnant on stage II. To see if stagnant lithosphere is a robust feature of stage II when the olivine barrier is strong, I calculated case LIHST where the initial temperature and internal heating rate is lowered from the

16 ETG 5-16 OGAWA:MAGMATISM AND MOVING PLATES values assumed in case ST660 (Table 1). The assumed internal heating rate Wm 3 occurs on stage II in case ST660 (see Figure 6); as in case LIHWK, I assumed that the internal heating rate is constant in time and uniformly distributed in this case. The lower initial temperature and internal heating rate implies that the magmatism of case LIHST is milder than the magmatism of case ST660. I present the result in Figure 9. [49] Figures 9c and 9g indicate that plates continuously move during the periods of t < 0.5 Gyr and 1.9 Gyr < t < 4.4 Gyr. A strong olivine barrier does not always lead to a stagnant lithosphere at low internal heating rate. The figures also indicate that the plates become stagnant during the subsequent period of t > 4.4 Gyr. At t > 9 Gyr, mantle overturn similar to the one observed in Figure 7 for t = 3.5 Gyr repeatedly occurs to induce the spikes in the plots of Figures 9b to 9e. The thermal and chemical state of the mantle of stage II in case ST660 is reproduced at t > 9 Gyr. [50] The continuous plate motion at t < 4.4 Gyr is a result of the smaller chemical buoyancy that accompanies the milder chemical stratification formed by the milder magmatism of case LIHST. The magmatism during the initial transient period (t < 0.1 Gyr, see Figure 9e) is not active and does not chemically differentiates the mantle so much as the corresponding magmatism of case ST660 does because of the lower initial temperature. Hence the lithosphere is not so chemically buoyant and a continuous plate motion takes place at t < 0.5 Gyr (Figure 9g for t = 0.3 Gyr). Even the resulting ridge volcanism (arrow a in Figure 9g) does not induce significant chemical stratification (Figure 9f for t = 0.3 Gyr) and the mantle remains chemically almost homogeneous at t = 0.5 Gyr when the plate motion stops. In such a chemically homogeneous mantle, only a slight excess of the lower mantle temperature is sufficient for making the lower mantle more buoyant than the upper mantle and mantle overturn occurs at t = 1.88 Gyr (Figure 9g). However, the magmatism induced by the overturn is not so vigorous as the magmatism due to overturn on stage II of case ST660, since the uprising lower mantle materials are not so hot (compare Figure 9g for t = 1.88 Gyr with Figure 7b for t = 3.5 Gyr). The magmatism does not make the lithosphere so chemically buoyant but still breaks the lithosphere to induce new plate margins. As a consequence, plate motion resumes at t = 1.9 Gyr and continues till t = 4.4 Gyr when the only ridge that is active at that moment merges with a subduction zone and disappears (Figures 9c and 9g for t = 2.15 and 3.04 Gyr). [51] On the later stage of case LIHST, however, mantle overturn does not induce a continuous plate motion because the ridge volcanism during the period of 1.9 Gyr < t < 4.4 Gyr (arrow b in Figure 9g for t = 3.04 Gyr) has induced a significant chemical stratification in the mantle (Figure 9f for t = 3.04 and 5.04 Gyr). Once the lower mantle becomes chemically much denser than the upper mantle, mantle overturn occurs only when the lower mantle is much hotter than the upper mantle as can be seen from the curves of T UM and T WM in Figure 9a at t > 9 Gyr; mantle overturn occurs at t = 9.5, 12.3, and 13.3 Gyr (see Figures 9b to 9e). Since the upwelling lower mantle materials are now hot, such mantle overturn induces vigorous magmatism similar to the one shown in Figure 7b for t = 3.5 Gyr. The vigorous magmatism makes the lithosphere chemically much buoyant and this is the reason why the magmatism due to mantle overturn at t > 9.5 Gyr does not induce continuous plate motion and instead makes the lithosphere stagnant. [52] Case LIHST shown in Figure 9 suggests that plates can continuously move even when the olivine barrier is strong provided that the chemical buoyancy that accompanies chemical stratification is not so strong. I confirmed this suggestion by calculating case WKCH2 where I weakened the compositional dependence of density by a factor of 2/3 from the value assumed in case LIHST (see Table 1). Namely, I obtained a result very similar to the one shown in Figure 5 (case LIHWK): (1) Plate motion continues throughout the calculated period of time (5 Gyr) and, at the end of the calculation, the continuous plate motion occurs as a part of the statistically steady state reached by this time. (2) Chemical stratification well develops at t < 2 Gyr when the mantle is rather hot and ridge volcanism is active, and survives convective stirring during the subsequent period of 2 Gyr < t < 5 Gyr when ridge volcanism is negligible or does not occur at all owing to the cooling mantle. At a higher internal heating rate (case WKCH3 in Table 1), however, I did not obtain a continuous plate motion and instead I obtained a mantle that behaves like the mantle of stage 0 in case ST660. The weaker compositional dependence of density of case WKCH3 makes the mantle-wide overturn and the resulting vigorous magmatism similar to the one observed in Figure 7 for t = 0.43 Gyr possible at the internal heating rate of Wm 3. [53] In contrast to the compositional dependence of density, I found that the mechanical strength of lithosphere (s I and s D in Figure 1d) did not so critically influence the numerical results (cases WKPL2 and WKPL3 in Table pffiffiffiffiffi 1). I lowered the mechanical strength by a factor of 10 from the value assumed in case ST660 in these cases by making larger (see Table 1 and equation (5)) and still I obtained a chemically stratified mantle that behaves like the mantle of stage I (stage II) shown in Figures 6 and 7 in case WKPL3 (WKPL2) where the internal heating rate is Wm 3 ( Wm 3 ), a value that occurs on stage I (stage II) in case ST660. Namely, I obtained a stagnant lithosphere and mantle overturn that intermittently occurs at the internal heating rate of Wm 3 (case WKPL2) but obtained a more continuous plate motion without mantle overturn at the higher internal heating rate of Wm 3 (case WKPL3). [54] In summary, the numerical models presented here suggest that (1) chemical stratification well develops in the mantle regardless of the strength of olivine barrier provided that the mantle is hot enough to induce vigorous magmatism and (2) plates continuously move when mass exchange between the upper mantle and the lower mantle is not so strongly impeded by the olivine barrier and the chemical buoyancy that accompanies the chemical stratification. 4. Discussion 4.1. Convective Regimes Chemically Stratified Regime [55] Chemical stratification well develops in all of the cases enumerated in Table 1 when the mantle is sufficiently hot as discussed above. This result suggests that the chemically stratified regime originally identified for a

17 OGAWA:MAGMATISM AND MOVING PLATES ETG 5-17 Figure 9. Same as Figure 5 but for case LIHST. See color version of this figure at back of this issue.

18 ETG 5-18 OGAWA:MAGMATISM AND MOVING PLATES highly idealized model of coupled magmatism-mantle convection system in paper 1 (see section 2.3) exists under more realistic circumstances, too. In particular, the chemically stratified regime is likely to exist for a mantle where mechanically strong plates continuously move and subduct. Moving plates affect how chemically stratified structure develops in the mantle in two ways: (1) Moving plates induce ridge volcanism, the major agent that induces chemical differentiation in the Earth s mantle [e.g., van Keken et al., 2002]. (2) Subducting slabs make basaltic crusts and their residue formed by ridge volcanism sink together deep into the lower mantle even under the influence of the basalt barrier [e.g., Gaherty and Hager, 1994]. Both of these processes are missing in the earlier models of paper 1 where the viscosity is constant and there are no moving plates and subducting slabs. Namely, ridge volcanism is not well simulated and only the basaltic crusts sink to the lower mantle leaving their chemically buoyant residue behind in the upper mantle in the models of paper 1. Both of the processes 1 and 2 are, however, selfconsistently simulated in the models presented here as shown for cases WK660, LIHWK (the earlier stage), ST660 (stage I), and LIHST (the earlier stage) in Figures 3, 5, 7, and 9, respectively. The chemically stratified regime is likely a robust feature of the coupled magmatism-mantle convection system in a sufficiently hot mantle. (However, see the caveat below.) [56] A crucial step in the development of chemical stratification observed here is the separation of subducted basaltic crusts and their chemically buoyant residue, which occurs in deep lower mantle (see, for example, the circle in Figure 3 for t = 0.81 Gyr). The separation is consistent with the results of earlier laboratory experiments [Olson and Kincaid, 1991] and some of the earlier numerical experiments [Christensen and Hofmann, 1994; Mambole and Fleitout, 2002; Davies, 2002] of mantle convection. The amount of subducted basaltic crusts that segregate from the convecting mantle observed here is, however, considerably larger than the amount observed in some of the earlier numerical experiments [Gurnis, 1986; Christensen and Hofmann, 1994; Davies, 2002]. There are three major reasons for the difference to arise: [57] 1. These earlier models are designed to simulate the mantle convection in the present Earth, where ridge volcanism is mild and forms only a thin basaltic crust of typically 6 km in thickness. This volcanism is much milder than the volcanism often observed in the models described in section 3. In case WK660, for example, the thickness of the basaltic crust is 15 to 30 km, an appropriate value for the early Earth [McKenzie, 1984; Davies, 1993]; I will apply the numerical results presented here mostly to the hot mantle of the early Earth. [58] 2. The Rayleigh number adopted here is Earth-like, i.e., 10 8 and is 2 3 orders of magnitude higher than the Rayleigh number assumed by Gurnis [1986] and Christensen and Hofmann [1994]. The higher Rayleigh number implies that the Stokes velocity of subducted basaltic crusts is much higher in the present numerical experiments than in these numerical experiments. The higher Stokes velocity implies stronger tendency toward segregation of subducted basaltic crust [Gurnis, 1986]. [59] 3. In an earlier model [Christensen and Hofmann, 1994], the mantle is heated mostly from the bottom boundary and hot plumes actively grow there to induce strong convective stirring at depth in the lower mantle. The lower mantle of the present model is, however, much more quiescent since the mantle is mostly heated by internal heat source as is the case for the Earth s mantle [Davies and Richards, 1992]. [60] Some cautions are, however, necessary here. As discussed in section 2.3, the chemically stratified regime arises as a result of dynamic balance between convective stirring and chemical differentiation due to magmatism. The efficiency of convective stirring depends on the complexity of plate motion [Gurnis and Davies, 1986; Gurnis, 1986; Christensen, 1989]. The convective stirring induced by a rather regular plate motion simulated here may be less efficient than the convective stirring induced by the more irregular plate motion of the Earth and the tendency toward development of chemical stratification may be overestimated in the models presented here. Furthermore, the efficiency of convective stirring depends on the spatial dimension [Schmalzl et al., 1996; Ferrachat and Ricard, 1998; van Keken and Zhong, 1999], too. Further extensive numerical experiments with various plate configuration in three dimensional space are important to establish the relevance of the chemically stratified regime to the Earth s mantle Thermal Convection Regime [61] In case LIHWK (and also WKCH2), where the internal heating rate is low (see Table 1) and still plates continuously move throughout the calculated period of time, the ridge volcanism declines as the upper mantle becomes colder (Figures 5a and 5e) and eventually the chemical differentiation due to magmatism becomes negligible (t > 2.5 Gyr in Figure 5f). As a result, the convective stirring due to moving plates and subducting slabs dominates the thermal and chemical state of the mantle and the mantle falls on the thermal convection regime (section 2.3 and paper 1) on the later stage in these cases. Though the thermal convection regime is separated from the chemically stratified regime by a bifurcation as shown in paper 1, Figure 5 suggests that the transition from the chemically stratified regime to the thermal convection regime only gradually occurs. This is because, once formed, a chemically stratified structure survives convective stirring due to moving plates and subducting slabs for billions of years without the help of further chemical differentiation due to magmatism at the assumed magnitude of compositional dependence of density as discussed in section Continuous Plate Motion [62] Plates continuously move regardless of the strength of the olivine barrier when the internal heating rate is moderate in the models presented here (cases WK660, Rev, WKPL3, and stage I of case ST660). At lower internal heating rate, however, whether or not plates continuously move critically depends on the strength of the olivine barrier. Plates continuously move at lower internal heating rate when the barrier is weak (cases WK660 and LIHWK) but become stagnant when the barrier is stronger (stage II of case ST660, the later stage of case LIHST, and case WKPL2) unless the chemical buoyancy that accompanies the chemical stratification in the mantle is mild (case WKCH2 and the earlier stage of case LIHST). [63] A stronger olivine barrier tends to make the lithosphere stagnant at low internal heating rate because a strong

19 OGAWA:MAGMATISM AND MOVING PLATES ETG 5-19 olivine barrier suppresses the agents that break the lithosphere to form mechanically weak plate margins in the chemically stratified mantle modeled here. As emphasized in section 2.2 and paper 3, the rupture strength of the lithosphere s I (Figure 1d) is higher than the stress induced by the weight of the lithosphere s R regardless of the chemical structure of the mantle (the plate-like regime) in all of the cases in Table 1 except cases WKPL2 and WKPL3. Even in cases WKPL2 and WKPL3 where s I = 38 MPa and s I < s R would hold if the mantle is chemically homogeneous (the weak plate regime of section 2.2), s I is still higher than s R since the lithosphere modeled here is chemically buoyant; basaltic crusts are chemically buoyant at depths less than the depth of basalt-eclogite transition. Plate margins cannot spontaneously develop under such circumstance and develop only when there are uprising flow of hot materials from deep mantle, like the hot plumes on stage I and the hot return flow of subducting slabs on stage II of case WK660 (Figure 3). Such uprising flow from the lower mantle is suppressed by the strong olivine barrier on stage II of case ST660, the later stage of case LIHST, and in case WKPL2. (Notice that the mantle overturn observed in these cases cannot induce continuous plate motion since the magmatism induced by the mantle overturn is so vigorous as to make the lithosphere chemically too buoyant to subduct.) 4.2. Implications for the Earth Regime of the Earth s Mantle [64] The existence of the chemically stratified regime under the influence of moving plates is important in understanding the apparent conflict [Tackley, 2000a; van Keken et al., 2002] between the whole mantle convection suggested from deep slab penetration [van der Hilst et al., 1997] and the chemical stratification in the mantle suggested from geochemical observations [Hofmann, 1997], seismic observations [e.g., van der Hilst and Karason, 1999] and energy balance estimates [e.g., Kellogg et al., 1999] of the Earth s mantle. Chemical stratification would well develop in the Earth s mantle even under the influence of stirring by moving plates and subducting slabs, if internal heating rate has been so high that the Earth s mantle has been on the chemically stratified regime for billions of years in its early stage of evolutionary history. Given the strong plume activity in the Archean and early Proterozoic suggested from the observations of Large Igneous Provinces [Isley and Abott, 1999; Eldholm and Coffin, 2000] and the high temperature in the Archean hot plumes suggested from the observations of komatiite (see Campbell and Griffiths [1992] for a review), it is a viable hypothesis that the mantle of early Earth is on the chemically stratified regime. [65] The chemical heterogeneity in the present mantle [Hofmann, 1997], however, does not necessarily imply that the magmatism is active enough to keep the mantle on the chemically stratified regime even in the present Earth [Sidorin and Gurnis, 1998]. Once well established, chemical stratification can survive convective stirring for billions of years without the help of further chemical differentiation due to magmatism if chemical buoyancy is sufficiently strong as observed in Figure 5 and has been emphasized by Davies and Gurnis [1986], Sidorin and Gurnis [1998], Hansen and Yuen [2000], Montague and Kellogg [2000], and Tackley [2002]. It still remains an open issue to identify the thermal and chemical regime of the present mantle Broad Hot Regions in the Lower Mantle [66] In all of the cases calculated here, subducting slabs, if occur, penetrate into the lower mantle and induce broad hot regions, where thermally induced positive buoyancy balances chemically induced negative buoyancy and mantle materials are neutrally buoyant (see Figure 4). An implication of the neutral buoyancy in the hot regions is that the broad seismically slow regions in the Earth s lower mantle [e.g., Su et al., 1994], which are similar in shape to the broad hot regions observed in the present numerical model, are not necessarily buoyant in spite of the inferred high temperature there [e.g., Forte and Mitrovica, 2001]. Indeed, a geodynamic model [Ricard et al., 1993] suggests that the Earth s geoid is fairly well explained by the density anomaly due to subducting slabs alone without assuming positive buoyancy for the seismically slow regions in the lower mantle. [67] Acknowledgments. The present numerical calculation was carried out on VPP-5000 at the computation center of Nagoya University. I would like to thank A. Lenardic, L. H. Kellogg, and an anonymous reviewer for their helpful comments. This work is financially supported by KAKENHI by MEXT of Japan. References Becker, T. W., J. B. Kellogg, and R. J. 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20 ETG 5-20 OGAWA:MAGMATISM AND MOVING PLATES Davies, G. F., and M. A. Richards, Mantle convection, J. Geol, 100, , de Smet, J. H., A. P. van den Berg, and N. J. Vlaar, The evolution of continental roots in numerical thermo-chemical mantle convection models including differentiation by partial melting, Lithos, 48, , Dupeyrat, L., C. Sotin, and E. M. Parmentier, Thermal and chemical convection in planetary mantles, J. Geophys. Res., 100, , Eldholm, O., and M. F. Coffin, Large igneous provinces and plate tectonics, in The History and Dynamics of Global Plate Motions, Geophys. Monogr. Ser., vol. 121, edited by M. A. Richards, R. G. Gordon, and R. D. van der Hilst, pp. 326, AGU, Washington, D. C., Erickson, S. G., and J. Arkani-Hamed, Subduction initiation at passive margins: The Scotian basin, eastern Canada as a potential example, Tectonophysics, 12, , Ferrachat, S., and Y. Ricard, Regular vs. chaotic mantle mixing, Earth Planet. Sci. Lett., 155, 75 86, Ferrachat, S., and Y. 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Ogawa, Department of Earth Sciences and Astronomy, University of Tokyo at Komaba, Meguro, Tokyo, , Japan. (masaki@chianti.c. u-tokyo.ac.jp)

21 OGAWA:MAGMATISM AND MOVING PLATES ETG 5-8

22 OGAWA:MAGMATISM AND MOVING PLATES Figure 3. (opposite) Distributions of (a) bulk composition x, (b) temperature T, and (c) normalized internal heating rate q norm, all indicated by color, at the elapsed time shown for case WK660. Also shown are the location of 660 km phase boundary (solid line in Figure 3a), the distribution of magma j (contour lines in Figure 3b), and the plots of horizontal velocity u s (black curve and black scale in Figure 3b) and degree of damage w s (red curve and red scale in Figure 3b) both along the surface boundary. In Figure 3a, basaltic materials (x = 0.1) are indicated by blue, while residue of magma (x > x init = 0.64) is indicated by yellowish green to orange. In Figure 3c, depleted materials (q norm < 1) are indicated by blue to green, while undepleted materials (q norm > 1) are indicated by green to orange. The contour interval for magma content j in Figure 3b is 5%. The scales for the plots of u s and w s are in cm yr 1 and nondimensional, respectively. ETG 5-9

23 OGAWA:MAGMATISM AND MOVING PLATES Figure 5. Model of case LIHWK. (a) to (e) Same as Figures 2a to 2e, respectively. Also shown are (f ) the distributions of bulk composition in the mantle and degree of damage along the surface boundary w s and (g) the distributions of temperature and magma in the mantle and surface velocity u s at the elapsed time shown. The contour interval for magma distribution is 5% and u s is in cm yr 1. ETG 5-11

24 OGAWA:MAGMATISM AND MOVING PLATES Figure 7. Same as Figure 3 but for case ST660. The contour interval for magma distribution is 10%. ETG 5-13

25 OGAWA:MAGMATISM AND MOVING PLATES Figure 7. (continued) ETG 5-14

26 OGAWA:MAGMATISM AND MOVING PLATES Figure 9. Same as Figure 5 but for case LIHST. ETG 5-17