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Mantle Convection and Global Sea Level: Implications for the Emergence of Plate Tectonics on the Earth Tetsuzo Seno and Satoru Honda Earthquake Research Institute, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan ABSTRACT We investigate the relationship between modes of mantle convection and the global sea level on the basis of new pieces of geological evidence; they are mantle potential temperature decrease by 200°C since 3 Ga, existence of sea water at 3.8 Ga, and start of regassing of water into the mantle around 1 Ga. We calculate a secular change of the sea level, using a parameterized convection approach, taking into account the continental growth, degassing/regassing of water, and change in thickness of the oceanic crust-harzburgite layer. Assuming β = 0.3, an index of the power law relation between Nusselt and Rayleigh numbers, we obtain a sea level higher by more than 3000 m in the Archean than present. This high sea level is inconsistent with the geological evidence of early Proterozoic emergence of continents. Previous studies on plate tectonics-like convection show that β is around 0.3. Thus, our study indicates that the assertion that plate tectonics has been operating for the past 4 b.y. is unlikely. To be consistent with the early Proterozoic emergence of continents, we present a model of the mantle convection which had been operated before plate tectonics, in which surface plates were stagnant before 2.0 Ga, and during 2.0-1.4 Ga, buoyant slabs were driven by convection into the asthenosphere. The sea level calculated from this model shows continent emergence during 2.8-1.9 Ga and 1.0-0 Ga. This two-stage continental emergence may have important implications for the evolution of life, banded iron formation, and ice age distribution. Introduction Plate tectonics has been functioning to exchange volatile between surface reservoirs and mantle (Schubert et al., 1989; McGovern and Schubert, 1989; Tajika and Matsui, 1992). The water and CO2 would be the most important volatile among others which affect surface environment and evolution of life. Although these agents affect surface environment through various factors, a global sea level change (or a continental freeboard defined in an opposite sense to the sea level) is one of the convenient measures by which we can investigate the effect of the surface water 1 on the environment. Hereinafter we call the global sea level, from which effects of the postglacial rebound, breakup and closing of continents, and local crustal movements are eliminated, simply the sea level. Since the sea level is closely related to mantle convection through various factors, such as the average sea floor depth, degassing or regassing of water, and continental crust volume (Wise, 1974; Turcotte and Burke, 1978; Schubert and Reymer, 1988), it provides a key to understand the interaction between the mantle activity and the surface layers through geological time. In this study, we explore this relationship more extensively than previous studies, taking into account newly obtained geological constraints. Schubert and Reymer (1988) noted importance of secular cooling of the Earth on the sea level through subsidence of the sea floor. They showed a few tens of percent growth of the continent area is necessary to maintain freeboard at a constant value since the early Proterozoic. Galer (1991) further took into account the oceanic crust thickness change due to the mantle potential temperature change in the past, and showed that this produces a large effect on the calculated sea level. Since the mantle temperature is likely to be high in the past, producing thicker crust at the ridge axis (Sleep and Windley, 1982; McKenzie and Bickle, 1988), a higher sea floor elevation and thus a higher sea level result. Galer (1991) considered the effects of three free parameters on the sea level, i.e. mantle potential temperature, sea floor spreading rate, and asymptotic plate thickness. He concluded that the potential temperature should not be higher by more than 150°C in the Archean than present to maintain freeboard at a constant value. Since the sea floor spreading rate and plate thickness are not independent from but vary along with the potential temperature, his conclusion on the Archean potential temperature is worth to be re-examined. In this study we reconstruct the sea level change in the past by a parameterized convection approach similar to, but different in several important aspects from the previous studies. The mantle potential temperature (denoted by Tm) is used to parameterize the sea floor spreading rate, surface plate velocity, maximum plate age, and degassing-regassing rates. We also take into account the temporal change in thickness of the oceanic crust and harzburgite layer with Tm, similar to Galer (1991), and various continental growth curves. The new aspect of our modeling is that these parameters, except for the continental growth curve, are determined by Tm simultaneously. Another new aspect of our study is that newly found geological evidence is used for the construction of the model. We use the mantle temperature constrained by recent petrologic experiments on the Archean-Proterozoic igneous rocks. In order to constrain the degassing and regassing rates, we use geological evidence of the existence of sea water at 3.8 Ga and recent petrologic experiments on slab dehydration. Finally we relax the constant 2 freeboard hypothesis (Wise, 1974; Schubert and Reymer, 1988), because it might be too rigorous than that actually observed (e.g., Moores, 1994). We first assume β = 0.3, which may be expected for the plate tectonics regime, and obtain significant inundation of continents during the Archean-early Proterozoic. This conflicts with the emergence of vast areas of continents during the early Proterozoic (Windley, 1977; 1995). This means that plate tectonics may not have worked during the whole geological time. We then examine the convection mode prior to plate tectonics and propose a new model, a series of stagnant lid convection, buoyant slab convection, and plate tectonics, from the past to the present. We calculate the sea level based on this model and show that continents emerge significantly during two stages, i.e. 2.8-1.9 Ga and 1.0-0 Ga. We will discuss implications of this episodic emergence on the surface environment. Factors Controlling the Sea Level In this section, we discuss factors which contribute the sea level change through geological time. They are the volume of continental plates, sea water volume, and mean sea floor elevation with respect to continents (Figure 1). We discuss each of these more thoroughly below, and show procedures of our sea level calculation. The basic equations which are used for the determination of the sea level are given in Appendix. Volume of Continental Plates. We assume that continental and oceanic plates are isostatically floating over the asthenosphere (Figure 1, Schubert and Reymer, 1988) to calculate the sea level. Thus time histories of the total volume of continental plates and their density versus depth profiles are required for the sea level calculation. We assume a constant thickness of 200 km for the continental lithosphere and assign a linear temperature versus depth profile with Tm at the bottom and with 0°C at the surface. The average density of the continental lithosphere at 0°C is determined in order that it explains the present freeboard of 750 m (Schubert and Reymer, 1988). The crustal thickness of the Archean cratons and shields is 40 km in average (Mooney et al., 1998) and has changed little since their emplacement (Condie, 1973; Durrheim and Mooney, 1994). The crust of early-middle Proterozoic age is thicker at present by several km than that of Archean age (Mooney et al., 1998), and might have grown by volcanism and underplating of magmas in the late Proterozoic and succeeding periods (Durrheim and Mooney, 1994). We will discuss its effect on the sea level, later. The present thickness of the lithosphere of Archean age revealed by seismic methods (240-300 km) is larger by 3 ca. 60 km than that of Proterozoic age (Durrheim and Mooney, 1994). The assumed lithosphere thickness, however, does not affect much the calculated sea level as far as the lithospheric thickness did not change over geological time, because we calculate the sea level backward in time after constraining the initial lithosphere density to be consistent with the present sea level. However, if the lithosphere thickness changed through geological time, it would affect the results. As for the Archean age lithosphere, there is a number of pieces of evidence indicating that the thermal structure and thus lithosphere thickness has not changed much (Burke and Kidd, 1978; England and Bickle, 1984; Richter, 1985). However, the thickness of the Proterozoic age lithosphere might have changed through volcanism, similarly to the crust of this age (Durrheim and Mooney, 1994). We will discuss its effect on the sea level, later. Based on the constant thickness assumption, we regard the continental growth curve as representing the continental lithosphere growth curve through geological time. Because the continental growth curve is still in dispute, we use two representative ones; for the slow growth model (e.g., Veizer and Jansen, 1979), the continent grew since 4 Ga up to 30 % and for the rapid growth model (e.g, Armstrong, 1968; Taylor and McLennan, 1995) up to 70 % of the present volume by 2.5 Ga. Sea Water Volume. It has been assumed that the volume of the sea water is constant in the previous studies (Wise, 1974; Schubert and Reymer, 1988; Galer, 1991). The parameterized convection models with sufficient efficiency of heat transport have large surface plate velocities and degassing rate in the early Earth, producing most of the present water volume by the end of the Archean (Schubert et al., 1989; McGovern and Schubert, 1989; Tajika and Matsui, 1992). However, there are some studies which claim that smaller surface velocity and degassing rate are necessary in the past, on the basis of the total amount of degassed 40Ar (Sleep, 1979; Tajika and Matsui, 1993). For the latter case, degassing of water to the surface layer would have been much slower. In this study, we calculate the sea water volume change by assigning present degassing and regassing rates as parameters to satisfy the requirement of presence of surface water at 3.8 Ga as evidenced by the sedimentary rocks (Nutman and Collerson, 1991) and the pillow basalts (Komiya et al., 1998) in Isua. Given the present values, degassing and regassing rates through geological time can be calculated by their relationship with the mantle temperature (Tajika and Matsui, 1992; McGovern and Schubert, 1989). Following Tajika and Matsui (1992), degassing rate Md may be given by 4 Md = fdC 0hdMw/M m, (1) where fd is the degassing fraction, which is the fraction of water that degasses to the surface to the total amount of water originally included in the degassing volume, C0 is the areal spreading rate of the ocean floor, hd is the thickness of the degassing volume, i.e., the oceanic crust-harzburgite layer thickness, in which melts generate (McKenzie and Bickle, 1988; Hirth and Kohlstedt, 1996), Mw is the total water mass within the mantle, and Mm is the mass of the mantle. We assume that the present total water contained within the mantle, Mw* (values with an asterisk * represent the present ones hereinafter), is two times the present ocean volume. This value is subject to at least factor 30% uncertainty (Jackson and Pollack, 1987), but does not affect the results much because there is a trade-off between Mw* and fd. C 0 and hd can be calculated from the mantle temperature as will be shown later. The regassing rate Mr is similarly written as (Tajika and Matsui, 1992) Mr = frC 0hoc , (2) where fr is the regassing fraction, defined as the fraction of water regassing into the mantle, and hoc is the thickness of the oceanic crust. Recent petrologic studies show that the regassing of water into the mantle was almost zero for the subducting slab younger than 50 Ma due to dehydration of the slab in the shallow portion (Maruyama and Okamoto, 1998). Since the maximum age of the oceanic plate (denoted by te, which we will show later how to calculate), is less than 40 Ma prior to 1.0 Ga, we assign zero to the regassing rate before 1.0 Ga. We assume that fd and fr are constant through geological time. We assign the present regassing rate, M r*, to be the same as the present degassing rate, Md*. Mr* might be different from Md*(e.g., Ito et al., 1983) and fr might not be even constant (Kasting and Holm, 1992). However, these do not affect the results since the contribution of regassing to the sea water volume is minor as far as regassing prior to 1.0 Ga is zero. The value of hoc is a function of Tm and read from Fig.3 of White and McKenzie (1989). The harzburgite layer thickness is assumed to be three times of hoc (Oxburgh and Parmentier, 1977). We calculate C0 as follows. We use the relationship between the surface plate velocity u0 and the Rayleigh number Ra (e.g., Turcotte and Oxburgh, 1967; Sleep and Langan, 1981; Christensen, 1986), u0/u0* = (Ra/Ra*)2β. (3) 5 Ra is defined by Ra = αρmgTmd3/(µiκ) (4) where α is the thermal expansion coefficient, µi is the viscosity of the interior of convection cell, κ is the thermal diffusivity, and ρm is the density of the asthenospheric mantle. β is a constant related to the heat transfer efficiency of convection. Plate tectonics is characterized by the mobile rigid upper surface. Such a behaviour may be realized by assuming weak plate boundaries (e.g., Jacoby and Schmelling, 1982). Gurnis (1989) and Honda (1997) found that β is around 0.3 for this regime in both steady and unsteady convections, which is similar to β of 2D uniform-viscosity convection (Turcotte and Oxburgh, 1967). Thus the plate tectonics-like convection cannot be distinguished from simple uniform viscosity convection or convection with a small viscosity contrast (See also Solomatov, 1995 and Sleep and Langan, 1981). The maximum age te of the sea floor may be estimated by te/t e* = u 0*/u0 = (Ra/Ra*)-2β. (5) For a triangular fractional sea floor area versus age distribution (Parsons, 1982), C0 = 2Ao/t e, where A o is the total area of the ocean. With (5), we obtain C 0/C 0* = (Ao/Ao*)(Ra/Ra*)2β. (6) Then C0 is determined by Ra and Ao. Ra is mainly controlled by Tm through the temperature dependent viscosity. We use the Arrhenius-type temperature dependence of the viscosity, written as µi = µi*exp(Ta/Tm-Ta/Tm*), (7) where Ta is the activation temperature; we use Ta of 64000 K (McGovern and Schubert, 1989). We constrain the mantle temperature by the recent laboratory experiments on the Archean-Proterozoic rocks. Komiya (1998) showed 6 that T m was higher by ca. 200°C at 3 Ga than present based on the FeO and SiO2 contents of the initial magmas estimated from MORB-type volcanic rocks found in the Archean-early Proterozoic accretionary prisms. We assume that T m was higher by 200°C at 3 Ga than Tm* ( = 1280°C, McKenzie and Bickle, 1988); cases for lower temperature at 3 Ga will also be examined. For the other periods, we assume a linear change of Tm. Considering previous work of parameterized convection, this is a reasonable approximation at least for the past 3 Gy. Given the values of hoc *, f d, M w*, M d*, and Mr*, the values of M d, M r, M w, and sea water volume Vo for geological time can be calculated backward in time. Sea Floor Elevation. The average elevation of the sea floor with respect to the continental plate directly influences the sea level (Figure 1). This factor can be decomposed into the following two sub-factors: one is the average depth of the sea floor with respect to the ridge crest (denoted by db), and the other is the elevation of the ridge crest itself with respect to the continental plate. db results from the subsidence due to cooling of the surface boundary layer of the oceanic upper mantle since its formation at the ridge axis (Figure 1). In this study, we estimate the subsidence based on the half space cooling model; the parameters used are tabulated in Table 1. The use of the half space cooling model over the plate model (McKenzie, 1969; Parsons and Sclater, 1975; Stein and Stein, 1992) is because of its easiness to parameterize the plate thickness with Tm. Since the maximum age of the ocean floor is younger than 80 Ma for most of geological time, we do not need the plate model practically. We then calculate the density versus depth profile of an oceanic plate with a given age using the temperature versus depth profile thus derived along with the densities of oceanic crust-harzburgite layer (Niu and Batiza, 1991) and normal mantle at 0°C (Table 1). It should be noted that the oceanic crust-harzburgite layer becomes thicker than the thermal boundary layer for the ancient time, and in this case, the plate thickness is defined by the chemical boundary layer. To obtain the average depth of the ocean floor with respect to the ridge axis, we use the triangular distribution of the fractional sea floor area versus age (Parsons, 1982) and integrate the depth over the maximum age te of the sea floor (See also Galer, 1991). This procedure is time consuming, but necessary because the average depth cannot be represented in an analytical form for the triangular area versus age distribution. The value te* is determined so as to give the average sea floor depth with respect to the ridge crest at present. The second factor which controls the average elevation of the sea floor is the ridge crest height with respect to the continental plate floating over the asthenosphere. This is important because the oceanic crust-harzburgite layer thickness increases considerably for the higher mantle temperature in the past (Sleep and Windley, 1982; McKenzie 7 and Bickle, 1988; Davies, 1992). We simply assume that the layer at the ridge axis has the mantle temperature Tm, and calculate the mass anomaly at the ridge axis with respect to the asthenosphere (See Appendix for the definition of mass anomaly). The asthenosphere density ρm is also varied with T m. Toward the past, starting from the present values, we obtain sea water volume, the mid-ocean ridge mass anomaly, db, and continental plate mass anomaly, and finally the sea level on the basis of isostasy (See Appendix). Results: β = 0.3 Throughout Firstly we assume that β = 0.3 in equations (3), (5) and (6) for the past 4 b.y. and calculate the sea level change. Figure 2 shows the temporal change of the sea level (a), thickness of oceanic crust (b), continent volume (c), continent area above sea level (c), ocean volume (d), surface plate velocity (e), surface heat flow (e), and degassing rate (f). (The regassing rate is not visible in this figure but almost the same as the degassing rate since 1.0 Ga). The continent volume, continent area above sea level, ocean volume, surface plate velocity, and surface heat flow are normalized by their present values. The continent area above sea level is calculated by assuming that the continental hipsometry for the geological past is the same as the present one (Harrison et al., 1981), which seems reasonable (England and Bickle, 1984). The mantle potential temperature at 3 Ga is 1480°C (denoted by Tm3Ga = 1480°C). The fraction of the continent volume grown up by 2.5 Ga is 70 % (denoted by fAc = 0.7). The degassing fraction fd is assigned to be 1 %. If it is larger than 1 %, the sea water volume vanishes at 3.8 Ga, inconsistent with the geological data. The value smaller than 1 % instead results in more surface water at 3.8 Ga, i.e. in more inundation. fd of 1 % produces 0.41*1010 kg/yr degassing rate at present, which is by two-order smaller than the geological estimate of Ito et al. (1983) (22*1010 kg/yr). The surface plate velocity and degassing rate become very large during the Archean-early Proterozoic, as already shown by many parameterized convection models using β = 0.3 (e.g., Christensen, 1986; McGovern and Schubert, 1989; Tajika and Matsui, 1992). This is the reason why the above small degassing rate at present is required. More seriously, the sea level becomes higher at least by more than 3000 m during the Archean than present. Accordingly the continent area above sea level diminished during Archean-Proterozoic time (Figure 2c, dotted line). This apparently contradicts the significant subareal distribution of continents during this time (Windley, 1977; 1995). We next examine how the choice of parameter values affects the above results. Figures 3a, b, c and d show the 8 results for fAc = 0.3, Tm3Ga = 1430°C, T m3Ga = 1380°C, and β = 0, respectively; other parameters are the same as for Figure 2 except for fd which is adjusted to produce some surface water at 3.8 Ga. Figure 3a shows that the continental growth curve does not affect much the sea level, which is different from the result of Schubert and Reymer (1988). This is because the effect of decreasing Tm on the oceanic crust-harzburgite layer thickness and the seafloor depth, which does not depend on the total continent area, dominates the sea level change. The cases of smaller Tm3Ga and β reduce the sea level during the Archean-early Proterozoic, but it is still higher at 2.5 Ga by 1000 m for T m3Ga = 1380°C and by 1500 m for β = 0 than present (Figures 3c and d). Therefore Tm3Ga and β should be reduced simultaneously for continents to emerge significantly during the Archean-Proterozoic. We next examine the effect of the temporal variation of the thickness of the Proterozoic-age crust and lithosphere. The present Proterozoic-age crust is thicker than the Archean one (40 km) by 10 km at most (Durrheim and Mooney, 1994). If this Proterozoic crust thickness has grown to the present one from its initial thickness of 40 km, the sea level at Proterozoic time would become 1.5 km higher than shown in Figure 2, giving a worse fit to the early Proterozoic continental emergence. This could be compensated if the Proterozoic-age lithosphere was thinner by 30 km at that time than present. Therefore the calculated sea level is difficult to be reconciled with the observed continental emergence during the early Proterozoic unless the lithosphere thickness at that time was much thinner (e.g., by 90 km) than present, which seems unlikely. The basic premise that β = 0.3 dates back to the Archean contradicts the early Proterozoic emergence of continents unless Tm3Ga was less than 1380°C. This result is different from that of Galer (1991) who permits 1430°C as a maximum value. This comes from the fact that we parameterize the plate thickness and sea floor spreading rate as functions of the mantle temperature. The case that β = 0 is tolerable when T m3Ga is less than 1430°C. We thus conclude that the plate tectonics dating back to 3 Ga is not likely and suggest that β should be smaller in the past if it is 0.3 at present. In the next section, we examine an alternative scenario for the evolution of mantle convection in the past than the case that β = 0.3 throughout. How Far did Plate Tectonics Date Back? An oceanic plate increases its buoyancy as the crust-harzburgite layer becomes thicker and the average age of the plate becomes younger as the mantle temperature becomes higher in the past. At present, the oceanic plate older than ca. 20 Ma has negative buoyancy with respect to the asthenosphere, but around 1 Ga, every part of the 9 oceanic plate becomes buoyant at the trench (Davies, 1992). Davies (1992) inferred that prior to this time the plate motion would be interrupted waiting for the plate to be cooled enough, and suggested that delamination of the mantle lithosphere, which leaves the oceanic crust stacked to continental margins, would rather be a dominant tectonic style over plate tectonics (See also Hoffman and Ranalli, 1988). He cast a doubt on the role of the phase transition of the basaltic oceanic crust to eclogite on the slab pull (Ringwood and Green, 1966) because kinetics of the transition is uncertain and the density excess of eclogite over normal mantle (80 kg/m3) is smaller than the density deficit of basalts (440 kg/m3). For uniform-viscosity convection, the mechanical energy, transformed from the heat from the bottom or generated within, is dissipated by viscous heating (Golitsyn, 1979; Hewitt et al., 1975). In convection, generally, this mechanical energy is spent both in the viscous dissipation and in another kind of work such as deformation of the surface boundary layer having high viscosity (McKenzie and Jarvis, 1980; Richter, 1984; Solomatov, 1995). Even if the oceanic plate loses its negative buoyancy at subduction zones, part of the mechanical energy may be used to force the plate to penetrate into the asthenosphere. We will show in the next section that convection possibly drags the slab into the asthenosphere while buoyancy of the slab is small. Because the buoyancy of a plate dug into the shallow portion of the asthenosphere is generally smaller than that of the whole slab, the buoyancy of the plate at the trench is not a problem. Once the plate being subducted into the asthenosphere, the basaltic crust will transform to the eclogite because the subducted crust contacts the asthenosphere directly and hydrous minerals within the crust promote the transformation (Ahrens and Schubert, 1975; Irifune and Ringwood, 1987). Therefore even prior to 1 Ga, subduction and surface tectonics similar to plate tectonics would have existed. Buoyant Slab Convection. Further back to the past, the depleted harzburgite layer becomes thicker as the mantle temperature becomes higher, and the subducting slab becomes hotter, then the slab becomes buoyant even after the basalt/eclogite transformation (Irifune and Ringwood, 1987). We will show that even in this case, while the buoyancy force is not large, convection can drive the slab into the asthenosphere (we call this buoyant slab convection). Figure 4b shows buoyant slab convection schematically, compared with plate tectonics (Figure 4a). This is still similar to plate tectonics in appearance, but we discriminate it from plate tectonics because β = 0.3 does not hold any more. Let u0 be the surface plate velocity and u1 the velocity of the convective flow beneath the plate (Figure 4b). The efficiency of conversion of heat into mechanical work is represented by d/Ht (the dissipation number, Golitsyn, 10 1979; Hewitt et al., 1975; McKenzie and Jarvis, 1980). Ht is the temperature scale height defined by Cp/αg, where C p is the specific heat, and g is the acceleration of gravity. The dissipation within the convection cell is of the order of this mechanical work (Solomatov, 1995), and we obtain µi(u1/d)2Ld = d/Ht(kTm/δ0)L = d(αg/C p)(kTm/δ0)L, (8) where k is the thermal conductivity, L is the convection cell size and δ0 is the thickness of the plate. This gives u12δ0d/κ 2 = Ra. (9) The mechanical work done by the convective dragging of the plate is µi(u1-u0)2L/d, which is generally smaller than the viscous dissipation (Compare with the first term of equation (8)), but of the same order when u0 is small. Because the surface velocity u0 is related δto 0 by δ0 = (κL/u0)1/2, (10) we obtain from (9) and (10) (u0/u0*)-1/2(u1/u1*)2 = Ra/Ra*. (11) This is an extension of equation (3) to the case when u0 and u1 are not equal. Letting ∆ms be the slab mass anomaly per unit slab length, the slab buoyancy force Fs in the dip direction per unit arc length is F s = ∆msgLssinθ, (12) where θ is the slab dip angle, and Ls is the slab length (Figure 4b). When the slab penetrates into the depth range of convection, Lssinθ = d, and we obtain, F s = ∆msgd. (13) 11 The viscous shear force at the plate base integrated over the plate size L, which we assume to be of the same order as the convection depth d, is F v = µi(u1 - u0)L/d ~ µi(u1 - u0) (14) From the force balance between Fs and Fv, we equate (13) and (14), normalizing them by u1*, and obtain, u0/u1* = u 1/u1*+ d∆msg/(µiu1*). (15) For the buoyant slab convection regime, we estimate average density anomalies of the crust and harzburgite layers over the depth range of 800 km as 0 and -50 kg/m3, respectively, based on the phase transformation data for the slab thermally equilibrated with the ambient mantle by Irifune and Ringwood (1987, 1993). As for the average thermal anomaly of the slab with respect to the ambient mantle, we take half of the temperature anomaly of the slab at the trench, since the edge of the slab would be thermally equilibrated with the ambient mantle. Combining the chemical and thermal mass anomalies of the slab, obtaining the net mass anomaly, solving equations (11) and (15) simultaneously, we obtain a set of u0 and u1 for the buoyant slab convection. When this set is not found, the slab is too buoyant to be driven into the asthenosphere, and the buoyant slab convection does not work. We will see in the next section that this transition occurs around 2 Ga for a representative set of parameters. On the other hand, the transition between buoyant and negative buoyant slabs can be identified from the sign of the slab mass anomaly; this occurs around 1.4 Ga. Results: Non-uniform Convection Case We now calculate the sea level for the buoyant slab convection discussed above and succeeding plate tectonics. Prior to the buoyant slab convection, plates couldn't subduct any more. We regard this stage as stagnant lid convection. This stagnant lid convection is different from that defined by Solomatov (1995) in the sense that the buoyancy of the products and residual materials of extensive melting controls the mechanical behavior in this case. Therefore we do not use the Solomatov's parameterization which is based on the temperature dependent viscosity, 12 but assign simply a constant surface velocity as a free parameter. Note also that even if convection enters the buoyant slab convection regime as the mantle cools, it would be necessary to break the surface lid in order to initiate the buoyant slab convection. It is still a difficult task to solve numerically this problem, because other factors such as volatile become important (Bercovici, 1998; Solomatov and Moresi, 1998). Because the transition time from the stagnant lid convection to the buoyant slab convection is determined as the time when equations (11) and (15) start to have solutions, it gives an earliest estimate for the initiation of the buoyant slab convection. However, we will show later that the early Proterozoic emergence of continents requires the transition time similar to our estimate. Figure 5 shows the sea level (a), oceanic crust thickness (b), continent crust volume (c), continent area above sea level (c), ocean volume (d), surface plate velocity (e), and surface heat flow (e), degassing and regassing rates (f) for our model. T m3Ga, oceanic crust-harzburgite layer thickness, and continental growth curve are the same as those for the plate tectonics case shown in Figure 2. The value of Md is 5.3*1010 kg/yr, which is the maximum one which satisfies the existence of the surface water at 3.8 Ga. This is still small but becomes closer to Ito et al. (1983)'s estimate. The value of viscosity at present, µi*, is assigned to be 1.7*10 22 Pa s. The fraction of the surface velocity of the stagnant plate to the buoyant slab convection velocity at their transition time (denoted by f s) is set to be 0.3. The effects of the choice of these parameters will be discussed later. We describe now the temporal change of the sea level and continent area above sea level calculated from the model from the past to the present (Figures 5a and c). The sea level gradually decreases for the first 2 b.y. because the sea floor deepening due to mantle cooling dominates the effects of the degassing of water and crustal growth. When the buoyant slab convection starts at 2 Ga, the surface velocity and degassing rate suddenly increase (Figures 5e and f), resulting in the sea level rise of about 800 m. It starts to decrease around 1.3 Ga along with the loose of the vigor of convection due to mantle cooling. The transition from the buoyant slab to the negative buoyant slab occurs around 1.4 Ga. According to the above sea level variation, there appear two stages of significant continental emergence. The continent surface more than 40 % of the present one appears first during 2.8-1.9 Ga and second during 1.0-0 Ga (Figure 5c). We show the cases with different choices of other parameters in Figure 6. Figures 6a, b, and c show the results for fs = 0.4, µi = 3.3*10 22 Pa s, and Tm3Ga = 1430°C, respectively, and Figure 6d shows the case of Tm3Ga = 1430°C and µi = 6.6*1021 Pa s; other parameters are the same as for Figure 5 except for fd which is adjusted to produce some surface water at 3.8 Ga. If fs is 0.4 and larger, the continent area above sea level during the late Archean-early 13 Proterozoic is less evident because the velocity rise at the initiation of the buoyant slab convection is subdued. This requirement of small fs may imply that the initiation of the buoyant slab convection occurred suddenly by brittle fracture of the stagnant plate. The viscosity µi controls the transition time between the stagnant lid convection and the buoyant slab convection through equation (15). The larger (smaller) is µi, the earlier (later) the transition. The case of µi = 3.3*1022 Pa s gives the transition time around 2.5 Ga, inconsistent with the early Proterozoic continental emergence. The case of Tm3Ga = 1430°C produces a similar effect to the larger µi, because µi becomes larger through equation (7) for the geological past. This can be counteracted by the smaller µi.; the case of Tm3Ga = 1430°C and µi = 6.6*1021 Pa s produces a significant emergent area during the Archean-early Proterozoic. Although this trade-off between µi and Tm3Ga and the uncertainty of the estimate of the slab buoyancy precludes us from gaining a unique picture of the past sea level change, it is notable that the non-monotonous sea level change and continent emergence can be obtained from a reasonable choice of the parameters. Discussion Continental Crust and Mantle Evolution. The late Archean is the time when the continental crust formed significantly, and melting of the subducting oceanic crust would have been a major process responsible for this (Campbell and Taylor, 1983; McCulloch, 1993; Taylor and McLennan, 1995). There are also early Archean accretionary complex in North America which indicates existence of subduction (Nutman and Collerson, 1991; Komiya et al., 1989). Then a question how the subduction and the crustal formation occurred in the stagnant lid convection regime would arise. We note that the thick stagnant layer is never stable, but sometimes destroyed by the instability due to the supply of heat from the deeper mantle (Ogawa, 1997). Some numerical experiments also show that a mechanical instability of the surface boundary layer occurs when the material has non-Newtonian or brittle rheology, which has been applied to the resurfacing of Venus at 0.5 Ga (Weinstein, 1996; Solomatov and Moresi, 1998). The episodic avalanche of the stagnant layer would inevitably have accompanied subduction and melting of the proto-oceanic crust. The episodic nature of the avalanche of the stagnant layer is favorable for the scarcity of silisic rocks compared with mafic rocks in the greenstone belts during the early Archean (Taylor and McLennan, 1995), the low Sr ratio of the continental crustal materials during the early Archean (Veizer and Compton, 1976), and the rapid episodic growth of continent segments (Reymer and Schubert, 1986). Imagine also that, if plate tectonics operated during the early Archean, a much more volume of acidic continental crust would 14 have formed due to larger subduction velocities at that time. The stagnant lid convection during the Archean-early Proterozoic is also consistent with the geochemistry of the source mantle for plumes sampled by komatiites and picrites. Campbell and Griffiths (1992) showed that the source mantle, i.e., the deep boundary layer, underwent a drastic change in geochemistry between the Archean and 2.0 Ga; most picrites younger than 2.0 Ga have originated from enriched (OIB-type) mantle, but Archean komatiites have depleted or neutral geochemistry. They interpreted this by the lack of mantle circulation by plate tectonics during the Archean, which is consistent with our model of the temporal change in mode of mantle convection. Degassing of Volatile The change in mode of convection proposed in this study predicts a much slower degassing rate of volatile from the Earth's interior. Compare the degassing rate of water of our preferred model (Figure 5d), for example, with that of the plate tectonics case (Figure 2d). The degassing of 40 Ar into the atmosphere would be similar, though the degassing process of 40Ar is slightly different from that of water. 40Ar is produced by the decay of 40K, and 40K is also transported to the continental reservoir through melting and accretion. However the total amount is mainly controlled by the time history of the sea floor spreading rate and melt generation depth, similarly to the degassing of water (Tajika and Matsui, 1993). Tajika and Matsui (1993) showed that convection with β = 0, i.e. having a constant spreading rate, is much more favorable for the total amount of degassed 40Ar than that with β = 0.3. Since the history of the spreading rate of our preferred model (Figure 5e) does not differ much from the constant velocity model (β = 0), our model would also be consistent with the total amount of degassed 40Ar. Implications for Environmental Evolution. The temporal variation of the emergence area of continents shown in Figure 5c might have implications for the evolution of life, banded iron formation, ice age distribution, and etc. According to our model of the convection mode change, emergence of continents had occurred firstly in the late Archean-early Proterozoic, and secondly in the late Proterozoic-Present. The cyanobacteria appeared in the late Archean and, for their prosperity, vast areas of shallow sea-shores would have been necessary. The eukaryote appeared in the latest period of the first emergence period. After the submergence around 2 Ga, there have been no significant event in the evolution of life until 1 Ga. Around 1 Ga, the continent surface above sea level starts to increase, which is the time of divergence of Metazoan phyla (Wray et al., 1996). See Moores (1994) who cited various kind of effects expected from the freeboard increase around 1 Ga. During the first period of emergence, the appearance of the photosynthesizing cyanobacteria increased the 15 atmospheric oxygen level. The shallow sea environment at this time would have been also favorable for the early Proterozoic huge banded iron formation (BIF) by promotion of mixing of Fe+2 in the deep ocean with O2 in the surface water through regression and transgression (Klein and Buekes, 1992). It should be noted that BIF appeared again when continents start to emerge during the late Proterozoic (e.g., Klein and Buekes, 1992). There had been two episodes of extensive glaciation during Precambrian time. The first one was the early Proterozoic (~2.4 Ga) represented by the North American, European, and African tillites depositions and the second one is the late Proterozoic (e.g., Windley, 1995). These ice ages are during the emergence periods of continents of our model. The wide subareal continent area would be a favorable factor for the extensive glaciation through the albedo-feedback. The decrease of δ13C of the sea water associated with the glaciation (Kaufman, 1997) would have resulted from the promoted circulation between the surface water and deep ocean, which would have oxidized the organic carbon burial in the deep ocean basins. Conclusions We calculate the secular change of the sea level for the past 4 b.y. assuming that β = 0.3 during this period. We assume a linear decrease of the mantle potential temperature by 200°C since 3 Ga, which is obtained from the petrologic studies of Archean-Proterozoic rocks. The Archean sea level becomes higher by at least 3000 m than present. This implies that our premise of β = 0.3 for the past 4 b.y. is wrong. Since the plate tectonics-like convection has β = 0.3, this further implies that the plate tectonics would have operated not for the whole 4 b.y. We next consider until what time plate tectonics could date back and what type of mantle convection operates before plate tectonics. A subducting slab gains its negative buoyancy with respect to the ambient mantle around 1.4 Ga. Even prior to this time, convection can drive the slab into the asthenosphere while the drag force overwhelms the buoyancy (we call this buoyant slab convection). Prior to 2 Ga, the convective drag force becomes smaller than the buoyancy force and stagnant lid convection would have worked. We calculate the sea level associated with the above temporal change of convection in the Earth. The calculated sea level rises up suddenly around 2 Ga when the buoyant slab convection starts, and with a peak around 1.3 Ga, it falls due to the cooling of the Earth since then. The continent emerges during 2.8-1.9 Ga and 1.0-0 Ga by more than 40 % of the present area; this episodic continent emergence might have significant implications for the evolution of life, such as appearance of cyanobacteria, eukaryote, and metazoa, and for the temporal distribution of 16 BIF and ice ages, all of which occurred in coincidence with these emergent periods. Appendix We describe here how we determine freeboard hf. The sea level change h sl - hsl* (asterisk denotes the present value) is related to the freeboard change as hsl - hsl* = h f* - h f. (A1) A negative value of hf means that the sea level is higher than the average continent height. hf can be determined by assuming that the continental plate, oceanic plate and sea water are in isostatic balance (Figure 1). It is convenient to divide the case into three (Figure A1): (a) the continent surface is higher than the mid-ocean ridge crest (denoted by m.o.r.), (b) the continent surface is lower than m.o.r. and higher than the oldest ocean floor, (c) the continent surface is lower than the oldest ocean floor. The last case might be thought unrealistic but it occurs, at least formally, prior to ca. 3 Ga when we assume β = 0.3 throughout the earth's history. Case (b) occurs prior to ca. 2.3 Ga for both the β = 0.3 and buoyant slab convection cases. The relative height of the continent surface to m.o.r., both of which are covered by sea water, is denoted by hrw (Figure A1). We define the mass anomaly of any specific lithospheric column, having vertical density profile ρ(z), with respect to the asthenospheric column, as ∆m = (ρ(z) - ρm)dz, (A2) where a and 0 indicate depths of the lithosphere's bottom and surface. Then hrw is determined by equating the mass anomaly at m.o.r. (denoted by ∆mmor) and that of the continental plate (denoted by ∆mc), as ∆mmor + h rw(ρw - ρm) = ∆mc. (A3) If hrw is negative, case (b) or (c) holds. Below we describe the determination of hf in each case. 17 Case (a) The water volume below m.o.r. is denoted by V2 and that between the continent surface and m.o.r. by V1 (Figure A1a). Then V 1 = hrwAo. (See text and Table 1 for the symbol notations such as Ao and others). We define db(t, t') as the depth of the ocean floor with age t with respect to that of age t'. Then db(t, t')(ρw - ρm) = ∆m(t') - ∆m(t), (A4) where ∆m(t) is the mass anomaly of the oceanic plate with age t. Then V2 = db(t, 0)dAo(t)/dt*dt, (A5) where dAo(t)/dt = C0(1 - t/te). Case (a) can be subdivided into three cases (Figure A1a): (1) Vo > V 1 + V 2, (2) V 1 + V 2 > V o> V 2, (3)V2 > V o. In case (1), -hf = (Vo - V1 - V2)/S. (A6) hf = [hrw - (Vo - V2)/Ao](ρw - ρm)/(-ρm). (A7) In case (2), In case (3), the plate age t3 is searched as satisfying, Vo = db(t, t3)dAo(t)/dt*dt. (A8) hf = [db(t3, 0) + hrw](ρw - ρm)/(-ρm). (A9) Then 18 Case (b) V2 denotes the sea water volume below the continent surface and V1 between the continent surface and m.o.r. Age tc is the plate age whose mass anomaly is equal to the mass anomaly of the continental plate (Figure A1b). Then V2 = db(t, tc)dAo(t)/dt*dt (A10) V1 = db(t, 0)dAo(t)/dt*dt - V2 - hrwAc. (A11) and Case (a) is subdivided into three cases (Figure A1b): (1) Vo > V 1 + V 2, (2) V 1 + V 2 > V o> V 2, (3)V2 > V o. In case (1), hf is determined by hf = -(Vo - V1 - V2)/S + h rw. (A12) In case (2), plate age t2 is searched numerically as satisfying, Vo = db(t, t2)dAo(t)/dt*dt +db(tc, t 2)Ac. (A13) hf = -db(tc, t 2). (A14) Then In case (3), plate age t3 is searched as satisfying, Vo = db(t, t3)dAo(t)/dt*dt. (A15) Then 19 hf = db(t3, t c)(ρw - ρm)/(-ρm). (A16) Case (c) V2 denotes the sea waver volume between the oldest ocean floor and continent surface, and V1 between the oldest sea floor and m.o.r. (Figure A1c). h c denotes the height of the oldest ocean floor with respect to the continent surface. Then hc is determined by hc(ρw - ρm) + ∆mc = ∆m(te). (A17) V2 = hcAc (A18) V1 = db(te, 0)A c + db(t, 0)dAo(t)/dt*dt. (A19) V1 and V2 are then and Case (c) can be subdivided into three cases (Figure A1c): (1) Vo > V 1 + V 2, (2) V 1 + V 2 > V o> V 2, (3)V2 > V o. In case (1), hf is determined by equation (A12). In case (2), plate age t2 is searched as satisfying, Vo = db(t, t2)dAo(t)/dt*dt + [hc +db(te, t 2)]Ac. (A20) hf = -hc - db(te, t 2). (A21) Then In case (3), 20 hf = -Vo/Ac. (A22) References Cited Ahrens, T. J., and Schubert, G., 1975, Gabbro-eclogite reaction rate and its geophysical significance: Reviews of Geophysics and Space Physics, v. 2, p. 383-400. Armstrong, R. 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H. ,1996, Molecular evidence for deep Precambrian divergences among metazoan phyla: Science, v. 274, p. 568-573. 25 Figure captions Figure 1 Factors which control the sea level. The continental and oceanic plates and the sea water are in an isostatic balance over the asthenosphere. The oceanic crust and harzburgite layer thickness determines the ridge crest depth with respect to the continent, and the cooling of the oceanic plate determines the depth of the sea floor with respect to the ridge crest, both of them influence the average sea floor depth. The ocean volume, determined by the degassing and regassing of water, and the continental plate volume also influence the sea level. Figure 2 The secular sea level change (a), oceanic crust thickness (b), continent volume (c, solid line), continental area above sea level (c, dotted line), ocean volume (d), surface heat flow (e, solid line), plate velocity (e, dotted line), and degassing rate (f) calculated based on the premise that plate tectonics is operating for all the geological time. It is assumed that β = 0.3, the mantle potential temperature Tm at 3 Ga (denoted by Tm3Ga) is 1480°C, and the fraction of the continent grown up by 2.5 Ga (denoted by fAc) is 70 % of the present area. The continent volume, continental area, ocean volume, plate velocity, and surface heat flow are normalized by their present values. The sea level is higher by more than 3000 m in the Archean than present. Figure 3 The secular sea level change (left), continent volume and continental area above sea level (middle), and plate velocity and surface heat flow (right) for the other choice of β, T m3Ga, and f Ac: (a) fAc = 0.3, (b) T m3Ga = 1430°C, (c) T m3Ga = 1380°C, and (d) β = 0. Other parameters are the same as in Figure 2. Figure 4 Schematic illustrations showing the two convection modes: (a) plate tectonics and (b) buoyant slab convection. In plate tectonics case, the plate velocity (u0) and the velocity of convection beneath the plate (u1) are the same, and the slab pull and ridge push forces are balanced by the viscous resistance forces. In the case of buoyant slab convection, u 1 is larger than u0, and the tangential shear force drives the plate toward the trench. This drag force is counteracted by the slab buoyancy force. Figure 5 The secular sea level change (a), oceanic crust thickness (b), continent volume (c, solid line), continental area above sea level (c, dotted line), ocean volume (d), surface heat flow (e, solid line), plate velocity (e, dotted line), and degassing rate (f) calculated based on a series of stagnant lid convection, buoyant slab convection and 26 plate tectonics. The continent volume, continental area, ocean volume, plate velocity, and surface heat flow are normalized by their present values. Tm3Ga = 1480°C, fAc = 0.7, µi = 1.7*1022 Pa s, and the fraction of the stagnant plate velocity to that of the buoyant slab convection at their transition (fs) is 0.3. The sea level and plate velocity suddenly rises at 2 Ga when buoyant slab convection starts, and with a peak at 1.3 Ga it falls since then. The continent emerges during 2.8-1.9 Ga and 1.0-0 Ga by more than 40 % of the present area. Figure 6 The secular variation of the sea level (left), continentl volume, continental area above sea level (middle), and plate velocity and surface heat flow (right) based on the series of the stagnant lid convection, buoyant slab convection and plate tectonics, for the other choice of fs, T m3Ga, and µi: (a) f s = 0.4, (b) µi = 3.3*10 22 Pa s, (c) T m3Ga = 1430°C, (d) Tm3Ga = 1430°C and µi = 6.6*10 21 Pa s. Other parameters are the same as in Figure 5. The smaller µi or Tm3Ga shifts the transition between the stagnant lid convection and the buoyant slab convection later. Figure A1 Relationship between the continental plate, mid-ocean ridge (m.o.r.), sea floor, and sea water in isostasy. hf denotes the freeboard. hrw denotes the height of the continent surface relative to m.o.r. when they are covered by water. (a) The continent surface is higher than m.o.r.. V 2 and V1 are the sea water volume below m.o.r. and between m.o.r. and the continent surface, respectively. Three cases of the sea level, i.e. higher than the continent surface (1), between the continent surface and m.o.r. (2), and below m.o.r. (3), are indicated. t2 is the age of the plate where the sea level intersects the sea floor. te is the oldest age of the ocean floor. (b) The continent surface is between m.o.r. and the oldest ocean floor. V1 and V2 are the sea water volume between m.o.r. and the continent surface and below the continent surface, respectively. Three cases of the sea level, i.e. higher than m.o.r. (1), between m.o.r. and the continent surface (2), and below the continental surface (3), are indicated. tc is the plate age where the continent surface intersects the ocean floor. t2 and t3 are the ages of the plate where the sea level intersects the sea floor. (c) The continent surface is below the oldest ocean floor. V1 and V 2 are the sea water volume between m.o.r. and the oldest ocean floor and below the oldest ocean floor, respectively. Three cases of the sea level, i.e. higher than m.o.r. (1), between m.o.r. and the oldest ocean floor (2), and below the oldest ocean floor (3), are indicated. t2 is the age of the plate where the sea level intersects the sea floor. hc is the height of the oldest ocean floor relative to the continent surface. 27 Freeboard Sea level Sea Degassing Continental plate db Ridge Oceanic crust Harzburgite layer Oceanic plate Asthenosphere Regassing Fig. 1 Fig. 2 Fig. 3a,b Fig. 3c,d (a) Plate tectonics u0 u1 u0 ~ u1 (b) Buoyant slab convection u0 u1 s Ls u0 < u1 d L Fig. 4 Fig. 5 Fig. 6a,b Fig. 6c,d (a) (1) hf (2) (3) Continent V1 V2 Mid-ocean Ridge hr w t3 te (b) (1) hr w (2) (3) V1 V2 te t3 tc t2 (c) (1) hr w (2) (3) V2 V1 t2 hc Fig.A1