Download Mantle Convection and Global Sea Level: Implications for the

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. L., 1968, A model for the evolution of strontium and lead isotopes in a dynamic earth: Rev.
Geophys., v. 6, p. 175-199.
Bercovici, D., 1998, Generation of plate tectonics from lithosphere-mantle flow and void-volatile self-lubrication:
Earth Planet. Sci. Lett., v. 154, p. 139-151.
Burke, K., and Kidd,W. S. F. , 1978, Were Archean continental geothermal gradients much steeper than those of
today?: Nature, v. 272, p. 240-241.
Campbell, I. H., and Taylor,S. R. , 1983, No water, no granites - no oceans, no continets: Geophys. Res. Lett.,
v. 10, p. 1061-1064.
Condie, K. C., 1973, Archean magmatism and crustal thickening: Geophys. Soc. Am. Bull., v. 84, p. 2981-2992.
Davies, G.F., 1992, On the emergence of plate tectonics: Geology, v. 20, p. 963-966.
Durrheim, R. J., and Mooney, W. D., 1994, Evolution of the Precambrian lithosphere:
Seismological and
geochemical constraints: J. Geophys. Res., v. 99, p. 15359-15374.
England, P. and Bickle M. , 1984, Continental thermal and tectonic regimes during the Archaean: J. Geol., v. 92,
p. 353-367.
Galer, S.J.G., 1991, Interrelationships between continental freeboard, tectonics and mantle temperature: Earth
Planet. Sci. Lett., v. 105, p. 214-228.
Golitsyn, G. S., 1979, Simple theoretical and experimental study of convection with some geophysical
applications and analogies: J. Fluid Mech., v. 95, p. 567-608.
Gurnis, M., 1989, A reassesment of the heat transport by variable viscosity convection with plates and lids:
Geophys. Res. Lett., v. 16, p. 179-182.
Harrison, C .G. A., Brass,G. W. , Saltzman, E. , Sloan, J., Southam, J., and Whitman, J. M. , 1981, Sea level
variations, global sedimentation rates and the hypsographic curve: Earth Planet. Sci. Lett., v. 54, p. 1-16.
Hewitt, J. M., McKenzie, D. P., and Weiss, N. O. 1975, Dissipative heating in convective flows: J. Fluid
21
Mech., v. 68, p. 721-738.
Hirth, G., and Kohlstedt, D. L.
1996, Water in the oceanic upper mantle: implications for rheology, melt
extraction and the evolution of the lithosphere: Earth Planet. Sci. Lett., v. 144, p. 93-108.
Hoffman, P. F., and Ranalli, G.,1988, Archean oceanic flake tectonics: Geophys. Res. Lett., v. 15, p. 1077-1080.
Honda, S., 1997, A possible role of weak zone at plate margin on secular mantle cooling: Geophys. Res. Lett., v.
24, p. 2861-2864.
Irifune, T., and Ringwood, A. E.,1987, Phase transformations in a harzburgite composition to 26 GPa:
Implications for dynamical behaviour of the subducting slab: Earth Planet. Sci. Lett., v. 86, p. 365-376.
Irifune, T., and Ringwood, A. E.,1993, Phase transformation in subducted oceanic crust and buoyancy
relationships at depths of 600-800 km in the mantle: Earth Planet. Sci. Lett., v. 117, p. 101-110.
Ito, E., Harris,D.,M., and Anderson, Jr.,A.T., 1983, Alteration of oceanic crust and geologic cycling of chlorine
and water: Geochimica et Cosmochimica Acta, v. 47, p. 1613-1624.
Jackson, M. J., and Pollack,H. N. , 1987, Mantle devolatilization and convection: Implications for the thermal
history of the earth: Geophys. Res. Lett., v. 14, p. 737-740.
Kasting, J. F., and Holm, N. G.,1992, What determines volume of the oceans: Earth Planet. Sci. Lett., v. 109, p.
507-515.
Kaufman, A. J., 1997, An ice age in the tropics: Nature, v. 386, p. 227-228.
Klein, C., and Beukes, N. J. , 1992, Proterozoic iron-formations: Proterozoic Crustal Evolution, ed. by K. C.
Condie, Elsevier, Amsterdam, v. , p. 383-418.
Komiya, T., 1998 (in Japanese), Reading the mantle temperature from mid-ocean ridge basalts: Kagaku, v. 68, p.
747-750.
Komiya, T., Maruyama, S. Masuda, T, Appel, P.W.U., and Nohda, S., 1998, Plate tectonics at 3.8-3.7 Ga: Field
evidence from the Isua accretionary complex of Western Greenland: J. Geol., v. , p. in press.
Maruyama, S., and Okamoto, K.,1998, Water transportation mechanism from the
subducting slab into the mantle transition zone: Chem. Geol., v. in press, p. .
McCulloch, M. T., 1993, The role of subducted slabs in an evolving earth: Earth Planet. Sci. Lett., v. 115, p.
89-100.
McGovern, P. J., and Schubert, G.,1989, Thermal evolution of the Earth: effects of volatile exhange between
atmosphere and interior: Earth Planet. Sci. Lett., v. 96, p. 27-37.
22
McKenzie, D. P., 1969, Speculations on the consequences and causes of plate motions: Geophys. J. R. astr. Soc.,
v. 18, p. 1-32.
McKenzie, D., and Jarvis, G.,1980, The conversion of heat into mechanical work by mantle convection: J.
Geophys. Res., v. 85, p. 6093-6096.
McKenzie, D., and Bickle, M. J., 1988, The volume and composition of melt generated by extension of the
lithosphere: Journal of petrology, v. 29 , p. 625-679.
Mooney, W. D., Laske, G., and Masters,T. G. , 1998, Crust 5.1: A grobal crustal model at 5｡x5｡: J. Geophys.
Res., v. 103, p. 727-747.
Moores, E. M., 1993, Neoproterozoic oceanic crustal thinning, emergence of continents, and origin of the
Phanerozoic ecosystem: A model: Geology, v. 21, p. 5-8.
Niu, Y., and Batiza,R., 1991, In situ densities of morb melts and residual mantle: Implications for buoyancy
forces beneath mid-ocean ridges: J. Geodyn., v. 99, p. 767-775.
Nutman, A. P., and Collerson,K. D. , 1991, Very early Archean crustal-accretion complexes preserved in the
North Atlantic craton: Geology, v. 19, p. 791-794.
Ogawa, M., 1997, A bifurcation in coupled magmatism-mantle convection system and its implications for the
evolution of the Earth's upper mantle: Phys. Earth Planet. Inter., v. 102, p. 259-276.
Oxburgh, E. R., and Parmentier, E. M., 1977, Compositional and density stratification in oceanic
lithosphere-causes and consequences: J. Geol. Soc. London, v. 133, p. 343-355.
Parsons, B., and Sclater, J. G. , 1977, An analysis of the variation of ocean floor bathymentry and heat flow with
age: J. Geophys. Res., v. 82, p. 803-827.
Parsons, B., 1982, Causes and consequences of the relation between area and age of the ocean floor: J. Geophys.
Res., v. 87, p. 289-302.
Reymer, A., and Schubert, G., 1986, Rapid growth of some major segments of continental crust: Geology, v. 14,
p. 299-302.
Richter, F. M. , 1984, Regionalized models for the thermal evolution of the Earth: Earth Planet. Sci. Lett., v. 68,
p. 471-484.
Richter, F. M., 1985, Models for the Archean thermal regime: Earth Planet. Sci. Lett., v. 73, p. 350-360.
Ringwood, A. E., and Green, D. H., 1966, An experimental investigation of the gabbro-eclogite transformation
and some geophysical implications: Tectonophysics, v. 3, p. 383-427.
23
Schubert, G., and Reymer, A. P. S. , 1985, Continental volume and freeboard through geological time: Nature, v.
316, p. 336-339.
Schubert, G., Turcotte, D. L., Solomon, S. C. and Sleep, N. H., 1989, Coupled evolution of the atmospheres and
interiors of planets and satellites: Origin and Evolution of Planetary and Satellite Atmospheres, Atreya,S. K.,
Pollack, J. B., and Matthews, M. S. eds., University of Arizona Press,Tucson, Ariz., v. , p. 450-483.
Sleep, N. H., 1979, Thermal history and degassing of the earth: some simple calculations: J. Geology, v. 87, p.
671-686.
Sleep, N. H., and Langan, R. T., 1981, Thermal evolution of the earth:
Some recent developments: Adv.
Geophys., v. 23, p. 1-23.
Sleep, N. H., and Windley, B. F. ,1982, Archean plate tectonics: constraints and inferences: J. Geol., v. 90, p.
363 - 379.
Solomatov, V. S., and Moresi, L.-N. ,1998, Convection with strongly temperature-dependent viscosity and brittle
facture: Generation pf plate tectonics, eposodic plate tectonics and stagnant lid convection: , v. , p. preprint.
Solomatov, V. S., 1995, Scaling of temperature- and stress-dependent viscosity convection: Phys. Fluids, v. 7, p.
266-274.
Stein, C. A., and Stein, S., 1992, A model for the global variation in oceanic depth and heat flow with
lithospheric age: Nature, v. 359, p. 123-129.
Tajika, E., and Matsui, T. ,1992, Evolution of terrestrial proto-CO2 atmosphere coupled with thermal history of
the earth: Earth Planet. Sci. Lett., v. 113, p. 251-266.
Tajika, E., and Matsui,T. , 1993, Evolution of seafloor spreading rate based on 40Ar degassing history: Geophys.
Res. Lett., v. 20, p. 851-854.
Taylor, S. R., and McLennan, S. M.,1995, The geochemical evolution of the continental crust: Rev. Geophys., v.
33, p. 241-265.
Turcotte, D. L., and Oxburgh, E. R., 1967, Finite amplitude convective cells and continental drift: J. Fluid Mech.,
v. 28, p. 29-42.
Turcotte, D. L., and Burke, K. , 1978, Global sea-level changes and the thermal structure of the earth: Earth
Planet. Sci. Lett., v. 41, p. 341-346.
Veizer, J., and Compston, W., 1976, 87Sr/86Sr in Precambrian carbonates as an index of crustal evolution :
Geochimica et Cosmochimica Acta, v. 40, p. 905-914.
24
Weinstein, S. A., 1996, Thermal convection in a cylindrical annulus with a non-Newtonian outer surface: Pure
Appl. Geophys., v. 146, p. 551-572.
White, R., and McKenzie D. , 1989, Magmatism at rift zones: the generation of volcanic continental magins: J.
Geophys. Res., v. 94, p. 7685-7729.
Windley, B.F., 1977, Timing of continental growth and emergence: Nature, v. 270, p. 426-428.
Windley, B. F., 1995, The evolving Continents: 3nd ed., John Wiley & Sons, v. , p. 526 pp..
Wise, D. U., 1974, Continental margins, freeboard and the volumes of continents and oceans through time:
Geology of Continental Margins, v. , p. 45-58.
Wray, G. A., Levinton, J. S., and Shapiro, L. 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