Scaling of time-dependent stagnant lid convection' Application to small-scale convection on Earth and other terrestrial planets

Transcription

1 JOURNAL OF GEOPHYSCAL RESEARCH, VOL. 105, NO. B9, PAGES 21,795-21,817, SEPTEMBER 10, 2000 Scaling of time-dependent stagnant lid convection' Application to small-scale convection on Earth and other terrestrial planets V. S. Solomatov Department of Physics, New Mexico State University, Las Cruces L.-N. Moresi Australian Geodynamics Cooperative Research Centre, CSRO Exploration and Mining Nedlands, Western Australia Abstract. Small-scale convection associated with instabilities at the bottom of the lithospheric plates on the Earth and other terrestrial planets occurs in the stagnant lid regime of temperature-dependent viscosity convection. Systematic numerical simulations of time-dependent, internally heated stagnant lid convection suggest simple scaling relationships for a variety of convective parameters and in a broad range of power law viscosities. Application of these scaling relationships to the Earth's oceanic lithosphere shows that for either diffusion or dislocation viscosity of olivine, convective instabilities occur in the lower part of the lithosphere between 85 and 100 km depth (the rheological sublayer). "Wet" olivine satisfies constraints on the heat flux and mantle temperature better than "dry" olivine, supporting the view that the upper mantle of the Earth is wet. This is also consistent with the fact that the rheological sublayer is located below the Gutenberg discontinuity which was proposed to represent a sharp change in water content. The viscosity of asthenosphere is (3-6)x 10 s Pa s, consistent with previous estimates. The velocities of cold plumes are relatively high reaching several meters per year in the dislocation creep regime. A low value of the heat flux in old continental cratons suggests that continental lithosphere might be convectively stable unless it is perturbed by processes associated with plate tectonics and hot plumes. The absence of plate tectonics on other terrestrial planets and the low he t transport efficiency of stagnant lid convection can lead to widespread melting during the thermal evolution of the terrestrial planets. f the terrestrial planets are dry, small-scale convection cannot occur at subsolidus temperatures. 1. ntroduction Small-scale convection was proposed as a possible explanation for the observed flattening of the oceanic floor, the heat flux, and the geoid in the old oceanic lithosphere [Parsons and McKenzie, 1978; Buck and Parmentier, 1986; Davaille and Jaupart, 1994; Doin et al, 1996]. A similar process, sometimes called delamination, was proposed for the continental lithosphere [Bird, 1979; Houseman et al., 1981; Yuen and Fleitout, 1985; $chmeling and Marquart, 1991; $eber et al., 1996; Jaupart et al., 1998]. The basic mechanism of both phenomena is probably similar, that is convective instability of the bottom of the lithosphere, and we will refer to both phenomenas small-scale convection (Figure 1). Copyright 2000 by the American Geophysical Union. Paper number 2000JB / 00 / 2000JB ,795 Small-scale convection plays an important role in determining the thicknesses of continental and oceanic lithospheres, the mantle temperature, the mantle viscosity, the convective velocities, and the intensity of mixing in the mantle. The interaction of small-scale convection with plate motion breaks the classical square root law of the cooling of the oceanic lithosphere, which can affect the scaling laws governing plate tectonics and global thermal evolution of the Earth. A quantitative description of small-scale convection requires both the constitutive laws for ductile creep and the fluid dynamical scaling relationships. n steady state ductile creep, solid mantle is usually considered to be a variable viscosity fluid [Karato and Wu, 1993] (Table 1): 1 (½)n-l(,)m RT Q r - C RT Q r/- - exp = exp,, (1) where A and c - (1/2A) - (h/b*) m are assumed to

2 21,796 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON The scaling of the stagnant lid regime has been uncertain, and a range of theoretical laws has been suggested. The scaling laws obtained by Morris and Canright [1984] and Fowler [1985] with the help of boundary layer theory proved to be very accurate in describing both two-dimensional [Moresi and Solomatov, 1995; Dumoulin et al., 1999] and three-dimensional [Reeset al., 1999b] steady state convection. Reeset al. [1998] Plumes extended Morris and Canright's [1984] and Fowleds [1985] theories to non-newtonian convection and supported the results with numerical calculations. Although detailed studies of steady state flow have Core-mantle boundary been crucial for understanding the fundamental features Figure 1. A general picture of plate tectonics with of stagnant lid convection, mantle convection is likely to small-scale convection and plumes. be time-dependent and may not be accurately described by scaling laws for steady state convection. The laboratory experiments by Davaille and Jaupart [1993] and be constant, u is the shear modulus, b* ls the Burgers numerical studies of time-dependent stagnant lid con- vector, T is the temperature, r is the second invariant of the deviatoric stress tensor, Q = E* + PV* is the activation enthalpy, P is the hydrostatic pressure, R is the gas constant, E* is the activation energy, V* is the activation volume, h is the grain size, n is the stress exponent, and m is the grain size exponent. Perhaps the most important part of this law is the dependence on temperature: Hot rocks flow more easily than cold rocks. Convection in fluids with such rheology occurs in the stagnant lid regime [Natal and Richter, 1982; Stengel et al., 1982; Christensen, 1984; Morris and Canright, 1984; Fleitout and Yuen, 1984a, 1984b; Jaupart and Parsons, 1985; Fowler, 1985; Buck, 1987; Ogawa et al., 1991; Davaille and Jaupart, 1993, 1994; Moresi and Solomatov, 1995; Solomatov, 1995; Doin et al., 1997; Dumoulin et al., 1999]. The most viscous part of the lithosphere is essentially rigid and convection involves only a thin layer at the bottom of the lithosphere. This nearly rigid part of the lithosphere participates in the global motion associated with plate tectonics (Figure 1). Stagnant lid convection was proposed to be the major convective style on other terrestrial planets [Ogawa et al., 1991; Solomatov and Moresi, 1996, 1997; Schubert et al., 1997] and on icy satellites [McKinnon, 1998]. vection with both Newtonian and non-newtonian vis- cosities [Doin et al., 1997; Tromperr and Hansen, 1998; Grasset and Parmentier, 1998; Reese et al., 1999a; Dumoulin et al., 1999] all suggested higher scaling exponents consistent with the scaling theory and boundary layer stability analysis [Solomatov, 1995]. The change in the scaling relationships upon the transition from steady state convection to time-dependent convection have been most convincingly demonstrated by Dumoulin et al. [1999]. Some discrepanciestill remain between the scaling laws suggested for time-dependent convection. These discrepancies can partially be attributed to the fact that in some studies, convection was in the transition between steady state and fully time-dependent convection, as it is clearly the case in Tromperr and Hansen's [1998] simulations. n other cases [Grasset and Parmentier, 1998] the differences could be due to a different heating mode (internal heating). nternally heated convection can, indeed, give different scaling laws [McKenzie et al., 1974; Kulachi and Emara, 1975, 1977; Jarvis and Peltlet, 1982; Schubert and Anderson, 1984; Parmentier et al., 1994; Sotin and Labfosse, 1999]. Unfortunately, both experimental [Davaille and Jaupart, 1993] and numerical [Grasset and Parmentier, 1998; Reese et Table 1. Viscosity Parameters Dislocation creep Diffusion creep Dry Wet Dry Wet A, s -x 3.5 x 10 ' ' 2.0 x 10 TM 8.7 x 10 x5 5.3 x 10 x n m E*, kj mol-x V*, cm a tool h, a cm ' ' - afor dislocation creep, h is controlled by dynamic recrystallization.

3 uncertain. SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON 21,797 al., 1999a] results for internally heated convection (con- 3. Nondimensional Parameters vection driven by cooling from above in Davaille and The equations for internally heated convection [e.g., Jaupart's [1993] experiments is very similar to internally Christensen, 1985] are usually nondimensionalized by heated convection) had errors too large to allow for a using the thickness d of the layer for the length scale, full inversion of the parameters. The scaling exponents were assumed rather than inverted from the data. Thus d2/n for the time scale, ature scale, where p is the density, n = k/pcp is the the scaling laws for internally heated convection remain thermal diffusivity, k is the thermal conductivity, and Another unresolved issue is the dependence of scaling relationships on the power law exponent n. The available numerical simulations usually provide a scaling law to temperature alone for convection with n = or n - 3, while the viscosity laws for rocks or ice can have different values of n [e.g., Karato and Wu, 1993; Durham et al., 1997; Goldsby and Kohlstedt, 1997]. Theoretical studies indicate that it might be possible to obtain a general scaling law for an arbitrary n [Solomatov, 1995; Reese et al., 1998]. Reese et al. [1999a] postulated such scaling laws for the heat flux and the velocity on the basis of various experimental and numerical data for both internal and bottom-heated convection. The goal of this study is to provide stronger constraints on the scaling relationships for time-dependent, internally heated, stagnant lid convection. To distinguish between alternative scaling laws, we increase the accuracy of the calculations. This is achieved by running the models for a longer time and by using suffi- Fowler, 1985; Davaille and Jaupart, 1993; Solomatov, ciently long boxes to eliminate the wall effect. We also 1995]. Also, it is convenient to define the Rayleigh number based on the temperature difference AT = Ti - To find scaling laws for a variety of parameters, as was between the interior temperature Ti of the convective done for steady state convection [Moresi and $olomatov, 1995; $olomatov and Moresi, 1996]. To find the depenlayer and the surface temperature To rather than based dence of scaling relationships on the power law exponent on the temperature scale phd 2/k. Therefore the scaln, systematic simulations are performed for n = 1, 2, ing analysis will be based on the following definition of and 3. As a result, we suggest scaling relationships for the Rayleigh number: a general temperature- and pressure-dependent power apgatd(n+2)/n law Arrhenius viscosity. These scaling relationships are Rai = bi/nni/n exp (-7Ti/n)' (5) applied to Earth and other terrestrial planets. 2. Viscosity where b and -/are constants, -/is related to the original formula through /= Q/RTi 2 and Ti is the interior temperature. Although we consider purely temperaturedependent viscosity, extrapolation to pressure-dependent viscosity can be done using previous numerical results on stagnant lid convection with pressure-dependent viscosity [Doin et al., 1997; Reese et al., 1999a; Dumoulin et al., 1999]. and phd2/k for the temper- cp is the thermal heat capacity at constant pressure. As a result, two nondimensional parameters appear in the equations: the logarithm of the viscosity contrast due phd 2 0 -ln(ar/) - k./_x, (3) which is known as the Frank-Kamenetskii parameter, and the surface Rayleigh number based on the internal heating rate H ap2 ghd(3n+2)/n (4) Ra t,o = k bi/nni/n, where g is the gravitational acceleration and a is the coefficient of thermal expansion. These definitions of the Rayleigh numbers are not very useful in the stagnant lid regime because convection beneath the lid is insensitive to the viscosity in the upper part of the lid [Morris and Canright, 1984; t is calculated a posteriori. The Nusselt number is defined as As in most previous studies, we consider an exponential viscosity instead of the Arrhenius relationship (1) where the heat flux F = phd. (Frank-Kamenetskii approximation) since the two laws Such definitions are essentially the same ones as for give almost identical results in the limit of large viscos- bottom-heated convection [Moresi and $olomatov, 1995; ity contrasts [Morris and Canright, 1984; Fowler, 1985; Dumoulin et al., 1999]. The only difference is that for $olomatov and Moresi, 1996; Doin et al., 1997; Ratcliff bottom-heated convection, AT is the temperature conet al., 1997; Reese et al., 1999a]: trast between the boundaries of the convective layer, which includes a small temperature drop in the bottom b boundary layer. - rn_ exp(-ft), (2) The interior temperature Ti is often defined as the average bottom temperature, which is somewhat influenced by the cold plumes accumulating at the bottom. Perhaps, a more meaningful (but not much different) definition of Ti is the maximum horizontally averaged temperature in the layer (Figure 2). This is the temperature that affects the instabilities at the bottom of the lid and controls the thickness of the lid in the stagnant lid convection regime. Fd

4 _.. 21,798 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON Stagnant lid Rheological sublayer.: :¾.&. :¾ ; ; ::'! : :.-..:- ::i:;: ' :. :... ": ; : :i 'i..,...'!::::i:,: ': ::::':.:... ::..:.-:. :.:...,.,,...:.::::.... :-.--::----::::..:.... i ::i; :::::::"::'"":"*: :"':""" ' ' " '.:;.i "">'... :.':::<' '"' :' -.:i.-" :::... 'q: 5.::.-': --"--.-':-..' '-:... '<?'" ":'""-:-*.5:' 7".: -':., 5' ' :'!?g} L '410'310' 10 ' 10 ' 10 '4 10 o Temperatu re Velocity Stress Viscosity Figure 2. The structure of stagnant lid convection and definition of various parameters. The bottom of the lid ( L is found from the velocity profile. The dotted line on the velocity graph corresponds to the maximum velocity gradient. The bottom of the lid is defined as the intersection of this line with the vertical axis. The dotted line on the temperature graph shows the steady state conductive temperature distribution. 4. Numerical Results We considered only the solutions which were in a well-. developed stagnant lid regime. They were selected using The finite element code CTCOM was used to perform systematic numerical simulations of stagnant lid the criterion that the surface mobility $ = 5o2Uo << 1 convection in a 4 x 1 box with free-slip boundary condi- [.Solomatov and Moresi, 1997], where 50 is the nondimensional thickness of the cold boundary layer and uo is tions. The domain is heated from within, and the botthe nondimensional surface velocity. The viscosity contom boundary is thermally insulated. A convergence trast A = exp(o/nu) in all selected cases is also subtest for n = 3, 0-60, Rao = 3 (Table 2) shows that stantially higher than A e4(n+l) suggested for the the errors due to spatial resolution depend on the ratio transition to stagnant lid convection [Solomatov, 1995; of the thickness 5rh of the unstable part of the lid (rhe- Moresi and $olomatov, 1995; $olomatov and Moresi, ological sublayer) to the cell size. This ratio is about unity for the 32 x 8 grid. For coarser grids the errors 1997; Reese et al., 1999a]. We also excluded steady and periodic solutions that would affect the scaling laws. All grow very quickly. A uniform grid of 128 x 32 ensures the remaining solutions are strongly time-dependent. that in all cases the thickness of the rheological sublayer Each case was prerun until the balance between the is spanned by at least two cells and the errors due to resolution do not exceed 0.5-1% for the Nusselt num- heat generation rate and heat loss rate was established within the accuracy of its determination (a fraction of ber, 2-3% for the root-mean-square (rms) velocity, and 3-5% for the rheological temperature difference. Table 3 shows that 4 x 1 box is sufficiently long to prevent wall effects. Wall effects are strong in 1 x 1 box and not completely eliminated in 2 x 1 box. Table 2. Convergence Test for Rao - 30, 0-60, and n - 3 a percent). Then the simulations were continued until the errors due to fluctuations drop below 0.01 ( 00.3%) for the Nusselt number. The absolute errors for the velocity vary from to 10 ( 03% on average) depending on the intensity of fluctuations. An example of calculation of time averages is given in Figure 3. Formation of instabilities at the bottom of the stagnant lid is shown in Mesh Nu rms arh 16 x x x x x athe errors due to fluctuations do not exceed 0.2% for Nu, 1% for u n, and 1% for a h. Table 3. Dependence on the Aspect Ratio for Ra - 10, 0-40, and n- 1 Aspect ratio Nu u ns arh Regime 1 x steady 2 x chaotic 4 x chaotic 8 x chaotic

5 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON 21, stagnant lid [Morris and Canright, 1984; Fowler, 1985; Davaille and Jaupart, 1993; Solomatov, 1995]- looo ATrh- arh0-1. > X 5o o,.ll ,,, To estimate arh from the numerical simulations, we can calculate ATrh and then calculate arh = OATrh. The value of ATrh depends on the definition. Davaille and Jaupart [1993] used the criterion that the convective heat flux becomes negligible at the bottom of the stagnant lid and estimated arh 2 for Newtonian viscosity (n = 1). n other studies [Davaille and Jaupart, 1994; Grasset and Parmentier, 1998], arh was determined from the requirement that the heat transport in the convective layer beneath the stagnant lid be equivalent to that of constant viscosity convection. This gives arh 2.2. ' We could define ATrh as the temperature contrast associated with the descending cold plumes. The value of arh obtained this way is between 2 and 4 depending on n and the exact definition of the temperature contrast in the plumes. Unfortunately, thermal diffusion quickly erodes temperature contrasts in the descending plumes and such measurements can only give a lower bound on ATrh. n fact, similar calculations for the case when the viscosity does not depend on temperature (including constant viscosity, n -- 1) and with a rigid upper boundary showed that the temperature contrasts in the descending plumes are only 40-80% of the actual temperature contrast in the thermal boundary layer. We can use a similarity with isoviscous convection and notice that the velocity changes approximately linearly near the rigid upper boundary. This suggests an Time Figure 3. An example of calculation of time averages is shown for rms velocity (Rao = 30, 0 = 60, n = 3). From top to bottom, the rms velocity Urms; time averaged value rms- t- f Urmsdt; and the surface heat flux (nondimensional). Figure 4. The results for n - 1, 2, and 3 are summarized in Table Scaling 5.1. Rheological Sublayer The cold thermal boundary layer of thickness 5 con- Figure 4. A sequence of frames is taken at equal time sists of the stagnant lid of thickness 5L and the rheologintervals (Rao = 30, 0 = 80, n = 3). t shows a typical sequence of formation of cold plumes: A slowly thickenical sublayer of thickness 5rh (Figure 2 and Appendix ing rheological sublayer (the top two panels) suddenly A). The temperature difference across the rheological develops an avalanche-like instability (the bottom two sublayer is the driving force for convection beneath the panels).

6 21,800 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON Table 4. Numerical Results 0 lzlah,o Nu Urms arh n=l x x x X x x x x x x x x 10-3 n= x x x X x x X X x x 10-3 n= x X x x x 10 - ' x x x x x 10 - ' alternative definition of the bottom of the stagnant lid which is based on the velocity profile (Figure 2). The values of ATrn are calculated as the temperature difference between the interior temperature Ti and the temperature at the bottom of the lid TL. The variation of arn with n (Table 4) shows an approximate agreement with the theoretical scaling suggested by the boundary stability analysis [Solomatov, 1995]: arh -, 1.2(n + 1). (8) 5.2. Nusselt Number All scaling relationships for the Nusselt number in the stagnant lid convection regime have the form Nu - ao -a Rai, (9) where a, a, and/3 are constants depending on n. Usually, a = 1 +/ except for one case with a = 1 [Fowler, 1985; Reese et al., 1998]. The factor that is responsible for a - in steady-state models and that theoretically could affect time-dependent convection as well is the magnitude of lateral variations in the thickness of the stagnant lid [Fowler, 1985]. Large variations in the lid thickness were observed in numeri- cal simulations of steady state convection with Newtonian viscosity [Moresi and $olomatov, 1995], while for non-newtonian viscosity the lid is fiat [$olomatov and Moresi, 1997]. n time-dependent convection the thickness of the lid becomes fairly uniform in all cases. This happens because the locations of the plumes are not fixed but vary randomly. Therefore we will assume a = 1 +/3. (10) Two different values of/3 have been suggested. The boundary layer theories of steady state stagnant lid convection predict [Morris and Canright, 1984; Fowler, 1985; Reese et al., 1998] n / - 2n + a' (11) while the scaling theory and the boundary layer stability analysis give [$olomatov, 1995] n / =. (12) n+2 Although all our solutions are strongly time-dependent, the thickness of the lid is not negligible. Therefore, instead of directly fitting our results with (9), we have to use a more general equation (A8) derived in Appendix A. t reduces to (9) in the limit Nu >> 1. An inversion for all three parameters, a, /, and arh, does not put meaningful constraints on arh (note that arh, which appears only because the lid is not asymptotically thin, does not play any role when Nu >> 1; see Appendix A). Using the results of section 5.1 we can assume that arh is between n + and 1.5(n + 1) and find only the two remaining parameters, a and/ (Table 5). The nonlinear inversion was performed using the Levenberg-Marquardt method [Berington and Robinson, 1992; Press et al., 1992] assuming that the absolute errors of the individual measurements are (due to fluctuations only). The value of/ is in good agreement with that sug- gested by (12). This justifies using the exact theoretical values for one-parameter fits (Table 5). The results of one-parameter fits are also shown in Figure 5. Previously suggested scaling laws for time-dependent stagnant lid convection are summarized in Table 6. The lower value of/ observed in Trompeft and Hansen's [1998] simulations (Table 6) was most likely due to a relatively slow convective flow. This imposes constraints on the spacing and timing of instabilities so that the scaling exponent can be somewhere between those corresponding to steady state convection and timedependent convection. This can be seen from Trompert

7 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON 21,801 Table 5. Fit for the Nusselt number Fit arh a / 2va n-1 Two-parameter fit q q Two-parameter fit q q One-parameter fit b q n-2 Two-parameter fit q q Two-parameter fit q q One-parameter fit b q n-3 Two-parameter fit q q Two-parameter fit q q One-parameter fit b q achi-square divided by the degrees of freedom. bassuming/ = n/(n + 2) and arh (n + 1). and Hansen's Figure 18, which shows that/ increases with the Rayleigh number and approaches the asymptotic value 1/3. Dumoulin et al. [1999] solved this problem by forcing the solution to be steady but did not find any agreement with Reese et al.'s [1998] scaling for n = 3. However, the A small value of/ in Grasset and Parmentier's [1998] viscosity in Dumoulin et al.'s simulations depends on simulations is due to two major factors: (1) / was assumed on the basis of constant viscosity convection simpressure, which makes an accurate comparison impossible. An approximate comparison can be done as folulations rather than found from inversion of data for lows. Since the lid is nearly fiat [Solomatov and Moresi, stagnant lid convection and (2) at low Nusselt numbers, 1997], we can assume that a = 1 +/ and that the flow convection was likely to be insufficiently chaotic which could also be partially due to wall effects in the 2 x 1 convection box (Table 3) used in Grasset and Parmentier's [1998] simulations. t is also possible that some of the high Nusselt number cases might include data corresponding to not well-developed stagnant lid convecis controlled by the viscosity in the interior rather than in the rheological sublayer [see also Fowler, 1985; 1993]. Equation (9) with the Rayleigh number defined at the bottom and with the effective Frank-Kamenetskii parameter for pressure-dependent viscosity [Reese et al., 1999a] gives an approximate agreement between Dution. These are difficult to eliminate because the tran- moulin et al.'s [1999] results and Reese et al.'s [1998] theory (Figure 7 and Table 7). sition to stagnant lid convection is not well constrained for rigid boundary conditions used by Grasset and Parmentier [1998]. n addition, in the runs with 0 = 100 the thickness of the rheological sublayer is about cell size, which can produce substantial errors (see section 4). A fit to only those data that presumably correspond to a developed time-dependent stagnant lid convection gives/, which is statistically indistinguishable from 1/3 (Figure 6 and Table 6) Nusselt Number: Steady State Scaling For comparison, we summarize previously suggested scaling laws for the steady state regime in Table 7. Equation (11) seems to agree well with steady state numerical solutions. A discrepancy between the theoretical laws and the numerical results for n = 3 [Rees et al., 1998] is mainly due to the fact that for n = 3 steady state solutions exist only in a very narrow parameter range and some of the solutions were time-dependent Velocity Like the Nusselt number, the nondimensional velocity in the actively convecting region scales as ui -- au O- a. Rai., (13) in which we will assume a, =/ (for the same reason as a - 1 +/ in the case of Nusselt number). The scaling theory for bottom heated convection [Solomatov, 1995] suggests that 2n /, -. (14) n+2 As in the case of Nusselt number, we start with twoparameter fits (Table 8) and assume 3% error for indi- vidual measurements.

8 21,802 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON Nusselt number "1= Data Figure,5. The results of the fit for the Nusselt number: the values calculated using the fitting formula (A8) (model) versus the numerical data (data). Equation (14) gives a value for/ u that is too large, well beyond the uncertainty range (Table 8 and Figure 8). On the other hand, it fits well the data obtained for bottom-heated convection for n - [Dumoulin et al., 1999] (Figure 9). Asymptotic boundary layer theories predict a different function u(n) [Fowler, 1985; Reese et al., 1998]. tion. A scaling described in early papers on internally heated convection with constant viscosity [e.g., McKenzie et al., 1974, equation 6] assumes that the convective flow consists of a broad upwelling and a narrow downwelling of width about the thickness of the thermal boundary layer. This theory gives very low values of for the Nusselt number, which are inconsistent with both the constant viscosity convection simulations of McKenzie et al. [1974] and the stagnant lid convection simulations presented here. Such inconsistencies are likely to be due to unrealistic assumption of an extremely asymmetric flow. This implies that the typical length of the velocity variations in the downwelling scales with the thickness of the cold plumes, which is not observed in our simulations. t was also believed that such a very broad upwelling is necessary to remove internally generated heat from the interiors. However, in strongly time-dependent convection the heat can be removed efficiently by random plumes which deliver cold material to all parts of the layer. We can derive an alternative scaling based on the assumption that most viscous dissipation is concentrated in the rheological sublayer and cold plumes rather than in the interior (we find that about half of the total viscous dissipation in our models is in the rheological sublayer). This implies a balance between the rate of viscous dissipation T](U/( rh)2v(( rh/d) in the rheological sublayer of characteristic thickness ( rh and the work agfv/cp done by buoyancy forces per unit time (V is the volume of the convective layer). Together with the scaling law for the Nusselt number (11) and the temperature drop in the rheological sublayer (7), this gives (n + 1)(n + 2)' (15) which fits reasonably well all n (Table 8 and Figure 10). The remaining discrepancy might be due to the fact that dissipation is not localized to such a degree. t is also difficult to imagine that this picture would be valid at very large Rayleigh numbers. At some point, plumes can become disconnected from the lid and dissipate most of their potential energy in the interior. This regime would be very similar to one described by Yuen et al. [1993]. n this case, viscous dissipation does not depend much on the viscosity of plumes and depends mainly on the viscosity of the interior. Therefore, the scaling law for the velocity might change at very high Rayleigh numbers Velocity of Cold Plumes The velocity Up of individual cold plumes is much higher than the average velocity ui. There is no scaling law for Up. To estimate this parameter, we calculated However, it is unlikely that any of the steady state rela- the ratio of the maximum velocity Umax in the layer to tionships are applicable to time-dependent convection, the time-averaged velocity ui as a function of time for a as is clearly the case for the Nusselt number (Tables 6 few cases for each n. During an avalanche-like instabiland 7). ity developed at the bottom of the lid, Umax gives the The answer probably lies in the difference between- vertical velocity of the current cold plume. The averbottom heated convection and internally heated convec- age of the peak values of Umax(t) excluding the small

9 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON 21,803 Table 6. Coefficients in the Scaling Law Nu- a O- Rai for Stagnant Lid Convection Determined by Various Studies: Time-dependent Solutions a c / Heating Mode Geometry Method Reference n cooling from above 0.3 x 0.3 x 0.2 m experiment 1 ß bottom heating... scaling theory bottom heating... stability criterion bottom heating 4 x 1 numerical bottom heating 4 x 1 numerical internal heating 4 x 1 numerical this work internal heating 2 x 1 numerical internal heating 2 x 1 numerical bottom heating 4 x 1 numerical bottom heating 4 x 4 x 1 numerical 4 n-œ ß bottom heating... scaling theory bottom heating... stability criterion internal heating 4 x numerical this work n=3 ß bottom heating... scaling theory bottom heating... stability criterion bottom heating 4 x numerical internal heating 4 x numerical 6, this work References are (1) Davaille and Jaupart [1993]; (2) Soloraatov [1995]; (3) Doin et al. [1997];(4) Trorapert and Hansen [1998]; (5) Grasset and Parraentier [1998]; (6) Rees et al. [1999a]; (7) Duraoulin et al. [1999]; (8) estimated from Grasset and Parraentier's [1998] data by considering only those runs which were in a well-developed time-dependent stagnant lid regime (Figure 6). Nusselt number (Grasset and Parmentier's data) o , i i i peaks at the bottom of the curves (Figure 11) gives an estimate for the velocity of the cold plumes Up. The ratio Up/Ui depends only weakly on the vigor of convection but depends very strongly on n (Figure 11): {3, n-1 U p. 8, n- 2 (16) ui 25, n Viscous Stress and Apparent Viscosity According to scaling theory and boundary layer stability analysis [Solomatov, 1995] the typical shear stresses ri in the interior is scaled as ri = arapgatrh rh. (17) Data We calculated ri as a horizontally averaged second in- Figure 6. Comparison of the asymptotic scaling rela- variant of the deviatoric stress tensor at 1/4 distance tionship with/ - 1/3 with the data from Grasset and from the bottom (Table 4). This ensures that in all Parmentier [1998]. The symbols correspond to different cases the stress is measured in the interior, away from values of 0:20 (circles), 30 (squares), 40 (diamonds), 50 the stagnant lid as well as from the bottom (Figure (triangles) and 100 (inverted triangles). Solid symbols indicate the data points which were used for inversion 2). Excellent agreement between this scaling and the by Grasset and Parmentier [1998]. The data for numerical results is found (Figure 12). The fact that and 0-30 are connected by solid lines to show the the coefficient ar is almost the same for all n deviation from the asymptotic scaling due to two fac- provides further support for the scaling (7) of the rhetors: insufficiently vigorous convection (low Nu) and ological temperature difference. insufficiently developed stagnant lid regime (high Nu). The apparent non-newtonian viscosity /i in the inte- Note that the data for 0-20 start deviating from the rior region is calculated as asymptotic curve earlier than for 0-30 and actually none of the points for 0-20 are in the correct asymptotic regime. r i - b ri x-n exp - uu ' (18)

10 21,804 SOLOMATOV AND MORES' TME-DEPENDENT STAGNANT LD CONVECTON Table?. Coefficients in the Scaling Law Nu- a O-aRai for Stagnant Lid Convection Determined by Various Studies: Steady State Solutions a a Heating Mode Geometry Method Reference bottom heating x boundary layer theory 1, bottom heating x boundary layer t"heory 2, bottom heating x numerical 4, bottom heating x numerical internal heating spher. shell numerical bottom heating x boundary layer theory 3, bottom heating x boundary layer theory 3, bottom heating x boundary layer theory 3, bottom heating x boundary layer theory 3, bottom.heating x numerical bottom heating 4 x numerical 9 n-1 n--2 n--3 References are (1) Morris and Canright [1984];(2) Fowler [1985];(3) Solomatov [1995];(4) Moresi and Solomatov [1995];(5) $olomatov and Moresi [1996];(6) Rees et al. [1998];(7) Dumoulin et al. [1999];(8) Rees et al. [1999b]; (9) Estimated from Dumoulin et al.'s [1999] data (Figure 7). 6. Summary of Scaling Relationships The results obtained for temperature-dependent exponential viscosity can be applied to the original temperature- and pressure-dependent power law Arrhenius viscosity (1) provided the following definitions of 0 and Rai are used [Reese et al., 1999a]: Steady state, n=3 (Dumoulin et al.'s data) O 3.5 og(ra/b) Figure 7. Dumoulin et al.'s [1999] data for pressureand temperature-dependent viscosity steady state convection (n - 3) are fitted with Nub -- a(rax/b), where Rax is defined at the temperature and pressure at the lower boundary, b - (OT-Op/Nu)/3, and 0T and 2 p are the coefficients in r/- r- exp(--0tt +OpZ). One run with a very weak convection (Nu = 2.05/ was excluded from the fit. The fitting coefficients are a and = 0.33 (Table 7). where ATE Pi VTo 0- RTi2 RTi2, (10) apgatd(n+2)/n Rai = cl/nnl/n exp [(E + PiV)/nRTi] ' (2o) Pi - pg5 (21) is the pressure at the bottom of the thermal boundary layer. With the new definitions of 0 and Rai, the scaling relationships can be summarized as d 2(nq-1) _.n (22) Nu- - ( n)0- +2 Rai +2, (23) AWrh (n + 1)o-lAw, (24) -i -- O.027apgATrhC rh, (25) 7i - cri 1- exp E + Pi V (26) RTi The two alternative scaling laws for the velocity are (scaling and scaling respectively)' ui - ( n), (27) (28) ui--( n-1) n ( ai) ' n(2nq-1) (n q-1)(nq-2)

11 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON 21, Uncertainties Due to Extrapolation 7.1. Extrapolation to Arrhenius Viscosity Uncertainties related to extrapolation from exponential viscosity to Arrhenius viscosity are relatively small. A second-order correction 0 m 1.20 can be used to take into account the inaccuracy of Frank-Kamenetskii approximation [Reese et al., 1999a]. This results in only K temperature difference Extrapolation to Pressure-Dependent Viscosity Extrapolation to pressure-dependent viscosity with the help of (19) and (20) is based on the numerical resuits of Doin et al. [1997], Reese et al. [1999a], and Dumoulin et al. [1999]. n addition to these definitions, 7.3. Extrapolation to Three Dimensions Three-dimensional studies [Tromperr and Hansen, 1998; Grasset and Parmentier, 1998; Reese et al., 1999b] indicate that the differences in the Nusselt number between the two-dimensional and three-dimensional results are small, in the range of 10-20%. Reese et al. [1999b] found a somewhat higher value of arh 3.7 compared to 2.6 in our 2-D simulations. The larger value of arh could be partially due to the fact that the solutions were steady and had larger variations in the lid thickness. This affects the horizontally averaged velocity profile and therefore the apparent location of the bottom of the stagnant lid Extrapolation to Earth's Parameter Range The application of the scaling relationships to prob- Dumoulin et al. [1999] proposed a correction coefficient lems of the Earth or other planets does not require for the pressure term in (19) for n = 3. However, Reese much extrapolation in 0. For parameters typical for et al. [1999a] found that for internally heated convec- the Earth, 0 is for diffusion creep (n = 1) and tion with n - 3 and with the pressure-induced viscosity for dislocation creep (n = 3 or 3.5). n our simcontrast of up to (at constantemperature and ulations the value of O/Nu (which is equivalento the stress), the actual heat flux is only --1% smaller than new definition (19) of 0) is for n = and for the one predicted by (19), which is negligible. n = 3. Extrapolation to different n also does not extend Velocity is significantly affected by pressure-depen- too far: the four rheologicalaws considered below have dent viscosity in a manner which is not accurately de- n = 1, 3, and 3.5. The pressure term in (19) contributes scribed by (27) and (28). The results obtained by Reese about - 1% to 0, while the numerical simulations tested et al. [1999a] indicate that the rms velocity is smaller this scaling for pressure-dependent viscosity up to 10% by a factor of 2 for viscosity variations of 4 orders of [Reese et al., 1999a]. These seemingly small contribumagnitude (due to pressure alone) compared to the val- tions to 0 correspond to several orders of magnitude in ues expected from (27) or (28). This can be treated as a the viscosity variations due to pressure. reduced effective depth scale for the velocity variations A more substantial pressure effect is due to the pres- [Rees et al., 1999a]. sure-induced viscosity contrast across the entire mantle. Table 8. Fit for the Velocity Fit arh a / X} n--1 Two-parameter fit Two-parameter fit One-parameter fit a One-parameter fit b n--2 Two-parameter fit Two-parameter fit One-parameter fit a One-parameter fit b Two-parameter fit Two-parameter fit One-parameter fit a One-parameter fit b n O O.O aassuming/ - 2n/(n + 2) and arh (n + 1). bassuming/ -- n(2n + 1)/(n + 1)(n + 2) and arh (n + 1).

12 21,806 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON 25O loo Velocity (scaling ) lations it is -, 1/4, and Dumoutin et at. [1999] reached - 1/10 for bottom-heated convection. Among all the parameters described above, velocity is the only one which depends on the depth of the convective layer and is the only one which is most likely to be affected by this extrapolation. Equations (27) and (28) provide some range for the velocity estimates. 8. Oceanic Lithosphere 8.1. Parameters The viscosity parameters are given in Table 1. The values of other physical parameters are as follows [Anderson, 1989; Karato and Wu, 1993]: p kg m -3, Cp J kg-1 tool-l, k W m -1 K -, c - 3 x 10-5 K -1, / - 80 GPa, and b* nm. The effective depth scale for small-scale convection due to pressure-dependent viscosity and phase transformations is assumed to be d = 500 km. As we discussed above this only affects the velocity Data Figure 8. The results of the fit for the rms velocity: the values calculated using the fitting formula (A9) with flu = 2n/(n + 2) and arh = 1.2(n + 1) versus the numerical data (see also Table 8). At sufficiently high pressure-induced viscosity contrasts the basic assumption of the scaling laws (19)-(26) that small-scale convection is controlled by the viscosity at the bottom of the lithosphere can become invalid. The Rayleigh number for the mantle is much higher than in the numerical models. This can be seen more clearly in terms of the ratio of the thermal boundary layer thickness (or, alternatively, rheological sublayer thickness) to the thickness of the mantle rather than directly in terms of the Rayleigh number. For the Earth this ratio is roughly 1/30, while in our numerical simu Stable Versus Unstable Equilibrium Figure 13 and Figure 14 show the results of calculations for "wet" olivine. For a given temperature, there are two solutions for stagnant lid convection: stable (solid lines) and unstable (dashed lines). This is a consequence of pressure-dependent viscosity [Fleitout and Yuen, 1984a, 1984b; Doin et at., 1997; Reese et at., 1999a]. f the thermal conditions are in the region below the dashed line, there is no convection. As soon as the system crosses this line, the lithosphere becomes convectively unstable, and thinning of the lithosphere by small-scale convection continues until a stable equilibrium is reached (solid lines) [Reese et at., 1999a]. Constraints on the heat flux F mw m -2 be- neath the oceanic lithosphere and the mantle temperature Ti K [Parsons and Sctater, 1977; Velocity, bottom heating (Dumoulin et al.'s data) Data Figure 9. The fit to Dumoutin et at.'s [1999] data for the rms velocity with Urms - Ui --O.062(Rai/O) ß.

13 / / / SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON 21,807 Lister et al., 1990; Stein and Stein, 1992] suggesthat small-scale convection is in the stable regime Estimates for Wet Olivine i i t is easy to satisfy the constraints on the temperature and the heat flux for wet olivine using the grain size h as the major variable. The obtained values h- mm and V* = 10 cm 3 mol - are within a reasonable range (Table 1). The thickness of the thermal boundary layer is simply 5- k(ti -To)/F 100 km (+10 km). The Velocity (scaling ) n=2 lo O0 o , * o Time Figure 11. Examples of fluctuations of the ratio of the maximum velocity to the average velocity in the convective part of the layer for n - (0-60, RaH,o -- 1), n -- 2 ( , RaH,o -- 10), and n - 3 ( - 80, Ra t,o - 30). thickness of the unstable part of the lithosphere (the 80 rheological sublayer) is - 15 km. t is located between 85 and 100 km depth. The temperature contrast across the rheological sublayer is 200 K. 30 The high velocity of the convective flow (3-10 cm y on average and 1-3 m y- for individual plumes (see equation (16)) is comparable and can be even higher Data than plate velocities. The high velocities of the cold plumes for nøn-newtønian viscosity are consistent with Figure 10. The results of the fit for the rms velocity: the values calculated using the fitting formula (A9) with previous models explaining rapid regional uplifts and flu = n(2n+l)/(n+l)(n+2) and arh (n+l) versus thinning of continental and oceanic lithospheres [Yuen the numerical data (see also Table 8). and Fleitout, 1985; Larsen et al., 1996, 1997]. The in-

14 21,808 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON lo -5 Stress., - n=l s O n= The stresses, MPa, are essentially controlled by the scale o pgatrh( rh and are not very sensitive to the uncertainties in various parameters Wet Versus Dry Fitting the Earth's temperature and heat flux is much more problematic for dry olivine. Diffusion creep gives the grain size of 0.1 mm (Figure 15) which is at the lowest end of the possible grain size range. Dislocation creep results in temperatures too high for any values of the activation volume (Figure 16). These results agree with early thermo-mechanical models of the lithosphere and asthenosphere [Schubert et al., 1976, 1978]. The fact that wet olivine gives a much better fit for small-scale convection is consistent with other evidence that the Earth is wet [Thompson, 1992; Bell and Rossman, 1992; Bai and Kohlstedt, 1992; Kohlstedt et al., 1996; Ohtani et al., 1997]. n addition, only small amounts of water are necessary to make the olivine effectively wet [Chopra and Paterson, 1984; Kavato and Wu, 1993; Hirth and Kohlstedt, 1996]. The requirement that the rheological sublayer must be wet is consistent with the idea that at km depth (Gutenberg discontinuity), there is a sharp change in water content as a result of removal of water due to partial melting [Kavato, 1986; Hirth and Kohlstedt, 1996; Karato and Jung, 1998]. The dry layer is located above the rheological sublayer and cannot affect small-scale convection crease of peak velocities of cold plumes as a result of power law viscosity and the absolute magnitudes of the velocities agree well with the results obtained by Larsen and Yuen [1997] for hot plumes. The viscosity ]i is (3-6)x10 8 Pa s (within a factor of 2) for either diffusion creep or dislocation creep. These values are consistent with other constraints on the viscosity of the asthenosphere which range from large grain size ( 0.1 m) is obtained for 0.1 MPa if 10 TM to 1020 Pa s [Rydelecks and Sacks, 1990; $igmunds- we use a "classical" law assuming that the dynamically son, 1991; Zandt and Carrigan, 1993; Fjeldskaar, 1994; Kaufmann and Wolf, 1996; Pollitz et al., 1998]. Similar estimates were obtained by Davaille and Jaupart [199a], Doin et al. [1997], and Dumoulin et al. [1999] Diffusion Versus Dislocation Creep We cannot distinguish between diffusion and dislocation creep based on our estimates alone. Seismic observations suggest that the upper mantle is anisotropic (for reviews, see Davis [1995], Silver [1998], Karato [1998], Montagner [1998], and Savage [1999]), which is a signature of dislocation creep [Karato and Wu, 1993; Karato et al., 1995]. Anisotropy is particularly strong in the upper 200 km [Karato and Wu, 1993]. This was one of the reasons that led to the hypothesis that the Data Lehmann discontinuity marks the bottom of anisotropic Figure 12. The results of the fit for the viscous layer [Kavato, 1992; Kavato and Wu, 1993; Gaherty and stresses. The coefficient in the fitting formula (17) is Jordan, 1995]. The unstable part of the lithosphere be for n - 1, for n - 2, and for n - 3. tween 85 and 100 km depth seems to lie within this anisotropic region. However, both the uncertainties in the depth of anisotropy and a poorly understood evo- lution of anisotropy in the convective mantle [see, e.g., Chastel et al., 1993] do not allow us to make a definite conclusion yet. n principle, we could exclude dislocation creep if the steady state grain size established by dynamic recrystallization turned out to be too big. ndeed, a very recrystallized grain size does not depend on temperature [Karato and Wu, 1993]. However, recent theories of dynamic recrystallization suggesthat the grain size does depend on temperature [Derby and Ashby, 1987;

15 _. _ ~. ß ß SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON 21,809 Earth, diffusion creep, "wet" / E O 4O 6O 8O O 5O 40 e>, 30 m ß 8 E 6-4 o - 2,' O 0,.,," '"' 220 ' i i! i i ß 190 o n _ o10 10 a ' Heat flux, mw/m 2 Heat flux, mw/m 2 Figure 13. Calculations of the temperature, the thickness of the rheological sublayer, the temperature drop across the rheological sublayer, the average velocity, the viscous stress, and the apparent viscosity for small-scale convection beneath the lithospheric plates on Earth for diffusion creep of wet olivine (Table 1). The dashed lines correspond to unstable solutions. The dotted lines indicate that the mantle is partially molten. The dependence of the results on the activation volume and the grain size is shown only for the temperature; the other parameters are not affected much by these variations. The velocity is estimated using the two alternative scalings (27) and (28). Shimizu, 1998]. Moreover, De Bresser et al. [1998] argue that this dependence is such that the steady state grain size controlled by dynamical recrystallization is very close to the boundary between diffusion creep and dislocation creep and can actually be found from the condition that the two creep mechanisms give the same viscosity. This happens almost automatically when we fit the heat flux and the temperature so that the two creep mechanisms remain indistinguishable Typical Length Scale of Lateral Variations Small-scale convection has different length scales. The smallest length scale is controlled by the spacing between plumes. t is of the order of the thickness of the convective layer and decreases weakly with the vigor of convection [Pavmentiev and Sotin, 2000]. n a more realistic, vertically stratified mantle (because of pressure-dependent viscosity and phase transformations) the spacing between plumes is probably of the order of the width of the low-viscosity zone (where the viscosity changes only by a factor of 10 or so). Seismic studies and viscosity models suggesthat the thickness of low-viscosity zone is only a few hundred kilometers [Hager, 1984; Grand and Helmberger, 1984; King, 1995; Mitrovica, 1996]. This is also a typical length scale for small-scale lateral variations in the gravity field which are presumably associated with small-scale convection [Haxby and Weissel, 1986]. The largest length scale is controlled by lateral variations in the thickness of the lid and can be as large as the horizon-

16 -- -,, o 21,810 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON Earth, dislocation creep, "wet" //\ Ocm /mol \ i i i i ' O3 10 ' 5 o ' '"' E ß 210 o m 200,'"' Q o [ 1018 i''"-, Heat flux, mw/m 2 Heat flux, mw/m 2 80 Figure 14. Same as Figure 13, but for dislocation creep of wet olivine. Earth, diffusion creep, "dry" Earth, dislocation creep, "dry" 2OOO 1800 ' '/ '!\ 0.2 mm OOO L /// \ \ '-.25 cm3/mol \, o.1 - o.o5 1900, Heat flux, mw/m Heat flux, mw/m 2 Figure 15. Temperature as a function of heat flux for diffusion creep of dry olivine. Figure 16. Temperature as a function of heat flux for dislocation creep of dry olivine.

17 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON 21,811 tal size of the lithospheric plates (thousands of kilo- oceanic and continental mantles. There are a few of meters). Long-wavelength geoid anomalies [Cazenave the most obvious factors which can make the continenet al., 1992, 1995] are probably associated with these tal lithosphere more viscous and convectively stable: large-scale variations in the lithospheric thickness. All (1) the mantle temperature, the subcontinental manthese complexities can be responsible for the varia- tle needs to be cooler than the suboceanic mantle only tions in anisotropy observed by Nishimura and Forsyth by K; (2) the grain size, if the diffusion creep [1988, 1989] and EkstrSm and Dziewonski [1998]. The is the dominant creep mechanism then about 30-50% situation is further complicated by the unknown pat- larger grains in the continental region are needed; (3) tern of small-scale convection in a more realistic, three- the water content; (4) partial melting; and (5) different dimensional case. On the one hand, the similarity be- mineralogy. tween convection beneath the stagnant lid and con- n the absence of small-scale convection, hot plumes stant viscosity convection indicates a possibility of sim- might be the major mode of convection beneath the ple two-dimensional rolls aligned perpendicular to the continents. The magnitude of the heat flux associated plate motion [Richter and Parsons, 1975; Haxby and with plumes [Davies, 1988; Sleep, 1990; Malamud and Weissel, 1986]. On the other hand, it is unclear how 15- Turcotte, 1999] seems to be consistent with the heat flux km-thick cold downwellings can form long stable sheets beneath the continents. rather than plumes. 10. Terrestrial Planets 9. Continental Lithosphere Other terrestrial planets and the Moon show little ev- Fast seismic velocities beneath continents indicate idence of plate tectonics (for discussion of possible plate that continental lithosphere is thick, probably between 200 and 400 km [Grand, 1994; Suet al, 1994; Polet and Anderson, 1995]. $clater et al. [1980] estimated that tectonics on Mars and Venus, see Sleep [1994], Acura et al. [1999], and Schubert and Sandwell [1995]). f subsolidus mantle convection occurs on other terrestrial the heat flux beneath the continents is around 25 mw planets, it must be predominantly in the stagnant lid m -2. More recent analyses of the heat flux beneath old continentalithosphere suggests that the heat flux from convection regime. Exploring the entire evolution of the terrestrial planets is beyond the scope of this paper, and the mantle is as low as mw m -2 [Pinet and Jau- here we only estimate the efficiency of small-scale conpart, 1987; Jones, 1988; Pinet et al., 1991; Gupta et al., 1991; Lesquer and Vasseur, 1992; Jaupart et al., 1998; Rudnick et al., 1998; Mareschal ½t al., 1999, 2000]. f the heat flux beneath the continental lithosphere is indeed so low, small-scale convection beneath the continental lithosphere is likely to be absent. Figures 13 and 14 show that at the heat fluxes below some critical vection by calculating the dependence of mantle temperature on the heat flux. To minimize uncertainties in the viscosity law and in the parameterization of convection, we can use the oceanic region of the Earth as a reference point. We start with the assumption that the rheology of Earth corresponds to dislocation creep of wet olivine (diffuvalue Fcr mw m -2, small-scale convection sion creep gives similar results) and then consider the cannot be in stable thermal equilibrium. This value is hardly affected by the various uncertainties. This can following two end-member cases: (1) other planets are as wet as the Earth and (2) they are dry. be seen from For dislocation creep of wet olivine, the activation kpgv*at volume V* is chosen to be V* - 10 cm 3 mol -, which Fcr - (n-b 2)RTi' (29) gives the correct heat flux and temperature for the which is approximately valid in the parameter range of interest. External forces associated with plate tectonics and hot plumes should be the primary cause for temporary destabilization of the lithosphere and delamination in the continental region. One of the possible explanations for the convective stability of the continental lithosphere is that the mantle below continents is depleted in iron and garnet and has a lower density [Jordan, 1975, 1978; Forte et al., 1995; Doin et al., 1996]. Composition alone may be insu cient to explain the stability of continental roots, and viscosity was proposed as another important facoceanic lithosphere on Earth. For dry olivine, V* is chosen to be at its low boundary which is 15 cm 3 mol - (Table 1). Figure 17 shows the dependence of the mantle temperature of the terrestrial planets on the mantle heat flux. The surface heat flux would be higher by the amount generated by the radioactive isotopes in the crust. Several conclusions can be drawn from these calculations. On smaller planets, convection can occur at somewhat lower temperatures and lower heat fluxes. This happens because on larger planets, pressure increases the viscosity at the bottom of the lithosphere tor [Pollack, 1986; Doin et al., 1997; Lenafdic and and makes convection more difiqcult (this trend would Moresi, 1999; De $met et al., 1999; Shapiro et al., 1999]. cease for very small bodies where the low gravity would According to our calculations to obtain low heat flux through the base of the continental lithosphere requires a viscosity contrast of only a factor of 10 between the eventually dominate the pressure-dependent viscosity). For an equilibrium heat flux corresponding to chondritic composition of the silicate mantles (assuming that

18 ... o. o o 21,812 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON Venus Mercury! dry. ' 1600 wet O 4O 6O Mars O 4O 60 Moon wet ß -.,, Heat flux, mw/m 2 Heat flux, mw/m 2 Figure 17. Mantle temperature as a function of the heat flux at the bottom of the lithosphere for Venus, Mars, Mercury and the Moon. The results are shown for dislocation creep. Diffusion creep gives similar results. The dotted lines indicate that the mantle is partially molten. At smaller values of the heat flux, the solutions are unstable (Figure 13) and are not shown. the concentration of uranium is 20 ppb [O'Nions et al., 1979]), stagnant lid convection seems to be possible on all planets. n the beginning of planetary evolution radiogenic heating was - 4 times higher and convection was likely to be even stronger (note, however, that the heat flux does not necessarily follow radiogenic heating and depends on the thermal history which can be that among all the terrestrial planets, Mercury has the highest ratio of the critical heat flux for the beginning of melting to the equilibrium heat flux due to radiogenic heat production (Table 9). This suggests that Mercury might have undergone the least degree of postaccretional melting. ts primordial crust has had a better chance of survival since its formation period than the quite complicated [e.g., Christensen, 1984; $olomatov crust of either Mars or Venus whose surfaces have been and Moresi, 1996; Reese et al., 1999a]). For the Moon the critical heat flux is especially low, 3 mw m -2. t should be compared with the Apollo measurements, which suggest - 18 mw m -2 for the mean heat flux modified by more recent magmatic events [Hartmann et al., 1999; Basilevsky et al., 1997]. This conclusion is consistent with the old age of the Mercurian surface [Strom and Neukum, 1988]. [Langseth et al., 1976]. Another conclusion is that the melting is difficult to avoid during some period of planetary evolution even 11. Conclusions for wet olivine rheologies (Table 9). On smaller planets, melting starts at smaller values of the heat flux. The solid lines in Figure 17 show the range in which the stagnant lid convection operates' at low values of the heat flux, the mantle is convectively stable, and at higher values, widespread melting begins. The estimates for the beginning of melting assume that the 1. n time-dependent stagnant lid convection the stagnant lid is at the boundary of convective instability. Small-scale convection depends mainly on the viscosity in the 15-km-thick layer at the bottom of the lithosphere (the rheological sublayer). This suggests simple scaling relationships for various parameters of small-scale convection with realistic temperature- and solidus (Tsol P-5.6P 2 [Reeset al., 1999a]) pressure-dependent power law Arrhenius viscosity. The is the same for all planets (we only consider the beginning of dry melting; wet melting would start even earlier). f planets are effectively dry, the critical heat flux velocity scaling depends on the heating mode, yet the differences are not significant in extrapolating to the mantle. for the beginning of widespread melting is substantially reduced so that melting begins almost simultaneously with the onset of convection. t is interesting to note 2. Constraints on the mantle temperature and the surface heat flux give the following estimates for smallscale convection on Earth: the viscosity of astheno-

19 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON 21,813 sphere is (3-6)x1018 Pa s, the average velocity is of stagnant lid convection is one of the possible causes 3-10 cm y-l, the velocity of cold plumes can reach for the postaccretional basaltic volcanism on Mars and 1-3 m y-l, the thickness of the rheological sublayer is Venus. The small size of the silicate mantle on Mercury 15 km, the temperature contrast across the rheological gives this planet a better chance of avoiding widespread sublayer is 200 K, and the stresses in the astheno- melting. The primordial crust formed on this planet sphere are 0.1 MPa. during the accretion period could have been preserved. 3. Among the four viscosity laws suggested by Karato 6. The new scaling relationships describing smalland Wu [1993] the wet olivine viscosity can easily sat- scale convection allow accurate prediction of the average isfy the constraints on the mantle temperature and the parameters of convection for any realistic pressure- and surface heat flux for either diffusion or dislocation creep. temperature-dependent power law viscosity (including The dry olivine viscosity requires either too low an activation volume (dislocation creep) or too small a grain size (diffusion creep). Although our present knowledge of mantle viscosity does not allow us to exclude the dry olivine viscosity based on fluid dynamic simulations alone, this is consistent with other indications that the Earth is wet. Since small-scale convection depends mainly on the viscosity in the rheological sublayer, it is not affected by a much stiffer layer above the Gutenberg discontinuity at km depth, which was proposed to represent a sharp change in water content. 4. The heat flux beneath the continental lithosphere can be below the critical heat flux for an equilibrium small-scale convection. This suggests that the continental lithosphere might be stable with respect to smallscale convection and that plumes can be the primary convective mode in the subcontinental mantle. The difference between the continental and oceanic litho- spheres could be due to small differences in the grain size, water content, temperature, and other parameters. 5. f other terrestrial planets and the Moon are as wet as Earth, they are likely to convect as well. The efficiency of heat transport for stagnant lid convection is much lower compared to plate tectonics, and global widespread melting is inevitable during some period of planetary evolution. f planetary interiors are dry, initi- oxygen fugacity, which only changes the apparent parameters in the Arrhenius law). However, there are several issues which deserve further investigation. First of all, scaling of stagnant lid convection is still largely restricted to temperature-dependent viscosity. Stagnant lid convection with pressure-dependent viscosity has been studied in a relatively narrow parameter range and without much theoretical support. This is a very important problem given large pressure-induced viscosity variations in the mantle and its effect on the velocity of both cold and hot plumes. Second, some scaling laws can change at higher Rayleigh numbers. The criteria for the transition to stagnant lid convection also need to be constrained better, especially at higher Rayleigh numbers and for the Arrhenius viscosity. These criteria are important for the understanding of initiation and cessation of plate tectonics and for the global thermal evolution of the Earth and terrestrial planets [Solo- matov and Moresi, 1996; Schubert et al., 1997]. Non- Boussinesq effects [e.g., van den Berg and Yuen, 1997] deserve a particular attention. For example, about half of the entire viscous dissipation associated with smallscale convection occurs at the bottom of the lithosphere. Variable thermal conductivity [Hofmeister, 1999] has just recently been recognized as an important factor for mantle convection [Dubuffet et al., 1999]. t might ation of convection is so difficult that it occurs only near affect small-scale convection as well. Finally, we would or above the solidus. The low heat transport efficiency like to mention grain-size-dependent viscosity (diffusion Table 9. Terrestrial Planets Earth Venus Mars Mercury Moon Surface temperature, K Acceleration of gravity, m s -2 Radius, km Mass, 1024 kg Mass fraction of silicate mantle a Radiogenic equilibrium heat flux (for H = 4.8 x 10 -x2 W kg-x), mw m Minimum heat flux for equilibrium small-scale convection wet, mw m -2 dry, mw m -2 Critical heat flux from the mantle for widespread melting wet, mw m -2 dry, mw m ß ß ß ß afrom Zharkov [1986].

20 21,814 SOLOMATOV AND MORES: TME-DEPENDENT STAGNANT LD CONVECTON creep), which remains one of the least studied problems [$olomatov, 1996]. Appendix A: Scaling Laws for a Thick Lid A1. Basic Equations Even for very vigorous, chaotic convection the lid thickness is not negligible in most numerical experiments, and convection occurs only in a portion of the convective layer beneath the stagnant lid. This fact is important in scaling internally heated convection [Grasset and Parmentier, 1998; Reese et al., 1999a]. Assuming that convection beneath the lid is driven by the rheological temperature scale ATrh, which is proportional to 7-1, we can write that the heat flux at the bottom of the lid is Nef f -- a k7- deft bl/nn apg7 1/n where a and/ are constants and deft - d- 5L Weft (A1) where _la(n+2)/n exp (- /Ti/n) ) ' (A2) is the thickness of the convecting layer beneath the lid. The effective heat flux which drives convection beneath the lid is Feff = pridefl. (A3) The relationship between the lid thickness 5L and the temperature at the bottom of the lid can be found from the steady state conductive profile in the lid, Fe 6 ph6 The average velocity in the convecting region is + (A4) _.(n+2)/n )) u i - au d ff n b l / 7 apg7 - aeff i --- i/ n (A5) where au and /3u are constants. Since the velocity is negligible in the stagnant lid, the rms velocity Urms in the entire region can be related to the rms velocity ui in the interior as d lti --- Urms--- deft The temperature TL at the bottom of the lid is (A6) TL -- Ti - ATrh, (A 7) where ATrh is the temperature drop in the unstable part of the lid (rheological temperature scale). A2. Nondimensional Scaling Laws The final scaling relationships for the Nusselt number and the nondimensional velocity (scaled by n/d) can easily be derived from (A 1)-(A 7): (1-2Nu - + 2arhO-1) 1- (n+2)/2n = a0-(l+ ) R a,o exp(/ O/nNu) (A8) Urms - au(1-2nu - + 2arhO- ) 3 (n+2)/2n xo - Ra exp(/ uo/nnu) H,0, (A9) where, by definition, Nu is related to non-dimensional Ti as Nu- Wi -1. (A10) The nondimensional theological temperature scale is ATrh - arh 0-1, (All) where arh is a numerical coefficient. The scaling law for the viscou stress is given by (17). n nondimensional form and for the thick lid, it is writ- ten as ri -- ar ATrhRaoSrh, (A12) 5rh -- (1-2Nu arhO-1) -1/2-- (1-2Nu-1) - /2. (A13) Acknowledgments. The authors thank D. Bercovici, J. T. Ratcliff, and D. A. Yuen for thoughtful and constructive reviews, T. M. Hearn for discussion of inverse problems, and O. Grasset and M. Parmentier for providing us with the original data from their 1998 paper. V.S. was supported by the NASA grant NAG5-6897, Alfred P. Sloan Foundation and NERC (U.K.). L.M. was partly supported by the Australian Geodynamics Cooperative Research Centre; this work is published with the permission of the director, AGCRC. References Acufia, M. H., et al., Global distribution of crustal magnetization discovered by the Mars Global Surveyor MAG[ER experiment, Science, 28, , Anderson, D. L., Theory of the Earth, Blackwell Sci., Boston, Mass., Bai, Q., and D. L. Kohlstedt, Substantial hydrogen solu- bility in olivine and implications for water storage in the mantle, Nature, 357, , Basilevsky, A. T., J. W. Head, G. G. Schaber, and R. G. Strom, The resurfacing history of Venus, in Venus - Geology, Geophysics, Atmosphere, and Solar Wind Environment, edited by S. W. Bougher, D. M. Hunten, and R. J. Phillips, pp , Univ. of Ariz. Press, Tucson, Bell, D. R., and G. R. Rossman, Water in the Earth's mantle: The role of nominally anhydrous minerals, Science, 255, , Bevington, P. R., and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed., 328 pp., McGraw-Hill, New York, Bird, P., Continental delamination and the Colorado Plateau, J. Geophys. Res., 8, , Buck, W. R., Analysis of the cooling of a variable viscosity fluid with applications to the Earth, Geophys. J.R. Astron. Soc., 89, , Buck, W. R., and E. M. Parmentier, Convection beneath young oceanic lithosphere: mplications for thermal structure and gravity, J. Geophys. Res., 91, , 1986.

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