comparison between tomographic structures and models of

Geophys. J . Int. (1996) 124,45-56

Comparison between tomographic structures and models of convection in the upper mantle

Pierre Vacher, Antoine Mocquet and Christophe Sotin Laboratoire de Geophysique et Planttologie, UFR des Sciences et techniques, 2 rue de la HoussiniLre, 44072 Nantes Cedex 03, France

Accepted 1995 June 30. Received 1995 June 30; in original form 1994 March 26

SUMMARY Griineisen’s and third-order finite-strain theories are used to compute the density and seismic-wave velocities of minerals. Assuming the mineralogical model of Ito & Takahashi (1987), seismic velocities of the upper mantle are calculated using the Hashin-Shtrikman averaging procedure. 1-D profiles are first obtained along adiabats, and compared to the IASP91 model. Different adiabats are considered in order to take into account the thermal effect of phase transitions. The best results are found with adiabats initiated at 1473, 1573 and 1613 K for a-olivine, B- and y-spinel, respectively. The incorporation of thermal effects resulting from phase transitions gives velocity jumps at discontinuities close to those of IASP91.

Next, a model of convection constructed by Dupeyrat, Sotin & Parmentier (1995), incorporating plate tectonics, is used to compute 1-D profiles and 2-D fields of seismic anomalies in the upper mantle. Averaged profiles show seismic-velocity gradients very close to those of IASP91, but individual values are much too high, suggesting that the mean temperature profile of the convection model is too cold by 400 K. When low- pass filtered to the resolution scale of presently available tomographic models, both the amplitude and shape of the computed seismic anomalies are consistent with the results of tomographic studies. The amplitude of the anomalies ranges between -2.7 and 3.8% for P-wave slownesses, and from - 3.3 to 4.5% for S-wave slownesses. These anomalies correspond to lateral temperature variations of -465 to 520 K. These calculations are used ( 1 ) as an aid to the interpretation of global tomographic models, for instance by computing spectra of lateral heterogeneities, and (2) to test the adequacy of the basic assumptions used in the computation of numerical models of mantle convection, and to build a theoretical temperature profile that would give the best fit to IASP91. In the uppermost mantle this theoretical model has a shape close to both the convection model and the 1473 K adiabat, but in the transition zone the profile is highly subadiabatic. The spectra obtained for the synthetic seismic anomalies resemble that of tomographic studies, with most of the energy contained within gravest angular orders 1, and a fast decrease of energy with increasing 1. The spatial filtering has clearly different effects on heterogeneities, depending on their respective wavelengths. It is suggested that the change of decreasing rate observed in tomographic models at 1 = 7 is closely related to the filter wavelength and may correspond at a lesser extent to a characteristic wavelength of mantle heterogeneities.

Key words: convection, seismic tomography, upper mantle.

one hand the numerical modelling approach permits the detailed study of the specific effects of a given set of parameters INTRODUCTION

Convective processes within the Earth’s mantle are usually (e.g. viscosity, heat sources, boundary conditions, phase investigated in two different and complementary ways, either changes, compressibility, etc.). The increasing capabilities of by computing theoretical models of mantle convection, or by computers allow us to account for more and more of the retrieving the present-day observable structure of the Earth complexity of mantle flow: fully spherical 3-D models have using gravimetric, topographic, and seismological data. On recently been published (Tackley et al. 1993); phase changes

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46 P. Vacher, A. Mocquet and C. Sotin

have been incorporated (Machetel & Weber 1991); calculations at large Rayleigh numbers have been carried out (Parmentier, Sotin & Travis 1994); the effect of variable viscosity has been investigated (Ogawa, Schubert & Zebib 1991; Tackley 1993); and the effect of plate tectonics has been assessed (Dupeyrat et ul. 1995). No model can describe the complete complexity of the Earth. However, each series of numerical calculation describes specific features that can be tested against observables if one can calculate these quantities from the models. On the other hand, the interpretation of seismological data can yield present-time structures of mantle flow. For example, degree 2 (e.g. Su & Dziewonski 1992) and to a lesser extent degrees 4 and 6 (e.g. Montagner & Romanowicz 1993) are predominant in available global tomographic models, when expanding the Earth’s lateral heterogeneities into spherical harmonics. Montagner & Tanimoto (1991) and Montagner & Romanowicz (1993) proposed that lateral heterogeneities evi- denced by global tomographic models were due to the super- position of two convective mechanisms: a surficial upper mantle convection driven by plate tectonics, overprinted on a larger- scale convection. Conversely, Snieder, Beckers & Neele ( 1991) suggested that these long-wavelength anomalies might be a spurious mapping of many small structures that cannot be resolved.

Recent studies have tried to compare results from different fields of geophysics: Machetel (1990) converted and filtered his thermal anomalies of the lower mantle into large-scale velocity anomalies with an experimental scaling law, and gave the same conclusions as Snieder et al. (1991). Jordan et al. (1993) recently constructed a low-pass-filtered thermal image of the Earth after a high-resolution thermal model, and used statistics to compare it with tomographic studies. Nataf & Ricard (1995) constructed synthetic seismic anomalies from a compilation of data on topography, crustal structure, thermal models of the lithosphere and mineral physics. The present paper is a first attempt to relate mineral physics, numerical simulation and seismic parameters quantitatively.

The computation of seismic parameters from mineral physics data has been used extensively in previous studies (Sammis, Anderson & Jordan 1970; Davies & Dziewonski 1975; Bass & Anderson 1984; Anderson & Bass 1986; Duffy & Anderson 1989; Ita & Stixrude 1992). These studies calculated velocities along a single adiabat, ‘representative’ of the mantle. In the first part of this paper, we remind the readers of the theories used in the calculation of seismic velocities. Results along different adiabats are subsequently presented, assuming the petrological model of Ito & Takahashi (1987) to be representative of mantle mineralogy. The use of different adiabats allows us to take into account the thermal effect due to phase transitions, which was neglected in previous studies. Then, temperatures obtained from numerical convection experiments (Dupeyrat et ul. 1995) are used to compute 1-D profiles and 2-D fields of P- and S-wave slowness anomalies.

These theoretical seismological models are compared with known results from seismology in order to check the validity of the basic assumptions used at the first stage of the convection experiments, and to compare the spatial pattern and amplitude of the synthesized density and seismic-velocity anomalies with those currently observed in seismic tomographic models. In particular, comparisons between synthetic and observed spherical harmonic spectra are used as a tool in the search for the characteristic dimension of mantle heterogeneities.

OUTLINE O F THE METHOD

Expressions for density and seismic velocities of minerals

The computation of seismic velocities and density from known thermodynamical conditions and mineralogies has been extensively explained by previous investigators (e.g. Davies & Dziewonski 1975; Anderson 1988; Duffy & Anderson 1989).

As a first step, available data on densities and thermo- elastical properties of mantle minerals are corrected for temperature effects at surface pressure conditions Po. Griineisen’s theory gives an expression of the molar volume of a mineral, as a function of temperature (e.g. Suzuki, Okajima & Seya 1979; Duffy & Anderson 1989). From this expression, internal energy and thermal expansion coefficients are easily computed as functions of temperature T, and the density of a mineral p(T) is evaluated through

where ~1 is the thermal expansion coefficient and To the reference temperature.

Using Anderson’s ( 1988) dimensionless logarithmic anhar- monic notation (DLA), the temperature dependence of an elastic modulus M (either bulk modulus K or shear modulus G) of a mineral is

where {M},= ( ~ ’ ~ ~ ) , - is assumed to be independent of

temperature (Anderson 1988). Finally, the adiabatic pressure derivatives of the elastic

moduli are given by

M’(T) = M’(&) exp + a(T‘) dT’ , (3) [ f 1 where the prime denotes differentiation with respect to pressure.

Duffy & Anderson (1989) compiled all the data that existed at that time, and the associated uncertainties on elastic and thermal parameters of mantle minerals at surface conditions. They found that uncertainties introduced by the approximations of the theory are much smaller than experimental ones: computed densities and elastic moduli are accurate to within 1 and 2 per cent, respectively.

The second step in the computation of seismic parameters is to correct the values at T and Po for pressure effects. This is done by finite-strain theory, where the volumic strain E is given by

(4)

Birch (1952), and the elastic energy U is expressed by a polynomial approximation in strain:

u = u, E + u2E2 + U3E3 + . . . . ( 5 ) Davies & Dziewonski (1975) have shown that (5) should include fourth order in strain in order to fit seismic parameters to reference models in the lower mantle. Since our study is restricted to the upper mantle, only third-order expansion in strain is considered. Pressure and seismic velocities V, and V,

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Convection and seismic tomography 47

can be expressed by (Davies & Dziewonski 1975; Duffy & Anderson 1989)

P = - 3Ke( 1 - 2 ~ ) ~ ’ ~ [ 1 + f(4 - K’)E] ,

v: = (1 - 245/2(~ + B&),

pv; = (1 - 245/2(~ + cE),

(6)

(7)

(8) where

A = K + $ G , B=5A-3KA, C = 5 G - 3 K G . (9)

Note that, since eq. (5) is derived from an adiabatic decom- pression of the mineral: P = -dU/dV where V is the volume, eqs (5) to (8) are only valid along an adiabat.

Density anti seismic velocities of the mantle

Two problems occur when computing seismic profiles of the mantle from single mineral profiles calculated along adiabats. First, the proportion of the different minerals must be known. We thus need a petrological model. Second, an accurate tool is required to derive the elastic properties of a rock from the elastic moduli of its constitutive minerals.

Discussions on mantle mineralogy and possible variations in chemical compositions are outside the scope of this study. Instead, the paper focuses on the effects of temperature variations on densities and seismic velocities. The mineralogical model proposed by Ito & Takahashi (1987) is therefore used without modification. This model, with 50 per cent of olivine and spinel phases and more than 40 per cent of garnet and majorite in volume fraction, is intermediate between the pyrol- ite (Ringwood 1975) and piclogite (Bass & Anderson 1984) end members. A few simplifications are assumed, due to the lack of experimental data for some minerals (Fig. 1).

(1) The majorite phase, a solid solution of garnets, is replaced by pyrope. Duffy & Anderson (1989) have computed

20

a“ 2-

10

0 0.2 0.4 0.6 0.8 1 .o Volume Fraction

Figure 1. Simplified mineralogical model used in this study (modified after Ito & Takahashi 1987). Ca-P CaO-rich phase; Cpx: clinopyroxene; Opx: orthopyroxene.

seismic velocities of Ca- and Mg-bearing majorite, approxi- mately 5 per cent higher than those of pyrope, but used thermal properties of pyrope to compute velocities for majorite. Here, we choose to compute only pyrope seismic parameters, since thermal and elastic data are self-consistent in this case.

(2) The so-called ‘diopsidic CaO-rich phase’ remains mysterious: Ito & Takahashi (1987) assumed that it might be a single phase at high-pressure conditions. Here it is replaced by diopside.

The next step is to compute the elastic properties of the mantle from the properties of individual minerals. Usual averaging methods generally give two bounds which bracket a ‘reasonable’ average. A Voigt-Reuss-Hill (VRH) averaging procedure is often used. In this method, the two bounds are calculated assuming uniform stress or uniform strain in the composite material. Alternatively, Hashin & Shtrikman (1963) proposed a method (HS) based on a sounder thermodynamical basis, namely energetic variational principles. For instance, to compute the average bulk modulus of n minerals, the two bounds K - and K , are related to the smallest and the largest values of individual minerals, respectively K , and K,:

vi and K , are the volumic proportion and the bulk modulus of the ith mineral, respectively. A parallel set of equations can be derived for the shear modulus. This method is rather laborious when compared to the VRH averaging method. Nevertheless, Watt, Davies & OConnell(l976) showed that the VRH average can be a poor approximation, which can even lie outside the two HS (Hashin-Shtrikman) bounds in some cases.

Both methods are tested by computing P-wave velocities and their standard deviations for the petrological model of Ito & Takahashi (1987), along an adiabat initiated at 1673 K and using the pressure profile of PREM (Dziewonski & Anderson 1981). The results are summarized in Table 1. Although the mean values are almost equal, the difference between the bounds is three to five times smaller when using the HS rather than the VRH method. Therefore, the HS procedure is used throughout this paper.

Estimates of temperature profiles

The appropriate adiabatic gradient is directly provided by finite-strain theory thus it does not need to be specified. However, it is essential to estimate the temperature profile associated with our computed velocities, especially to gain an understanding of the effect of phase transitions and to be able to use temperature fields obtained from convection models (see the following sections).

Classical values of the adiabatic gradient within the uppermost mantle are close to 0.5 K km-’ (Scott 1992). These values

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48

Table 1. P-wave velocities (km s-') calculated with the model of Ito & Takahashi (1987) along an adiabat at 1673 K, using the pressure profile of PREM (Dziewonski & Anderson 1981). VRH: Voigt- Reuss-Hill average; HS: Hashin-Shtrikman average; u: difference between upper and lower bounds for both methods.

P. Vacher, A. Mocquet and C. Sotin

Depth (km) VRH f u H S f u

150 8.031 f 0.027 8.032 f 0.008 3 10 8.528 0.022 8.528 f 0.006 500 9.658 f 0.013 9.659 f 0.003 670 10.105 5 0.019 10.107 f 0.005 871 10.857 & 0.071 10.858 0.014

1071 1 1.264 f 0.064 11.264 f 0.013

decrease with depth, down to values of 0.3 K km-' at the base of the upper mantle (Turcotte & Schubert 1982). More accurate values must be derived for the present study.

Following Chopelas & Boehler (1992),

where (dT/aP), is the adiabatic gradient and C, the heat capacity at constant pressure. Constant-temperature dimensionless differentiation with respect to volume gives

(E),=[ a I n v l T + ( W ) , - 1 .

This formula can be simplified because the second term on the right-hand side can be neglected at temperatures above the Debye temperatures of minerals (Chopelas & Boehler 1992), and because (a In a/a In V), - 5.5 for most minerals in the mantle (Chopelas & Boehler 1989). Hence, (15) becomes

(15) a In(aT/aP), a In Cp

and by integration,

or, equivalently,

The values of heat capacities C, are taken from Saxena & Shen (1992), and all remaining parameters are computed from the theory described previously. The adiabatic increase of temperature with depth, (aT/az),, is deduced from (18) using the pressure profile of PREM (Dziewonski & Anderson 1981). The adiabatic gradient of the multicomponent mantle is expressed as an average of the gradients of individual minerals, weighted by their respective volume fraction. Fig. 2 shows the values of (aT/az), with their standard deviations as a function of depth and temperature at the foot of the adiabat under consideration. We find again that, for a given temperature at the foot of an adiabat, the gradients decrease with increasing depth. The solid curve shows the temperature profile of the convection model of Dupeyrat et al. (1995, see description below). These models do not take into account the temperature increase due to increasing pressure. Hence their temperature profiles are 'virtual', and valid at surface pressure conditions. This property enables us to use them directly as a series of temperatures T,(z), which correspond to the feet of the adiabats valid at the respective depths z. The gradients to be used at each depth of a given temperature profile are thus readily obtained. These gradients increase up to 0.35 K km-' at 100 km depth, decrease to values lower than 0.2 K km-' at a depth of 500 km, and increase again down to the base of the upper mantle.

Phase transitions

Previous studies of the upper mantle based on finite-strain theory focused on the effects of phase transitions on the elastic parameters of the mantle: they accounted well for the variations in the elastic properties of olivine and its spinel phases, but computed their densities and velocities along the same representative adiabat (Bass & Anderson 1984; Anderson & Bass 1986; Duffy & Anderson 1989; Ita & Stixrude 1992). Hence, they neglected the consequences of phase transitions on the temperature profile. Both transitions, a-olivine to p-spinel and

Temperature (K)

Figure 2. Isocurves of adiabatic gradients plotted as a function of depth and temperature at the foot of the adiabat (values and 2u standard deviations are in K km-'). The temperature profile derived from convection experiments (Dupeyrat et al. 1995) with a plate velocity of 5 cm a-' is also shown (heavy solid curve).

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a-spinel to y-spinel, which are respectively assumed to be responsible for the seismic discontinuities at 410 and 520 km, are exothermic: the 'representative' adiabat of the mantle is thus shifted by phase transitions towards higher temperatures. This 'Verhoogen effect' (Jeanloz & Thompson 1983) results in an increase of the adiabat temperature AT:

where L is the latent heat, T the temperature, AV the volume variation and y the Clapeyron slope of the phase transition. Cp is an average heat capacity (Turcotte & Schubert 1982, p. 193). This effect is taken into account by using different adiabats for the different parts of the upper mantle. If the adiabat of cc-olivine (depth range: 0-400 km) is given, then the Verhoogen effect gives the corresponding adiabats to be used for the 400-520 km and 520-660 km depth ranges. The diffi- culty is to choose the feet of the adiabats carefully, so that thermal jumps at discontinuities are maintained. We took y = 3 MPa K-' for a-p transitions (Katsura & Ito 1989) and y = 4.8 MPa K-' for p-y transitions (Ashida, Kume & Ito 1987). The remaining parameters involved in the calculation of AT are computed using our theory. Computed temperature jumps are A T = 100 K and 40 K at the feet of the adiabats, for a-p and p-y transitions respectively, and they do not depend on the choice made for the foot of the adiabat of a-olivine. These temperature shifts are large enough to influence significantly the seismic profiles, especially the amplitude of the velocity jump at a depth of 410 km.

Results along adiabats

Fig. 3 shows the P- and S-wave velocity profiles (top and bottom panels, respectively) obtained for three different temperatures: T, = 1473, 1573 and 1673 K. Mineral velocities in the depth ranges 410-520km and 520-660km are subsequently calculated along adiabats fixed by the AT described above. The profiles are compared with the IASP91 reference model (Kennett & Engdahl 1991). This model is preferred to the PREM model (Dziewonski & Anderson 1981), because it makes approximations consistent with the modelling described here, i.e. no difference in viscosity between the lithosphere and the asthenosphere. This approximation results in the lack of the low-velocity zone. Furthermore, it does not include a priori assumptions concerning the adiabatic behaviour of the mantle.

In the depth range 0-120km, our modelling of P-wave velocities cannot fit the reference profile for two main reasons: the high velocity of the lithosphere induces a strongly over- adiabatic boundary layer; and the mineralogy is much too complex. Between 120 and 410 km, the correlation between calculations along adiabats and the IASP91 model is quite good. The adiabatic initiated at 1473 K fits the reference model rather well (values and gradients of velocity are maintained), except'for a small deviation between 140 and 240 km. These correlations are quantified in Fig. 4, where the deviations of the calculated profiles from IASP91 are plotted, in per cent, as a function of depth. The misfit is lower than 0.2 per cent for the 1473 K adiabat down to a depth of 410 km, except for a small region from 160 to 220 km in depth, where the 1573 K adiabat gives the best fit.

The transition zone shows a more complex behaviour. The gradients of the calculated profiles are much smaller than those of IASP91 (2.8 x s-I) . This s- ' instead of 3.36 x

10.5

9.5

I------IASP)II 5.6 i 4 5.2

4.8

4.4

t 1571 K

depth (km)

Figure 3. Solid curves: averaged profiles of P-wave (top) and S-wave (bottom) velocities, calculated along a series of three adiabats. Indicated temperatures correspond to the feet of the adiabats used for the first 400 km. Adiabats change in the transition zone according to Verhoogen's effect (see text). Reference profiles IASP91 (Kennett & Engdahl 1991) are shown for comparison (dashed curves).

-iasp-1473 - - -iasp-1573 . . . . . iaw-1673

n 100 20 300 40 50 600 70 depth (km)

Figure 4. Discrepancy between calculated profiles and IASP91 for a series of three adiabats (P-wave velocities only).

discrepancy is partially explained by the smoothness of the IASP91 model, which does not include any discontinuity at 520 km depth. The misfit might also be explained by a non- adiabatic behaviour of the transition zone. Indeed, Fig. 4 clearly shows that the three considered adiabats successively

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50 P. Vucher, A. Mocquet and C . Sotin

give the best fit. An interpretation in terms of temperature pofiles within the transition zone will be discussed in the last section.

The velocity jump at 410 km depth in the computed pro- He reaches a slightly higher value than the one of IASP91: 0.40 km s f l (4.4 per cent) instead of 0.33 km s-' (3.6 per cent). This is certainly caused by the choice we made to neglect the width of the phase transition. The influence of the Verhoogen effect is essential here: if it were neglected, the velocity jump discrepancy would be doubled. At a depth of 520km, the computed velocity jump is 0.05 km s-' (0.05 per cent), but 0.07 km s-l (0.07 per cent) if the Verhoogen effect is neglected. It is thus clear that a careful interpretation of seismic discontinuities must rely on the Verhoogen effect (Jeanloz & Thompson 1983).

The correlation between our S-wave velocity profiles and IASP91 (bottom panel in Fig. 3) is worse than for P-wave velocities: gradients are smaller than the one of IASP91 down to 200 km, and then higher down to 410 km. In the transition zone, the computed gradient is 0.9 x s-' versus 2.1 x s-' for the reference model. The velocity jump at a depth of 410 km is 0.3 km s- l (6 per cent), whereas the IASP91 model reports a jump of 0.2 km s - l (4 per cent). In fact, there are two main explanations for this bad correlation. From a theoretical point of view, finite-strain theory is based on volumic deformation, which is not appropriate to study the behaviour of geologic materials submitted to shear strain. The second reason is the lack of precise data concerning shear moduli of minerals and their derivatives (Duffy & Anderson 1989). Both theoretical and experimental studies are hence needed before any interpretation of S-wave velocity profiles can be attempted in terms of chemical and temperature variations. For example, Karato (1993) showed recently that the temperature dependence of seismic-wave velocities arising from anelasticity could almost double the temperature V, derivatives computed from anharmonicity alone.

MODELS O F CONVECTION

Description of the models

Numerical experiments on thermal convection of an incompress- ible and isoviscous fluid heated from below were carried out to investigate the effect of plate tectonics on the pattern of convection in the upper mantle (Dupeyrat et al. 1995). The flow is modelled in a box perpendicular to the ridge axis. Aspect ratios of up to 16 were used to account for the large width of the plates. Rayleigh numbers range from 5 x lo4 to lo6, and were chosen such that both the depth of melting and the heat flux at the surface are consistent with Earth-like values. The top surface is rigid and moves,at a constant velocity. The Peclet number is close to the one required for the plate velocity to be equal to the mean velocity at the free-slip top surface of an isoviscous fluid convecting at the same Rayleigh number, in order to model the displacement of plates. The vertical planes are symmetry boundaries, where the downwelling flow is a simple way to model subduction zones. At a plate velocity of 5 cm a-', convection is strongly dominated by plate tectonics, and induces the formation of two convective cells, one on each side of the ridge (top panel of Fig. 5 ) .

Results: averaged profiles

In Fig. 6, P- and S-wave velocity profiles are compared with IASP91. The results are quite puzzling at first glance. Computed velocities are strikingly higher than those of IASP91, by 0.15-0.35 km s-' (2-4 per cent). This suggests that the mean temperature profile of the convection model is much too cold, by 400 K. Computed velocities decrease from the surface down to a depth of 70 km, showing that the temperature effect due to the thermal boundary layer is more important that the pressure increase in that depth range. Below 80 km depth, computed gradients are very close to those of the reference model down to 520 km. This correlation is quantified in Fig. 7. Synthetic P-wave velocities (white dots: calculated after convection model; black dots: calculated along adiabats) are plotted as a function of IASP91 velocities, for a depth range of 120 to 410 km. Both straight lines are well correlated to IASP91: the slopes are 0.985 and 1.038 for the 1473K adiabat and the convection models, respectively. The slope of the straight line provides information on the temperature gradient in the mantle. The adiabatic temperature gradient gives a slope of the straight line slightly lower than one. The subadiabatic gradient in the convection model of Dupeyrat et al. (1995)

h

i Y v

- 3 Y v

P

11

10.5

10

9.5

9

8.5

8

7.5

7

6

5.6

5.2

4.8

4.4

4

I " ' ' I ' " ' I ' " 1 I ' 1 " I " " I " I '

IAsp91 - - - _ - -

0 100 200 3 0 400 500 600 700

I . " " ' " ' " ' " ' , " " " ' " " " " ' " " ~

t - i " ~ " " ~ " " ~ " " ~ " " ~ " " ~ " ' A 0 100 200 300 400 500 600 700

depth (km)

Figure6. Averaged profiles of P-wave (top) and S-wave (bottom) velocities, calculated with the temperature profile of Dupeyrat et al. (1995).

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(a) Temperature field

(b) Results

Figure 5. (a) Temperature field of Dupeyrat et al. (1995). (b) Temperature variations, density anomalies, P- and S-wave slowness anomalies are shown at the resolution of convection models (top panels), and filtered down to the resolution of tomographic studies (bottom panels). Each slice is 670 km deep, and centred with respect to the ridge axis. The total length is 10 700 km.

Dow




9 . 4 L ' , , I , , , I , , , I , , , I , , , , , I , , , , ' ,

t 9 -

8.8 -

8.6 -

8.4 1

8.2 1

8 t . . . L ~ . ~ 1 ~ ~ ~ 1 ~ ~ . 1 ~ ~ " ~ . . l , ~ ~ ~ i

8 8 2 8.4 8.6 8.8 9 9.2 9.4 Vp IASP91 (kmls)

Figure 7. Synthetic P-wave velocities (black dots: calculated along an adiabat initiated at 1473 K; white dots: calculated after the convection model of Dupeyrat et al. 1995), plotted as a function of IASP91 velocities (Kennett & Engdahl 1991). The numbers give the slope of the linear fits (both have correlation coefficient higher than 0.99).

yields a slope for the straight line slightly larger than one. However, the present-day experimental uncertainties of the elastic parameters do not allow us to discriminate between adiabatic and subadiabatic thermal gradients.

We interpret the offset between the straight lines and the diagonal as a temperature shift at the base of the lithosphere between IASP91 and either model. A 1473 K adiabat provides a good fit, suggesting that the temperature at the base of the lithosphere is around 1200 "C. On the other hand, the convection model provides much too low a temperature (about 950 "C). This discrepancy is discussed below.

The velocity jump at a depth of 410 km is overestimated: 0.42 versus 0.33 km s- ' for P-wave velocity, and 0.31 versus 0.20 km s-' for S-wave velocity. We know from the previous section that this discrepancy is almost entirely due to the Verhoogen effect, which is not incorporated in this modelling. In fact, temperature variations due to phase transitions were not incorporated in the convection experiments of Dupeyrat et al. (1995). Therefore we decided to disregard the Verhoogen effect in the profiles of Fig. 6, in order to be consistent with the convection model. The velocity jump at a depth of 520 km is small (0.05 km s-l) and disappears in the S-wave profile.

A critical stagnation and a decrease in the computed velocities appear below 500 km. At a depth of 670 km, computed velocities are much smaller than the reference one: 0.35 km s-l (3.4 per cent) and 0.3 km s-l (5.3 per cent) for P- and S-wave velocities, respectively. This is due to the high temperature at the base of the convection models of Dupeyrat et at. (1995): 2200 K (Fig. 2). This value is imposed by the model, which assumes that the upper mantle is a layer heated from below, and which obtains such a high temperature in order to obtain a surface heat flux in agreement with that measured at the surface of the Earth. Incorporating internal heating and variable viscosity may lower the value of the temperature at the upper mantle-lower mantle boundary, but such a study is clearly outside the scope of the present paper (see discussion).

2-D models

The resolution scale of convection models and present-day tomographic images of the Earth differ considerably. In the

model of Dupeyrat et al. (1995) (top panel of Fig. 5), the grid spacing is about 20 km. On the other hand, recent tomographic studies develop the Earth's lateral structure in spherical harmonics up to degree 36 (Zhang & Tanimoto 1991; Su & Dziewonski 1992), or parametrize the Earth with blocks of 5" by 5" (e.g. Inoue et at. 1990).

One way to low-pass filter a model is to expand it into spherical harmonics, and to truncate the expansion at low angular degrees and orders (Hager & Clayton 1989; Jordan et al. 1993). Instead of expanding the temperature field provided by Dupeyrat et al. (1995) into spherical harmonics, an equivalent procedure is used, derived from Montagner's regionalization method (e.g. Montagner 1986; Nataf, Nakanishi & Anderson 1986). The filter is a moving average operator, defined using correlation lengths L and L' in the horizontal (x) and vertical (z) directions, respectively. The filtered slowness anomaly S(x, z) is calculated by

where S(xo,zo) is the slowness at the centre (x,,z,) of the moving window. We choose a lateral correlation length L of 250 km, and the vertical correlation length L' varies with depth following the values of Mocquet & Romanowicz (1990). It is fixed at 100 km down to 88 km depth. Below this depth, L' increases linearly to reach a value of 290km at a depth of 670 km.

In Fig. 5, computed results for a plate velocity of 5 cm aK1 are presented at the same resolution scale as convection models (top panel) and after spatial filtering (bottom panel). The statistics of these results are listed in Table 2. Temperature variations with respect to the mean temperature profile are very large, ranging from -1000 to 1163 K, but, in fact, the major part of the upper mantle has a temperature close to the averaged profile: actually, the average of the variations is equal to 0 and the mean amplitude is 183 K. Stronger anomalies are very localized in space: for instance the plume under the ridge (AT= 1163 K) reaches the surface and then spreads ont while cooling. Cold structures (AT= - 1000 K) sink down to 670 km close to the edges of the model, and then go back horizontally over 1000 km to the centre of the box.

The next strips show corresponding anomalies of density and seismic slownesses. All results are strongly correlated to the thermal pattern. Unfiltered values are very high, about one order of magnitude higher than the values observed in tomographic studies: anomalies in density range between - 3.84 and 2.70 per cent, and slowness anomalies range from -5.25 to 8.22 per cent and -6.33 to 9 per cent for P- and S-waves, respectively. However, this also holds for density, slowness and temperature variations: most of the mantle displays low- amplitude anomalies, and only a few localized peaks appear, concentrated at the ridge axis and at the edges of the model. Averaged values of the anomalies are 0.46, 0.96 and 1.16 per cent, for density, P- and S-wave slownesses, respectively. Note that slowness anomalies are two or three times higher than density anomalies. The plume under the ridge is very narrow, but it rises up through the entire upper mantle, and takes a mushroom-like shape near the surface.

When smoothing the models down to the resolution of the most recent tomographic studies, the amplitude ranges of the anomalies decrease down to values much closer to those of seismic studies: -2.69 to 3.80 per cent and -3.25 to 4.53 per

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52 P. Vacher, A. Mocquet and C . Sotin

Table 2. Statistics of the results. Amplitude at 20 means that 95 per cent of the anomalies have an amplitude lower than the given values. (a) Same resolution scale as convection experiments (b) filtered results with a lateral correlation length of 250 km.

Temperature Density anomalies P-wave slowness S-wave slowness variations (K) (”/) anomalies (%) anomalies (YO)

( 4 (b) ( 4 (b) (a) (b) (a) ( b)

Minimum - lo00 - 465 - 3.84 - 1.80 - 5.25 - 2.69 -6.33 - 3.25 Maximum 1163 520 2.70 1.34 8.22 3.80 900 453 Amplitude at 2u 183 130 0.46 0.34 0.96 071 1.16 0.86

cent for P- and S-wave slownesses, respectively. Density anomalies range from -1.80 to 1.34 per cent. These anomalies correspond to end-point temperature variations equal to - 465 and 520 K. The averaged values show a slight decrease, thus reflecting the homogeneization of the models induced by the spatial filtering process. The shape of the structures also changes: the ascending plume under the ridge vanishes in terms of density and slowness anomalies; for instance, the 3 per cent 2000km long anomaly under the ridge axis shrinks, after filtering, down to a length of 1600 km, and hardly reaches a depth of 200 km. Note that the relationship between the age of the sea-floor and its slowness remains unaltered by the filtering process (Fig. 5 ) .

Amplitude spectra

A usual way to help in the interpretation of the spatial distribution and strength of seismic velocity anomalies is to expand the models in spherical harmonics and to draw conclusions about the decrease of the amplitude spectra with respect to the angular order 1. Fig. 8 thus shows the amplitude spectra of P-wave slowness anomalies before (black dots) and after (white dots) filtering at four selected depths. Although constructed after Fourier transform of the models presented in Fig. 5, spectra are plotted as a function of an equivalent harmonic degree (up to 1 = 36) rather than spatial frequency, in order to make a direct comparison with tomographic studies.

The general shapes of the amplitude spectra are consistent with those of usually observed spectra on a global scale: most of the energy is contained within gravest orders, and the amplitude decreases with increasing 1. The highest peak is always found for 1 = 1 as a consequence of considering symmetric boxes centred at the ridge axis.

Three different types of spectra are obtained. First, at depths of 100 and 500 km, the 1 = 1 peak is followed by an oscillatory behaviour where the amplitude is concentrated within odd degrees. This is also due to the symmetry of the boxes, which is the only source of heterogeneity. Note that even degrees are slightly excited at a depth of 500km: the centre of the convective cells lead to homogeneization of the mantle at this depth, so all degrees are excited. Second, the spectrum at a depth of 400 km is quite different: the mantle looks rather homogeneous, odd degrees never display an amplitude higher than 0.25 per cent, compared with 1.3 per cent for the 1 = 1 peak at a depth of 100 km. However, the ridge displays a strong (6 per cent) slowness anomaly in the 2-D plot (Fig. 5 ) . This might be an effect of the constructive interferences previously described by Su & Dziewonski (1992): a strong anomaly arises from the summation of terms with small individual amplitudes, but which are in phase. This feature occurs between

depths of 180 and 450 km. Third, the spectrum at a depth of 650 km has zero amplitude for l > 12. We have already mentioned that our results are biased at the base of the upper mantle, and the constant temperature there prevents any strong anomaly from showing up.

The filtering process has a variety of effects on heterogeneities, depending on their respective wavelengths: after filtering, the 1 = 1 peaks remains dominant and rises up to 0.4 per cent at a depth of 400 km. Conversely, small wavelength anomalies (1 > 6) decrease in amplitude. The implications of this effect on the spectra of global tomographic models are discussed at the end of this paper.

DISCUSSION

Consistency of the results

In order to relate seismic heterogeneities to temperature variations within the upper mantle, global tomographic models often use a priori relationships between density and seismic velocities. Masters et al. (1982), among others, took

(-) dln V, =0.8 and (*) =0.4. dln Vs d In Vs

These relations come from experimental studies (e.g. Anderson et al. 1968; Isaak et al. 1989). Joint inversions of seismic and geodynamic data performed recently by Forte, Woodward & Dziewonski (1994) on a global scale yield much smaller (d In p / d In G)p values, of the order of 0.1 within the uppermost 400 km of the mantle. These authors proposed that partial melting and/or lateral chemistry variations in the upper mantle (for example, the effect of continent-ocean chemical differences) could account for this low value. Alternatively, as mentioned previously, Karato (1993) showed that seismic dissipation mechanisms could contribute to the temperature derivatives of shear-wave velocity, to an extent similar to the effect of pure anharmonicity, thus reducing (d In p / d In &)p to values ranging from 0.2 to 0.3 in the upper mantle.

None of these effects is taken into account in the convection models. The mantle is chemically homogeneous and perfectly elastic within each depth range, and only temperature variations are considered. Therefore, a mutual consistency between density, P- and S-wave velocity variations in our models must be associated with (d In V,/d In Vs)p and (d In p/d In Vs)p values close to 0.8 and 0.4, respectively.

Fig. 9 shows the values of (d In p / d In &)p and (d In V,/ d In V& of our models, as a function of depth. The first ratio varies from 0.365 to 0.417, with a mean value equal to 0.392. The second one ranges from 0.774 to 0.862, with a mean value of 0.808. These results are very close to the usual experimental

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1.4

1.2

1

0.8

0.6

0.4

h

4 s 0.2

5 0

- a h a -

1.4 5 v) 1.2

i: ?z 0.8

1 ,

0.6

0.4

0.2

0

I

0 6 12 18 24 30 36 0 6 12 18 24 30 36

I 1

Figure8. Power spectra of slowness anomalies as a function of equivalent harmonic degree. Black dots and white dots show the spectra of unfiltered and filtered anomalies, respectively. Four depths are considered (100, 400, 500 and 650 km).

\ - dlnVp/dlnVs -- dlnpIdlnVs

J

- 0.86

I 0.84

1 0.82

0.8

: 0.78

0.36 i . , . , ’ ’ ” , ’ ’ ” ’ . ” . ’ ” , ’ ’ , , ’ ’ 0.76 0 100 200 300 400 500 600 700

depth (km)

Figure 9. Values of (d In p/d In Vs)p (solid curve) and (d In Vp/d In Vs)p (dashed curve) plotted as a function of depth.

values; we can thus consider that the variations of density, P- and S-wave velocities are self-consistent in our models.

Constraining temperature profiles

The results shown previously are biased at the base of the upper mantle. In the models of Dupeyrat et al. (1995), the temperature at the base of the upper mantle is fixed to a value equal to 2200 K. It even exceeds 2400 K after the addition of the adiabatic gradient (Fig. 10). This temperature is imposed such that first melt starts at about 80 km beneath the ridge axis and such that the surface heat flux is equal to 75 mW m-’. This value is much higher than the temperature of the transformation of y-spinel to perovskite plus magnesiowustite (1873 K) which occurs at 660 km. The numerical experiments

600 8 0 1 0 0 1200 1400 1600 1 8 0 2000 2200

Temperature (K)

Figure 10. Temperature profile of Dupeyrat et al. (1995), after addition of the adiabatic gradient. The series of three adiabats used in the calculations are also shown. Black dot: temperature of the dissociation of y-spinel; white dot: temperature obtained at the base of the lithosphere by Davaille & Jaupart (1994). By considering the deviation between our synthetic seismic profiles and the IASP91 model (Kennett & Engdahl 1991), a theoretical temperature profile is constructed. It is well constrained in the depth range 100-410 km (solid curve) and more hypothetical in the transition zone (dashed curve).

carried out for an isoviscous fluid heated from below predict a thermal boundary layer with a large temperature gradient, which is not present in the tomographic data. Two models can be proposed in order to match the seismic data. First, the imposed temperature at a depth of 660km can be altered

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54 P. Vucher, A . Mocquet and C . Sotin

significantly if one takes into account the volumetric heating. Although the amount of heat produced by radiogenic decay in the upper mantle represents less than a quarter of the total heat, it may be large enough to heat up the mantle and to lower the calculated temperature at the 660 km depth. The second possibility is to consider whole mantle convection (e.g. Jordan et al. 1993). The thermal boundary layer at the 670 km discontinuity does not exist any more. However, we have not yet been able to carry out numerical experiments at high enough Rayleigh numbers to account for the observed heat flow and depth of melting beneath mid-ocean ridges.

An application of our code to a single-layer convection model would provide strong arguments to prove whether convection processes are stratified or not. Unfortunately, there is a lack of mineral physics data pertinent to the lower mantle. At high pressures, Davies & Dziewonski (1975) have shown that fourth-order finite-strain theory, and therefore second derivatives of elastic moduli, is required. These second derivatives of the elastic moduli are not yet available, or only very imprecisely (Hofmeister 1991). Further work in mineral physics is thus required in order to use fourth-order finite-strain theory.

Fig. 10 also shows the series of three adiabats considered previously (Fig. 3), starting at 1473, 1573 and 1673 K. The black dot shows the temperature imposed by laboratory experiments on the P-spinel- perovskite + magnesiowustite transformation ( 1600 "C), and the white dot shows the temperature obtained at the base of the lithosphere (1523 K at 80 km depth) by laboratory experiments in fluids with temperature- dependent viscosity (Davaille & Jaupart 1994). These two points agree with the adiabats shifted at discontinuities by the Verhoogen effects: the adiabats initiated at 1473 K give temperatures of 1510 K at 80 km depth and 1868 K at the base of the upper mantle. Considering the 'best-fitting' adiabat at each depth in Fig. 4, we can build a theoretical temperature profile which gives seismic velocities as close as possible to the IASP91 model (assuming that the model of Ito & Takahashi 1987 is valid). This theoretical profile shows a strongly subadiabatic shape in the transition zone, and high temperature jumps at discontinuities: AT= 250 and 150 K at depths of 410 and 520 km, respectively. These sharp temperature discontinuities argue in favour of the conclusion derived by Montagner & Tanimoto ( 1991) from their S-wave global tomographic model, by which they proposed that the upper mantle was a region of interaction between two types of overprinted convective processes, the first one being surficial and driven by plate tectonics, the second one involving deeper processes. The interaction between both types of convection would induce a stratification of the mantle within the transition zone. On the other hand, numerical convection experiments including phase transition effects (e.g. Peltier & Solheim 1992) do not see such a stratification, because, conversely to the endothermic spinel to postspinel dissociation, the phase transitions at 410 and 520 km depth are exothermic; therefore, they do not act against convection (Sotin & Parmentier 1989). Hence we believe that, if a temperature jump exists at a depth of 410 km, its amplitude might be much smaller than the one we found here. A chemical differentiation between the uppermost mantle and the transition zone might explain part of this temperature variation: a transition zone with a higher proportion of garnet would have smaller seismic velocities, and then the theoretical temperature profile would decrease and give a smaller temperature jump.

We would like to emphasize that the shape of the tempera-

ture profile of Dupeyrat et al. (1995) in the uppermost mantle is consistent with the theoretical one we found. The sigmoidal shape of the profile, which induces a subadiabatic behaviour is too pronounced, but in good agreement with the profile that fits the IASP91 model. But, as we have already mentioned, the mean temperature is much too low. This may be related to the fact that the models were conducted at constant viscosity. Recently, Davaille & Jaupart (1994) suggested that the thermal boundary layer of a convective fluid with a highly temperature- dependent viscosity would be overlain by a conductive lid. The temperature difference in the thermal boundary layer would result in viscosity variations of less than two orders of magnitude. Applied to the Earth, this variation corresponds to a temperature increase equal to 150K. Furthermore, Davaille & Jaupart (1994) proposed that the temperature at the base of the lid would be 1520 K. Since the temperature at the 660 km discontinuity should be around 1850K (It0 & Takahashi 1987), the temperature difference between the top and the bottom of the convecting fluid can be as small as 300 K. With such a small temperature difference, the internal heating becomes important relative to the cooling from above. The temperature profile can be slightly subadiabatic as shown by Parmentier et al. (1994). The hot thermal boundary layer at the 660 km discontinuity may be very thin, and may explain the shape of the theoretical profile displayed in Fig. 10. We are presently carrying out new numerical experiments that describe the convection pattern of the upper mantle with variable viscosity, internal heating and various spreading rates. Such a c'udy is far beyond the aim of the present paper.

Scale of mantle heterogeneity

A direct comparison between our amplitude spectra and those of tomographic studies would be useless since ours are computed from 2-D theoretical slices. However, the spectra before and after filtering can be used as a tool in the current discussion about the characteristic scale of mantle heterogeneities. It is well known that filtering an amplitude spectrum changes its decreasing rate. In our case, this change is quantified by plotting the power spectra on a log-log scale. Fig. 11 (top panel) shows the power spectra of the P-wave slowness anomalies integrated over all depths. Su & Dziewonski (1992) plotted this type of diagram for their SS residuals and for the model of Zhang & Tanimoto (1991). They found a change in slope between degrees 6 and 8, and concluded that 2500-3500 km is a limit for the characteristic dimension of mantle heterogeneity. We have plotted our spectra for each degree up to 1 = 7, and then only odd degrees up to 1 = 19, since even degrees do not show a significant signal. The effect of the filtering, as we have already mentioned, is quite different for different wavelengths. For small odd degrees ( 1 < 7), the power spectra before and after filtering are similar, but for I > 7 the filtered power spectrum decreases much faster than the unfiltered one with increasing 1. This result suggests that the change in slope observed in tomographic model spectra might be an effect of filtering rather than a characteristic scale of mantle heterogeneity.

The models cited previously used different filtering processes. Su & Dziewonski (1992) smoothed their residuals by a moving cap of radius 5", and Zhang & Tanimoto (1991) used an equal area (5" x S o ) block discretization. In order to test the effect of filtering, we used the spherical harmonic coefficients up to

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0 -

1 i 0

0

1

0 "d I m , , , I " I 1 10

10 I I I I I I I I I

Su et Dziewonski, 1992

- -0.79 -0.87

-

0.1 : -1.68 ' - -2.64 % !

t -3.54 -

I = 36 provided by Su & Dziewonski (1992) and we constructed a map of SS residuals on a 1" by 1" cell grid. We subsequently filtered this map with equal-area moving caps of different sizes, and once again performed the inversion. By doing so, we obtained new spherical harmonic coefficients for each cap size. Unfortunately, the accuracy of the coefficients provided by Su & Dziewonski restricted our expansion to E = 14. The bottom panel of Fig. 11 shows the spectra with one residual for each degree (black dots and solid line), with a moving cap of radius 5" (white dots and short-dashed line) and with a moving cap of radius 9" (white squares and long-dashed line). For 1 < 7 the filtering. has almost no effect on the spectra, since the slopes only vary from -0.73 to -0.87. For 1 > 7, however, the slopes change critically from - 1.68 to -3.54. Therefore, we suggest that the change in decreasing rate between 1 equals 6 and 8 observed in tomographic studies may be related to the smoothing process introduced in the models. Indeed, there is a change in slope (by a factor of 2) on the black dot curve, which represents a 'perfect tomography'. Hence this change in slope must be present in the Earth, but we believe that it is greatly amplified by the filtering of the models.

CONCLUSION

Finite-strain theory was used to compute density and slowness anomalies in the upper mantle. Results along adiabats are different from those of previous studies. First, three separate feet of adiabats are considered to find the temperature profile that fits as closely as possible the IASP91 model (Kennett & Engdahl 1991) in the depth range 0-450 km for a given mineralogy. Contrary to previous studies, a series of adiabats are considered in the transition zone in order to take thermal effects due to phase transitions into account. Calculations of this Verhoogen effect lead us to the use of adiabats starting 100 K and 140K higher than the one for a-olivine, for the P-spinel and y-spinel regions, respectively. Synthetic seismic velocity gradients display a nearly adiabatic behaviour in the depth range 100-410km, but the transition zone looks highly subadiabatic.

Seismic profiles constructed after temperature fields provided by the convection model of Dupeyrat et al. (1995) bring new constraints on these convection models-internal heat sources must be introduced in the modelling scheme in order to warm up the temperature profile: the latter is too cold by at least 400 K. At the base of the upper mantle, however, the temperature must be decreased to a value closer to the one imposed by experiments on phase transitions (1600 "C). Nevertheless, the shape of the temperature profile seems to be in good agreement with the IASP91 model, at least in the depth range 100-410 km. This is one of the promising results of this paper.

Another interesting result is that synthetic 2-D fields of filtered slowness anomalies show amplitudes similar to those of global tomographic models of the upper mantle, from -4 to 4 per cent, corresponding to temperature variations of - 500 to 500 K. These encouraging results lead us to the conclusion that a fully quantitative comparison of numerical simulations and localized seismic observations is feasible.

The amplitude spectra of unfiltered and filtered synthetic anomalies show different evolutions, suggesting that the filtering process affects the decreasing rate of the spectra: indeed, the filtered power spectrum decreases much faster than the unfiltered one for 1 > 7. Synthetic tests performed on the model of Su & Dziewonski (1992) show that this result still holds for global tomographic models. The change in slope between t equals 6 and 8 reported in tomographic studies may exist in the Earth's mantle, but we suggest that it is greatly magnified by filtering effects.

ACKNOWLEDGMENTS

This research was funded by INSU of CNRS under ATP Tomographic Nos 93TOM01 and 94TOM02. This work is part of PVs PhD thesis, supported by the French ministry of research. AM is grateful to 0. Jaoul for introducing him to Gruneisen theory. Constructive reviews from anonymous reviewers are acknowledged for improving a first version of this paper.

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