The geoid constraint in global geodynamics: viscosity structure, mantle heterogeneity models and boundary conditions

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1 Geophys. J. Znt. (1997) 131, 1-8 The geoid constraint in global geodynamics: viscosity structure, mantle heterogeneity models and boundary conditions Catherine Thoraval and Mark A. Richards Department of Geology and Geophysics, Uniuersity of California, Berkeley, CA 94720, USA. Accepted 1997 February 22. Received 1997 February 3; in original form 1996 May 20 SUMMARY The Earth s non-hydrostatic gravity field, or geoid, provides a first-order constraint on mantle density structure and dynamics. Geodynamic models for the geoid have proliferated since the advent of seismic mapping of mantle heterogeneity structure (tomography) because the geoid offers perhaps the best-measured independent constraint on mantle density heterogeneity. However, dynamic geoid models involve a number of questionable physical assumptions and uncertainties whose effects need to be evaluated before geodynamic inferences based upon the geoid can be considered sound. Troubling issues include the appropriate surface boundary conditions (free-slip, no-slip, plates?) and parametrization of radial viscosity variations (how many layers can be resolved?), in addition to lateral viscosity variations, possible chemical layering of the mantle, phase transitions, etc. There are also uncertainties in the density heterogeneity models used as input to dynamic geoid models, most of which are derived from seismic tomography and require weakly constrained, empirical conversion factors to go from seismic velocity variations to density variations. Here we address several of the most straightforward problems inherent in geoid modelling, namely the issues of viscosity structure resolution, uncertainties in appropriate boundary conditions, and differences among mantle heterogeneity models. A robust feature of all models is a lower-mantle viscosity at least a factor of 30 greater than that of the upper mantle, but there is little resolution with regard to finer details such as lithospheric or uppermost mantle ( low-viscosity zone ) viscosity. Ironically, free-slip boundary conditions result in the best fits to the geoid in all cases, but all boundary conditions exhibit predictable trade-offs with the uppermostmantle viscosity. Models with a single viscosity layer representing the lower mantle yield similar dynamic topography estimates of the order of m in amplitude, regardless of the finer details of upper-mantle viscosity structure, boundary conditions or input heterogeneity models. Comparing mantle heterogeneity models based on two independent seismological determinations (Harvard and Berkeley models) and on the history of subduction, we find that these models are virtually indistinguishable regarding inferences of mantle viscosity structure and amplitude of dynamic topography, and in terms of the effects of different boundary conditions. Uncertainties concerning which type of boundary condition is appropriate are much more important than which mantle heterogeneity model is chosen. Given other uncertainties in modelling the geoid, particularly the strong effects due to lateral viscosity variations for intermediate (< km) wavelengths, we conclude that the class of dynamic geoid models explored so far cannot reliably elucidate the details of upper-mantle viscosity structure. Key words: boundary conditions, density heterogeneity, dynamic topography, geoid, viscosity. 1 INTRODUCTION It has been more than a decade since seismologists provided the first global maps of velocity heterogeneity in the mantle *Permanent address: UMR39/CNRS, 18 av. E. Belin, Todouse Cedex, France. (Dziewonski, Hager & O Connell 1977; Dziewonski 1984; Woodhouse & Dziewonski 1984). An immediate result of these studies was the recognition that the long-wavelength velocity heterogeneity structure of the deep mantle is highly correlated with the shape of the Earth s non-hydrostatic geoid (e.g. Hager et d. 1985; Richards & Hager 1988), with geoid highs correlated with low-velocity anomalies in the lower mantle and vice versa RAS 1

2 2 C. Thoraval and M. A. Richards At about the same time, theoretical geodynamic models were developed which explained this correlation in terms of mantle flow and surface deformations induced by internal density heterogeneities (Richards & Hager 1984; Ricard, Fleitout & Froidevaux 1984; Forte & Peltier 1987). These developments led quickly to successful geoid models based on steadily improving seismic heterogeneity models (Hager et al. 1985; Ricard, Vigny & Froidevaux 1989; Hager & Richards 1989; Forte & Peltier 1991). Within the framework of these geodynamic/seismic Earth models, the mantle has been treated as a layered spherical shell with radially varying viscosity, and various authors have also considered the possibility of layered mantle flow, with the flow barrier usually placed at 670 km depth. There is now a general consensus that whole-mantle flow models fit the observed geoid better than layered models, with about a factor of increase in viscosity between the upper and lower mantle (see e.g. King & Masters 1992). However, these models do not rule out layered flow as a possibility, and many of them also generally predict a very high amplitude of associated dynamic topography which does not appear to be observed (Thoraval, Machetel & Cazenave 1995). There are also fundamental problems regarding the particular model formulations used. (1) It has been found from most models that free-slip upper surface boundary conditions give the best fit to the geoid, and such boundary conditions have been employed, even though the Earth s surface is broken into piecewise continuous plates which may more closely approximate a no-slip condition than free-slip. (2) The input density heterogeneity models, and the conversion factors chosen to go from velocity to density variations, are not uniquely determined. (3) It is not clear what effects various parametrizations of the radial viscosity profile have on the models, and it is equally unclear what resolution the geoid/tomography models offer with regard to radial viscosity variations. Several authors have performed inversions to constrain the viscosity profile within the Earth (Ricard et al. 1989; Forte & Peltier 1991; Ricard & Wuming 1991; King & Masters 1992; Forte, Woodward & Dziewonski 1994; Corrieu, Ricard & Froidevaux 1994). Different success criteria have been used: fitting the geoid or gravity field, fitting plate motions, and/or minimizing the amplitude of the dynamic topography. All studies agree that the lower mantle is more viscous than the upper mantle. Depending on the parametrization of the viscosity profile, one may prefer a smooth increase of viscosity throughout the lower mantle or favour a stronger viscosity jump at 670 km depth. However, the viscosity profile inferred is strongly dependent on the a priori information (Forte et al. 1994; Corrieu et al. 1994). There is also a trade-off between the values of the conversion factor from seismic velocities to density anomalies and the magnitude of viscosity contrasts. (4) Lateral viscosity variations are usually ignored, owing mainly to the difficulty of modelling them, and these viscosity variations may have a major effect on geoid and flow models at intermediate to short wavelengths, that is wavelengths less than about km or spherical harmonic degrees greater than 4 (Richards & Hager 1989; Zhang & Christensen 1993). All of these problems cast doubt on the reliability of the results obtained so far from using seismic-velocity heterogeneity maps to infer mantle dynamics. In this paper, we address simultaneously the first three problems described above by forming model comparisons involving alternative boundary conditions, input density heterogeneity models and viscosity structure parametrizations. The results confirm that our weak understanding of how to treat the upper surface boundary conditions is a serious shortcoming in the current generation of global geodynamic models, apparently outweighing, for example, differences in large-scale mantle structure among candidate density heterogeneity modeis. Furthermore, we find that modelling the geoid offers little resolution of upper-mantle/lithosphere viscosity structure, but we confirm that whole-mantle flow models require a substantial increase in viscosity from the upper to the lower mantle. These results must be considered optimistic, since we have not included other possible physical effects such as layered flow, mantle phase transitions and lateral viscosity variations. 2 DESCRIPTION OF MODEL COMPARISONS We employ mantle flow models based on the simultaneous solution of the mass and momentum conservation equations, and Poisson s equations for the gravitational potential. We include the effects of compressibility, following the PREM model (Dziewonski & Anderson 1981), and mantle viscosity is assumed to be Newtonian (linear) and to vary only in the radial direction. We ignore the effects of mantle phase transitions. For further details, see Thoraval, Machetel & Cazenave (1994) and Corrieu, Thoraval & Ricard (1995). A thorough analysis of the effects of compressibility is given by Panasyuk, Hager & Forte (1996). Three types of surface boundary conditions have been investigated: free-slip, no-slip and plates. For free-slip conditions it is assumed that the shear stress vanishes at the upper surface of the mantle. For no-slip conditions it is assumed that horizontal velocities are zero at the upper surface. The plates condition allows for the piecewise continuous motion of the lithospheric plates. These plates can move in response to the shear tractions due to mantle flow, subject to the condition that the torques on each plate sum to zero (Hager & O Connell 1981; Ricard & Vigny 1989). For the plates condition, flow and surface deformations arise due to both plate motions and the driving density contrasts themselves (with no-slip conditions). Geoid and surface topography calculated assuming free-slip, no-slip or plates boundary conditions do not depend on the absolute value of mantle viscosity, but only on viscosity contrasts. The amplitude of plate velocities calculated within the plates condition depends both on the viscosity contrasts and the absolute value of viscosity. There is a problem with the plates condition due to the non-integrable stress singularity which arises at plate boundaries (Hager & O Connell 1981; Lithgow-Bertelloni & Richards 1995), but this problem does not seriously affect the results which follow. We have followed the analysis of Ribe (1992) in treating the membrane stresses in the surface plates. We have performed model comparisons using three different models for mantle density heterogeneity: S12WM13 (Su, Woodward & Dziewonski 1994), SAW12D (Li & Romanowicz

3 The geoid constraint in global geodynamics ) and the slab model (Ricard et al. 1993). The first two of these models are recent global tomographic models of shearvelocity heterogeneity up to spherical harmonic degree and order 12. The values of seismic velocity to density-variation conversion factors [(c In p)/(2 In y)ip have been arbitrarily fixed to 0.2 in the lower mantle and 0.3 in the upper mantle. By making these choices we are assuming a best-case scenario for the resolution of other model parameters-that is that we have perfect knowledge of how to convert velocity variations to density variations. The values we have chosen for the conversion factors are within the range of accepted values, according to high-pressure measurements (Karat0 1993). No attempt has been made to adjust these parameters to improve model fits. We ignore velocity or density variations in the top 300 km of the mantle, because velocity structure in the uppermost mantle is probably dominated by chemical heterogeneities (deep continental roots, or tectosphere ), which may not represent large net density contrasts (Jordan 1978; Forte, Dziewonski & O Connell 1995). Partial melting beneath island arcs, ridges and hotspots also affect uppermost-mantle seismic velocities, but the associated density contrasts are difficult to - Free Slip Plates -2 -I o a estimate. The slab model is a synthetic model for present-day mantle density heterogeneities derived from Cenozoic and Mesozoic plate-motion reconstructions under the assumption that the subducted slabs sink vertically through the mantle (Richards & Engebretson 1992; Ricard et a/. 1993). Because geoid and topography kernels are independent of absolute viscosity, we express mantle layer viscosities as the ratio of layer viscosity to upper-mantle viscosity. We have investigated two parametrizations of the radial viscosity profile. The first parametrization consists of three layers: an 80 km thick lithosphere at the top of the mantle, an upper mantle from 80 to 670 km depth, and a lower mantle below 670 km. This parametrization is chosen to investigate the trade-off between lithosphere viscosity and boundary conditions. We vary the lithosphere/upper-mantle viscosity ratio from 0.01 to 100, and the lower-/upper-mantle viscosity ratio from 10 to 100. The reasons for choosing these ranges will become apparent as we discuss the modelling results below. In the second parametrization, we consider an additional layer from 80 to 220 km depth (a low-viscosity zone, LVZ). This parametrization is chosen to investigate the ability of No Slip I g F SIabModel ia ia a a a F SAWlZD 5 s I.a I= S12-WM13 ia I.a 10 ia a o a 1 2 WG(llithllM) LOG(ll,il%M1 LOG(@luMj Figure 1, Variance reduction obtained from models of the observed non-hydrostatic geoid (degrees 2-8) for the first viscosity structure parametrization, that is three layers, including a lithosphere 80 km thick, the upper mantle and the lower mantle. From left to right: free-slip, no-slip and plates boundary conditions. From top to bottom: the slab model, the seismic SAWl2D model and the S12-WM13 seismic model. The x-axis is the logarithm of the viscosity ratio between the lithosphere and the upper mantle [Iog(qh1/yUM)]. The y-axis is the logarithm of the viscosity ratio between the lower mantle and the upper mantle [log(qlm/qum)].

4 4 C. Thoraval and M. A. Richards 3 2 Free Sliv Plates No Slip g -: I SlabModel I I I $ I gj I I ~G(?~J?UM) LoG(?dqUM) LOG(qli/bf) 0 SIZ-WM13 Figure 2. Variance reduction obtained from models of the observed non-hydrostatic geoid (degrees 2-8) for the second viscosity structure parametrization, that is four layers, including a lithosphere 80 km thick, and LVZ down to 220 km depth, the upper mantle and the lower mantle. From left to right: free-slip, no-slip and plates boundary conditions. From top to bottom: the slab model, the seismic SAWl2D model and the S12-WM13 seismic model. The lower mantle is 50 times more viscous than the upper mantle. The x-axis is the logarithm of the viscosity ratio between the lithosphere and the upper mantle [log(qiit/qum)]. The y-axis is the logarithm of the viscosity ratio between the LVZ and the upper mantle (WVL"Z/~"M)I. geoid models to resolve more complicated viscosity structure in the upper mantle. In this case, the lower-/upper-mantle viscosity ratio is fixed at 50, and the LVZ/upper-mantle and lithosphere/upper-mantle viscosity ratios both range from to For all of these cases, we map the variance reduction of the model geoid with respect to the observed non-hydrostatic geoid up to spherical harmonic degree 8 (Lerch, Klosko & Patch 1983; Nakiboglu 1982). 3 RESULTS The main results of our study are given in Figs 1 and 2, where we show maps of model variance reduction in the geoid for the three different mantle density heterogeneity models, and for the three different types of boundary conditions employed. The vertical and horizontal axes in each plot represent the logarithm (base 10) of the layer viscosity ratios considered. In the three-layer case (Fig. l), these are the lower-/upper-mantle and lithosphere/upper-mantle viscosity ratios. In the four-layer case (Fig. 2) the lower-/upper-mantle viscosity ratio is fixed at 50, and the axes give the LVZ/upper-mantle and lithosphere/ upper-mantle viscosity ratios. Each map gives contours of geoid variance reduction, in per cent, with positive values shown in white and other regions in grey. We note that all of these fits could have been improved by allowing variations in the velocity/density conversion factors used, but such variations would not affect our main conclusions. 3.1 Lower-mantle and lithosphere viscosity The results of Fig. 1 show that the most robust feature common to all models is a viscosity increase of at least a factor of 30 between the upper and lower mantle. Postglacial rebound studies provide independent estimates of mantle viscosity that are compatible with the lower bound of the range of viscosity contrasts we infer (Mitrovica 1996). The need for a viscosity increase with depth persists regardless of the density heterogeneity model or boundary conditions used. By contrast, the viscosity of the lithosphere, or uppermost mantle (0-80 km

5 The geoid constraint in global geodynamics 5 depth), is not resolved at all. Free-slip conditions require a stiff lithosphere to fit the geoid, while no-slip conditions require a low-viscosity uppermost mantle. Another way to view this is to recognise the obvious similarity between free-slip conditions with a stiff lithosphere and no-slip conditions with a weak lithosphere. This trade-off is illustrated in Fig. 3, where we compare good-fitting geoid and dynamic topography kernels for the free-slip and no-slip cases. Since the power spectrum of the observed geoid decreases strongly with increasing harmonic degree, the variance reduction achieved depends mostly on the lowest degrees (2 and 3). Fig. 3 shows that the geoid kernels at degree 2 are almost the same for the two cases. These kernels also show that the influence of shorter wavelengths is larger for no-slip conditions than for free-slip conditions. Given that the density heterogeneity models have generally less red spectra than the geoid, we can immediately recognize why the best fits are achieved for free-slip conditions. However, Ravine & Morgan (1993) have shown that accounting for mixed boundary conditions (free-slip for oceanic regions and no-slip for continental areas) might increase the 6oW 55w 3 5aM L aM 35w Geoid redness of the geoid spectrum. It is interesting to note that the plates boundary condition results in variance reduction maps (Figs 1 and 2) that indicate best-fitting lithospheric viscosities similar to those obtained using no-slip conditions. Fig. 4 compares the observed non-hydrostatic geoid with the best-fitting model geoids obtained from the three different density heterogeneity models, assuming free-slip upper-surface boundary conditions. The slab model gives the best overall fit, consistent with the findings of Ricard et al. (1993), but the overall shape of each model is, of course, similar. 3.2 Parametrization of the uppermost-mantle viscosity Fig. 2 shows variance reduction maps for the cases with a LVZ where we have fixed the lower-/upper-mantle viscosity ratio at a factor of 50. Here the trade-off between lithosphere and LVZ viscosity ratios is obvious; in the free-slip case, a wide range of lithosphere and LVZ viscosities is permitted, but we have little resolution of either. In the plates and no-slip cases, the,%?face topography Free Slip Y 3 %j 5aM F No Slip Figure 3. Geoid and surface topography kernels for free-slip conditions (top panel) and no-slip conditions (bottom panel). The viscosity contrast between the upper and lower mantle is 50. The viscosity of the uppermost mantle or lithosphere (down to 80 km) is a factor of 20 greater than that of the upper mantle for the free-slip case, and 50 times smaller for the no-slip case. Solid line for degree 2, dashed line for degree 4 and dotted line for degree 8.

6 6 C. Thoraval and M. A. Richards,L 0" 60' 120" 180" 240' 300" 0" UJ 60 ' h0 0" 0' -60 ' -6U hl 60' 120' 180' 240' 300' 0' "/ 60 ' 60 a 0' 0" ' "I 0" 60' 120' 180' 240' 300' 0' L/ 60 ' 60 ' 0' 0' a0 * -60" J I 0" 60' 120" 180' 240' 300' 0' u/ 60 ' 60 ' 0' 0" " 60" 120' 180" 240" 300' 0' Figure4. (a) Observed non-hydrostatic geoid up to degree 8 compared with the best-fitting model geoid obtained using free-slip conditions with the three density heterogeneity models; (b) slab model?lm/~um = 79, qla/qum = 50,72 per cent variance reduction; (c) SAW12D model qlm/qum = 79, qat/qum = 50, 62 per cent variance reduction; (d) S12-WM13 model qlm/qum = 63, qat/qum = 50,54 per cent variance reduction. In (b) and (c) the values of the conversion factors are 0.2 in the Iower mantle and 0.3 in the upper mantle. No density anomalies are accounted for within the top 300 km. Negative values are shaded and contours are drawn every 20 m. ranges are somewhat more restricted, but the trade-offs are still present. In the plates case we obtain the best fit for a high-viscosity LVZ and a low-viscosity lithosphere, which is, of course, absurd. For the free-slip case, similar maps are obtained, with a shift of the lithosphere towards even lower viscosities. None of these results appears very promising with regard to the ability of geoid models to resolve upper-mantle viscosity structure. 3.3 Dynamic topography Fig. 5 shows contours of surface dynamic topography rms amplitude for the three-layer models. The white region of each plot shows the regions of best geoid variance reduction derived from Fig. 1. In all cases, the amplitude of dynamic topography obtained is of the order of m for the best-fitting geoid models. The overall amplitude of dynamic topography does not appear to be a strong function of the density heterogeneity model employed, or of the boundary conditions used. Observed long-wavelength dynamic topography has an amplitude of about 400 m or less (see LeStunff & Ricard 1995), so that the amplitudes computed in our models appear too large by about a factor of 2. However, these model amplitudes could be reduced in two ways: (1) a low-viscosity layer at the base of the mantle (D"), which is dynamically reasonable, can increase the model geoid/topography ratio (Hager & Richards 1989), thereby reducing the required amplitude of dynamic topography; (2) unmodelled internal density anomalies due to the deformation of phase boundaries (or chemical boundaries) in the mantle may also significantly reduce the amplitude of dynamic topography (Thoraval et al. 1995). 4 CONCLUSIONS Our model results illustrate five major points. (1) Good fits to the geoid may be obtained for a wide range of viscosity profiles, depending upon how radial viscosity variations are parametrized. The only robust feature we find is that the lower mantle must be, on average, about a factor of 30 more viscous than the upper mantle. (2) The viscosity of the uppermost mantle depends strongly on the boundary conditions employed and tends to balance the effects of free-slip or no-slip conditions: free-slip conditions are balanced by a more viscous uppermost mantle, and no-slip conditions require a low-viscosity zone. (3) Models of the long-wavelength geoid derived from the various density heterogeneity models do not constrain the details of viscosity stratification in the upper mantle. A large range of upper-mantle viscosity contrasts, varying up to four orders of magnitude, are permissible, even if we restrict our attention to free-slip boundary conditions. (4) The long-wavelength dynamic surface topography amplitude ( m) is not a strong function of either the uppersurface boundary conditions of the upper-mantle viscosity structure. (5) The various seismic and slab models for mantle heterogeneity are virtually indistinguishable in terms of their robust implications for mantle viscosity structure, at least within the parametrizations we have used. Our results suggest that great caution should be exercised in using the geoid as a constraint on some aspects of mantle RAS, GJ1 131, 1-8

7 Free Sip Plates No Slip The geoid constraint in global geodynamics I li 1-5 S12-WM u 10 Figure 5. RMS amplitude of the predicted surface topography (degrees 2-8) for the three-layer parametrization (Fig. 1). The density distribution used is the slab model. The boundary conditions are free-slip (left panels), no-slip (middle panels) and plates (right panels). The x-axis is [l~g(q,~~/q~~)]; the y-axis is [log(qlm/qum)]. RMS amplitude contours are in metres of dynamic topography. The variance reduction of the observed geoid is superimposed: the white and the shaded areas correspond, respectively, to variance reduction above and below 60 per cent. dynamics, especially viscosity structure. We believe that an important task facing geodynamicists is to constrain better the true nature of the upper-surface boundary conditions that are appropriate for geodynamic models. Although it may appear obvious that the best boundary condition should include piecewise continuous plate motions, it is troubling that freeslip boundary conditions consistently yield better fits to the geoid. Since this apparent contradiction derives mainly from the need to 'redden' the geoid spectrum derived from both seismic and geodynamic mantle heterogeneity models, the real problem may derive from the spectral content of the heterogeneity models themselves, rather than from plate-like boundary conditions. If our conclusions seem pessimistic, it should be kept in mind that we have ignored other important modelling uncertainties associated with, for example, phase changes, layered or partially-layered convection and lateral viscosity variations. Further studies are needed to understand more fully the full range of modelling trade-offs involved in geodynamic modelling based on various models of mantle heterogeneity structure. ACKNOWLEDGMENTS This work was partially supported by NSF grant EAR to MAR. We thank Scott King and an anonymous reviewer for helpful comments on the original manuscript. REFERENCES Corrieu, V., Ricard, Y. & Froidevaux, C., Converting mantle tomography into mass anomalies to predict the Earth's radial viscosity, Phys. Earth planet. Inter., 84, Corrieu, V., Thoraval, C. & Ricard, Y., Mantle dynamics and geoid Green functions, Geophys. J. lnt., 120, Dziewonski, A.M., Mapping the lower mantle: determination of lateral heterogeneity in P-velocity up to degree and order 6, J. geophys. Res., 89, Dziewonski, A.M. & Anderson, D.L., Preliminary reference Earth model, Phys. Earth planet. Inter., 25, Dziewonski, A.M., Hager, B.H. & O'Connell, R.J., Large-scale heterogeneities in the lower mantle, J. geophys. Res., 82, Forte, A.M. & Peltier, W.R., Plate tectonics and aspherical Earth structure: the importance of poloidal-toroidal coupling, J. geophys. Res., 92,

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