Gravity above subduction zones and forces controlling plate motions

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007jb005270, 2008 Gravity above subduction zones and forces controlling plate motions Y. Krien 1 and L. Fleitout 1 Received 15 July 2007; revised 10 February 2008; accepted 21 March 2008; published 20 September [1] Short- and intermediate-wavelength gravity and geoid anomalies are used to provide constraints on the mechanical structure of subduction zones and on the forces involved. This study is based on 2D Cartesian dynamically self-consistent models with Newtonian or power law rheologies. We show that both strong decoupling of the two convergent plates (shear stresses of the order of 10 7 Pa) and weakened bending lithosphere are necessary to reproduce the observed geoid and gravity data. Good fits are found for relatively low failure stresses (30 50 MPa). For all models providing reasonable predictions of gravity, only a small fraction of the downgoing slab weight is transmitted to the surface plates. About 10% of the energy is dissipated in the contact zone between the two plates, 10% to 20% in the bending region, and more than 70% in the sublithospheric mantle. The basal tractions (on the order of N/m) induce a net motion of the plates, with the subducting lithosphere moving faster than predicted by the no-net-motion principle. A marked positive geoid anomaly is predicted above subduction zones at intermediate wavelength (l = km) in the case of pure whole mantle convection. Such large geoid highs are not observed. Introducing partial layering (i.e., mass anomalies hampering mantle flow through the transition zone) is necessary to reconcile model predictions and observations for these wavelengths. Citation: Krien, Y., and L. Fleitout (2008), Gravity above subduction zones and forces controlling plate motions, J. Geophys. Res., 113,, doi: /2007jb Introduction [2] Understanding the forces that drive or regulate plate tectonics has been one of the main concerns of geodynamics during the last four decades. Indeed, both the long-term thermal evolution of the mantle and the global lithospheric stress field depend critically upon the amplitude and nature of the various forces applied on plates, in particular in subduction zone areas. Studies based on a global inversion of the forces equilibrating the plates [e.g., Forsyth and Uyeda, 1975; Chapple and Tullis, 1977] conclude in favor of a dominant role played by slab pull, counterbalanced by viscous resistance of the mantle. The resulting force transmitted by the slab could be moderate, of the order of the ridge push. In more recent studies, other indices point toward the large magnitude of the forces required to deform the lithosphere in subduction zone areas, either through bending [Conrad and Hager, 1999, 2001; Becker et al., 1999; Bellahsen et al., 2005; Buffett, 2006; Buffett and Rowley, 2006] or through friction along plate boundaries. Buffett [2006] concludes that 40% of the excess weight of slabs is necessary to bend the plates. Conrad et al. [2004] propose that the entire excess weight of slabs is transmitted to the subducting plate for the seismically uncoupled 1 Laboratoire de Géologie, CNRS UMR 8538, École Normale Supérieure, Paris, France. Copyright 2008 by the American Geophysical Union /08/2007JB005270$09.00 subduction zones. It also seems difficult to change the plate velocities rapidly in a system dominated by the coupling of slab pull and mantle resistance [Lithgow-Bertelloni and Richards, 1998]. Moreover, the simple physical laws which link velocity, viscosity and heat flow in a uniform viscosity convecting fluid do not seem to apply to the Earth s mantle. This may indicate a stiff upper boundary layer, i.e., sizable forces are needed to deform the lithosphere near the plate boundaries. [3] During the last 20 years, models have tried to link in a self-consistent way mass anomalies deduced from tomography or plate history with flow and stresses in the mantle, and with the observed large-wavelength geoid pattern. Although these models have usually been successful in reproducing acceptable geoid anomalies, they had trouble including the plate boundaries in a satisfactory way. Some of them replaced the system of mobile discrete plates by a layer with a moderate viscosity [e.g., King, 1995; Moresi and Gurnis, 1996; Thoraval and Richards, 1997; Moucha et al., 2007; Mitrovica and Forte, 2004]. This type of models usually predicts a good fit to the observed geoid. However, as shown by Karpychev and Fleitout [1996], mass heterogeneities do not produce the same geoid anomalies below discrete plates as below a lid with a uniform and moderate viscosity. In other models [e.g., Hager and O Connell, 1979, 1981; Cadek and Fleitout, 1999, 2003], the plate velocities have been imposed. This type of approach also predicts adequately the large-wavelength geoid but gives no information on the forces involved in plate dynamics. The 1of20

2 velocities can also be computed from the torques applied on the plates, as proposed by Ricard and Vigny [1989]. This approach has the advantage that it incorporates plates and predicts surface velocities relatively similar to the ones observed, although it usually does not include any explicit plate boundary forces. In this latter approach, the slabs pull the two plates symmetrically, and there is no notion of subducting or overriding lithosphere. Typically, the Eurasian and Pacific plates are predicted to have similar velocities unless deep cratonic roots slow down the continental plate [Ricard et al., 1991; Zhong, 2001]. Is such a symmetrical pull an artifact of the simplified mechanical structure used in these models in the subduction zone? Would it disappear with more realistic mechanical properties? From a global model predicting plate velocities (but without continental roots), Conrad and Lithgow-Bertelloni [2002] concluded that upper mantle slabs exert an asymmetrical pull, with most of the excess weight of the slab transmitted to the subducting plate. More generally, the mass heterogeneities linked with subduction zones represent important mass anomalies in the upper and lower mantle, probably playing a key role in plate motions [Becker and O Connell, 2001], and one may wonder whether the complicated mechanical structure linked with the slab itself and the plate boundary may alter the results of these global models based essentially on a layered viscosity approach. Spherical 3D finite elements models [e.g., Zhong and Davies, 1999] including plates and lateral viscosity variations in the mantle have shown the importance of including mobile plates and viscous subducting slabs, but they do not include the small scale features of a subduction zone which may induce asymmetry. [4] Better insight into the gravity signal in subduction zone areas has been gained quite recently thanks to detailed studies including more realistic mechanical properties in the margin region [e.g., Zhong and Gurnis, 1992, 1994, 1998; Billen and Gurnis, 2001, 2003; Billen et al., 2003]. However, these studies are not yet fully successful in reproducing all the surface observables. Zhong and Gurnis models [Zhong and Gurnis, 1994, 1998] nicely predict the trench topography, but they also display a large dynamic topography depression below the arc. Billen et al. [Billen and Gurnis, 2001, 2003; Billen et al., 2003] propose a low viscosity wedge at depths, extending approximately from 20 km to 200 km below the arc to eliminate this depression not visible in gravity data. With this new type of model, the gravity above these regions is well fit. However, the trench becomes very deep, and a large positive geoid anomaly is predicted at medium wavelength ( km) above the trench area. Applying a cut-off to the trench topography provides in this case a realistic gravity anomaly, but is not really satisfactory from a dynamic point of view. One of the purposes of the present paper will be to investigate further the questions above, using rheologies that yield the best matches to the observed gravity at a scale of a few hundred to a few thousand kilometers. With a simplified 2D Cartesian approach, used as a first exploratory step toward future 3D models, we will not discuss the large-wavelength (l > 4000 km) geoid anomalies. [5] We first give a brief overview of the main features of gravity and topography over subduction zones (section 2). We then compare the data with predictions obtained using a very simple model of subduction, in which the whole slab excess weight is transmitted to the surface plates. This test, along with a simple analytical scaling relationship between gravity and stress level, provides preliminary insights into the question of the amount of slab weight that is supported by surface plates, as well as the level of stress in plate convergent boundaries (section 3). Newtonian and non- Newtonian numerical models are then implemented to further explore the mechanical properties of subduction zones that can lead to a relatively good fit to gravity data at short-wavelengths (sections 4, 5 and 6). The question of surface plate motions and the magnitude of the forces controlling plate tectonics is addressed in section 7. We investigate the effect of mantle layering on the gravity field for intermediate-wavelength ( km) in section 8. Finally, the distribution of energy dissipation is discussed in section Data [6] The main features of topography and gravity over convergent boundaries do not change much from one subduction zone to another. Topography is generally characterized by trenches about 2 5 km deep and 100 km wide, associated with a forebulge on the downgoing plate, ranging in vertical amplitude from 100 m to 1 km, approximately. Gravity displays a 200 mgal low over the trench, and a high over the arc. While the long-wavelength component of the geoid presents a maximum over subduction zones, a low of about ten meters over the trench is also observed (Figure 1). [7] Here, we choose to compare predictions given by our models with surface data (mainly gravity and geoid) over the Aleutians, principally because this region does not seem to suffer a complex geodynamic setting, like recent ridge subduction, but also for geometrical reasons: slab dip is close to the mean value, as calculated by a recent statistical analysis of subduction zone parameters [Lallemand et al., 2005], and the slab does not seem to go deeper than 410 km, limiting the impact of deep and often unknown processes on the dynamic response. Model parameters (slab density and geometry, maximum depth, plate thickness, see section 4) are thus chosen to roughly mimic this subduction zone. However, one should keep in mind that our goal is not to present a very accurate dynamic model of a specific region, but rather to compare our predictions with data representative of most subduction zones. 3. Preliminary Considerations From a Naive Model and a Simple Scaling Relationship Between Gravity and Stress Level [8] We first present a very simple model of subduction zone (Figure 2) in which gravity is calculated for a slab with its whole excess weight transmitted to the surface plates. We suppose that the subducting limb acts as a stress guide: the deformation is localized in a small region above the slab. This model is used to provide some preliminary insight into the amplitude of the depression that can be induced in the plate boundary area, and on the gravity anomaly linked to the slab s excess weight. We have assumed that the slab is 80 km-thick with a density anomaly dr = 80 kg/m 3. It 2of20

3 Figure 1. Topography, gravity, and geoid anomaly across the Tonga (West-East profile at 23 S), Japan (West-East profile at 39 N), and Aleutian (summation of 10 North-South profiles from 181 E to 190 E) trenches. Data are taken from altimetry and ship depth soundings [Smith and Sandwell, 1997], and smoothed with a Gaussian filter. Profiles have been shifted up for comparison. For the geoid, a largewavelength linear trend joining the two extremities of the signal has been substracted. extends from 80 km to 670 km depth. Because of vertical equilibrium, there will necessarily be a regional surface topographic depression exactly compensating the slab s excess weight, as long as the slab pull is localized. Depending upon the flexural properties of the surface plates, this depression will extend across a more or less broad area within ±200 km of the trench. Here, the depression has been concentrated on a 100 km wide region, where it corresponds to a mass deficit of about kg/m 2, i.e., a gravity depression around 1600 mgal (Figure 2, bottom). Buoyant shallow mass anomalies within the lithosphere (sediments, thickened crust, serpentine) may alter the topography. However, isostatically compensated shallow density anomalies cannot change the integral of the gravity on a regional scale, so they have been ignored in this naive model. The deep slab also creates a gravity and geoid high at intermediate-wavelength which can be computed analytically or numerically. The surface plates on the two sides of the trench have been assumed here to be 5000 km long. As seen in Figure 2, the observed global gravity depression at the 100 km-scale is far less than that predicted by our naive model. This indicates that the slab does not act as a stress guide, and instead exerts a broad suction on the two plates, or that a strong fraction of the slab s weight is supported from below (see also, for example, the models of Moresi and Gurnis [1996]). [9] A somewhat similar conclusion can also be deduced qualitatively from a simple scaling relationship between gravity and deviatoric stress in the lithosphere. In a number of cases (isostasy for mass anomalies at shallow depth, mass anomalies in a uniform viscosity mantle, flexure...), one can write: R u ¼ g 0g 2pGktL 1 where tl represents the deviatoric stresses integrated over the thickness L of the lithosphere, g 0 =9.81ms 2 the standard acceleration of gravity, g the amplitude of the observed gravity anomaly and k its characteristic wave number. The Runcorn number R u [Ricard et al., 1984] is often not exactly equal to 1, so that it is possible to forge cases where the gravity anomaly is zero while the deviatoric stresses are not. In nature, however, the random ð1þ superposition of various processes characterized by different Runcorn numbers of order 1 is expected to yield deviatoric stresses and gravity not perfectly correlated but still yielding Runcorn numbers of order 1. The observed gravity anomalies as seen in section 2 amount to about 200 mgal for a wavelength of 400 km. A first guess is therefore that the deviatoric stress level (integrated over the thickness of the lithosphere) should be on the order of Nm 1. This is one order of magnitude smaller than the weight of the slab. The simple scaling law deduced from the Runcorn number seems then to indicate that most of the slab weight is not transmitted to the plates. [10] More elaborate numerical models are needed to explore further which mechanical properties of the subduction zone can lead to a better fit to gravity data. The first numerical models that we will present here will have a strong but realistic increase of viscosity with depth, helping to support the slab s weight at depth. However, we will see, through various numerical computations, that this is not sufficient and that the top 100 km around the subduction zone also has to be weakened to minimize the portion of the slab s weight that is supported from above. We will also try to use the relative values of the gravity over the trench and arc areas to place bounds on the coupling between the two plates. As in the naive model, the larger scale geoid anomalies reflect the mass anomalies at depth that are related to the slab. This will be discussed in section Numerical Model [11] We modeled instantaneous viscous flow in twodimensional Cartesian subduction zones (Figure 3) using ZEBULON, a multipurpose finite element code [Besson and Foerch, 1998; Besson et al., 1998]. Practically, a quasi incompressible viscoelastic behavior (with Young s modulus E = 1, N) is used, but our runs are done in a regime where the elastic deformation is negligible compared to the creeping one. Modifying the elastic parameters does not affect the results presented here (see also the Appendix A, B, and C). Using semi-analytic solutions, we checked that the pressure was accurately computed. The lower and upper surfaces are free-slip (the appendix provides an estimate of the error induced by the choice of free-slip instead of free-surface boundary conditions). Reflecting 3of20

4 Figure 2. (top) Schematic representation of our naive model. The whole weight of the slab is supposed to be transmitted to the surface plates. The topographic depression is assumed to be concentrated on a region 100 km wide. Gravity and geoid are the summation of two contributions: deep mass anomalies related to the dense slab, and surface dynamic topography. (bottom) Gravity and geoid profiles predicted, as well as contributions related to the slab and surface topography. Data over the Aleutians are also plotted, for comparison. conditions are used on the sidewalls. Element (6-node triangles) sizes range from 4 km in the plate contact region to 150 km in the lower mantle. Subduction takes place in the middle of a 10,000-km-long model box (a reduction of the width by 4000 km is found to have little influence on the flow dynamics). As will be discussed in section 7, the relative velocity of the two plates depends, among other factors, upon the viscous drag below the plates. Here, a symmetric pull will induce equal velocities (of opposite sign) for the two plates. [12] The flow is driven by a prescribed negative buoyancy associated with the subducting slab and variations in 4of20

5 Figure 3. (top) Model setup and (bottom) magnified view in the vicinity of the subduction zone. The effective viscosity plotted here in a logarithmic scale corresponds to model 13 (see Table 2), with the inclusion of a weak low viscosity wedge. WC, BR, SL, and LVW stand for weak channel, bending region, slab, and low viscosity wedge respectively. The hachured region (bottom) indicates the location of density anomalies implemented to simulate crustal thickness variations in the overriding plate. The thin dashed grey lines (top) delineate the prescribed slab position for the models with a deep penetrating limb (Models 23!28 and 31!36). The slab is supposed to thicken by a factor 3 in the lower mantle because of velocity reduction [e.g., Ricard et al., 1993]. The prescribed density anomaly associated with the deep slab is the same than for its upper part. lithospheric age. In the subducting plate, density anomalies increase from the ridge to the trench assuming a square root dependence of the lithospheric density as a function of age. The prescribed density anomaly prior to subduction and in the subducted plate (80 kg/m 3 ) is deduced from an assumed topographic differential of 2800 m between the ridge and seafloor for a 80 km-thick lithosphere. As numerous studies prefer a much lighter slab, we also investigated models with a density anomaly of 40 kg/m 3 (not presented in this paper). Our conclusions still hold in this case. However, we think that the value of 80 kg/m 3 is appropriate in the top upper mantle. It may then decrease somewhat with depth because of the pressure dependence of the thermal expansion coefficient. [13] The exact density structure of the overriding plate is very poorly constrained. The forearc is serpentinized in a number of areas, while peridotite seems to persist in others [Hyndman and Peacock, 2003; Seno, 2005]. Below the arc, the mantle is expected to be hotter and there might be some water coming from the deserpentinization of the slab mantle. Because of so much uncertainty, we opted for a very simple density model: we assumed that the whole lithospheric mantle of the overriding plate has an asthenospheric density, i.e., we assumed that thermal and petrological effects exactly counterbalanced. A 25 km-thick lighter crust has been added (Figure 3). We find that this finally provides an acceptable topography. We do not expect the gravity effects linked to a different mass distribution in the overriding lithosphere, compatible with the observed topography, to exceed a few tens of milligals. [14] In some models, partial layering at 670 km is taken into account. As the physical processes that may influence the flow across the boundary between the upper and lower mantle remain relatively unknown and difficult to quantify, we choose to model them using a single coefficient which characterizes the reduction of mass exchange across the discontinuity with respect to the whole mantle flow. Following Cadek and Fleitout [1999, 2003], we apply surface anomalies at 670 km depth proportional to the ones needed to achieve a perfectly layered circulation. The proportionality coefficient l, called the layering coefficient, varies between 0 to 1. These bounds correspond respectively to a whole and a perfectly layered mantle circulation. [15] The asthenosphere and deep mantle are assumed to be Newtonian. For the lithosphere, both linear and power law viscous rheologies with a high exponent have been investigated. The former was motivated by the fact that many of the previous studies on a similar topic used a Newtonian formulation [e.g., Conrad and Hager, 1999, 2001; Billen and Gurnis, 2001, 2003; Billen et al., 2003]. The latter is more realistic, and is a good way to approach the brittle behavior of the top lithosphere. The relation between the second invariant of stress (s) and viscous strain 5of20

6 Table 1. Model Parameters Variable Name Value Reference viscosity, h Pas Lithosphere thickness 80 km Gravitational acceleration 10 ms 2 Slab density anomaly 80 kgm 3 Slab dip 60 Reference density 3300 kg/m 3 Water density 1000 kg/m 3 rate (_ v ) tensors can be expressed with the classical equation: s ¼ K _ 1 n v ; where K is a factor representing the temperature, pressure, and compositional dependence of viscosity. For linear rheologies, a thin lubricating layer (sometimes referred to as the slab channel or subduction channel ) is implemented in the contact zone between the subducting and overriding plates. This low viscosity channel formulation can be viewed as a proxy for the weak hydrated faulted crust, sediment layers, or a layer of mantle serpentinized by the water coming from crustal dehydration [Seno, 2005; Hyndman and Peacock, 2003]. It is consistent with a ð2þ number of geophysical data (low seismic velocities, reduced frictional stress, and high electrical conductivity) along the upper boundary of the subducting plate [e.g., Guillot et al., 2001]. It has been shown to reproduce the basic features of trench morphology [Davies, 1989]. We also used a fault in some initial models, as proposed by several authors [e.g., Zhong and Gurnis, 1994, 1998; Billen and Gurnis, 2001, 2003; Billen et al., 2003], but were then unable to reproduce the observations above the overriding plate (large depression in the arc-back arc region) unless an additional weak layer was implemented along the fault. Moreover, the fault favors unconstrained stress in the tip region, leading to an unrealistic trench depth if the grid resolution is high. Several methods, such as a prescribed yield stress or trench depth truncation [e.g., Billen and Gurnis, 2001], have been used so far to overcome this problem but we find the weak channel formulation more satisfactory. The horizontal distance between the weak channel end points is about 180 km (Figure 3), consistent with what has been inferred for most subduction zones [Lallemand et al., 2005], in particular for the Aleutians, which will be used, as mentioned before, to compare our predictions with real data. For models with a highly nonlinear rheology, the lithosphere is considered as a uniform layer. A selfgenerated low viscosity decoupling zone between the two plates forms spontaneously. Table 2. Mantle Rheology: see Table 3 a Mantle Rheology n K h wc /h 0 LVV LVW h br /h 0 h sl /h 0 l Crust Slab Depth, km Model 1 MR no no Model 2 MR no no Model 3 MR1 1 1 no no Model 4 MR no yes Model 5 MR no no Model 6 MR no no Model 7 MR2 1 1 no no Model 8 MR no no Model 9 MR no no Model 10 MR1 1 1 no no Model 11 MR no no Model 12 MR no no Model 13 MR no no Model 14 MR no no Model 15 MR no no Model 16 MR no yes Model 17 MR no no Model 18 MR no no Model 19 MR no no Model 20 MR yes no Model 21 MR no no Model 22 MR no no Model 23 MR no no Model 24 MR no no Model 25 MR no no Model 26 MR no no Model 27 MR no no Model 28 MR no no Model 29 MR no no Model 30 MR no no Model 31 MR no no a n: Stress Exponent in equation (2). K, preexponent in equation (2) for non-newtonian rheologies. h wc and h br, prescribed weak channel and bending region viscosities for Newtonian models. Other parts of lithosphere (except for the wedge when there is one) are assumed to have a viscosity of h sl, slab viscosity. LVV: yes means that the model includes lateral viscosity variations in the asthenosphere (viscosity under the overriding plate is increased by a factor 10, except in the asthenospheric region of LVW). LVW, yes means that a low viscosity wedge has been included. Viscosity is decreased by a factor 100 and 10 in the lithospheric and asthenospheric regions of LVW respectively. l, layering coefficient. Crust, density anomaly related to the thickened crust with the island arc (in kgm 3 ). Slab depth, maximum depth of the sinking slab. 6of20

7 Table 3. Mantle Viscosity Away From Plate Boundaries (10 20 Pas) for Rheologies MR1, MR2, and MR3 (See Table 2) Depth, km MR1 MR2 MR > [16] A low viscosity region in the corner above the subducting slab is also occasionally included. It may represent the relatively hot mantle area with a high water content coming from the decomposition of serpentine in the subcrustal slab mantle [Rupke et al., 2004; Arcay et al., 2005]. About 150 km wide, it extents from a depth of 50 km to 150 km (Figure 3). [17] To assess the success of each model, geoid and gravity profiles are plotted and compared with a mean transect across the Aleutian trench, derived from the summation of ten North-South profiles from 181 E to 190 E. Model parameters values are listed in Tables 1, 2, and Constraints Brought by Gravity Anomalies: Newtonian Rheology 5.1. Effects of Decoupling [18] Three Newtonian cases are computed for different weak channel viscosities (Models 1, 2, and 3 in Table 2). We assumed a relatively weak asthenosphere (h asth =0.1h 0 ), underlying a strong lithosphere (h lith =10 3 h 0 ). Viscosity jumps by a factor 10, 3, and 50 at 200 km, 410 km, and 670 km depth respectively (Rheology MR1, Table 3). In Model 1, the overriding plate is strongly coupled to the subducting lithosphere. Large stresses are transmitted directly upward from the slab to the upper plate, producing a large basin over the island arc region, resulting in significant free air gravity (400 mgal) and geoid (40 m) low (Figure 4). Decreasing the channel viscosity (Model 2) enhances the plate decoupling and hampers the stress transmission to the overriding lithosphere. The upper plate down-warping tends to vanish. The slab pulls mainly the subducting plate, resulting in increased outer rise and trench depth. The negative gravity and geoid anomaly over the arc are reduced significantly compared to the previous case. Yet, unrealistically high values of gravity and geoid over the forebulge and low values over the trench are obtained. These effects are even more pronounced if the channel viscosity is reduced further (Model 3): the arc depression completely disappears, but the trench and forebulge signals are much too large. This transfer of part of the slab s weight from the arc region to the trench area when the coupling between the two plates decreases is in good agreement with what was found in previous studies with a somewhat different density and mechanical pattern [Billen and Gurnis, 2003; Billen et al., 2003] What About a Low Viscosity Wedge? [19] A low viscosity wedge contributes to plate decoupling, and is therefore expected to reduce the arc anomaly. We investigated its effects starting with model 1, and decreasing the mantle wedge viscosity (Model 4). The right-hand part of the down-warping is reduced, as expected (Figure 4). However, the anomalous negative gravity close to the plate boundary remains, and becomes even larger, as the transmitted stresses now concentrate preferentially above the wedge across the rather strong part of the plate contact area. This shows that the plates need to be decoupled along the whole contact region in order to reproduce the gravity data. These results may appear inconsistent with previous studies, in which the inclusion of a low viscosity wedge allowed to match the upper plate topography and geoid [Billen and Gurnis, 2001, 2003; Billen et al., 2003]. However, these models also included a weak channel linking the top of the wedge to the surface. The association of a wedge with this weak channel is a way to fully decouple the subducting from the overriding plate, and gives then good fits to the gravity over the arc. However, as shown by our previous results (section 5.1), a simple weak decoupling channel along the whole plate contact region is sufficient to eliminate the overriding plate depression. It should be noted however that our computations include an asthenosphere with a viscosity of Pas. A stronger asthenospheric mantle would enhance coupling between the slab and upper plate because of the relatively large stresses expected in the asthenospheric corner. This might lead to noticeably different conclusions. Models with a more viscous asthenosphere will be investigated in section 5.5. However, we think a viscosity of the order of Pas to be a realistic value for the asthenosphere below oceans or tectonically active areas. Such a low viscosity is indeed necessary for small-scale convection to bring the appropriate heat flow to the base of the lithosphere [Dumoulin et al., 1999; Fleitout and Yuen, 1984] Influence of Mantle and Slab Viscosity [20] Other scenarios can also be considered to explain the lack of a large negative gravity anomaly above the overriding plate. First, one can imagine that a significant amount of the dense slab is supported from below thanks to a highly viscous mantle. Figure 4 (lower panel) shows the effect of a large viscosity increase in the deep mantle. Starting from a model where the surface plates are relatively tightly coupled (Case 1), we find that the upper plate depression as well as forebulge and trench are significantly affected if the viscosity below 410-km depth is increased by a factor of 10 (Model 5). For instance, in our computations, gravity is reduced by about 40%, 25% and 15% above the arc, trench, and outer rise regions respectively. However, decoupling between the plates is still needed and leads to unreasonnable gravity and geoid anomalies over trenches and outer rises if the channel is strongly weakened (Models 6 and 7), as shown previously. [21] Slab viscosity is also expected to play a role [Billen et al., 2003], as a strong subducted limb pulls more on the lithosphere than a weak one. Figure 5 (upper panel) displays results obtained when the slab is weakened by a factor 10 (i.e., the slab is only 10 times more viscous than the surrounding mantle). One reaches the same conclusions as for the highly viscous mantle case. Even for a moderate viscosity contrast between the slab and the surrounding mantle, the slab induces a localized pull on the surface plates, and the predicted gravity anomalies are different from the observed ones. 7of20

8 Figure 4. (top) Surface gravity and geoid anomalies for different degrees of plate coupling (different weak channel (WC) viscosities). (middle) Effect of a low viscosity wedge (LVW) on surface gravity and geoid anomaly profiles without a marked weak channel (WC). (bottom) Gravity and geoid anomalies for a highly viscous mantle (Rheology MR2) and several weak channel viscosities. 8of20

9 Figure 5. (top) Gravity and geoid anomalies for a weakened slab and several weak channel viscosities. (middle) Gravity and geoid anomalies for several bending region (BR) viscosities. (bottom) Surface gravity and geoid anomaly: effect of a low viscosity wedge for a stronger asthenosphere. 9of20

10 Figure 6. Surface topography for several bending region (BR) viscosities Weak Bending Plates? [22] The results obtained so far strongly support a weak plate mechanical coupling. However, an exclusive stress transmission from the slab to the subducting lithosphere seems to result in unreasonable trenches and outer rises. Neither strong mantle nor weak slab models proved their ability to reproduce the principal features of observed surface gravity. We hence investigated the effect of a weakened bending lithosphere. Figure 5 (middle panel) shows profiles of gravity and geoid anomaly with a weak plate coupling for different viscosities of the flexural region. Surface topography is also displayed for comparison with real data (Figure 6). Trench and forebulge are significantly reduced when viscosity is decreased, resulting in realistic values if the bending lithosphere is sufficiently weakened. A small down-warping of the overriding plate appears, as the viscosity contrast between the weak channel and bending region decreases. Additional weakening of the flexural region (or increased channel viscosity) would reinforce this effect and result in a strong and unrealistic depression over the arc region as seen before. This is the case of Model 14, which displays a gravity low of about 200 mgal above the overriding plate (not shown here). [23] Given the simplicity of our approach, weak subduction zones models match surprisingly well the data at shortwavelength. In model 13, much of the energy (about 79%) is dissipated in the asthenosphere or sublithospheric mantle. Weak channel and lithospheric contributions amount to about 8% and 13% respectively (see Table 4). [24] These conclusions can be explained qualitatively with a simple schematic representation of forces in the subduction zone area (Figure 7). The downward pull exerted by the subducting slab on the plates above has to Table 4. Relative Contribution to Energy Dissipation for Each Region: Weak Channel (WC), Bending Region (BR), and the Other Parts of Lithosphere, Low Viscosity Wedge (LVW), Slab, Upper Asthenosphere (80!200 km), Lower Asthenosphere (200!410 km), Transition Zone (410!670 km), Lower Mantle, and Ridge Region a WC BR + Plates LVW Slab 80! ! !670 Lower Mantle Ridge V sub V up Model 1 4.5% 10.3% 3.8% 31.1% 18.4% 7.9% 19.4% 3.4% 1.2% Model % 14.9% 3.3% 21.4% 14.4% 7.7% 18.9% 3.4% 1.0% Model % 29.7% 1.5% 10.9% 10.8% 7.3% 17.8% 3.3% 0.7% Model 4 4.6% 10.8% 3.0% 30.9% 18.4% 8.0% 19.7% 3.5% 1.2% Model 5 4.9% 11.6% 2.9% 27.4% 14.9% 8.4% 27.4% 1.9% 0.6% Model % 14.9% 2.0% 17.8% 11.4% 8.3% 28.6% 2.1% 0.5% Model % 23.8% 0.6% 11.7% 8.3% 8.2% 30.0% 2.6% 0.4% Model 8 1.2% 7.5% 4.1% 31.3% 12.2% 12% 28.8% 2.6% 0.4% Model 9 4.8% 9.2% 3.7% 27.8% 11.0% 11.9% 28.5% 2.6% 0.4% Model % 18.4% 2.6% 19.7% 8.4% 11.1% 26.7% 2.8% 0.4% Model % 16.4% 2.1% 7.0% 17.8% 13.4% 30.0% 5.3% 0.9% Model % 14.5% 2.5% 7.2% 16.3% 14.7% 30.8% 5.1% 0.8% Model % 13.0% 2.6% 7.7% 14.4% 15.9% 32.7% 4.7% 0.7% Model % 15.5% 2.9% 10.1% 8.1% 13.1% 30.3% 3.5% 0.4% Model % 15.3% 11.3% 6.8% 15.1% 15.0% 26.0% 4.8% 0.2% Model % 17.5% 2.4% 6.8% 16.9% 17.5% 27.7% 5.0% 0.2% Model % 4.0% 29.8% 16.7% 11.5% 27.0% 2.7% 0.5% Model % 3.6% 13.9% 11.3% 12.0% 28.1% 3.6% 0.5% Model % 3.2% 9.5% 13.1% 12.4% 28.6% 4.3% 0.7% Model % 12.3% 2.7% 7.5% 18.9% 17.0% 28.7% 5.1% 0.6% Model % 13.6% 2.7% 7.9% 14.6% 16.2% 34.1% 2.3% 0.6% Model % 13.9% 2.6% 8.0% 14.6% 16.1% 34.9% 1.5% 0.5% Model % 10.5% 1.9% 23.7% 10.7% 12.2% 22.7% 10.8% 0.6% Model % 11.2% 2.0% 26.2% 11.1% 12.6% 24.6% 4.9% 0.5% Model % 11.5% 2.0% 27.6% 11.1% 12.6% 25.6% 2.7% 0.4% Model % 4.5% 0.9% 24.2% 4.6% 5.4% 9.5% 47.2% 0.4% Model % 6.3% 1.1% 27.0% 6.2% 7.1% 14.2% 33.5% 0.4% Model % 7.3% 1.3% 31.5% 7.2% 8.1% 18.5% 21.9% 0.2% Model % 3.2% 9.9% 13.9% 12.5% 29.7% 2.1% 0.6% Model % 2.2% 25.2% 11.2% 9.5% 23.1% 5.0% 0.5% Model % 1.1% 26.5% 5.3% 4.8% 11.8% 37.9% 0.3% a The last two columns give the plates horizontal velocity in the absolute reference frame. 10 of 20

11 Figure 7. Schematic representation of the shear and vertical forces exerted by slab pull (7 and 8), slab suction (5 and 6), and surface topography (forebulge (1), trench (2), arc depression (3)), as well as shear and normal stresses along the plate boundary (4). Torque and vertical stress equilibrium of the overriding and subducting plate (dark and light grey regions respectively) require upper limits for stresses in the plate contact region and for the force transmitted to the subducting lithosphere by the dense slab. be compensated by a negative topography over the arc and trench regions. The vertical equilibrium of the total light and dark grey areas on figure can be written: (6) + (7) = (1) + (2) + (3). Vertical equilibrium of the overriding plate implies that a decrease of the shear and normal stresses in the plate contact region (4) is compensated by a reduction of the upper plate depression (3). On the subducting plate however, this will induce a larger downward vertical force, resulting in a clockwise rotation torque. These additional vertical force and torque can only be compensated by an increased surface topography (deep trench (2) and high forebulge (1)) unless the downward pull exerted by the subducting slab is reduced (7 and 8) Models With a Stronger Asthenosphere [25] Results presented so far have been obtained assuming a relatively weak asthenosphere. A stronger mantle underlying the lithosphere is likely to enhance coupling with the slab, and may change somewhat some of the previous conclusions. In particular, a more significant downward pull exerted by the slab on the upper plate through the mantle wedge might be expected. Figure 5 (lower panel) displays surface gravity and geoid anomaly obtained with a strong asthenosphere (viscosity has been multiplied by a factor 10 compared to previous models), a weak bending region and a reduced plate coupling. Without a low viscosity wedge (Model 15), numerical predictions remain broadly reasonable at short-wavelength, even if the signals over the arc and back-arc regions of the upper plate are lower than observed. The inclusion of a low viscosity wedge (Model 16) significantly reduces this discrepancy. These results confirm, as proposed in previous studies [Billen and Gurnis, 2001, 2003; Billen et al., 2003], that a localized weakening of the mantle wedge might be necessary to explain the observed signals over subduction zones if the mantle underlying the lithosphere is relatively viscous. [26] In summary, assuming a Newtonian rheology in the various regions of the mantle, we found that the gravity anomalies over the subduction area could be fit only if both the decoupling region between the two plates and the bending zone are rather weak. These models are relatively satisfactory as far as the fit to gravity is concerned. As will be seen later, some details of the velocity pattern are less realistic. 6. Constraints Brought by Gravity Anomalies: Non-Newtonian Rheology [27] We investigate the case where the whole region between 0- and 80-km depth (as well as the small deeper part of the bending region, see Figure 3) has a power law rheology. No a priori weak zone is introduced. Below, the rheology is linear and corresponds to MR1. We first plotted gravity and geoid anomalies predictions for three models (17, 18, 19) with decreasing values of the pre-exponent K (K = , , respectively), and n = 10 (Figure 8). As found previously for the Newtonian cases, relatively strong subduction zones models (i.e., high values of K) lead to unrealistically large trenches and depressions over the arc region. The models with the weakest rheology fit relatively well the gravity data at short-wavelength, and also allow for a significant stress transmission from the slab to its attached plate. The plate asymmetry is then more pronounced compared to the Newtonian cases (see section 7). The pattern of energy Figure 8. Gravity, geoid anomaly, and surface velocity profiles for cases with a non-newtonian rheology (models 17, 18, and 19). For comparison, we also plotted surface velocity for Newtonian models with and without lateral viscosity variations under the plates (models 13 and 20). 11 of 20

12 Figure 9. Effective viscosity in a logarithmic scale for Model 19. dissipation is very similar, although the contribution of the bending region slightly increases. For Model 19, about 72% and 28% of the potential energy released is dissipated in the sublithospheric mantle and lithosphere respectively (see Table 4). [28] We also tested models with n = 3 and n = 20 (not presented here). The former was not fully successful in reproducing plate-like behavior, as weakening of the lithosphere in the margin area was not sufficiently localized, while the latter gave results very similar to the ones presented above. It should be noted finally that a thin weak zone along the plate contact naturally emerges from our models with high exponent (n = 10) power law (or pseudovisco-plastic ) rheologies (Figure 9). 7. Forces, Plate Motions and Velocity Asymmetry [29] One of the purposes of this paper is to put constraints on the magnitude of the forces controlling plate motions using gravity as a stress gauge. What is the magnitude of the forces transmitted by the slab to the plates? Is it on the order of the slab excess weight [Conrad and Lithgow- Bertelloni, 2002], or is the slab mainly supported from below [Forsyth and Uyeda, 1975]? Another question concerns a plausible net force in the same direction as that of the oceanic plate. This net force is not always clearly evidenced in mechanical models of subduction zones. It is assumed in most statistical analyses on plate velocities where slab pull is larger than slab suction [Conrad and Lithgow-Bertelloni, 2002; Forsyth and Uyeda, 1975]. (N.B., the terms slab pull and slab suction do not have exactly the same meaning in the various publications. Here, slab pull will designate the total driving force exerted by the subducted slab on the oceanic plate, and slab suction the total driving force exerted on the overriding plate.) [30] On Figure 10, the shear and vertical stresses at a depth of 80 km (s xz80 and s zz80 ) are plotted for the models 13 and 19 which provide a good fit to the gravity data in the case of a Newtonian and a non-newtonian rheology respectively. The shear stress s xz has a moderate amplitude except close to the subduction zone. In the domain of moderate amplitude, the dominant component is the viscous drag, in first approximation proportional to the plate velocities according to couette flow type of models. Just above the sinking slab, s xz has a strong amplitude. It contributes to the slab pull. We name it s xzsp. Then it reverses sign below the overriding plate and contributes to the slab suction (s xzsuc ). The horizontal stress s xx averaged over the plate thickness L relates to s xz80 through the relation: which yields by integration: d ð dx Ls xxþ ¼ s xz80 ð3þ Z 5000 s xx j5000 s xx j 5000 ¼ 5000 s xz80 dx ð4þ L [31] Let us decompose s xz80 like in Figure 10 as the sum of s xzdrag + s xzsp + s xzsuc. In first approximation, s xzdrag can be written s xzdrag = v x f (h) where f (h) is the function of the viscosity stratification which relates in a Couette flow with return flow the horizontal velocity to the shear stress at the base of the plate. Note that in the limit where the ridges have a low viscosity, s xx j 5000 s xx j 5000 = 0. One can then write: Z Z 5000 Z 5000 v x f ðhþdx þ s xzsp dx þ s xzsuc dx ¼ [32] We see from equation (5) that the plate velocities can have a nonzero integral ( R v x dx 6¼ 0) (i.e., a no net velocity condition, equivalent to the no net rotation concept for the case of a spherical geometry) either if f(h) varies spatially or if R s xzsp dx + R s xzsuc dx 6¼ 0. The first case has already been proposed [e.g., Ricard et al., 1991; Zhong, 2001] for explaining the difference of speed between subducting and overriding plates and the nonzero net rotation of the lithosphere. Indeed, we checked that increasing the viscosity below the overriding plate induces a velocity asymmetry (Model 20, Figure 8). [33] However, what we want to emphasize in the present paper is that R s xzsp dx + R s xzsuc dx can very well be different from zero. This is not very clear if one considers only the Newtonian cases. When there is a strong coupling between the slab and the plates (e.g., case 1), R s xzsp dx is indeed much larger than R s xzsuc dx but the predicted ð5þ 12 of 20

13 Figure 10. (top) Schematic cross section of a subduction zone. F sw : slab weight, F sp : shear tractions induced by slab pull under the subducting plate, F suc : suction force (driving component of the shear tractions under the overriding plate), F vrs : viscous resisting force exerted along the slab boundary. F drag sub and F drag up are the viscous drag under the subducting and upper plate respectively. L sp and L suc delineate the regions for which s xz = s xzsp and s xz = s xzsuc respectively (see text). F sp = R R L sp s xzsp dx and F suc = L suc s xzsuc dx. s zzsub is the vertical stress at 80 km depth in the L sp region. (bottom) Shear and vertical stresses at the base of the lithosphere (s xz80 and s zz80 respectively) for models 13 and 19. gravity does not fit the data. In the Newtonian case, in order to reproduce the gravity, we had to decrease the plate coupling up to the point where the two quantities R s xzsp dx and R s xzsuc dx become almost equal. The subducting and overriding plates move at almost the same speed (Model 13, Figure 8). This is not true for non-newtonian models, even for low values of K. The slab-pull is much larger than the suction, resulting in a nonzero net surface horizontal velocity. [34] It is well known that the forces exerted by a mass anomaly on two adjacent plates can be asymmetric only in cases where there are lateral viscosity variations. The oceancontinent contrast has often been invoked to explain the long-standing problem of the plates westward drift [e.g., Ricard et al., 1991; Becker, 2006]. This degree 1 component of plate velocities might indeed be partly linked to cratonic roots. However, we claim that it may also simply arise because the slabs are stiffer than the asthenosphere, and are thus able to induce a net pull on the plates. [35] On Figure 10, s zz reaches a large amplitude in a narrow area just above the slab. We name this part s zzsub. Its integrated value ( Nm 1 for model 13 and Nm 1 for model 19) makes it a very sizable actor in the rate of convergence between the two plates. However, s zzsub is not involved in the global horizontal equilibrium 13 of 20

14 Figure 11. Effect of layering on gravity and geoid anomalies with Newtonian models, for a slab going down to (top) 410 km, (middle) 670 km, or (bottom) 1850 km depth. For comparison, we also plotted the data (dashed black curves) over the Aleutian, Japan (West-East profile at 39 N), and Mariana (West-East profile at 19 N) trenches. The geoid profiles have been detrended. 14 of 20

15 (equation (3)), thus it does not affect the net horizontal surface velocity of the system. s zzsub contributes to the deviatoric stress necessary for the deformation in the bending and decoupling zone. If the bending region and decoupling zone were very ductile, then the horizontal stresses s xx, prior to subduction, would be approximately equivalent to s zzsub. When a sizable deviatoric stress is necessary to deform the bending and contact area, then, the value of s xx is smaller and the plate velocities are reduced. It is why the bending resistance is sometimes presented as the equivalent of a force [Buffett, 2006]. In all the cases where the predicted gravity anomaly is compatible with the observations, we find this force to be no more than a few N.m Gravity Field for Intermediate Wavelength and Mantle Layering [36] The previous sections emphasized the fact that the sinking slabs could not be supported from above, i.e., are not compensated by a large dynamic negative topographic anomaly in the trench-arc area. This was deduced from the gravity signal on a few hundred kilometers wide zone in the trench neighborhood. At a larger scale, the gravity effect of the positive mass anomaly associated with the slab (Figure 2) should therefore appear, unless some other process induces negative compensating mass anomalies. Indeed, all our models with moderate gravity depression in the trench-arc area also present a geoid high at intermediate wavelength not compatible with the observations. The inadequacy between observed and predicted medium-wavelength geoid anomalies above subduction zones has already been noticed in previous studies: the geoid profiles computed by Billen et al. [2003] with somewhat different rheological parameters are also characterized by such broad geoid highs above the subduction zone, although the slab density anomaly is there slightly lower than in the models proposed here. Billen and Gurnis propose to decrease further the slab density (by a factor 1.3). Even then, the models predict a geoid high (approximately 20 m) at intermediate-wavelength not compatible with observations. [37] Mass anomalies linked to phase transitions may reduce the slab s buoyancy at depth. The physical mechanisms responsible for a deflection of the phase transitions are still debated, so that we choose to parameterize the deflection by a layering coefficient (see section 4 and Cadek and Fleitout [1999] for further details). We imposed mass anomalies hampering the flow at the boundary between upper and lower mantle, and plot surface gravity and geoid for three cases: a whole mantle convection (l = 0), a perfectly layered circulation (l = 1), and an intermediate model in which the vertical flow through the transition is divided by a factor 2 (l = 0.5). We present here the results for layering with Newtonian models (non-newtonian models give very similar conclusions). As the magnitude of the layering effect is expected to depend on the maximum slab s depth, calculations are performed first for a slab going down to 410 km (Models 13/21/22, Figure 11, top), then for another reaching the 670-km boundary (Models 23/24/25, Figure 11, middle), and finally for a slab penetrating deeply into the mantle (Models 26/27/28, Figure 11, bottom). [38] The effect of layering is two-fold. The phase boundary deflection and associated density anomalies first directly contribute to the gravity and geoid signals. These anomalies also balance the negative buoyancy of the slab, and thus reduce the net slab pull. These effects can be seen at short-wavelength in surface gravity profiles where the forebulge-trench peak-to-peak amplitude is shown to be reduced with increasing layering coefficient. At longer wavelength, the large positive km length-scale component of gravity and geoid over the slab decreases. These effects are enhanced if the slab penetrates deeply into the mantle, eventually leading to a large negative gravity and geoid signal over the back-arc region for a layered convection model including a dense limb into the deep mantle. Comparison with data (black dashed curves, Figure 11), though not straightforward because of uncertainties in density heterogeneities distribution in subduction zones, and simplicity of our models, always shows a better agreement at intermediate wavelength ( km) for layering coefficients greater than 0.5. Layering seems to be essential for explaining the absence of large (>50 m) geoid anomalies over trenches at those wavelength. This indicates that mantle convection should be partially layered. Note that our previous results still hold if we take layering into account, as we compared essentially short length scale features, and considered a slab going down only to 410 kmdepth. [39] Although we assume here for simplicity that all the layering takes place at 670 km depth, we have no objection against some kinetic or thermal effects linked to the 410 km phase change, or to the broader phase transformations affecting the high pressure pyroxene phases. This would not affect our global conclusions concerning the effect of partial layering on geoid and gravity anomalies. 9. Where is the Energy Dissipated? [40] Table 4 provides the percentage of dissipated energy in each region for all the models presented in this paper. For all cases which give a reasonable fit to gravity and geoid data, only a small amount of energy is dissipated in the flexural zone (10 20%) and in the contact region (<10%). Most of it is dissipated in the deformation of the viscous mantle below. Is there a contradiction between the fact that most of the energy is dissipated in the intermediate and deep mantle and the observation that plate velocities can vary very rapidly? We do not think so. Quick changes in plate velocities indeed contradict models where the main pulling force on plates comes from viscous coupling with the flow driven by deep mantle anomalies. The predicted velocity in our models is very sensitive to the mechanical properties in the subduction zone area, and the deviatoric stresses are moderate there. This means that a small variation of these stresses will modify the response of the system a lot. For example, we checked that introducing a 15 km-thick submarine plateau (dr = 420 kg/m 3 ) on the subducting lithosphere along the plate contact region, down to 80 km, was reducing the convergence velocity by a factor 2 both in the Newtonian and non-newtonian cases. Inversely, a very short slab (incipient subduction) is able to induce a large plate velocity: 12 cm/a for both plates in case 13, 6 and 3 cm/a for the subducting and overriding plates respectively 15 of 20

16 in case 19 if the slab is shortened to a 200 km depth. Notice also in Table 4 that the velocities of the subducting plate depend only moderately upon the length of the slab: for l = 0.5, V sub is equal to 13.2, 13.4, and 16.3 cm/a for the Newtonian cases (21, 24, 27), and 13.2, 14, and 16.4 for the non-newtonian models (Cases 29, 30, 31). (N.B. absolute values of the velocities only depend on the reference viscosity. If h 0 is multiplied by a factor a, the stresses and gravity are unaffected while the velocities are multiplied by 1/a). 10. Conclusions and Discussion [41] On the basis of relatively simple subduction zone models, and systematic investigation of a variety of parameters, we argue that only a small portion of the dense slabs is supported from above. Models with weak mechanical coupling at plate margins as well as limited strength in the bending region are favored: strong subduction zones would result in very large trenches, forebulges, and marked gravity depressions over the upper plate. Relatively good agreements with gravity and geoid data are obtained in models where most of the energy (about 70 80%) is dissipated in the sublithospheric mantle. The plate contact region and the bending lithosphere are found to dissipate about 10% and 10 20% respectively of the descending lithosphere s potential energy release. We propose here a rather low stress level in the top subducting area. The horizontal slab pull on the oceanic plate is predicted to be of the order of the ridge push in good agreement with a deviatoric stress state close to zero at all ages in the oceanic lithosphere [Fleitout and Froidevaux, 1983]. This view in favor of low strength of the lithosphere in the subduction area and of low forces finally applied by the subduction on plates fits also well with a limited level of deviatoric stresses all over the Earth s outer shell. Indeed, a high level of deviatoric stresses in the subduction zone area would probably imply large horizontal compressive stresses in the localized areas where a thickened crust (continent, thick oceanic plateau) slows down the convergence velocity between plates. The crust in such areas would thicken until the average vertical stress counterbalances the horizontal stress (moment law). However, on Earth, the topography rarely exceeds 4 km (in Tibet for example), which puts an upper bound on the intraplate horizontal stresses of the order of N/m [Houseman et al., 1981; Fleitout, 1991]. [42] The laws linking internal viscosity to surface plate velocities are the basis for the parameterized convection models which predict the thermal evolution of the Earth since the Archean. The physics of the bending region of subduction zones have been proposed to be the main ingredient of the mantle convective regime. Here, this bending region is rather weak, and the velocity of the slab at depth is controlled by the viscosity of the deep mantle. If surface velocities and velocities at depth were tightly linked, this may lead in parameterized convection models to the 1 2 usual velocities proportional to h 3. However, we think that surface plate velocities and slab velocities are not always identical at present (cases of detached slabs) and that they were very different from each other in a hotter archean mantle with thicker crusts (in the case we ran with a 15 km thick oceanic crust, the surface velocity was diminished by a factor 2 and the velocity at depth little affected). The role played by the buoyancy of the light crustal petrological phases in the case of a low stress-level subduction, as proposed here, had certainly a growing importance in the past billion years [Van Hunen and Van den Berg, 2008] and would deserve further quantification. [43] The stress on the contact zone between the two plates is important for our understanding of the earthquake cycle in subduction zones. Our model predicts the average stress to be of the order of 10 MPa (about N/m once integrated over the whole contact zone). We do not claim this stress to be uniform in space and time: it is expected to vary during the seismic cycle. One can notice that this value of N/m is somewhat less than predicted from heat flow measurements in the Tonga area [Von Herzen et al., 2001] but more than the stress-drop of the large earthquakes in the top part of the contact zone. [44] Billen et al. [Billen and Gurnis, 2003; Billen et al., 2003] proposed a low viscosity and low density wedge replacing most of the overriding plate along with a weak channel along the top part of the fault to explain the gravity signal over the back-arc region. We find here that a relatively thin decoupling zone, induced by faulting or localized viscous deformation of weak material (such as hydrated sediments or serpentine), is sufficient to explain the observations if there is a weak asthenospheric layer (10 19 Pas) underlying the lithosphere. However, we agree that a weak wedge is necessary to reduce the pulling effect of the slab on the upper plate if the asthenospheric layer is relatively viscous (10 20 Pas). If it exists, we believe this wedge to be confined to depths larger than say 50 km. The heat flow on the overriding plate is rather low (30 40 mw/m 2 ) between the trench and the arc [Van den Beukel and Wortel, 1988; Von Herzen et al., 2001; Gutscher and Peacock, 2003]. Then it raises over the volcanic arc to reach values of the order of 80 mw/m 2. Substitution of most of the overriding plate by a wedge with a viscosity equal or lower than that of the decoupling channel seems rather unlikely as shown by the thermal models. On the basis of these heat flow measurements and on the high Q cold nose found by seismic tomography, Abers et al. [2006] propose a decoupling channel down to 80 km depth, quite similar to what is modeled here. [45] As told before, acceptable gravity and geoid anomalies can be predicted using Newtonian rheologies. In these models, however, because the bending region has to be uniformly weakened, the subducted plate is not tightly coupled to the oceanic surface plate and the pull on the two plates is almost symmetrical. Another unrealistic aspect of these models is that the oceanic plate deforms before the subduction. With a non-newtonian rheology, the surface velocity stays uniform until the trench on each plate, and the pull on the plates is clearly asymmetrical. Once transposed to a 3D spherical geometry, this asymmetrical pull is thought to be an important mechanism for the westward drift of plates. We believe the non-newtonian models to mimic better the subduction dynamics. In these models, the decoupling zone between the plates forms naturally. We do not claim here that there are no intrinsic petrological rheological heterogeneities on the interface between the two plates. Indeed, serpentinization may contribute to the 16 of 20

17 low strength of such a decoupling zone. However, the non- Newtonian rheology used here allows to form this zone very easily technically while maintaining a larger global strength in the bending region. We tested that more sophisticated rheologies with strength increasing with depth in the top lithosphere, then decreasing in the ductile lower lithosphere yielded very similar results. The only noticeable difference was the shape of the decoupling zone in the very top lithosphere which was more elongated with a subhorizontal low dip. [46] The simple 2D models presented here do not include deep anomalies other than directly linked to the subduction and are not appropriate to bring further constraints based on the statistical analysis of the plate velocities. In particular, 3D spherical models should bring more light on the debate concerning slab pull versus active viscous drive induced by deep mass anomalies. However, as the models presented here imply a strong increase of viscosity with depth and partial layering, the deep mass anomalies are expected to create long-wavelength geoid anomalies without inducing large stresses in the plates or applying large forces on the plates. These conclusions are very similar to the ones proposed in the large-wavelength geoid models of Cadek and Fleitout [1999, 2003]. Note also that we predict here plate velocities rather independent of the slab length for slabs deeper than 400 km. The plates seem mainly driven by shallow buoyancy effects. [47] Layering at 670 km is shown to contribute significantly to the observed signals, highly decreasing the km length-scale positive component of gravity and geoid anomaly, especially for a slab penetrating deeply into the mantle. In order to diminish this geoid high, Billen et al. [2003] proposed to reduce the slab density at depth. This proposition is in good agreement with the concept of partial layering, as it means indeed a global diminution of the mass anomalies at depth near the slab. Kinetically delayed phase transition in the cold slab or Clapeyron slope effects contribute to the partial layering. There are other potential mechanisms involving phase changes and linked to volume change (Krien and Fleitout, submitted to Journal of Geophysical Research, 2008), petrology, or thermal effects which happen on a zone broader than just the slab and can contribute to counterbalance the positive thermal density anomalies at depth. Appendix A: Free Surface Versus Free Slip Boundary Condition [48] The computations performed up to now in this paper involve a free-slip boundary condition. A free surface would be more relevant. One may worry that the employed surface boundary condition could alter the results presented in this paper. With the free-slip boundary condition, on the line of altitude z = 0, one imposes a zero vertical velocity and zero shear stress. In the realistic free surface case, on the plane z = 0, below a topography h, the vertical velocity w writes: ða1þ where u is the local horizontal velocity. This introduces an error on the vertical stress (so also on the vertical Figure A1. Effect of a free surface boundary condition on predicted topography. Case 1: same rheology than model 13, with a free surface; Case 2: similar to case 1, except that dr s has been multiplied by a factor 2; Case 3: similar to case 1, except that the internal density anomalies have been divided by 2; Case 4: similar to case 1 except that h br has been multiplied by a factor 4. The topography predicted in cases 2 and 3 has been multiplied by a factor 2 for comparison. These two cases are expected to be close to the free slip model. topography) which scales like 2hkw (where h and k are the characteristic viscosities and wave numbers). The fact that the shear stress is in reality imposed on a distorted surface is also potentially the cause of topography errors (this effect is well known for example as a cause of buckling or boudinage). The shear-stress on the plane z = 0 writes: s xz ¼ ða2þ [49] This leads in turn to topography anomalies of the order of kls xz where L is the lithosphere thickness. Note that both of these effects are nonlinear (see equations (A1) and (A2)), while the surface stress s zz due to a mass anomaly m at a depth d in the free-slip case is a linear function of the amplitude of this mass anomaly: s zz = mf(k, d). The difference in surface topography between free-surface and free-slip models will therefore write as m 2 g(k, d) at first order. If one performs the numerical computation with the natural mass anomalies at depth divided by a factor a with a free surface, one obtains, when a 1, a surface topography which is that for the free-slip case divided by a (only the first order terms remain). [50] Imposing a free slip boundary condition is easy and has the advantage that instantaneous solutions, i.e., solutions corresponding exactly to the imposed initial mechanical structure can be obtained and compared. With a free surface, the notion of instantaneous solution disappears. Indeed, the solution depends, upon other things, on the surface vertical velocity. The surface velocity for a freesurface system is not an instantaneous quantity: it depends necessarily on the previous life of the system as it takes a 17 of 20

18 Figure B1. Velocity vectors in the immobile reference frame for Model 13. certain time to readjust. As the mechanical properties of the matter are advected, the mechanical structure of the model varies with time, which makes comparisons more difficult. In the present case, in order to evaluate the importance of free surface versus free-slip surface boundary condition, we run various variants of case 13 in a large deformation mode for a long period of time and compare the results. In order to have deviatoric stresses which are not four orders of magnitude lower than the background hydrostatic pressure, we introduce in our computations only perturbation density anomalies with respect to the hydrostatic case (dr). As a consequence, the surface topography does not generate automatically vertical stresses. To take this topography into account on the distorted surface of our system, we impose s zz = dr s gh as boundary condition, where h is the topography and dr s = r surf r water. By multiplying dr s by an arbitrary factor, one can change the amplitude of the surface topography and test its effect (the free-slip case corresponds to dr s very large). The surface topography for four test models is presented on Figure A1. The crust below the arc has been suppressed and the excess density of the oceanic plate is constant (60 kg/m 3 ). Case 1 has exactly the same rheology as model 13 but the computation is performed for a free surface and the results are plotted after 60,000 years of subduction. Case 2 is similar to case 1 but dr s is multiplied by a factor 2, so that the topography is expected to be roughly half that of case 1 (but the free surface effect divided by 4). On Figure A1, the predicted topography has been multiplied by 2 for comparison. In case 3, all the internal density anomalies have been divided by 2 and dr s is the same as for case 1. Because the velocities are twice smaller, the results are plotted for a time of 120,000 years so that the global deformation is similar to that of cases 1 and 2. The surface topography is also multiplied by 2 for comparison. Case 4 is similar to case 1 but the viscosity h br has been multiplied by 4. The first three models have very similar topographies. Cases 2 and 3 are expected to be closer to free-slip models. Their difference with case 1 is much smaller than the difference associated with small rheological variations (case 4). Therefore we believe our conclusions to depend little upon our choice of free-slip versus free-surface boundary condition. For stiffer rheologies, the sensitivity to the surface boundary condition somewhat increases (as it is a nonlinear effect), but the predicted topography, although slightly different for very short wavelength is much too large for leading to realistic gravity anomalies. Note also that the curves for cases 2 and 3 are almost indistinguishable. Case 3 has a Deborah number twice smaller than case 2. The fact that the predicted topography is very similar for these two cases confirms that the elastic part of the viscoelastic deformation plays a minor role on the points discussed in this paper. Appendix B: Slab Geometry and Flow Pattern [51] The instantaneous models presented here are meaningful only if the pattern of cold anomalies associated with the slab is compatible with the predicted velocity. Here, for all the models providing good fits to gravity data, the calculated velocity is almost parallel to the slab in the immobile upper plate reference frame (e.g., model 13 and Figure B1). These models are then relatively consistent 18 of 20