3.3. Forces Balance
 A long list of authors so far addressed the dynamical aspects of subduction/convection system from kinematic data [e.g., Forsyth and Uyeda, 1975; McKenzie, 1977; Davies, 1980; Carlson, 1995; Conrad and Hager, 1999; Conrad and Lithgow-Bertelloni, 2002; Becker et al., 1999; Buffett and Rowley, 2006; Faccenna et al., 2007; Di Giuseppe et al., 2008], but the quantification of the different contributions in the subduction system remains elusive and dependent upon assumptions and somehow arbitrary scaling parameters.
 The analysis presented here, although not pretending to give exact quantification of the forces at work, first uses the mode of trench migration as a proxy for the competition between the different contributions. The slab/mantle system is here considered to result from the competition between arcward and oceanward forces acting on a slab, or more specifically, the folding and unfolding torques acting at the pivot made by the slab hinge. To better understand the role of the main forces, we will assume a constant radius of curvature Rc, slab length L, and dip α. We neglect the interaction of the slab with both the 660 km discontinuity and the lower mantle, which are extremely difficult to model because slab imagery at this depth is often unclear and difficult to interpret. We consider the change in slab dip to be negligible, so that trench motion equals horizontal slab migration.
 The correlation between the age of the subducting plate A and its absolute motion Vsub indicates that slab pull is the main force driving the motion of the subducting plate. Resisting forces are partitioned between mantle and lithosphere [Conrad and Hager, 1999]. We neglect friction and suction at the top of the slab since they are at least 1 order of magnitude smaller [Turcotte and Schubert, 1982; Conrad and Hager, 1999]. We assume that the upper mantle behaves passively as it is excited by slab subduction and migration. The asthenospheric mantle viscously resists to facewise translation of the slab, i.e., form drag. Finally, we distinguish three forces and related torques (see Figure 7).
 [Turcotte and Schubert, 1982], κ being the thermal diffusivity, and A being the slab's age in seconds. The coefficient 0.25 accounts for the decreasing of the excess mass of the slab with depth [Carlson et al., 1983]. Fsp exerts a bending torque Msp at the pivot that contributes to the downward folding of the slab at the hinge. Except for buoyant subducting plates, this torque transmits an oceanward pressure along the interface between the plates. Msp is the product of Fsp with the lever arm between the hinge and the gravity center of the slab (Figure 7),
Keeping L and α constant, the bending torque Msp is proportional to the age of the subducting lithosphere at trench A since Δρ is a direct function of Tth.
 2. Within the lithosphere, the main resisting force is related to the bending of the lithosphere at trenches. We use an elastic rheology for the slab as a proxy because of the lack of constraints to adjust an “efficient bending” viscosity for a viscous slab at trench. For a semifinite elastic slab, the bending moment Mb is the flexural rigidity of the plate D divided by its mean radius of curvature Rb [Turcotte and Schubert, 1982],
where E and ν are the mean Young modulus and Poisson ratio, respectively, and
is the elastic thickness in meters with A (expressed in Ma) [see McNutt, 1984]. The torque Mb resists the bending of the lithosphere and thus exerts an arcward pressure along the interface between the plates. Assuming a constant Rb and a constant α, the moment Mb then increases with A3/2.
 3. There is an additional force Fa that accounts for the mantle reaction to slab facewise translation, including toroidal and poloidal flows. Some equations have been proposed by Dvorkin et al.  or Scholz and Campos  on the basis of the translation of an ellipsoid through a viscous fluid [Lamb, 1993]. In this study, we propose a simpler analytical solution adapted from work by Panton  who considers the motion of an oblate spheroid (disk) into an infinite viscous fluid. The anchoring force Fa is thus estimated as
and the viscous shear force Fsh is estimated as
where e < 1 is the ratio between the disk radius a and its thickness taken as 1/4 if we consider a “circular” slab with L = 800 km and Tth = 100 km. Here ηm is the average viscosity of the displaced mantle, and U is the translation rate with normal and tangential components UN and UT with respect to the slab surface (see Figure 7),
 The viscous drag acting on both sides of the descending slab does not produce any torque at the slab hinge. We will thus only consider the anchoring force in this balance. To be compared with the first two forces, Fa is normalized to the slab's width taken as equal to L, since we consider a “circular slab” in this equation. Here a should thus be replaced by a/L which is equal to 1/2. The simplified equation of the anchoring force thus becomes
and the anchoring torque becomes
 For a given mantle viscosity ηm, dip angle α, and slab length L, this torque scales with the facewise translation rate UN = Vtsinα. The force is directed either arcward or oceanward depending on the sense of slab motion, since it resists slab forward or backward motion.
 Assuming no stress transmission through the plates' interface (no significant deformation of the supposedly weak arc), we balance the oceanward/folding and arcward/unfolding torque components of these three forces (Figure 8), taking into account that in steady state, the sum of the moments should be zero. We write that
Ma being positive in the case of slab rollback, null for a fixed trench, and negative in the case of an advancing slab. These situations correspond to low, moderate, and high Vsub, respectively. The three examples in Figure 8 correspond to those illustrated in Figure 5.
Figure 8. Sketch showing the balance of torques for the three examples illustrated in Figure 5. The slab pull torque Msp increases with the age of the subducting plate A, as does Vsub if we consider that it is the main driving force for the subducting plate (see explanations in section 3.3). The mantle reaction to slab migration generates an anchoring torque Ma which tends to unbend the plate when the trench rolls back and to bend it when the trench is advancing.
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 Let us consider three subduction systems (Figure 8) with the same length L = 700 km, dip α = 50°, and radius of curvature of subducting slabs Rb = 350 km and values for the other parameters as listed in Table 3. We compare in Table 4 the behavior of five cases, including the three ones mentioned in Figure 8, with various ages of subducting lithospheres, 20, 60, 80, 100 and 140 Ma and respective trench absolute velocities Vt, −2.5, 0, +2.5 and +5 cm a−1. According to Cloos , we can estimate the bulk density of an oceanic plate from those of a 7 km thick oceanic crust and a lithospheric mantle whose thickness is deduced from the total thermal thickness Tth (see equation (2)). This allows us to calculate the excess mass Δρ for each slab, taking mean densities for the crust, lithospheric mantle, and asthenosphere as given in Table 3.
Table 3. Parameters Used in the Calculations and Notationsa
|Thermal diffusivity||κ||10−6 m2 s−1|
|Gravity acceleration||g||9.81 m s−2|
|Young modulus||E||155 GPa|
|Asthenosphere viscosity||ηm||1019–1021 Pa s|
|Crustal density||ρc||2900 kg m−3|
|Lithospheric mantle density||ρlm||3300 kg m−3|
|Asthenosphere density||ρa||3230 kg m−3|
Table 4. Estimates of Forces and Related Torques for Various Slab Ages and Trench Migration Ratesa
| ||A (Ma)||Te (km)||Tth (km)||Δρ (kg m−3)||D (1022 N m)||vt (cm a−1)||Fsp (1012 N)||Fa (1012 N)||Msp (1018 N m)||Ma (1018 N m)||Mb (1018 N m)|
 We observe in Table 4 that Mb is lower than Msp for young slabs, except the very young ones for which Msp can become negative, and exceeds Msp for old ones. Using our values (see Table 3), the change happens for slabs aged about 80 Ma for which Mb ≅ Msp. Slab pull forces Fsp vary from 2.2 to 13.7 1012 N per unit length of trench, leading to a slab pull moment Msp from 0.5 to 3.1 1018 N m in the range of slabs aged between 20 and 140 Ma at trench, whereas the bending moment Mb varies from 0.3 to 4.9 1018 N m. The anchoring force Fa both depends on slab facewise translation velocity Vtsinα and mantle viscosity ηm. Table 4 is thus only indicative of tendencies but should be adapted depending on some key parameters like ηm, which is a matter of debate between low values, 1019 to 1020 Pa s, and higher ones, 1021 to 1022 Pa s [Winder and Peacock, 2001; Cadek and Fleitout, 2003; Mitrovica and Forte, 2004; Enns et al., 2005]. If we use 1021 instead of 1020 Pa s, then our anchoring moments will increase by a factor of 10. All other parameters may also vary, like L, α, Rb, Δρ, or the slab width. Ultimately, we have chosen to consider a passive mantle that opposes to slab facewise migration, but we have evidence of mantle flow in various regions, especially around slab lateral edges, as in Tonga. In this case, we must count with an additional force exerted by the mantle onto the slab.
 From the balance of forces and earlier observations, we can say that arcward trench motion (advancing) is marked by high resistance of the plate to bending at the trench together with a quickly subducting plate, and that the opposite occurs for trench rollback. This explains the negative correlation between the age of the lithosphere at trench A and Vt [Heuret and Lallemand, 2005], i.e., trench rollback for young subducting plates and vice versa together with the positive correlation between A and Vsub (Figure 3). Such observations highlight the role of the bending resistance that scales with the cube of the square root of A, whereas the slab pull scales with A only. It is therefore dominant for old (and fast) subducting plates [Conrad and Hager, 1999; Becker et al., 1999; Faccenna et al., 2007]. Di Giuseppe et al.  also pointed out the competition between slab stiffness and slab pull. They concluded that advancing-style subduction is promoted by thick plate, a large viscosity ratio, or a small density contrast between plate and mantle. On the basis of 2-D numerical models with a viscosity structure constrained by laboratory experiments for the deformation of olivine, Billen and Hirth  also concluded that the slab stiffness has a stronger affect on dynamics than small differences in slab density due to plate age.
 In our model, we have pointed out how the lithosphere and the mantle together control trench migration. We consider that the classification of transects into three groups (extensional, neutral, and compressional) as a function of Vsub and Vup is another positive argument in favor of the HS3 reference frame.
 One important point is that back arc rifting or spreading is not necessarily related to trench rollback, even in the SB04 reference frame (see Izu-Bonin, Kermadec, or Andaman). These behaviors are the result of a specific combination of Vsub and Vup that favors either extension or compression.