3.3. Forces Balance
[23] 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 LithgowBertelloni, 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.
[24] 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 R_{c}, 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.
[25] The correlation between the age of the subducting plate A and its absolute motion V_{sub} 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).
[27] [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]. F_{sp} exerts a bending torque M_{sp} 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. M_{sp} is the product of F_{sp} with the lever arm between the hinge and the gravity center of the slab (Figure 7),
Keeping L and α constant, the bending torque M_{sp} is proportional to the age of the subducting lithosphere at trench A since Δρ is a direct function of T_{th}.
[28] 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 M_{b} is the flexural rigidity of the plate D divided by its mean radius of curvature R_{b} [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 M_{b} resists the bending of the lithosphere and thus exerts an arcward pressure along the interface between the plates. Assuming a constant R_{b} and a constant α, the moment M_{b} then increases with A^{3/2}.
[29] 3. There is an additional force F_{a} that accounts for the mantle reaction to slab facewise translation, including toroidal and poloidal flows. Some equations have been proposed by Dvorkin et al. [1993] or Scholz and Campos [1995] 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 [1996] who considers the motion of an oblate spheroid (disk) into an infinite viscous fluid. The anchoring force F_{a} is thus estimated as
and the viscous shear force F_{sh} 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 T_{th} = 100 km. Here η_{m} is the average viscosity of the displaced mantle, and U is the translation rate with normal and tangential components U_{N} and U_{T} with respect to the slab surface (see Figure 7),
[30] 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, F_{a} 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
[31] For a given mantle viscosity η_{m}, dip angle α, and slab length L, this torque scales with the facewise translation rate U_{N} = V_{t}sinα. The force is directed either arcward or oceanward depending on the sense of slab motion, since it resists slab forward or backward motion.
[32] 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
M_{a} 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 V_{sub}, respectively. The three examples in Figure 8 correspond to those illustrated in Figure 5.
[33] 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 R_{b} = 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 V_{t}, −2.5, 0, +2.5 and +5 cm a^{−1}. According to Cloos [1993], 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 T_{th} (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 Notations^{a}Quantity  Symbol  Value (Range) 


Thermal diffusivity  κ  10^{−6} m^{2} s^{−1} 
Gravity acceleration  g  9.81 m s^{−2} 
Young modulus  E  155 GPa 
Poisson ratio  ν  0.28 
Asthenosphere viscosity  η_{m}  10^{19}–10^{21} Pa s 
Crustal density  ρ_{c}  2900 kg m^{−3} 
Lithospheric mantle density  ρ_{lm}  3300 kg m^{−3} 
Asthenosphere density  ρ_{a}  3230 kg m^{−3} 
Table 4. Estimates of Forces and Related Torques for Various Slab Ages and Trench Migration Rates^{a}  A (Ma)  Te (km)  T_{th} (km)  Δρ (kg m^{−3})  D (10^{22} N m)  v_{t} (cm a^{−1})  F_{sp} (10^{12} N)  F_{a} (10^{12} N)  M_{sp} (10^{18} N m)  M_{a} (10^{18} N m)  M_{b} (10^{18} N m) 


Trench rollback  20  18.8  58.3  22.0  9.3  −2.5  2.20  2.00  0.50  0.14  0.26 
Fixed trench  60  32.5  100.9  42.2  49.0  0  7.31  0  1.64  0  1.40 
Advancing trench  80  37.6  116.5  46.0  74.5  2.5  9.20  −2.00  2.07  −0.14  2.13 
Advancing trench  100  42.0  130.3  48.5  103.8  2.5  10.85  −2.00  2.44  −0.14  2.97 
Advancing trench  140  49.7  154.2  51.8  172.0  5.0  13.70  −4.00  3.09  −0.28  4.91 
[34] We observe in Table 4 that M_{b} is lower than M_{sp} for young slabs, except the very young ones for which M_{sp} can become negative, and exceeds M_{sp} for old ones. Using our values (see Table 3), the change happens for slabs aged about 80 Ma for which M_{b} ≅ M_{sp}. Slab pull forces F_{sp} vary from 2.2 to 13.7 10^{12} N per unit length of trench, leading to a slab pull moment M_{sp} from 0.5 to 3.1 10^{18} N m in the range of slabs aged between 20 and 140 Ma at trench, whereas the bending moment M_{b} varies from 0.3 to 4.9 10^{18} N m. The anchoring force F_{a} both depends on slab facewise translation velocity V_{t}sinα and mantle viscosity η_{m}. Table 4 is thus only indicative of tendencies but should be adapted depending on some key parameters like η_{m}, which is a matter of debate between low values, 10^{19} to 10^{20} Pa s, and higher ones, 10^{21} to 10^{22} Pa s [Winder and Peacock, 2001; Cadek and Fleitout, 2003; Mitrovica and Forte, 2004; Enns et al., 2005]. If we use 10^{21} instead of 10^{20} Pa s, then our anchoring moments will increase by a factor of 10. All other parameters may also vary, like L, α, R_{b}, 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.
[35] 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 V_{t} [Heuret and Lallemand, 2005], i.e., trench rollback for young subducting plates and vice versa together with the positive correlation between A and V_{sub} (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. [2008] also pointed out the competition between slab stiffness and slab pull. They concluded that advancingstyle subduction is promoted by thick plate, a large viscosity ratio, or a small density contrast between plate and mantle. On the basis of 2D numerical models with a viscosity structure constrained by laboratory experiments for the deformation of olivine, Billen and Hirth [2007] also concluded that the slab stiffness has a stronger affect on dynamics than small differences in slab density due to plate age.
[36] 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 V_{sub} and V_{up} is another positive argument in favor of the HS3 reference frame.
[37] One important point is that back arc rifting or spreading is not necessarily related to trench rollback, even in the SB04 reference frame (see IzuBonin, Kermadec, or Andaman). These behaviors are the result of a specific combination of V_{sub} and V_{up} that favors either extension or compression.