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## Two-plate interaction along a weak subduction interface

Even if models of an isolated SP are useful to capture meaningful features of subduction, adding an OP to the system is crucial for more realistic modeling of natural subduction zones. In this context three main questions arise: (i) how does the presence of the OP inuence the kinematics of the SP? (ii) what controls the interplate stress state along the subduction interface? (iii) what drives the deformation and motion of the OP (e.g. Krien & Fleitout, 2008; van Dinther et al., 2010; Gerya, 2011; Duarte et al., 2013; Garel et al., 2014)? In recent years, several authors have used 3-D analog and numerical models of the free class to address the question of the mechanism of deformation of the OP, focusing, in particular, on the controlling factor for backarc extension. In general, it is found that backarc extension is strongly correlated with trench retreat (e.g. Duarte et al., 2013; Meyer & Schellart, 2013; Schellart & Moresi, 2013; Chen et al., 2016). A possible mechanical interpretation of this result is that slab rollback induces a toroidal mantle ow that exerts shear stresses on the base of the OP that in turn lead to backarc opening (gure 1.7a). The rate of extension in the backarc zone depends on whether the OP is free to move or is xed at its end on the opposite side from the trench (Chen et al., 2015). Interestingly, backarc extension is also observed in the 2-D (toroidal ow absent by denition) numerical model of Holt et al. (2015a) when the OP is positively buoyant. For such a case, if the poloidal ow suddenly becomes weaker due to interaction of the slab with a viscosity increase at 660 km depth, a shift from extension to compression in the backarc zone may occur.

### Viscous dissipation of energy at subduction zones

One possible explanation for the failure of the cooling model presented above lies in the small value of the mantle relaxation time τR, which makes the process of secular cooling highly sensitive to any uctuations from the equilibrium state. In the context of the calculations presented in the preceding section, the low value of τR might be ascribed to the exponent β, which we assumed to have the value 1~3 in accordance with the scaling law (1.20). However, there is no good reason to suppose that β = 1~3 is representative of mantle convection. First, the prediction β = 1~3 assumes that there are no volumetric heat sources within the mantle. As we have seen, this is not a good approximation for whole-mantle convection given the radioactive decay of uranium, thorium and potassium. Nev- ertheless, as recent studies seem to suggest, internal heating should not change

much the 1/3 power dependence of the Nusselt number on the Rayleigh number (e.g. Sotin & Labrosse, 1999; Vilella & Deschamps, 2018). Second, convection models with the horizontal convection pattern like the one of gure 1.10 (e.g. Grigné et al., 2005) are characterized by a ‘rectangular’ seaoor age distribution (Labrosse & Jaupart, 2007). The latter arises from the fact that all the plates start to subduct after traveling the same time, thus inducing a constant spatial distribution of seaoor ages in the whole domain of the model (gure 1.12(A)-(B)). On the contrary, the peculiar characteristic of mantle convection is to have generated a plate-tectonics system with a ‘triangular’ seaoor age distribution (Sclater et al., 1981; Rowley, 2002; Cogné & Humler, 2004). A simplied representation of such a system is depicted in 1.12(C). Because young plates also can now subduct, the spatial distribution of seaoor ages in the whole model domain has a peak for short seaoor ages and decreases linearly as the seaoor age increases (gure 1.12(D)).

Taking into account this feature, Labrosse & Jaupart (2007) obtained τR ≈ 10 Gy in their empirical cooling model. Third, the assumption of an isoviscous system neglects the dissipation of energy that occurs at subduction zones where highly viscous lithospheric plates must bend and then slide along the subduction interface. The resistance to deformation at such plate boundaries might partly decouple the dynamics of the lithosphere (i.e. the upper TBL) from the mantle convection that takes place below it. The plate speed and the corresponding surface heat ow would then be less sensitive to any variations in the properties of the mantle, thereby reducing the eective value of β.

#### Boundary-integral representation

By combining dierent types of singular solutions we can build a useful repre- sentation of Stokes ow, called the boundary-integral representation. Unlike the classical partial dierential equations, which describe the spatial gradient of the velocity over the whole uid domain V , the boundary-integral representation expresses the velocity at any point in V in terms of the velocity ui and the stress σij on the surface S bounding the uid domain. This representation of the Stokes ow is particularly convenient as it reduces the dimensionality of the problem by one (we solve line integrals in a 2D domain or surface integrals in a 3D domain).

Thus, it makes possible a powerful numerical technique, called the boundaryelement method, which does not require the discretization of the whole ow do- main (Pozrikidis, 1992).

**2-D Boundary-integral representation of two uid drops immersed in a uid half-space**

In the light of the boundary-integral representation (2.10), we derive here the integral representation of the system depicted in gure 2.5 that represents the basis of the subduction model we have developed for our work. In gure 2.5 we have two viscous drops immersed in a innitely deep ambient uid, bounded at the top by a free-slip surface. The force triggering the motion is the negative/positive buoyancy of the two drops associated with their higher/lower density surplus with respect to the ambient uid. All the other parameters describing the system are listed in the caption of gure 2.5. We begin by writing the boundary-integral representation for each of the three uid domains appearing in the model. According to eq. (2.10), it is: for the uid drop 1:

**Table of contents :**

List of Figures

List of Tables

**1 Introduction **

1.1 Subduction

1.1.1 Subduction modeling

1.1.2 Subduction of an isolated plate

1.1.3 Two-plate interaction along a weak subduction interface

1.2 Rayleigh-Bénard convection

1.2.1 Onset of R-B convection

1.2.2 Steady-state boundary layer analysis

1.3 Thermal evolution of the Earth

1.3.1 Parameterized cooling model

1.3.2 Viscous dissipation of energy at subduction zones

1.4 Thesis outline

**2 Stokes ow **

2.1 Singular solutions

2.1.1 Eect of a free-slip wall

2.2 Boundary-integral representation

2.2.1 2-D Boundary-integral representation of two uid drops immersed in a uid half-space

2.3 Thin viscous-sheet theory

2.3.1 Exact governing equations

2.3.2 Midsurface kinematics and constitutive relations for N and

2.3.3 Viscous dissipation and bending length

**3 Mechanics of subduction **

3.1 Model setup

3.2 BEM formulation

3.3 Unsteady subduction

3.4 Thin-sheet analysis: SP kinematics

3.4.1 Instantaneous solutions: SP only

3.4.2 Instantaneous solutions: SP+OP

3.5 Thin-sheet analysis: OP deformation

3.6 Geophysical application: evaluation of the interface viscosity of the central Aleutian slab

3.7 Discussion

3.8 Conclusions

**4 Energetics of subduction and large-scale mantle convection **

4.1 Rates of viscous dissipation of energy

4.2 Scaling analysis

4.2.1 Subduction of an isolated SP

4.2.2 Subduction below an OP

4.3 Unsteady subduction

4.4 Parameterized model of mantle convection

4.4.1 Thermal convection dominated by mantle viscous dissipation

4.4.2 Thermal convection below a strong deforming boundary layer

4.5 Inuence of the lengthscales `b vs. Rmin on dissipation partitioning

4.6 Conclusions

**5 Laboratory modeling of mantle convection **

5.1 Drying of colloidal systems

5.2 Preliminary results

5.2.1 Observations

5.2.2 In situ measurements

**6 Conclusions **

Appendix A Stretching rate of a thin-viscous sheet below a lubrication layer bounded by a free-slip surface

Appendix B Spreading gravity current below a free-slip surface

Appendix C Numerical implementation

**Bibliography **