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Fluid mechanics of free subduction on a sphere. Part 1. The axisymmetric case

Published online by Cambridge University Press:  27 October 2021

Alexander Chamolly
Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France Institut Pasteur, Université de Paris, CNRS UMR3738, Developmental and Stem Cell Biology Department, F-75015 Paris, France
Neil M. Ribe*
Lab FAST, Université Paris-Saclay, CNRS UMR7608, F-91405 Orsay, France
Email address for correspondence:


To understand how a spherical geometry influences the dynamics of gravity-driven subduction of the oceanic lithosphere on Earth, we study a simple model of a thin and dense axisymmetric shell of thickness $h$ and viscosity $\eta _1$ sinking in a spherical body of fluid with radius $R_0$ and a lower viscosity $\eta _0$. Using scaling analysis based on thin viscous shell theory, we identify a fundamental length scale, the ‘bending length’ $l_b$, and two key dimensionless parameters that control the dynamics: the ‘flexural stiffness’ $St = (\eta _1/\eta _0)(h/l_b)^3$ and the ‘sphericity number’ $\varSigma = (l_b/R_0)\cot \theta _t$, where $\theta _t$ is the angular radius of the subduction trench. To validate the scaling analysis, we obtain a suite of instantaneous numerical solutions using a boundary-element method based on new analytical point-force Green functions that satisfy free-slip boundary conditions on the sphere's surface. To isolate the effect of sphericity, we calculate the radial sinking speed $V$ and the hoop stress resultant $T_2$ at the leading end of the subducted part of the shell, both normalised by their ‘flat-Earth’ values (i.e. for $\varSigma = 0$). For reasonable terrestrial values of $\eta _1/\eta _0$ ($\approx$ several hundred), sphericity has a modest effect on $V$, which is reduced by $< 7\,\%$ for large plates such as the Pacific plate and by up to 34 % for smaller plates such as the Cocos and Philippine Sea plates. However, sphericity has a much greater effect on $T_2$, increasing it by up to 64 % for large plates and 240 % for small plates. This result has important implications for the growth of longitudinal buckling instabilities in subducting spherical shells.

JFM Papers
© The Author(s), 2021. Published by Cambridge University Press

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Audoly, B. & Pomeau, Y. 2010 Elasticity and Geometry: From Hair Curls to the Non-Linear Response of Shells. Oxford University Press.Google Scholar
Bayly, B. 1982 Geometry of subducted plates and island arcs viewed as a buckling problem. Geology 10, 629632.2.0.CO;2>CrossRefGoogle Scholar
Bessat, A., Duretz, T., Hetenyi, G., Pilet, S. & Schmalholz, S.M. 2020 Stress and deformation mechanisms at a subduction zone: insights from 2-D thermomechanical numerical modelling. Geophys. J. Intl 221, 16051625.CrossRefGoogle Scholar
Billen, M.I., Gurnis, M. & Simons, M. 2003 Multiscale dynamics of the Tonga–Kermadec subduction zones. Geophys. J. Intl 153, 359388.CrossRefGoogle Scholar
Bird, P. 2003 An updated digital model of plate boundaries. Geochem. Geophys. Geosyst. 4, 3.CrossRefGoogle Scholar
Buckingham, E. 1914 On physically similar systems; illustrations of the use of dimensional equations. Phys. Rev. 4, 345376.CrossRefGoogle Scholar
Capitanio, F.A. & Morra, G. 2012 The bending mechanics in a dynamic subduction system: constraints from numerical modelling and global compilation analysis. Tectonophysics 522–523, 224234.CrossRefGoogle Scholar
Coltice, N., Husson, L., Faccenna, C. & Arnould, M. 2019 What drives tectonic plates? Sci. Adv. 5, eaax4295.CrossRefGoogle ScholarPubMed
Crameri, F., Tackley, P.J., Meilick, I., Gerya, T.V. & Kaus, B.J.P. 2012 A free plate surface and weak oceanic crust produce single-sided subduction on Earth. Geophys. Res. Lett. 39, L03306.CrossRefGoogle Scholar
Frank, F.C. 1968 Curvature of island arcs. Nature 220, 363.CrossRefGoogle Scholar
Funiciello, F., Faccenna, C., Heuret, A., Lallemand, S., Di Giuseppe, E. & Becker, T.W. 2008 Trench migration, net rotation and slab-mantle coupling. Earth Planet. Sci. Lett. 271, 233240.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1991 Low Reynolds Number Hydrodynamics, 2nd edn. Kluwer Academic.Google Scholar
Houseman, G.A. & Gubbins, D. 1997 Deformation of subducted oceanic lithosphere. Geophys. J. Intl 131, 535551.CrossRefGoogle Scholar
Karato, S.-I. 2008 Deformation of Earth Materials. An Introduction to the Rheology of Solid Earth. Cambridge University Press.CrossRefGoogle Scholar
Kim, S. & Karrila, S.J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth- Heinemann.Google Scholar
Lallemand, S., Heuret, A. & Boutelier, D. 2005 On the relationships between slab dip, back-arc stress, upper plate absolute motion, and crustal nature in subduction zones. Geochem. Geophys. Geosyst. 6, 9.CrossRefGoogle Scholar
Laravie, J.A. 1975 Geometry and lateral strain of subducted plates in island arcs. Geology 3, 484486.2.0.CO;2>CrossRefGoogle Scholar
Li, Z. & Ribe, N.M. 2012 Dynamics of free subduction from 3-D boundary element modeling. J. Geophys. Res. 117, B06408.Google Scholar
Mahadevan, L., Bendick, R. & Liang, H. 2010 Why subduction zones are curved. Tectonics 29, TC6002.CrossRefGoogle Scholar
Manga, M. & Stone, H.A. 1993 Buoyancy-driven interaction between two deformable viscous drops. J. Fluid Mech. 256, 647683.CrossRefGoogle Scholar
McKenzie, D.P. 1969 Speculations on the consequences and causes of plate motions. Geophys. J. R. Astron. Soc. 18, 132.CrossRefGoogle Scholar
McKenzie, D.P. 1977 The initiation of trenches: a finite amplitude instability. In Island Arcs, Deep Sea Trenches and Back-Arc Basins (ed. M. Talwani & W.C. Pitman III), Maurice Ewing Series, vol. 1, pp. 57–61. American Geophysics Union.CrossRefGoogle Scholar
Morra, G., Regenauer-Lieb, K. & Giardini, D. 2006 Curvature of island arcs. Geology, 34, 877880.CrossRefGoogle Scholar
Morra, G., Chatelain, P., Tackley, P. & Koumoutsakos, P. 2009 Earth curvature effects on subduction morphology: modeling subduction in a spherical setting. Acta Geotech. 4, 95105.CrossRefGoogle Scholar
Morra, G., Quevedo, L. & Müller, R.D. 2012 Spherical dynamic models of top-down tectonics. Geochem. Geophys. Geosyst. 13, Q03005.CrossRefGoogle Scholar
Novozhilov, V.V. 1959 The Theory of Thin Shells. Noordhoff.Google Scholar
Padmavathi, B.S., Amaranath, T. & Palaniappan, D. 1995 Motion inside a liquid sphere: internal singularities. Fluid Dyn. Res. 15, 167176.Google Scholar
Parsons, B. & Sclater, J.G. 1977 An analysis of the variation of ocean floor bathymetry and heat flow with age. J. Geophys. Res. 82, 803827.CrossRefGoogle Scholar
Pozrikidis, C. 1990 The deformation of a liquid drop moving normal to a plane wall. J. Fluid Mech. 215, 331363.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Press, W.H., Teukolsky, S.A., Vetterling, W.T. & Flannery, B.P. 1992 Numerical Recipes in Fortran 77, 2nd edn. Cambridge University Press.Google Scholar
Rayleigh, L. 1945 The Theory of Sound, 2nd edn. Dover.Google Scholar
Ribe, N.M. 2010 Bending mechanics and mode selection in free subduction: a thin-sheet analysis. Geophys. J. Intl 180, 559576.CrossRefGoogle Scholar
Schettino, A. & Tassi, L. 2012 Trench curvature and deformation of the subducting lithosphere. Geophys. J. Intl 188, 1834.CrossRefGoogle Scholar
Schmeling, H., et al. 2008 A benchmark comparison of spontaneous subduction models—towards a free surface. Phys. Earth Planet. Inter. 171, 198223.CrossRefGoogle Scholar
Scholz, C.H. & Page, R. 1970 Buckling in island arcs. EOS (Am. Geophys. Union Trans.) 51, 429.Google Scholar
Tanimoto, T. 1997 Bending of a spherical lithosphere – axisymmetric case. Geophys. J. Intl 129, 305310.CrossRefGoogle Scholar
Tanimoto, T. 1998 State of stress within a bending spherical shell and its implications for subducting lithosphere. Geophys. J. Intl 134, 199206.CrossRefGoogle Scholar