Thin-shell dynamics of subduction

During the past 15 years I and my colleagues have studied the dynamics of free (gravity-driven) subduction using a twofold approach: numerical simulations using the boundary-element method (BEM), and interpretation of the solutions using the theory of thin viscous shells. The basic model comprises a shell with thickness h and viscosity η1 subducting in a mantle with viscosity η2. The mantle has a finite depth H (in 3-D Cartesian geometry) or an outer radius R0 (in spherical geometry). The key length scale governing subduction is the 'bending length' lb, the sum of the slab length and the lateral extent of the  seaward flexural bulge. A dimensionless 'flexural stiffness' St = (η1/η2)(h/lb)3 determines whether the subduction rate is controlled by η1 or η2. 3-D BEM simulations closely reproduce laboratory experiments, and reveal the physical mechanisms underlying the different modes of subduction observed.   In spherical geometry, subduction is controlled by St and a 'dynamical sphericity number' Σ = (lb/R0) cotθt, where θt is the angular radius of the trench. Spherical BEM solutions demonstrate the 'sphericity paradox' that the effect of sphericity on flexure is greater for small (more nearly flat) plates than for large ones  (e.g. hemispherical). Another surprising result is that state of stress in a doubly-curved slab is dominated by the longitudinal normal ('hoop') stress. BEM predictions of hoop stresses in slabs with positive and negative Gaussian curvature agree well with earthquake focal mechanisms in the Mariana slab. Linear stability analysis shows that a slab under compressive hoop stress is unstable to longitudinal buckling, which may explain the peculiar geomery of the Mariana slab. Finally, I will describe a new hybrid boundary-integral/thin-shell approach to coupling mantle flow with the deformation of a thin shell having non-Newtonian rheology.