Hoop Stresses in Free Subduction on a Sphere

Because Earth's tectonic plates are doubly curved shells, their mechanical behavior during subduction can differ significantly from that of flat plates. We use the boundary-element method to study free (gravity-driven) subduction in 3-D spherical geometry. The model comprises a shell with thickness h and viscosity η₁ subducting in a viscous planet with radius R₀. Our focus is on the magnitude of the longitudinal normal membrane stress (`hoop stress'), which has no analog in Cartesian geometry. Scaling analysis based on thin-shell theory shows that the resultant (integral across the shell) of the hoop stress obeys the scaling law T_φ ∼ (η₁h W/R₀) max(1, cotθ) where θ is the colatitude and W is the velocity of the shell normal to its midsurface that is associated with bending. We find that the state of stress in the slab is dominated by the hoop stress, which is 3-7 times larger than the downdip stress. Because the hoop stress is compressive, it can drive longitudinal buckling instabilities. We perform a linear stability analysis of a subducting spherical shell to determine a scaling law for the most unstable wavelength, which we compare with observed shapes of trenches in the Pacific ocean.