Scaling of Free Subduction on a Sphere

Desirous to understand the dynamics of gravity-driven subduction of oceanic lithosphere on Earth, we study a simple model of a thin axisymmetric shell of thickness h and viscosity η₁ sinking in a spherical body of fluid with radius R and a lower viscosity η₀. 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 = (η₁/η₀)(h/l_b)³ and the 'sphericity number' α = (l_b/R) cot(θ₀), where θ₀ is the angular radius of the subduction trench. To validate the scaling analysis, we obtain numerical solutions using the boundary-element method, based on new analytical point-force Green functions that satisfy free-slip boundary conditions on the sphere's surface. By comparing our solutions with those for the `flat-Earth' limit, we show that sphericity reduces the subduction rate by a factor (of up to ~ 4) that increases with increasing St and α. We shall close with a brief discussion of the implications for the dynamics of selected terrestrial subduction zones.