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

Abstract

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.