Fluid mechanics of free subduction on a sphere, 1: The axisymmetric case

TL;DR

This paper investigates how spherical geometry affects the dynamics of subducting oceanic plates on Earth, revealing that sphericity modestly reduces sinking speed but significantly increases hoop stress, influencing buckling instability growth.

Contribution

The study introduces a scaling framework and numerical methods to quantify the impact of Earth's sphericity on subduction mechanics, highlighting its effect on stress and stability.

Findings

Sphericity reduces sinking speed by up to 7% for large plates.

Sphericity increases hoop stress by up to 64% for large plates.

Sphericity effects are more pronounced on stress than on sinking velocity.

Abstract

To understand how spherical geometry influences the dynamics of gravity-driven subduction of 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' $\Sigma= (l_b/R_0)\cot\theta_t$, 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 $\Sigma = 0$). 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.