Lagrangian structures and stirring in the Earth’s mantle

Abstract

In this paper we investigate three Lagrangian methods that have been recently proposed to quantify mixing in chaotic and time-aperiodic geophysical flows. The analytical method proposed by Haller [e.g., G. Haller, Chaos 10 (2000) 99–108] has a strong mathematical foundation and seeks to determine the location of stable (i.e., most attracting) and unstable (i.e., most repelling) material lines. The main hyperbolic lines describe the spatial organization of chaotic mixing. The finite-size Lyapunov exponents estimate the local mixing properties of the flow using the finite dispersion of particles, while the finite-time Lyapunov exponents have been often used to locate dynamically distinguished regions in geophysical flows. Mantle stirring is induced by the repeated action of stretching and folding, thus requiring to follow the strain history of a fluid element along a trajectory. We calculate the trajectories of more than half a million passive tracers forward and backward in time. The Eulerian velocity field is computed using a finite element code for solid state convection. We focus on a thermochemical model with a chemically denser layer at the base of the Earth’s mantle. This case allows us to test the ability of the Lagrangian techniques to detect the location of a dynamical barrier that inhibits mass exchanges and delimits domains characterized by different efficiency of stirring. We find that the Lagrangian techniques provide a satisfactory description of the main structures governing stirring, and enlighten different and complementary aspects: the methods based on the Lyapunov exponents provide a clear picture of mantle domains characterized by different strength of stirring, while the method proposed by Haller identifies the skeleton of the main structures around which stirring is organized. Our paper builds toward a more rigorous analysis of the stirring processes in the Earth’s mantle, which is required to understand the existence of geochemical reservoirs under a dynamical prospective.