Mantle circulation models with variational data assimilation: inferring past mantle flow and structure from plate motion histories and seismic tomography

Abstract

S U M M A R YMantle convection models require an initial condition some time in the past. Because this initial condition is unknown for Earth, we cannot infer the geological evolution of mantle flow from forward mantle convection calculations even for the most recent Mesozoic and Cenozoic geological history of our planet. Here we introduce a fluid dynamic inverse problem to constrain unknown mantle flow back in time from seismic tomographic observations of the mantle and reconstructions of past plate motions using variational data assimilation. We derive the generalized inverse of mantle convection and explore the initial condition problem in high-resolution, 3-D spherical mantle circulation models for a time period of 100 Myr, roughly comparable to half a mantle overturn. We present a synthetic modelling experiment to demonstrate that mid-Cretaceous mantle structure can be inferred accurately from fluid dynamic inverse modelling, assuming present-day mantle structure is well-known, even if an initial first guess assumption about the mid-Cretaceous mantle involved only a simple 1-D radial temperature profile. We also demonstrate that convecting present-day mantle structure back in time by reversing the time-stepping of the energy equation is insufficient to model the mantle structure of the past. The difficulty arises, because such backward convection calculations ignore thermal diffusion effects, and therefore cannot account for the generation of thermal buoyancy in boundary layers as we go back in time. Inverse mantle convection modelling should make it possible to infer a number of flow parameters from observational constraints of the mantle.

Profiling the robustness, efficiency and limits of the forward-adjoint method for 3-D mantle convection modelling

“…It has also been shown that this technique can be extended to address the problem of having an incomplete picture of Earth's historical mantle structure (Bunge et al 2003).…”

“…The adjoint method for mantle circulation models (MCMs) has previously been utilised using both synthetic (Bunge et al 2003; and tomography derived (Horbach et al 2014) true state forward-adjoint mantle models. Further studies using a simplified adjoint calculation have also been conducted (Ismail-Zadeh et al 2004;Spasojevic et al 2009), which use a subset of the full adjoint equations.…”

Ensemble Kalman filter for the reconstruction of the Earth's mantle circulation

“…by Bunge et al (2003), would take into account model errors, numerical approximations, errors on plate reconstructions and on the estimation of the current tomography-derived temperature field to provide the best fit given all sources of information.…”

“…So far, variational data assimilation dominates over other methods to estimate 25 mantle circulation (Bunge et al, 2003;Horbach et al, 2014;Ghelichkhan and Bunge, 2016). To simplify the problem, they minimize the misfit between the final temperature field of the mantle circulation model and the one deduced from seismic tomography.…”

“…On the contrary, the alternative method, variational data assimilation, requires the development of an adjoint code that needs further development for each additional complexity added to the forward model (see Kalnay et al, 2007, for a comparison of EnKF and 4D variational methods). For the mantle circulation 25 problem, this results in a series of derivation of the adjoint considering different approximations (Ismail-Zadeh et al, 2003;Bunge et al, 2003;Ghelichkhan and Bunge, 2016;Worthen et al, 2014). The ability of a data assimilation scheme to adapt to different numerical codes is a particularly important issue for mantle convection since models are in constant evolution, with current developments including the implementation of chemistry, nonlinear rheologies, elasticity, phase transition and compressibility (see e.g.…”