Modelling 4.5 billion years of Earth’s thermal evolution: Insights from core-mantle coupling, lithospheric viscosity, grain-size-dependent rheology, and surface boundary conditions

Abstract

We investigate Earth’s evolution through thermally coupled core-mantle models spanning 4.5 billion years. These models employ a spherical pseudo-spectral approach to solve the conservation equations for mass, momentum, and energy within a compressible, self-gravitating mantle. The methodology incorporates time-dependent treatments for core-mantle coupling, dislocation and diffusion creep mechanisms, internal heating, and thermal conductivity. Using 3-D numerical simulations, we evaluate the sensitivity of mantle cooling, viscosity structure, and inner-core growth to variations in lithospheric viscosity, diffusion viscosity, mechanical surface boundary conditions, and initial core-mantle boundary and core liquidus temperatures. Results underscore the central role of lithospheric viscosity, particularly near an effective value of ∼ 10²² Pa s, in producing a mantle cooling pattern consistent with petrological constraints, characterized by net warming prior to ∼ 3 billion years ago (Ga) followed by long-term cooling, as predicted by low-Urey-ratio thermal evolution models. Notably, one model with lithospheric viscosity allowed to vary between 10¹⁸ and 10²⁴ Pa s exhibits nonlinear rheological feedbacks that trigger an early-stage thermal rebound. This behavior results from a relatively abrupt increase in lithospheric viscosity which redirects the mantle onto a sustained warming trajectory that departs from the expected monotonic cooling. This example also demonstrates how nonlinear parameter interactions can produce non-monotonic thermal evolution. However, lithospheric viscosity alone cannot fully account for present-day observations of heat flux, inner-core radius, and depth-dependent viscosity profiles. We find that varying the activation enthalpy ratio for grain-growth-controlled diffusion viscosity modifies the radial viscosity structure while leaving the overall cooling pattern intact. Furthermore, surface boundary conditions permitting viscous coupling between rigid surface plates and underlying mantle flow — specifically in our plate-like (PL) model — yield the most acceptable mantle cooling rates and dynamic evolution. This PL configuration also facilitates more realistic coupling between surface kinematics and internal convection, allowing plate velocities to emerge from the flow dynamics rather than being imposed. The PL model exhibits patterns which are similar to independently estimated present-day mantle viscosity profiles, including features such as the lithosphere-asthenosphere gradient and the viscosity jump at the 660 km discontinuity. The PL model also exhibits persistent large-scale lateral temperature anomalies, consistent with previous billion-year convection studies, and illustrates how plate-like surface coupling promotes the emergence and maintenance of hemispheric-scale heterogeneity. Findings confirm that initial core-mantle boundary temperature and liquidus temperature at the inner-core boundary significantly influence inner-core growth rates. To isolate the effects of the initial thermal state of the core, we adopt a simplified initialization in which the liquidus temperature is set equal to the core-mantle boundary (CMB) temperature at time zero — this condition is imposed only at initialization. For the PL model, initializing both the CMB temperature and the inner-core liquidus temperature at 5600 K optimizes predictions of present-day inner-core radius and suggests an inner-core onset around 2.0 ‒ 1.5 Ga, aligning with previous independent estimates. This study emphasizes that a robust modeling of Earth’s core-mantle thermal and dynamic history requires careful calibration of lithospheric viscosity and grain-size-sensitive mantle viscosity, surface boundary dynamics, and initial temperatures at the CMB and core liquidus. All model predictions are empirically anchored and evaluated against a wide thermal evolution. By enabling a broad exploration of the parameter space — incorporating complex rheology and a viscosity range spanning 12 orders of magnitude — these tools facilitate sensitivity analyses and the refinement of parameter constraints. However, the complexity of the coupled core-mantle system highlights the need for continued refinement of computational techniques to fully capture planetary evolution. Furthermore, while primarily designed for Earth, the methodology can be adapted — with appropriate modifications — to study telluric planets in our solar system and Earth-like exoplanets, advancing our understanding of planetary evolution across diverse contexts.

Keywords

Mantle convection modeling / Inner-core growth / Lithospheric viscosity / Mantle viscosity / Grain-size-dependent rheology / Thermal evolution of Earth / Mechanical surface boundary conditions

CRediT authorship contribution statement

Petar Glišović: Writing - review & editing, Writing - original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Alexander Braun: Writing - review & editing, Supervision, Resources, Project administration, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research was supported through the Stephen Cheeseman Geoselenic Research Project at Queen’s University at Kingston, ON, Canada. PG acknowledges support for this work provided by Institut de Physique du Globe de Paris (IPGP), France and Research Centre in Earth System Dynamics (GEOTOP), Univeristé du Québec à Montréal. Advanced computing resources were provided by the Digital Research Alliance of Canada (Alliance), the organization responsible for digital research infrastructure in Canada, and the Centre for Advanced Computing at Queen’s University. AB and PG jointly designed the study. PG developed the scientific concepts and modelling techniques. PG created the code, performed the modelilng and the analysis of results. PG and AB jointly wrote the manuscript.

Data Availability

All model output data, and initial temperature fields used to generate the results presented in this study are publicly available at https://doi.org/10.5281/zenodo.17070050.

Code Availabil ity

The source code is written in C++ and requires compilation on a high-performance computing (HPC) environment with specific architecture and dependencies. Due to this limitation, a precom-piled executable cannot be provided. However, the source code may be shared under a specific agreement upon reasonable request to the corresponding author, together with guidance on the compi-lation and runtime environment required to reproduce the simulations.

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