  Thermal history calculations provide important insights into the interior
evolution of planets, but incorporate simplified dynamics from the systems they
represent. Planetary interiors typical incorporate complex rheologies, viscous
layering, lateral heterogeneities, and time delays in processes, which have not
been traditionally represented by parameterised approaches. Here we develop
numerical models for planetary evolution, incorporating the physical complexity
of Earth's interior, and use them to generate statistically-based Nu-Ra
scalings. These encapsulate the main effects of tectonic transitions, geometry,
and depth-dependent rheology, and time-sensitivity. We find an exponent $\beta$
of ~0.26 best describes the Nu-Ra relationship for evolving mobile lid systems,
and $\beta$ ~0.12 for stagnant-lid systems. Systems with time dependent
subduction have $\beta$ varying between ~0.26 during the Hadean, when external
factors such as impacts facilitate tectonics, to ~0.12 during the Archaean,
when the system is dominated by long periods of quiescence, and systems driven
by external forcings (eg. due to impacts in the first 100Myr of Earth's
history) may exhibit much higher exponents. We also find a time-lag between Ra
(which primarily depends on mantle temperature) and Nu (normalised surface heat
flow) of around 200-300Myr, suggesting a significant delay between mantle
thermal configuration, and its surface manifestation. These results provide an
approach for the rapid characterisation of tectonic, volcanic, and atmospheric
evolution of terrestrial exoplanets.
