3D thermal convection with variable viscosity: can transient cooling be described by a quasi-static scaling law?

Abstract

The present numerical study describes the transient cooling process of a variable viscosity fluid in the conductive lid regime. The quasi-static hypothesis, assuming that each time-step can be considered as a steady-state heat transfer, is further investigated. A first period of the cooling is essentially transient. Two stages are observed: (1) onset time for convection and (2) convective adjustment period. Onset of convection is seriously delayed when compared to isoviscous thermal convection. The numerical experiments are well described by a boundary layer analysis that we propose, predicting that the dimensionless convective onset time τ o can be scaled as a function of a viscous temperature scale Δ T v linked to the rheological characteristics of the thermal boundary layer [Davaille, A., Jaupart, C., 1993. Transient high-Rayleigh-number thermal convection with large viscosity variations. J. Fluid Mech., 253, 141–166.], the internal Rayleigh number Ra i and the temperature drop Δ T across the fluid layer: τ o ∝Ra i −2/3 ΔT ΔT v 8/3 Due to the slow diffusive heat transfer of the first thermal instabilities through the conductive lid, a convective adjustment stage follows the onset of convection. It is shown that this duration Δ τ II is linked to the square of the conductive lid thickness δ: Δτ II ∝δ 2 Application to the thermal evolution of planetary interiors indicates that this initial transient period is shorter than a few hundreds of million years. The subsequent cooling in the conductive lid regime is shown to be quasi-static. The conductive lid, however, is not equivalent to the elastic lithosphere, as thermal history models usually assume. Cooling numerical experiments described in the present paper are in good agreement with the scaling law describing steady-state heat transfer below a conductive lid.