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

The present numerical study describes the transient cooling process of a va riable viscosity fluid in the conductive lid regime. The quasi-static hypot hesis, assuming that each time-step can be considered as a steady-state hea t transfer, is further investigated. A first period of the cooling is essen tially transient. Two stages are observed: (1) onset time for convection an d (2) convective adjustment period. Onset of convection is seriously delaye d when compared to isoviscous thermal convection. The numerical experiments are well described by a boundary layer analysis that we propose, predictin g that the dimensionless convective onset time tau(o) can be scaled as a fu nction of a viscous temperature scale Delta T-v linked to the rheological c haracteristics of the thermal boundary layer [Davaille, A., Jaupart, C., 19 93, 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 Delta T across the fluid layer: tau(o) proportional to Ra-i(-2/3)(Delta T/Delta T-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 Delta tau(II) is linked to t he square of the conductive lid thickness delta: Delta tau(II) proportional to delta(2) Application to the thermal evolution of planetary interiors indicates that this initial transient period is shorter than a few hundreds of million yea rs, The subsequent cooling in the conductive lid regime is shown to be quasi-st atic, The conductive lid, however, is not equivalent to the elastic lithosp here, as thermal history models usually assume, Cooling numerical experimen ts described in the present paper are in good agreement with the scaling la w describing steady-state hear transfer below a conductive lid. (C) 2000 El sevier Science B.V. All rights reserved.