Convective regimes of internally heated steady thermal convection of temperature-dependent viscous fluid

TL;DR

This study investigates steady thermal convection with temperature-dependent viscosity driven by internal heating, classifying regimes and analyzing stability, with implications for understanding complex convective behaviors.

Contribution

It provides a detailed classification of convective regimes with temperature-dependent viscosity and compares solutions for Frank-Kamenetskii and Arrhenius viscosities.

Findings

Identification of sluggish, mobile, and stagnant lid regimes.

Nusselt number scales as the 1/6 power of Rayleigh number in most regimes.

Steady solutions become unstable at high Rayleigh numbers due to plume growth.

Abstract

We study dynamical regimes of thermal convection with temperature-dependent viscosity driven by homogeneous internal heating. Two-dimensional steady-state convective solutions with the Frank-Kamenetskii viscosity are obtained by the Newton method for a number of different values of the Rayleigh number and the strength of the dependence of the viscosity on temperature. By classifying the solutions with the top surface mobility, we find the sluggish lid regime between the mobile and stagnant lid regimes. The solutions of the sluggish regime are characterized by a large viscosity contrast through the boundary layer below the conductive lid and a rapid increase of the Nusselt number with respect to the Rayleigh number. For most solutions in the mobile and stagnant lid regimes, the Nusselt number is proportional to the 1/6 power of the Rayleigh number, which can be derived by taking into account the effect of thin and strong downwelling plumes. Time evolution calculations show that the steady solutions become unstable for large Rayleigh numbers, where additional downward plumes grow on the background convective flows. This can be explained by the timescale of the Rayleigh--Taylor instability for the thermal boundary layer between the conductive lid and the convective core, which is shorter than the timescale of horizontal advection by the background flow. In addition, steady convective solutions with Arrhenius-law viscosity are calculated for several values of the parameters. The obtained regime diagram qualitatively agrees with that of the Frank-Kamenetskii viscosity, while there is a slight difference in the locations of the regime boundaries.