Numerical simulations of three-dimensional thermal convection in a fluid with strongly temperature-dependent viscosity

Abstract

Numerical calculations are presented for the steady three-dimensional structure of thermal convection of a fluid with strongly temperature-dependent viscosity in a bottom-heated rectangular box. Viscosity is assumed to depend on temperature T as exp (− ET), where E is a constant; viscosity variations across the box r (= exp (E)) as large as 105 are considered. A stagnant layer or lid of highly viscous fluid develops in the uppermost coldest part of the top cold thermal boundary layer when r > rc1, where r = rc1 ≡ 1.18 × 103Rt0.309 and Rt is the Rayleigh number based on the viscosity at the top boundary. Three-dimensional convection occurs in a rectangular pattern beneath this stagnant lid. The planform consists of hot upwelling plumes at or near the centre of a rectangle, sheets of cold sinking fluid on the four sides, and cold sinking plume concentrations immersed in the sheets. A stagnant lid does not develop, i.e. convection involves all of the fluid in the box when r < rc1. The whole-layer mode of convection occurs in a three-dimensional bimodal pattern when r > rc2 = 3.84 × 106Rt−1.35. The planform of the convection is rectangular with the coldest parts of the sinking fluid and the hottest part of the upwelling fluid occurring as plumes at the four corners and at the centre of the rectangle, respectively. Both hot uprising plumes and cold sinking plumes have sheet-like extensions, which become more well-developed as r increases. The whole-layer mode of convection occurs as two-dimensional rolls when r < min (rc1, rc2). The Nusselt number Nu depends on the viscosity at the top surface more strongly in the regime of whole-layer convection than in the regime of stagnant-lid convection. In the whole-layer convective regime, Nu depends more strongly on the viscosity at the top surface than on the viscosity at the bottom boundary.