Secondary Instabilities Developed in Upwellings of High Rayleigh Number Convection.

The purpose of this work is to show how these secondary instabilities can develop as a consequence of the interaction of the shear flow developed by the large-scale circulation and the rising plume. This will be conducted first within the framework of a constant viscosity fluid. First, we will employ a two-dimensional axisymmetric spherical-shell model within the framework of a Boussinesq fluid with an aspect-ratio of around six. Second, we have used a three-dimensional Boussinesq model with an aspect-ratio of 3x3x1. Two-dimensional simulations are carried out from 3*E⁷ to Ra=10¹⁰. Secondary plume bifurcation takes place at a Ra a little bit higher than 10⁷ in 2-D. As Ra moves above 10⁸, the tendency to plume branching increases and is accompanied by multiple foldings. In 3-D we have gone up to 5*E⁷ in a 5x5x1 box and did not find any signs of plume bending. Then at Ra=10⁸ plumes are found to be bent severely by the large scale circulation produced at this high Ra and have gone to Ra=10⁹. We may expect some sort of layered convection to take place in 3-D configuration for Ra between 3*E¹⁰ and 10¹¹. We have demonstrated here within the framework of a constant viscosity model that the secondary instabilities can develop in a self-consistent manner in both 2-D and 3-D large aspect-ratio convection with bifurcation Rayleigh numbers of O(10⁷⁾ and O(10⁸⁾ respectively.