Extreme multiplicity in cylindrical Rayleigh-Benard convection: II. Bifurcation diagram and symmetry classification

TL;DR

This paper presents a detailed bifurcation analysis of Rayleigh-Benard convection in a cylinder, identifying multiple steady states, their stability, and symmetry classifications using advanced numerical methods.

Contribution

It introduces a comprehensive bifurcation diagram for cylindrical convection, classifies steady states by symmetry, and traces their origins from the conductive state.

Findings

Identified 17 branches of steady states with various symmetries.

Mapped bifurcation connections including pitchfork bifurcations.

Determined the stability and symmetry classification of convection patterns.

Abstract

A large number of flows with distinctive patterns have been observed in experiments and simulations of Rayleigh-Benard convection in a water-filled cylinder whose radius is twice the height. We have adapted a time-dependent pseudospectral code, first, to carry out Newton's method and branch continuation and, second, to carry out the exponential power method and Arnoldi iteration to calculate leading eigenpairs and determine the stability of the steady states. The resulting bifurcation diagram represents a compromise between the tendency in the bulk towards parallel rolls, and the requirement imposed by the boundary conditions that primary bifurcations be towards states whose azimuthal dependence is trigonometric. The diagram contains 17 branches of stable and unstable steady states. These can be classified geometrically as roll states containing two, three, and four rolls; axisymmetric patterns with one or two tori; three-fold symmetric patterns called mercedes, mitubishi, marigold and cloverleaf; trigonometric patterns called dipole and pizza; and less symmetric patterns called CO and asymmetric three-rolls. The convective branches are connected to the conductive state and to each other by 16 primary and secondary pitchfork bifurcations and turning points. In order to better understand this complicated bifurcation diagram, we have partitioned it according to azimuthal symmetry. We have been able to determine the bifurcation-theoretic origin from the conductive state of all the branches observed at high Rayleigh number.