A model for Rayleigh-B\'enard magnetoconvection

TL;DR

This paper develops and analyzes a model for three-dimensional Rayleigh-Bénard magnetoconvection in low-Prandtl-number fluids with rigid boundaries under a vertical magnetic field, exploring stability, bifurcations, and flow structures near the onset.

Contribution

The paper introduces a detailed model for magnetoconvection near onset, analyzing flow structures, bifurcations, and the effects of magnetic field and Prandtl number on stability.

Findings

Straight rolls become unstable and bifurcate into 3D structures.

Flow patterns transition from steady to periodic and quasiperiodic states.

Critical Rayleigh number and frequency depend on Prandtl number and magnetic field.

Abstract

A model for three-dimensional Rayleigh-B\'{e}nard convection in low-Prandtl-number fluids near onset with rigid horizontal boundaries in the presence of a uniform vertical magnetic field is constructed and analyzed in detail. The kinetic energy $K$, the convective entropy $\Phi$ and the convective heat flux ($Nu-1$) show scaling behaviour with $\epsilon = r-1$ near onset of convection, where $r$ is the reduced Rayleigh number. The model is also used to investigate various magneto-convective structures close to the onset. Straight rolls, which appear at the primary instability, become unstable with increase in $r$ and bifurcate to three-dimensional structures. The straight rolls become periodically varying wavy rolls or quasiperiodically varying structures in time with increase in $r$ depending on the values of Prandtl number $Pr$. They become irregular in time, with increase in $r$. These standing wave solutions bifurcate first to periodic and then quasiperiodic traveling wave solutions, as $r$ is raised further. The variations of the critical Rayleigh number $Ra_{os}$ and the frequency $\omega_{os}$ at the onset of the secondary instability with $Pr$ are also studied for different values of Chandrasekhar's number $Q$.