Effects of a small magnetic field on homoclinic bifurcations in a low-Prandtl-number fluid

TL;DR

This study investigates how small magnetic fields influence homoclinic bifurcations in low-Prandtl-number Rayleigh-Bénard convection, revealing different bifurcation behaviors depending on magnetic field orientation and strength through simulations and models.

Contribution

It provides new insights into the effects of magnetic fields on bifurcation phenomena in low-Prandtl-number fluids, supported by direct numerical simulations and low-dimensional models.

Findings

Weak vertical magnetic fields induce both forward and backward homoclinic bifurcations.

Stronger magnetic fields lead to stationary flow patterns at secondary instability.

Horizontal magnetic fields break symmetry and do not produce homoclinic bifurcations.

Abstract

Effects of a uniform magnetic field on homoclinic bifurcations in Rayleigh-B\'{e}nard convection in a fluid of Prandtl number $Pr = 0.01$ are investigated using direct numerical simulations (DNS). A uniform magnetic field is applied either in the vertical or in the horizontal direction. For a weak vertical magnetic field, the possibilities of both forward and backward homoclinic bifurcations are observed leading to a spontaneous merging of two limit cycles into one as well as a spontaneous breaking of a limit cycle into two for lower values of the Chandrasekhar's number ($Q\leq 5$). A slightly stronger magnetic field makes the convective flow time independent giving the possibility of stationary patterns at the secondary instability. For horizontal magnetic field, the $x\leftrightharpoons y$ symmetry is destroyed and neither a homoclinic gluing nor a homoclinic breaking is observed. Two low-dimensional models are also constructed: one for a weak vertical magnetic field and another for a weak horizontal magnetic field. The models qualitatively capture the features observed in DNS and help understanding the unfolding of bifurcations close to the onset of magnetoconvection.