Transitions near the onset of stationary rotating magnetoconvection: role of magnetic Prandtl number

Abstract

We investigate the instabilities and associated bifurcation structure near the onset of rotating magnetoconvection of low Prandtl number fluids by performing three dimensional direct numerical simulations. Previous studies considered zero magnetic Prandtl number ($\mathrm{Pm}$) limit for the investigation of bifurcation structure near the onset of convection. Here we numerically investigate the effect of $\mathrm{Pm}$ on the bifurcation structure. The classical Rayleigh-B\'{e}nard convection setup in the presence of horizontal magnetic field and rotation about the vertical axis are considered for the study. The control parameters, including the Taylor number ($\mathrm{Ta}$), Chandrasekhar number ($\mathrm{Q}$), reduced Rayleigh number ($\mathrm{r}$), and magnetic Prandtl number ($\mathrm{Pm}$) are varied in the ranges $0 < \mathrm{Ta}\leq 500$, $0 < \mathrm{Q}\leq 1000$, $0.8\leq \mathrm{r} \leq 2$ and $0 < \mathrm{Pm} < 1$ by considering Prandtl numbers $\mathrm{Pr}= 0.025$ and $0.1$. The investigation reveals the presence of supercritical, subcritical and hybrid transitions to convection. These transitions leads to infinitesimal and finite amplitude fluid patterns at the onset of convection. The finite amplitude solutions can be both stationary and time dependent. The bifurcation structures associated with these flow patterns at the onset are studied in detail. For very small $\mathrm{Pm}$, the bifurcation structure is found to be qualitatively similar to the ones observed in the $\mathrm{Pm}\rightarrow 0$ limit. However, as $\mathrm{Pm}$ is increased, several new solutions appear at the onset and the resulting bifurcation structures are greatly modified.