Supercritical and subcritical rotating convection in a horizontally periodic box with no-slip walls at the top and bottom

Abstract

The study of instabilities in the convection of rotating fluids is one of the classical topics of research. However, in spite of more than five decades of research, the instabilities and related transition scenario near the onset of rotating convection of low Prandtl number fluids is not well understood. Here we investigate the transition scenario in rotating Rayleigh-B\'{e}nard convection with no slip boundary conditions by performing 3D direct numerical simulations (DNS) and low dimensional modeling. The governing parameters, namely, the Taylor number ($\mathrm{Ta}$), Rayleigh number ($\mathrm{Ra}$) and Prandtl number ($\mathrm{Pr}$) are varied in the ranges $0< \mathrm{Ta}\leq 8\times 10^3$, $0 <\mathrm{Ra} < 1\times 10^4$ and $0 <\mathrm{Pr} \leq 0.35$, where convection appears as stationary cellular pattern. In DNS, for $\mathrm{Pr} < 0.31$, supercritical or subcritical onset of convection appears, according as $\mathrm{Ta} > \mathrm{Ta_c}(\mathrm{Pr})$ or $\mathrm{Ta} < \mathrm{Ta_c}(\mathrm{Pr})$, where $\mathrm{Ta_c}(\mathrm{Pr})$ is a $\mathrm{Pr}$ dependent threshold of $\mathrm{Ta}$. On the other hand, only supercritical onset of convection is observed for $\mathrm{Pr}\geq 0.31$. At the subcritical onset, both finite amplitude stationary and time dependent solutions are manifested. The origin of these solutions are explained using a low dimensional model. DNS show that as $\mathrm{Ra}$ is increased beyond the onset of convection, the system becomes time dependent and depending on $\mathrm{Pr}$, standing and traveling wave solutions are observed. For very small $\mathrm{Pr}$ ($\leq 0.045$), interestingly, finite amplitude time dependent solutions are manifested at the onset for higher $\mathrm{Ta}$.