Supercritical and Subcritical Convection: Insights from Direct Numerical Simulation and Low-Dimensional Models in Rotating Systems*

We explored the transition scenarios in rotating Rayleigh-B\'{e}nard convection with no-slip boundary conditions through 3D direct numerical simulations (DNS) and low-dimensional modeling. The governing parameters: Taylor number ($\mathrm{Ta}$), Rayleigh number ($\mathrm{Ra}$), and Prandtl number ($\mathrm{Pr}$) are varied within 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 manifests as stationary cellular patterns. For $\mathrm{Pr} < 0.31$, the DNS results indicate that the onset of convection can be supercritical or subcritical, depending on whether $\mathrm{Ta} > \mathrm{Ta_c}(\mathrm{Pr})$ or $\mathrm{Ta} < \mathrm{Ta_c}(\mathrm{Pr})$, where $\mathrm{Ta_c}(\mathrm{Pr})$ is a $\mathrm{Pr}$ dependent threshold of $\mathrm{Ta}$. Conversely, for $\mathrm{Pr} \geq 0.31$, only supercritical onset of convection is observed. At the subcritical onset, both finite amplitude stationary and time-dependent solutions emerge, which are explained using a low-dimensional model. Furthermore, this low dimensional model is reduced to a one dimensional form, which also qualitatively captures the stationary scenario. The DNS results also show that as Ra increases beyond the onset of convection, the system becomes time-dependent, and depending on Pr, both standing and traveling wave solutions are observed. Notably, for very small $\mathrm{Pr} (\leq 0.045)$, finite amplitude time-dependent solutions appear at the onset for higher $\mathrm{Ta}$.