Zero-Prandtl-number convection with slow rotation

Abstract: We present the results of our investigations of the primary instability and the flow patterns near onset in zero-Prandtl-number Rayleigh-B\'enard convection with uniform rotation about a vertical axis. The investigations are carried out using direct numerical simulations of the hydrodynamic equations with stress-free horizontal boundaries in rectangular boxes of size $(2\pi/k_x) \times (2\pi/k_y) \times 1$ for different values of the ratio $\eta = k_x/k_y$. The primary instability is found to depend on $\eta$ and $Ta$. Wavy rolls are observed at the primary instability for smaller values of $\eta$ ($1/\sqrt{3} \le \eta \le 2$ except at $\eta = 1$) and for smaller values of $Ta$. We observed K\"{u}ppers-Lortz (KL) type patterns at the primary instability for $\eta = 1/\sqrt{3}$ and $ Ta \ge 40$. The fluid patterns are found to exhibit the phenomenon of bursting, as observed in experiments [Bajaj et al. Phys. Rev. E {\bf 65}, 056309 (2002)]. Periodic wavy rolls are observed at onset for smaller values of $Ta$, while KL-type patterns are observed for $ Ta \ge 100$ for $\eta =\sqrt{3}$. In case of $\eta = 2$, wavy rolls are observed for smaller values of $Ta$ and KL-type patterns are observed for $25 \le Ta \le 575$. Quasi-periodically varying patterns are observed in the oscillatory regime ($Ta > 575$). The behavior is quite different at $\eta = 1$. A time dependent competition between two sets of mutually perpendicular rolls is observed at onset for all values of $Ta$ in this case. Fluid patterns are found to burst periodically as well as chaotically in time. It involved a homoclinic bifurcation. We have also made a couple of low-dimensional models to investigate bifurcations for $\eta = 1$, which is used to investigate the sequence of bifurcations.