Zero-Prandtl-number convection with slow rotation

TL;DR

This study investigates the onset and pattern formation in zero-Prandtl-number rotating Rayleigh-Bénard convection, revealing dependence on geometric and rotational parameters, with diverse flow patterns including wavy rolls, KL-type patterns, and bursting phenomena.

Contribution

The paper provides detailed numerical analysis of primary instability and flow patterns in zero-Prandtl-number convection with rotation, highlighting the effects of aspect ratio and Taylor number, and introduces low-dimensional models for bifurcation analysis.

Findings

Wavy rolls observed at primary instability for certain aspect ratios and low Taylor numbers.

Küppers-Lortz patterns appear at specific aspect ratios and higher Taylor numbers.

Bursting and chaotic patterns occur, involving homoclinic bifurcation phenomena.

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.