The onset of zonal modes in two-dimensional Rayleigh-B\'enard convection

TL;DR

This paper investigates the stability of steady convection rolls in 2D Rayleigh-Bénard convection across a wide range of Prandtl and Rayleigh numbers, revealing complex stability boundaries and the emergence of zonal modes.

Contribution

The study provides a detailed stability analysis of convection rolls in 2D Rayleigh-Bénard convection over twelve orders of magnitude in Prandtl number, identifying conditions for zonal mode emergence and stability transitions.

Findings

Zonal modes emerge only after the steady convection roll state becomes unstable to odd perturbations.

The stability boundary exhibits intricate features and multiple stability loss and regain regions.

In the low-Prandtl limit, convection rolls become unstable almost immediately, leading to nonlinear oscillations.

Abstract

We study the stability of steady convection rolls in 2D Rayleigh--B\'enard convection with free-slip boundaries and horizontal periodicity over twelve orders of magnitude in the Prandtl number $(10^{-6} \leq Pr \leq 10^6)$ and five orders of magnitude in the Rayleigh number $(8\pi^4 < Ra \leq 3 \times 10^7)$. The analysis is facilitated by partitioning our modal expansion into so-called even and odd modes. With aspect ratio $\Gamma = 2$, we observe that zonal modes (with horizontal wavenumber equal to zero) can emerge only once the steady convection roll state consisting of even modes only becomes unstable to odd perturbations. We determine the stability boundary in the $(Pr,Ra)$-plane and observe remarkably intricate features corresponding to qualitative changes in the solution, as well as three regions where the steady convection rolls lose and subsequently regain stability as the Rayleigh number is increased. We study the asymptotic limit $\Pr \to 0$ and find that the steady convection rolls become unstable almost instantaneously, eventually leading to non-linear relaxation osculations and bursts, which we can explain with a weakly non-linear analysis. In the complementary large-$\Pr$ limit, we observe that the stability boundary reaches an asymptotic value $Ra = 2.54 \times 10^7$ and that the zonal modes at the instability switch off abruptly at a large, but finite, Prandtl number.