From zonal flow to convection rolls in Rayleigh–Bénard convection with free-slip plates

Rayleigh-Benard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimensional (2-D) RB convection and the other one three-dimensional (3-D) RB convection with a rotating axis parallel to the plate, which for strong rotation mimics 2-D RB convection. For the 2-D simulations, we explore the parameter range of Rayleigh numbers Ra from 10(7) to 10(9) and Prandtl numbers Pr from 1 to 100. The effect of the width-to-height aspect ratio Gamma is investigated for 1 <= Gamma <= 128. We show that zonal flow, which was observed, for example, by Goluskin et al. (J. Fluid. Mech., vol. 759, 2014, pp. 360-385) for Gamma = 2, is only stable when Gamma is smaller than a critical value, which depends on Ra and Pr. The regime in which only zonal flow can exist is called the first regime in this study. With increasing Gamma, we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger Gamma, in a third regime, only convection roll states are statistically stable and zonal flow is not sustained. How many convection rolls form (or in other words, what the mean aspect ratio of an individual roll is), depends on the initial conditions and on Ra and Pr. For instance, for Ra = 10(8) and Pr = 10, the aspect ratio Gamma(r) of an individual, statistically stable convection roll can vary in a large range between 16/11 and 64. A convection roll with a large aspect ratio of Gamma(r) = 64, or more generally already with Gamma(r) >> 10, can be seen as 'localized' zonal flow, and indeed carries over various properties of the global zonal flow. For the 3-D simulations, we fix Ra = 10(7) and Pr = 0.71, and compare the flow for Gamma = 8 and Gamma = 16. We first show that with increasing rotation rate both the flow structures and global quantities like the Nusselt number Nu and the Reynolds number Re increasingly behave like in the 2-D case. We then demonstrate that with increasing aspect ratio Gamma, zonal flow, which was observed for small Gamma = 2 pi by von Hardenberg et al. (Phys. Rev. Lett., vol. 15, 2015, 134501), completely disappears for Gamma = 16. For such large Gamma, only convection roll states are statistically stable. In-between, here for medium aspect ratio Gamma = 8, the convection roll state and the zonal flow state are both statistically stable. What state is taken depends on the initial conditions, similarly as we found for the 2-D case.