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

Rayleigh-Bénard (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 from to and Prandtl numbers from to. The effect of the width-to-height aspect ratio is investigated for. We show that zonal flow, which was observed, for example, by Goluskin et al. (J. Fluid. Mech., vol. 759, 2014, pp. 360-385) for, is only stable when is smaller than a critical value, which depends on and. The regime in which only zonal flow can exist is called the first regime in this study. With increasing, we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger, 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 and. For instance, for and, the aspect ratio of an individual, statistically stable convection roll can vary in a large range between and. A convection roll with a large aspect ratio of, or more generally already with, 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 and, and compare the flow for and. We first show that with increasing rotation rate both the flow structures and global quantities like the Nusselt number and the Reynolds number increasingly behave like in the 2-D case. We then demonstrate that with increasing aspect ratio, zonal flow, which was observed for small by von Hardenberg et al. (Phys. Rev. Lett., vol. 15, 2015, 134501), completely disappears for. For such large, only convection roll states are statistically stable. In-between, here for medium aspect ratio, 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.