From zonal flow to convection rolls in Rayleigh-B\'enard convection with free-slip plates

Rayleigh-B\'enard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimension (2D) RB convection and the other one three-dimension (3D) RB convection with a rotating axis parallel to the plate. We explore the parameter range of Rayleigh numbers Ra from $10^7 to $10^9$ and Prandtl numbers $Pr$ from $1$ to $100$. We show that zonal flow, which was observed, for example, by Goluskin \emph{et al}. \emph{J. Fluid. Mech.} 759, 360-385 (2014) for $\Gamma=2$, is only stable when $\Gamma$ is smaller than a critical value, which depends on $Ra$ and $Pr$. 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. For the 3D simulations, we fix $Ra=10^7$ and $Pr=0.71$, and compare the flow for $\Gamma=8$ and $\Gamma = 16$. We demonstrate that with increasing aspect ratio $\Gamma$, zonal flow, which was observed for small $\Gamma=2\pi$ by von Hardenberg \emph{et al}. \emph{Phys. Rev. Lett.} 15, 134501 (2015), 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 2D case.