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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-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 $Γ=2$, is only stable when $Γ$ is smaller than a critical value, which depends on $Ra$ and $Pr$. With increasing $Γ$, we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger $Γ$, 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 $Γ=8$ and $Γ= 16$. We demonstrate that with increasing aspect ratio $Γ$, zonal flow, which was observed for small $Γ=2π$ by von Hardenberg \emph{et al}. \emph{Phys. Rev. Lett.} 15, 134501 (2015), completely disappears for $Γ=16$. For such large $Γ$ only convection roll states are statistically stable. In between, here for medium aspect ratio $Γ= 8$, 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.