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

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Wang, Q., Chong, K. L., Stevens, R. J. A. M., Verzicco, R., & Lohse, D. (2020).
From zonal flow to convection rolls in Rayleigh–Bénard convection with free-slip plates.* Journal of
Fluid Mechanics,* *905*: A21. doi:10.1017/jfm.2020.793.

Cite as: http://hdl.handle.net/21.11116/0000-0007-6BB0-A

##### Abstract

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 Ra from 107 to 109 and Prandtl numbers Pr from 1 to 100 . The effect of the width-to-height aspect ratio Γ is investigated for 1⩽Γ⩽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 Γ=2 , is only stable when Γ 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 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. 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=108 and Pr=10 , the aspect ratio Γr 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 Γr=64 , or more generally already with Γr≫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=107 and Pr=0.71 , and compare the flow for Γ=8 and Γ=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 Γ , zonal flow, which was observed for small Γ=2π by von Hardenberg et al. (Phys. Rev. Lett., vol. 15, 2015, 134501), 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 2-D case.