Multiple States in Turbulent Large-Aspect-Ratio Thermal Convection: What Determines the Number of Convection Rolls?

Wall-bounded turbulent flows can take different statistically stationary turbulent states, with different transport properties, even for the very same values of the control parameters. What state the system takes depends on the initial conditions. Here we analyze the multiple states in large-aspect ratio (Γ) two-dimensional turbulent Rayleigh-Bénard flow with no-slip plates and horizontally periodic boundary conditions as model system. We determine the number n of convection rolls, their mean aspect ratios Γₑ=Γ/n, and the corresponding transport properties of the flow (i. e. , the Nusselt number Nu), as function of the control parameters Rayleigh (Ra) and Prandtl number. The effective scaling exponent β in Nu∼Ra^β is found to depend on the realized state and thus Γₑ, with a larger value for the smaller Γₑ. By making use of a generalized Friedrichs inequality, we show that the elliptical shape of the rolls and viscous damping determine the Γₑ window for the realizable turbulent states. The theoretical results are in excellent agreement with our numerical finding 2/3≤Γₑ≤4/3, where the lower threshold is approached for the larger Ra. Finally, we show that the theoretical approach to frame Γₑ also works for free-slip boundary conditions.