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 (Gamma) twodimensional turbulent Rayleigh-B ' enard 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 Gamma(r) = Gamma/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 beta in Nu similar to Ra-beta is found to depend on the realized state and thus Gamma(r), with a larger value for the smaller Gamma(r). By making use of a generalized Friedrichs inequality, we show that the elliptical shape of the rolls and viscous damping determine the Gamma(r) window for the realizable turbulent states. The theoretical results are in excellent agreement with our numerical finding 2/3 <= Gamma(r) <= 4/3, where the lower threshold is approached for the larger Ra. Finally, we show that the theoretical approach to frame Gr also works for free-slip boundary conditions.