Multiple states in turbulent large-aspect ratio thermal convection: What determines the number of convection rolls?

Recent findings suggest that wall-bounded turbulent flow 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 ($\Gamma$) two-dimensional 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 $Nu \sim Ra^\beta$ 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 instability and viscous damping determine the $\Gamma_r$-window for the realizable turbulent states. The theoretical results are in excellent agreement with our numerical finding $2/3 \le \Gamma_r \le 4/3$, where the lower threshold is approached for the larger $Ra$. Finally, we show that the theoretical approach to frame $\Gamma_r$ also works for free-slip boundary conditions.