Connecting flow structures and heat flux in turbulent Rayleigh-B\'enard convection

The aspect ratio ($\Gamma$) dependence of the heat transfer (Nusselt number $Nu$ in dimensionless form) in turbulent (two- dimensional) Rayleigh-B\'enard convection is numerically studied in the regime $0.4 \le \Gamma \le 1.25$ for Rayleigh numbers $10^7 \le Ra \le Ra^{9}$ and Prandtl numbers $Pr =0.7 $ (gas) and $4.3$ (water). $Nu (\Gamma )$ shows a very rich structure with sudden jumps and sharp transitions. We connect these structures to the way the flow organizes itself in the sample and explain why the aspect ratio dependence of $Nu$ is more pronounced for small $Pr$. Even for fixed $\Gamma$ different turbulent states (with different resulting $Nu$) can exist, between which the flow can or cannot switch. In the latter case the heat transfer thus depends on the initial conditions.