Counter-gradient heat transport in two-dimensional turbulent Rayleigh-B\'{e}nard convection

We present high-resolution numerical investigations of heat transport by two-dimensional (2D) turbulent Rayleigh-B\'{e}nard (RB) convection over the Rayleigh number range $10^8 \leqslant Ra\leqslant 10^{10}$ and the Prandtl number range $0.7\leqslant Pr \leqslant10$. We find that there exist strong counter-gradient local heat flux with magnitude much larger than the global Nusselt number $Nu$ of the system. Two mechanisms for generating counter-gradient heat transport are identified: one is due to the bulk dynamics and the other is due to the competitions between the corner-flow rolls and the large-scale circulation (LSC). While the magnitude of the former is found to increase with increasing Prandtl number, that of the latter maximizes at medium $Pr$. We further reveal that the corner-LSC competitions lead to the anomalous $Nu$-$Pr$ relation in 2D RB convection, i.e. $Nu(Pr)$ minimizes, rather than maximizes as in three-dimensional cylindrical case, at $Pr\approx2\sim3$ for moderate $Ra$.