Counter-gradient heat transport by large-scale circulation in two-dimensional turbulent convection

We investigate numerically the counter-gradient heat transport in two-dimensional turbulent Rayleigh-Bénard convection, which derives its existence from the descending hot plumes and ascending cold plumes with the large-scale circulation. Both qualitative and quantitative evidence is presented for the Rayleigh numbers from 104 to 1011 and the Prandtl number 1.0 in a domain of aspect ratio 2 with periodic boundary conditions in the horizontal direction. The counter-gradient heat transport increases with Rayleigh numbers, and becomes comparable to the gradient heat transport. The gradient heat transport has a power law dependence on the Rayleigh number with scaling exponent 1/3, while a phase transition of the counter-gradient heat transport can be identified around the critical Rayleigh number $2\times10^7$ . The Reynolds numbers of the gradient and counter-gradient heat transports exhibit identical power law dependence on the Rayleigh numbers with scaling exponent 1/2. The gradient heat transport is partially compensated by the counter-gradient heat transport with increasing Rayleigh numbers, which attenuates the effects of the large-scale circulation on the global heat transfer especially for high Rayleigh numbers. We emphasize that the counter-gradient heat transport also exists in the three-dimensional turbulent Rayleigh-Bénard convection, although the plume and flow structures therein are quite different