Bolgiano-Obukhov scaling in two-dimensional Rayleigh-Bénard convection at extreme Rayleigh numbers

Understanding the behavior of the kinetic energy spectrum and flux in two-dimensional (2D) turbulent thermal convection remains a challenge. In this paper, using high-resolution direct numerical simulation of Rayleigh-Bénard convection for Rayleigh numbers ${10}^{11}$–${10}^{14}$ and unit Prandtl number, we show that 2D turbulent convection exhibits Bolgiano-Obukhov scaling. At small wave numbers, where buoyancy feeds energy to the velocity field, kinetic energy exhibits inverse cascade. Consequently, the kinetic energy spectrum scales as $k^{-11/5}$ and the kinetic energy flux shows $k^{-4/5}$ scaling at small wave numbers. Buoyancy is weakened at large wave numbers, and this leads to a constant enstrophy cascade and $k^{-3}$ kinetic energy spectrum, similar to 2D hydrodynamic turbulence. However, the entropy spectrum exhibits a bispectrum with the upper branch varying as $k^{-2}$. We also observe constant entropy flux in the inertial range. Finally, we also draw a connection between the entropy flux in the dissipation range and the entropy dissipation rate in the bulk.