Critical wave speed governs heat transfer in buoyancy-driven turbulence under hybrid spatiotemporal modulation

Thermal forcing in natural environments, such as Earth’s surface, exhibits complex spatiotemporal variations due to daily and seasonal cycles. This motivates our study of Rayleigh–Bénard convection with hybrid spatiotemporal modulation at the thermal boundary, achieved by applying a travelling thermal wave to a bottom plate with modulated wavenumber $k$ and frequency $f$ . At low frequencies, spatial modulation dominates, organising coherent thermal plumes. At high frequencies, the rapid propagation of the thermal wave smooths out the plumes, thereby reducing convective efficiency. We find that the emergence of the ‘smoothing’ effect is governed by the ratio between the wave speed ( $c = f/k$ ) and the pseudo-speed of thermal diffusion, $c_{\textit{diff}} = 4\pi k/\sqrt {\textit{RaPr}}$ , a scale-dependent measure of thermal damping. By comparing these speeds, we identify distinct regimes: (i) a spatially modulated-dominated regime ( $c\lt c_{\textit{diff}}$ ), in which the slow movement of the boundary thermal wave allows coherent thermal plumes to follow the wave, maintaining coherence in both time and space; and (ii) a travelling-wave-dominated regime ( $c\gt c_{\textit{diff}}$ ), where the fast-moving thermal wave disrupts the spatial coherence of thermal structures near the boundary layer. These findings establish a new framework for understanding the interplay of spatial and temporal modulation, advancing our knowledge of heat transfer in systems with complex boundary conditions.