Quantum-Inspired Simulation of 2D Turbulent Rayleigh-Bénard Convection

Turbulent thermal convection governs heat transport in systems ranging from stellar interiors to industrial heat exchangers. Two-dimensional Rayleigh-Bénard convection serves as a paradigm for these flows, reproducing key features such as thin boundary layers, large-scale circulation, and sustained plume dynamics. While Matrix Product State (MPS) methods have demonstrated significant compression of isothermal turbulent fields, their application to buoyancy-driven flows with active thermal coupling has remained unexplored. We apply MPS to two-dimensional Rayleigh-Bénard convection with dynamical simulations up to Ra = 10¹⁰. An a priori decomposition of DNS snapshots up to Ra = 10¹¹ shows that the bond dimension χ required to represent the flow fields grows without saturation, in contrast to the plateauing of χ reported for velocity fields in isothermal 2D turbulence. Crucially, however, dynamical simulations solving the governing equations directly in the compressed MPS format at fixed χ show that the χ required to recover statistical observables, such as the Nusselt number, scales significantly more favorably with Ra than the a priori complexity suggests. At Ra = 10¹⁰, a relative error of 1.8\% in the mean Nusselt number is achieved with a nearly 9-fold reduction in degrees of freedom, using a χ comparable to that required at Ra = 10⁹. Spectral analysis confirms the progressive recovery of spatial and temporal scales with increasing χ. These findings establish MPS as a scalable tool for simulating thermally driven turbulence, suggesting the method may remain viable for investigations of the ultimate regime at substantially higher Ra.