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The Burgers-Rayleigh-B\'enard system is a one-dimensional toy model of a convective cell that is amenable to a thorough theoretical analysis. Despite its simplicity, it has a lot in common with realistic Rayleigh-B\'enard thermal convection: a supercritical pitchfork bifurcation at the onset of convection; the organization of the flow into boundary layers and a bulk zone; Nusselt and Reynolds numbers scaling laws obeying the ultimate regime in the asymptotic high Rayleigh limit. However, differences with realistic natural convection are also present: the persistence of flow stationarity, the absence of turbulence and the possibility of thermally insulating shock states.
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Calzavarini et al. (2023) studied this question.
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