  The dynamics of heat transfer in a model system of Rayleigh-B\'enard (RB)
convection reduced to its essential, here dubbed Burgers-Rayleigh-B\'enard
(BRB), is studied. The system is spatially one-dimensional, the flow field is
compressible and its evolution is described by the Burgers equation forced by
an active temperature field. The BRB dynamics shares some remarkable
similarities with realistic RB thermal convection in higher spatial dimensions:
i) it has a supercritical pitchfork instability for the onset of convection
which solely depends on the Rayleigh number $(Ra)$ and not on Prandlt $(Pr)$,
occurring at the critical value $Ra_c = (2\pi)^4$ ii) the convective regime is
spatially organized in distinct boundary-layers and bulk regions, iii) the
asymptotic high $Ra$ limit displays the Nusselt and Reynolds numbers scaling
regime $Nu = \sqrt{RaPr}/4$ for $Pr\ll 1$, $Nu=\sqrt{Ra}/(4\sqrt{\pi})$ for
$Pr\gg1$ and $Re = \sqrt{Ra/Pr}/\sqrt{12}$, thus making BRB the simplest
wall-bounded convective system exhibiting the so called ultimate regime of
convection. These scaling laws, derived analytically through a matched
asymptotic analysis are fully supported by the results of the accompanying
numerical simulations. A major difference with realistic natural convection is
the absence of turbulence. The BRB dynamics is stationary at any $Ra$ number
above the onset of convection. This feature results from a nonlinear saturation
mechanism whose existence is grasped by means of a two-mode truncated equation
system and via a stability analysis of the convective regime.
