  Natural convection is ubiquitous throughout the physical sciences and
engineering, yet many of its important properties remain elusive. To study
convection in a novel context, we derive and solve a quasilinear form of the
Rayleigh-B\'enard problem by representing the perturbations in terms of
marginally-stable eigenmodes. The amplitude of each eigenmode is determined by
requiring that the background state maintains marginal stability. The
background temperature profile evolves due to the advective flux of every
marginally-stable eigenmode, as well as diffusion. To ensure marginal stability
and to obtain the eigenfunctions at every timestep, we perform a
one-dimensional eigenvalue solve on each of the allowable wavenumbers. The
background temperature field evolves to an equilibrium state, where the
advective flux from the marginally-stable eigenmodes and the diffusive flux sum
to a constant. These marginally-stable thermal equilibria (MSTE) are exact
solutions of the quasilinear equations. The mean temperature profile has
thinner boundary layers and larger Nusselt numbers than thermally-equilibrated
2D and 3D simulations of the full nonlinear equations. We find the Nusselt
number scales like $\rm{Nu} \sim\rm{Ra}^{1/3}$. When an MSTE is used as initial
conditions for a 2D simulation, we find that Nu quickly equilibrates without
the burst of turbulence often induced by purely conductive initial conditions,
but we also find that the kinetic energy is too large and viscously attenuates
on a long viscous time scale. This is due to the thin temperature boundary
layers which diffuse heat very effectively, thereby requiring high-velocity
advective flows to reach an equilibrium.
