Phenomenological Theory for Phase Turbulence in Rayleigh-Bénard Convection

Abstract

We present a phenomenological theory for phase turbulence (PT) in Rayleigh-Bénard convection, based on the generalized Swift-Hohenberg model. We apply a Hartree-Fock approximation to PT and conjecture a scaling form for the structure factor $S(k)$ with respect to the correlation length $ξ_2$. We hence obtain {\it analytical} results for the time-averaged convective current $J$ and the time-averaged mean square vorticity $Ω$. We also define power-law behaviors such as $J \sim ε^μ$, $Ω\sim ε^λ$ and $ξ_2 \sim ε^{-ν}$, where $ε$ is the control parameter. We find from our theory that $μ= 1$, $ν\ge 1/2$ and $λ= 2 μ+ ν$. These predictions, together with the scaling conjecture for $S(k)$, are confirmed by our numerical results.