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Phenomenological Theory for Spatiotemporal Chaos in Rayleigh-Benard Convection

We present a phenomenological theory for spatiotemporal chaos (STC) in Rayleigh-Benard convection, based on the generalized Swift-Hohenberg model. We apply a random phase approximation to STC and conjecture a scaling form for the structure factor $S(k)$ with respect to the correlation length $ξ_2$. We hence obtain analytical results for the time-averaged convective current $J$ and the time-averaged vorticity current $Ω$. 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 $λ= 2 μ+ ν$ for phase turbulence and that $μ= 1$, $ν\ge 1/2$ and $λ= 2 μ+ 2 ν$ for spiral-defect chaos. These predictions, together with the scaling conjecture for $S(k)$, are confirmed by our numerical results. Finally we suggest that Porod's law, $S(k) \sim 1/ξ_2 k^3$ for large $k$, might be valid in STC.