Phenomenological Theory for Spatiotemporal Chaos in Rayleigh-Benard Convection

Abstract

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 ${\xi }_2$. We hence obtain analytical results for the time-averaged convective current $J$ and the time-averaged vorticity current $\Omega$. We also define power-law behaviors such as $J\sim {\epsilon }^{\mu }$, $\Omega \sim {\epsilon }^{\lambda }$ and ${\xi }_2\sim {\epsilon }^{-\nu }$, where $\epsilon$ is the control parameter. We find from our theory that $\mu =1$, $\nu \ge 1/2$ and $\lambda =2\mu +\nu$ for phase turbulence and that $\mu =1$, $\nu \ge 1/2$ and $\lambda =2\mu +2\nu$ 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, $S(k)\sim 1/{\xi }_2{k}^3$ for large $k$, might be valid in STC.

arXiv comment19 pages, 3 figures and 2 tables