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 ${\xi }_2$. We hence obtain {\it analytical} results for the time-averaged convective current $J$ and the time-averaged mean square vorticity $\Omega$. 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 $\lambda =2\mu +\nu$. These predictions, together with the scaling conjecture for $S(k)$, are confirmed by our numerical results.

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