Phenomenological Theory for Phase Turbulence in Rayleigh-B\'{e}nard Convection

We present a phenomenological theory for phase turbulence (PT) in Rayleigh-B\'{e}nard 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.