Phenomenological Theory for Phase Turbulence in Rayleigh-Bénard Convection

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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 ξ₂. We hence obtain 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 εμ, Ω ελ and ξ₂ ε⁻ν, where ε is the control parameter. We find from our theory that μ= 1, ν≥ 1/2 and λ= 2 μ+ ν. These predictions, together with the scaling conjecture for S(k), are confirmed by our numerical results.