Self-Consistent Field Theory for the Convective Turbulence in a Rayleigh-Benard System in the Infinite Prandtl Number Limit

Abstract

The kinetic energy spectrum \(E_{u}(k)\) for three dimensional convective turbulence in a Rayleigh-Benard system,where k is the wave vector, was shown to scale as \( k ^{-13/3}\) on heuristic grounds in the recent work of Pandey, Verma and Mishra in the infinite Prandtl number limit. They also presented clear numerical evidence of this scaling. This limit is very similar to the spherical model of critical phenomena and hence amenable to exact treatment in a self-consistent field theory. We find that self-consistency gives \(E_u (k)\propto R^{22/15}k^{-13/3}(R\) is the Rayleigh number) but the inevitable presence of sweeping adds a part which is proportional to \(k^{-7/2}\). This can account for the slight k-dependence of the compensated spectrum of Pandey et al. We also estimate the anisotropy in the spectrum and find that the second order Legendre function has a strength of 15 % relative to the isotropic part. In two spatial dimensions the scaling exponent of the energy spectrum is still 13/3 but the anisotropy is larger.