Thermal turbulence at very small Prandtl number

The equations of thermal turbulence are derived for the case of small Prandtl number, a case of a relatively simple realizable turbulent flow. The power spectrum of velocity is described using the transfer functions of Heisenberg and of Kovasznay. It is seen that these do not give uniformly good approximations, since they force spectral energy to flow from low to high wave numbers, though the Kovasznay approximation may be useful for large Rayleigh numbers. A general comparison of the present study with the Malkus theory of convection indicates disagreement which probably results from the dominance of nonlinear terms in the low-Prandtl-number limit.