A randomly stirred model for Bolgiano–Obukhov scaling in turbulence in a stably stratified fluid

Abstract

A randomly stirred model, akin to the one used by DeDominicis and Martin for homogeneous isotropic turbulence, is introduced to study Bolgiano–Obukhov scaling in fully developed turbulence in a stably stratified fluid. The energy spectrum E ( k ), where k is a wavevector in the inertial range, is expected to show the Bolgiano–Obukhov scaling at a large Richardson number Ri (a measure of the stratification). We find that the energy spectrum is anisotropic. Averaging over the directions of the wavevector, we find E ( k ) = K 0 ε θ 2 / 5 k − 11 / 5 , where ε θ is the constant energy transfer rate across wavenumbers with very little contribution coming from the kinetic energy flux. The constant K 0 is estimated to be of O(0.1) as opposed to the Kolmogorov constant, which is O(1). Further for a pure Bolgiano–Obukhov scaling, the model requires that the large distance ‘stirring’ effects dominate in the heat diffusion and be small in the velocity dynamics. These could be reasons why the Bolgiano–Obukhov scaling is difficult to observe both numerically and experimentally. This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.