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 [Formula: see text], where ε_θ is the constant energy transfer rate across wavenumbers with very little contribution coming from the kinetic energy flux. The constant K₀ 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)'.

Keywords: mode-coupling theories; randomly stirred models; stably stratified fluids; turbulence; universal numbers.