A theoretical study of small scale turbulence in stratified turbulent shear flows

This report examines the postulate of local isotropy in stratified homogeneous turbulence from a theoretical point of view. The study is based on a priori analysis of the evolution equations governing single-point turbulence statistics that are formally consistent with the Navier-Stokes equations. The Boussinesq approximation has been utilized to account for the effect of buoyancy – a simplifying assumption that constitutes an excellent approximation in the case considered here. The study concludes that the hypothesis of local isotropy is formally inconsistent with the Navier-Stokes equations in homogeneous stratified turbulence. An estimate is provided that suggests that local isotropy may constitute only a physically justifiable approximation in the limit of a clear-cut separation between the time scales associated with the imposed buoyancy and the turbulent eddy-turnover time scale. This is unlikely to happen in most flows, at least those not too far from equilibrium. The results also suggest that the dynamical dependence of the small-scale turbulence on large-scale anisotropies associated with imposed density stratification is significantly stronger than that caused by an imposed mean straining. This report has in a revised form been published in SIAM Journal of Applied Mathematics, 2003, Vol. 64, No. 1, pp. 309-321.