A statistical model of fluid-element motions and vertical diffusion in a homogeneous stratified turbulent flow

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Abstract
The vertical displacements Z(t) of fluid elements passing through a source z = 0 at t = 0 in a horizontal mean flow with stably stratified statistically stationary turbulence (with buoyancy frequency N and velocity time-scale T), under the action of random pressure gradients and damping by internal wave motions, are investigated by a model Langevin-like equation, and by a general Lagrangian analysis of the displacements, of the density flux and of the energy of fluid elements. Solutions for the mean-square displacement , and the autocor-relation of the velocity are calculated in terms of the spectrum [Fcy ](s) of the pressure gradient. We use model equations for the momentum of fluid elements and for the exchange of density fluctuations between fluid elements, taking the elements’ diffusion timescale to by γ−1 times the buoyancy timescale N−1, where γ is a measurable parameter.In the case of moderate-to-strong stable stratification (i.e. NT [gsim ] 1), we find the following. (i) When there is no change of the fluid elements’ density (γ = 0), the mean-square displacement and Rw(t) are discussed in some detail and shown to be consistent with this model.

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