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

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 Z²̄(t) , the mean-square velocity w²̄ , 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 Z²̄ of marked particles ceases to grow when t [gsim ] N −1 , and its asymptotic value is proportional to w²̄/N² , where the constant of proportionality, ζ²̄(∞) , is O (1) and a decreasing function of ( NT ) −1 . This result is also shown to be a general consequence of the finite potential and kinetic energy in the stationary turbulence. (ii) If there is a small diffusive interchange of density between fluid elements (i.e. γ [Lt ] 1), the marked particles’ mean-square displacement has a slow linear growth (i.e. Z²̄²̄/N²(1+O(γ²)tN) ). (iii) Such molecular processes must also dilute the initial concentration of contaminants (e.g. dye or smoke) in those fluid elements that diffuse above the limit in (i). (iv) The mean-square fluctuation of density is proportional to the product of the asymptotic mean-square displacement of marked particles and the square of the mean density gradient (- G)\;( i.e.ρ²= G²Z²̄=1/2ζ²̄(∞ ,γ = 0) G²w²̄/N²) . (v) The flux F ρ of density in a turbulent flow can be expressed exactly as the sum of two terms, the first 1/2(dZ²̄/dt)\, G , being caused by the growth of the displacements of fluid elements, and the second Ż(ρ + ρ^) being caused by the mixing between fluid elements. In stable flows, it is shown that the second element is dominant, and Fρ~²̄Nρₒg while the first is smaller by O (γ). Previous laboratory and field measurements of w²̄, Z²̄, ρ² and R w (t) are discussed in some detail and shown to be consistent with this model.