Vertical dispersion by stratified turbulence

We derive a relation for the growth of the mean square of vertical displacements, delta z, of fluid particles of stratified turbulence. In the case of freely decaying turbulence, we find that for large times (delta z(2)) goes to a constant value 2(E-P(0) + aE(0))/N-2, where E-P(0) and E(0) are the initial mean potential and total turbulent energy per unit mass, respectively, a < 1 and N is the Brunt-Vaisala frequency. In the case of stationary turbulence, we find that (delta z(2)) = <delta b(2)>/N-2 + 2 epsilon(P)t/N-2, where epsilon(P) is the mean dissipation of turbulent potential energy per unit mass and <delta b(2)> is the Lagrangian structure function of normalized buoyancy fluctuations. The first term is the same as that obtained in the case of adiabatic fluid particle dispersion. This term goes to the finite limit 4E(P)/N-2 as t -> infinity. Assuming that the second term represents irreversible mixing, we show that the Osborn & Cox model for vertical diffusion is retained. In the case where the motion is dominated by a turbulent cascade with an eddy turnover time T >> N-1, rather than linear gravity waves, we suggest that there is a range of time scales, t, between N-1 and T, where <delta b(2)> = 2 pi C-PL epsilon(P)t, where C-PL is a constant of the order of unity. This means that for such motion the ratio between the adiabatic and the diabatic mean-square displacement is universal and equal to pi C-PL in this range. Comparing this result with observations, we make the estimate C-PL approximate to 3.