Vertical dispersion by stratified turbulence

Abstract

We derive a relation for the growth of the mean square of vertical displacements, δz, of fluid particles of stratified turbulence. In the case of freely decaying turbulence, we find that for large times 〈δz2〉 goes to a constant value 2(EP(0) +aE(0))/N2, whereEP(0) andE(0) are the initial mean potential and total turbulent energy per unit mass, respectively,a< 1 andNis the Brunt–Väisälä frequency. In the case of stationary turbulence, we find that 〈δz2〉 = 〈δb2〉/N2+ 2εPt/N2, where εPis the mean dissipation of turbulent potential energy per unit mass and 〈δb2〉 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 4EP/N2ast→ ∞. 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 timeT≫N−1, rather than linear gravity waves, we suggest that there is a range of time scales,t, betweenN−1andT, where 〈δb2〉 = 2πCPLεPt, whereCPLis 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 πCPLin this range. Comparing this result with observations, we make the estimateCPL≈ 3.