Turbulent fluxes of passive and active scalar in unsteady strongly stratified homogeneous turbulence are analyzed using the rapid distortion theory (RDT) [1]-[2]. Analytical solutions of the RDT equations demonstrate that the initial conditions, i.e., the initial cross correlation between passive scalar and active scalar (density), and the initial potential energy, make difference between the passive and active scalar flux in the vertical direction, giving difference betweenthe turbulent diffusion coefficients for passive and active scalars. This is in contrast to the previous literature which usually assumed the same diffusicon coefficient for the passive and active scalars.
On the ohter hand, our results show that the zero initial correlation between passive and activescalars leads to complete correlation between the passive and active(density) scalar in a long time,in contrast again to the previous discussions in the literature that initial strong correlation is necessary for the final strong correlation.
The difference between the passive scalar flux and active scalar(density) flux appears in the passive scalar flux in its components slowly oscillating at frequency N (N: Brunt-Vaisala frequency), while the active scalar flux has only the twice as rapidly oscillating components with frequency 2N.
It is also found that the correlation coefficient between passive and active scalars is not a good measure of identifying the agreement or disagreement of the turbulent diffusion coefficients for these scalars, since if the passive scalar variance is initially large, final correlation coefficient becomes small even when the turbulent diffusion coefficients are equal for the active and passive scalars.
The above results suggests the importance of the ′unsteady′ analysis and the ′initial conditions′ in the diffusion problem in stratified turbulence,which has been largely ignored in the previous studies.
Effects of molecular diffusion of passive scalar (Schmidt number Sc) and density (Prandtl number Pr) show that if Sc>2/(1+1/Pr), the slowly oscillating mode with half frequency N decays more slowly than the fast mode with frequency 2N. Then the difference between the passive and activescalar fluxes will become even more significant in a long time. This would be important in practical applications since in most practical conditionsin the atmosphere/ocean Pr>0.7 and Sc>>1 hold so that the condition Sc>2/(1+1/Pr) is usually satisfied.
[1] H. Hanazaki, ″Linear processes in stably and unstably stratified rotating turbulence,″ J. Fluid Mech. 465, 157--190 (2002).
[2] H. Hanazaki, ″Effects of initial conditions on the passive and active scalar fluxes in unsteady stably stratified turbulence,″ Phys. Fluids (2003, in press). |
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