coefficient by as much as 40% (Sheng, 1980). 



Turbulence Parameterization 



A semi-empirical theory of vertical mixing is used in this study. 

 The effect of stratification, as measured by the Richardson number, Ri, 

 on the intensity of vertical turbulent mixing is parameterized by a 

 number of empirical stability functions: 



Avo ^i (Ri); 



Kv = 



Kwn ^. (Ri); Ri 



^vo 



-g 8p_ 

 p 3z 



.W'i^f 



_1 



(20) 



where A^^ and K„q, are the eddy coefficients in the absence of any 

 density stratification and <^ and <^ are stability functions. 

 Traditionally, these stability functions have been determined 

 empirically by comparing model output with measured data. As shown in 

 Figure 3a, great discrepancy exists among the various empirical forms 

 of the stability functions. In addition, the critical Richardson 

 numbers, at which turbulence is completely damped by buoyancy, given by 

 these formulas are much too high (10) compared to the measured value of 

 0.25 (Erikson, 1978). To unify this discrepancy, stability functions 

 may be determined from a second-order closure model of turbulence. 

 Assuming a balance between turbulence production and 



dissipation, i.e., the so-called "super-equilibrium" condition 

 (Donaldson, 1973), we can obtain a simpler set of algebraic 

 relationships between the turbulent correlations and mean flow 

 gradients. As shown in Figure 3b, such a stability function leads to a 



Blumberg 



Kent a Pritchord 



Bowden 8 Hamilton 

 Munk & Anderson 



(b) 



\^ 



f2>'"i 



-4.0 



\ 





-3.2 



/Ri)\ 



1 



-2.4 

 -1.6 





 Ri 



.2 .4 .6 



Figure 3. Stability Function vs. Richardson Number: 



(a). Empirical Formulations, 



(b). "Superequilibrium" Formulation Derived from 

 Reynolds Stress Model, 



274 



Sheng 



