Turbulence Parameterization 



At this point, it is appropriate to discuss the parameterization of 

 turbulence in the present model. 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: 



V = Ayo ,^^(Ri); Ky = Kyp ^.^(Ri); D^ = 0^^ ,^^(Ri ) ; (2.44) 



where 



Ri =-iS^ ^ 

 Up(l+p) 9° 



(2.45) 



where AyQ, K^q, and D^q are the eddy coefficients in the absence of any 

 density stratification and ((> , <^ , and <^ are stability functions. Typically 

 they are written as: 



,^^ = (l+a^Ri)"^! ; ^^= {Uo^Ri)'^2 ; .^3 = (l+^gRi )"'3 (2.46) 



where the constants o , o , a , m , m , and m^ are generally determined 

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

 2.4(a), great discrepancy exists among the various empirical forms of the 

 stability functions. Much of the discrepancy resulted from (1) the difference 

 in numerical schemes used; (2) the different turbulent regime of the water 

 bodies studied; and (3) the difference in resolution of the model and the 

 measured data. In addition, the critical Richardson numbers, at which 

 turbulence is completely damped by buoyancy, given by these formulas are much 

 too high 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 (Appendix D). 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 2.4(b), such a stability function leads to a critical 



41 



