37 
eddy viscosity. The eddy viscosity is assumed to have the form A v = L v V v where l v is 
the length scale (eddy size) of vertical mixing, and v v is the scale of the whirling velocity. 
A basic building block is v. Karman’s (1930) formula for length scale. It is devised for a 
shear boundary layer and characterizes the length scale A in terms of the relative 
variation of shear y/{z) = -+ (§-)" . i-©- A(z) = k 
¥ 
which means 
¥ 
= k— . 
A 
For practical use, the definition is regularised by means of some maximum value A raax to 
read 
1 
ÔW 
1 
77 
=- 
- + 
A K ¥ A mx 
The v. Karman formula was made without particularly taking into account density- 
stratified flow. For a more adequate consideration of stratification, the formula is 
generalised by a simple modification proposed by Laikhtman (1979). Let Ri f be the flux 
Richardson number and form the expression y/(z) = Ri f ). Then the modified 
formula is as above, but with ¥ replaced by ¥*. 
The flux Richardson number, in turn, is expressed in terms of the gradient Richardson 
number 
Ri z =“ 
gÊB. 
& dz 
P¥ 
and the Prandtl number 
A turbulence model is invoked to give the Prandtl number in terms of the gradient 
Richardson number. Use is made of Mellor & Yamada (1974). 
1 0.725*0.688 
Pr Ri g + 0.186 + (Ri g 1 2 -0.316Ri g +0.034ô) 1/2 
On inserting, we obtain an expression for the flux Richardson number in terms of the 
gradient Richardson number. The local length scale formula is complete. It is diagnostic 
(in algebraic terms of mean flow quantities), and it is to be understood as a steady state 
formula. However, when invoked in a dynamic model, it proves to be highly sensitive. As 
a means to control mixing, it easily generates significant disturbances that may give rise 
to considerable artificial fluctuation of the flow. Therefore, stabilisation is required before 
such a length scale formula can be used. 
This can be done by smoothing with respect to space and time. In terms of physics, 
spatial integration amounts to a non-local closure scheme. Besides, an integral formula 
is more in tune with the distributed and continuous nature of mixing.