39 
pycnoclines, the disconnecting effect is multiple. 
In a stochastic view, G u and 0 l are distribution functions of the probability of parcel 
disintegration, and the derivatives are probability densities. From this point of 
view we are dealing with the mixing reach distribution. 
Having made the above preparations, we now construct a non-local formula for mixing 
length. An integral of local contributions to parcel dissolution is formed. 
The water column reaches from bottom z = -H to top z-g. Thus, with respect to z 
within that range, maximum values of the probing distance 5 are attained for z + S = g 
(upper segment) and z-S = -H (lower segment), respectively. 
The mixing length is the mean free path, i.e. the distance at which a water parcel is 
dissolved. Accordingly, we put 
í~ z f ana A í * 
L-,(z) = 2k J <s[ dS= 2k J S 
K 
0 V J 
H+i Í psgl \ 
L[(z) = 2k 
0 
H+z 
A (z + 5) 
dô^lK [ô 6 l (z,ô)dô 
J A ( v — 
e u {z,5)dô 
A(z - 5) 
The normalising coefficient 2k has been introduced to make the integral definition 
conform to standard facts. In particular, it has to be compatible with v. Karman’s local 
formula itself. Therefore, as a special case, we briefly consider a logarithmic boundary 
layer as it is known from open channel flow over rough bottom and without wind forcing. 
We take it for granted that 
y/{z)= ■ ■ Um - ■ L v (z) = k(z + H) , V v (z)=U, 
k(z + H) 
where JJ, denotes the friction velocity. 
In the vicinity of the bottom, the result should be determined by downward probing. That 
is, in fact, the case as can be easily checked. On the other hand, upward probing 
produces a result that is much too high. This example suggests that in order to combine 
upward and downward probing in a general setting, we should take the minimum. It might 
be speculated that mixing is in some way isotropic, and the space limitation is due to the 
proximity of the wall. Length scale is limited by the distance to the closest wall. 
Accordingly, the desired result is obtained by combining the upper and lower 
contributions as follows. 
L v (z)=min(Z “ (z),L “ (z)) 
This non-local diagnostic formula serves as our definition of mixing length for vertical 
exchange. Incidentally, partial integration shows that the above formulae are equivalent 
to 
L u v (z)=2k \9 u (z,5)d8, L[{z)=2k \d l {z,8)d5 
0 0