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In Prandtl’s picture of (the action of) turbulence, mixing is accomplished by the formation, 
travel and dissolution of water parcels. Physically, the term ’’vertical mixing” signifies 
vertical motion in which a parcel dissolves along its path, intermingling its properties with 
the surrounding water body. Mixing comes about as water parcels are moved about in 
turbulent excursions (fluctuations of the local flow). Let us sketch an idealised conception 
of the disintegation of a migrating water parcel. 
Once a parcel has formed and starts moving, it loses portions of its mass on its travel 
path. Suppose that a water parcel has formed somewhere, and observe its migration. 
We imagine following it as it undergoes turbulent excursions. All along its path, it is 
exposed to erosion causing its gradual decay. Some contribution to its disintegration is 
made at each point it passes. 
We suppose that, on the scale of vertical mixing, the water mass properties are 
horizontally homogeneous so that we may confine ourselves to purely vertical motions. 
Furthermore, we assume that dissolution occurs exclusively in one-way upward or 
downward excursions. From the place where it has formed, a water parcel travels until its 
complete disintegration. 
While travelling up or down, a parcel is subject to dissolution as a continuous process. At 
any location, some percentage of the remaining mass is affected. We suppose that (in 
steady state) the local loss of integrity is governed by the local mixing length. To describe 
the parcel integrity along the path, let us introduce two continuous functions 6 U , 0 1 (for 
upper and lower water body, respectively) of (vertical) travel distance 5, with reference 
position z as parameter. Formally our hypothesis reads 
The reader will notice a formal resemblance to the definition of local length scale. Given 
the local length scale A as from the above diagnostics, it is easy to find the integrity 
functions 6 U , 6‘ as solutions to a one-sided boundary value problem. The boundary is at 
the travel start 5 = 0 where integrity is normalised to be 
0 u (z,0) = 1. e l (z,0) = 1 
Integration gives 
For any z , these functions decrease monotonically with increasing distance S. They 
tend to zero where a barrier (surface, bottom) is encountered on the way. When an 
intermediate obstacle (pycnocline) is encountered, they suffer a marked drop due to the 
fact that the region beyond is scarcely involved in mixing. In the case of stacked