Turbulent fluxes - Turbulence in the Atmosphere - page 84

positive eddy diffusivity K such that uw =− K∂U/∂z . The shape of the U profile,
Figure 3.1 , also suggests that K is smaller near the channel walls and larger near
the center.

4.5 The “mixture length”
In this chapter we've seen that the turbulent flux of a quantity - the covariance of
turbulent velocity and the turbulent part of the quantity being transported - has a
central role in t urbulent flows. In the example of Section 4.2 the vertical flux of
temperature wθ dictates the evolution of the daytime temperature profile.
It is natural to try to relate the turbulent part of the quantity being transported to
the corresponding mean gradient, e.g.,

d ∂ θ ∼−

∂z ,

(4.31)

with d a length scale related to the turbulent fluid displacements in z . Then the
flux is

dw ∂ K ∂ θw ∼−

∂z ∼− ∂z ,

(4.32)

with K an “eddy diffusivity.”
The notion of a length scale d related to turbulent mixingwas evidently developed
independently by Taylor and Prandtl. In a 1913 experiment Taylor measured the
vertical profiles of temperature and humidity above the cold water of the Grand
Banks of Newfoundland, finding that the rate at which the cooling wave penetrated
into the atmosphere could be accounted for “by a process like molecular conduction
but much more vigorous.” In reflecting on his early attempts to understand the
“effective conductivity due to turbulence,” Taylor ( 1970 )wrote:
To complete a theoretical analogy betweenmolecular and turbulent transfer it is necessary to
think up some length connected with turbulence which is analogous to the mean free path of
molecules. I was driven to imagine a purely hypothetical process to represent the collisions
which terminate each molecular free path, and in 1915 I put out the idea that coherent
fluid masses move a certain distance up or down vertically carrying all their transferable
properties and then mix with the surroundings in which they find themselves.
The … idea … of a mixture length was used by Prandtl [in a 1925 paper] who afterwards
told me that he had never heard of my 1915 paper.

A key notion in Taylor's 1915 paper is “the average height through which an eddy
moves from the layer at which it was at the same temperature as its surroundings,
to the layer with which it mixes.” In his 1925 paper Prandtl introduced the term
“Mischungsweg,” in English “mixture length,” by which it became known. Taylor
( 1970 ) made clear that he had considered these notions about what had come to be
known as the mixture length to be quite crude: