The analogy between turbulent and molecular diffusion coefficients is reflected
by the fact (see Chap. 2) that both have L
2 T
−1 dimensions, indicative of the product
of a velocity and a length. The mixing length relates to the full-length scales of
turbulence, being a function of the system geometry (e.g. characteristic dimension)
as well as of the type of thermal stratification of the atmosphere. In this context, the
friction velocity
À
uà is a scalar of turbulent velocity obtained from Eq. (3.25):
u
2
à ¼ Àu 0
1 u 0
3
ð3:25Þ
or by combining Eq. (2.15) with Eq. (2.17):
u
2
à ¼ l
@u
@z
ð3:108aÞ
Limitations of some of the assumptions of the mixed layer theory should be
pointed out (see for e.g. Chap. 2 and later in this chapter on the application of the
eddy covariance WPL correction method). For example, scalar quantities transported can affect the transport process and eddy length scales can be about the same
order of magnitude as those associated with the distribution of the quantity to be
transported. Accordingly, it may not be accurate to express the vertical flux in terms
of the local gradient of the property. The use of turbulent diffusion coefficients that
serve to simplify the complex flow, while simultaneously retaining the equations of
second moments, discussed above, makes it possible the analysis of terms related to
the dissipation and transport of kinetic energy. In this way, some information
inherent to the covariance terms is restored (Shaw 1995d).
As an example of the modeling of the components for equations of higher order,
the return-to-isotropy term (term V in Eq. 3.76) can be simplified by the equation:
1
q
p0
@u 0
i
@x k
þ
@u 0
k
@x i
"
#
¼ À
u Ã
k
u
0
i u
0
j À
d ij
3
u
2
Ã
ð3:109Þ
where k is a length scalar and u à the friction velocity. Equation (3.109) indicates
that for normal forces i = j, the momentum budget is proportional to the difference
between the variance of velocity and 1/3 the friction velocity.
Similarly, for the term for viscous dissipation of linear momentum, analogous to
Eq. (3.68), the equation can be written (Shaw 1995d):
2v
@u 0
i
@x k
@u 0
j
@x k
¼ À
u
3
Ã
l
d ij
ð3:110Þ
The Eq. (3.110) is based on the principle that the dissipation rate is of the order
of u
3
à =l, where l is the characteristic length scale of the energetic eddies.
When large eddies predominate in the transport processes, further developments
of numerical techniques for the simulation of large-scale turbulent flows (LES),
62
3 Characterization of Turbulent Flow in the Surface Boundary Layer
by the fact (see Chap. 2) that both have L
2 T
−1 dimensions, indicative of the product
of a velocity and a length. The mixing length relates to the full-length scales of
turbulence, being a function of the system geometry (e.g. characteristic dimension)
as well as of the type of thermal stratification of the atmosphere. In this context, the
friction velocity
À
uà is a scalar of turbulent velocity obtained from Eq. (3.25):
u
2
à ¼ Àu 0
1 u 0
3
ð3:25Þ
or by combining Eq. (2.15) with Eq. (2.17):
u
2
à ¼ l
@u
@z
ð3:108aÞ
Limitations of some of the assumptions of the mixed layer theory should be
pointed out (see for e.g. Chap. 2 and later in this chapter on the application of the
eddy covariance WPL correction method). For example, scalar quantities transported can affect the transport process and eddy length scales can be about the same
order of magnitude as those associated with the distribution of the quantity to be
transported. Accordingly, it may not be accurate to express the vertical flux in terms
of the local gradient of the property. The use of turbulent diffusion coefficients that
serve to simplify the complex flow, while simultaneously retaining the equations of
second moments, discussed above, makes it possible the analysis of terms related to
the dissipation and transport of kinetic energy. In this way, some information
inherent to the covariance terms is restored (Shaw 1995d).
As an example of the modeling of the components for equations of higher order,
the return-to-isotropy term (term V in Eq. 3.76) can be simplified by the equation:
1
q
p0
@u 0
i
@x k
þ
@u 0
k
@x i
"
#
¼ À
u Ã
k
u
0
i u
0
j À
d ij
3
u
2
Ã
ð3:109Þ
where k is a length scalar and u à the friction velocity. Equation (3.109) indicates
that for normal forces i = j, the momentum budget is proportional to the difference
between the variance of velocity and 1/3 the friction velocity.
Similarly, for the term for viscous dissipation of linear momentum, analogous to
Eq. (3.68), the equation can be written (Shaw 1995d):
2v
@u 0
i
@x k
@u 0
j
@x k
¼ À
u
3
Ã
l
d ij
ð3:110Þ
The Eq. (3.110) is based on the principle that the dissipation rate is of the order
of u
3
à =l, where l is the characteristic length scale of the energetic eddies.
When large eddies predominate in the transport processes, further developments
of numerical techniques for the simulation of large-scale turbulent flows (LES),
62
3 Characterization of Turbulent Flow in the Surface Boundary Layer