Geoscience Reference
In-Depth Information
the four heights in Equation (2.33) need not all be different; thus levels 4 and 3 could be
the same as 2 and 1, respectively. In the case of the vertical momentum flux, one obtains
in the same way
u
1
)
2
w
u
=−
Cd(
u
2
−
(2.34)
where Cd is the transfer coefficient for momentum, also called the drag coefficient; in
the case of the vertical sensible heat flux, one has similarly
w
θ
=−
Ch(
u
2
−
u
1
)(
θ
4
−
θ
3
)
(2.35)
where Ch is the heat transfer coefficient, also called the Stanton number.
In many ap
p
lications the lowest reference level of the wind speed is taken at the
surface where
u
0. When in addition the vertical water vapor flux refers to that at the
ground surface, namely
E
, Equation (2.33) assumes the common form
=
E
=−
Ce
ρ
u
q
(2.36)
where
u
is the wind speed at a certain reference height above the ground and
q
is
the difference between the mean specific humidity at two other reference heights (one
of which may also be at the water or ground surface level), whose values will, again,
affect the magnitude of Ce. In the same way, for the surface shear stress, Equation (2.34)
becomes
u
2
τ
0
=
Cd
ρ
(2.37)
and, for the surface sensible heat flux, Equation (2.35) becomes
H
=−
Ch
ρ
c
p
u
θ
(2.38)
Recall that the difference between
T
and
is often small in the lower layers of the surface
layer, where most measurements are made. Therefore in many situations, when the height
difference of the temperatu
re
measurements is onl
y
a few meters, in expressions like
(2.35) and (2.38) the use of
T
is allowed instead of
θ
θ
.
2.5.2
Some specific implementations: flux-profile functions
The dimensionless transfer coefficients Ce, Cd and Ch, and their dependence on other
dimensionless variables, have been the subject of much research. Major progress was
made in the thirties by means of mixing length theory, as a result of contributions by
Prandtl, von Karman, and Taylor in the framework of the turbulent diffusion approach;
this led initially to the formulation of the logarithmic profile equations for the mean
wind speed, the potential temperature, the specific humidity and other admixtures of
the flow (see Monin and Yaglom, 1971; Brutsaert, 1982; 1993) and subsequently to
further developments by Monin and Obukhov and others. In this section a few similarity
approaches are reviewed that have been useful in the practical estimation of surface
fluxes.
