Geoscience Reference
In-Depth Information
Neutral atmospheric surface layer
It is now generally agreed, and almost accepted by definition, that in the dynamic sublayer,
and under neutral conditions in the whole atmospheric surface layer, the concentration
of any admixture of the flow is a logarithmic function of height above the ground. Many
different derivations of this relationship have appeared in the literature but the simplest is
no doubt that given by Landau and Lifshitz (1959) in the 1944 edition of their topic (see
also Monin and Yaglom, 1971). The derivation is based strictly on dimensional analysis
an
d
on the observation that in plan-parallel flow an increase in velocity in the
z
-direction,
(
du
dz
), is evidence of a downward momentum flux and a sink at the surface. Thus, the
mean velocity gradient in a fluid of density,
/
ρ
, is determined by the shear stress at the
wall,
d
0
); in the last variable the (zero-plane)
displacement height
d
0
is introduced to account for the uncertainty of the position of the
wall in the case of irregular and uneven surfaces. These variables can be combined into
a single dimensionless quantity as follows,
τ
0
, and the distance from the wall, (
z
−
u
∗
dz
)
=
k
(2.39)
(
z
−
d
0
)(
du
/
where
u
∗
is defined as in Equation (2.32). Experimentally, this combination
k
has been
found to be nearly invariant and close to 0.4 under many different conditions; it is referred
to commonly as von Karman's constant. The logarithmic profile follows upon integration
of Equation (2.39).
In general, this logarithmic profile can be written as
ln
z
2
−
u
k
d
0
u
2
−
u
1
=
(2.40)
z
1
−
d
0
where the subscripts 1 and 2 refer to two levels within the neutral surface layer. This
result produces immediately the drag coefficient, as it appears in Equation (2.34), namely
Cd
2
. Equation (2.39) can also be integrated as follows
=
{
k
/
ln [(
z
2
−
d
0
)
/
(
z
1
−
d
0
)]
}
ln
z
u
k
−
d
0
u
=
(2.41)
z
0
where
z
0
is an integration constant, whose dimensions are length; it is usually referred to
as the momentum roughness parameter or the roughness length. Its value depends on the
conditions at the lower boundary of the region of validity of Equation (2.39). Graphically,
it may be visualized as the zero velocity intercept of the straight line resulting from a
semi-logarithmic plot of mean velocity data versus height in a neutral surface layer (see
Figure 2.9). Equation (2.41) leads to the drag coefficient, as it appears in Equation (2.37),
namely Cd
2
.
Dimensional arguments, similar to those leading to the profiles of the mean wind
speed, produce for the mean specific humidity gradient
=
{
k
/
ln[(
z
−
d
0
)
/
z
0
]
}
/ρ
E
dz
)
=−
k
(2.42)
u
∗
(
z
−
d
0
)(
dq
/
