Geoscience Reference
In-Depth Information
with densely placed obstacles such as natural vegetation with average height
h
0
, the
momentum roughness
z
0
can be assumed to be of the order of
h
0
/
10,
d
0
of the order of
h
0
/
100 or smaller. The scalar roughness
parameters
z
0h
and
z
0v
continue to be the subject of research (see, for example, Brutsaert
and Sugita, 1996; Qualls and Brutsaert, 1996; Sugita and Brutsaert, 1996; Cahill
et al
.,
1997).
2to2
h
0
/
3, and
z
0h
and
z
0v
of the order of
h
0
/
Monin-Obukhov similarity in the surface layer
Neutral conditions occur only seldom in the atmospheric boundary layer. Therefore, it is
practically always necessary to include the effect of the stability, i.e. the density stratification,
of the atmosphere in the formulation of the profile equations and of the corresponding
transfer coefficients. One of the more common ways of doing this is based on the Monin-
Obukhov (1954) approach, which assumes that the effect of the density stratification of
the flow can be represented by the production rate of turbulent kinetic energy, resulting
from the work of the buoyancy forces; it can be shown (see Monin and Yaglom, 1971;
Brutsaert, 1982) that near the ground this rate is given by (
g
/
T
a
)[(
H
/
c
p
ρ
)
+
0.61
T
a
E
/ρ
].
The dimensionless variables in Equations (2.39) and (2.42) have the variables (
z
−
d
0
) and
u
∗
in common. Accordingly one can hypothesize that in a stratified turbulent flow any
dimensionless characteristic of the turbulence depend only on the following: the height
above the virtual surface level, (
z
−
d
0
); the shear stress at the surface,
τ
0
; the density,
ρ
and the turbulent energy production rate by the buoyancy. These four quantities, which can
be expressed in terms of three basic dimensions, viz. time, length and air mass, can be
combined into one dimensionless variable. This variable, which was proposed by Monin
and Obukhov (1954) (originally for
d
0
=
0), is
z
−
d
0
L
ζ
=
(2.45)
where
L
is known as the Obukhov stability length, defined by
−
u
3
∗
k
(
g
/
T
a
)(
w
θ
0
+
0
.
61
T
a
w
q
0
)
L
=
(2.46)
in which
T
a
is a mean reference temperature (in K) of the air near the ground and the
subscript 0 refers to near-surface values of the fluxes, so that by definition these fluxes
represent (
H
), respectively. In the original formulation of
L
the turbulent
water vapor flux term did not appear; although in many cases the effect of the water vapor
on the density stratification can be neglected, it is still is advisable to include it whenever
possible.
With this hypothesis the dimensionless gradients of the mean wind, of the temperature
and of the humidity, can be written as
/
c
p
ρ
) and (
E
/ρ
k
(
z
−
d
0
)
u
∗
du
dz
=
φ
m
(
ζ
)
(2.47)
ku
∗
(
z
−
d
0
)
w
θ
0
d
dz
=
φ
h
(
ζ
)
−
(2.48)
ku
∗
(
z
−
d
0
)
w
q
0
dq
dz
=
φ
v
(
ζ
)
−
(2.49)
