Environmental Engineering Reference
In-Depth Information
K
M
0
u
−
1
z
0
u
∼
.
(7.4)
∗
In turn,
K
M
0
can be estimated by matching the roughness layer with the surface
layer. In neutral stratification, (7.1a) provides an estimate for
K
M
0
∼
u
∗
h
0
, and (7.4)
reduces to the classical formula:
z
0
u
∼
h
0
.
(7.5)
In stable stratification (SS), the familiar formulation for
K
M
in the surface layer is
K
M
=
L
)
−
1
, where
C
u
≈
3 is an empirical dimensionless constant
(Zilitinkevich and Esau, 2007). It corresponds to the log-linear velocity profile (used
below in data analysis) and gives
K
M
ku
z
(1
+
C
u
z
/
2
−
∗
L
at
z>L
. Hence, when
h
0
exceeds
L
,
the roughness-layer eddy viscosity scale becomes
K
M
0
∼
∼
u
∗
u
∗
L
, so that (7.4) gives
z
0
u
∼
L
.
(7.6)
To cover neutral and stable regimes, we interpolate between (7.5) and (7.6) taking
the sum of reciprocals:
z
−
1
0
u
(
C
L
)
−
1
, where
C
and
C
are dimension-
less coefficients (to give priority to the smaller limit:
h
0
or
L
). Then, designating the
roughness length in neutral stratification as
z
0
u
(
(
C
h
0
)
−
1
=
+
C
h
0
), the interpolation formula
=
becomes
z
0
u
z
0
=
1
L
,
(7.7)
1
+
C
SS
h
0
/
C
is an empirical dimensionless coefficient.
In unstable stratification (US),
K
M
C
/
where
C
SS
=
F
1
/
3
b
C
−
1
U
z
4
/
3
, where
C
U
1.7
(Kader and Yaglom, 1990; Zilitinkevich, 2006). In strongly unstable stratification
z
=
ku
∗
z
+
≈
F
1
/
3
b
z
4
/
3
(which corresponds to the
z
1
/
3
velocity
profile used below in data analysis). Then the roughness-layer eddy viscosity scale
becomes
K
M
0
∼
>
−
L
this reduces to
K
M
∼
F
1
/
3
b
h
4
/
3
0
, and (7.4) gives
L
)
1
/
3
,
z
0
u
∼
−
h
0
/
h
0
(
(7.8)
so that
z
0
u
increases with increasing instability (as predicted in Coelho and Hunt,
1989; Zilitinkevich et al., 2006a,b).
A reasonable interpolation formula linking the neutral and the strongly unstable
regimes is just the sum of the right hand sides of (7.5) and (7.8):
C
US
h
0
−
1
/
3
z
0
u
z
0
=
1
+
,
(7.9)
L
where
C
US
is one more empirical dimensionless coefficient.
The suggested theory is evaluated with data measured over very rough surfaces
including vegetation and urban canopy (Table 7.1). For the stable stratification data
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