Geoscience Reference
In-Depth Information
2.7
Evaporation measurements on Lake Ontario have revealed that the mass transfer coefficient in
Equation (2.36) is Ce
=
1
.
1
×
10
−
3
,
on average, under neutral conditions, with measurements at
10 m above the surface. If the momentum roughness of a water surface can be assumed to be
z
0
=
0
.
02 cm, estimate the scalar roughness for water vapor,
z
0v
.
2.8
In practical applications, the wind
pr
ofile in the lower atmosphere is sometimes approximated by
a power-type equation, as follows:
u
=
C
p
u
∗
(
z
/
z
0
)
m
. In this equation,
C
p
and
m
are constants,
whose values can be assumed to be around 6 and (1
/
7), respectively, under neutral conditions.
(a) Derive an expression for the drag coefficient, Cd, with this eq
ua
tion. (b) Calculate its magnitude
if the surface roughness is
z
0
=
0
.
02 cm and the wind speed,
u
, is measured at 10 m above the
surface. (
c
) Ne
xt
, consider the mass transfer coefficient, Ce, as given by Equatio
n
(2.36) in which
−
q
=
q
s
−
q
,
q
s
is the value of the specific humidity at the surface and
q
is the value at
height
z
. Under conditions of light winds above open water, it can be assumed that Ce
=
Cd. With
this assumption, derive the power-type equation for the specific humidity profile (
q
s
−
q
), which
is the analog of that for wind, given above.
2.9
The stability of the lower atmosphere is commonly characterized by means of the dimensionless
variable,
ζ
=
(
z
−
d
0
)
/
L
,
defined in Equatio
n
(2.45). An alternative variable to do this is the
Richardson number, defined as Ri
=
(
g
/
T
a
)[(
d
θ/
dz
)
/
(
du
/
dz
)
2
]. Derive the relationship between
ζ
and
Ri
,
in terms of
φ
m
and
φ
h
,
defined in (2.47) and (2.48). Assume that the water vapor flux
term,
w
q
,in
L
is negligible.
2.10
Derive an expression for the specific humidity profile, similar to Equation (2.43), but applicable to
stable conditions. Note that (2.43) is valid only for neutral conditions. Assume that the flux profile
relationship for stable conditions is given by (2.49) with (2.57). Check your result by comparing
with (2.52) and (2.58).
2.11
During a field experiment above a grassy surface, the following mean values were measured over a
1 h period: the temperature at 1.5 m above the ground,
T
1
.
5
=
31
.
29
◦
C; at 3.0 m above the ground,
T
3
=
30.87
◦
C; and the wind speed at 2.0 m,
u
2
=
3ms
−
1
. The surface roughness was estimated
to be
z
0
=
0
.
01 m, and the displacement height,
d
0
,
was found to be negligible. (a) Calculate, by
iterat
ion, t
he friction velocity,
u
∗
(m s
−
1
)
,
and the sensible heat flux,
H
Wm
−
2
. The evaporation
term
w
q
in
L
can be assumed to be negligible. (b) If the net radiation is
R
n
=
392Wm
−
2
,
calculate the rate of evaporation from
L
e
E
=
R
n
−
H
, first in W m
−
2
and then in mm month
−
1
.
2.12
In a neutral atmosphere, a northerly wind (i.e. blowing to the south) has a velocity,
u
=
8ms
−
1
,
at
2 m above prairie terrain; the surface parameters are
z
0
=
50 m. (a) Calculate
the
x
- and
y
-components and the direction of the “free stream wind,” (i.e.
u
b
and
0
.
09 m and
d
0
=
0
.
h
b
)by
using
A
=
4.5 and
B
=
1.5 in Equation (2.65). Assume that the ABL has a thickness,
h
b
=
800 m.
(b) If the rate of evaporation is 0
.
6mmh
−
1
, the air temperature is 15
◦
C, and the relative humidity
is 70% at 2 m, what is the specific humidity at
z
=
h
b
=
h
i
=
800 m above the ground? Assume
D
=
0 in (2.71). Hint: combine (2.71) with (2.44).
v
b
at
z
=
2.13
Derive an expr
es
sion for (
θ
1
−
θ
m
) over very rough terrain as the analog of (2.68) in which
θ
s
is replaced by
θ
1
; the latter is the potential temperature at a height,
z
1
,
within the atmospheric
surface layer. Hint: subtract (2.55) from (2.68) and substitute (2.70) for
C
.
2.14
The magnitude of the solar declination angle depends on the day of the year; it varies between
roughly
±
23.28
◦
on the solstices and 0
◦
on the equinoxes. (a) What is the zenith angle of the Sun,
