THE NEAR-SURFACE LAYER OF THE OCEAN
increase of the viscous sublayer thickness. With increasing friction
velocities, drops with radii greater than about 1 mm lead to waves that cause
a rather rough flow, while waves produced by smaller drops do not disturb
the flow smoothness.
Smooth surface waves contribute to the surface roughness in a different
way compare to random roughness elements. According to Motzfeld’s
(1937) experiment in a wind tunnel, the drag coefficient at a height of 0.2 m
over surface waves is about seven times smaller than that over a surface with
rough elements of the same size.
et al. (1997) estimated that
under the assumption of a logarithmic wind profile this implies a reduction
of the roughness length of the rain-induced wavelets by a factor of 0.4 when
compared to the roughness length of random rough elements of the same
height. This effect leads to some increase in the wave height above which
the flow becomes rough shown by the curve labeled
,
0
w red
h
r in Figure
2-22a. Taking into account the Marshall-Palmer drop-size distribution (1.71)
an upper limit of the mean height of the wavelets is estimated from the
formula,
,
,
0
0
0
0
0
0
exp 2
/
exp 2
c
c
w red
w red
r
r
h
P
h n
r dr
n
r dr
f
f
§
·
/
/
¨
¸
¨
¸
©
¹
³
³
(2.123)
Formula (2.123) implies that the rain-induced waves do not decay and
uniformly cover the sea surface. Equation (2.123) has been resolved
substituting (2.113) into (2.122) and subsequently approximating
3/ 4
,
0
10
2
0
1
exp
/
w red
h
r
r r X
E E
E
,
(2.124)
where 0 0.0129
E
mm, 1 1.60686
E
mm
1/4 , and 2 1.0978
E
. The resulting
relationship is;
^
`
7 / 4
,
0
1
7 / 4
1
1
2
2
7/4 ,2
2
7/4 ,
2
2 e x p 2
.
w red
c
c
c
h
R
r
r
r
r
r
X
X
E
E X
E
*
/
/
*
/
/
/
(2.125)
Figure 2-22b shows the roughness length calculated for the rain-induced
wavelets using a coarse estimate given by Lettau (1969) as
1.19
0
,
0.058
r
w r e d
z
h
|
.
For comparison purposes, the roughness length 0C
z of the wind-induced
surface roughness under neutral conditions according to Charnock’s (1955)
formula is also given on this composite plot. The above estimate for the raininduced roughness length is the maximum possible value, requiring an
132
Schl ssel
ü
increase of the viscous sublayer thickness. With increasing friction
velocities, drops with radii greater than about 1 mm lead to waves that cause
a rather rough flow, while waves produced by smaller drops do not disturb
the flow smoothness.
Smooth surface waves contribute to the surface roughness in a different
way compare to random roughness elements. According to Motzfeld’s
(1937) experiment in a wind tunnel, the drag coefficient at a height of 0.2 m
over surface waves is about seven times smaller than that over a surface with
rough elements of the same size.
et al. (1997) estimated that
under the assumption of a logarithmic wind profile this implies a reduction
of the roughness length of the rain-induced wavelets by a factor of 0.4 when
compared to the roughness length of random rough elements of the same
height. This effect leads to some increase in the wave height above which
the flow becomes rough shown by the curve labeled
,
0
w red
h
r in Figure
2-22a. Taking into account the Marshall-Palmer drop-size distribution (1.71)
an upper limit of the mean height of the wavelets is estimated from the
formula,
,
,
0
0
0
0
0
0
exp 2
/
exp 2
c
c
w red
w red
r
r
h
P
h n
r dr
n
r dr
f
f
§
·
/
/
¨
¸
¨
¸
©
¹
³
³
(2.123)
Formula (2.123) implies that the rain-induced waves do not decay and
uniformly cover the sea surface. Equation (2.123) has been resolved
substituting (2.113) into (2.122) and subsequently approximating
3/ 4
,
0
10
2
0
1
exp
/
w red
h
r
r r X
E E
E
,
(2.124)
where 0 0.0129
E
mm, 1 1.60686
E
mm
1/4 , and 2 1.0978
E
. The resulting
relationship is;
^
`
7 / 4
,
0
1
7 / 4
1
1
2
2
7/4 ,2
2
7/4 ,
2
2 e x p 2
.
w red
c
c
c
h
R
r
r
r
r
r
X
X
E
E X
E
*
/
/
*
/
/
/
(2.125)
Figure 2-22b shows the roughness length calculated for the rain-induced
wavelets using a coarse estimate given by Lettau (1969) as
1.19
0
,
0.058
r
w r e d
z
h
|
.
For comparison purposes, the roughness length 0C
z of the wind-induced
surface roughness under neutral conditions according to Charnock’s (1955)
formula is also given on this composite plot. The above estimate for the raininduced roughness length is the maximum possible value, requiring an
132
Schl ssel
ü