1 wavelength scaling of ocean surface friction coefficient in wind sea and mixed sea paul a. hwang...
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Wavelength Scaling of Ocean Surface Friction Coefficient in
Wind Sea and Mixed Sea
Paul A. HwangRemote Sensing Division, Naval Research Laboratory
Héctor García-Nava and Francisco J. Ocampo-Torres Departamento de Oceanografía Física, Centro de
Investigación, Científica y de Educación Superior de Ensenada, Ensenada, Baja, California, México
2
Dimensionally consistent similarity relation of the ocean surface friction coefficient:
(wink) (wink) reference wind velocity
Comparison with field dataWind seaMixed sea
3Schlichting, 1955~68+ [1951]
2 20 *
18
u u
udR
100
dimensionless roughness
s
Rk
Smooth pipe
Rough pipe
Hagen-Poiseuille1839-40 Blasius
1913
Prandtl1933-35
Nikuradse1929
2 20 *
12 fu C U
Rough plate
U lR
1000
Cf
s
lksU k
Smooth plate
Prandtl/Schlichting1934 Schultz-Grunow
1940
Law of friction and velocity distribution 0.250.3164 / (20.5)
1/ 2.0 log 0.8 (20.30)
ud
ud
2*2
*
; f
U y uyf Cu U
4
While in almost all branches of fluid mechanics, the drag coefficient is expresses in dimensionally consistent expressed of the flow/fluid properties, in air-sea interaction, the most common expression is a linear function of wind speed. It violates a basic cardinal
rule of fluid dynamics: dimensional consistency.
310 1010C a bU
Law of friction and velocity distribution
5
2
/ 20
2
2**
1 ln
1 ln
p
c
Ck z
z
2
100
2
2**
101 ln
101 ln
p
p
p
c
kC
k z
kz
* 0
1/ 2
1 lnz
z
U zu z
C
zc=gz0/u*2
kpz0 z0*=zc** p
2=gkp **=pu*/g Why 10?
/ 2 / 2exp | exp 0.043;zkz U U The main source of the problem may be the arbitrary reference wind speed (10 m), what’s needed is the free-stream velocity.
*/ 2
cap
c
uC A
g
Ac = 1.220102,ac = 0.704
6
Wavelength rangeD [3.3, 5.7]M [8.6, 19.0] A [28.6, 75.8] J [56.4, 101.6]
The range of wavelengths/water depth:D: Donelan (1979) [3.3, 5.7]/12m (Lake Ontario)M: Merzi and Graf (1985) [8.6, 19.0]/3m (Lake Geneva)A: Anctil and Donelan (1996) [28.6, 75.8]/2/4/8/12m (Lake Ontario)J: Janssen (1997) [56.4, 101.6]/18m (HEXOS, North Sea MPN Dutch coast )
Steady windNo swellkp available
1/ 22 2
i ii
i ii i
x yQ
x y
Q=0.95 Q=0.74
Hwang 2004
0.5
0
2 2
/ 2 20 **
1 ln
1 1ln ln
z
p c
zCz
Ck z z
0.815103
0.825
3 10
0.989/ 23
/ 2 0.947
3 / 2
1.289 10 , 0.89, 0.024,
1.145 10 , 0.89, 0.025,
1.163 10 , 0.88, 0.025,
1.039 10 , 0.85, 0.028
p
p
p
p
UQ S
g
U Q Sc
UQ S
gC
U Q Sc
0.704
*2
0.696
2 *
,
1.220 10 , 0.95, 0.017,
1.086 10 , 0.94, 0.019.
p
p
uQ S
g
u Q Sc
0.423*3
0.442
3 *
0.391103
10
0.593
3 10
6.059 10 , 0.74, 0.028,
6.010 10 , 0.77, 0.026,
1.632 10 , 0.55, 0.034,
1.476 10 , 0.60, 0.033,
5.5
p
p
p
p
uQ S
g
u Q Sc
UC Q Sg
U Q Sc
4 0.522
10
3 0.524*
32 10 , 0.60, 0.032,
2.839 10 , 0.79, 0.025.
U Q S
u Q S
Hwang 2005
Comparison of calculated C10 using wavelength scaling similarity with measurements. Once the similarity relation is found, it is easy to compute C10 for practical application.
10
**/ 2
10 #
cac
a
AC
A
0 0*
0.5/ 2exp
pk z z
C
/ 2
10
0*
0*
ln /ln 10 /
U
p
UR
U
zk z
210 / 2 UC C R
2 2 2* 10 10 / 2 / 2u C U C U
#=pU10/g, **=pu*/g 1/ 2
* 0
1 lnzz
U zCu z
2 2
/ 20 0*
1 1ln lnp
Ck z z
9Hwang 2005
Geernaert et al. 1987Dobson et al. 1994Banner et al. 1999
10
Mixed sea: IntOA Experiment (García-Nava et al. 2009)Hwang, García-Nava, and Ocampo-Torres 2011
(JPO, in press)
11
(0.142, 0.932)
(0.178, 0.800)
(0.192, 0.598)
(rms, R2)
1/
pM
S d
S d
12 2
2 2
/ /w pw s pspM
w s
Equivalent momentum weighting
3 apparent choices of reference frequencies. Weighting by equivalent momentum consideration.
12
** * /pu g # 10 /pU g
Works very well for mixed seas also.
12 2
2 2
/ /w pw s pspM
w s
1/
pM
S d
S d
13Swell also contributes, a swell index is suggested.
pMs
pw
I
12 2
2 2
/ /w pw s pspM
w s
1/
pM
S d
S d
14
3 2
3 2
10
3 210
0.0291 0.0405 0.0180 0.00156
1.094 1.526 0.741 0.2340.00129
0.985 1.374 0.851 0.191
c s s s
c s s s
s s s
A I I I
a I I IA
a I I I
10
**/ 2
10 #
cac
a
AC
A
15
The similarity relation of drag coefficient as a function of dimensionless frequency and swell index.
3 2
3 2
10
3 210
0.0291 0.0405 0.0180 0.00156
1.094 1.526 0.741 0.2340.00129
0.985 1.374 0.851 0.191
c s s s
c s s s
s s s
A I I I
a I I IA
a I I I
10
**/ 2
10 #
cac
a
AC
A
16
Now the similarity relation of drag coefficient can be applied to both wind sea and mixed sea.
17
2 2 4 2 2 4 2 2 4# 10 * / 2 ** *
# 10 * / 2 ** *
/ , / , // , / , /
e g U e g U e g uU g U g u g
# ** **x x t Hwang 2006
18
10pU
UR
g
*pU
uR
g
22 2210* *
/ 2 102 2 2/ 2 10 / 2
UUu uC C R
U U U
0.50 / 2exppk z C
{ } 10
{ } / 2
/ 2
Pierson & Moskowitz 19641.25p FD
p FD
c U
c U
U U
/ 2
10
0*
0*
ln /ln 10 /
U
p
UR
U
zk z
19
0.5
0
2
/ 20
1 ln
1 ln
z
p
zCz
Ck z
0.50 / 2exppk z C ** **
*/ 2
cap
c
uC A
g
Ac = 1.220102,ac = 0.704
Hwang 2006
20
• Friction coefficient of the ocean sea surface: dimensionally consistent similarity relation referenced to free stream reference wind velocity–Wind sea–Mixed sea
• U/2 is U Hwang, P. A., H. García-Nava, and F. J. Ocampo-Torres, 2011:
Dimensionally consistent similarity relation of ocean surface friction coefficient in mixed seas. J. Phys. Oceanogr., doi:
10.1175/2011JPO4566.1 (in press).
Wave age dependence of the dynamic roughness (Charnock coefficient). Field data are from Toba et al. (1990). Solid curves is wavelength scaling. Both original and corrected
versions of the Bass Strait data are shown.
0.52 0.5 2 0.7040* ** / 2 ** **exp exp 0.0122z C
Hwang 2005
22
1( 1)
1
ab
cR bP AB
apb
* *re R
1/ bb
a
ARB
arb
1 1
( 1)
1
bb
bcQ B R b
bqb
in dis netdS Q Q Qdt
Hwang & Wang 200423
* * *
* * *
a p
b q
e Ax Pt
Bx Qt
Burling 1959; Hasselmann et al. 1973; Donelan et al. 1985; Dobson et al. 1989; Babanin and Soloviev 1998
2 2
2 2
2 *
2 *
* 2 * 2 *
* 2 * 2 *
ln2 0 *
2 1 2 *
ln2 0 *
2 1 2 *
exp( )2 ln
exp( )2 ln
a p
b q
x
x
e A x P t
B x Q t
A xa x
B xb x
1( 1)
1
ab
cR bP AB
apb
1 1
( 1)
1
bb
bcQ B R b
bqb
1 11
* *p pw
neta
P p
3* *
** *
*
2
2
w net
a net
p
dE gQdt
e Udee Ptdt
*
2 2
* 4rms
Ug
ge
U
24
2.3
2.3*0.10 0.10
p
Uc
* *32 2w dis agQ Ue
3 3.3* *, with 0.20aU e
25Mature Young
* *re R
Phillips 1985Felizardo and Melville 1995
Hanson and Phillips 1999 26
27
28
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