1

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 sea

Mixed sea

3Schlichting, 1955~68+ [1951]

2 20 *

1

8u u

udR

100

dimensionless

roughness

s

R

k

Smooth pipe

Rough pipe

Hagen-Poiseuille1839-40 Blasius

1913

Prandtl1933-35

Nikuradse1929

2 20 *

1

2 fu C U

Rough plate

U lR

1000

Cf

s

l

ksU k

Smooth plate

Prandtl/Schlichting1934 Schultz-Grunow

1940

Law of friction and velocity distribution 0.25

0.3164 / (20.5)

1/ 2.0 log 0.8 (20.30)

ud

ud

2*2

*

; f

U y uyf C

u 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**

1ln

1ln

p

c

Ck z

z

2

100

2

2**

101ln

101ln

p

p

p

c

kC

k z

k

z

* 0

1/ 2

1lnz

z

U z

u 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

ca

pc

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/ 2

2 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 **

1ln

1 1ln ln

z

p c

zC

z

Ck z z

0.815

103

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

UQ S

c

UQ S

gC

UQ S

c

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

uQ S

c

0.423

*3

0.442

3 *

0.391

10310

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

uQ S

c

UC Q S

g

UQ S

c

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

z

k z

210 / 2 UC C R

2 2 2* 10 10 / 2 / 2u C U C U

#=pU10/g, **=pu*/g 1/ 2

* 0

1lnz

z

U zC

u z

2 2

/ 20 0*

1 1ln ln

p

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 0.234

0.00129

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 I

A

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 0.234

0.00129

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 I

A

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

U

Uu uC C R

U U U

0.50 / 2exppk z C

{ } 10

{ } / 2

/ 2

Pierson & Moskowitz 1964

1.25p FD

p FD

c U

c U

U U

/ 2

10

0*

0*

ln /

ln 10 /

U

p

UR

U

z

k z

19

0.5

0

2

/ 20

1ln

1ln

z

p

zC

z

Ck z

0.50 / 2exppk z C ** **

*/ 2

ca

pc

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

a

bcR b

P AB

ap

b

* *re R

1/bb

a

AR

B

ar

b

1 1

( 1)

1

b

bb

cQ B R b

bq

b

in dis net

dSQ Q Q

dt

Hwang & Wang 2004

23

* * *

* * *

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 x

a x

B x

b x

1( 1)

1

a

bcR b

P AB

ap

b

1 1

( 1)

1

b

bb

cQ B R b

bq

b

1 11

* *p pw

neta

P p

3* *

** *

*

2

2

w net

a net

p

dEgQ

dt

e U

dee Pt

dt

*

2 2

* 4rms

U

g

ge

U

24

2.3

2.3*0.10 0.10

p

U

c

* *

32 2w dis agQ Ue

3 3.3* *, with 0.20aU e

25

Mature Young

* *re R

Phillips 1985Felizardo and Melville 1995

Hanson and Phillips 1999 26

27

28