Page 1

663

Journal of Oceanography, Vol. 66, pp. 663 to 671, 2010

Keywords:

⋅ ⋅ ⋅ ⋅ ⋅ Gas transfer

velocity,

⋅ ⋅ ⋅ ⋅ ⋅ wind speed,

⋅ ⋅ ⋅ ⋅ ⋅ wind wave,

⋅ ⋅ ⋅ ⋅ ⋅ significant wave

height.

* Corresponding author. E-mail: dlzhao@ouc.edu.cn

Copyright©The Oceanographic Society of Japan/TERRAPUB/Springer

where u∗ is the friction velocity of the air, ρa is air den-

sity. U10 is the wind speed at 10 m height above the sea

surface in neutral stratification condition; CD is the drag

coefficient. Many studies have shown that air-sea ex-

change is regulated by turbulence associated with wind

and wind waves at the air-sea interface (Jähne et al., 1987;

Komori et al., 1993). However, it is often difficult to find

a suitable parameter that is robust enough to describe tur-

bulence intensity in natural environmental conditions.

Alternatively, wind speed has been mostly chosen as the

parameter since wind is the primary forcing of the air-sea

boundary layer and easy to obtain from routine observa-

tional data.

In order to extrapolate fluxes over long time and

space scales, gas transfer velocities are usually assumed

to be a function of wind speed alone (Liss and Merlivat,

1986; Wanninkhof, 1992; Nightingale et al., 2000a, b;

Sweeney et al., 2007). These relationships show a wide

range of scatter, especially at high wind speed, and give

rise to large discrepancies in the estimation of air-sea gas

fluxes. Such a scatter could be caused by the uncertain-

ties in the measurement of gas transfer velocities and in

the determination of the wind speed. It could also be

caused by other factors that influence gas transfers, but

have not been taken into account. For instance, in addi-

A Practical Bi-parameter Formula of Gas Transfer

Velocity Depending on Wave States

DONGLIANG ZHAO1,2* and LIAN XIE2

1Physical Oceanography Laboratory, Ocean University of China, Qingdao 266100, China

2Department of Marine, Earth and Atmospheric Sciences, North Carolina State University,

Raleigh, NC 27695, U.S.A.

(Received 1 April 2010; in revised form 10 July 2010; accepted 2 August 2010)

The parameter that describes the kinetics of the air-sea exchange of a poorly soluble

gas is the gas transfer velocity which is often parameterized as a function of wind

speed. Both theoretical and experimental studies suggest that wind waves and their

breaking can significantly enhance the gas exchange at the air-sea interface. A rela-

tionship between gas transfer velocity and a turbulent Reynolds number related to

wind waves and their breaking is proposed based on field observations and drag coef-

ficient formulation. The proposed relationship can be further simplified as a function

of the product of wind speed and significant wave height. It is shown that this bi-

parameter formula agrees quantitatively with the wind speed based parameterizations

under certain wave age conditions. The new gas transfer velocity attains its maxi-

mum under fully developed wave fields, in which it is roughly dependent on the square

of wind speed. This study provides a practical approach to quantitatively determine

the effect of waves on the estimation of air-sea gas fluxes with routine observational

data.

1. Introduction

Various air-sea fluxes including momentum, heat,

moisture and gas play a key role in air-sea interaction,

and global climate change. The gas flux at the air-sea inter-

face is typically expressed as the product of the gas trans-

fer velocity kL, solubility s, and the difference of the par-

tial pressure of the gas such as CO2 between air and wa-

ter:

Fk s P

L

P

=−

()

( )

1

CO2wCO2a

where PCO2w and PCO2a are the partial pressure of CO2 in

water and air, respectively. The air-sea momentum flux

or wind stress at the sea surface (τ) can be expressed as:

τρ=

( )

2

aD

C U10

2

CuU

D=

( )

3

∗

2

10

2

Page 2

664D. Zhao and L. Xie

tion to wind, it is believed that wind waves and their

breaking may also directly influence the air-sea bound-

ary-layer processes (Monahan and Spillane, 1984; Jähne

et al., 1987; Ocampo-Torres and Donelan, 1995). Thus,

the effect of wave field on air-sea gas transfer should be

considered in the parameterization of gas transfer veloc-

ity (Wanninkhof, 1992; Zhao et al., 2003; Woolf, 2005).

In the ocean, observations have shown that CD is not

a constant but highly variable. Jones and Toba (2001)

presented a comprehensive review on various effects that

can cause the scattering in the measurements of CD. It

has been assumed that the only systematic variation is

with wind speed (Wu, 1980; Smith, 1980; Yelland et al.,

1998).

Toba et al. (2006) suggested that the dynamical con-

ditions at sea can be described by two nondimensional

parameters in terms of wind waves: wave age β∗ and

windsea Reynolds number RB or RH. The wave age (β∗ =

g/u∗ωp) expresses the state of wind wave development.

Here g is the acceleration due to gravity and ωp is the

angular frequency at the spectral peak of wind waves.

The wave age can also be defined in terms of U10, as β =

g/ωpU10. With the development of wind waves, the wave

age and significant wave height (SWH) increase with

fetch. A fully developed wave field has β = O(1), which

is usually less than 1.2 (Pierson, 1991; Jones and Toba,

2001).

The so-called windsea Reynolds numbers RB and RH,

regarded as the fundamental parameters that control the

behavior of air-sea transfers, are defined as:

RuR u H

∗

Bp aHsa

==

( )

4

∗

2

ω νν

;

where Hs is the SWH of wind waves, νa is the air kin-

ematic viscosity. Zhao and Toba (2001) collected a large

amount of data, including a variety of wave states and

wind speeds up to 20 m s–1, and tested statistically a

number of parameterizations. They showed that RB and

RH are the best parameters among those tested to describe

the whitecap coverage. Zhao et al. (2003) proposed a for-

mula for gas transfer velocity as a function of RB:

kR

LB

0.63

=

( )

50 13.

where kL is normalized to Schmidt number (Sc) of 660 in

unit of cm h–1. Woolf (2005) assumed that the contribu-

tion of waves to gas transfer velocity can be explicitly

separated into two parts. From his equations (2), (4), (5)

and (11), kL can be expressed as:

kRu

LH

0.96

=×+

( )

6

−

∗

3 26 10.53 89.

4

where kL is in unit of cm h–1 for Sc = 660, and u∗ in

m s–1. The first term on the right hand side of Eq. (6)

represents the enhancement by breaking waves through

bubble-mediated transfer. The second term is the contri-

bution of non-breaking waves through the direct effect

of wind shear on gas transfer.

However, RB and RH is difficult to determine from

routine observational data due to the lack of information

about u∗ and ωp, which severely limits the practical ap-

plication of Eqs. (5) and (6). In this paper, replacing RB

and RH, a new parameter RHU which can be easily ob-

tained from routine observational data is introduced to

parameterize the gas transfer velocity. By adjusting the

wave age, this approach is shown to be consistent in mag-

nitude with the current parameterizations of gas transfer

velocity under certain wave age conditions. In the condi-

tion of fully developed wave field, it also provides an

upper limit of gas transfer velocity that approaches a quad-

ratic dependence of wind speed.

2. New Parameterization in Terms of RHU

In case of windsea, Toba et al. (2006) indicated that

the existence of similarity laws implies that it is suffi-

cient to select only one of the wave-property variables

(ωp or Hs), together with u∗, in order to completely de-

scribe the dynamical system. In terms of physical con-

stants, the acceleration due to gravity g and the kinematic

viscosity of air νa can be chosen to construct

nondimensional variables. It does not need to consider

the surface tension since it is only related to very high

frequency waves. Therefore, Toba et al. (2006) con-

structed two fundamental nondimensional variables, RB

and RH, to represent the dynamical processes near the air-

sea interface. It is also quite reasonable to assume that u∗

is equivalent to U10, so RH is proportional to U10Hs. On

the other hand, Woolf (2005) suggested that the dissipa-

tion rate is proportional to U10Hs if the energy input to

waves that is related to the cube of wind speed. It is obvi-

ous that the dissipation rate dominates the turbulence near

the air-sea interface. Therefore, in order to parameterize

the gas transfer velocity from routinely available obser-

vations, a new parameter RHU is introduced as:

R U H

10 HUsa

=

( )

7

ν .

Similar to RB and RH, RHU can be considered as a turbu-

lent Reynolds number describing the turbulent intensity

near the air-sea interface. The relationship between RB

and RHU can be determined in two ways. For clarity, a

parameter b = RB/RHU = u∗2/U10Hsωp is defined. The first

method to quantify parameter b is to directly determine

its value from observational data by the least square ap-

proach, in which case, the result is affected by the se-

lected data. The second approach to derive b is from CD

Page 3

A Practical Bi-parameter Formula of Gas Transfer Velocity Depending on Wave States 665

parameterizations and wind-wave growth relations, which

are widely applied in wave studies.

Although many observations focus on the sea sur-

face roughness or wind stress, only a few of them have

measured wave parameters simultaneously. Some repre-

sentative data obtained from field observations which

contain information on waves are adopted in our analy-

sis, as shown in Fig. 1. Without reduction in correlation

coefficient (0.9), the relationship between RB and RHU can

be expressed as:

RR

BHU

=×

( )

8

−

9 5 10.

3

.

Due to the high correlation coefficient, it is reasonable to

conclude that RB is in a linear relationship with RHU, and

parameter b can be taken as a constant. The parameter b

will be discussed further below.

Based on observational data from laboratory and field

programs, a large number of wind-wave growth relation-

ships as a function of nondimensional fetch have been

proposed. It is shown that these relationships are gener-

ally consistent with the Toba-3/2 power law (Toba, 1972)

after eliminating the fetch (Guan et al., 2004). Toba-3/2

power law is expressed as:

gH

u

BgT

u

B

ss

∗

∗

=

=

( )

9

2

3 2

/

0 062.,

where Ts is the significant wave period, and B is an em-

pirical constant. The relationship between Ts and ωp can

be written as ωp = 2π/(1.05Ts) (Mitsuyasu, 1968). Equa-

tion (9) and the definitions of drag coefficient and wave

age are used to rewrite b as b1 in terms of drag coeffi-

cient and wave age. Therefore, the relationship between

RB and RHU can be expressed as:

Rb R

1BHU

=

( )

10

where b1 = 1.11CD3/4β –1/2. Based on the field observa-

tional data from JMA (Japan Meteorological Agency)

buoys, Zhao (2002) suggested that wave age is related to

the nondimensional SWH via a 3/5 power law:

β =

( )

112 56.

10

2

3 5

/

gH

U

s

Eq. (11) and the definition of drag coefficient are used to

104

105

106

RHU

107

108

102

103

104

105

106

RB

Johnson et al.(1998)

Dobson et al.(1994)

Geernaert et al.(1987)

Janssen(1997)

Sugihara et al.(2007)

Lafon et al.(2007)

Eq.(8)

Fig. 1. Relationship between RB and RHU derived from the observational data. The solid line is Eq. (8) determined by the method

of least square.

Page 4

666D. Zhao and L. Xie

rewrite b as b2 in terms of CD and β. Thus RB can be de-

scribed by RHU:

R b R

2BHU

=

( )

12

where b2 = 4.79CDβ –2/3. Although the proportionality fac-

tors b1 and b2 in Eqs. (10) and (12) are very different in

form, it will be shown later that they are equivalent to

each other in magnitude. It is also interesting to note that

Fig. 2. Comparison of relationships between RB and RHU derived by 3/2 and 3/5 power law. From (a) to (c), CD parameterizations

used in calculations are Wu (1980), Smith (1980) and Yelland et al. (1998), respectively. Equation (8) is denoted as a solid

line in the figures for comparison.

Page 5

A Practical Bi-parameter Formula of Gas Transfer Velocity Depending on Wave States667

if it is taken b1 = b2, a relationship of CD can be obtained

as CD = 2.9 × 10–3β2/3, which predicts that CD increases

with the development degree of wind waves.

In order to quantitatively compare Eqs. (10) and (12),

CD must be specified first. Three representative formulas

parameterized in terms of wind speed proposed by Wu

(1980), Smith (1980) and Yelland et al. (1998) are em-

ployed in our calculations. At the same time, SWH must

also be specified in the analysis. It is assumed that SWH

can not be greater than that of a fully developed wave

field that is specified by wind speed alone and independ-

ent of fetch. Following Carter (1982), the maximum of

SWH is taken as:

HU

sm=

( )

130 025.

10

2

.

Substituting Eq. (13) into Eq. (11), wave age β ≈ 1.1,

which agrees with the limitation suggested by Pierson

(1991).

In order to compare Eqs. (10) and (12), wind speed

U10 is specified varying from 1 to 20 m s–1, Hs increases

from 0.1Hsm to Hsm for each U10, in which Hsm is deter-

mined by Eq. (13). Then RHU can be calculated for νa =

1.53 × 10–5 m2s–1 at 20°C. CD is calculated from U10 for

each of the three formulae proposed by Wu (1980), Smith

(1980) and Yelland et al. (1998), and u∗ is determined

from Eq. (3). Wave age β is calculated from Eq. (11),

which will then be used to determine ωp from the defini-

tion of β. Finally, RB can be calculated from u∗ and ωp.

The comparisons between Eq. (10) and Eq. (12) are shown

in Fig. 2 for the three CD formulas stated above. Equa-

tion (8) is also shown in Fig. 2 for comparison. The rep-

resentative values of b1 and b2 are depicted in Table 1. It

can be seen that Eqs. (10) and (12) are quite consistent in

magnitude, no matter which CD formula is applied. This

indicates that the two methods give similar results for a

practical range of wave ages in determining the relation-

ship between RB and RHU. It is also shown that both Eqs.

(10) and (12) determined by this method agree with Eq.

(8), especially at higher wind speeds. As shown in Table

1, the values of b1 and b2 vary within a relatively small

range, and their average values are comparable in magni-

tude.

The coefficients of b1 and b2 are complicated param-

eters related to wind and wind waves. It is beyond the

scope of this paper to discuss the details of these com-

plex relationships. For simplicity, we assume that b1 and

b2 can be approximately taken as a constant. Taking this

constant as the average value of the six average values

for b1 and b2 shown in Table 1, the relationship of RB and

RHU can be expressed as:

RR

BHU

=×

( )

14

−

8 2 10.

3

.

Substituting Eq. (14) into Eq. (5), the gas transfer veloc-

ity can be parameterized by RHU as:

kR

LHU

0.63

=×

( )

15

−

6 3 10.

3

.

By substituting the value of νa at 20°C, Eq. (15) can be

further simplified as a function of (U10Hs):

kU H

(

10Ls

=

)

( )

166 81.

0 63

.

where Hs, U10 and kL are in units of m, m s–1 and cm h–1,

respectively. Equation (16) shows that gas transfer ve-

locity is proportional to the product of wind speed and

SWH. For a given wind speed, it predicts that gas trans-

fer velocity increases with SWH. In the open ocean, SWH

can vary from several centimeters to a few tens of meters

for different wave states. As a result, it leads to a signifi-

cant difference in gas transfer velocity parameterizations

between those that consider wave effect and those in

which wave effect is neglected.

It must be kept in mind that the proportionality fac-

tor in Eq. (16) is highly uncertain. This uncertainty re-

mains to be reduced by more observational data. Never-

theless, as will be discussed in the next section, Eq. (16)

quantitatively agrees with various existing

parameterizations by adjusting the wave age. Thus, Eq.

(16) can be a practical way to consider the wave effect in

the estimation of air-sea gas fluxes.

By using wind-wave growth relationships, the coef-

Authors

b1 (×10–2)

b2 (×10–2)

Max.Min.Aver.Max.Min.Aver.

Wu (1980)

Smith (1980)

Yelland et al. (1998)

1.96

1.79

1.83

0.504

0.418

0.369

0.904

0.867

0.851

2.20

1.96

2.01

0.361

0.281

0.238

0.816

0.760

0.745

Table 1. The values of b1 and b2 calculated from Eqs. (10) and (12) with three formulas proposed by Wu (1980), Smith (1980) and

Yelland et al. (1998).

Page 6

668D. Zhao and L. Xie

ficient b related RB and RHU can be written as some kind

of combination of drag coefficient CD and wave age β,

such as b1 and b2, in which the parameters of U10 and Hs

are not explicit. Although U10 and Hs can vary drastically

in the field, the values of CD ([0.5~1.5] × 10–3) and β

(0.1~1.2) are limited in a relatively narrow range, and

their combinations will further reduce the variations of

b1 and b2. As a result, U10 and Hs have little effect on the

coefficient of b. It is also hopeful that a good average

value of b is obtained in Eq. (14), so Eq. (16) is a robust

representation of gas transfer velocity at various wave

states.

3. Discussion

Figure 3 shows the comparison of Eq. (16) at vari-

ous wave ages with some of the existing parameterizations

in terms of wind speed. Some observational data are also

shown for reference. It is evident that Eq. (16), at wave

age β = 0.4, 0.6, 0.7, 1.0 and 1.2, is consistent with the

relationships proposed by Liss and Merlivat (1986),

Nightingale et al. (2000b), Wanninkhof (1992), Kuss et

al. (2004) and Jacobs et al. (1999), respectively. Such

agreements can be explained as follows.

The relationship of Liss and Merlivat (1986) was

based on a combination of data obtained from a lake ex-

periment and a wind/wave tank study. Due to the short

fetches in lake and laboratory settings, young wave field

with small wave ages is expected to be applicable to the

relationship found in their study. At the same wind speed,

their SWH is less than that at sea, leading to a low gas

transfer velocity according to Eq. (16). The relationship

of Liss and Merlivat (1986) gives the smallest values com-

pared with the others for the same wind speed. Thus, the

relationship of Liss and Merlivat (1986) is expected to

agree with Eq. (16) at a small wave ages (e.g., β = 0.4)

(Fig. 3).

Enhanced transfer might be expected in the open

ocean in response to the occurrence of breaking waves in

a more fully developed wave field. Wave age in the open

ocean varies in a broad range, and is usually greater than

that of lake and laboratory due to longer fetch. The rela-

tionship of Nightingale et al. (2000b) is a best fit to pub-

lished dual tracer data obtained from the coastal and open

ocean. In their figure 13 of Nightingale et al. (2000a), it

is clearly shown that gas transfer velocity increases with

SWH, which supports our argument. However, since no

wave data in numerical form is provided in their paper,

the value of SWH can only be roughly estimated from

their figure. It is found that their wave ages range from

0.5 to 0.9. Thus, it is not surprising that their relationship

0481216 20

U10(m s-1)

0

50

100

150

200

250

kL(cm h-1)

Liss & Merlivat(1986)

Nightingale et al.(2000b)

Wanninkhof(1992)

Kuss et al.(2004)

Jacobs et al.(1999)

Sweeney et al.(2007)

This study(β=0.4)

This study(β=0.6)

This study(β=0.7)

This study( β=1.0)

This study(β=1.2)

Ho et al.(2006)

Wanninkhof et al.(2004)

Borges et al.(2004)

McGillis et al.(2001)

Nightingale et al.(2000a)

Oost (1999)

Fig. 3. Comparisons of gas transfer velocity of Eq. (16) at wave age of 0.4, 0.6, 0.7, 1.0 and 1.2 with other parameterizations in

terms of wind speed. Some observational data are also plotted in the figure.

Page 7

A Practical Bi-parameter Formula of Gas Transfer Velocity Depending on Wave States 669

agrees well with Eq. (16) at a wave age of β = 0.6. The

quadratic relationship between wind speed and gas trans-

fer velocity proposed by Ho et al. (2006) obtained in the

Southern Ocean is also well consistent with Eq. (16) at

wave age β = 0.6 (not shown in the figure). As an inter-

pretation of bomb 14C measurements, Sweeney et al.

(2007) proposed a relationship between gas transfer ve-

locity and wind speed. As shown in Fig. 3, their result is

slightly greater than Eq. (16) at β = 0.6.

The relationship of Wanninkhof (1992) is not directly

associated with any particular experiments, but based on

a modeled fit to the oceanic uptake of bomb-derived ra-

diocarbon. It is surprising that his relationship is highly

consistent with Eq. (16) at wave age β = 0.7. The obser-

vational data used by Kuss et al. (2004) was obtained in

the eastern Gotland Sea (Baltic Sea). They did not pro-

vide wave information. Their relationship is consistent

with Eq. (16) at wave age β = 1.0.

The relationship of Jacobs et al. (1999) was based

on the data obtained in the North Sea during the air-sea

gas exchange program ASGAMAGE. The detailed infor-

mation about wave state was given by Oost et al. (2002).

From their figure 10, it can be seen that their wave ages

range mainly from 0.6 to 1.6, which indicates that the

wave conditions in their study are near the fully devel-

Fig. 4. Comparison of Eq. (17) as an upper limit of gas transfer velocity corresponding to the fully developed waves with some

other parameterizations.

0481216 20

U10(m s-1)

0

50

100

150

200

250

kL(cm h-1)

Liss & Merlivat(1986)

Nightingale et al.(2000b)

Wanninkhof(1992)

Jacobs et al.(1999)

Wanninkhof & McGillis(1999)

This study(fully developed)

oped waves. It is not surprising that the relationship of

Jacobs et al. (1999) agrees well with Eq. (16) at wave

age β = 1.2.

It is true that when the gas transfer velocities were

measured in the field, especially for cases where dual

tracer methods were used, they require some time during

which wave age might vary. The same situation happened

with regard to wind speed and any other environmental

factors. Thus here assigned a certain value of wave age

to the results of previous studies is just trying to describe

the general wave states.

In the condition of fully developed wave field, sub-

stituting Eq. (13) into Eq. (16), we can obtain the upper

limit of gas transfer velocity as:

kU

L=

( )

170 75.

10

1 89 .

.

The comparison of Eq. (17) with some other

parameterizations is shown in Fig. 4. It shows that the

observational data are smaller than those predicted by Eq.

(17). It is also worth noting that Eq. (17) approximates to

a quadratic dependence of wind speed that has been sup-

ported by past studies (Jacobs et al., 1999; Kuss et al.,

2004; Ho et al., 2006).

Page 8

670D. Zhao and L. Xie

In practical application, the lower value from Eqs.

(16) and (17) should be used in the estimation of air-sea

CO2 flux, and the formulae are for Sc = 660. They can be

generally written as:

k

U H

(

10

Sc

U Sc

L

s

=

)

()

()

( )

18

−

−

min

./

./

.

.

.

.

6 81 660

0 75660

0 63

0 5

10

1 89

0 5

where “min” indicates gas transfer velocity will be cho-

sen as the lower value of the two formulae.

We admit that there is great large uncertainty on the

coefficient of b = RB/RHU because it is determined with

limited observational data, and some empirical relations

that may only be valid in ideal conditions. For example,

the drag coefficient of CD introduced in this study con-

tains the very controversial problem of wind-dependence

of sea surface roughness. This ambiguity will certainly

affect the accuracy of b. However, this approach provides

a practical way to include the wave effect on gas transfer

processes, and it can be improved in the future when more

reliable observational data is available. With this bi-pa-

rameter formula of gas transfer velocity, it is hoped that

the uncertainty in the estimation of CO2 fluxes through

the air-sea interface can be reduced to some extent.

4. Conclusions

The scope of this paper is limited to the considera-

tion of the influence of wave state on gas transfer veloci-

ties. Although there is no doubt that other factors may

affect the estimated transfer velocities, this study shows

that variation in wave state is likely to be a major factor.

With the application of Eq. (17) as an upper limit of gas

transfer velocity, and SWH taken from the routine obser-

vational data, such as buoys, satellite altimeters and wave

models, Eq. (16) can be used to estimate the gas transfer

velocity no matter whether the wave state is wind wave

or swell. With the combination of Eqs. (16) and (17), Eq.

(18) is proposed as the final parameterization for kL that

can be used in general conditions with various Schmidt

numbers. Although a thorough validation of this bi-pa-

rameter formula of gas transfer velocity remains to be

carried out using observational data that include the in-

formation of wave state, it is shown that it can reconciles

the differences among several existing parameterizations

obtained at different wave states. This approach provides

a practical, yet more accurate way to estimate air-sea gas

flux by taking into account the effect of waves.

Acknowledgements

The authors would like to thank Professor Yoshiaki

Toba and the two anonymous reviewers for their valu-

able suggestions and constructive comments which have

significantly improved the original manuscript. The au-

thors would also like to acknowledge the financial sup-

port of the National Basic Research Program of China

(2009CB421201; 2005CB422301) and the National Natu-

ral Science Foundation of China (40676014; 40830959;

41076007). The second author also acknowledges the

support from the Changjiang Scholar Program of the Edu-

cation Ministry of China and the U.S. Department of En-

ergy Grant #DE-FG02-07ER64448.

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