Journal of Oceanography, Vol. 66, pp. 663 to 671, 2010
⋅ ⋅ ⋅ ⋅ ⋅ Gas transfer
⋅ ⋅ ⋅ ⋅ ⋅ wind speed,
⋅ ⋅ ⋅ ⋅ ⋅ wind wave,
⋅ ⋅ ⋅ ⋅ ⋅ significant wave
* Corresponding author. E-mail: email@example.com
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-
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
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-
Fk s P
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:
664 D. 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.,
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,
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
B p aHsa
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:
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:
3 26 10.53 89.
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:
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
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:
9 5 10.
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:
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:
R b R
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:
Eq. (11) and the definition of drag coefficient are used to
Johnson et al.(1998)
Dobson et al.(1994)
Geernaert et al.(1987)
Sugihara et al.(2007)
Lafon et al.(2007)
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.
666D. Zhao and L. Xie
rewrite b as b2 in terms of CD and β. Thus RB can be de-
scribed by RHU:
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.
A Practical Bi-parameter Formula of Gas Transfer Velocity Depending on Wave States 667
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:
Substituting Eq. (13) into Eq. (11), wave age β ≈ 1.1,
which agrees with the limitation suggested by Pierson
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-
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:
8 2 10.
Substituting Eq. (14) into Eq. (5), the gas transfer veloc-
ity can be parameterized by RHU as:
6 3 10.
By substituting the value of νa at 20°C, Eq. (15) can be
further simplified as a function of (U10Hs):
k U H
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-
Yelland et al. (1998)
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).
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
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)
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
Liss & Merlivat(1986)
Nightingale et al.(2000b)
Kuss et al.(2004)
Jacobs et al.(1999)
Sweeney et al.(2007)
This study( β=1.0)
Ho et al.(2006)
Wanninkhof et al.(2004)
Borges et al.(2004)
McGillis et al.(2001)
Nightingale et al.(2000a)
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.
A Practical Bi-parameter Formula of Gas Transfer Velocity Depending on Wave States669
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
04812 16 20
Liss & Merlivat(1986)
Nightingale et al.(2000b)
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:
17 0 75.
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).
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:
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.
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.
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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