Measurement and Parameterization of Momentum and Scalar Turbulent Transfer
Coefficients
C.W. Fairall
1. Introduction
Turbulent fluxes characterize a major part of ocean-atmosphere exchange. While
they may be measured directly with appropriate sensors placed on a suitable platform
over the ocean, most applications require estimates distributed over space and time. The
principal use of direct in situ flux observations is to advance indirect methods so that
fields of fluxes can be determined from variables, such as wind speed and sea-surface
temperature, that are available on the required space/time scales. The indirect methods
are known as bulk flux algorithms. Bulk flux algorithms have been in use for nearly a
century. They now form the basis for the ocean-surface boundary condition in virtually
all climate and NWP models, retrieval of turbulent fluxes from satellite observations, and
have been used extensively to estimate the heat balance of the oceans from historic
weather observations from volunteer observing ships (WCRP 2000). Advances in
understanding of the physical processes involved in air-sea exchange and in observing
technologies has promoted steady improvements in the sophistication and accuracy of
these algorithms.
2.
Theory
In bulk algorithms the turbulent fluxes are represented in terms of the bulk
meteorological variables of mean wind speed, air and sea surface temperature, and air
humidity:
1
/
21
/
2


w
x
c

X

C
S

X
xc
dS
x
(1)
where x can be u, v wind components, the potential temperature, , the water vapor
specific humidity, q, or some atmospheric trace species mixing ratio. Here cx is the bulk
transfer coefficient for the variable x (d being used for wind speed) and Cx is the total
transfer coefficient; ΔX is the sea-air difference in the mean value of x, and S is the mean
wind speed (relative to the ocean surface), which is composed of a magnitude of the
mean wind vector part U and a gustiness part Ug:
2 2

X

X

X
(
z
);
S

U

U

U
G
.
sea
g
(2)
In [2] z is the height of measurements of the mean quantity X(z) above the sea surface
1

(
U
U
)2 is the gustiness factor. The gustiness term in [2]
(usually 10 m) and G
g/
represents the near-surface wind speed induced by the BL-scale; it prevents the transfer
coefficients from becoming singular at low wind speeds.
Properly scaled dimensionless characteristics of the turbulence at reference height z
are universal functions of a stability parameter,   z / L , defined as the ratio of the
reference height z and the Obukhov length scale, L. Thus, the transfer coefficients in [1]
have a dependence on surface stability prescribed by Monin-Obukhov similarity theory:
1



1
/
2
c
1
/
2
xn
c
(
)

, c
 ,
(3)
xn
1
/
2
ln(
z
/
z
)
1

(
c
/
)

(
)
ox
xn
x
where the subscript n refers to neutral ( = 0) stability, x is an empirical function
describing the stability dependence of the mean profile, and zox is a parameter called the
roughness length that characterizes the neutral transfer properties of the surface for the
quantity, x (see also Fairall et al. [2003] for details).
1
/
2
x
3.
Instrumentation
The neutral transfer coefficients (or, equivalently, the roughness lengths) are
determined by direct observations of the fluxes and associated mean bulk variables
required in [1]. Mean bulk variables may be measured with relatively slow response
instruments optimized for accurate means. Turbulence variables require a flat frequency
response to fluctuations out to about 10 Hz (somewhat dependent on the height of the
sensor and the conditions). A variety of techniques have been used to estimate the fluxes
(Smith et al. 1996) but the eddy-correlation method is considered the standard. In the
eddy-correlation method u’ and x’ are measured and the flux is estimated as a simple time
'x' 
w
'x'). The inertial-dissipation method (IDM)
average of their products (i.e., w
has also seen broad application. IDM is based on the high-frequency part of the variance
spectrum. Its principal advantage is insensitivity to motion (hence, no motion
corrections). However, an empirical stability function is required to obtain the flux so it
is not considered an unbiased standard. Velocity turbulence is typically measured with a
sonic anemometer or a multiport pressure system; humidity turbulence is usually
measured with a fast infrared absorption hygrometer; temperature turbulence is usually
obtained from sonic anemometer speed-of-sound or from micro-thermal wires. Aircraft
and ship platforms dominate the transfer coefficient database, and these require motion
corrections to obtain the true air velocity (Edson et al., 1998). An example of a cluster of
sensors for a ship-based flux observing field program is shown in Fig. 1.
4. Computation of Transfer Coefficients
The reduction of an ensemble of observations of turbulent fluxes and near-surface
bulk meteorological variables to estimates of the mean 10-m neutral transfer coefficient is
a problem of some subtlety. The straightforward approach is to convert each observation
to Cx10n
w
'
x
'
C


,
x
10
n
(4)
U

X
G
ln(
10
/
z
)
ln(
10
/
z
)
10
n
10
n
o
ox
then average to obtain
w
'x
'

C
 
.
x
10
n
(5)
U

X
G
10
n
10
n
The 10-m neutral values of the mean profile are computed as
u
u
*10
* z
U

ln(
)

U
(
z
)

[ln(
)


(
z
/
L
)]
10
n
u
(6a)
z
10
o
 


2


x
x
*10
* z

X


ln(
)


X
(
z
)

[ln(
)


(
z
/
L
)]
10
n
q
,
(6b)
z
10
ox
where x can be   or q . Note, the sign difference between [6a] and [6b] follows from
X being defined as Xs  X(z) in Equation [1].
Because artificial correlation may confuse attempts to compute the mean transfer
coefficients via [5], Fairall et al. [2003] computed estimates of mean transfer coefficients
as a function of wind speed. Here the fluxes are averaged in wind speed bins and the
mean transfer coefficient is the one that correctly returns the mean or median flux

w
'x
'

C
 
C

,
x
10
n
x
10
nb
(7)

w
'x
'
b
where the subscript b refers to values computed with the bulk algorithm.
The accuracy of a set of transfer coefficients from a particular field program is
extremely difficult to assess. It is clear that uncertainty in the coefficient is the combined
result of the flux and bulk mean variable inaccuracies. Sampling uncertainty, sensor bias,
frequency attenuations, platform flow distortion, and inadequate motion corrections all
degrade the results (Fairall et al. 1996; 2000; McGillis et al. 2001). The actual
computation of the transfer coefficient also relies on the application of various 2nd-order
physical corrections (Fairall et al. 2003): choice of Ψ functions, a cool-skin correction
SST, reduction in the water vapor pressure of seawater by salinity, use of a gustiness
parameter, the dilution effect (Webb et al. 1980), or flow distortion corrections for mean
wind speed (Yelland et al. 1998). High-quality estimates of the scalar transfer
coefficients require conditions where X is large. This both improves the signal-to-noise
ratio of the flux observation and reduces the measurement fractional error in X .
5.
Turbulent Flux Parameterizations
Turbulent fluxes may be estimated from specifications (observations) of the basic
mean bulk variables in [1] by specifying the height and stability-dependent transfer
coefficients, Cx, or the roughness lengths, zox. The momentum (drag) coefficient is known
to vary significantly with mean wind speed while the scalar coefficients have weak wind
speed dependence. The computation of fluxes should also account for the 2nd-order
effects mentioned above, but these are often ignored or assumed to be imbedded in the
transfer coefficient. The observing technologies have advance sufficiently in recent years
that ignoring the 2nd-order effects makes a noticeable difference. The drag coefficient is
often represented as a simple wind speed dependent formula,
C
F
(U
d10
n
10
n)
(8)
or the velocity roughness length is represented as a Charnock plus smooth flow form
(Smith 1988)
2
u
zou
* 
g
u
*
(9)
3
Where u* is the friction velocity, ν the kinematic viscosity of air, α the Charnock
parameter, and β the smooth flow parameter. The sensible heat and latent heat transfer
coefficients may be given as a constant or the roughness length parameterized by the
zouu*
velocity roughness length Reynolds number, Rr 

z0x F(Rr )
(10)
A bewildering variety of algorithms and approaches are available (see Brunke et al.,
2003 for twelve examples). Brunke et al. (2003) compared the algorithms to a ship-based
data set with about 7200 hours of observations and found mean bias magnitudes of 1 to
10 E-3 out of 65 E-3 Ntm-2 for stress and 0.5 to 20 out of 102 Wm-2 for the sum of latent
and sensible heat flux. In Fig. 2 we show the wind speed dependence of momentum and
moisture transfer coefficients for five sample algorithms; mean estimates from the
database used by Brunke et al. (2003) are included. This same basic approach, with a few
modifications to account for the frozen surface, can be applied to turbulent flux
parameterization over ice (see Brunke et al. 2008 for examples from four climate and
weather models).
6.
Issues
Fig. 2 illustrates a few points about bulk flux algorithms. Given a sufficiently large
data base of direct observations, the coefficients in the algorithm can be adjusted to fit the
mean of the observations at each wind speed. We see that the algorithms, with one or
two exceptions, tend to agree for wind speeds between 2-14 ms-1 where (not coincidently)
there is a lot of data. Clearly, more observations are needed at wind speeds greater than
14 ms-1. Another major issue is hidden by Fig. 2, namely the real physical variability of
fluxes in space and time at a given mean wind speed. It is known that the sea state affects
the fluxes, principally the stress. Clearly, a steady wind blowing straight into large swells
will generate more surface stress than the same wind going with the swells (observations
suggest the difference is about a factor of two). Despite decades of work, wave effects on
surface fluxes remain the single most important and confounding problem. Much of the
work on wave-flux interactions has keyed on supplementing mean wind speed with wave
characterizations such as significant wave height, hs, and/or wave phase speed, cp
(Drennan et al. 2005). One approach is to allow the Charnock parameter to be a function
of wave age
F(cp /u*)
(11)
Another assumes roughness length scales with wave height
2
z
/
h

F
(
c
/
u
)
or
F
(
h
g
/
c
)
ou
s
p
*
s
p
(12)
To date these approaches have been ‘promising’ on limited datasets but a simple,
universal form that reconciles all conditions has not been found.
4
Another interesting topic is the possible difference in sensible heat and moisture
transfers. While scalar transfer in the atmospheric is dominated by turbulent flux,
molecular diffusion must contribute heavily to the transfer at mm-scales near the
interface. Because the molecular diffusion coefficients for water vapor and heat are 20%
different, we expect their bulk transfer coefficients to be slightly different (order of 5%)
but not sufficiently different to be unambiguously detectable by today’s observations. At
very high wind speeds evaporation of sea spray will effectively enhance moisture flux
and reduce sensible heat flux. Because the spray effects do not scale as [1] the effects
cannot be accounted for simply by adding wind speed dependence to the bulk transfer
coefficients (Fairall et al. 1995). Progress on developing simple bulk algorithms to
characterize spray contributions continues (e.g., Andreas et al. 2008). However, direct
measurement of this effect is even more difficult than wave effects.
References
Andreas, E.L., P.O.G. Persson, and J.E. Hare, 2008: A bulk turbulent air–sea flux
algorithm for high-wind, spray conditions. J. Phys. Ocean., 38, 1851-1896.
Brunke, M. A., C W. Fairall, and X. Zeng, 2003: Which bulk aerodynamic algorithms are
least problematic in computing ocean surface turbulent fluxes? J. Clim., 16, 619-635.
Brunke, M. A., M. Zhou, X. Zeng, and E.L. Andreas, 2009: An intercomparison of bulk
aerodynamic algorithms used over sea ice with data from the SHEBA experiment J.
Geophys. Res., to appear.
Drennan , W.M., P.K. Taylor, and M.J. Yelland, 2005: Parameterizing the sea surface
roughness. J. Phys. Oceanog., 35, 835-848.
Edson, J.B., A. A. Hinton, K. E. Prada, J.E. Hare, and C.W. Fairall, 1997: Direct
covariance flux estimates from moving platforms at sea. J. Atmos. Oceanic Tech., 15,
547-562.
Edson, J. B. and C. W. Fairall, 1998: Similarity relationships in the marine atmospheric
surface layer for terms in the TKE and scalar variance budgets. J. Atmos. Sci., 55, 23112338.
Fairall, C.W., J. Kepert, and G.J. Holland, 1995: The effect of sea spray on surface
energy transports over the ocean. The Global Atmospheric Ocean System, 2, 121-142.
Fairall, C.W., E.F. Bradley, D.P. Rogers, J.B. Edson, and G.S. Young, 1996: Bulk
parameterization of air-sea fluxes for TOGA COARE. J. Geophys. Res.,101, 3747-3767.
5
Fairall, E.F. Bradley, J.S. Godfrey, J.B. Edson, G.S. Young, and G.A. Wick, 1996: Cool
skin and warm layer effects on the sea surface temperature. J. Geophys. Res., 101, 12951308.
Fairall, C.W., A.B. White, J.B. Edson, and J.E. Hare, 1997: Integrated shipboard
measurements of the marine boundary layer. J. Atmos. Oceanic Tech., 14, 338-359.
Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk
parameterization of air-sea fluxes: Updates and verification for the COARE algorithm. J.
Clim., 16, 571-591.
Fairall, C. W., Ludovic Bariteau, A.A. Grachev, R. J. Hill, D.E. Wolfe, W. Brewer, S.
Tucker, J. E. Hare, and W. Angevine 2006: Coastal effects on turbulent bulk transfer
coefficients and ozone deposition velocity in ICARTT. J. Geophys. Res., 111, D23S20,
doi:1029/2006JD007597.
McGillis, W. R., J. B. Edson, J. D. Ware, J. E. Hare, and C. W. Fairall, 2001: Direct
covariance CO2 fluxes across the air-sea interface. J. Geophy. Res., 106, 16,729- 16,746.
Persson, P. O. G., J. E. Hare, C. W. Fairall, W. D. Otto, 2005: Air-sea interaction
processes in warm and cold sectors of extratropical cyclonic storms observed during
FASTEX. Q. J. Roy. Met. Soc., 131, 877-912.
Smith, S. D., C. W. Fairall, G. L. Geernaert, and L. Hasse, 1996: Air-sea fluxes: 25 years
of progress. Bound.-Layer Meteorol., 78, 247-290.
WCRP, 2000: World Climate Research Programme, Final report of the Joint
WCRP/SCOR Working Group on Air-Sea Fluxes (SCOR Working Group 110):
Intercomparison and Validation of Ocean-Atmosphere Energy Flux Fields. WCRP-112,
WMO/TD-No. 1036, Case Postale No. 2300, CH-1211 Geneva 2, Switzerland. 303 pp.
Yelland, M., B. I. Moat, P. K. Taylor, R. W. Pascal, J. Hutchings, and V. C. Cornell,
1998: Measurements of the open ocean drag coefficient corrected for air flow disturbance
by the ship. J. Phys. Oceanogr., 28, 1511-1526.
6
Figure 1. Cluster of sonic anemometers, fast humidity sensors, and mean T/RH sensors
on the jackstaff of the NOAA Ship Ronald H. Brown for GASEX-3. The PSD sonic is
in the center. The sleeved fast-sample LI-7500 is the unit on the right with the large blue
ventilation tube. The two open path units are on the left. Each sonic is mounted to its
own motion-measuring system.
7
-3
3
x 10
CE10n
2.5
2
1.5
1
0.5
0
5
10
5
10
15
20
25
30
15
20
U10n (m/s)
25
30
-3
3
x 10
Cd10n
2.5
2
1.5
1
0.5
0
Figure 2. Turbulent transfer coefficients as a function of 10-m neutral wind speed. The
circles (with 1-sigma median limits) are the observed medians within wind-speed bins as
described by Equation (7). Upper panel is the moisture coefficient; lower panel is the
momentum coefficient. The parameterizations shown are COARE algorithm (red), NCEP
reanalysis (green), NCEP operational (cyan), ECMWF (blue), Large and Pond (magenta).
8