pression with ε = 1 from the results of their experiments. Quite recently, Taylor and Yelland (2001) have de-
They showed that the Charnock relation has ε = 0 and a termined a formula for roughness length using published
formula proposed by Toba (1979) and Toba and Koga data from the North Sea near the Dutch coast, Lake On-
(1986) has ε = –1. Scaling z 0 by rms wave height σ , tario and the Baltic Sea near the island of Lolland, Den-
Donelan et al. (1993) derived mark. The formula is
z0 U z0 H
= 6.7 × 10 −4 10 (6 ) = A S (7)
σ Cp HS Lp
from data obtained from Lake Ontario, from the North where H S is the significant wave height, Lp peak wave-
Sea near Dutch coast, and from an exposed site in the length, A = 1200 and B = 4.5. However Cheng and
Atlantic Ocean off the coast of Nova Scotia. This for- Mitsuyasu (1992) suggested that the drag coefficient of
mula is also a function of the inverse wave age, like the the water surface is not much affected by the swell steep-
above formulae. ness, based on their wind-wave flume experiment.
These formulae with ε ≠ 0 and the Donelan et al. We believe that the roughness length does not de-
(1993) formula for the dimensionless roughness length pend on the spectral peak frequency, the wave steepness,
are functions of the dimensionless spectral peak frequency or the wave age. We explore another approach that does
ω p or the inverse wave age. Data points of the not use wave state parameters such as the spectral peak
dimensionless roughness length gz 0/u∗2 obtained from frequency or the wave age. The roughness length calcu-
field observations are plotted far apart from those obtained lated from the Charnock relation (3) does not depend on
from laboratory experiments in the scatter diagram of gz0/ the wave state. However figure 10(a) of Yelland and
u∗2 as a function of ω pu∗/g (=u∗/Cp) of Toba et al. (1990, Taylor (1996) indicates that the Charnock constant var-
their figure 14) or Donelan et al. (1993, their figure 2). It ies with the wind speed. Laboratory experimental data
is impossible to determine a regression curve for all the do not always obey the Charnock relation, as shown be-
data. This is because the spectral peak frequency of labo- low. Kondo et al. (1973) postulated that the roughness
ratory data is about ten times as large as that of field data, length is related to the sea-surface irregularities associ-
although the roughness length has almost the same mag- ated with the high-frequency waves. However this rela-
nitude irrespective of whether field data or laboratory data tion of the roughness length to the sea-surface irregulari-
are used. In other words, the 10-m neutral drag coeffi- ties has not yet been confirmed as far as we know. Wu
cient obtained from laboratory experiments is close to that (1980) refined the Charnock relation into
in the ocean, as described below. This indicates that the
roughness length obtained from laboratory data is the al-
αu∗ µu∗ β − 2
most same as that from field data because there is a unique z0 = (8)
relationship between the 10-m neutral drag coefficient and g Γ
the roughness length as defined by Eq. (2). Taylor and
Yelland (2001) pointed out that a clear wave age depend- with β = 2.25 using his wind-wave flume experiment,
ence has proved hard to detect in the open ocean and the where µ and Γ are the viscosity and surface tension of
variations of roughness length predicted by wave-aged- water, respectively. According to this formula, the rough-
based formulae have not been observed. These consid- ness decreases with increasing the surface tension. This
erations suggest that the spectral peak frequency or the contradicts the experimental results of Mitsuyasu and
wave age are not appropriate variables to describe the Honda (1982). Keller et al. (1992) related the roughness
roughness length. There is direct evidence that the rough- length only to the friction velocity. They obtained
ness length does not depend on the peak frequency.
Kusaba and Masuda (1988) conducted a wind-wave flume
experiment to determine an expression for roughness
(u∗ ≤ 0.16 m / s) ( 9)
length. Figure 3 in Kusaba and Masuda (1988) shows that 0.00098u∗
(u∗ > 0.16 m / s)
the roughness lengths and the friction velocities at mean
wind speeds of 10 m/s and 12.5 m/s do not vary with fetch, from a wave-tank experiment. They did not relate the
while the peak frequencies decrease with fetch. Their fig- roughness length to the spectral peak frequency. How-
ure 5(b) shows that gz02/u∗2 for mean wind speeds of 10 ever, their expression does not have a dimensionless form.
m/s and 12.5 m/s is almost constant irrespective of the A wind-wave flume experiment carried out by Mitsuyasu
values of ωpu∗/g. and Honda (1982) showed that the friction velocity of
820 K. Ueno and M. Deushi
wind and the aerodynamic roughness length over me-
chanically-generated regular waves in aqueous surfactant
solution are smaller than those in tap water and that sur-
factant suppresses wind waves. This suggests that the
surface tension is important in determining the wind ef-
fect over water. Honda and Mitsuyasu (1980) obtained 10-2
the drag coefficient over pure water and aqueous sur-
factant solution from a wind-wave flume experiment.
They suggested that the drag coefficient depends on the
physical properties of the fluid and not on the wave state. 0.5 0.6 0.7 0.8 0.9 1 2
As described above, some of the Kusaba and Masuda
(1988) data do not show a dependence of the roughness
length on the wave age. Fig. 1. Dependence of the roughness length on the inverse wave
Furthermore, theoretical studies, such as those of age. The diamonds and crosses show our experimental data
Makin et al. (1995) and Makin and Kudryavtsev (1998), and the Toba (1972) data respectively. The thick solid line,
suggested that the roughness length is independent of the thin solid line and the dashed line show the Charnock
wave age. Makin et al. (1995) developed a model of the relation with the Charnock parameter of 0.0185, the regres-
lower part of the boundary layer above the sea. Their sion curve for the Toba (1972) data and the regression curve
model shows that the dependence of the drag on wave for our data.
age depends sensitively on the relation between the high
wavenumber tail (decimeter and meter waves) and wave
age. If the tail is wave-age independent, the drag or the
roughness length appears to be virtually independent of roughness length. Our experimental data (diamonds) and
wave age. Makin and Kudryavtsev (1998) developed a the Toba (1972) data (crosses) of roughness length gz0/
new scheme to couple waves with the atmosphere. The u∗2 analyzed in the following sections are plotted as a
results of the new scheme show that the waves shorter function of the inverse of the dimensionless wave age in
than 1 m support 50% of the form drag and about 10% of Fig. 1. The horizontal thick solid line, the thin solid line
this is supported by gravity-capillary waves with length and the dashed line show the Charnock relation with a
λ < 0.017 m. This result shows that most of the wave- Charnock parameter value of 0.0185, the regression curve
induced momentum is transferred to centimeter-to-meter gz0/u∗2 = 0.020(ωpu∗/g)–0.33 for the Toba (1972) data, and
waves. This also suggests that the roughness length is the regression curve gz 0/u∗2 = 0.0059( ωpu∗/g)0.37 for our
independent of wave age if the high wavenumber tail is data. The correlation coefficients between the logarithm
wave-age independent. of the dimensionless roughness length ln(gz0/u∗2), and the
We, therefore, explore a new formula for the rough- logarithm of the inverse of the dimensionless wave age
ness length similar to that of Keller et al. (1992) formula ωpu∗/g is 0.39 for our data and 0.21 for the Toba (1972)
(9) as a function of the surface tension, the viscosity and data. The correlation coefficient between the logarithm
the friction velocity in dimensionless form. of the dimensionless roughness length and the logarithm
The purposes of this paper are to: of the inverse of the dimensionless wave age for all data
(1) express the dimensionless aerodynamic rough- is 0.33. However, the correlation coefficient between
ness of water surface as a function of the dimensionless ln(gz 0/u∗2) and ln(u∗) for our data is 0.85. As for our data
friction velocity (Section 2); and the Toba (1972) data, the wave age is not appropriate
(2) determine the parameter of the function such as as a physical quantity to determine the roughness length.
the power exponent using laboratory experimental data The reason why we compare the correlation coefficient
(Section 3); between ln(gz 0/u ∗2) and ln( ω p u ∗/g) with that between
(3) examine whether or not the formula for the ln(gz 0/u∗2) and ln(u∗) instead of ln(z 0) and lnu∗ is given in
roughness length obtained from laboratory experimental Appendix.
data is valid in the ocean by comparing the drag coeffi- Surface tension is one of the physical quantities that
cient calculated from the new formula for the roughness determine the roughness length, as shown by Mitsuyasu
length with that obtained from field data (Section 4). and Honda (1982). This suggests that capillary waves are
related to the roughness length. The frequency spectrum
2. Dimensional Analysis of capillary waves depends on the surface tension, the
We explore physical quantities to determine the water density, the friction velocity and the viscosity, as
roughness length. To begin with we examine whether or shown by Lleonart and Blackman (1980). Donelan and
not the wave age is a physical quantity related to the Pierson (1987) proposed the high-wave number spectrum
A New Empirical Formula for the Aerodynamic Roughness of Water Surface Waves 821
on the assumption that the shortwave energy density re- b
ν 2 gz0 γ 3 u3
flects a balance between direct wind forcing and dissipa- = β 4 ∗ . (14)
tion due to breaking and to the viscosity of water. The γ2 gν gν
high frequency spectrum depends on the viscosity. There-
fore, the kinematic viscosity of water is one of the physi- We determined b by fitting Eq. (14) to the experimental
cal quantities to determine the roughness length. The lin- data.
ear dispersion relation for surface waves is expressed by The roughness length for strong wind seems to obey
the Charnock relation, as shown in the next section. We
Γ 3 have the Charnock relation
ω 2 = gk + k (10)
2 =β (15)
where ω is the angular frequency of a component wave, k u∗
the wave number, ρw water density. The properties of short
surface waves are related to Γ, ρw and g. The accelera- by assuming a = 2/3 and b = –2/3. This suggests that the
tion due to gravity, g, is one of the physical quantities roughness length does not depend on the surface tension
that determine the roughness length. Therefore we assume and the viscosity owing to intense wave breaking by
that physical quantities that determine the roughness strong wind.
length are the surface tension Γ, water density ρw, the Wu (1968) proposed the roughness length for low
kinematic viscosity of water ν , and the friction velocity wind speeds
of wind u∗.
Using Buckingham’s pi-theorem, we obtain the in-
dependent dimensionless combinations ν2gz0/γ2, u∗3/gν z0 = β 2 , (16)
and γ3/gν4, where γ ≡ Γ/ρw. We do not scale z 0 by u∗2/g u∗
because there is a possibility of spurious self-correlation
through the use of the common scaling parameter u∗. We which is derived by setting a = –2/3 and b = –1/3. The
have formula for roughness length at light winds (less than
about 8 m/s) is different from that at wind speeds larger
than about 8 m/s, similar to as the Honda and Mitsuyasu
ν 2 gz0
= f u∗ / gν , γ 3 / gν 4 ,
) (11) (1980) formula (17) described below, since the drag co-
efficient has a different form according to wind speed.
Furthermore we assume that 3. Determination of the Exponents a and b from
Laboratory Experimental Data
a b A water surface is aerodynamically smooth under a
u3 γ 3
f ( 3
u∗ / gν , γ / ν
) = β ∗ 4
(12) light wind and aerodynamically rough under a strong one.
The wind velocity range is conventionally divided into
two parts according to the roughness of water surface.
for simplicity, where β, a and b are numerical constants. The boundary of the light wind and the strong wind is 6
As shown later, the roughness length at moderate winds m/s or 8 m/s according to Yelland and Taylor (1996) or
seems to be proportional to the third power of friction Honda and Mitsuyasu (1980), respectively. Furthermore,
velocity. Making the further assumption that b = 0 ini- we have divided the wind velocity range under which a
tially, we obtain water surface is aerodynamically rough into a moderate
wind range and a strong one. The boundary between the
moderate wind range and the strong one is shown in Sub-
ν 2 gz0 u3 section 3.2 to depend on the kinematic viscosity and sur-
= β ∗ . (13)
γ2 gν face tension. As for pure water, the friction velocity of
the boundary increases from 0.25 m/s to 1.26 m/s with
the decreasing in water temperature from 25° to 7.6°. We
Fitting this equation to our experimental data measured
first examine the roughness in the moderate wind range.
under the experimental condition of the almost same ν
Secondly, we examine the strong wind range and we ex-
and γ , we get a = 1 as discussed in the next section. How-
amine the light wind range last.
ever, the other data points deviate slightly from Eq. (13).
Therefore we tried
822 K. Ueno and M. Deushi
Table 1. Values of the kinematic viscosity and the surface tension used for the calculations.
ν (10 – 6 m2 /s) Γ (10 – 2 N/m) Water temperature (°C)
EXP1 1.4 7.4 4–12
Pure water 1.25 7.4 12
Aqueous surfactant solution 1.25 3.2 12
Mitsuya and Honda (1982)
Pure water 1.4 7.4 4.7–17.0
Aqueous surfactant solution 1.4 2.7 3.8–11.0
Keller et al. (1992) 0.9 7.2 18.5–29.5
Honda and Mitsuyasu (1980) 1.3 7.4 2.9–17.0
Yelland et al. (1998) 1.3 7.4
Wu (1968) 1.0 7.4
3.1 Moderate wind 10-4
The dimensionless roughness length ν 2gz0/γ2 is plot-
ted as a function of dimensionless cube of friction veloc-
ity u∗3/gν in Fig. 2. The squares denote data obtained from 10-5
an experiment using a wind-wave flume conducted in the
Meteorological Research Institute of the Japan Meteoro-
logical Agency (referred to as EXP1 in Table 1). The wind- 10-6
wave flume is 16 m long, 0.5 m wide and 1.3 m high. The
water depth in the flume was kept at 0.4 m during the
experiment. The friction velocity and the roughness length 10-7
over pure wind waves (solid diamonds) and mechanically-
generated regular waves (solid squares) with a frequency
of 2 Hz are measured at fetches of 6 m and 12 m. The 10 103 104 105 106
measurements of the friction velocity were done with the
eddy correlation method using X-wire anemometry at tem-
peratures from 4°C to 12°C. The roughness lengths were
Fig. 2. Dimensionless roughness length versus dimensionless
determined by z0 = zexp(–κu(z)/u∗), derived from the loga- cube of friction velocity. The diamonds and squares show
rithm wind profile (1). The data points are plotted in Fig. our experimental data obtained using the narrow wind flume:
2, assuming that the kinematic viscosity of water ν is solid diamonds: pure wind waves; solid squares: regular
1.4 × 10–6 m2/s at 7.6°C and the surface tension Γ is 7.4 × waves. The triangles represent our experimental data ob-
10 –2 N/m at 10°C. Table 1 summarizes the values of the tained using the wide wind-wave tank: solid triangles: pure
kinematic viscosity and the surface tension used for the water; open triangles: aqueous surfactant solution. Circles
calculations. show the Mitsuyasu and Honda (1982) data: solid circles;
In Fig. 2 the triangles represent the data from an ex- pure water: open circles; aqueous surfactant solution. The
periment done using another wider wind-wave tank in the thick solid line, the dashed line, the thick dot-dashed line,
the thin dot-dashed and the thin solid line show Eq. (18),
Meteorological Research Institute (called EXP2 in Table
the relationship calculated from the Honda and Mitsuyasu
1). The conspicuous feature of this tank is that the wind
(1980) relation for the drag coefficient (17), the relation-
direction relative to the propagation direction of the regu- ship calculated from the Yelland et al. (1998) drag coeffi-
lar waves is variable from 0° to 60°. We report only the cient relation (26), the Keller et al. (1992) relation (9) and
data on the wind direction of 0° relative to the propaga- the Wu (1968) relation (16) respectively.
tion direction of the mechanically-generated waves. We
have performed an experiment under oblique wind con-
ditions and we will publish the results in the future. This
wind-wave tank is 6 m long, 6 m wide and 1.1 m high length were determined from the wind profile using Eq.
and the water depth is 0.5 m. The vertical wind profile (1). The measurements were made at a water temperature
over mechanically-generated regular waves with a fre- of about 12°C using tap water (solid triangles) and water
quency of 2.7 Hz were measured at a fetch of 3 m using a containing surfactant (sodium lauryl sulphate,
Pitot-static tube. The friction velocity and the roughness C12H25OSO3Na) at a concentration of 2.22 × 10–2% (open
A New Empirical Formula for the Aerodynamic Roughness of Water Surface Waves 823
10-4 expressed by the thin dot-dashed line in Figs. 2 and 3
under the assumption that Γ = 7.2 × 10–2 N/m and ν = 9 ×
10–7 m2/s for pure water at 25°C. The dashed line in Figs.
10-5 2 and 3 is the dimensionless roughness length calculated
from the Honda and Mitsuyasu (1980) formula for the
0.00177U100.244 (U10 ≤ 8 m / s) (17)
(U10 > 8 m / s)
10-8 4 using z 0 = 10exp(– κ/ CD ) under the assumption that the
10 105 106 107 108 kinematic viscosity ν is 1.3 × 10–6 m2/s and the surface
3 4 0.2 3
( /g ) u* /g tension Γ is 7.4 × 10 –6 N/m. In Fig. 2 Wu’s light wind
dependence (16) with β = 0.06, revised by Bourassa et
Fig. 3. Dimensionless roughness length versus dimensionless al. (1999) (thin solid line), is plotted under the assump-
cube of friction velocity ( γ3/gν4)0.2u ∗3/g ν. The data sym- tion that the kinematic viscosity is 1.0 × 10–6 m2/s and
bols are the same as Fig. 2. The thick solid line, the dashed the surface tension Γ is 7.4 × 10–2 N/m. The dimensionless
line, the thick dot-dashed line and the thin dot-dashed show roughness length calculated from the Honda and
the moderate wind dependence (21), the relationship calcu-
Mitsuyasu (1980) formula (17) is very close to that cal-
lated from the Honda and Mitsuyasu (1980) relation for the
culated from the Wu (1968) formula (16) at low wind
drag coefficient (17), the relationship calculated from the
Yelland et al. (1998) drag coefficient relation (26) and the speeds.
Keller et al. (1992) relation (9) respectively. The data points, except for one solid triangle, one
open circle and one solid circle, are almost close to the
thick solid line irrespective of whether pure wind waves,
regular waves, pure water or aqueous surfactant solution
are concerned. The Honda and Mitsuyasu (1980) relation
triangles). The kinematic viscosity ν at a temperature of
(dashed line) in the dimensionless friction velocity range
12.6°C and the surface tension Γ at a temperature of 16°C
larger than about 1400 is close to the thick solid line.
for an aqueous solution with a surfactant concentration
These observations indicate that these two dimensionless
of 1.85 × 10–2% were measured as 1.25 × 10–6 m2/s and
quantities ν 2gz 0/ γ 2 and u∗ 3/gν are appropriate for this
3.7 × 10–2 N/m, respectively. The surface tension has a
analysis. The deviation of these three data points from
weak temperature dependence. The kinematic viscosity
the solid thick line seems to be related to the transient
has a weak dependence on the surfactant concentration.
surface state, from an aerodynamically smooth surface at
Neglecting the temperature dependence of the surface
light winds to an aerodynamically rough one at moderate
tension and the concentration dependence of the kinematic
viscosity, we assume that the kinematic viscosity is
To determine the proportional constant β and the
1.25 × 10 –6 m2/s and the surface tension of an aqueous
exponent a of Eq. (13), the method of least squares was
solution with surfactant concentration of 2.22 × 10–2% is
applied only to the data denoted by the diamonds and
3.2 × 10–2 N/m determined by an interpolation of this
squares because the kinematic viscosity and the surface
measurement and the data obtained by Mitsuyasu and
tension were almost the same and the data determined
with the eddy-correlation method are more accurate than
In Figs. 2 and 3 the solid circles (pure water) and the
those obtained from the wind profile. The equation
open circles (aqueous surfactant solution) denote the data
ln(ν2gz0/γ2) = lnβ + aln(u∗3/gν ) was fitted to the loga-
for regular waves in table 2 of Mitsuyasu and Honda
rithm of the data. We have
(1982). For pure water, the kinematic viscosity and the
surface tension are assumed to be 1.4 × 10–6 m2/s rather
than the values of 1.8 × 10–6 m 2/s and 7.4 × 10–2 N/m. ν 2 gz0 u3
The water temperature at which the kinematic viscosity = 6.46 × 10 −11 ∗ . (18)
is 1.8 × 10–6 m2/s is about 0°C. The experiments were
conducted at water temperatures ranging from 3.8°C to
17.0°C (Kusaba, personal communication). Therefore the The correlation coefficient between ln( ν 2 gz 0 / γ 2 ) and
value of 1.4 × 10 –6 m2/s at about 8°C is used rather than ln(u∗3/gν) is 0.98. The value 1.0117 of the exponent is
1.8 × 10–6 m2/s. The Keller et al. (1992) formula (9) is replaced by 1 for convenience. We modified the propor-
824 K. Ueno and M. Deushi
tional constant β of Eq. (13) so that β × 10000 1 = 6.46 × 10-1
10 –11 × 100001.0117 for u∗3/gν of 10000. We obtained a β T M
value of 7.19 × 10 –11. We have -2 B
ν 2 gz0 u3 10-3
= 7.19 × 10 −11 ∗ , (19)
z0 ( m )
which is shown by the thick solid line. The value of this
formula for u∗3/gν of 1000 is 2.7% less than that given 10-5
by Eq. (18). The value of this formula for u∗3/gν of 100000
is 2.7% larger than that given by Eq. (18).
Some open circles, the open triangles and the Keller 10 100 101
et al. (1992) formula (9) at low wind speeds deviate u* ( m/s )
slightly from Eq. (19) in Fig. 2. We use Eq. (14) instead
of Eq. (13). In order to determine the parameter b by the Fig. 4. Roughness length versus friction velocity. The data sym-
method of least squares, Eq. (14) is transformed into bols are the same as Fig. 2. The crosses show the Toba
(1972) data. The thick solid line was calculated from Eq.
(21) with Γ = 7.35 × 10–2 N/m and ν = 1.14 × 10–6 m2/s.
ν 2 gz gν γ3 The thin solid lines show Eq. (21) with Γ = 7.2 × 10–2 N/m
log 2 0 3 = log β + b log 4 . (20)
γ u∗ gν and ν = 0.9 × 10–6 m2/s (denoted by T), Γ = 7.4 × 10 –2 N/m
and ν = 1.4 × 10 –6 m 2/s (denoted by M) and Γ = 2.7 × 10–2
N/m and ν = 1.4 × 10–6 m2/s (denoted by B). The dashed
We use all our data and the Mitsuyasu and Honda (1982) line shows the relationship calculated from Eq. (17) respec-
data except for one data point for tap water and two data tively. The thick and thin dot-dashed line show the Charnock
points for aqueous surfactant solution, probably in the relation with α = 0.0185 and the Keller et al. (1992) rela-
light wind range. We used three data points calculated by tion (9) respectively. The dot-dot-dashed line shows the
the formula (9) of Keller et al. (1992) with u∗ = 0.13 m/s, Kunishi and Imasato (1966) relation (22).
0.145 m/s, 0.16 m/s and eight data points calculated by
the Honda and Mitsuyasu (1980) formula (17) with U 10 =
8 m/s, 11 m/s, 14 m/s, 17 m/s, 20 m/s, 23 m/s, 26 m/s, 29
m/s. Using the method of least squares, we get a b value the water surface is aerodynamically smooth in the range
of 0.194 with a standard deviation of 0.060. We use 0.2 of (γ3/gν4)0.2u∗3/gν less than 140,000. The Yelland and
as the value of b for simplicity. We obtained Taylor formula (1996) shows that the water surface is
aerodynamically smooth at wind speeds less than 6 m/s.
0.2 Our experimental data, denoted by squares and diamonds
ν 2 gz0 γ3 3
= 7.08 × 10 −13 4 . (21) in Figs. 2 and 3, show no clear evidence of an aerody-
γ2 gν gν namically smooth surface. The wind speed of the transi-
tion from a smooth surface to a rough surface is not defi-
The dimensionless roughness length ν2gz0/γ2 is plotted nite. The lower boundary of the wind speed range where
as a function of (γ3/gν4)0.2u∗3/gν in Fig. 3. Equation (21) the moderate wind dependence (21) is valid is at most a
is denoted by the thick solid line. (γ3/gν 4)0.2u∗3/gν value of 140,000.
The roughness length is sensitive to a change in the
surface tension and the kinematic viscosity. Therefore the 3.2 Strong wind
roughness length is strongly dependent on the water tem- In Fig. 2 the Keller et al. (1992) formula (9) (thin
perature through the temperature dependence of the kin- dot-dashed line) for high wind speeds deviates from the
ematic viscosity. The kinematic viscosity of water changes moderate wind dependence (21) above (γ3/gν 4)0.2u∗3/gν
from 1.8 × 10–6 m 2/s at a water temperature of 0°C to of 140,000. This suggests that scaling u∗3 by gν and z 0 by
1.0 × 10–6 m2/s at a water temperature of 20°C. The rough- γ2/gν 2 is not appropriate for the Keller et al. (1992) for-
ness length for a water temperature of 20°C is 9.3 times mula (9) above a friction velocity of 0.16 m/s.
as large as that for a water temperature of 0°C. We express the dimensional diagram of the rough-
The roughness length calculated from the Honda and ness length versus the friction velocity to avoid confu-
Mitsuyasu (1980) formula (17) has a minimum at (γ3/ sion from the dimensionless presentation. Figure 4 shows
gν 4)0.2u∗3/gν of 140,000, corresponding approximately to the dependence of the roughness length on the friction
U10 = 8 m/s. According to Honda and Mitsuyasu (1980) velocity. The thick and thin dot-dashed lines show the
A New Empirical Formula for the Aerodynamic Roughness of Water Surface Waves 825
Charnock relation (3) with α = 0.0185 determined by Wu the roughness length in the strong wind range except for
(1980) and the Keller et al. (1992) formula (9), respec- the friction velocity are probably neither the kinematic
tively. Wind-wave flume data of Kunishi and Imasato viscosity of water nor the surface tension but only grav-
(1966) are quoted in figure 12 of Toba and Ebuchi (1991). ity. We obtain the Charnock relation from dimensional
The equation derived from visual fits to their plots is ex- analysis.
pressed by The roughness length for moderate wind is expressed
by the moderate wind dependence (21) (solid lines) and
that for strong wind is expressed by the Charnock rela-
1.20 × 10 −3 u∗ .92
(u∗ ≤ 1.48 m / s) (22) tion (3). Therefore, the roughness length is expressed by
2.06 × 10 u∗
(u∗ > 1.48 m / s),
which is represented by the dot-dot-dashed line. The 0.2
γ 3 γ 2 u∗ 3
Keller et al. (1992) relation (9) and the Kunishi and 7.08 × 10 −13 4
gν g ν
Imasato (1966) relation (22) display similar behavior such
that the roughness length increases almost proportionally
10 gν gν 3
with a cube of friction velocity almost up to a bend of u∗ ≤ 2.61 × 10 3
Eqs. (9) and (22) and roughly approximate the roughness γ γ2
length determined from the Charnock relation (3) from z0 = (23)
the bend. In Fig. 4, crosses denote the roughness length 0.0185
calculated from the drag coefficient of Toba’s 1961 wind- g
wave tunnel experiment at 15°C in table 1 of Toba (1972). 0.2
The data points, except for five points, lie near the thick u∗ > 2.61 × 1010 3 .
dot-dashed line (the Charnock relation) and the thick solid
line calculated from the moderate wind dependence (21)
with Γ = 7.35 × 10–2 N/m and ν = 1.14 × 10–6 m2/s.
The data points spread widely below the Charnock The Kudryavtsev and Makin (2001) model shows that
relation. This is because the roughness length for the the Charnock parameter gz0/u∗2 increases with U10 from
moderate wind depends on the kinematic viscosity and 7.5 m/s to 20 m/s and is constant over 20 m/s. The mod-
the surface tension. As an example, the thin solid lines erate and strong wind dependence (23) is consistent with
are drawn to show the roughness lengths calculated from the result of Kudryavtsev and Makin (2001). However
the moderate wind dependence (21) with Γ = 7.2 × 10 –2 the Kunishi and Imasato relation (22) also deviates from
N/m and ν = 9 × 10–7 m 2/s (top, denoted by T), Γ = 7.4 × the Charnock relation with increasing the friction veloc-
10 –2 N/m and ν = 1.4 × 10–6 m2/s (middle, denoted by M) ity. The upper limit of the friction velocity range where
and Γ = 2.7 × 10–2 N/m and ν = 1.4 × 10–6 m2/s (bottom, the moderate and strong wind dependence (23) is valid
denoted by B). The top, middle and bottom thin solid lines may be 2 m/s.
correspond to the Keller et al. (1992) relation (9), the
EXP1 data and the roughness length for aqueous sur- 3.3 Light wind
factant solution of the Mitsuyasu and Honda (1982), re- We investigated the roughness length at 10-m wind
spectively. The Toba data points, one square of our data, speeds less than 8 m/s or at the friction velocities less
one solid circle of Mitsuyasu and Honda (1982) and the than 0.26 m/s. Honda and Mitsuyasu (1980) obtained the
Kunishi and Imasato (1966) regression curve at the fric- same drag coefficient at 10-m wind speeds less than 8
tion velocity of 1.5 m/s are near the Charnock relation. m/s for pure water and aqueous surfactant solution. The
This and the bends of the Keller et al. (1992) relation (9) same drag coefficient means the same roughness length
and the Kunishi and Imasato (1966) relation (22) suggest under neutral conditions. This indicates that the rough-
that the Charnock relation is the upper limit of the rough- ness length for a 10-m wind speed less than 8 m/s does
ness length. The intersecting point of the Charnock rela- not depend on the surface tension. We conjecture that the
tion and the moderate wind dependence (21) is the bound- reason may be: capillary waves generated by a light wind
ary of the moderate wind range and the strong one. are too small to make the water surface rough; when the
The Keller et al. (1992) relation is close to the water surface is aerodynamically smooth, the roughness
Charnock relation at a friction velocity of about 0.2 m/s, length is determined by the kinematic viscosity of air and
but deviates from the Charnock relation (3) with increas- the friction velocity and not by the surface tension.
ing the friction velocity. However, we adapt the Charnock In Fig. 5 the dashed line and the thin solid lines ex-
relation as the roughness in the strong wind range. The press the roughness length calculated from the Honda and
reason is as follows: the physical quantities to determine
826 K. Ueno and M. Deushi
10-3 where the kinematic viscosity of air ν a. Assuming that
νa = 1.5 × 10–5 m2/s at an air temperature of 20°C, we
show Eq. (25) by a dot-dot-dashed line in Fig. 5 for com-
parison. The roughness length defined by the light wind
10-4 dependence (24) is a little longer than that expressed by
z0 ( m )
Eq. (25) with νa = 1.5 × 10–5 m 2/s and one tenth as large
as that calculated from Eqs. (16) with Γ = 7.4 × 10–2 N/m
3.4 Comparison of laboratory data and field data
We compared the roughness length obtained from
laboratory experiments with that calculated from the CD
10 100 to U 10 relationship obtained from a field observation.
u* ( m/s ) Yelland et al. (1998) presented a formula for the neutral
10-m drag coefficient
Fig. 5. Roughness length versus friction velocity. The data sym-
bols are the same as Fig. 3. The thick solid line shows Eq. 1000CD = 0.50 + 0.071U10 (6 m / s ≤ U10 ≤ 26 m / s). (26)
(24). The dashed line shows the relationship calculated from
the Honda and Mitsuyasu (1980) drag coefficient relation
(17). The thin solid lines show the Wu (1968) relation (16) In Fig. 3 the thick dot-dashed line denotes the roughness
with Γ = 7.4 × 10–2 N/m (denoted by T) and Γ = 2.7 × 10–2 length calculated from the Yelland et al. (1998) drag co-
N/m (denoted by B). The dot-dashed line and the dot-dot- efficient (26) under the assumption that the kinematic
dashed line show Eqs. (9) and (25) respectively. viscosity ν is 1.3 × 10–6 m2/s and the surface tension Γ is
7.4 × 10 –2 N/m. The roughness length calculated from
Eq. (26) agrees well with laboratory experimental results,
especially with that (dashed line) calculated from Eq. (17).
Mitsuyasu (1980) formula (17) and the Wu relation (16) This suggests that the roughness length does not depend
with Γ = 7.4 × 10–2 N/m (the upper thin line denoted by on the wave state parameter such as the peak frequency
T) and Γ = 2.7 × 10–2 N/m (the lower thin line denoted by or the steepness. This is because the roughness of ocean
B), respectively. It is not apparent that in Fig. 5 the rough- data is almost the same as that of laboratory data, although
ness length for pure water is of the same magnitude as the wave age of ocean waves is about ten times as large
that for aqueous surfactant solution. The upper limit of as that of very young wind waves in a laboratory. How-
the light wind range is not clear. The data points scatter ever, there may be a possibility that the coincidence of
especially around a friction velocity of 0.2 m/s. As for the roughness length between the ocean data and the labo-
the Keller et al. (1992) relation the boundary between ratory data is due to the large difference of the wave age
the moderate wind range and the strong wind one is 0.16 and the scale of the boundary layer. We obtained a for-
m/s. The moderate wind range for the Keller et al. (1992) mula for the roughness length calculated from the Yelland
lies in the light wind range for the Honda and Mitsuyasu et al. (1998) drag coefficient (26)
(1980) relation (17). The physical condition for determin-
ing whether or not a low friction velocity is in the light
wind range is not clear. However, we tentatively assume 3 0.2 u 3
ν 2 gz0 −12 γ ∗
that the roughness length (m) at low friction velocities = 5.92 × 10 4
(m/s) is expressed by γ2 gν gν
(0.18 m / s ≤ u∗ ≤ 1.26 m / s) (27)
z0 = 0.0057 / u∗
by the method of least squares. From this and the expo-
which has the same function form as the Wu relation (16).
nent of the moderate wind dependence (21), we infer that
This is denoted by the thick solid line passing the open
the value of the exponent 3a of u∗ is between 2.5 and 3.0.
circle near the solid symbols in Fig. 5. The roughness
length for an aerodynamically smooth flat plate is ex-
For practical purpose, we determine a function of
the drag coefficient on the wind velocity at a height of z
z0 = 0.11 (25) when the roughness length z 0 is expressed by the moder-
u∗ ate wind dependence (21). From the logarithmic wind
A New Empirical Formula for the Aerodynamic Roughness of Water Surface Waves 827
profile (1) and the definition of the drag coefficient, we 0.003
have T B
= exp − (28) 0.002
z0 = ACD/ 2 u 3
where A ≡ β(γ3/gν 4)0.2γ2/g2ν 3. Eliminating z0, we obtain
3 0 10 20 30
CD z κ
κ = Aκ 3u 3 exp − C . (30) U10 ( m/s )
Fig. 6. Drag coefficient versus 10-m wind speed. The thin solid
Regarding Eq. (30) as an equation in the unknown vari- lines show the drag coefficient calculated from the moder-
able CD/κ2, we have ate wind dependence (21) using the kinematic viscosities
of 1.08 × 10–6 m2/s (denoted by T), 1.24 × 10–6 m 2/s (de-
noted by M) and 1.4 × 10–6 m2/s (denoted by B). The thick
CD / κ 2 = F κA1 / 3u / z1 / 3 . ) (31) solid line: the kinematic viscosity of 1.18 × 10 –6 m2/s. The
thin dot-dot-dashed line shows the drag coefficient for aque-
ous surfactant solution with Γ of 2.7 × 10–2 N/m and ν of
This drag coefficient for the moderate wind dependence
1.4 × 10–6 m2/s. The solid line with circles shows the drag
(21) is denoted by CDM. First we calculate U10(u∗) = (u∗/ coefficient calculated from the Charnock relation. The dot-
κ)ln(10/Au∗3) and CD(u∗) = (u∗/U10)2 with changing u∗. ted line was calculated from Eq. (24). The thin solid line
We determine a CD to U10 relationship using κ = 0.4, ν = with triangles was calculated from Eq. (17). The dashed line
1.24 × 10–6 m2/s and γ = 7.4 × 10–5 Nm2/Kg. Secondly we is the Yelland et al. (1998) relation. The thick dot-dot-dashed
transform this relationship to a CD/κ2 to κA1/3u/z1/3 rela- line shows the Eymard et al. (1999) relation. The circles
tionship or Eq. (31). Finally, we approximate Eq. (31) by with the standard error are the RASEX data from Vickers
the polynomial: and Mahrt (1997).
κA1 / 3
10 3 CD / κ 2 = 1.99 + 2.59 × 10 −1 u As the roughness length for the moderate wind and the
z1 / 3 strong wind is expressed by the moderate and strong wind
κA1 / 3 dependence (23), the drag coefficient is expressed by
−2.25 × 10 −2 1 / 3 u
3 CD = min(CDM , CDC ), (35)
κA1 / 3
+3.60 × 10 −5 1 / 3 u . (32)
z where min(a, b) is the minimum value of a and b.
In order to examine the validity of the new formula
for the roughness length in the open ocean, the neutral
As for the Charnock relation we have 10-m drag coefficient CD is compared with observational
results. The drag coefficients CDM (thin solid lines) cal-
CD / κ 2 = F κ (α / gz )
u. (33) culated from Eqs. (2) and the moderate wind dependence
(21) with the surface tension Γ of 7.4 × 10–2 N/m and the
three kinematic viscosities ν of 1.08 × 10–6 m2/s at 17.1°C
The drag coefficient for the Charnock relation is denoted (top, denoted by T), 1.24 × 10–6 m2/s at 11.9°C (middle,
by CDC. We approximate this expression by the polyno- denoted by M) and 1.4 × 10 –6 m2/s at 7.6°C (bottom, de-
mial: noted by B) are plotted as a function of the 10-m wind
speed U10 in Fig. 6. The thick solid line is calculated us-
10 3 CD / κ 2 = 5.12 + 71.9(α / gz )
κu ing the kinematic viscosity of 1.18 × 10–6 m2/s at 13.7°C.
The solid line with circles shows the drag coefficient CDC
(0.039(gz / α )1/ 2
/ κ ≤ u ≤ 0.198( gz / α )
/κ . (34) calculated from the Charnock relation (3) with α of 0.0185
828 K. Ueno and M. Deushi
or the moderate and strong wind dependence (23) at high of proportionality of the moderate wind dependence (21)
wind speeds. The drag coefficient CDM calculated from must be approximately doubled.
the moderate wind dependence (21) depends on the wa- At strong winds the Vickers and Mahrt (1997) data,
ter temperature through the temperature dependence of the Eymard et al. (1999) relation and the Drennan et al.
the kinematic viscosity of water. The dotted line is calcu- (1999) relation do not follow the relationship of the drag
lated from the light wind dependence (24). The thin dot- coefficient calculated from the Charnock relation or the
dot-dashed line for aqueous surfactant solution is calcu- moderate and strong wind dependence (23) at high wind
lated from the moderate wind dependence (21) using the speeds. However, the drag coefficient from the second
surface tension Γ of 2.7 × 10–2 N/m and the kinematic Storm Wave Study experiment, SWS-2 in figure 11 of
viscosity ν of 1.4 × 10 –6 m2/s. Taylor and Yelland (2001) behaves as that calculated from
The circles with the standard error show the Vickers the moderate and strong wind dependence (23). This
and Mahrt (1997) data of the neutral 10 m drag coeffi- SWS-2 drag coefficient at wind speeds more than 18 m/s
cient. The drag coefficient calculated from the moderate is close to that calculated from the Charnock relation with
wind dependence (21) (thick solid line) and the light wind α = 0.0185.
dependence (24) (thick dotted line) agrees well with the The drag coefficient calculated from the moderate
Vickers and Mahrt (1997) data of the drag coefficient, and strong wind dependence (23) does not depend on the
except for the roughness length at a 10-m wind speed of wave state. However, Vickers and Mahrt (1997) reported
16 m/s. that variation of the neutral drag coefficient in RASEX
The Honda and Mitsuyasu (1980) relation (17) (thin (Risø Air Sea experiment) is dominated by variation of
solid line with triangles) agrees well with the Yelland et wave age, frequency bandwidth of the wave spectra and
al. (1998) relation (26) (thick dashed line). The slopes of wind speed. On the other hand, Taylor and Yelland (2001)
these relations are a little less than ours. We conjecture pointed out that most of the Vickers and Mahrt (1997)
that the sea surface temperature of the observation area drag coefficients were similar to open ocean values (e.g.
where the Yelland et al. (1998) relation (26) (thick dashed Yelland et al., 1998) despite the short fetch and predomi-
line) was obtained ranges from 0°C to 12°C from their nantly young waves. Furthermore, Janssen (1997) re-
ship’s track. The bottom thin solid line (denoted by B) ported that the effect of sea-state on wind stress is much
for the water temperature of 7.6°C is not very close to smaller than the experimental noise level from a statisti-
the Yelland et al. (1998) relation (26) (thick dashed line) cal error analysis of HEXMAX (Humidity Exchange over
at wind speeds ranging from 8 m/s to about 15 m/s. How- the Sea Main Experiment) data. The moderate and strong
ever, this line crosses the Yelland et al. (1998) relation wind dependence (23) derived from the laboratory ex-
(26) (thick dashed line) at 23 m/s. periment for young, pure wind waves and regular waves
The thick solid line for a water temperature of 13.7°C explains some of the drag coefficient data observed in
is very close to the Eymard et al. (1999) relation (thick the ocean where the wave age is about ten times as large
dot-dot-dashed line) for water temperatures ranging as that of very young wind waves in a laboratory. These
mainly from 10 to 15°C in the 10-m wind speed range suggest that the drag coefficient does not depend on the
from 7 m/s to 15 m/s. The solid line with circles calcu- wave state, as suggested in the previous section.
lated from the Charnock relation (3) deviates from the The moderate and strong wind dependence (23) for
Eymard et al. (1999) relation above a 10-m wind speed the roughness length predicts reasonable values of the
of 15 m/s. However, Yelland et al. (1998) showed that drag coefficient in the open ocean. This suggests that Eq.
the airflow distortion correction reduces the drag coeffi- (23) for the roughness length is valid in the open ocean.
cient. The correction due to the ship’s distortion of the The light wind dependence (24) explains the Vickers and
airflow may reduce the Eymard et al. (1999) drag coeffi- Mahrt (1997) drag coefficient. However, the light wind
cient. If the airflow correction is applied, their value may dependence (24) is a tentative assumption. It is still open
distribute near the middle thin solid line denoted by M whether or not the roughness length at low friction ve-
with a kinematic viscosity of 1.24 × 10 –6 m2/s at 11.9°C locities depends on the surface tension. The boundary
at wind velocities weaker than 18 m/s and the solid line between the light wind range and the moderate one is not
with circles at wind velocities stronger than 18 m/s. definite. the roughness length at low and high friction
The Drennan et al. (1999) relation for pure wind velocities therefore needs more experimental study.
waves (the dot-dashed line) is close to the top thin solid
line at a water temperature of 17.1°C. However, the 5. Conclusion
Drennan et al. (1999) relation was derived from the data A new formula for expressing the aerodynamic
obtained at an average water temperature of about 5°C in roughness of water surface waves at moderate wind speeds
Lake Ontario and at water temperatures ranging from 9°C is stated as a function of the surface tension, the water
to 13°C in the open ocean, which means that the constant density, the kinematic viscosity of water, the friction ve-
A New Empirical Formula for the Aerodynamic Roughness of Water Surface Waves 829
−αx Science, Sports and Culture, Japan (No. 12304025). We
v would like to thank Dr. Tadao Kusaba of Kyusyu Univer-
sity for giving us some advice on the usage of surfactant
and information on the experimental conditions. We are
θ grateful to Dr. Takeshi Uji of the Meteorological Research
(a) Institute for his encouragement and critical comments on
−αx this manuscript.
θ The correlation coefficient between ln[ν2gz0/γ2] and
x ln[(u∗3/gν )], or between lnz0 and lnu∗ is 0.992. This is
(b) larger than the correlation coefficient between ln(z0/u∗2)
and lnu∗. However, we compared the correlation coeffi-
Fig. 7. Spurious correlation in the case that the dependent vari- cient between ln(z 0/u∗2) and ln( σpu∗) with that between
able contains the explanatory variable.
ln(z0/u∗2) and lnu∗ instead of that between lnz 0 and lnu∗.
The reason is as follows.
Let x and y be an explanatory variable and a depend-
locity of wind and the acceleration of gravity. This does ent variable, respectively. Suppose that the regression
not include such wave state parameters as the spectral function is y = ax + b. The correlation coefficient between
peak frequency or the steepness of waves unlike almost x and y is calculated from
all parameterizations that have been proposed to date. The
Σ( xi − x )( yi − y )
aerodynamic roughness at strong wind speeds is expressed
by the Charnock relation. This new formula for rough- r= . (A1)
Σ( xi − x ) Σ( yi − y )
ness length above the dimensionless cube of friction ve-
locity ( γ3/gν4)0.2u∗3/gν of 140,000 (approximately corre-
sponding to the 10-m wind speed of 8 m/s) is
The angle between x = (x1 – x , x2 – x , ..., xn – x ) and
y = (y1 – y , y2 – y , ..., yn – y ) is assumed to be θ.
γ 3 γ 2 u∗ 3
r = cos θ . (A2)
7.08 × 10 −13 4
gν g ν
Let a new dependent variable v be y – αx. The correlation
10 gν coefficient r′ between v and x is calculated from
u∗ ≤ 2.61 × 10 3
Σ( xi − x )(vi − v )
r′ = = cos θ ′. (A3)
0.0185 Σ( xi − x ) Σ(vi − v )
u∗ > 2.61 × 1010 3 . The angle between x = (x1 – x , x2 – x , ..., xn – x ) and
v = (v1 – v , v2 – v , ..., vn – v ) is assumed to be θ ′. The
regression function is v = (a – α)x + b. We illustrate the
relation between θ and θ′ schematically in the case that
This formula predicts some of the drag coefficients ob- a > 0, α > 0 and a – α > 0 for example in Fig. 7(a). In this
served in the ocean. The drag coefficient calculated from case the angle θ ′ is larger than θ . The correlation between
this formula at moderate wind speeds depends on the sur- x and y is better than that between x and v = y – αx. If α <
face tension of water and the water temperature through 0, then the correlation between x and y is worse than that
the temperature dependence of the kinematic viscosity of between x and v = y – αx as shown in Fig. 7(b).
water. The drag coefficient at high wind speeds does not The correlation between y = lnz 0 and x = lnu∗ is al-
depend on the surface tension or the kinematic viscosity ways better than that between v = ln(z 0/u∗2) = lnz 0 – 2lnu∗
of water. and x = lnu∗ if a > 0, α > 0 and a – α > 0. When compared
with the correlation between ln(z 0 /u ∗2 ) and lnσ pu∗ the
Acknowledgements dependent variable ln(z 0/u∗2) is more appropriate than the
This study was partially supported by Grant-in-Aid dependent variable lnz0.
for Scientific Research from the Ministry of Education,
830 K. Ueno and M. Deushi
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