THE SEA S U R F A C E IS A E R O D Y N A M I C A L L Y R O U G H

EVEN U N D E R L I G H T W I N D S

JIN WU

Air-Sea Interaction Laboratory, Graduate College of Marine Studies, University of Delaware, Lewes, DE 19958, U.S.A.

(Received in final form 19 October, 1993)

Abstract. The sea surface is generally considered to be aerodynamically rough at high winds (U > 7 m/s), where the roughness length increases with wind velocity; below this velocity, the atmo- spheric surface layer enters a transition region and then becomes aerodynamically smooth as the wind velocity further decreases. The sea surface is shown, however, to reach its smoothest condition at a wind velocity of about 5 m/s, and then become rough again at lower velocities. In the latter case, the roughness length increases as the wind velocity decreases in accordance with the surface-tension relation governing wind-wave interactions.

1. Introduction

The atmospheric surface layer has been considered to be aerodynamically smooth at low winds, with the momentum flux being transmitted through the viscous shear stress acting on the sea surface (Wu, 1968; Liu et al. , 1979). As wind velocity increases, the surface goes through a transition region to become aerodynamically rough; the form drag on roughness elements consisting of surface waves is then responsible for momentum transfer (Charnock, 1955; Wu, 1968). Studies on the air-sea transfer of momentum have been concentrated on relatively high winds; many formulae on the transfer coefficient have also been proposed (Garratt, 1977; Smith, 1980; Wu, 1980). The wind over the world's oceans, however, is commonly light (Harrison, 1989), especially in tropical regions (Sadler et al. , 1987), Quantifi- cation of the air-sea momentum transfer under light winds is therefore important for linking global atmospheric and oceanic models.

The increase of the wind-stress coefficient towards lower wind velocities featured in the aerodynamically smooth regime (Liu et al. , 1979) has gained acceptance only recently (Smith, 1988; Wu, 1988). The delay is largely due to the lack of data. Experiments of Geernaert et al. (1988) and Bradley et al. (1991) conducted under mostly light winds are, therefore, especially valuable. Although the wind- stress coefficient is found in these experiments to increase as the wind velocity decreases, they have much greater values than those predicted by the smooth-flow formula (Liu et al., 1979; Wu, 1988). In addition, even under light winds, the atmospheric surface layer is shown to be aerodynamically rough. These findings are actually consistent with a concept proposed much earlier by Wu (1968) that there exists a surface-tension relation governing wind-wave interactions at light winds. The data are also shown to follow reasonably well the nondimensional

Boundary-Layer Meteorology 69: I49-158, 1994. �9 1994 Kluwer Academic Publishers. Printed in the Netherlands.

150 JJN Wt,

parametric function established on the basis of Weber number (Wu, 1969a), grouped from the surface tension, wind velocity and a length scale representing the growth of the atmospheric surface layer. There are, Of course, other parameters influencing wind-wave interactions at low winds, such as atmospheric-stability conditions and surface waves; we concentrate on the most important variations of the roughness length, and therefore the wind-stress coefficient, with wind velocity.

2. Relevant Studies

Wu (1968) proposed two regimes of wind-wave interaction. The equilibrium state between wind and waves was assumed to be governed by surface tension at low winds with capillary waves serving as roughness elements, and by gravity at high winds with gravity waves as roughness elements. The latter justifies the use of gravitational acceleration as the dynamical parameter in the Charnock (1955) formulation for high winds,

= a , (1)

where z0 is the roughness length of the sea surface; u , = (r /p) ~/2 is the friction velocity of the wind, r being the wind stress acting on the sea surface and p the air density; g is the gravitational acceleration; and a is the so-called Charnock constant. Subsequently, the Froude-number scaling of wind-stress coefficients at various fetches under different wind velocities was deduced by Wu (1969a) from this relation and the logarithmic wind profile,

C~/2= K/ln(1/aCzf2); F = U~/(gZ) t/z , (2)

where C~ = r / p U 2 is the wind-stress coefficient, U~ the wind velocity at elevation Z, K the von Karman constant, and F the Froude number. The anemometer height Z is used here as a physical parameter to indicate the growth with the fetch and wind velocity of the constant momentum-flux layer of turbulent airflow. Quantitat- ively, this height was suggested to vary with the fetch Reynolds number (Wu, 1971a),

Z = 7.35 x 10-7R~/S(m) (3)

in which R e- = U~L/v is the fetch Reynolds number, with L being the fetch and v the kinematic viscosity of air. The anemometer height Z = 10 cm is generally adopted in the laboratory, and Z = 10m fo r open seas. This scaling was also shown to group well laboratory and field data (Wu, 1969a). The scaling function, Equation (2), with values of Z = 10 m and a = 0.0185 obtained from the compi- lation of field data (Wu, 1980) is shown in Figure 1, featuring the increase of wind-stress coefficient with wind velocity.

A nondimensional relationship governing wind-wave interactions at low winds was suggested as (Wu, 1968)

T H E S E A S U R F A C E IS A E R O D Y N A M C A L L Y R O U G H

I I I I

151

I O

o

o ' 3

r-

~2 O

O

2

I e -

~Ecl . [5] with w = 0.18

' ~ _ - - - - - Eq. [7 ]

0 ' " I I I 1 I

0 2 4 6 8 10 12

Wind Velocity, Ulo (m s -1) Fig. 1. Various formulations of wind-stress coefficients at low winds.

Zo / (T /pwu2 . ) = w , (4)

where T is surface tension, pw the density of water, and w the proportionality constant. Contrary to that in the gravity regime, the roughness length increases as the wind-friction velocity decreases. The roughness length, of course, is deter- mined not only by the height of roughness elements, but also even more impor- tantly by their spatial distribution and density (Schlichting, 1968). For example, the roughness length is small for a surface with densely packed uniform roughness elements; it reaches a maximum value at a certain spatial density of these elements.

152 J~N Wu

The general equation for correlating the wind-stress coefficients in this low-wind case was expressed as (Wu, 1969a)

C1/2 = K/ ln(CzW2/w) , W = Uz / (T /pwZ) u2 (5) Z

where W is in the form of the Weber number, and the proportionality constant w = 0.18 was found from our laboratory experiment (Wu, 1968). Corresponding to this value, the Weber-number scaling law with Z = 10 m is shown in Figure 1; it shows an increase of wind-stress coefficient toward lower wind velocities.

In relatively high winds, as discussed earlier, the atmospheric surface layer is aerodynamically rough and wind-wave interactions are governed by gravity forces. Oceanic data of the wind-stress coefficient have been represented with simple empirical formulae (Garratt, 1977; Smith, 1980; Wu, 1980; Large and Pond, 1981); among them the following was proposed by Wu (1980):

Clo = ~-/pv~lo = (0 .80 + 0 . 0 6 5 G o ) • 10 -3 , (6)

where Cm is the customarily used wind-stress coefficient, and Uto (m/s) is the wind velocity at the standard anemometer height of 10m above the mean sea surface. The above formula is seen in Figure 1 to almost coincide with the Froude- number correlation curve in the aerodynamically rough regime with Um> 7 m/s. Note that the formulae proposed in four investigations cited above differ only slightly, especially in the rate of Clo varying with Urn.

When the atmospheric surface layer becomes aerodynamically smooth at low winds, the wind-stress coefficient follows (Wu, 1988)

C~-o ~,2 -- ( l / K ) ln(Cl/o2UtoZ/v) + 5.5, Z = 10 m . (7)

This variation of Cm with /-710 is also shown in Figure 1. Note that the above expression for smooth flows was derived from results over a solid surface. For the air-sea interface, another formula was proposed by Liu et al. (1979); the latter, providing somewhat larger wind-stress coefficients than Equation (7), will be discussed in the next section. As illustrated in Figure 1, the wind-stress coefficient for aerodynamically smooth flow increases as the wind velocity decreases. The expression for high winds, Equation (6), is seen to intersect the smooth-flow curve at Um= 2.4 m/s, differing little from the critical wind velocities proposed to identify the smooth-flow regime (Kondo, 1975; Wu, 1981).

As mentioned earlier, data at low winds have been collected recently by Geerna- ert et al. (1988) and Bradley et al. (1991). Geernaert et al. 's measurements were from a tower 2 km off the California coast. A hot-film system was used to deter- mine the wind stress with the dissipation technique. Neutral wind-stress coef- ficients, with [Z/L[ < 0.05, where L is the Monin-Obukhov stability length, were obtained by them. Their results ,are reproduced in Figure 2a. Bradley et al. conducted their measurements onboard the R / V Franklin in the Western Equa- torial Pacific Ocean. The wind stress was determined by the dissipation method with a sonic anemometer. Their data without adjustment to neutral stability are

THE SEA SURFACE IS AERODYNAMCALLY ROUGH 153

also reproduced in Figure 2a, while adjusted values, as indicated by them, were only slightly smaller for their experimental conditions.

3. Further Analyses

The data of Bradley et al. (1991) were mostly at wind velocities below 4 m/s; the quite complementary data of Geernaert et aI. (1988) start from about 3 m/s and up. Overall, the wind-stress coefficients obtained in these two investigations have the same trend, although the data of Geernaert et al. have somewhat larger values than those of Bradley et al. over their overlapping region. Differences in experimental sites and techniques could contribute to the discrepancies. In order to quantify further these results, which are comparable in magnitude and even in scatter, the data are band averaged. For wind velocities below 1m/s, the band width of 0.2 m/s is selected, of 0.4 m/s for wind velocities up to 3 m/s, and of 0.6 m/s up to 6 m/s; there are few data points beyond 6 m/s. The average value and standard deviation of each band are shown in Figure 2b. Standard deviations are seen to be much smaller than the systematic trend indicated by the data. Taken together, both sets of data and the band-averaged results show consistently that the wind-stress coefficient decreases as wind velocity increases to reach a minimum value in the neighborhood of U10 = 5 m/s.

Various formulations of the wind-stress coefficient are also shown in Figure 2b; these include the Weber-number correlation curve shown in Equation (5) with w - 0.18, the wind-stress coefficient for aerodynamically smooth flows according to Equation (7), and the smooth line of Liu et al.'s (1979) model calculated by Bradley et al. (1991). First of all, the smooth-flow line adopted from the solid boundary, Equation (7), responds inadequately to the increase of C1o toward lower U10. The smooth line deduced for the air-water interface by Liu et al.

depicts a correct trend for only a very small portion of the data with U10 < 2.2 m/s, beyond which it actually predicts an increase of Cm with Urn. Even for this small portion over very light winds (Urn < 2.2 m/s), the predicted wind-stress coefficient is seen clearly to be much smaller than given by the data. The Weber-number correlation in Figure 2b not only represents the data better over the entire low- wind region, but also approximates the rising trend more closely at very light winds.

Over a solid surface, as flow velocity decreases, the resistance coefficient ap- proaches asymptotically the value indicated by the smooth-flow formula. This, however, is not the case for the atmospheric surface layer over a sea surface, because of ever-present waves. Accepting for the moment that the surface layer is, as suggested earlier, aerodynamically rough at light winds, the wind profile is the same as for rough flow:

U J u . = (l/K) l n ( z / z o ) . (8)

We can then obtain the roughness length from the above expression to calculate

154 JIN Wu

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~ { - - - - Eq. [73

0 .5 r I I t I i 0 1 2 3 4 5 6 7

Wind Velocity, Ulo (m /s )

Fig. 2. Neutral wind-stress coefficients obtained by Geernaert et aL (I988) and Bradley e! al. (1991) (a), and their band averaged values along with standard deviations (b). The lines representing various

formulations are also shown in (b).

the roughness Reynolds number R~ =- U,Zo/V, characterizing the atmospheric sur- face layer; see Figure 3a obtained from the averaged data in Figure 2b.

The dividing roughness Reynolds numbers for various regimes over the sea surface are also shown in Figure 3a. The atmospheric surface layer was shown earlier (Kondo, 1975; Wu, 1981) to be aerodynamically rough for wind velocities

T H E S E A S U R F A C E IS A E R O D Y N A M C A L L Y R O U G H 155

Fig. 3.

.~. 10 3 o

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WTnd Velocity. U10 ( m / s )

(b)

10 2 i 0 6 7

Roughness Reynolds numbers and critical fetches at various wind velocities. The results in (a) are obtained from averaged data shown in Figure 2b.

above 7 m/s; this is confirmed by the results shown in Figure 3a. The data between velocities of 5-7 m/s, however, do not appear to drop below R r = 2.0 as indicated earlier with much more extensive data for this velocity region (Wu, 1981). None- theless, it is interesting to see that the sea surface reaches its smoothest condition at wind velocities near 5 m/s; this feature was noticed by others (Donelan, 1990).

[56 JIN W u

More importantly, the surface layer indeed becomes aerodynamically rough again at lower wind velocities.

At low winds, the wind-stress coefficient increases as wind velocity decreases. Such a trend, a reverse of that at higher wind velocities, was thought to be due to the fact that the atmospheric surface layer became aerodynamically smooth at low winds (Liu et al . , 1979; Wu, 1988). This, of course, was derived from studies over a solid surface, The extensive data over the sea surface under low winds that have become available are plotted in Figure 2b, where it is seen that they are even greater than those provided by the smooth-flow formula. In other words, the up-turn of the wind-stress coefficient toward lower wind velocities associated with aerodynamically smooth flows is not sufficient to accommodate the actual increase indicated by the data.

There are, of course, differences between boundary layers over a solid surface and an air-sea interface. As mentioned earlier, roughness elements over a wavy sea surface grow with wind velocity above 5 m/s but the sea surface becomes rough again for wind velocities below 5 m/s. This feature has been studied systematically in the laboratory (Wu, 1968); the wind-disturbed water surface offers a minimum roughness length at an intermediate wind velocity, where wind-wave interactions are in transition from a surface-tension regime to a gravity regime. From there towards both lower and higher wind velocities, the roughness length and therefore the wind-stress coefficient increase; the present interest is, of course, in the low- wind region. We reason that the sea surface is covered by capillary waves at very light winds, and that their spatial distribution becomes denser as wind velocity decreases (Wu, 1971b).

The correlation curve according to the Weber-number scaling, Equation (5) with w = 0.18, is seen in Figure 2b to represent the data well for the core region of winds from 1.5 to 4m/s , but under-predict the data at very low winds (U10 < 1.5 m/s). The discrepancy between the data and the curves may be caused by other factors influencing wind-wave interactions. Among them, wave age per- haps should be examined first. Wind waves grow with both fetch and wind velocity; an empirical formula was proposed earlier (Wu, 1969b)

c~ U = 0.05(gL/U2) ~ , (9)

where c is the phase velocity of the dominant wave. The critical fetch, Lc, required for waves to be fully developed, with their phase velocity reaching the wind velocity ( c / U = 1), is shown in Figure 3b. It is seen clearly that the dominant waves at light winds are fully developed at rather short fetches. Consequently, the wind-stress coefficients in various experiments were generally obtained under very different wave ages. Much still needs to be studied to understand this complex phenomenon to quantify further the wind-stress coefficients at very light winds. Another factor is that the air-sea transfer at very light winds is also likely to be influenced by convective processes (Businger, 1973); the latter actually dominate

T H E SEA S U R F A C E IS A E R O D Y N A M C A L L Y R O U G H

a i r - s e a i n t e r ac t ion for w ind ve loc i t ies at or b e l o w 0 . 5 - 1 m/s

Be l j aa r s , 1991).

157

( G o d f r e y and

4. Concluding Remarks

The wind-s t ress coeff ic ient at low winds is a cr i t ical inpu t for g loba l a t m o s p h e r i c

and ocean ic mode l s , as the wind ove r wor ld ' s oceans is gene ra l ly re la t ive ly l ight .

V a l u a b l e da t a u n d e r neu t r a l a tmosphe r i c - s t ab i l i t y cond i t ions o f fe red by G e e r n a e r t

et al. (1988) and B r a d l e y et al. (1991) a re fu r the r ana lyzed and i n t e r p r e t e d . T h e

mos t d is t inct f ea tu re of the i r m e a s u r e d wind-s t ress coeff icients be ing g rea t e r than

the smoo th - f low values is exp la ined . T h e s e va lues and the i r var ia t ions of wind

ve loc i ty a re cons i s ten t wi th the su r face - t ens ion r e l a t ion concep t p r o p o s e d ea r l i e r

(Wu , 1968, 1969a). T h e s e resul ts ind ica te tha t the sea sur face is a e r o d y n a m i c a l l y

r o u g h u n d e r r e la t ive ly l ight winds (U10 < 5 m/ s ) , which m a y also have impl ica t ions

on hea t and mass t ransfers .

Acknowledgement

I a m gra te fu l for the sponso r sh ip of m y w o r k p r o v i d e d by the M a r i n e M e t e o r o l o g y

P r o g r a m , Office of N a v a l R e s e a r c h u n d e r G r a n t N00014-93-1-0345.

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158 JaN Wv

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