analysis of sea-surface drag parameterizations in open

ANALYSIS OF SEA-SURFACE DRAG PARAMETERIZATIONS IN OPENOCEAN CONDITIONS

KARL F. RIEDERScripps Institution of Oceanography, University of California at San Diego, La Jolla, California,

USA

(Received in final form 18 June, 1996)

Abstract. Data from the Surface Waves and Processes Program (SWAPP) are employed to test currentsea-surface drag parameterizations in open ocean conditions. General trends in the data indicate thatdrag increases with increasing wind speed and wave height, and decreases with wave age. However,scatter in the data limits the use of these parameters and other wave dependent parameterizationsfor modelling efforts. Upon close inspection, it is found that during the onset of three wind eventsanalyzed separately, each of these parameters correlate well with the drag coefficient. However, thedependence of the drag coefficient on each of these parameters varies markedly from event to event.The disparity appears most closely linked to the turning rate of the wind, indicating that temporal anddirectional effects may play an important role. A temporal lag of O(4) hours between the rise of thewind and subsequent rise in the drag coefficient is also noticed, further pointing out the complexityof the wind-stress system.

1. Introduction

As the link between the easily measured wind velocity and the more difficultdirect measurement of wind stress, the drag coefficient is the key parameter forthe determination of the momentum transfer between atmosphere and ocean. Thistransfer of momentum drives a wide range of atmospheric and oceanic phenomena,including the oceans’ surface currents, the oceanic and atmospheric mixed layers,and surface gravity and capillary waves. Consequently, the drag coefficient playsan important role in coupled atmosphere-ocean circulation models that are usedto forecast weather and climate and are often applied to scientific issues such asglobal warming and El Nino. However, despite the large amount of work over thelast few decades, research has yet to establish a universal theory by which the drag,and hence the wind stress, may be accurately estimated from bulk measurements.

The wind stress is most often estimated via a simple “bulk formula”

� = �CdjUjU (1)

where � is the wind stress, � the air density, Cd the drag coefficient, and Uis the mean wind velocity. Sea-surface drag can be equivalently described by aroughness length, z0, which, for neutrally stratified atmospheric boundary layers,can be related to the drag coefficient by assuming a logarithmic profile of the meanwind speed with height:

z0 = z= exp[�=(Cd)1=2] (2)

Boundary-Layer Meteorology 82: 355–377, 1997.c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

356 KARL F. RIEDER

where � is the Von Karman constant and z is the measurement height from meansea level. (The roughness length is mathematically the zero-crossing height of anextrapolated logarithmic wind profile).

Over land, the roughness length has been successfully related to the averageheight of roughness elements. However, the extension of this relation to the rough-ness length over the sea is conceptually difficult to make since at the atmosphere-ocean interface, the roughness elements are moving and these are unevenly distrib-uted over the phase of the long waves (Kitaigorodskii, 1973).

Attempts to model the sea-surface drag coefficient and roughness length haveboth wind and wave dependencies (Geernaert, 1990). It was first suggested byCharnock on dimensional grounds that the roughness length over the sea surfaceshould vary with wind speed (Charnock, 1955). Since that time, many empiricalformulae for the drag coefficient have been constructed by researchers using directmeasurements of the wind stress (e.g. Donelan, 1982; Geernaert et al., 1986; Largeand Pond, 1981; Wu, 1982). In general, a weak dependence on wind speed wasfound; however, large variations in the magnitude of the drag coefficient werecommon, and results vary significantly from study to study.

Several attempts have been made to integrate wave measurements into the for-mulation of the roughness length. It is not known which scales of waves play thegreatest role in the creation of the sea-surface roughness. Therefore these formu-lations vary in derivation. As a rule, they perform uniformly poorly, except underspecial circumstances. Donelan (1982) proposed that the drag has two forms, onedependent on the form drag due to the long waves and another due to the viscousdrag of the short waves; Byrne (1982) postulated that the roughness length wasproportional to the integral wave velocity over the wind wave frequencies; Kitaig-orodskii (1973) derived an equivalent roughness length in the moving referenceframe of the surface gravity waves – the resulting expression puts the roughnesslength in terms of an exponentially weighted wave energy density; Hsu (1974)related the surface roughness to the steepness of only the dominant waves; andJanssen (1989) expressed the roughness length in terms of the growth of the waves.The search for a proper roughness formulation has been based conceptually onwave spectra bereft of swell, and these models have not been rigorously tested inmore complex conditions. While it is clear that the sea-surface roughness must bemainly due to the existence of surface gravity waves, research has yet to identify aunique set of wave parameters on which practical models may be based.

Most recently, effort has been put into representing the roughness length viathe wave age (cp=u�, the ratio of the celerity of the peak wind wave frequencyto the friction velocity), which is a measure of the state of development of thewaves. While studies during conditions in which swell was absent indicated thatthe roughness length varied inversely with wave age (e.g. Donelan et al., 1993;Smith et al., 1992), a study where swell was present did not indicate a clearrelation (Dobson et al. 1993). Other studies have suggested that for waves thatare in dynamic equilibrium, roughness and wave age may be directly proportional

ANALYSIS OF SEA SURFACE DRAG PARAMETERIZATIONS 357

(Toba et al., 1990), contradicting the above noted trend. Overall, it seems clear thatdespite the amount of research done, a consensus has not been found.

To further complicate the issue, it has been recently found that the wind stress andmean wind are not necessarily collinear (Geernaert, 1988; Geernaert et al., 1993;Zemba and Friehe, 1987), invalidating a basic assumption of the bulk formula.Most commonly, stress estimates calculated by the eddy correlation method havereinforced the assumption of wind and stress alignment by failing to calculatethe crosswind stress. Stress estimates made using the inertial dissipation methoddo not have directionality, and a priori alignment with the wind is assumed; thecalculated drag coefficient therefore applies to the magnitude of the whole stressvector (Geernaert, 1990). The data considered here have already been shown toinclude large angles between the wind and the stress. These angles have beenrelated to the direction of waves across a wide range of frequencies (Rieder et al.,1994) and to the direction of wave breaking (Rieder et al. 1995). In this work wewill simply define the drag coefficient as the coefficient of proportionality betweenthe square of the wind speed and the component of the wind stress in the directionof the mean wind, thus maintaining mathematical sense.

Geernaert (1996) has made a first attempt at modelling the direction indepen-dently. However, it has yet to be agreed upon how to make an estimate of the totalwind stress vector and apply previous drag coefficient parameterizations to thatend.

This paper re-investigates the important parameters that affect the sea-surfacedrag. In particular, we are interested in testing the current theories and roughnessparameterizations in open ocean conditions, where a mix of sea (i.e. locally generat-ed waves) and swell exists. Wind and wave measurements taken in the North Pacif-ic 500 kilometres off Point Conception, California will be employed to this end.

2. The Experimental Program and Data Analysis

Data were collected for this study during the Surface Waves and Processes Pro-gram (SWAPP), which took place from February 24 through March 18, 1990approximately 500 km off Point Conception, California in 4000 metres of water.Measurements of the surface gravity waves, mixed-layer structure, and air-seafluxes were made from the research platform FLIP to study air-sea and surfacewave-upper ocean boundary-layer interactions (Weller et al. 1991)

Sea surface velocities were measured by four 195 kHz Doppler sonars, directedat 45� increments in azimuth. A “quick-look” analysis of the measured velocitiesprovide estimates of the first three Fourier components of the directional spectraevery 12.8 minutes within the frequency range of 0.05–0.5 Hz (Smith and Bullard1995).

Direct calculations of the momentum flux were made possible by high frequency(10 Hz) 3-component measurements of the wind velocity. Two sonic anemometers

358 KARL F. RIEDER

were deployed on the aft and port booms of FLIP at 6 and 8 m above mean sealevel respectively. Wind stress could then be calculated directly by the Reynoldsaveraged equation:

� = ��[hu0w0ii+ hv0w0ij]; (3)

where u0, v0, w0 are the downwind, crosswind, and vertical fluctuating wind veloc-ities.

The sonic anemometer data were reduced to 2 Hz and wind stress estimates weremade every 30 minutes. As shown in Figure 3 of Rieder et al. (1994), virtuallyall contributions to the wind stress come from frequencies between 0.01 and 1 Hz(periods of 100 and 1 s, respectively). Reducing the data to 2 Hz produces an errorin the wind stress of less than 1%. Trends and fluctuations with periods greater than500 s were filtered out to ensure that the stress estimates were not biased by anyslowly varying change in the wind speed. The estimates were required to satisfyseveral quality controls, as in Rieder et al. (1994). First, wind speeds of less than3 m s�1 were not considered. Second, to remove any periods for which the windstress is ill-defined (due to a partial sample, for example), estimates of the downwindstresses were required to be different from zero with a statistical confidence of 99%(assuming a normal distribution). Third, conditions during which the superstructureof FLIP sheltered or shadowed the sonic anemometer were removed. Only datafrom the sonic anemometer on the aft boom of FLIP ever passed the third test.Finally, the motion of FLIP was analyzed (Smith and Rieder, 1997) and found to besmall in producing possible errors in the wind stress calculation. Re-analysis showsthat during a representative period (March 11, 20:29), vertical (0.03 m s�1) andhorizontal (0.22 m s�1 rms) velocities induced by rotation and tilt of FLIP producedapparent stresses almost two orders of magnitude smaller than the measured windstresses.

Also measured were air temperature, sea temperature, and humidity, which wereused to determine atmospheric stability (c.f. Large and Pond, 1981). Atmosphericstability has been shown by previous research to have dramatic effects on themean wind profile, and to be of significant importance in the determination of boththe magnitude and direction of the wind stress (Geernaert, 1988). Here, we willinvestigate only near-neutral conditions, so that we may avoid the use of stabilitycorrections on the profile of the mean wind. Conditions are considered near-neutralif jz=Lj < 0:05, where z is the measurement height andL is the Obukhov length, ascale height based on buoyancy flux and wind stress. This generally excluded datajust prior to the onset of the three wind events.

3. Experimental Results

The wind speed, root-mean-square wave and sea height, inverse wave age, anddrag coefficient, Cd, over the course of the ten day period of SWAPP are plotted


in Figure 1. The drag coefficient is found by extrapolating the wind speed to 10m equivalent height (scientific standard) using the direct measurements of windstress and the assumption of a logarithmic profile. The rms wave height is foundby summing over the total wave height spectrum, and taking the square root. Therms sea height is determined from a sum over only the high frequency sea waves,where the low frequency cut-off is taken as the peak in the velocity spectrum. Thecelerity of this wave frequency is used in the calculation of wave age. We will seethat this choice does not qualitatively affect the results.

The peak of the velocity spectrum produces the best estimate of the lower limitof the saturation region. To illustrate this, Figure 2 shows the wave height spectrafor four periods, each separated by three hours, starting on March 7, 1990 at 20:00when strong winds began. As seen in the first period, low frequency swell existed asthe high frequency sea waves began to grow. As the sea waves developed, the peakof wind wave spectrum steadily moved to lower frequencies. The arrows indicatethe low frequency cut-offs of the wind wave spectrum, as chosen by the peak inthe velocity spectrum.

Three strong wind events occurred over a ten day period (see Figure 1a). Non-shaded regions indicate the beginning of each event; these will be discussed furtherlater. The first, starting on March 4, 1990, began with a strong increase of the windfrom calm conditions (the wind speed rising to 10 m s�1). Growth of the wavescan be seen clearly following this wind increase (Figure 1b). The root mean squarewave height reached a peak of nearly 1.0 m (approximately 4 m significant waveheight). The onset of the wind occurred with little accompanying swell, indicatedby the large inverse wave age (Figure 1c), and the small difference between waveand sea heights. During the second event, the wind speed increases very abruptly,doubling in speed in minutes, resulting in strong wave growth. In contrast, the thirdevent shows a more gradual wind speed increase during conditions of pre-existingswell.

The variation of Cd over the course of the ten day period can be seen inFigure 1d. Only data passing all criteria are shown. There are long time periodsduring which no wind stress estimates passed these criteria, commonly due to theshadowing of FLIP. General trends shown in Figure 1 indicate that drag increaseswith increasing wind speed, rms wave height, and inverse wave age (decreasingwave age). Furthermore, individual peaks in the drag coefficient are seen to bemirrored in the other time series, particularly in the inverse wave age. A secondarypeak in the Cd time series can be seen for each of the three wind events (followinga primary peak which accompanied the initial rise of the wind). Similar peaks areseen in the inverse wave age for the first and second events and in the wind speedrecord for the first and third events.

Despite the fact that these general trends seem apparent by examination of thetime series, a closer inspection shows that high drag coefficients do not alwayscorrespond to high wind speeds, high wave height and/or small wave age. Also,there are periods when Cd varies strongly from half-hour to half-hour, while there

360 KARL F. RIEDER

Figure 1. Time series of the (a) wind speed, (b) rms wave and sea heights, (c) inverse wave age, and(d) drag coefficient during the ten days of SWAPP analyzed. General trends of increasing wind speed,wave height, and inverse wave age (decreasing wave age) can be seen to accompany increasing drag.However, large scatter is evident from half-hour period to period. Less scatter occurs during the onsetof the three main wind events, shown in the non-shaded regions.

are other periods where the drag coefficient steadily increases or steadily decreases.This can be seen clearly during the long second wind event, from March 7, 12:00to March 9, 12:00. During this forty-eight hour period, wind speeds remained rela-tively strong, rapidly increasing over a six hour period and then slowly decreasing


Figure 2. Wave height spectra for four periods, each separated by three hours, starting on March 7,20:00. As winds increase, high frequency sea waves can be seen to grow apart from the low frequencyswell. Arrows indicate the low frequency cut-off of the wind wave spectrum, chosen as the peak inthe wave velocity spectrum.

during the remaining 42 hours. This caused a quick increase and slow decrease ofthe wave energy. (However, note that the waves continue to grow following thepeak of the wind speed). During the increase of the wind and the quickest growth ofthe waves, a distinct increase inCd can be seen. Then, as the wind slowly decreasesin speed, the drag coefficient drops and then varies wildly. This variation will surelycomplicate the modelling problem.

A final note about the time series concerns the apparent temporal lag betweenthe increases and decreases of the wind and the resulting increases and decreasesin the drag coefficient. Three clear instances of this can be seen: the beginning ofthe first wind event, the beginning of the second wind event, and the secondarywind increase during the third event. During the first two wind events there was asudden increase in the wind, and an accompanying increase in the drag coefficient.However, the peak in the wind speed occurs prior (on the order of 4 hours) to thepeak in the drag coefficient. Similarly, as the wind speed is falling and quicklybegins to increase again during the third wind event, near the end of March 11, afalling and quick rebound of the drag can be seen. But again there is a temporal lagof several hours.

362 KARL F. RIEDER

Figure 3. Drag coefficient versus 10 m wind speed. Large scatter and no correlation can be seen inthe data. The mean drag coefficient is equal to 1.21� 10–3.

We next look at the wind speed, sea state, and wave dependencies of the dragcoefficient. This will be done by examining the current drag coefficient parameter-izations and testing how well they match its variability. The variance and temporallag issues will then be investigated in greater detail to understand better the rela-tionship between the wind and waves and the roughness of the sea surface.

3.1. WIND SPEED DEPENDENCE

The calculated drag coefficient is plotted vs. the 10 metre wind speed in Figure 3.Most noteworthy is the large amount of scatter in the points, typical in such plots. Inthe present case, the correlation is insignificant. This low correlation is remarkablesince clear general trends were apparent in the time series. The wind speed alonedoes not reduce the variance of the drag coefficient over the range of wind speedshere. The mean drag coefficient is equal to 1.21 � 10�3 with a standard deviationof 0.36 � 10�3 or 30% of the mean.

3.2. SEA STATE DEPENDENCE

The non-dimensional roughness length, or Charnock constant (z0� = gz0=u2�),

versus wave age has been plotted in Figure 4. A trend of decreasing drag withincreased wave age is weakly suggested in the data. A line indicating a power law


Figure 4. Non-dimensional roughness length versus wave age. Previous research suggested a powerlaw fit with n = �1 as depicted by the straight line. These data do not support this.

of n = �1, as suggested by Smith et al. (1992) and Donelan et al. (1993), is drawnfor comparison with previous results. The data here do not support these findings.There is significant scatter, and no significant correlation. Dobson et al. (1993) sawsimilar scatter in their data from the Grand Banks experiment during times whichswell dominated, despite attempts at its removal. The implied spectral similarityassumed in the use of wave age may not be appropriate for swell intensive seas,such as those seen during SWAPP and the Grand Banks experiment.

3.3. WAVE DEPENDENCE

The models of Hsu (1974), Byrne (1982), and Kitaigorodskii (1973) were previ-ously examined by Geernaert et al. (1986) using a data set collected in the NorthSea. Hsu (1974) postulated that the non-dimensional roughness length varies withthe wave steepness of the dominant wind waves. Expressing this in terms of thephase velocity of deep water surface gravity waves, the roughness length can beexpressed as:

z0 �1

2�H

(c=u�)2 ; (4)

364 KARL F. RIEDER

where H and c are the height and phase speed of the dominant wind wave. Byrnetheorized that the roughness length was proportional to the integral wave orbitalvelocity over the wind wave frequencies:

z0 �

Z1

!0

S(!)!2 d!; (5)

where S(!) is the wave energy density at frequency !. By assuming that thewave height can be interpreted as some scale for the protrusions of a completelyrough immobile wall, Kitaigorodskii derived an equivalent roughness length in themoving reference frame of the surface gravity waves. The resulting expression putsthe roughness length in terms of an exponentially weighted wave energy density:

z0 �

�2Z1

0S(!) exp

��2k!u�

�d!

�1=2

: (6)

Geernaert et al. found that comparing to the North Sea data set, the formulationsof Hsu and Kitaigorodskii performed better than a simple wind-speed dependentdrag law, while Byrne’s model performed poorly.

As the models of Hsu and Byrne are based on the height and velocity of thewind waves, it is necessary to separate the wind wave spectrum from the swell.This was done by analyzing the waves above a low frequency cut-off determinedas the peak of the velocity spectrum (as in calculation of the rms sea height andthe wave age). This separation, however, cannot remove the effect of the longperiod swell on the wind waves. We are interested to find whether the existenceof swell invalidates the use of these models, and whether the trends in the dataare maintained. We find correlation coefficients of r2 = 0.01 for both the Hsu andKitaigorodskii wave parameters, indicating insignificant correlation and a clearfailure of these two models for this data set. The correlation coefficient betweenthe measured roughness length and the Byrne modelled roughness length of r2

= 0.03 is poor as well, as seen by the large amount of scatter in Figure 5. Themeasured roughness length varies by several orders of magnitude, but no suchvariation is seen in the modelled roughness, due to the lack of variability in thesum of the wind wave velocity spectrum in the open ocean. (No frequency bandof the 1-d wave spectrum varies by the many orders of magnitude necessary.) Theresulting modelled drag coefficients can not vary much from a constant level, incontrast to the large deviations seen in the data.

3.4. VARIATION IN DRAG

The attempt to explain Cd variations by traditional parameterizations is so fardiscouraging. By inspection of the time series and the general trends in the data,one would expect that the surface drag could be estimated via the wind speed, waveenergy, and/or the wave age. However, when analyzing this open ocean data set,


Figure 5. Byrne modelled roughness length versus the measured roughness length. The smallvariability in the Byrne modelled roughness does not match the orders of magnitude of variabilityobserved in the measured roughness length.

no strong correlations are found. While these models have been shown to performsuccessfully in fetch-limited or wind wave dominated conditions, the existence ofswell apparently invalidates the results.

The above analysis tested whether there is a one-to-one correlation betweenhigh winds (for example) and large drag, the answer being no. But do changes inthe wind correspond to changes in the drag? Or at the very least, are the periodsof variability in the wind (in direction and magnitude) the same periods when thedrag is varying? To test this, we have plotted the variability of the wind, downwindand crosswind components, over the course of the ten day period, along with thevariability of Cd (Figure 6). The variability is measured by the magnitude of thefirst difference in the vector wind and drag coefficient, smoothed by a running meanfor plotting. We see that drag variability often corresponds to wind variability, butnot always. Periods of more constant wind correspond to periods when the dragis varying less and visa-versa. In fact, peaks in the wind’s variability are oftenmirrored by peaks in the drag’s. The fact that the crosswind variability is importantindicates that periods of turning winds are periods of change in the drag as well.This is of interest, because the previously analyzed models contained no directionalor temporal information. But this clearly does not explain the whole story.

The drag coefficient is most steady during conditions when the seas are activelygrowing due to a strong increase in the wind. Therefore, to look further, we inspect

366 KARL F. RIEDER

Figure 6. Variability in the vector wind and drag coefficient. The periods of large variability in thedrag coefficient often correspond to those of the vector wind. The matching of the drag’s variabilitywith that of the crosswind variability indicates that periods of turning winds are periods of increaseddrag.

the “beginning period” of each of the three wind events to determine to what extentthe short term variability limits the potential to model the drag coefficient. The“beginning period” of each wind event extends from the initial wind increase tothe peak in the rms sea height (see the non-shaded areas in Figure 1). Within theseperiods, the usable data will come from the portions of these periods where all thedata have passed the quality criteria.

Figures 7, 8, and 9 show the variation of Cd with wind speed, rms wave height,and wave age over each of these three periods. The first, second, and third windevents are represented by the circles, stars, and crosses respectively. Consideringthese three periods together, we see that there is an improved correlation betweenCd and each of the three measures. The correlation coefficients are r2 = 0.07, r2 =0.17, and r2 = 0.24 (negative correlation) betweenCd and the 10 m wind speed, rmswave height, and wave age, indicating that surface roughness increases with windspeed and wave height, and decreases with wave age. These correlations are stillquite low, but do represent an improvement to those for the entire data set:r2 = 0.01,r2 = 0.08, and r2 = 0.15 respectively. If we now inspect each period individually, wesee that the trends are more apparent. This is particularly true for the comparisonwith the rms sea height, where each data subset can be well represented by a linearbest fit. However, the empirical regression with wind speed, wave height, and waveage of each period are different from the others (see Figure 8). In particular, thesecond time period, depicted by the stars, shows a much stronger variation of


Figure 7. The drag coefficient versus 10 m wind speed for the beginning of the three wind eventsanalyzed (circles, stars, and crosses represent the first, second, and third wind events respectively).Significantly higher correlations are seen between these data as compared to the full data set, indicatingthat during these times, drag increases with increasing wind speed. A significant correlation (r2 =.46) for the subset of data which includes the first and third wind events is illustrated by the best-fitregression line.

drag with wind speed and wave height as compared with the other two periods.This is exemplified by the correlation between Cd and the 10-m wind speed forthe combination of the first and third wind events, depicted by the regressionline through the data (regression lines are drawn only for statistically significantcorrelations). The correlation coefficient of r2 = 0.46 is a marked increase over anyprevious comparison. These results indicate that significant variability occurs overa wide range of temporal scales, from the averaging period (half-hour) to synoptictime scales.

The next logical step is to consider whether a combination of these parametersmay better describe the variability of the drag coefficient than any single parametercan. To properly consider a multivariate regression, the drag coefficient is assumedto be dependent on parameters that are independent of each other. Clearly, whenconsidering the variables tested so far, wind speed cannot be used, since it is wellcorrelated to both the wave age and the rms wave height. However, the wave ageand wave height are not well correlated. Indeed, they are well suited since they

368 KARL F. RIEDER

Figure 8. Drag coefficient versus rms wave height for the beginning of the three wind events analyzed(circles, stars, and crosses represent the first, second, and third wind events respectively). Significantlyhigher correlations are seen between these data as compared to the full data set, indicating that duringthese times drag increases with wave height. Moreover, each period individually shows stronger linearcorrelation, as suggested by the three best-fit lines. In particular, event two shows a stronger increasein drag with wave height.

represent variability in the sea, swell, and wind. We therefore express the dragcoefficient as a linear combination of these variables:

Cd = �+ �1Hrms + �2u�=cp (7)

where �, �1 and �2 are least-squares fit coefficients, equal to�5.3� 10�6, 0.0010,and 0.017 respectively. The measured drag coefficient is plotted versus this mod-elled drag in Figure 10; the first, second, and third wind events are again depictedby the circles, stars, and crosses respectively, and the rest of the data by the dots. Ingeneral, small values of the drag coefficient are over-estimated while large valuesare under-estimated. Additionally, much scatter remains: this multivariate model isable to describe only 28% of the variance in the drag coefficient. This result shouldnot be surprising, considering the poor correlation of each variable independently.Other parameters (e.g. wave steepness, dominant wave height, and wind gustiness)in various combinations were tested, none producing a significantly better result.It should also be noted that simple models using the directionality of the wind, sea,and swell were tested, but failed to make much improvement; a concerted effort


Figure 9. Drag coefficient versus wave age for the beginning of the three wind events analyzed (circles,stars, and crosses represent the first, second, and third wind events respectively). Significantly highercorrelations are seen between these data as compared to the full data set, indicating that during thesetimes drag decreases with increasing wave age.

to include directionality and the complex interactions between the various wavefrequencies is a major task for future work.

To investigate what may account for the large difference in drag between thethree wind events, a variety of wind and wave parameters were considered. Firstly,a comparison of the wave height spectra was made; the spectra for the first andlast period of each of the three wind events are shown in Figure 11. For display,the spectra of the first wind event have been offset downward by a factor of 10,and those of the third have been offset upward by a factor of 10. Significant wavegrowth occurred over each of the three periods, as shown by the increase in waveenergy above the high frequency saturation region. Comparing the three periods,a large difference is noticed between the first event and the latter two. While littleswell accompanied the first event, strong swell was present at the beginning of thesecond and third, as indicated by the peaks at low frequencies. As well, a contrastbetween the first two and third events can be made. Each of the first two had a cleardouble peaked structure, and the largest growth occurred principally between thepeaks. In contrast, the third had growth over the entire range of frequencies. Whilequalities of the first and third events may be identified as unique in comparisonwith the others, no such quality was found concerning the second. The directional

370 KARL F. RIEDER

Figure 10. Measured vs. multivariate modelled drag coefficient (circles, stars, and crosses representthe first, second, and third wind events respectively, while the rest of the data is represented by thedots). Low values of the drag coefficient are generally over-estimated by the model, while large valuesare under-estimated. The model accounts for approximately 28% of the variance of the measureddrag.

characteristics of the waves were also investigated, but did not yield any significantclue either.

To look at other possibilities, we calculated some of the important wind para-meters for the three events (Table I). While the wind speeds are similar for allof the wind events, both the mean Cd and rate of increase of Cd are much largerfor event number 2. The drag increases approximately five times more quickly ascompared to the other two wind events. Of the other factors considered, we findthat the turning rate of the wind to be most nearly a match. For event #2, thewind was turning quickly, as can be seen in the last column of Table I. The windturned steadily 24� in 5.5 hours: a rate more than four times as large as either ofthe other two events. Interestingly, it is during turning winds of steady speed, asopposed to increasing winds of steady direction, that the surface drag increases themost rapidly. Most importantly, however, these results indicate the importance ofdirectional and temporal characteristics of the wind and the waves in modellingof the drag coefficient. In drag coefficient parameterizations to date, directionalityand temporal change in the wind or waves have not been included.


Figure 11. Wave height spectra for the beginning and end of the three periods analyzed. Spectra forwind event #1 have been reduced by a factor of 10 and those for event #3 have been increased by 10.It is seen that the second and third events occurred during conditions of strong swell, indicated by thelarge peaks at lower frequencies. Also noticed is the double-peaked structure of the first and secondperiods and the resulting strong wave growth of the frequencies between the peaks.

Table ISome of the important wind parameters for the three events analyzed

Wind event1 2 3

Date, 3/4/1990, 3/7/1990 3/10/1990Time 15:30–20:30 19:30–01:30 18:30–05:30Mean U 9.45 m s�1 10.25 m s�1 11.00 m s�1

Mean dU /dt .24 m s�1 hr�1 .07 m s�1 hr�1 .24 m s�1 hr�1

Mean Cd 0.93 � 10�3 1.55 � 10�3 1.11 � 10�3

Mean dCd/dt 5.6 � 10�5 hr�1 25.1 � 10�5 hr�1 2.9 � 10�5 hr�1

Mean d�wind=dt 0.7� hr�1 4.3� hr�1 0.8� hr�1

3.5. TEMPORAL LAG BETWEEN THE WIND AND DRAG

We noted earlier that there was a temporal lag observed between the increase of thewind and the increase in the drag coefficient, and that this lag was on the order ofseveral hours. This reinforces the possible importance of temporal information in

372 KARL F. RIEDER

drag modelling. Also, it raises an interesting notion about which wave frequenciesare responsible for creating the drag of the ocean surface. By investigating how thevarious frequencies of waves respond to an increase in wind during these events(the higher frequencies will respond more quickly than low frequencies), we cansee which waves have the same temporal response as the drag.

There were three particular time periods when this temporal lag was evident:during the onset of the first and second wind events, and the secondary windincrease during the third wind event. For these three periods, we have plottedthe magnitudes of the 6–7, 4–5, and 2–3 second waves, the wind, and the dragcoefficient The wave magnitudes have been normalized by the saturation valuegiven by the Phillips (1958) spectra, where the Phillips constant is set to 5.5 �10�3.

Figure 12 depicts the first wind event, when winds doubled in just over an hour,slowed slightly in the following hour, and then increased again and maintainedspeed over the course of the next day. The 2–3 second waves respond quickly bynearly reaching saturation during the initial wind increase, while the 4–5 secondwaves do not reach saturation until just after the secondary peak in the wind. Thelowest frequency waves respond slower still, reaching a peak level more than 4hours after the final peak in the wind. Looking at Cd, we see that its magnitudeis more variable. It reaches its highest level just following the secondary peak inthe wind, at approximately the same time as the 4–5 second waves. As well, thevariation inCd mirrors most closely that of the 4–5 second waves, flattening duringthe decrease of the wind between the first and second peaks.

How the waves and drag respond to an abrupt increase in the wind is shown inFigure 13. During the second event, the wind increases rapidly and then maintains anearly constant value. The highest frequency waves are saturated prior to the windincrease, while the 4–5 and 6–7 second waves undergo notable growth. The 4–5second waves reach saturation approximately 3 hours after the peak in the wind,and the 6–7 second waves reach a peak approximately 5 hours after. Cd increasessteadily to a peak approximately 4 hours after the peak in the wind, in between thesaturation of the 4–5 and 6–7 second waves.

Figure 14 shows the response of the waves and drag to the sudden re-intensifi-cation of the wind during the third wind event. While a strong increase of the dragoccurs approximately 6 hours after the ramp-up of the wind, similar to the othertwo events, it is not accompanied by any notable increase in saturation level of anyof the three wave frequencies.

In these three periods, there seems to be a nearly-constant time lag of 4–6 hoursbetween the wind speed and the response of the drag coefficient. While it is notobvious that the increase of the drag is consistently linked with any particular wavefrequency, the response of the drag is most similar to that of waves with periodsbetween 4 and 6 seconds. The existence of this time lag further underscores thecomplexity of the wave and wind stress system.


Figure 12. (Top) normalized wave energy, (bottom) wind speed, and (bottom circles) drag coefficientduring the onset of the first wind event. Wave energies are normalized by the Phillips (1958) saturationspectrum. A distinct temporal lag of about four hours between the peak in the wind and the peak inthe drag is noticed. The variation of the drag can be seen to most closely match that of the 4–5 secondwaves.

4. Conclusions and Discussion

Several conclusions arise from this open ocean study:

� The drag coefficient,Cd, exhibits large temporal variability, which occurs overvaried time scales, from fractions of hours to several days.

� The historical drag modelling parameters of wind speed, wave age, and variouswave measures do not match variability, despite the fact that each seems tofollow the general trends in Cd and at times mirror local peaks and spikes.

� Roughness length models are unable to describe the range of variability ofCd.� Periods of variability in the downwind and crosswind components of the wind

often match periods of variability in Cd, indicating that both increasing andturning winds affect surface drag.

� Cd steadily increases and is more correlated to historical parameters duringgrowing seas, but the response of the drag coefficient varies from event toevent; the turning rate of the wind appears to be linked to higher drag increase.

� Multivariate regressions of Cd offer an improvement over single parametermodels.

374 KARL F. RIEDER

Figure 13. (Top) normalized wave energy, (bottom) wind speed, and (bottom circles) drag coefficientduring the onset of the second wind event. Wave energies are normalized by the Phillips (1958)saturation spectrum. Again, a temporal lag of approximately 4 hours can be seen. The variation ofthe drag seems to most closely match that of the 6–7 second waves.

� A temporal lag of O(4) hours exists between the peak in wind speed and thepeak inCd; lag and variation of the drag are matched best, but not consistently,by 4–6 second waves; higher frequency waves respond more quickly, lowerfrequency waves more slowly.

These results suggest that the modelling of the drag coefficient is a far morecomplicated task than previously assumed. Swell adds a level of complexity thatis not included in the historical formulations of the sea surface drag. Having beenbased upon spectral similarity found in fetch-limited seas, these models do nothave general application; the simplifying assumptions are clearly unjustifiable inthe open ocean.

While surface roughness does seem to increase with wind speed (or wave heightor inverse wave age), it does so at a rate and in a manner that is apparently highlydependent on the wind and wave history. The resulting effect is that there is nota one to one correlation between a given wind speed, for example, and a givendrag coefficient. This, added to the inherent short term variability that appearsimpossible to model, effectively disables the use of any one of these parametersalone to accurately estimate the surface drag. Other terms that hold temporal anddirectional information, such as the turning rate of the wind and the directonalityof the wave spectra, must play important roles in the determination of the overall


Figure 14. (Top) normalized wave energy, (bottom) wind speed, and (bottom circles) drag coefficientfollowing the re-increase of wind speed during the third wind event. Wave energies are normalizedby the Phillips (1958) saturation spectrum. A temporal lag is again noticed, despite the lack of anynoticeable response by the waves.

surface roughness. These parameters need to be determined and incorporated intothe models in order to allow them more universal application. Also, the observedtime lag between wind and drag should be further studied; it may hold clues to animproved parameterization.

Drag modelling for the open ocean is clearly more complex than for fetch-limited seas. Fetch-limited or equilibrium conditions provide a far more constanthistory and therefore result in a more consistent wind and wave dependency. In theopen ocean, the solution becomes much more difficult. The existence of waves fromdifferent parts of the ocean creates conditions in which the waves often have littleto do with the local wind. It is this lack of direct cause and effect that makes openocean drag coefficient modelling and prediction so difficult. While fetch limitedconditions do provide an excellent first start for modelling efforts, open oceanconditions need to be more seriously considered in order to be adapted for use inpredictive models. Clearly, these conditions, in which a mixture of waves co-exist,are the most poorly understood; however, these are the conditions that are mostcommon globally and are therefore those of greatest importance.

The overall patterns in these data indicate that increased drag occurs duringperiods of turning winds and growing seas, particularly when accompanied bylarge swell. These periods are also times of probable large standing wave ener-

376 KARL F. RIEDER

gy; perhaps, standing waves are important to overall surface roughness. Wavestraveling downwind are nearly aerodynamically smooth; standing waves, at leastconceptually, should represent a more effective roughness surface. As well, stand-ing waves perhaps could be more easily linked to formulations that are used toestimate roughness over the land. This is a subject for future research.

Finally, it is important to note that there are numerous other parameters unrelatedto the surface roughness that may contribute to the scatter in the drag coefficient. Thestructure of the boundary layer as controlled by large-scale atmospheric features,for example, must have an important influence on the wind stress. These parameters,however, have not been adopted into any drag coefficient modelling.

Acknowledgments

I am indebted to Dr. Jerry Smith who was critical in helping to direct the researchand edit the manuscript. Deep thanks go to Drs. Bob Weller and Al Plueddemannwho were responsible for the collection of the wind stress data set. I would also liketo acknowledge Dr. Gary Geernaert for his constructive criticism of the manuscriptand Dr. Richard Seymour for his continued advice and support.. This research wassupported by the Office of Naval Research and the Mineral Management Service,under contracts N00014-90-J-1285, N00014-93-1-0359, and N00014-93-1-1157.

References

Charnock, H.: 1955, ‘Wind Stress on a Water Surface’, Quart. J. Roy. Meteorol. Soc. 81, 639–640.Dobson, F. W., Smith, S. D., and Anderson, R. J.: 1993, ‘Measuring the Relationship Between Wind

Stress and Sea State in the Open Ocean in the Presence of Swell’,Atmosphere-Ocean 32, 237–256.Donelan, M. A.: 1982, The Dependence of the Aerodynamic Drag Coefficient on Wave Parameters,

First International Conference on Meteorology and Air/Sea Interaction of the Coastal Zone, TheHague, Netherlands, American Meteorological Society.

Donelan, M. A., Dobson, F. W., Smith, S. D., and Anderson, R. J.: 1993, ‘On the Dependence of SeaSurface Roughness on Wave Development’, J. Phys. Oceanogr. 23, 2143–2149.

Geernaert, G. L.: 1988, ‘Measurements of the Angle Between the Wind Vector and Wind StressVector in the Surface Layer Over the North Sea’, J. Geophys. Res. 93, 8215–8220.

Geernaert, G. L.: 1996, ‘On Modeling the Wind Stress Direction Based on Thermal Advection andSurface Waves’, in Mark Donelan (ed.), Proceedings of A Symposium on the Air-Sea Interface:Radio and Acoustic Sensing, Turbulence, and Wave Dynamics, in press.

Geernaert, G. L., Hansen, F., Courtney, M., and Herbers, T.: 1993, ‘Directional Attributes of theOcean Surface Wind Stress Vector’, J. Geophys. Res. 98, 16571–16582.

Geernaert, G. L.: 1990, ‘Bulk Parameterizations for the Wind Stress and Heat Fluxes’, Surface Wavesand Fluxes: Volume 1 – Current Theory, Kluwer Academic Publishers. 336 pp.

Geernaert, G. L., Katsaros, K. B., and Richter, K.: 1986, ‘Variation of the Drag Coefficient and ItsDependence on Sea State’, J. Geophys. Res. 91, 7667–7679.

Hsu, S. A.: 1974, ‘A Dynamic Roughness Equation and Its Application to Wind Stress Determinationat the Air-Sea Interface’, J. Phys. Oceanogr. 4, 116–120.

Janssen, P. A. E. M.: 1989, ‘Wave-Induced Stress and the Drag of Air Flow over Sea Waves’, J. Phys.Oceanogr. 19, 745–754.


Large, W. G. and Pond, S.: 1981, ‘Open Ocean Momentum Flux Measurement in Moderate to StrongWinds’, J. Phys. Oceanogr. 11, 324–336.

Phillips, O. M.: 1958, ‘The Equillibrium Range in the Spectrum of Wind-Generated Ocean Waves’,J. Fluid Mech. 4, 426–434.

Rieder, K. F., Smith, J. A., and Weller, R. A.: 1994, ‘Observed Directional Characteristics of theWind, Wind Stress, and Surface Waves on the Open Ocean’, J. Geophys. Res. 99, 22589–22596.

Rieder, K. F., Smith, J. A., and Weller, R. A.: 1995, ‘Some Evidence of Co-Linear Wind Stress andWave Breaking’, J. Phys. Oceanogr. in press.

Smith, J. A. and Bullard, G. T.: 1995, ‘Directional Surface Wave Estimates from Doppler SonarData’, J. Atmos. Oceanic Technol. 12, 617–632.

Smith, J. A. and Rieder, K. F.: 1997, ‘Wave Induced Motion of FLIP’, Ocean Engineering 24, 95–110.Smith, S. D. et al.: 1992, ‘Sea Surface Wind Stress and Drag Coefficients: The HEXOS Results’,

Boundary-Layer Meteorol. 60, 109–142.Toba, Y., Iida, N., Kawamura, H., Ebuchi, N., and Jones, I. S. F.: 1990, ‘Wave Dependence on

Sea-Surface Wind Stress’, J. Phys. Oceanogr. 20, 705–721.Weller, R. A., Donelan, M. A., Briscoe, M. G., and Huang, N. E.: 1991, ‘Riding the Crest: A Tale of

Two Wave Experiments’, Bull. Amer. Meteorol. Soc. 72, 163–183.Wu, J.: 1982, ‘Wind-Stress Coefficients Over Sea Surface From Breeze to Hurricane’, J. Geophys.

Res. 87, 9704–9706.Zemba, J. and Friehe, C. A.: 1987, ‘The Marine Atmospheric Boundary Layer Jet in the Coastal

Ocean Dynamics Experiment’, J. Geophys. Res. 92, 1489–1496.

analysis of sea-surface drag parameterizations in open

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