LIU Min, YANGBailin, JIANan, and ZOU Zhongshui

Dependence of Estimating Whitecap Coverage on Currents and Swells

LIU Min1), YANGBailin2), JIANan3), and ZOU Zhongshui4), *

1) Ocean College, Hebei Agriculture University, Qinhuangdao 066000, China 2) School of Resources and Materials, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China 3) PLA Information Engineering University, Zhengzhou 450001, China 4) School of Marine Science, Sun Yat-Sen University, Zhuhai 519082, China

The shipboard measurements of whitecap coverage () and the meteorological and oceanographic information from two cruises in the South China Sea and Western Pacific are explored for estimating. This study aims at evaluating how to improve the parameterizations ofwhile considering the effects of currents and swells on wave breaking. Currents indeed affectin a way that winds with following currents can decrease, whereas winds with opposing currents can increase. Then, 10-m wind speed over sea surface (10) is calibrated by subtracting the current velocity from10when the propagating directions of winds and currents are aligned. By contrast, when the direction is opposite,10is calibrated by adding the parallel velocity component of the opposing current to10. The power fits ofdependence on the10-related parameters of10, friction velocity, wind sea Reynolds number in terms of this calibrated-10obtain better results than those directly fitted to10. Different from the effect of currents on, wind blowing along the crest line of swells may contribute to the increase in. The conclusions suggest that10should be calibrated first before parameterizingin areas with a strong current or some swell-dominant areas.

whitecap coverage; currents; friction velocity; wind sea Reynolds number; swells

1 Introduction

Wave breaking is important for air-sea interaction processes, which can be expressed as whitecap coverage () for easier observation and better description. The parameterizations ofare of great interest and traditionally fitted to 10-m wind speed over sea surface (10) (Monahan, 1971; Hanson and Phillips, 1999; Stramska and Petelski, 2003). As certain differences exist among10, onlyparameterizations are used in various studies. A considerable effort has been exerted to find other parameters, such as field statistics and meteorological factors, for newparameterizations. Till now, the overall parameters used for estimatingcan be divided into two kinds: one is the10-related parameters, such as friction velocity (*) and wind sea Reynolds number (H) (Wu, 1979; Guan, 2007; Yuan, 2009), the other is named secondary factors or10absence parameters, including current velocity (current), thermal stability of the lower atmosphere, seawater temperature, and fetch (Wu, 1988; Xu, 2000; Callaghan, 2008; Salisbury, 2013).

Anguelova and Webster (2006), Ren(2016), and Brumer(2017) provided a chronological listing of whitecap fraction datasets during the period of 1971–2004, 1963–2013, and 2007–2016, respectively. The10-relatedpower law functions can be influenced by currents due to wave current interaction (Khojasteh, 2018). When the along-wind current velocity is large enough,Haus (2007) found that the relative winds shifted by the currents, thereby affecting wave growth rates.Pearman(2014) showed that the effect of the current field on the swell is negligible. Nevertheless, for the wind wave high-frequency tail of wave spectral, steepening on the opposing current may contribute to thewave breaking dissipation or observed decay. Novel airborne observationsaccomplished by Romero(2017)over areas withstrong wavecurrent interactionsshowedmaximum vertical vorticity with a largebecause of strengthened wave break- ing.

Currents indeed have an influence on, which may change the ‘effective-10’ for wave growth. For example, although the10is in the range of 7–8ms−1, Kraan(1996) found no visual whitecaps and suggested that the absence of whitecaps was caused by strong tidal currents of up to 1ms−1. Callaghan(2008) believed that a directionalalignment of wind and currentaccompanied by an increase incurrentproduceda marked increase in. Meanwhile, the measurements ofin the presence of the recorded magnitude and direction ofcurrents are scarce. Therefore, the manner in which currents work is unclear.

Apart from currents, swells are also under our consideration if they have the same effect as the currents on(Zheng, 2019a, 2019b, 2020). Woolf (2005) proposed the importance of swells for. Sugihara(2007) and Callaghan(2008) indicated that whitecaps tend to be suppressed by the presence of swells, especially under the condition of opposing swells. Hwang(2013) believed that swells could promote whitecaps.

On the basis of the data measured by ship, we attempt to further explore how currents and swells affectand determine if they influencein the same way. The following are the remaining parts of the paper. Section 2 introduces the data used in this study. Section 3 discusses how the magnitudes and directions of currents and swells influenceand proposes the improved power fits ofto10-related parameters on the basis of the current calibrated-10. Section 4 describes some important conclusions.

2 Whitecap Observation

Meteorological and oceanographic data were collected from two cruises byfrom Ocean University of China: the South China Sea cruise from December 5, 2013 to January 4, 2014 and the Western Pacific Ocean cruise from November 7, 2015 to January 7, 2016. The two projects focused on the study areas around 102˚–115˚E, 1˚S–21˚N and 135˚–161˚E, 1˚S–36˚N (Fig.1).

Fig.1 A map showing the geography of the zone and trajectory of the scientific cruises. The blue stars and red pluses represent the South China Sea cruise and the Western Pacific Ocean cruise, respectively.

The field statistics of waves and currents were continually recorded by the X-band Radar of WaMoSII and measured by the Waverider and Conductance Temperature Depth at some fixed stations. Meteorological parameters were obtained by the automatic weather station, 16m in height above the sea surface. The observed wind speed values were corrected to10on the basis of the logarithmic wind profile formula. The measurements ofwere manually taken by a camera on the top of the vessel, and thenwas extracted from photographs using the automated white- cap extraction (AWE) method proposed by Callaghan and White (2009) and the improved AWE algorithm proposed by Jia and Zhao (2019).

Here is a brief description of the improved AWE algorithm, that is, how the photos of the sea surface are converted to values of. For traditional AWE, each photo is first converted into a grayscale image with pixels ranging from 0 to 1, and the percentage increase in pixels is used to obtain potential thresholds for separating whitecaps with background water. To reduce the influence of strong sun- light, the light distribution and brightness contrast of the measured photo are adjusted by Jia and Zhao (2019) to obtain a precise intensity threshold. Avalue is then acquired after computing the ratio of white pixels to black pixels on the basis of the intensity threshold. Data are divided into two subsets: the deflection angle between the propagating directions of wind and current smaller (greater) than 90˚ is denoted as |wind−current|<90˚ (>90˚), indicating the alignment (encounter) of wind and current. Considering the effect of currents on,10is calibrated on the basis of the above deflection angles. Using the least square method,10and current calibrated-10dependent parameterizations are individually fitted to each dataset and combined. Two fit statistics of correlation coefficients (2) and root mean square error (RMSE) are introduced to evaluate the parameterizations. Parameterization equations with higher2and lower RMSE provide better results.

3 Results and Discussion

3.1 Influence of Currents on W

10is the most traditional and effective way of estimating. To discover the effect of currents on, Fig.2 shows two original whitecap images taken during the Western Pacific Ocean cruise under the same10condition. Other meteorological information, wave parameters, andlisted in Table 1 are different for images (a) and (b). Although10is identical, the wave heights of mixed wave, swell, and wind wave for image (a) are much higher than those for image (b), which may be due to the effect of currents(Haus, 2007). Moreover, additional whitecaps can be seen from image (a), and a high(0.3486 in %) is obtained after averaging several hundreds of whitecap images taken at the same station asimage (a). However, averagedis only equal to 0.1474 forimage (b).

The datasets analyzed here display nosignificantcurrentdependence (not shown), reminding us to find a new way to study the influence of currents on. The datasets are divided into two subsets to refit the(10) power laws: winds with following currents and winds with opposing currents. The parameterizations ofare displayed in Table 2 and Fig.3 where winds with followingcurrents labeled by a dot-dashed line obtain a small, whereas winds with opposing currents obtain a great. The direction difference between winds and currents(represented by |wind−current| wherewindandcurrentare directions of wind and current, respectively) can impact thepowerlaws, and the opposing (following) currentsare suggested to increase(decrease). |wind−current| should be considered for the continued improvement of whitecap para- meterizations.Intrigued by these findings, we introducecurrentinto whitecap parameterizations when considering the direction deviation of currents with winds.

Fig.2 Two original whitecap images taken during the western Pacific Ocean cruise under the same wind speed condition. Meteorological information, wave parameters, and the whitecap ratio W (%) for (a) and (b) are listed in Table 1.

Table 1 Summary of the meteorological and oceanographic conditions

Notes:10, wind speed at 10m height;wind, wind direction;current, current speed;current, current direction;s, significant wave height;ss, wave height of swell;sw, wind sea;, whitecap coverage. The geographical locations for Figs.2(a) and 2(b) are also illustrated.

Table 2 Parameterizations of W (%) as a function of U10 and Ucurrent calibrated-U10 (marked by ΔU)

Notes: Numbers 1–3 (4–6) given in the first column are used as references in the legends in Fig.3 (Fig.4). The second column presents the data used when obtaining the formula, and |wind−current| are the deflection angles between the propagating directions of winds and currents. The Δin Numbers 5 and 6 represent the difference between10and the velocity of following currents and the sum of10and the parallel velocity component of opposing currents, respectively. The Δin Number 4 is a combination of the two Δin Numbers 5 and 6.2and RMSE represent the correlation coefficients and RMSE, respectively. The increase/decrease rate in the last column is the change rate of2and RMSE by fitting to Δinstead of10.

10onlyparameterization, the most traditional and effective way of estimating, is improved by including wave field related parameters in this study. Considering the effect of currents on wave breaking, adjusting the parameters to include thecurrentin the10basedparameterization is performed in this section according to the value of |wind−current|. That is,10is calibrated with different methods separately for winds with following or opposing currents. When the value of |wind−current| is smaller than 90˚,10is calibrated by subtracting thecurrentfrom10because the directional alignment of winds and currents accompanied by small wave steepness can weaken the wave breaking. On the contrary, when |wind−current| is greater than 90˚, Δequals10is calibrated by adding the parallel velocity component of opposing currents. The calibrated-10marked by Δare as follows:

∆=10−current, when |wind−current|<90˚, (1)

∆=10+currentcos(180−|wind−current|),

when |wind−current|>90˚. (2)

Δbasedparameterizationsare refitted and displayed in Fig.4, which shows the same conclusions as those in Fig.3. For the directional alignment of wind and current situations,2increases from 0.5995 to 0.7107 and RMSE decreases from 0.1461 to 0.1242 when comparing the results in Rows 2 and 5 of Table 2. For the opposing current situation in Rows 3 and 6 of Table 2, we obtain the consistent conclusion that2increases up to 0.8048 accompanied by a decreasing RMSE. For the fit with the combined dataset, Δbasedparameterization provides a better fit than that fitted to10. Moreover, thelast column in Table 2 presents that the improvement of goodness of fit for the following current situation is the most significant; the increase rate of2is up to 18.55%, and the decrease rate of RMSE is down to −15.00%. In summary, Δbased parameterizations show tighter correlations and better interdataset agreement than10only parameterizations for the methods of calibrating10in Eqs. (1) and (2).

Fig.3 Dependence of W (%) on U10. The curves symbolize the best fit to the different datasets of Numbers 1–3 in Table 2: solid line, dot-dashed line, and dashed line are fitted by all winds, winds with followingcurrents, and winds with opposingcurrents, respectively. Stars and dots represent averagedW (%) when the winds and currents have a consistent and opposite direction, respectively.

Fig.4 Dependence of W (%) on ΔU. ΔU (as in Eq. (1)) represents the difference between U10 and the velocity of following currents for star data and dot-dashed line. ΔU (as in Eq. (2)) represents the sum of U10 and the parallel velocity component of opposing currents for dot data and dashed line. ΔU for solid line is a combination of the two ΔU above.

Inspired by the occurrences above, the contour map, which includes the combined effect of the difference in the magnitudes and directions of winds and currents, is displayed in Fig.5. Specifically, the contour suggests the effect of the magnitude and direction of currents on. We already fit theto10–currentregardless of |wind−current| (not shown). It shows that the greater10–currentmakes, the higher, as expected, which can also be inferred from Fig.5. Certainly, the improved parameterizations of, as a function of10–current, are still worse than those offitted to Δin Fig.4. Therefore, we confirm again that the currents significantly influenceand must be considered to calibrate-10before fittingto10-related parameters.

Fig.5 W (%) as a function of ΔU and |Dwind−Dcurrent|. Black dots represent the original observations.

3.2 Adjustment of u* and RH with Calibrated-U10

Many parameterizations of, as a function ofu, have been used in previous studies (Lafon, 2007; Sugihara, 2007;Schwendeman and Thomson, 2015; Brumer, 2017).ucan be obtained through Eq. (3):

whereis the wind stress, andis the air density.10represents a 10m drag coefficient over sea surface. The10used for calculatingufrom previous studies here is displayed in Fig.6 and summarized in Table 3.Compared with the fit statistics of formula from 1 to 14,10from Sheppard(1972) should be the best choice, and thisucan be applied for further analysis hereafter (Row 6 in Table 3). The fit to the calculatedumost closely follows that proposed by Schwendeman and Thomson (2015), as shown in Fig.6.

The calculatedubased on the10from Sheppard(1972) is applied to obtain(u) parameterization. Following the same approach discussed in Section 3.1, the parameterizations of, as a function ofu, are determined by fitting the two subsets of data as defined, and the results are illustrated in Fig.7. Plots of theu, as a function of Δ, are shown here to illustrate the important influence of currents on. For example,2increasing from 0.7668 to 0.7958 and RMSE decreasing from 0.2157 to 0.2018 confirm the effect of currents onubased parameterizations. Overall, better fits are found whenuis expressed as a function of Δinstead of10, as in the case of10only parameterizations in Table 2.

Table 3 Parameterizations of W (%) as a function of u* and RH

Note: Numbers given in the first column are used as references to calculateuin the legends in Fig.6.

Fig.6 W (%), as a function of u*calculated using a different CD10 formula concluded in Table 3, corresponds to lines 1–14. Lines 15, 16, 17, and 18 represent W (u*) power law fit summarized in Schwendeman and Thomson (2015), Sugihara et al. (2007), Lafon et al. (2007), and Brumer et al. (2017), respectively.

Zhao and Toba (2001) first put forward a kind of dimensionless parameter defined asHand suggested thatHismore related withthan with10alone. However, significant wave heights are usually selected to computeH, even though it was originally applied exclusively for windsea circumstances (Goddijn-Murphy, 2011). In this study, we use Eq. (4) to calculateH:

whereswis the wave height of the windsea, andwis the viscosity of seawater. Most air temperatures covered therange of 22℃–30℃ during the two cruises. Thus, in Eq. (4),wis fixed at the value of 1.0098×10−6m−2s when the temperature of seawater is 22℃. Aswis dependent ontemperature and the salinity of seawater (Monahan and Zietlow, 1969; Monahan and O’Muircheartaigh, 1986;Sharqawy, 2010),Hbased parameterizations can have a good fit if thewis variable. Fig.8 shows theplotted against theHin terms of10and Δ, and the fit statistics of2and RMSE are listed in Table 4. The power law fit of, in terms of Δ, obtains a greater2of 0.7499 and a smaller RMSE of 0.2483 than thein terms of10. Only 17 out of 128 data points have a current velocity of more than 1.00m·s−1, which may lead to a relatively low increase or decrease rate for2and RMSE by fitting Δin Tables 2 and 4, respectively. The currents do have an effect on. If many measurements are obtained from a cruise passing strong current areas, then the conclusions can be further confirmed.

Fig.7 Dependence of W (%) onu*. u*–U10 (dots) and u*–ΔU (stars) refer to the parameters of u* calculated usingU10 and ΔU in Table 2, respectively. Thecorresponding best fit tou*is shown by the solid line and dashed line.

To sum up, the parameterizations of, as a function of10,u, andH, are all in better agreement with observations when fitted to Δthan those directly fitted to10. We suggest that currents are important for the parameterizations of, as a function of10-related parameters, because currents can change the ‘effective wind speed’ for wave growth.

3.3 Influence of Swell on W

We consider wave current interaction and thus investigate the statistical distributions of wave height, wave direction for wind, wind sea, and swell, as illustrated in Fig.9. The distributions of wind and wind sea are consistent with each other, whereas the direction of swell is scattered. The directional overlap between wind and wind sea suggests that enhanced wave breaking or increasedwith opposing winds and currents is likely a result of opposing wind waves and currents and the wave current interaction between them. Wave breaking may be strength- ened when laminar flow changes into turbulent associated with horizontal shear instability. For horizontally sheared currents, MacIver(2006) provided evidence that opposing(following) waves bend toward the currentnormal(parallel) andincrease (decrease) in height based on laboratory experiments. Similarly, for uniform currents, wave heights increase(decrease), and wave wavelength isshortened (lengthened), leading to large (small) wave steepness when waves move against an opposing(following) current (Haus, 2007). The scattered swells in Fig.9 urge us to reconsider if swells are insignificant for, especially in swell-dominant waters whenis parameterized usingHor if swells influencethe same way as currents.

Fig.8 Dependence of W (%) on RH. RH–U10 (dots) and RH–ΔU (stars) mean the parameters of RH calculated usingU10 and ΔU, respectively. The corresponding best fit toRHis represented by the solid line and dashed line.

Table 4 Parameterizations of W (%) as a function of u* and RH.

Notes:u(H)–10andu(H)–Δmean the parameters ofu(H) calculated using10and Δ, respectively. The increase/decrease rate in the last column is the change rate of2and RMSE by fitting to Δinstead of10.

Fig.9 Measurements ofprobability distributions by the X-band radarof WaMoS II: wind, wind sea, and swell.

To implore the reason whyis suppressed by swells shown in Fig.10,, as a function of10, and the deflection angle between wind and swell (|wind−swell|) are displayed in Fig.11. The most striking feature of Fig.11 is the conspicuously high values of10and |wind–swell| that are approximately 11ms−1and 90˚, respectively. Lowvalues ofare found where the propagating directions of winds and swells are parallel. Sugihara(2007) found no certain relationship betweenand the deflection angle between the propagating directions of wind waves and swells. Fig.11 summarizes that the perpendicular (parallel) winds to the propagating direction of swells can increase (decrease). An explanation for a greatwhen the deflection angle is approximately 90˚ may rest on that wave breaking occurs easily on the wave crest through disturbance when the wind blows along the crest line of swells.

Fig.10 Parameterizations of W (%) as a function of the wave height of swell (Hss) and wind sea (Hsw).

3.4 Discussion

Kraan(1996) found no visual whitecaps at10of 7–8ms−1and suggested that it is caused by strong tidal currents. Given the absence of direction information, we guess that such tidal currents and their observedwindare in the same direction, which contributes to this extremely smallaccording to the theory we proposed above. Callaghan(2008) observed a sharp increase inwhen10decreases steadily. Whencurrentincreases rapidly,windremains the same, butcurrentchanges abruptly. We disagree with Callaghan(2008), who believed that the directional alignment of winds and currents and increase incurrentco-produce an increase in. Here, we propose that the enlarged difference in magnitude and direction between winds and currents increases.

Fig.11 W (%) as a function of U10 and |Dwind–Dswell|. The black dots represent the original observations.

Moreover, we consider that the effect of |wind–current| onis caused by wave current interaction because of the basically consistent directions of winds and wind seas. In areas with strong currents, such as the western boundary current, the calibration of10is necessary. Swells indeed affectbut in a different way compared with currents. We suppose that the propagating direction of swells perpendicular to winds contributes to a great. We cannot avoid scarce observations, including current information in previous studies. The theory here can be a good explanation for the phenomenon in Kraan(1996) and Callaghan(2008) and should be verified with additional data in further studies.

4 Conclusions

Using the ship-based observations from two cruises in the South China Sea and Western Pacific, we present an analysis of the influence of currents and swells on whitecap fraction.

First, according to the deflection angle between the propagating directions of winds and currents smaller or greater than 90˚, data are divided into two subsets.10-dependent parameterizations are fitted using the two subsets individually and combined to find that the following (opposing) currents can decrease (increase). On the basis of the result,10is calibrated by subtractingcurrentfrom10in the condition of following currents, whereas the opposite situation10is calibrated by adding the parallel velocity component of opposing currents to10. The power law fits in terms of Δand provides an increasing R2and decreasing RMSE, illustrating better estimates ofthan those directly fitting to10. Second, Δis introduced touandH, leading to an improvement on theparameterizations.

Overall, swells can cause a negative impact on. Nevertheless, wave breaking is enhanced when the winds blow along the crest line of swells. Considering that we lack information about wave velocity, swells withparameterizations are yet to be studied and may be discussed in the future.

Acknowledgements

This work was financially supported by the Hebei Agricultural University Research Project for Talented Scholars (No. YJ201835), the National Natural Science Foundation of China (No. 41806028), the China Postdoctoral Science Foundation (No. 2019M65206), and the Fundamental Research Funds for the Central Universities (No. N182303031). The efforts of the researchers who obtained and published the data adopted in this study are much appreciated. We thank the crew, scientists, and students infor the help in the process of collecting observation data.

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(Edited by Xie Jun)