1. Introduction
Wind stress over the ocean is an important physical parameter, and it plays a significant role in the numerical simulation of both atmosphere and ocean systems, which requires an accurate parameterization. While numerous parameterization schemes for wind stress have been proposed in previous studies (Large and Pond 1981; Taylor and Yelland 2001; Oost et al. 2002; Drennan et al. 2003; Fairall et al. 2003), there are still large uncertainties and differences in these parameterizations due to the complexity of the physical properties behind wind stress (Babanin and Makin 2008).
Since then, the drag coefficients have differentiated behaviors from previous accessible literature. In the case of CD behaviors, a large scatter of CD is presented in plots of CD and U10 at a given wind speed, especially at low to moderate wind speeds (Toffoli et al. 2012; Högström et al. 2015; Zou et al. 2017), while CD values become concentrated with increasing wind speed, and scatter still exists at wind speeds above 20 m s−1 (Chen et al. 2022). A variety of physical processes are hidden in this scatter, such as wind field, atmospheric stratification, mixed sea state, swell, wind-wave misalignment, wave breaking and other possible aspects (Babanin and Makin 2008; Andreas et al. 2012; Porchetta et al. 2019; Patton et al. 2019; Sauvage et al. 2023). To reduce this large scatter, we should incorporate these physical processes into the CD parameterization. Toffoli et al. (2012) used the 132 measurement records in the Lake George Experiment to show that the combined effect of wind, relative humidity, wave steepness and water depth can account for most of the scatter. Patton et al. (2019) used large eddy simulations to evaluate the influence of swells on CD and showed that the predicted results are roughly concentrated near the 1:1 scale line with observations after incorporating swell direction into the parameterization proposed by Andreas et al. (2012).
Another phenomenon is that the drag coefficient and roughness length may be very different between coastal regions and open oceans. In situ experiments indicated that coastal waters have higher surface drag or roughness relative to the open ocean (DeCosmo et al. 1996; Taylor and Yelland 2001), which is also supported by modeling evidence with wind speeds up to 20 m s−1 provided by Jiménez and Dudhia (2018), who suggested that the roughness length formulation needs to be modified in shallow waters. The enhanced surface drag in shallow waters is likely associated with the shoaling wave effect and the increased wave steepness. In addition, at higher wind speeds (U10 > ∼20 m s−1), the drag coefficient will saturate and even attenuate (Powell et al. 2003; Jarosz et al. 2007). Note that due to the scarcity of sample size and the limitations of the method itself, this attenuation has a certain uncertainty and needs to be treated with caution (Richter et al. 2021). It is evident that the critical wind speed with saturated drag coefficient over coastal regions is much lower than that over open oceans (Powell et al. 2003; French et al. 2007; Holthuijsen et al. 2012; Zhao et al. 2015; Hsu et al. 2017), which is linked to the evolution of surface waves due to water depth variations. A wave boundary layer model illustrated that U10c decreases with decreasing depth (Xu and Yu 2021). Furthermore, Chen et al. (2022) concluded that the statistical average U10c is 20 m s−1 for coastal regions and 30 m s−1 for open oceans through combination with previous observations. However, thus far, the modelers still use the same CD or z0 scheme derived from the open oceans to the shallow waters.
Coastal regions are the most active areas of human activities in the world and are heavily affected by tropical cyclones and storm surges. To simulate the atmosphere and ocean dynamics over coastal regions more accurately, a more appropriate wind stress parameterization needs to be established, which requires field observations with high standard quality as data support. Offshore tower-based platforms can provide stable and reliable sites for in situ observations, which can ensure the acquisition and accumulation of air–sea fluxes and surface wave datasets. The purpose of the present study is to propose a wave-state-based drag coefficient parameterization suitable for coastal regions, from low to high wind speeds, through in situ observations on three offshore tower-based platforms. The remainder of the study is organized as follows. The field observations and data handling are described in section 2. Our observational results and comparisons with previous studies are presented in section 3. A new wave-state-based drag coefficient parameterization is proposed in this section. Finally, our conclusions are summarized in section 4.
2. Field observations and data processing
a. Field observations
The air–sea momentum flux and surface wave datasets obtained from three offshore tower-based platforms are used to investigate the wind stress (Fig. 1). A fixed tower has remarkable advantages in measuring high-frequency turbulence relative to buoy- or ship-based systems. To date, there are few offshore towers suitable for obtaining flux observations over the global ocean (Edson et al. 2007; Zhao et al. 2015; Porchetta et al. 2019). Detailed information about the Bohe observation tower (BHOT; 21°26.5′N, 111°23.5′E) in the northern South China Sea and the Dongou observation tower (DOOT; 27°40′N, 121°21′E) in the western East China Sea has been described in our previous papers (Chen et al. 2018, 2019, 2020a,b). Here, we provide some critical parameters. The water depths are approximately 16 and 28 m, and the distances from the coastline are approximately 6.5 and 24 km for BHOT and DOOT, respectively. Both towers deployed the same eddy covariance system to continuously record the 10-Hz three-dimensional velocities u, υ, and w, ultrasonic virtual temperature, and other meteorological variables, such as air temperature and pressure. Here, the system consists of a CSAT3A three-dimensional ultrasonic anemometer and other sensors produced by Campbell Scientific, Inc. The same bottom-supported acoustic wave buoy was used to observe surface waves at both sites. The wave buoy (AWAC series) is manufactured by Nortek company, Norway, and the sampling frequency is set at 2 Hz, and continuous measurements are made every hour for 1024 s to obtain wave information. The datasets used in this study were collected in the experiments in 2015 (EXP1) and 2018 (EXP2) for BHOT and in 2017 (EXP3) for DOOT, which were published by Chen et al. (2018, 2019, 2020a,b). The heights of our systems above the mean sea surface during EXP1 and EXP3 are 17 and 26 m, respectively; during EXP2, three-level systems were deployed at heights of approximately 8, 12, and 15 m, respectively.
(a) Regional map and locations of the CBOT, BHOT, and DOOT and photos of the (b) CBOT, (c) BHOT, and (d) DOOT. The dates shown in the plots represent the observational periods. The black numbers represent the heights of our eddy covariance systems above the mean sea surface.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0081.1
The Chengbei observation tower (CBOT; 38°15.6′N, 118°59.5′E) is our newly developed platform suitable for comprehensive experiments. It located in the southern Bohai Sea of China, 18 km away from the nearest coastline, and the water depth is approximately 19 m at this site. On a small platform with a distance of approximately 10 m from the mean sea surface is a 15-m-high tower; the main body of this tower is a hollow triangular steel structure, and the distance between both steels is 1.2 m. We conducted a series of experiments including observations of air–sea fluxes and surface waves from 14 November 2021 to 6 January 2022 (EXP4) and from 16 October 2022 to 3 January 2023 (EXP5). These two periods were selected mainly to measure the air–sea fluxes and surface waves with higher wind conditions under the influence of northeast to northwest cold air. Our eddy covariance system was mounted on a stable horizontal boom toward the north and extended outward 2.5 m from the tower structure, which can minimize the flow distortion of the structure. The distance between the system and the mean sea surface is 12 m. Near this tower, a bottom-supported acoustic wave buoy similar to those at the BHOT and DOOT is also deployed to measure wave properties, such as wave height, period, and direction. All devices are consistent with those introduced by our previous papers. We collected a total of 250 days of data over five experiments, which were further analyzed and processed.
b. Analysis method and data processing
Notably, several previous studies have argued that MOST is invalid under swell- dominated conditions (Drennan et al. 1999; Högström et al. 2013; Zou et al. 2018). To minimize the site-dependence of parameterization and provide a more general formulation, we need to convert the wind speed at different sites and various heights to the reference height of 10 m in the framework of MOST. Therefore, it is necessary to analyze the uncertainty or accuracy of this transformation to U10N. Here, we use the three-layer observations during EXP2 to transform the wind speed at each level to the 10-m height. During EXP2, the sea state is dominated by swell (Chen et al. 2020a). The comparisons of U10N are shown in Fig. 2. The slopes and R2 values of the linear fit between U10N1, and U10N3 and U10N2 are 1.01 and 0.99, and 1.00 and 0.99, respectively. This means that compared to other heights, U10N values at certain heights generally fall on the 1:1 scale line and match well with each other, excluding some particular values, even for wind speeds below 5 m s−1. Notably, this is compared without data quality control. In addition, the average deviations between them are less than 5%. These results indicate that the transformation of wind speed is practicable and meaningful for the analysis of the drag coefficient at various heights and sites. For EXP2, the wind speed U10N2 transformed through the middle layer is used in the subsequent analysis. Hereafter, we use CD and U10 to represent the drag coefficient and wind speed under near-neutral conditions for simplicity.
Comparison of the wind speed U10N at heights of 8.4 m (U10N1), 12.4 m (U10N2), and 15.3 m (U10N3). The black dashed lines in both plots represent the 1:1 scale line.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0081.1
Then, a strict quality control procedure for all datasets is adopted to identify the data points that can be used for further analysis, as shown in Table 1. The first step (QC1) is despiking. Data are treated as spikes with values greater than 3 times the standard deviation for each run, which are omitted from the data segment and interpolated linearly using adjacent data points. The data segment with spikes beyond 5% of the entire time series is discarded in this step. For all experiments, 324 groups of data are excluded from 6000 groups. Then, the double coordinate rotation method is applied to the tilt correction, rotating the X axis to the mean wind direction, with mean vertical and lateral velocities being zero. After subtracting the mean value of u, υ, w, and Tυ or θυ, the corresponding fluctuations are obtained. Then, the covariances 〈u′w′〉, 〈υ′w′〉, and
The number of data runs in the process of quality control for observational datasets during all experiments. QC1 represents despiking; QC2 stands for selecting data according to the wind direction, removing data runs from behind the ultrasonic anemometer; QC3 represents the elimination of data with a stability parameter z/L greater than 2 and less than −2; QC4 means removing data runs with U10 < 1 m s−1; and QC5 represents outlier elimination referring to the data selection method in Oost et al. (2000, 2002). The corresponding numbers in columns QC1 to QC5 are the data runs removed from the total.
The high-frequency turbulence, especially 3D velocities observed by ultrasonic anemometers, with the wind from behind the instrument, referred to as the wind shadow zone, is possibly affected by the structure of the tower and the instrument itself, resulting in increased uncertainty in the data (Porchetta et al. 2019). Therefore, in the second step (QC2), we strictly eliminate the data runs in the wind shadow zone, which depends on the orientation, such as 0°, and the data in the 90°–270° direction range are removed. After this procedure, approximately 36% of the data runs are eliminated, especially for the CBOT, with more than 55%; however, the remaining data belong to the northwest to northeast wind range, which ensures a relatively long fetch. When stability correction is applied [Eq. (4)], extremely large stability parameters z/L may produce unrealistic values, such as the negative U10 at low wind speeds. In fact, several studies have suggested that the absolute z/L should be limited to less than 2 in Eq. (4) (Zou et al. 2017). Therefore, the data runs with z/L > 2 and z/L < −2 are excluded (QC3), and 104 sets of data are discarded for all experiments. Additionally, the wind direction has a large uncertainty at wind speeds below 1 m s−1 (Anfossi et al. 2005). Hence, 46 groups are further removed from all measurements (QC4).
Finally, we perform a further outlier inspection on the remaining datasets with reference to the data filtering method in Oost et al. (2000, 2002), as shown in Fig. 3. We also provide a second-order polynomial fit to
Friction velocity
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0081.1
3. Results and discussions
Wind stress is usually induced by wind, which is the most direct factor. Over the ocean, surface waves significantly modulate the momentum transfer between the atmosphere and the ocean, and they are also largely dependent on surface winds. In other words, winds, wind stresses, and surface waves have complex interactions with each other. However, the earliest studies on wind stress focused only on wind speed (Garratt 1977; Wu 1980; Large and Pond 1981). Later, researchers realized the importance of surface waves and began to parameterize the drag coefficient CD or the roughness length z0 with wave parameters, such as wave age and wave steepness (Taylor and Yelland 2001; Oost et al. 2002; Drennan et al. 2003). Although the wind information is hidden in these wave parameters, it should be mentioned that swell is not related to local wind at all. Several studies have used a wave boundary layer model (WBLM) to estimate wind stress and then coupled it with ocean and atmospheric models (Janssen 1989; Moon et al. 2004; Du et al. 2017). Wind stress is a combination of wind-generated turbulent stress and wave-induced stress in WBLM. Therefore, we will start with the relationship between the drag coefficient CD and wind speed U10 to determine what can be improved and then provide a new parameterization of CD, including wind and wave states based on observations.
a. Relationship between drag coefficient and wind speed
(a) Friction velocity
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0081.1
Subsequently, the behavior of Eq. (10) at wind speeds up to 30 m s−1 is presented in Fig. 5. Previous observed drag coefficients over coastal regions are shown in Fig. 5 (Vickers et al. 2013; Zhao et al. 2015; Bi et al. 2015; Chen et al. 2022), also adding COARE 3.5 (Edson et al. 2013) and other fitting functions in previous studies (Wu 1980; Large and Pond 1981; Donelan 1982; Yelland and Taylor 1996). At wind speeds less than 20 m s−1, the average values of CD are greater than those estimated by the COARE 3.5 flux algorithm and other functions established through flux measurements over open oceans (Figs. 4b and 5), which is similar to the result reported by Jiménez and Dudhia (2018). The RMSE between COARE 3.5 and our observations is 2.31 × 10−3, which is greater than Eq. (10). As Edson et al. (2013) evaluated, COARE 3.5 is valid in the U10 range of 0–25 m s−1. When extended to 30 m s−1, it does not exhibit saturation. However, our formula [Eq. (10)] can well predict the saturation and attenuation of CD and agrees with previous observations at other coastal sites. In addition, the critical wind speed for CD saturation is close to the statistical average value of 20 m s−1 concluded by Chen et al. (2022) in coastal regions (see their Fig. 7b). Starting with the relationship between friction velocity and wind speed, we derive a formulation that only depends on wind speed, suitable for a wind speed range of 1–30 m s−1, and most importantly, appropriate for coastal regions. It should be pointed out, although our parametric model predicts the attenuation of drag coefficients, there is still uncertainty in the behavior of drag coefficients at high wind speeds due to the lack of large direct measurement dataset. This attenuation of CD is still controversial or not settled, which still requires more observational data for evaluation and examination.
The drag coefficient CD vs the wind speed U10. The gray dots and red curves are consistent with those in Fig. 4, but extended to 30 m s−1. Previous results are presented for comparison with our parameterization. The black dashed curve is COARE 3.5. The blue circles refer to the observations on BHOT during the passage of Typhoon Mujigae in 2015 reported by Chen et al. (2022). The other symbols represent the average values over a certain U10 bin size given by other papers in coastal regions. The other lines represent the fitted lines by other papers in open oceans.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0081.1
b. Wave-state-based parameterization on drag coefficient
Although the formulation [Eq. (10)] can predict the behaviors of CD overall and simulate the saturation of CD, it is evident that the observed CD values disperse at a given wind speed, especially at low wind speeds. This finding illustrates that other factors or physical processes modulate wind stress in addition to winds (Babanin and Makin 2008). Surface waves are a remarkable factor that has been widely confirmed through observations and numerical simulations (Janssen 1989; Högström et al. 2015; Zou et al. 2018).
Surface waves are generally produced by wind, wind-seas are generated locally, and swells are generated remotely. In general, the surface wave state is distinguished by the wave age, β = Cp/U10, where Cp is the peak wave phase speed. Wave age β > 1.2 is the prevailing swell condition, and β < 1.2 belongs to the wind-sea dominated condition. To identify the influence of wind-sea and swell on the wind stress, the variations in
The friction velocity
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0081.1
Except for the maturity or magnitude of the surface waves, the misalignment between wind and wave directions also affects CD and possibly leads to large scatter, especially under swell conditions (Drennan et al. 2003; Patton et al. 2019; Hsu et al. 2019). To investigate the angle influence, we further divided the wave off-wind angle θ into two bins in the case of swell (β ≥ 1.2) and wind sea (β < 1.2). Bins 1 and 2 correspond to 0° ≤ θ < 90° and 90° ≤ θ ≤ 180°, which means that the surface wave is aligned with and opposed to the wind direction, respectively. This classification is consistent with Porchetta et al. (2019), which may have some limitations; however, it has statistical significance in analyzing the influence of directionality on CD. Figure 7 shows the probability density distribution of CD in different situations. The comparisons indicate that CD in conditions of 90° ≤ θ ≤ 180° is statistically larger than that in 0° ≤ θ < 90° for swell-dominated conditions (Fig. 7a), which is similar to the analysis by Porchetta et al. (2019). They concluded that the roughness length increases with increasing θ in this situation. This conclusion is consistent with the observed result reported by Donelan et al. (1997); a larger CD will occur in the case of a counterswell. However, following swells travel faster than the local wind and can produce a negative wave stress component and cause a reduction in wind stress (Chen et al. 2020a), resulting in a smaller CD.
Probability density distribution of drag coefficient CD, which is the natural log, (a) β ≥ 1.2, (b) β < 1.2. The black and red solid lines indicate that the wave off-wind angle θ is less than 90° (0° ≤ θ < 90°) and greater than 90° (90° ≤ θ ≤ 180°), respectively. The vertical lines represent the mean value for each situation.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0081.1
For the wind-sea dominated conditions, it is clear that the probability density distribution of CD in cases of 90° ≤ θ ≤ 180° exhibits a shift to the left relative to that in 0° ≤ θ < 90° (Fig. 7b), which differs from the swell-dominated conditions. Notably, there are only 96 sets of data in the case of 90° ≤ θ ≤ 180°, approximately 3% of the total. This indicates that in the wave age as the division standard, the situation with 90° ≤ θ ≤ 180° is extremely rare. Although this is uncommon, it may occur in the rear sector of hurricanes, where opposing swells are present (Holthuijsen et al. 2012). We calculate the average β value for the case of 90° ≤ θ ≤ 180°, β = 1.09, which belongs to a mixed wind-sea and swell condition according to the definition suggested by Högström et al. (2013). In mixed seas, the surface wave spectrum is bimodal, opposing swell, with a larger spectrum peak than the adjacent wind-sea spectrum peak, which may exist under nominal wind-sea conditions (β < 1.2). Still, dividing wind sea and swell by wave age is an artificial and empirical method, which may be somewhat inadequate. Nevertheless, surface waves aligned with wind direction will strengthen the wind sea, while surface waves opposite to wind direction will weaken the wind sea. Since wind sea supports most of the stress in the case of prevailing wind sea, the enhanced or reduced wind sea possibly determines the larger or lower drag coefficient.
(a) The variations in drag coefficient CD with wind speed U10 and (b) the percentage of scatter degree of CD estimated by the wave-state-based parameterization relative to that of the observation. In (a), the red and blue dots represent the observed values and the values estimated by parameterization, respectively. The red and blue rectangles represent the mean values of CD with a U10 bin size of 1 m s−1, corresponding to the observation and estimation, and the standard error bars are also shown. The circles in (b) correspond to the U10 bin sizes in (a). The gray curve in (a) is the wind-based function [Eq. (10)].
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0081.1
To further investigate the applicability of this wave-state-based parameterization on CD, we provide a comparison between the estimated CD by Eqs. (10) and (12) and the observed CD in a tropical cyclone reported in our recent paper (Chen et al. 2022), as shown in Fig. 9. As Chen et al. (2022) stated, CD presents an asymmetric distribution in the right-rear and right-front quadrants of a tropical cyclone (TC), again shown in Fig. 9a. It is clear that the CD calculated by the wave-state-based formulation [Eq. (12)] shows an asymmetric pattern, whereas the wind-based formulation [Eq. (10)] does not. Although an asymmetric pattern is simulated by wave-state-based parameterization to some extent, there are still some differences from the observations, such as smaller values in the right-rear quadrant and larger values in the right-front quadrant. Notably, surface waves were not observed by Chen et al. (2022), and the ERA5 reanalysis product was used, which has shown that qualitative analysis is feasible. Furthermore, other factors affecting the drag coefficient should be incorporated into the parameterization, such as the gustiness and sea spray induced by wave breaking (Babanin and Makin 2008; Holthuijsen et al. 2012). Therefore, more physical processes need to be considered, and more comprehensive experiments need to be performed to examine and update the wave-state-based parameterization [Eq. (12)]. Additionally, the observation results of Hsu et al. (2019) and Zhou et al. (2022) in the open ocean statistically indicated the asymmetric characteristics of drag coefficients in different quadrants of tropical cyclones, and they emphasized that this behavior depends on the sea state. Although our scheme suitable for coastal waters can simulate the asymmetry of drag coefficients, it cannot be directly applied to the observations with wind speeds exceeding 30 m s−1 in the open ocean. Therefore, new efforts are needed to develop a unified parameterization suitable for different water depths through the data accumulations.
The spatial distributions of the drag coefficient CD in the motion-relative quadrants of a tropical cyclone (TC). (a) Observations, (b) estimation by the wind-based formulation [Eq. (10)], and (c) estimation by the wave-state-based parameterization [Eq. (11)]. Plot (a) is shown in Fig. 9b in Chen et al. (2022), which is given here for convenient comparison. The black and gray arrows are the moving and rotating directions of the TC; RF and RR are the right-front and right-rear quadrants of the TC, respectively; and the circles represent the distance from the TC center. These representations are the same as those in Chen et al. (2022).
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0081.1
4. Conclusions
In this study, we combined five in situ experimental datasets of surface waves and air–sea momentum fluxes based on three tower-based platforms over different coastal regions to investigate the impacts of surface winds and waves on wind stress. Based on the wind-based formulation, we further proposed a new wave-state-based parameterization on CD suitable for coastal regions.
To avoid the widely spread behaviors of CD at low wind speeds, we started with friction velocity
The parameterization on CD is applicable for coastal regions, can predict the saturation of the drag coefficient at the statistical average of 20 m s−1 wind speed, and can also estimate the scatter of drag coefficients at a given wind speed. Most importantly, it can simulate the asymmetric drag coefficient in motion-relative quadrants of a TC. Although the newly proposed parameterization of CD is promising and can be conveniently applied to atmosphere model, the basis function for the parametric model is assumed based on visual inspection; a more general machine learning approach might be used to learn the basis function from the data itself. Additionally, the parameterization is a fit based only on data from three coastal sites, and it is currently unclear how well it would generalize to other locations around the world. Besides, the model exhibits a decrease in drag coefficient under high wind speeds with very few data. Therefore, more comprehensive field observations from different sites are still needed to accumulate data to validate this new scheme and investigate the impacts of other physical processes, such as gustiness, sea spray and sea current impacts on wind stress.
Acknowledgments.
This research was jointly supported by the National Natural Science Foundation of China (NSFC) under Grant 42276024, and the Basic Scientific Fund for National Public Research Institutes of China under Grant 2022Q01, and NSFC 41821004, and the Science and Technology of Laoshan Laboratory under Grant LSKJ202201602, and NSFC 42249902. The authors are grateful to the platform maintenance personnel. This work is a contribution to the UN Decade of Ocean Science for Sustainable Development (2021–2030) through both the Decade Collaborative Centre on Ocean-Climate Nexus and Coordination Amongst Decade Implementing Partners in P. R. China (DCC-OCC) and the approved Programme of the Ocean to climate Seamless Forecasting system (OSF).
Data availability statement.
Data used in this study can be accessed at https://doi.org/10.17632/ydw6y46j9g.1, and TC data can be accessed at https://doi.org/10.17632/sj776jbk7m.3.
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