Swell Modulation on Wind Stress in the Constant Flux Layer

The impact of swell on wind stress is investigated through direct three‐layer flux measurements taken by a fixed tower in the marine atmospheric boundary layer. Observations confirm that the assumption of constant momentum flux layer is valid under swell‐dominated conditions around the reference height of 10 m. The swell can modulate the total wind stress to be less than the turbulent stress derived from the first‐order closure method, and the extent of this modulation decreases with height. The critical layer that represents the top of the layer affected by stronger swells is estimated to reach 45‐m altitude, and the depth of this layer decreases as the swells weaken and the wind speed increases. Furthermore, a simple swell correction scheme for the total stress calculation is developed, showing good performance against observations.


Introduction
Over the ocean, surface waves play a crucial role in the determination of the wind stress in the marine atmospheric boundary layer (MABL), which has been widely explored through numerous observations Högström et al., 2015;Smedman et al., 2009) and numerical studies (Babanin et al., 2018;Janssen, 1989;Jiang et al., 2016). Mechanisms for wind stress modulation have two forms, wind-sea and swell, and their impacts on wind stress are fundamentally different (Chen et al., 2019;Hanley & Belcher, 2008). Wind-seas propagating slower than winds decelerate the overlying airflow, which causes an increase in the wind stress, while following-wind swells traveling faster than winds accelerate the overlying airflow, which induces a decrease in the wind stress.
Swells are the most common over the ocean, which can strongly modulate atmospheric turbulence and affect the wind stress. Högström et al. (2015) observed prominent peak at swell frequency in the turbulent velocity co-spectra at a height above 10 m in the MABL. Additionally, Chen et al. (2020) reported that this spectral peak can induce a sharp variation in the co-spectral Ogive curves at a height up to 17 m above the sea surface. Both studies provided observational evidence for the modulation of wind stress by swells. Other recent studies have shown that swells can reduce the wind stress through the form of swell-induced negative stress (Hanley & Belcher, 2008;Wu et al., 2017). When the swells are strong enough, an upward momentum flux from ocean to atmosphere is generated and a wind jet is seen in ©2020. The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

10.1029/2020GL089883
Key Points: • Multilevel momentum flux measurements confirm the assumption of constant flux layer at around 10 m above sea surface in swell conditions • Swell impact on the wind stress decreases with height, and an observation-based scheme is developed to account for the swell effect • The height of the critical layer affected by stronger swells is predicted to reach 45 m, and this height decreases with the wind speeds Supporting Information: • Supporting Information S1 Correspondence to: F. Qiao, qiaofl@fio.org.cn the bottom of MABL, as documented by field observations and numerical simulations (Nilsson et al., 2012;Smedman et al., 2009). Chen et al. (2019) showed that swells can influence the wind stress up to 26-m height above the sea surface through field observations. Moreover, it is predicted by a one-dimensional boundary layer model that swells can affect the MABL structure (Hanley & Belcher, 2008;Song et al., 2015). However, the actual height of the critical layer with the swell impact is difficult to be determined. Previous MABL models were generally developed based on the assumption of constant flux layer to simulate vertical wind profiles and wave-driven wind speed under swell conditions (Semedo et al., 2009;Zou et al., 2018).
The assumption of constant flux layer is that the total wind stress τ does not vary with height z in the near-surface MABL: Note that the viscous component τ visc can be neglected because it is nonzero only close to the surface; u * is the friction velocity. τ turb is the turbulent stress, which can be parameterized according to the first-order closure method: here, K m is the turbulent eddy viscosity and can be approximated through K m = κzu * ; κ=0.4 is the von Kármán constant. The τ wave is the wave-induced stress, which is assumed to decay exponentially with height (Wu et al., 2017): here τ wave0 is the wave-induced stress at the sea surface, and τ wave0 = τ − τ turb0 , where τ turb0 is the surface turbulent stress; G is the decay coefficient; k is the peak wavenumber of surface wave, which can be calculated by the dispersion relation (2π/T) 2 = gk tanh (kh); here, T is the peak period, h is the water depth, and g is the acceleration of gravity. Combining Equations 1-3, the wind speed as a function of z is derived: here, z 0 ¼ 0:011 u 2 * g þ 0:11 υ u * is the aerodynamic roughness length. Note that Chen et al. (2020) showed the direct evidence that the observed stress contains the swell-induced component τ wave , although Voermans et al. (2019) revealed that their observed stress by eddy covariance system (ECS) is the turbulent component. Danard (1981) estimated that the height of the constant flux layer is approximately 10 m over the water surface. Notably, 10 m is the typical surface layer height that is used in the parameterization of wind stress or drag coefficient in the ocean, climate, and wave models (Babanin & Mcconochie, 2013), as well as in the widely used COARE algorithm for flux calculations. However, Smedman et al. (2009) demonstrated that the stress magnitude decreases significantly with height near the sea surface due to the swell impact based on multilevel tower observations. Mahrt et al. (2018) also observed a substantial decrease of the stress magnitude with height under swell conditions. They pointed out that the nonstationarity of the wind or the shallow MABL may be responsible for the large stress divergence.
The purpose of this study is to investigate whether the constant momentum flux assumption is always invalid in the MABL when swell is dominant and, based on this, to study the swell impact on wind stress. We also aim to quantify the swell effect on the total wind stress and how this effect varies with height and wind speed using observations. This study is organized as follows. The observational data and analysis method are described in section 2. Results and discussions are presented in section 3. A simple correction of swell effect for the estimated wind stress is proposed and presented in this section. Finally, conclusions are summarized in section 4.

Data Collection and Analysis Method
The data used in this study was collected by a fixed observation system, BoHe observation tower (BHOT), over the coastal regions in the northern South China Sea (SCS) as shown in Figure 1a. The mean water depth is approximately 16 m at this tower (21°26.5′N, 111°23.5′E). This BHOT is approximately 6.5 km from the coastline and consists of a main tower and two auxiliary towers. Our ECSs were mounted at three levels on the northeast auxiliary tower from 17 March to 14 May 2018. This auxiliary tower with a diameter of 0.4 m is approximately 6 m away from the main tower. The ECSs were mounted 1.5 m away from the side of the auxiliary tower. This distance is enough to minimize the flow distortion caused by the tower (Chen et al., 2018;Gill et al., 1967). The three levels of ECSs were 8.4, 12.4, and 15.3 m, respectively, above the sea surface ( Figure 1b).
Each ECS consists of a three-dimensional ultrasonic anemometer (UA) with orientation of~61°north by east, an infrared gas analyzer, and high-frequency temperature and humidity sensors, which can continuously collect atmospheric turbulence data with a sampling frequency of 10 Hz. These data can be used to calculate air-sea fluxes. Additionally, a propeller type conventional anemometer (CA) was installed at a distance of 1 m on the right side of the upper ECS to observe the mean wind speed and direction, which can be used to validate the mean wind speed and direction from the UA. During the observational periods, a bottom-supported wave buoy was deployed to simultaneously measure surface waves near the tower.
We focus on the air-sea momentum flux measured by the UA in this study. Based on the eddy covariance method, the wind stress can be calculated directly through: here, U′ and w′are the horizontal and vertical velocity fluctuations (Babanin & Makin, 2008) . The Reynolds average time of 1 h was chosen. Note that the nonstationary motions were removed from 1-h turbulence data before the calculation of the covariance following the method as in Chen et al. (2018), to guarantee the stationarity and avoid the possible data contamination.
Additionally, we compared the wind directions measured by the upper UA and CA for further quality control. The wind direction and speed measured by the UA compared to those measured by the CA generally fall in the 1:1 line (Figures 1c and 1d). The direction and speed measured by UA within ±20°of the UA orientation match the best, shown by the red dots in Figures 1c and 1d. Therefore, for flux calculations, the data with wind direction between 41°and 81°were selected.
After data quality control, 463 flux runs are obtained, and the wind speeds vary from low to moderate, mostly below 9 m/s ( Figure S1a in the supporting information), which further minimize the potential flux contamination caused by the tower motion. During the period of interest, swells always occur and travel in the southeast direction with an average of approximately 120°because of the northwest coastline (Chen et al., 2018(Chen et al., , 2020, see their Figure 2). Besides, the wave age β = C p /Ucosθ is mostly greater than 1.2 ( Figure S1b), where C p is the peak wave phase speed and θ is the wave off-wind angle. The swell significant wave height H swell s is generally higher than that of wind-sea H windsea s ( Figure S1c). Both results suggest that the swells dominate the wave field (Högström et al., 2015). The simultaneous surface wave and flux data are analyzed to investigate the impact of swell on the wind stress.

Constant Flux Layer
Because the wind stress at low wind speeds is very small, we present the friction velocity u * rather than the stress. The friction velocities at three levels are compared in Figure 2, showing a significant agreement among all levels. The bias between u * middle and u * lower and that between u * upper and u * lower are close to 0. This result suggests that the assumption of constant flux layer at around 10 m is validated in our study. The vertical gradient of the wind stress, ∂τ/∂z, is on the order of 10 −4 m/s 2 , and the averaged stress divergence approaches 0, which further confirm this constant flux layer. Of note, Babanin and Makin (2008) argued that the constant flux concept is impossibly satisfied at low wind speeds (U 10 < 4 m/s) and the constant flux layer height is less than 10 m in this condition following Komen (1994). Their conclusion was inferred from a simple comparison between the friction velocity calculated through the logarithmic profile method and that directly obtained from the limited single level ultrasonic measurement. However, our three-layer momentum flux measurements avoid this limitation and show that the wind stress around 10 m satisfies the condition of constant flux, even at low wind speeds (Figure 2). Again, this height (10 m) is similar to the theoretical surface layer height proposed by Danard (1981).
Our result indicates that the total wind stress does not vary with height even in the presence of swells, which differs significantly from that of Mahrt et al. (2018) and Smedman et al. (2009) who showed large stress divergence due to swells. As the stress divergence may be caused by factors such as the nonstationarity of the wind and the thin boundary layer (Mahrt et al., 2018), the marine condition in our study is likely very different from that in these two studies.

Swell Impact on Wind Stress
According to classic turbulence theory (Tennekes & Lumley, 1972), the mean velocity profile follows the log-law when the measurement is taken sufficiently away from wall or boundary where the wind is independent of the types of the stresses. Similarly, when the wave-induced stress is negligible (i.e., τ wave = 0) in the MABL, τ = τ turb , and consequently, u * = u *turb , here, u * turb is the turbulent friction velocity, Equation 4 is reduced to For comparison purposes, u * in the absence of wave-induced stress is expressed as u * turb hereafter. Note that the wind stress calculated based on the COARE algorithm is consistent with that estimated using Equation 6 because the wave effect is not considered . We can estimate the turbulent stress using the COARE algorithm and observed mean wind speeds, although the wave-induced stress is not negligible at the instrument measurement height in our study. The difference between the estimated turbulent stress and directly measured total stress is thus equivalent to the wave-induced stress.
As mentioned earlier, quantifying the swell impact on the wind stress is the focus of this study. Data with wind speeds at lower layer greater than 9 m/s are excluded due to quality control and a total of 448 sets of flux runs are obtained. For the total wind stress, we use the observations at the height of 8.4 m based on the premise condition of constant flux layer. Comparison between the turbulent stress estimated based on the COARE algorithm and the observed stress are shown in Figure 3, where u * obs is the observed friction velocity. It is evident from Figure 3a that u * turb deviates significantly from u * obs at the lower level, with a positive bias of 0.031 m/s. The extent of this deviation decreases with height, but the estimated u * turb is still greater than the actual observation at middle and upper levels (Figures 3b and 3c). It is the swells that cause this deviation of the estimated stress from the observed one. The swells can induce a negative wave-induced stress, τ wave < 0 (Hanley & Belcher, 2008;Semedo et al., 2009), which results in the total wind stress being less than the turbulent stress τ < τ turb . Besides, the magnitude of swell-induced stress decays exponentially with height (Hristov et al., 2003;Jiang et al., 2016), which causes the turbulent component to approach the total stress as the height increases.
The wave-induced stress can be calculated through the wave spectrum, which has been widely used to correct the turbulent stress to be consistent with the actual total stress (Babanin et al., 2018;Chen et al., 2019;Zou et al., 2018). Here, we apply the same method into our observations and find that this correction is feasible ( Figure S2). We develop a simple parameterization to correct the calculated turbulent stress τ turb with only one wave parameter. According to Equation 3, the magnitude of τ wave can be expressed in terms of the calculated turbulent stress τ turb in the absence of wave effect: where α is a correction factor. Coefficients G and A can be determined using observations. Note that Equation 7 in our study is similar to Equation 20 in Voermans et al. (2019), although they parameterized the wave-induced stress using the significant wave height H s and the observed turbulent stress. Considering the swell-induced stress is negative, the total stress is formulated as The coefficients, G and A, can be determined using the observed wind stress at lower and upper levels in our study, and the best fit values are G = 3 and A = 0.2. This value of coefficient A is equal to that of  Figure 3d shows the comparison of the corrected friction velocities by our parameterization and the observed ones, showing a good agreement. This result supports that the wind stress corrected by our parameterization (i.e., Equations 7 and 8) with the observation-derived coefficients can be used to estimate the total wind stress at a typical height of 10 m. Furthermore, this parameterization can be used to modify the COARE algorithm and correct the wind stress under following-wind swell conditions.
It should be noted that the swell with off-wind angle less than 90°is the most prevalent in our observations; therefore, α is negative. However, the counter-wind swells can cause increasing wind stress (Donelan et al., 1997), so that the observed stress is greater than the estimated turbulent stress by the COARE algorithm and α should be positive in such conditions. When applying our method for correcting the wave-induced stress in the COARE algorithm, the wind and wave states need to be determined first. Thus, more experiments under different sea states, including counter-wind swell condition, are needed to confirm the form of wave-induced stress for a universal application.

Critical Layer Height
The extent of swell influence on wind stress decreases with height, which implies that the swell-induced stress will approximately approach zero (τ wave →0) at sufficient level where the total wind stress equals the turbulent stress: τ = τ turb . The maximum height or the critical layer height below which swells affect the total wind stress can be estimated based on three-layer stress observations. After the turbulent stress is calculated through Equations 2 and 6, the total wind stress at each level can be determined using the corresponding best fit equation shown in Figure 3c. The three-layer friction velocity profiles are displayed in Figure 4. Based on the nearly linear relationship between the friction velocity u *turb and the lower-level wind speed U, the corresponding wind speed U for each profile is determined. The wave age β is also shown in Figure 4. The critical layer height for swells has a minimum value of 25 m based on our observations (Figure 4), and 2% of τ wave0 is the cutoff value in Equation 3 at this minimum height.
The critical layer height for swells can reach 45 m or more and decreases with the wind speed. This height scale increases with the wave age (Figure 4), suggesting that the perturbations induced by stronger swells are more likely to be caught at a certain height above the sea surface. Chen et al. (2019) observed swell-induced disturbances in the turbulent velocity spectra at a height of 26 m in the MABL for wind speeds <5 m/s. Furthermore, Smedman et al. (2009) reported large influence of swells on the wind velocity at 30 m above the sea surface under both cross-wind swell (|β | >2.8) and following-wind swell (|β | >1.8) conditions. They also showed more pronounced impacts of stronger swells on the wind field than weaker swells. Both are quite similar to our estimation of critical layer height. Additionally, it should be noted that the critical heights derived from the wind stress profiles are similar to those shown in Figure 4.

Conclusions
Multilevel observations of momentum fluxes above the sea surface show that the wind stress around the typical surface layer height of 10 m satisfies the assumption of constant flux layer at low to moderate wind speeds. Swells have a significant effect on the wind stress, which act to reduce the total wind stress to be less than the turbulent stress derived from the first-order closure method or the COARE algorithm without considering the wave effect. We found that the swell-induced stress can be used to correct the bias between the turbulent and total stress using a simple correction parameterization based on observations. This parameterization only requires the calculated turbulent stress using the COARE algorithm and the peak wavenumber of surface wave to determine the total wind stress under swell conditions. Furthermore, the swell influence on the wind stress is found to decrease with height, which can be used to estimate the critical layer height affected by swells. We found that the maximum height that swells can affect is expected to reach 45 m, and this critical height decreases with the decreasing wave age and the increasing wind speed. Our results provide the first evidence for the concept of a near constant flux layer with a height of~10 m, and provide a promising method to determine the critical layer affected by swells. This method along with the wave-induced stress parameterization require further verification through the multilayer flux measurements under different wind conditions and sea state in future studies.

Data Availability Statement
Data supporting this study are available by any users from http://data.fio.org.cn/qiaofl/CS-GRL-2020 website.