Abstract

Surface wind stress is a crucial driver of upper-ocean processes, impacting air–sea gas flux, wind-wave development, and material transport. Conventional eddy covariance (EC) processing requires imposing a fixed averaging window on the wind velocity time series in order to estimate the downward flux of momentum. While this method has become the standard means of directly measuring the wind stress, the use of a fixed averaging interval inherently constrains one’s ability to resolve transient signals that may have net effects on the air–sea interactions. Here we utilize the wavelet transform to develop a new technique for directly quantifying the wind stress magnitude from the wavelet coscalogram products. The time averages of these products evaluated at the scale of maximum amplitude are highly correlated with the EC estimates (R2 = 0.99; 5-min time windows), suggesting that stress is particularly sensitive to the dominant turbulent eddies. By taking advantage of the new method’s high temporal resolution, transient wind forcing and its dominant scales may be explicitly computed and analyzed. This technique will allow for more general investigations into air–sea dynamics under nonstationary or spatially inhomogeneous conditions, such as within the nearshore region.

1. Introduction

As wind blows over the sea surface, momentum is exchanged between the two fluids. This exchange, the wind stress τ, is defined as the rate of change of horizontal air-side momentum per unit surface area of water. The wind stress is a crucial driver of air–sea interaction dynamics, affecting processes such as material transport (Klein and Coste 1984) and air–sea gas exchange (Jähne et al. 1979). In the open ocean, wind speed is the primary determinant of wind stress (Edson et al. 2013), though the aerodynamic smoothness produced by swell waves has been indicated to reduce air–sea drag (Potter 2015). Along coastal margins, stress may change dramatically over short distances depending on land effects (Sun et al. 2001), shoaling wave fields (Shabani et al. 2014; MacMahan 2017), and current shear (Ortiz-Suslow et al. 2015). General studies on the topic have shown that the relationship between wind speed and momentum flux in nearshore and surfzone regions is modulated by wave and bathymetric effects (Gao et al. 2009; Yang et al. 2018). It follows that a proper approach to describing the wind stress in coastal areas should be able to account for the great spatial variability of these effects.

Distilling a rich and multidimensional dataset into a single vector or set of zero-dimensional scalars is often done to conveniently summarize complexities in the record. Given little more than an anemometer and a temperature probe, the wind speed time series may be processed in this way via the application of a bulk formula (Fairall et al. 2003) or the inertial dissipation technique (e.g., Edson et al. 1991). If one has access to a sonic anemometer (capable of recovering the 3D wind vector at high frequency) and an inertial measurement unit (IMU; to correct the velocity record for platform motion), one may obtain the wind stress τ using the eddy covariance (EC) technique (see Edson et al. 1998). Each of these approaches makes use of fluid mechanical theory in order to boil down our record of the atmospheric boundary layer flow into familiar, easily digestible quantities, such as the friction velocity u*, the aerodynamic drag coefficient given neutral conditions CDN, and the mean wind speed at 10 m above the surface given neutral conditions U10N (Bourras et al. 2014, e.g.). These are related in the following way: , and , where ρ is the air density. A corollary can be drawn to ocean surface gravity waves, where a multidimensional wave field is often summarized into a single significant wave height , peak wave period , and peak wave direction . These statistical quantities are convenient and useful, but they will always be limited in their ability to describe transient behavior. Nonstationary forcing has been shown to be quite prevalent and to considerably modify geophysical dynamics even far from coastal margins (Drennan et al. 1999a). However, the full effect of such forcing is poorly understood, motivating the present study.

The early 1980s saw the development of the wavelet transform, a nonstationary signal processing technique (Morlet et al. 1982; Grossmann and Morlet 1984). An expansion of its modes of application in the areas of image and signal processing during the 1990s brought some promising results related to the field of geophysical fluid dynamical analysis (Liu et al. 1995a,b; Peng et al. 1995; Donelan et al. 1996). However, these earlier works were limited in that their investigations were largely qualitative and wavelet processing was not used to provide a quantifiable metric comparable to the more widely used Fourier transform. More recently, advancements in spatial sensing capabilities and improvements in geophysical fluid model resolution have highlighted some of the limitations of pure Fourier and statistical analysis as applied to one-dimensional quantities. The present work is rooted in a desire for renewed focus on adaptive processing techniques of highly variable signals, specifically targeting the calculation of wind stress in strongly varying regimes. To improve the spatiotemporal resolution of air–sea momentum flux and to expand our understanding of inhomogeneous wind forcing, a series of wind velocity observations were made in a laboratory wind-wave flume and along the northern coastline of Monterey Bay in California from a small, heavily instrumented research vessel. Two wavelet-based techniques are presented here, one being a simple extension of past (e.g., Peng et al. 1995) work and the other being entirely new. The new technique is validated with the laboratory measurements, while the field data serve as a case study and demonstration of the method’s applicability.

The methods employed here are given in section 2. Results are presented in section 3 and are discussed as part of their broader impact in section 4.

2. Methods

In this section, methods for estimating wind stress magnitude from the wavelet transforms of the wind velocity vector are described. The techniques were tested on sets of laboratory and field measurements in order to provide both validation and a larger scientific context, respectively. The MATLAB codes used for the data processing described in this section are included as supplemental material.

a. Description of observations

Laboratory observations were made in the Surge Structure Atmosphere Interaction (SUSTAIN) facility at the University of Miami’s Rosenstiel School of Marine and Atmospheric Science. The SUSTAIN facility contains a wind-wave tank with interior dimensions of 23 m × 6 m × 2 m. A Campbell Scientific IRGASON sonic anemometer was mounted 16 m downstream of the tank inlet and fixed 46 cm above the mean water level (which was set to 77 cm above the tank bottom). The IRGASON is a sophisticated hybrid sonic anemometer/infrared gas analyzer with the ability to measure the three-dimensional wind velocity vector at 20 Hz with errors of 0.08 m s−1 in the horizontal and 0.04 m s−1 in the vertical. A representation of the setup is given in Fig. 1. The fan was controlled to produce 49 different wind speed conditions with 10-m neutral wind speeds ranging from 5 to 25 m s−1; 26 of the cases lay in the wind speed range 5–10 m s−1 recorded during the field observations. Each condition was sampled at 20 Hz for 6 min, allowing for the evaluation of a clean, stationary 5-min segment of data for analysis. These short time windows are reasonable due to 1) the relatively short time scales of the dominant motions inside the wind-wave tank and 2) the flume’s stable and controlled conditions.

Fig. 1.

Mock-up of wind-wave tank within the SUSTAIN laboratory: (a) side view, (b) downwind view, and (c) IRGASON anemometer. The water level was set to 77 cm above the tank bottom, and the anemometer was positioned such that the sampling volume was halfway between the mean water level and the ceiling of the tank, at 1.23 m above the tank bottom.

Fig. 1.

Mock-up of wind-wave tank within the SUSTAIN laboratory: (a) side view, (b) downwind view, and (c) IRGASON anemometer. The water level was set to 77 cm above the tank bottom, and the anemometer was positioned such that the sampling volume was halfway between the mean water level and the ceiling of the tank, at 1.23 m above the tank bottom.

Field observations were made in northern Monterey Bay as part of the Coastal Land–Air–Sea Interaction (CLASI) campaign, which took place during June 2016. Monterey Bay (Fig. 2a) is a large west-facing bay that is well exposed to North Pacific wind and swell systems; furthermore, the bay’s coastline provides a variety of environments in which to observe and model nearshore air–sea interaction [see, e.g., the spatial varability in sea surface roughness as inferred from satellite-based synthetic aperture radar (SAR); Figs. 2c and 2d]. The present work focuses on the observations made from 1530 to 2300 UTC 8 June 2016 in the waters south of Santa Cruz, California. The area of study featured substantial inhomogeneity in the wind and wave fields over waters that were still safely navigable via small boat. On this day, a northwesterly low-level jet flowed into the bay and wrapped around the southern face of the Santa Cruz Mountains. This set up a strong west–east gradient in the wind field and local wind sea that strengthened throughout the day as the jet fully developed and pushed eastward into the bay. The measurements shown in this work are representative of a series of six ≈30-min transects over which wind and wave conditions varied to create a fairly wide parameter space over the course of this single day. It should be noted that given the single moving platform of the small boat, it is impossible to fully decouple the spatial inhomogeneity from the temporal nonstationarity. The case study is therefore presented as is, the momentum flux variance along the vessel track existing all the same.

Fig. 2.

(a) Map of Monterey Bay, with relevant locations labeled, (b) view of the RHIB, (c) a Constellation of Small Satellites for the Mediterranean Basin Observation (COSMO-SkyMed) SAR intensity image of northern Monterey Bay, and (d) a series of image intensity along an example vessel transect (the magenta line, with green and red squares marking the start and stop positions, respectively). In (b), the red star indicates the location of the IRGASON, which sits atop a 2.75-m mast at 5.4 m above the mean water level, while the red circle indicates the location of the Senix water surface elevation sensor at 2.65 m above the mean water level. Photo in (b) courtesy of Ryan Yamaguchi.

Fig. 2.

(a) Map of Monterey Bay, with relevant locations labeled, (b) view of the RHIB, (c) a Constellation of Small Satellites for the Mediterranean Basin Observation (COSMO-SkyMed) SAR intensity image of northern Monterey Bay, and (d) a series of image intensity along an example vessel transect (the magenta line, with green and red squares marking the start and stop positions, respectively). In (b), the red star indicates the location of the IRGASON, which sits atop a 2.75-m mast at 5.4 m above the mean water level, while the red circle indicates the location of the Senix water surface elevation sensor at 2.65 m above the mean water level. Photo in (b) courtesy of Ryan Yamaguchi.

Data were acquired from a 7.9-m-long reinforced hull inflatable boat (RHIB) shown in Fig. 2b. The sampling, data acquisition, and processing techniques used for the EC portion of this study were based on coastal observations reported in Ortiz-Suslow et al. (2015). The three-dimensional wind velocity vector was collected at 20 Hz using the same Campbell Scientific IRGASON mounted 5.4 m above the mean water level (the red star in Fig. 2b). Examination of recent studies of the wind stress profile in similar environmental conditions to those described here (Högström et al. 2013; Wu et al. 2018) suggest that wave-coherent motions should not dominate the signal at this measurement height. Based on the flux footprint calculations of Högström et al. (2008), it is estimated that the upstream distance containing 80% of the measured flux was on the order of 100 m, allowing for a wind stress estimate that may be considered to be fairly local. The low profile of the RHIB, the minimal superstructure, and the height of the IRGASON provided a measurement relatively free from potential flow distortion effects when the vessel was oriented and moving into the flow (following from the flow distortion computations of Yelland et al. 2002). Water surface elevation measurements were made underway via a triplet of Senix ToughSonic REMOTE 30 altimeters (the red circle in Fig. 2b), reporting distance to the free surface at 20 Hz. The beamwidth (−3 dB) of each sensor is 7.5° off axis, resulting in an approximate footprint diameter of 70 cm at mean water level. The boat’s linear accelerations and angular velocities were sampled at 20 Hz using a hybrid six-degrees-of-freedom inertial measurement unit composed of a Columbia Research model SA-307HPTX three-axis accelerometer and three Systron Donner QRS11-00050-630 rate gyros. The velocity vector time series and wave gauge data were corrected for platform motion using these data streams and the algorithm used by Anctil et al. (1994). Manufacturer-reported accuracy for the sensors used here is provided in Table 1.

Table 1.

Manufacturer-reported RMS sampling errors for the instruments used in this study.

Manufacturer-reported RMS sampling errors for the instruments used in this study.
Manufacturer-reported RMS sampling errors for the instruments used in this study.

Water surface elevation data were subjected to Fourier transforms and were Doppler corrected following the methods of Collins et al. (2017). As this method requires Earth-referenced wave direction as an input parameter, a technique that estimates the intrinsic wave propagation direction [the wavelet directional method of Donelan et al. (1996)] was first employed to provide such a quantity.

b. The EC technique

The EC technique (e.g., Edson et al. 1998) has long been the standard for computing the stress applied by the atmosphere to the ocean. This method follows from the Reynolds-averaged Navier–Stokes (RANS) equation, a form of the Navier–Stokes equation in which stationary velocity components have been decomposed into mean and turbulent portions. Assuming that advection, viscous stresses, and body forces may be neglected, the wind stress vector is computed from the covariances of the turbulent along- and crosswind velocities ( and , respectively) with the vertical wind component , hence the term eddy covariance (Tennekes and Lumley 1972):

 
formula

In Eq. (1), is the density of air, and a prime indicates that the quantity is the turbulent portion of the Reynolds decomposition (i.e., = = = 0). The angle brackets indicate that a time average of the contained quantity has been computed over some appropriate interval. Over the open ocean, intervals typically range from 20 min to 1 h (Large and Pond 1981; Edson et al. 2013). In coastal areas of rapidly changing conditions, windows are typically shorter [e.g., 15 min for the tower-based measurements of Shabani et al. (2014)]. When making these types of measurements from aboard a small vessel, which may move to sample different regions of a spatially inhomogenous area, EC windows have been as short as 5 min (Ortiz-Suslow et al. 2015).

c. The wavelet transform

Wavelet analysis has frequently been employed to study transient characteristics of geophysical fluids. In contrast to its well-known cousin the Fourier transform, the wavelet transform is able to expand the dimensionality of the input signal, recovering its amplitude, frequency, and phase as it varies in time at the cost of some resolution in frequency (Morlet et al. 1982; Grossmann and Morlet 1984). Wavelet transforms perform far better than successive short time Fourier transforms at the task of describing the dynamics of signals that change rapidly and nonperiodically (Liu and Miller 1996). Often (Hayashi 1994, e.g.), applications of wavelet analysis to wind velocity time series have been limited to qualitative studies of the transformed signal’s characteristics or their time-averaged, scale-integrated properties. The latter application has generally followed the form of integrating Fourier cospectra over all frequencies, allowing one to obtain the covariance (and therefore the friction velocity; e.g., Large and Pond 1981). The wavelet-based techniques described here begin with the application of wavelet transforms to the individual components of the wind velocity time series. For the present work, this will be accomplished in Fourier space. The discrete Fourier transform (DFT) of a time series u is defined thusly:

 
formula

Conversely, the inverse transform shall be represented as . The Morlet wavelet (Morlet et al. 1982; Grossmann and Morlet 1984) is commonly used for nonstationary signal analysis. Functionally, it has the form of a sinusoid within a Gaussian envelope. It has been shown to be a strong choice for the decomposition of geophysical fluid mechanics time series (Donelan et al. 1996) and is therefore used here. This wavelet [Eq. (3)] and its Fourier transform [Eq. (4)] are given below (following Ashmead 2012):

 
formula
 
formula

where , a and b are constants of normalization and dimensionality, respectively; and c is the frequency of the mother wavelet, here taken as 5 Hz in order to capture as many scales of motion as possible. For all wavelet transforms described in this work, = 0.01 Hz and = 5 Hz, yielding octave minima and maxima of −7 and 3, respectively. The transforms were computed with 16 voices (wavelet parlance for further frequency subdivisions) per octave, resulting in 176 total frequency bins. Combining Eqs. (2)(4), we compute the time–frequency evolution of an input signal u as follows:

 
formula

where , the output of the wavelet transform, is often described using the term scalogram. Given a real-valued, one-dimensional input time series, a scalogram is a complex-valued, two-dimensional array that is defined over both the time and frequency (i.e., scale, hence the name) of variation of the input signal. It has the units of the input time series, which in this case is meters per second. For an example, fluctuating velocity components are shown in Figs. 3a–c. The real portions of the corresponding wavelet scalograms are given alongside them in Figs. 3d–f.

Fig. 3.

(left) Five-minute snippets of wind velocity vector fluctuation time series field data, and (right) the real portion of the corresponding transformed signals (colors; m s−1). Here, (a) , (b) , and (c) correspond to along-stream, cross-stream, and vertical fluctuating components, respectively, of the velocity vector in a right-hand convention.

Fig. 3.

(left) Five-minute snippets of wind velocity vector fluctuation time series field data, and (right) the real portion of the corresponding transformed signals (colors; m s−1). Here, (a) , (b) , and (c) correspond to along-stream, cross-stream, and vertical fluctuating components, respectively, of the velocity vector in a right-hand convention.

The coscalograms between the along-stream (or across stream) and vertical wind velocity components are defined as the real part of the product [the asterisk (*) indicates the complex conjugate] of their individual scalograms (transformed signals):

 
formula
 
formula

For the special case of wind velocity wavelet scalograms, the sign of these coscalograms indicates the direction of vertical momentum flux (<0, downward; >0, upward): for flux in the along-stream direction and for flux in the cross-stream direction. In Peng et al. (1995), these coscalograms were integrated with respect to time and scale (frequency) in order to estimate the total momentum flux:

 
formula

Following Eq. (8)—but integrating only over frequency, an extension of Eqs. (6) and (7) from Peng et al. (1995)—we produce the time series of wind stress as estimated as an “integrated product” of the wavelet coscalograms:

 
formula

The Peng et al. (1995) analysis was focused mainly on relating the peak spectral component of momentum flux to the phase of the swell, exploring the role of swell in mediating said flux. Similarly, the authors of Liu et al. (1995a) sought to investigate the effects of wave groups on air–sea momentum flux. Many of the graphics in that work will appear to be quite similar to Figs. 3 and 4. However, that paper stopped short of validating a new approach for computing air–sea momentum flux under nonstationary conditions. For the present work, we take that step and offer a new technique for estimating wind stress from the wavelet coscalograms.

Fig. 4.

Illustration of the (top) along-vertical and (bottom) cross-vertical wavelet coscalograms (using the transformed signals shown in Fig. 3). Color bars indicate coscalogram values (m2 s−2).

Fig. 4.

Illustration of the (top) along-vertical and (bottom) cross-vertical wavelet coscalograms (using the transformed signals shown in Fig. 3). Color bars indicate coscalogram values (m2 s−2).

d. The peak wavelet amplitude technique

This section deals with the peak wavelet amplitude (PWA) technique. With the PWA, we analogize and (Fig. 4) to the covariances and , respectively. By following the style of Eq. (1), one obtains an output array from the wavelet coscalograms with dimensions of newton per square meter, shown in Fig. 5a. By inspecting this wavelet coscalogram product, one can see the maximum magnitude at each time step is quite close to the EC stress magnitude (0.0516 N m−2 here). For a deeper investigation of this feature, frequency–amplitude slices are plucked out of the array (Fig. 5a). The scale of maximum amplitude is identified, and that amplitude is extracted (Fig. 5b). This process is repeated for each time step of the coscalogram product [Fig. 5c; Eq. (10)],

 
formula

It is shown alongside moving window 1-min and static window 5-min EC computations. Note that the moving 1-min EC computation actually shifts the apparent location of the transient forcing near the 125-s mark in addition to smoothing over much of the temporal variability.

Fig. 5.

Progression from the wavelet coscalogram product array to the time series of PWA stress magnitude (field data; same snippet as in Figs. 3 and 4). (a) The removal of a frequency–amplitude “slice” from the product array. (b) The identification and extraction of the wavelet coscalogram product amplitude at the dominant frequency. (c) The process shown in (a) and (b) is repeated along the temporal dimension of the array. The EC1 and EC5 markers denote the 1-min EC (moving window, 10-s translation per time step) and the 5-min EC for this time series.

Fig. 5.

Progression from the wavelet coscalogram product array to the time series of PWA stress magnitude (field data; same snippet as in Figs. 3 and 4). (a) The removal of a frequency–amplitude “slice” from the product array. (b) The identification and extraction of the wavelet coscalogram product amplitude at the dominant frequency. (c) The process shown in (a) and (b) is repeated along the temporal dimension of the array. The EC1 and EC5 markers denote the 1-min EC (moving window, 10-s translation per time step) and the 5-min EC for this time series.

3. Results

Generally, the time-averaged PWA products were found to strongly correlate with the EC stress estimates computed from the laboratory data. The integrated product was shown to qualitatively capture the same flume-generated gust event as the PWA, though at a lower magnitude. The field observations provide less clarity, however, as neither PWA nor the integrated product appear to better correlate with EC than the other. Adding to this confusion is the inescapable fact that EC may not even be appropriate for the types of short-window, fast-vessel, and nearshore operations that motivated the development of the wavelet techniques in the first place, thus serving as a poor “ground truth.” In any case (based on the laboratory measurements), the wavelet techniques appear to have skill at describing air–sea momentum flux and are ultimately far more revealing of local dynamics in spatially inhomogeneous regions than the traditional approaches have been.

A single comparison between wind stress magnitude computed via EC, PWA, and the signed components of the integrated products forming the terms of Eq. (9) is shown in Fig. 6. The time-averaged values shown in Fig. 6b were computed for all laboratory cases and are shown in Fig. 7a, with Fig. 7b showing a different time series example for reference. Note that all three approaches are qualitatively similar for low wind speed magnitude, though the latter diverges from EC and PWA for increasing wind speed. Figure 8 plots stress computed from EC against stress computed from PWA. The exceptional agreement between EC and PWA (coefficient of determination = 0.99; root-mean-square difference = 0.01 N m−2) computed from the laboratory data indicates that the latter is functionally equivalent to EC during conditions of stationary wind forcing. A single gust was generated in the laboratory as a case study transient signal. For this case, the fan frequency was kept constant to produce a flume wind speed of 4 m s−1. A brief (30 s total) gust was created by linearly ramping the flume speed up to 8 m s−1 and then back down to 4 m s−1, where the condition stayed for the remainder of the observational period (Fig. 9a). This gust was easily detected in the wavelet coscalogram product (Fig. 9b) and was highly localized in the PWA and integrated product stress estimates (Fig. 9c), though the latter tended to underestimate the forcing as it did in the stationary cases. EC wind stress magnitudes varied greatly as a function of window size; furthermore, the position of the EC averaging window itself had quite the effect on the magnitude. The 1-min estimate was centered on the gust, while the two adjacent 2-min estimates straddled it. This resulted in the gust’s energy being focused on the 1-min EC window but split between the two 2-min windows. Of course, the very nature of the gust violates the stationarity assumption invoked for EC. Such behavior gives fixed-window processing techniques a distinct disadvantage as compared to these wavelet techniques.

Fig. 6.

From the same example wind velocity time series: (a) Integrated wavelet product, along-stream and cross-stream components—the integrals of Eq. (9). Dashed lines indicate EC values, with the same color convention. (b) Comparison of PWA, integrated product, and EC stress magnitudes as computed in Eqs. (10), (9), and (1), respectively.

Fig. 6.

From the same example wind velocity time series: (a) Integrated wavelet product, along-stream and cross-stream components—the integrals of Eq. (9). Dashed lines indicate EC values, with the same color convention. (b) Comparison of PWA, integrated product, and EC stress magnitudes as computed in Eqs. (10), (9), and (1), respectively.

Fig. 7.

Comparison of wind stress computed from the laboratory dataset using three different methods: eddy covariance, PWA, and integrating the wavelet coscalograms over all frequencies. (a) Comparison over all 49 cases; (b) comparison over the lowest wind speed case, stretched into a time series.

Fig. 7.

Comparison of wind stress computed from the laboratory dataset using three different methods: eddy covariance, PWA, and integrating the wavelet coscalograms over all frequencies. (a) Comparison over all 49 cases; (b) comparison over the lowest wind speed case, stretched into a time series.

Fig. 8.

Comparison of wind stress magnitude computed via EC and wind stress magnitude computed via PWA, laboratory observations. The red line indicates a 1:1 relationship. The coefficient of determination resulting from treating as a linear fit to is 0.99, significant at the 0.05 level (p = 0.039). The color bar indicates (m s−1).

Fig. 8.

Comparison of wind stress magnitude computed via EC and wind stress magnitude computed via PWA, laboratory observations. The red line indicates a 1:1 relationship. The coefficient of determination resulting from treating as a linear fit to is 0.99, significant at the 0.05 level (p = 0.039). The color bar indicates (m s−1).

Fig. 9.

Representations of a gust created inside the wind-wave tank. (a) Time series of the along-stream velocity component, with U1 and U2 indicating the mean and gust speeds, respectively. (b) Wavelet coscalogram product array for the whole velocity vector time series (color bar indicates N m−2). (c) Stress magnitude as computed by EC for 1-, 2-, and 5-min windows and by the wavelet techniques over the whole time series. In (c), U1 and U2 correspond to the stress magnitudes computed for steady values of the mean and gust speeds, respectively.

Fig. 9.

Representations of a gust created inside the wind-wave tank. (a) Time series of the along-stream velocity component, with U1 and U2 indicating the mean and gust speeds, respectively. (b) Wavelet coscalogram product array for the whole velocity vector time series (color bar indicates N m−2). (c) Stress magnitude as computed by EC for 1-, 2-, and 5-min windows and by the wavelet techniques over the whole time series. In (c), U1 and U2 correspond to the stress magnitudes computed for steady values of the mean and gust speeds, respectively.

Figure 10a shows wind stress computed from the field data (N = 32) using three separate techniques and the COARE 3.0 algorithm (Fairall et al. 2003) as functions of the 10-m neutral wind speed. Figure 10b shows the “R” values (Ortiz-Suslow et al. 2015) over the same wind speed range. The stress estimates are directly compared in Fig. 11. There is substantially more scatter in these estimates than in those produced from the wind-wave tank data, though the integrated product’s underestimation of the wind stress can still be clearly seen for high wind speed.

Fig. 10.

From field observations (5-min windows; N = 32): (a) Wind stress as a function of 10-m neutral wind speed, computed via EC, PWA, integrated wavelet coscalograms, and the COARE 3.0 algorithm. (b) The R parameter (Ortiz-Suslow et al. 2015) for the same data points, representing the value of the observed air–sea frictional drag relative to that computed from COARE.

Fig. 10.

From field observations (5-min windows; N = 32): (a) Wind stress as a function of 10-m neutral wind speed, computed via EC, PWA, integrated wavelet coscalograms, and the COARE 3.0 algorithm. (b) The R parameter (Ortiz-Suslow et al. 2015) for the same data points, representing the value of the observed air–sea frictional drag relative to that computed from COARE.

Fig. 11.

Comparison of wind stress magnitude computed via EC and wind stress magnitude computed via the wavelet techniques, field observations (5-min windows; N = 32). The red line indicates a 1:1 relationship. Error bars indicate the interdecile ranges of the wavelet-derived quantities over the EC windows (i.e., no “bin averaging” has been performed). The coefficients of determination are 0.70 and 0.61 for EC-PWA and the EC-integrated product, respectively.

Fig. 11.

Comparison of wind stress magnitude computed via EC and wind stress magnitude computed via the wavelet techniques, field observations (5-min windows; N = 32). The red line indicates a 1:1 relationship. Error bars indicate the interdecile ranges of the wavelet-derived quantities over the EC windows (i.e., no “bin averaging” has been performed). The coefficients of determination are 0.70 and 0.61 for EC-PWA and the EC-integrated product, respectively.

Figure 12 shows a transition in the nature of the stress signal over the course of an example transect. At the start of the transect, the vessel was in a portion of the bay that was sheltered from the long-period swells traveling eastward and from the strong winds generating the wind sea. Here, the wind stress was largely governed by low-frequency pulses. By the end of this transect, the PWA stress magnitude became higher and showed greater variance as a result of the increased wave energy (a 66% increase in root-mean-square water surface elevation from A to B). In this wavier domain, all EC-determined stresses were found to increase. Throughout the entire transect, high-amplitude transient eddies are shown by the wavelet coscalograms to exist and to have a marked effect on the time-averaged wind stress, though none of the EC estimates are truly able to resolve their presence.

Fig. 12.

Observations over a single 35-min westward transect. (top) Coscalogram product (color bar is in N m−2); (middle) series of wind stress magnitude; (bottom) 2-min moving root-mean-square water surface elevation (white) and mean wind speed (orange). Wind stress includes the 1-s PWA (violet circles) and EC at 1-, 2-, and 5-min resolution (blue, green, and yellow squares, respectively). The red solid boxes indicate the locations of the segments corresponding to Fig. 13 (box A) and Fig. 14 (box B).

Fig. 12.

Observations over a single 35-min westward transect. (top) Coscalogram product (color bar is in N m−2); (middle) series of wind stress magnitude; (bottom) 2-min moving root-mean-square water surface elevation (white) and mean wind speed (orange). Wind stress includes the 1-s PWA (violet circles) and EC at 1-, 2-, and 5-min resolution (blue, green, and yellow squares, respectively). The red solid boxes indicate the locations of the segments corresponding to Fig. 13 (box A) and Fig. 14 (box B).

Visual inspection of the individual cases (sections A and B from Fig. 12; shown in Figs. 13 and 14) reveals the individual eddies that constitute the overall estimate. Specifically, for Fig. 13b, contributions from the wind sea, swell, and large-scale atmospheric eddy frequency bands appear quite clearly in the mean cospectrum: net downward momentum flux occurs at the wind-sea frequencies. For Fig. 14b, the wind speed is higher and overall wave energy is increased. In this case, the wind-sea band is far more apparent in the coscalogram and the downward momentum flux is much more tightly clustered at the wind-sea peak.

Fig. 13.

Section A (5 min in time) from Fig. 12. (a) The variation in wind stress, and (b) the ocean surface wave frequency spectrum. The marker color/shape convention of (a) matches the one used in Figs. 12 and 9. The wave spectrum in (b) is colored by the time-averaged value of for each frequency.

Fig. 13.

Section A (5 min in time) from Fig. 12. (a) The variation in wind stress, and (b) the ocean surface wave frequency spectrum. The marker color/shape convention of (a) matches the one used in Figs. 12 and 9. The wave spectrum in (b) is colored by the time-averaged value of for each frequency.

Fig. 14.

Section B (5 min in time) from Fig. 12. The same layout is used as in Fig. 13. The black dashed line represents the wave spectrum computed for Fig. 13.

Fig. 14.

Section B (5 min in time) from Fig. 12. The same layout is used as in Fig. 13. The black dashed line represents the wave spectrum computed for Fig. 13.

4. Discussion and conclusions

One feature of the present study that is difficult to ignore is the shortness of the eddy covariance windows (typically 5 min, though as short as 1 min). The use of 5-min windows is not without precedent (Ortiz-Suslow et al. 2015); however, the 2- and 1-min windows may be on the order of the temporal scales of eddies that are important in air–sea momentum flux. This would result in the interpretation of longer-duration features as nonstationarities that are removed in the detrending process; it almost certainly would result in long-period forcing aliasing, seen anecdotally within Fig. 5c. For the observations made here, it seems that the smallest trustworthy EC window is 5 min, even though in those we were unable to capture gusts (as expected) or reliably describe spatial inhomogeneity in our coastal region of interest. This leads us to the further development and application of nonstationary analysis to these signals.

The agreement between and the time average of the PWA time series might appear to be counterintuitive. After all, EC computed over a segment of a velocity time series includes some integrated quantity produced from all of the information contained in that time series. The important detail arises from the fact that in computing stress via PWA, one does not average in frequency and pluck out the peak amplitude from the whole time series; rather, one identifies the dominant eddy at each time step and averages those as appropriate. In the estimation of wind stress via EC, we compute the friction velocity . This quantity’s sensitivity to the dominant scales of variation in the time series is implicit but taken for granted; that is, at any given time, the product of two fluctuating velocity components will be largely determined by the dominant scale of motion at that time. With PWA, the dimensionality of the input time series is expanded and the dominant deviations are used explicitly. In that sense, the structure of how stress is estimated has not truly been changed—it has merely been made a bit more transparent.

We reiterate that summarizing the wind stress using EC is akin to summarizing a wave field using parameters like , , and . Such a summary can be useful and is certainly convenient, but the product does not include critical dynamics that may have a substantial net effect on the system’s evolution. Although past analyses have captured the multiscale variability in the wind velocity field by using the wavelet transform (Meneveau 1991; Hudgins et al. 1993; Hayashi 1994; Saito and Asanuma 2008), PWA allows for the explicit, quantitative description of individual atmospheric forcing events within a nonstationary record. One of the principal assumptions of EC is that the wind velocity time series is stationary. As a result, EC should not be expected to appropriately resolve any such nonstationary events. In contrast, PWA produces a stress estimate that is both properly localized and of appropriate magnitude without any a priori information about the feature’s duration or shape. Perhaps the most intuitive approach to this problem would be to integrate the wavelet coscalograms and over all frequencies, resulting in “net” descriptions of the along-stream and cross-stream fluxes. However, as shown in Fig. 7, the “integrated product” approach produces stress estimates near to those of EC only for low wind forcing (and may underestimate gust forcing, as in Fig. 9). In any case, both techniques appear to capture the relevant short time features in the wind velocity time series, so future studies may indeed incorporate either or both approaches with modifications to the wavelet transform parameters.

The principal benefit of the wavelet-oriented approach as compared to short time or moving window eddy covariance coupled with Fourier analysis depends upon the nature of the forcing. Inspection of Fig. 10 indicates that COARE has underestimated air–sea momentum flux in the coastal region of study for U10N < 7 m s−1. It is likely that wave-related effects that may be second order in the open ocean become significant in this low wind speed regime (Gao et al. 2009; Potter 2015). These effects contribute to the observed variability as the craft moves alongshore: a recent study of wind stress in Monterey Bay that used shore-tower-based measurements indicated that the location of observation affected the relationship between wind speed and wind stress, even for onshore wind (Yang et al. 2018). One way to investigate these processes further is to examine the individual transects. During the transect from which the data comprising Fig. 12 were gathered, the RHIB moved from one distinct wave regime to another (near abscissa > 3500 m). This transition was not instantaneous, however, as the wind speed and RMS water surface elevation were seen to gradually increase as the PWA array showed a combination of increased energy at the wind-sea band and intermittent pulses surrounding the swell frequency. Short time Fourier transforms would have worked to alias the swell frequency pulses and distort the estimation of their temporal separation; shortening the window would improve resolution but reduce the quality of the spectral estimate at low frequencies. A moving window EC would smooth over their effect altogether, producing an output that blends the effects of wind sea, swell, and larger-scale atmospheric forcing in a manner that may miss transient signals. These effects range from changes to the direction of the stress vector (Sjöblom and Smedman 2002; Zhang et al. 2009; Shabani et al. 2016) to attenuation of turbulent momentum transfer at high frequencies under certain conditions (Drennan et al. 1999b; Sullivan et al. 2008; Kahma et al. 2016). As such, this type of blending should be avoided if one desires to resolve these distinct and independent sources of air–sea momentum flux variance. Traditionally, those seeking to describe spatial inhomogeneity in wind and wave fields would need to use a ship that stayed on station for extended periods of time. If the inhomogeneity were temporally nonstationary, then one would have had to make use of a series of fixed platforms. Both of these approaches are expensive and cumbersome to execute in practice. The power of the wavelet-based techniques lies in the way that they allow scientists to measure transient phenomena and describe inhomogeneity of wind forcing over water in regions that had previously not been amenable to such analyses (e.g., the highly variable region shown in Fig. 2c).

The dynamics of atmospheric forcing and wind-wave coupling near coastal margins figure directly into the way in which humans interact with these environments. As remote sensing techniques allow for recovery of ocean surface wave information nearer and nearer to shore, interest in describing these highly variable regions has dramatically increased. The methods introduced in this work and resulting glimpses into nonstationary wind forcing have the capacity to positively impact our observations and analyses of air–sea momentum flux, providing an opportunity for higher-resolution studies of coastal ocean surface transport and wind-wave interactions.

Acknowledgments

The authors extend their thanks to the RSMAS researchers and support staff who assisted in data collection during the CLASI campaign and laboratory experiments—Neil Williams, Mike Rebozo, Cédric Guigand, Sanchit Mehta, and Andrew Smith. Finally, the authors are appreciative of the scientists whose wisdom and/or code proved essential to this work, including, but not limited to, Mark Donelan, Will Drennan, and Bertrand Chapron. This work was supported by ONR Grant N000141612196.

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Footnotes

a

Current affiliation: Division of Ocean and Climate Physics, Lamont-Doherty Earth Observatory, Columbia University, Palisades, New York.

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