a. Experimental setup

The measurements described here were collected from the Floating Instrument Platform (R/P FLIP) moored approximately at the center of the triangle formed by the islands of San Nicholas, Santa Catalina, and San Clemente off the coast of Southern California (33°13.202′N, 118°58.767′W), in water 1160 m deep from 7 to 22 November 2013, during the ONR Southern California 2013 (SoCal2013) experiment. A telescopic mast was deployed at the end of the port board boom approximately 18 m from the 9-m-diameter hull of R/P FLIP (Figs. 1 and 2). The telescopic mast consists of four sections that slide into each other. Each section is approximately 3 m long. While the top section (section 4 in Fig. 2) was tightly coupled to the boom, the extension of each section underneath the boom (sections 1, 2, and 3, section 1 being the lowest) was adjustable. Each section of the mast was equipped with an ensemble of instruments deployed to measure the properties of the airflow above the ocean. Sections 1, 2, 3, and 4 have a 2-, 3-, 4-, and 5-in. square hollow section, respectively. A set of rigging lines connected sections 1, 2, and 3 to the hull of R/P FLIP to constrain the horizontal displacement of the mast.

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Experimental setup on R/P FLIP. Details of the telescopic mast are presented in Fig. 2. The following instruments were mounted on the booms of the R/P FLIP: IMU/GPS (SPAN-CPT, Hemisphere), laser wave gauges (ILM-500), net radiometer (CNR1), and wind lidar (Windcube).

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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(left) The telescopic mast with (right) details of the different sections. The following instruments were mounted on the mast: sonic anemometers (CSAT3 and Gill R3-50), temperature–humidity sensors (HC2S3), IMU (AHRS400, MTi300), hygrometer CO₂ sensor (LICOR7500), and laser wave gauges (ILM-500).

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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The top section (section 4) of the mast, lengthened by a 2-m vertical pole (with a 2-in. square cross section), was equipped at its highest point with a 3D sonic anemometer (Campbell CSAT3) and with an open-path infrared hygrometer CO₂ sensor (LICOR 7500). These two instruments were mounted on a horizontal pole (with a 2-in. cross section) perpendicular both to the mast and the boom, 1 m away from the mast. At the intersection between the mast and the horizontal pole were mounted a 6-degrees-of-freedom inertial measurement unit [IMU; MEMSIC Attitude Heading Reference System (AHRS) 400] and a relative humidity–temperature sensor (Campbell HC2S3). All these instruments were located about 3 m above the walkway of the boom. As for section 4, each moving section 1–3 of the mast was equipped, at its base, with a 3D sonic anemometer CSAT3, an IMU, and a humidity–temperature sensor (HC2S3). The main differences between the moving sections and section 4 are that the moving sections are not equipped with a hygrometer CO₂ sensor, the IMU model is different (Xsens MTi 300), and the IMU is mounted directly behind the anemometer. In addition to these instruments, the lowest section of the mast was equipped with a 3D sonic anemometer (Gill R3-50) mounted upside down directly below the lowest point of the mast. Hence, the measuring volumes of the Gill and the lowest CSAT3 were separated by 0.85 m in the vertical direction and 1 m in the horizontal direction. Aside the base of the Gill, a single-point laser altimeter (MDL ILM-500) was mounted to monitor the distance between the bottom of the mast and the sea surface. This laser altimeter was also used as a wave gauge. This setup ensured that all four CSAT3 anemometers were strictly one above the other.

The total extension of the mast was controlled by an electrical winch, while the relative extension of each section was adjusted with a set of rigging lines, manually controlled from the hull of R/P FLIP. The extension of the mast was set to bring the lowest instruments as close as possible to the mean sea level (MSL) while preserving the sensors’ integrity and distributing the instruments along the vertical axis. Thus, while the highest anemometer remained at 14.8 m (±0.2 m) above MSL, the lowest anemometer was positioned at an adjustable height, as low as 2.6 m from the MSL (see Fig. 3b).

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Environmental conditions of the marine atmospheric boundary layer: (a) (black line) and wind direction (from; red line). (b) Heights of the sonic anemometers relative to the mean sea level. (c) Air temperature at (red line) and sea temperature at m (blue line). (d) Stability parameter with L defined in Eq. (4). All the data points are 30-min averages.

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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Reference values of humidity and temperature were provided by Campbell HC2S3 sensors. The hygrometer CO₂ instrument (LICOR 7500), which is also equipped with a pressure sensor, was used to measure the fast fluctuations of the water vapor density and the absolute pressure of the air. Each sonic anemometer measured the three components of the wind.

The temperature–humidity sensors, the hygrometer CO₂ sensor, and the sonic anemometers were sampled at 20 Hz. The laser altimeter is internally sampled at 12 kHz and was averaged down to 100 Hz. The MEMSIC IMU was sampled at 20 Hz, while the Xsens IMUs were sampled at 50 Hz.

In addition to the instruments mounted on the mast, an array of five laser wave gauges (MDL ILM-500, sampled at 100 Hz) located on the starboard, port, and face booms of R/P FLIP were deployed to measure the directional spectrum of the sea surface displacement. A Global Navigation Satellite System (GNSS) and Inertial Navigation System (INS; NovAtel SPAN-CPT) was deployed to measure the motion of R/P FLIP. This device combines GNSS and INS solutions to deliver position, velocity, and altitude angles with very high accuracy (the accuracy in absolute position is only a few centimeters). The combined solution was sampled at 100 Hz.

A temperature chain (PME T-Chain; 17 thermistors sampled at 0.8 Hz) was deployed from the starboard boom of R/P FLIP to monitor the temperature of the first 50 m of the water column. Finally, a wind lidar system (Leosphere Windcube WLS7) was deployed on the starboard boom to measure the vertical profile of the wind every 20 m from 50 up to 400 m above the MSL. The four-beam lidar provided a triangulated measure of the wind components every 4 s.

b. Data postprocessing and analysis

The sonic velocities and the air–sea interface displacements were corrected to account for the motion of R/P FLIP. Those corrections were computed combining solutions from both the IMUs and the SPAN-CPT. The combined solution was used to correct the wind components following Edson et al. (1998). The sea surface elevations measured by the laser wave gauges were corrected for the vertical displacements of the instruments according to the solutions provided by the SPAN-CPT instrument.

After corrections, the wind vector was projected onto a reference frame aligned with the mean direction of the wind using the double-rotation method (Wilczak et al. 2001). This method ensures that the mean vertical velocity is null over 30-min records. Turbulent fluxes of momentum, heat, and water vapor were computed using the covariance method over the same 30-min windows. This time window ensures sufficient statistical reliability in the wind stress and spectral estimates. To avoid the influence of nonstationary effects on the flux estimates, linear trends in the wind velocity were removed before the computation of the fluctuations. The spectral functions (spectrum, cross-spectrum, phase, and squared coherence) used to derive the products presented below were obtained from fast Fourier transforms (FFTs) using 100-s sliding windows (with a 50-s overlap) over each 30-min record. With signals sampled at 20 Hz, the frequency resolution of the spectral functions is 0.01 Hz. For each anemometer n, the mean wind speed at the height was computed over the same 30-min record. Using , , the frequency f, and the dispersion relationship for linear gravity waves in deep water, we defined the spectral wave age and the normalized (or spectral) height as follows:

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Using the same dataset, Grare et al. (2016) showed wind speed discrepancies between measurements from the CSAT3 and the Gill R3-50. They found that differences in the mean wind speed could reach up to 4%, a few degrees in the mean wind direction, up to 5% for the standard deviation of the vertical wind component , and up to 20% for the friction velocity. These differences were mainly correlated to the azimuthal angle of the wind relative to the instruments. A portion of these differences were attributed to the transducer’s shadowing attenuation occurring on the CSAT3. After correction for this effect, the differences were reduced but the dependence on azimuthal angle of the wind remained. Based on the results from Grare et al. (2016), we expect to observe discrepancies between the CSAT3 and the Gill R3-50 measurements of the amplitude of the wave-induced velocities to be less than 5%; such discrepancies remained within the scatter of the data, and we believe that they do not alter the conclusions of this manuscript. However, since the discrepancies for the friction velocity could reach up to 20%, discrepancies for the wave-induced momentum flux could be more than 40%. Therefore, analysis and results of the wave-induced momentum flux are not presented in this manuscript. Data from the CSAT3s presented here were corrected for the transducer’s shadowing attenuation using the method described in the appendix of Grare et al. (2016).

Finally, 30-min records when the sonic anemometers were located in the wake of the hull of R/P FLIP or in the wake of the mast were discarded from the analysis. The wake from the hull of R/P FLIP was assumed to be contained within a ±45° segment about the port boom. Similarly, the wake from the mast was assumed to be contained within a ±45° segment about the direction the CSAT3s were pointing to.

To further analyze the data, the wave-induced velocities were computed. Following Hussain and Reynolds (1970), under the influence of surface waves, we can decompose the fluctuating components of the wind as follows:

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where is the mean value, is the wave-induced component, and is the turbulent component. In the reference frame aligned with the mean direction of the wind, is the streamwise component, is the crosswind component, and is the vertical component. To extract the wave-induced components, we used the technique developed by Veron et al. (2008, 2009), which uses the cross-spectral functions between the wind components and the vertical displacement of the sea surface. See also Wu et al. (2018).

c. Environmental conditions

Figure 3 shows the variation of statistical parameters of the marine atmospheric boundary layer (MABL) during the SoCal2013 experiment. The horizontal axis represents the date in days from 11 to 21 November 2013 using the coordinated universal time (UTC). Data points are 30-min averages. The data presented hereinafter do not include measurements performed prior to 11 November 2013 because all the instruments described in the previous section were not fully operational until then. Figure 3a shows the mean wind speed at 10 m and the direction of the wind ϕ. The wind speed was calculated by linear interpolation of the wind speed measured by the two anemometers located directly below and above 10 m, that is, using sonic anemometers at level 4 and level 5. The wind direction ϕ was the average of the wind direction measured by all five anemometers. The difference in wind direction between anemometers was less than ±3°. The wind speed varied from 0 to 11 m s⁻¹, mostly coming from the west–northwest.

Figure 3b shows the heights of the sonic anemometers during the course of the experiment. These heights are relative to the MSL. The mast was retracted when the waves could potentially hit and damage the lowest anemometer (e.g., on 15 and 16 November 2013).

Figure 3c shows the variations of the air and the sea temperatures. The air temperature was measured at (see Fig. 3b) and the sea temperature at m. Except on 13 and 14 November 2013, the sea was warmer than the air, which is favorable for unstable conditions. Figure 3d shows the variations of the stability parameter , where L is the Monin–Obukhov length (Monin and Obukhov 1954) measured at the height by the anemometer collocated to the lowest humidity–temperature sensor. The L is defined as follows:

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where all variables are measured at the same height . The is the virtual temperature, the friction velocity, g is the gravity, and the overbar represents an average with respect to time. Figure 3d confirms that the conditions were unstable for most of the experiment.

Figure 4 presents the time variations of the mean wind speed profiles measured both by the array of sonic anemometers and the wind lidar. The color plot in Fig. 4a represents the variations of the mean wind speed as a function of both the time and the altitude. Data from the array of sonic anemometers correspond to the lower half of the color plot in the range 2.5 < z < 15 m, while data from the lidar correspond to the upper half of the color plot (60 < z < 500 m). In Fig. 4b, vertical profiles of the normalized mean wind speed are plotted every 8 h with

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where and are the local minimum and maximum values, respectively, of at a given time t; that is, . This figure shows the variability of the mean wind speed in both time and height. It also highlights the extent of the surface layer that contains enough scatterers to return laser pulses from the lidar. In the first days of the experiment, lidar measurements were only available in the first two hundred meters above the MSL, while a few hours before the windiest event, which occurred on 15–16 November 2013, there is a thickening of this layer that extended up to 500 m some time after the wind event passed by our location. Figure 4b shows the variability of the vertical profiles during the course of the experiment. It also shows that many profiles display a concave or convex curve, meaning that these profiles deviate from the straight line of the logarithmic boundary layer. This highlights the limits of the logarithmic shape assumption and how statistical values based on best-fit log profiles (such as friction velocity, roughness length, and wind speed at 10 m) can be sensitive to the height of the measurement.

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(a) Mean wind speed as a function of time and height measured by both the sonic anemometers on the telescopic mast (2.5 < z < 15 m) and the lidar (60 < z < 500 m). (b) Vertical profiles of the normalized mean wind speed plotted every 8 h with , where and are the minimum and maximum values, respectively, of the vertical profile of the mean wind speed ; that is, . The extent of the range [0, 1] is reported in the bottom right corner of the panel. For both panels, data are 30-min averages.

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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For each 30-min record, the omnidirectional frequency spectrum and the directional frequency spectrum of the surface elevation were computed. An example of these wave spectra (at 0230 UTC 11 November 2013) is reported in Fig. 5. The omnidirectional spectrum was computed for each individual single-point wave gauge (red curve in Fig. 5a), while the directional spectrum was computed from the array of wave gauges using the WAFO MATLAB library (WAFO Group 2000) for frequencies in the range 0.03–0.4 Hz. The omnidirectional spectrum can also be computed from the azimuthal integration of the directional spectrum (black curve in Fig. 5a).

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Example of the omnidirectional and directional spectra of the wave field from 30-min records starting at 0230 UTC 11 Nov 2013. (a) Omnidirectional spectra from measurements of the laser wave gauge located at the bottom of the telescopic mast (red) and by the azimuthal integration of the directional spectrum (black). (b) Directional spectrum from the array of laser wave gauges. (c) Smoothed directional spectrum used to partition the wave field. For the directional spectra, the direction refers to the direction the waves move to and the frequencies correspond to concentric circles logarithmically spaced. The colored boundaries delimit the main partitions of the wave field: southern swell (purple), northern swell (black), local swell (green), and wind waves (orange). The solid colored dots correspond to the mean frequency and direction of each partition.

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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For the directional spectrum (Figs. 5b,c), the frequency scale corresponds to concentric circles that are logarithmically spaced, and the azimuthal direction corresponds to the direction of propagation of the waves. The colored contours delimit the different partitions of the wave field. This example emphasizes the utility of the directional spectrum, as in the omnidirectional spectrum only three peaks are distinguishable, while four wave partitions are clearly visible in the directional spectrum. The partitioning of the wave field was performed running a script based on the methods described in Hanson and Phillips (2001) and Portilla et al. (2009). As suggested in Portilla et al. (2009), we applied to the directional spectra a 2D noise-removal filter before partitioning the spectrum using a “watershed” algorithm.¹ In Fig. 5b, the raw directional spectrum is plotted while its filtered form is plotted in Fig. 5c.

The example presented is characteristic of the conditions encountered during the experiment. The wave field was formed by the superposition of four main components:

• Wind waves (delimited by the orange contour) directly associated with local winds, with frequencies generally higher than 0.1 Hz.

• Local north–northwest swell (delimited by the green contour), residual of wind waves generated upwind of our location [say, O(100) km], which propagated to our position after the wind died. These waves were passing by for short periods of time (less than 2 days) with frequencies ranging between 0.08 and 0.15 Hz.

• Old (remote) north–northwest swell (delimited by the black contour) generated by storms in the North Pacific Ocean, passing by our location for up to four days, with the longest waves (down to ~0.06 Hz) arriving before the shorter ones (up to ~0.1 Hz).

• Swell from the Southern Hemisphere (delimited by the purple contour) generated by remote storms in the Southern Hemisphere, with frequencies in the range of 0.07–0.1 Hz.

For each 30-min record and for each wave component defined by an ensemble of pairs () bounded by the contour , we computed the following:

• The mean frequency²

  

  

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• The mean direction

  

  

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• The mean energy

  

  

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  Each pair () is reported in Figs. 5b and 5c as a solid colored point.

Figure 6a shows the omnidirectional spectrogram of the surface displacement. The is computed from the directional spectrum, and the color scale is logarithmic. The plot shows that throughout the duration of the experiment, different wind-wave events occurred, and swell was always a significant component of the wave field.

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Characteristics of the wave field during the experiment. (a) Spectrogram of the surface displacement. (b) Significant wave height (black line) and wave age (blue line). (c) Frequencies and directions of the main partitions of the wave field: wind waves (orange arrows), northern swell (black arrows), local swell (green arrows), and southern swell (purple arrows). (d) Relative energy of the main partitions of the wave field. See Eq. (8) for the definition of . Colors are as in (c).

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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Figure 6b shows the significant wave height and the mean wave age , where is the phase speed at the peak of the wave elevation spectrum. The significant wave height was computed from the directional spectrum as follows:

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The significant wave height remained small, less than about 1.5 m, for most of the experiment except on 15 November 2013 and the beginning of 16 November 2013, when it reached 2.5 m, corresponding to a high-wind event. Hence, the wave age was large (greater than 2) for most of the experiment.

The mean frequency and direction of each partition of the wave field are reported in Fig. 6c. The relative mean energy of each partition is shown in Fig. 6d, where is the total wave energy per unit surface area, integrated over all frequencies and directions.

These two bottom panels show that the wave field was quite complex during this experiment, with a superposition of local wind waves, young and old swells from the north–northwest, and long swell from the south where the relative energy contribution of each wave component varies strongly over time.

The plots presented in Figs. 3 and 6 show dropouts in the data (e.g., on 17 November 2013). These dropouts either were the result of the loss of GPS signal from the SPAN-CPT inertial unit because of temporary technical issues or were the result of periods of low wind speeds when the laser wave gauges had low returns from the glassy sea surface. Cases with a percentage of returns less than 12.5% were excluded from our analysis. This 12.5% threshold was selected because we found that below this threshold the temporal and spatial distributions of the returns were too scattered to properly reconstruct the wave profile.

a. Coherence and phase

Figures 7 and 8 show 11 days of the squared coherence and phase shift between the waves and the wind measured by each anemometer. The left panels show the coupling with the horizontal fluctuations , while the right panels show the coupling with the vertical fluctuations . For the horizontal velocity (and similarly for the vertical velocity), the squared coherence is the magnitude of the squared cross-spectral density normalized by the product of the wave spectrum and the velocity spectrum . The phase shift is the argument of . The top panels refer to the highest anemometer while the bottom panels refer to the lowest one. The right vertical axis is the frequency, and the left vertical axis is the phase speed of the waves . For each panel, the black line represents the mean wind speed at the height of the corresponding anemometer . The black line also corresponds to the speed of the waves, which have the corresponding anemometer located at their critical height. The critical height was introduced by Miles (1957) in his quasi-laminar inviscid model built on the concept of resonant interaction between wind and waves. In this model, the equations of motion governing a small perturbation (assumed to be linearly correlated to the surface waves) of a two-dimensional shear flow in an inviscid and incompressible fluid leads to the inviscid form of the Orr–Sommerfeld equation. This equation has a singularity where the wind speed equals the phase speed c of the waves, that is, at the height of the critical layer, which implies that the rate at which energy and momentum are transferred to a wave of speed c depends on quantities at the critical height (see also Lighthill 1962; Hristov et al. 2003; Young and Wolfe 2014). Wave-coherent fluctuations at phase speeds greater than the wind speed (i.e., above the black line in Fig. 7) were measured below the critical layer associated with those waves. The color scale for the squared coherence is logarithmic. For all the anemometers, the squared coherence between the waves and the vertical velocity is greater than the squared coherence between the waves and the horizontal velocity. This observation is consistent with Grare et al. (2013a). We observe a glimpse of the vertical decay of the wave-induced fluctuations as, for both horizontal and vertical fluctuations, the squared coherence is larger closer to the surface. This is particularly true at frequencies associated with wind waves since for the lowest anemometers, we observes patches of high coherence in the proximity of the black line (i.e., at frequencies where the phase speed was close to the wind speed), while the squared coherence remained low at these frequencies at higher elevations. For the top anemometer, high levels of squared coherence (say greater than 0.2) are only observed at frequencies smaller than about 0.2 Hz, while for the lowest anemometer, high coherence is observed at higher frequencies (up to 0.5 Hz) and also when c is close to . Fluctuations associated with both the swell and the wind waves are present in the data. This is most clearly illustrated in Figs. 7e and 7j on 11 November and between 20 and 21 November by the presence of distinctive patches of high coherence. Figure 7 also shows that most of the wave-induced fluctuations measured in this experiment were found below the critical layers as the squared coherence remains small below the black lines (i.e., above the critical layer).

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The squared coherences and between the surface displacement η and the wind components and . (a)–(e) Horizontal velocity. (f)–(j) Vertical velocity. (from bottom to top) Velocities measured by anemometers 1 to 5. The right y axis is the frequency, and the left y axis is the linear phase speed of the waves. The black line is the 30-min mean wind speed at the height of each anemometer. It is also the phase speed of the waves for which the anemometer is at their critical height.

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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The phase shift and between the surface displacement η and the wind components and . The order of the panels, the axes, and the black lines are as in Fig. 7. The phase is relatively constant below the critical layers (above the black lines), the horizontal component being out of phase (about 180°) with the waves while the vertical component is in quadrature (about 90°). Close to the surface, the lowest anemometers are able to measure coherent signals across the critical layers where a sharp change of the phase is observed, especially for the vertical component which shifts from +90° to −45°.

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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Figure 8 shows that the phase between the waves and the wind is relatively constant below the critical layers, the horizontal component being out of phase (about 180°) with the waves while the vertical component is in quadrature (about 90°). For most of the data, the phase above the critical layers is very noisy and appears random. This is a direct effect of the low level of coherence square observed in Fig. 7 above the critical layers where is generally smaller than 0.05. Indeed, the normalized random error of the estimates of the phase depends directly on the level of coherence square (Bendat and Piersol 2010); for example, for 100-s FFT windows along the 30-min record, when . Close to the surface, the lowest anemometers measured signals with a level of coherence high enough to observe the sharp change of the phase across the critical layers, especially for the vertical component, which shifts from +90° to −45° (purple patches on 15–16 November and 19–20 November below the black curve). This is in accordance with the critical layer theory (Miles 1957) and with the measurements from Hristov et al. (2003) and Grare et al. (2013a). Away from the surface, the coherence above the critical layers is too weak to accurately calculate the phase between the wind modulations and the waves. Therefore, the sharp change of phase across the critical layers cannot be observed with the highest anemometers.

b. Selection of the wave-coherent fluctuations

To study the variation of the wave-induced velocities with wind forcing and height, we have selected fluctuations of the wind coherent with the waves as done in Hare et al. (1997) and Grare et al. (2013a), that is, according to the level of squared coherence between the surface elevation and the velocities. To minimize the effect of spurious noise and to ensure reliable wave-induced signals, only the frequencies for which the squared coherence was above a conservative threshold of 0.2 were included in the final dataset. The value of this threshold is set arbitrarily in order to gather the maximum number of data points available for the analysis while limiting the scatter in the data [this threshold was set to 0.1 in Hare et al. (1997) and Grare et al. (2013a)]. The value of 0.2 provides a good compromise for the current dataset. We have also restricted the selection to frequencies in the range 0.06–0.5 Hz to ensure rejecting wind fluctuations correlated with the heave and the tilt motions of FLIP (around 0.04 and 0.02 Hz, respectively; see Smith and Rieder 1997), which do not correspond to the physical coupling between the airflow and the wavy surface.

c. Double dependence and parameterization

Hare et al. (1997) examined the structure of the atmospheric wave-induced variables as a function of the wind forcing (or the wave age ) and the normalized height . They formed their approach guided by the expression of the wave growth parameter β forced by wind through the form drag only:

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where E is the wave energy per unit surface area, the overdot is time derivative, and the form drag is equal to the correlation between the wave-induced pressure perturbation at the surface and the slope of the waves. Using results from Plant (1982), Al-Zanaidi and Hui (1984), and Jacobs (1987), they assumed that the single most important parameter to describe wind-wave interaction is the dimensionless forcing . They justified the vertical dependence of the pressure and velocity fields on the normalized height from the results of Snyder et al. (1981) and Jacobs (1987), who showed that the wave-induced pressure–slope correlation has an exponential decay with α close to 1. Assuming that the wave-induced motions are linearly related to the surface displacements , the amplitude of these motions are proportional to the magnitude of the wave displacement described by the omnidirectional wave spectrum . Using the wavelength λ and the phase speed c of the waves as the typical length and velocity scales, they ended up with the following relationship between the nondimensionalized wave-induced velocities and some function , which depends on both the wind forcing and the dimensionless height :

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where is the complex cross spectrum between the vertical surface displacement and the wind component , is the streamwise component, is the crosswise component, and is the vertical component. Building from the results of Hristov et al. (2003) and Grare et al. (2013a), who have highlighted the strong influence of the critical layer on both the amplitude and the phase of the wave-induced velocity fields, here we support the hypothesis that the local wave age should be used in lieu of the wind forcing to better characterize the effect of the critical layer. Furthermore, in order to satisfy the kinematic conditions at the air–sea interface, the wave-induced components of the airflow should match, at the surface, the orbital velocity of waves of amplitude O() at leading order. Therefore, wave-induced velocities can be directly nondimensionalized by the orbital velocity . Using this new scaling, and focusing on the amplitude of the wave-induced velocities, we obtain a new relationship where the dimensionless wave-induced velocity depends only on the local wave age and the dimensionless height :

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As the wave field was complex in our experiment with waves propagating in different directions, each wave component is likely to generate its own wave-induced field of velocity in its direction of propagation. To account for the three-dimensional properties of the wave-induced airflow, we regrouped the streamwise and crosswise wave-induced velocity components by computing the amplitude of the horizontal wave-induced velocity . Thenceforth, refers either to the horizontal or the vertical velocity ( or w, respectively).

Figure 9 shows the variations of the amplitude of the dimensionless wave-induced horizontal ( in the left panels) and vertical ( in the right panels) velocities as a function of the dimensionless height for each measurement level. The amplitude of the dimensionless wave-induced velocities were estimated from the norm of the spectral transfer function between wind and waves normalized by the angular frequency :

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Data from the five sonic anemometers are reported from top to bottom, with Figs. 9a and 9b representing the highest anemometers (level 5) and Figs. 9i and 9j representing the lowest anemometer (level 1). Data points are colored by bins of the spectral wave age according to the color bar at the top of the figure. In Figs. 9k and 9l, data from all anemometers have been combined and bin averaged per dimensionless height bins for each wave age bin. The error bars represent the standard deviation of the data within a dimensionless height bin. For each panel, the solid black curve is the exponential function . For all anemometers, data show that when the wave age becomes large, the vertical profiles of the wave-induced velocity amplitude tend toward the exponential decay . However, for smaller values of the wave age (less than 2), the scatter of the data is quite large; for example, for the vertical component measured by the lowest anemometer (Fig. 9j), data with a wave age between 1 and 2 (i.e., for data measured close to the critical layers) spread over almost one decade when is about 0.5. This highlights the influence of the critical layers on the wave-induced velocities whose amplitude decreases abruptly in the vicinity of the critical layer. Note that for low wind cases the wave-induced fluctuations can contribute to a large fraction of the total fluctuations, up to 90% (not shown here).

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(a)–(j) The amplitude of the dimensionless wave-induced velocities (left) and (right) as a function of the normalized height for each level. Data are colored depending on spectral wave age ranges . (k),(l) Averaged profiles from data from all levels, bin-averaged by spectral wave age with the standard deviation within each bin drawn with the horizontal error bars. In each panel, the solid black line is the exponential function .

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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The double dependence of the wave-induced velocities on both the wave age and the dimensionless height, is presented in the Fig. 10, where the variations of and as a function of are plotted in Figs. 10a and 10b, while the variations with are reported in Figs. 10c and 10d. In this figure, data have been bin averaged individually for each level, where blue is the highest anemometer, and purple is the lowest anemometer. Hereinafter, bin-averaged data with fewer than 10 data points within the bin are not plotted. Raw data are reported in Fig. S1 in the online supplemental material.

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The bin-averaged amplitude of the (left) horizontal and (right) vertical dimensionless wave-induced velocities, (a),(b) as a function of the spectral wave age and (c),(d) as a function of the normalized height , for each level. Error bars correspond to the standard deviation of the data within each bin. The functions in (a) and (b) and in (c) and (d) are plotted with dashed black lines.

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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Figures 10a and 10b show that the amplitudes of the wave-induced velocities decrease for wave ages close to 1. Figures 10c and 10d show that wave-induced velocities decay with normalized height following an exponential law. We show here that for a given wave age, the amplitude of the wave-induced velocities is inversely proportional to the height above the surface. However, in the bottom panels (especially in Fig. 10d), for a given normalized height (e.g., ), the amplitude of the wave-induced velocity is smaller when measured at low levels than when measured at higher levels. This trend is due to the fact that for a given , where , the wave age at the lower level is smaller than the one at the higher level; that is . Therefore, data from the lower anemometer are more affected by the critical layer than are those at higher levels. As shown in Figs. 10a and 10b, in the proximity of the critical layer, the amplitude of the wave-induced velocities are reduced especially when . This critical layer influence explains why the bin-averaged data shown in Fig. 10d present values that are smaller at lower levels than at higher levels.

To decouple the double dependence on and , we propose to parameterize the variations of the wave-induced fluctuations with two independent functions and , such that

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where and are constants close to 1. As demonstrated in our analysis, we can assume that the vertical decay can be described with an exponential function:

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where is a constant close to 1. The subscript i is equal to either u or w depending on whether the parameterization is applied to the horizontal or the vertical wave-induced velocity ( or , respectively). For clarity, the subscript will be omitted hereinafter. To characterize the dependence on wave age, we first assume that the decay in amplitude of the wave-induced fluctuations can be described using the inverse wave age :

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The use of this function is motivated by Miles’s (1957) critical layer theory to describe the forcing of the waves by the wind. In this theory, the wave-induced velocities drop rapidly close to the critical layer, where the mean wind speed equals the phase speed of the waves, that is, where . Although there is no exact analytical expression for the wave-induced velocities, Lighthill (1957, 1962), Kudryavtsev et al. (2001), and Kudryavtsev and Makin (2004) have shown that to a first approximation , where is the slope of the water surface. The constants A and α are estimated by minimizing a squared-difference cost function, and their values are reported in Table 1. The coefficient of determination between the data and the different parameterizations is also reported in this table. The best-fit regression produces values of A quite close to unity ( for the horizontal component and for the vertical component) when A is unconstrained; however, the exponential decay coefficients and change by about 10%–15% between the case where A is unconstrained and the case where A is set to one.

Table 1.

Best-fit coefficients of proposed wave-induced velocity parameterizations.

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Figure 11 shows the amplitude of the wave-induced fluctuations scaled by the functions and plotted as functions of the wave age and the normalized height in the case where A is set to one. Data are bin averaged for each anemometer subset of records. Raw data are presented in the Fig. S2. In Figs. 11a and 11b, the data have been corrected for their dependence on with for the horizontal wave-induced velocity and for the vertical velocity. In these panels the collapse of the data from the five different anemometers is remarkable. Some discrepancies remain as gets close to 1. This is due to the small number of data points within the bins for this range of ; for example, for level 5 (blue curve) there are only 19 data points within the bin , leading to a standard deviation (STD) for this bin 40 times greater than the median value of the STD for all bins. Another indicator of the performance of this scaling is the scatter of the data around the bin-averaged curves. Raw data presented in Figs. S1a, S1b, S2a, and S2b show that the scaling to account for the exponential decay with height drastically reduced the spreading of the data. Note that in the top panels, the dashed orange line represents the function . The agreement with the data is very good for high , where the best-fit curve lies within the error bars of the bin-averaged data. However, when is close to one, the measurements deviate from the function , reaching a finite minimum while goes to zero. This deviation is not surprising since if the amplitude of the wave-induced fluctuations were to go to zero, the correlation between velocity fluctuations and wave elevation would also go to zero, and we would not have been able to sample them. Nevertheless, since we observed wave-induced fluctuations large enough to be detected when is equal to one, it means that the proposed parameterization [Eq. (17)] is wrong and that a more realistic function needs to be used to better characterize the variations of the wave-induced velocities in the proximity of the critical layer. The divergence between the data and the current model is emphasized in Figs. 11c and 11d, where the bin-averaged data, plotted as a function of the normalized height, are scaled by the function . In these panels, the data from the lowest levels (1 and 2) deviate from the best-fit functions (orange lines) because, as we discussed before, wave-induced velocities measured at the lower levels are more subject to the effects of the critical layers than at higher levels. Therefore, the scaling by the function , which underestimates the amplitude of the wave-induced velocities for data points where , leads to an overcorrection of the wave age dependence when is close to 1. This is clearly illustrated in Figs. 11c, 11d, and S2. These overcorrected points are responsible for large values of the bin-averaged data for both the horizontal and vertical components (see Figs. 11c,d).

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The bin-averaged amplitude of the (left) horizontal and (right) vertical dimensionless wave-induced velocities (a),(b) corrected for their dependence on kz plotted as a function of the spectral wave age and (c),(d) corrected for their dependence on plotted as a function of the normalized height for each level. The functions in (a) and (b), in (c), and in (d) are plotted with dashed orange lines. Coefficients and are set to one, and the value of the coefficients and are reported in Table 1. The function is plotted with dashed black lines in (c) and (d). Error bars correspond to the standard deviation of the data within each bin.

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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To better parameterize the wave age dependence for small values of , we propose the following function to account for the minimum values of the amplitude at , using constants β and ζ:

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The coefficient β defines the minimum values of the wave-induced velocity at the location of the critical layer, while the coefficient ζ adjusts how quickly the wave-induced fluctuations decrease approaching to the critical layer. The values of the coefficients for the best-fit functions minimizing the squared-difference cost function as well as the coefficients of determination are reported in Table 1. We want to stress that, if the coefficients β and ζ are set to one, the Taylor expansion of the function at the first order is equivalent to the function F when tends to one. Figure 12 shows the amplitude of the wave-induced velocities scaled by the functions and [Eqs. (18) and (16), respectively] as functions of and . With this new parameterization, the collapse of the bin-averaged data from the five anemometers is excellent, each of them lying within the error bars of the other individual anemometer subsets of data. Raw data presented in Fig. S3 also show a reduced scatter of the data around the best-fit functions and . The vertical dependence with is similar to what was observed before with the previous scaling with the decay coefficient equal to 0.85 for the horizontal velocity and for the vertical component. This result shows that both horizontal and vertical components follow the same exponential decay with . Values of and smaller than one means that the amplitude of the wave-induced velocities deviate from the exponential decay in described by a potential airflow above waves forced at the surface by the orbital velocities . Several factors can explain this. The first obvious reason is that the presence of turbulence especially strong close to the surface can alter the decay rates of the wave-induced components through nonlinear interactions between the wave-induced and turbulent components of the airflow. Another reason can be due to the presence of waves traveling in different directions, their resulting wave-induced components working against each other rather than adding up. Finally, the last reason would be the presence of current in the ocean responsible for a Doppler shift in the dispersion relationship. This creates a bias when converting frequencies into wavenumbers, therefore affecting both the vertical decay and the kinematic conditions at the surface, that is, and .

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The bin-averaged amplitude of the (left) horizontal and (right) vertical dimensionless wave-induced velocities (a),(b) corrected for their dependence on kz plotted as a function of the spectral wave age and (c),(d) corrected for their dependence on plotted as a function of the normalized height , for each level. The functions in (a), in (b), in (c), and in (d) are plotted with dashed orange lines. The coefficients of the functions above are all unconstrained, and their values are reported in Table 1. The functions in (a) and (b) and in (c) and (d) are plotted with dashed black lines. Error bars correspond to the standard deviation of the data within each bin.

Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0121.1

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In Eq. (18), we introduced two degrees of freedom as has two adjustable coefficients (β and ζ). It is remarkable that for both velocity components, the coefficient ζ is close to or equal to 1 ( for the horizontal velocity and for the vertical velocity). By imposing , the coefficients A, α, and β are equal (with less than ±1%–2% difference) to those found when ζ is unconstrained. This implies that the wave age dependence can be parameterized with only one coefficient β to characterize the minimum value of the wave-induced amplitude at the critical height.