## 1. Introduction

### a. Background and motivations

The problem of airflow over surface gravity waves is old, with an ongoing record of publications on the topic that started nearly 100 years ago (Jeffreys 1925, 1926; Miles 1957; Phillips 1957; Janssen 1991; Belcher and Hunt 1993; Hristov et al. 2003; Janssen and Bidlot 2023). A nuanced understanding of both the growth of waves and the statistics of atmospheric velocity and pressure over a wavy sea surface is continuing to evolve (Ayet and Chapron 2022). A common central theme in wind-over-wave theories is the relative phase of atmospheric pressure and the sea surface. Under assumptions of potential flow, the pressure is 180° out of phase with the sea surface resulting in no net energy flux or wave growth [see Young (1999), section 4.3.1.1., for a derivation]. For waves to grow through a pressure-driven process, the phase of the atmospheric pressure must be shifted downwind of the wave crest such that there is a low pressure anomaly aligned with upward wave orbital motions (leeward side), resulting in net negative pressure work on the surface. As described in Janssen (1999), this pressure–velocity correlation is expected to persist in the atmosphere above the air–water interface and serves as a sink of kinetic energy in the atmospheric turbulence budget. In this way, the wave-coherent pressure work is an important link between the wave-incoherent turbulent motions that otherwise would be expected to follow boundary layer theory (e.g., Edson and Fairall 1998) and the wave-coherent motions that lead to wave growth.

### b. TKE budget in the wave-affected atmospheric boundary layer

*τ*

_{tot}= −

*ρ*〈

_{a}*u*′

*w*′〉 is total (turbulent plus wave-coherent) stress, Π

*is the wave-induced KE transport,*

_{w}*ϵ*is the TKE dissipation rate, 〈⋅〉 represents a time average,

*u*′ =

*u*− 〈

*u*〉 is the fluctuating horizontal velocity, and

*w*′ is the fluctuating vertical velocity. Fluctuations can be decomposed into the into wave-coherent and wave-incoherent (i.e., turbulent) parts such that

^{1}Janssen (1999) argued that the triple velocity product is expected to be small, and proposed a form for the magnitude and vertical profile of the wave-induced pressure work as

*k*is the wavenumber of surface waves,

*z*is the height above water, and

*e*

^{−2}

*vertical decay rate comes from the assumption of potential flow.*

^{kz}^{2}We note that this assumption of vertical form has been disputed, particularly very close to the sea surface. A discussion in Donelan et al. (2006) compares the exponential decay for the pressure (not the pressure work) with the output of the model of Kudryavtsev et al. (2001), finding significant differences very close to the surface. Recognizing these limitations, we continue to follow Janssen (1999) and the suggested vertical decay scale therein, where the wave-induced KE transport gradient becomes

*S*

_{in}(

*f*,

*θ*) is the spectral energy input to the surface waves,

*f*is the frequency,

*θ*is the relative wind-wave direction, and

*k*is assumed to follow the linear dispersion relation.

### c. Wind-wave input and wave growth rates

*β*and the surface wave energy spectrum

*E*(

*f*,

*θ*),

*E*(

*f*,

*θ*) = (1/2)

*ρ*(

_{w}gG_{ηη}*f*,

*θ*), where

*ρ*is the water density and

_{w}*G*(

_{ηη}*f*,

*θ*) is the wave elevation spectrum. Plant (1982) fit numerous existing wave growth datasets, finding a growth rate of

*ω*= 2

*πf*is the radial frequency, and

*c*=

*ω*/

*k*is the frequency-dependent wave phase speed. Janssen (1989, 1991) showed that the drag over the water depends on sea state, which in turn modifies the growth rate. The new proposed growth rate was

*B*is the so-called Miles’ constant which is a function of the nondimensional critical height, and fraction of wave to total stress

*B*is often achieved through iteration as described in Komen et al. (1996) and outlined here in appendix A. In practice, Eq. (7) only deviates from the Plant (1982) formulation at high and low wave age. While the energy input of the very short wind-waves is important for wave growth in community wave models, the contribution of the short waves to the pressure work via Eq. (3) is small due to the rapid decay with height.

Janssen (1999) showed that when winds are large (on the order of 15 m s^{−1}), the wave-coherent pressure work is expected to account for roughly 10%–20% of the local TKE budget (e.g., scaled to *ϵ* and 〈*u*〉 (Ayet and Chapron 2022, Fig. 5 therein).

While there are many existing field measurements of stress, mean winds, and TKE dissipation rate over the open ocean, and some measurements of wave-coherent pressure, there are (to the authors’ knowledge) no reported field measurements of wave-coherent pressure work which serves as the mechanistic link between the downward flux of energy that makes waves grow, and the loss of energy from atmospheric turbulence.

### d. Wave-coherent pressure and velocity: Similarity functions and phase

*p*and

*w*,

*G*(

_{ηη}*f*), and the cross-spectra and phase of

*p*and

*w*with the sea surface, respectively:

*G*is the cross-spectrum of surface wave elevation and atmospheric pressure,

_{ηp}*G*is the cross-spectrum of sea surface elevations and vertical velocity, and Φ is the spectral phase. More on Eq. (8) can be found in appendix B. Therefore, a model for |

_{ηw}*G*|, Φ

_{ηp}*, |*

_{ηp}*G*|, and Φ

_{ηw}*would similarly predict the magnitude and vertical shape of wave-coherent pressure work, and be relevant to the atmospheric TKE equation.*

_{ηw}*kz*, spectral inverse wave age

*U*(

*z*)/

*c*, and nondimensional atmospheric forcing

*N*should approach 1 at the surface to be consistent with the no-slip boundary condition.

_{ηw}Both Hare et al. (1997) and Grare et al. (2018) found the vertical velocity similarity function decayed exponentially as exp(−*αkz*), where *α* = 1 is the decay scale from potential flow theory. However, fits to the data by Grare et al. (2018) suggest decay rates less than unity, with best fit *α* in the range 0.62–0.82. Grare et al. (2018) reported correlation coefficients with four different forms for *N _{ηw}* as functions of

*U*(

*z*)/

*c*and

*kz*, finding highest correlation with a four parameter fit. The phase between sea surface elevation and vertical wind velocity Φ

*has been shown to exhibit a sharp transition across the critical layer, or the height at which the wave speed and wind speed match, 〈*

_{ηw}*u*(

*z*)〉 =

*c*(Hristov et al. 2003; Grare et al. 2013a, 2018). This is often shown as a spectral critical layer in deep water,

*f*=

_{c}*g*/(2

*π*〈

*u*〉), which allows for comparing the local wind speed to the measured spectral phase. Hristov et al. (2003) and Grare et al. (2013a, 2018) showed phase shifts between 90° and 180° across the critical layer frequency in different field measurements made from R/P

*FLIP*. These phase shifts have been used as support for the critical layer mechanism (sometimes called the matched layer mechanism) for wind-wave growth. Hristov et al. (2003) solved the Rayleigh equation numerically to compare against measured Φ

*, finding constant values on either side of the critical layer which agreed well with their measurements.*

_{ηw}The phase between the sea surface elevation and atmospheric pressure remains a topic of much interest, as it directly links to the growth rate of wind waves. Field and laboratory experiments (Snyder et al. 1981; Hare et al. 1997; Donelan et al. 2005; Grare 2009; Grare et al. 2013b) have sought to measure the pressure–sea surface phase from both static, and wave-following platforms finding wave-coherent pressure phases between Φ* _{ηp}* = 120° and Φ

*= 180°. However, the phase of pressure at the wavy sea surface remains of considerable interest. For example, the recent two-phase DNS study of Wu et al. (2022) reports much larger phase shifts than have been measured in the field (on the order of 90°), and suggested the importance of mechanisms related to wave steepness as well as wave age.*

_{ηp}### e. Structure of this work

Here, we focus on the magnitude and spectral shape of wave-coherent atmospheric pressure work, motivated by its relevance to the turbulent kinetic energy (TKE) budget of the atmospheric boundary layer. To do this, we will use measurements from a fixed Eulerian reference frame which offers many logistical advantages. While recognizing that these fixed-frame measurements cannot capture important statistics below the wave crests which may deviate from their above-crest statistics (Grare et al. 2013b), they can still be used to evaluate the pressure work relevant to the Eulerian TKE equation in the atmosphere.

Section 2 describes the field site, the data collection, and the data processing. Section 3 presents the results, including measured phase and magnitude of wave-coherent pressure and velocity, and the resulting estimates of pressure work. These measurements are compared against the similarity functions and decay scales suggested from previous studies, and to the Janssen (1999) model for pressure work. Section 4 discusses the implications of these measurements for the atmospheric wave-affected boundary layer, and avenues for future work on the topic. A summary is presented in section 5.

## 2. Methods

### a. Site description and measurement overview

Measurements were made from an open-lattice steel tower (Fig. 1) deployed in roughly 13-m water depth in Buzzards Bay, Massachusetts. Buzzards Bay is a 48 km × 12 km basin open on the southwest side to Rhode Island Sound. The average depth is 11 m, with a tide range of 1–1.5 m, depending on the neap–spring cycles. Winds in Buzzards Bay are commonly aligned with the long axis of the bay (from the northeast or southwest), and are often strong in the fall and winter. The tower was deployed near the center of the bay at 41.577 638°N, 70.745 555°W for a spring deployment lasting from 12 April 2022 to 13 June 2022. Wind speeds were measured up to 16 m s^{−1}, and were large and sustained during a 3-day event in early May that will receive additional focus here (Fig. 2).

(a) 20-min mean windspeeds measured from the four vertical levels. Here, boom-1 represents the lowest level boom shown in Fig. 1. (b) Significant wave heights and (c) peak wave periods from BBASIT are shown above.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

(a) 20-min mean windspeeds measured from the four vertical levels. Here, boom-1 represents the lowest level boom shown in Fig. 1. (b) Significant wave heights and (c) peak wave periods from BBASIT are shown above.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

(a) 20-min mean windspeeds measured from the four vertical levels. Here, boom-1 represents the lowest level boom shown in Fig. 1. (b) Significant wave heights and (c) peak wave periods from BBASIT are shown above.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

Atmospheric measurements included three primary instrument booms that housed paired sonic anemometers (RM Young 81000RE) and high-resolution pressure sensors (Paros Scientific). The pressure sensor intakes were terminated with static pressure heads (Nishiyama and Bedard 1991), which reduce the stagnation pressure contribution to the measured pressure. The tower booms were aligned at 280° such that the northeast and southwest winds would be unobstructed by the tower’s main body. A fourth sonic anemometer (Gill R3) was extended above the tower such that it was open to all wind directions and clear of wake by the tower structure. A single point lidar (Riegl LD90-3i) was mounted to the highest boom and measured the water surface elevation underneath the anemometer and pressure sensors to within a few centimeters horizontally. All instruments were time synchronized with a custom “miniNode” flux logger that aggregated the data streams from each instrument. The miniNode is an in-house constructed datalogger used to time synchronize multiple serial data streams, to control power (duty cycle) instruments, and to communicate with a shoreside user. It is composed of a BeagleBone computer with several custom-made data/power boards and is housed in a water-tight Delrin cylinder equipped with Subconn bulkhead connectors to connect to instrumentation. The version used in this study updates similar miniNode dataloggers used in Coupled Boundary Layer Air–Sea Transfer (CBLAST; among many other previous studies) which required synchronization of data streams from multiple different sensors (Edson et al. 2007). The heights of each instrument above the deployment mean water level are shown alongside a photograph of the tower in Fig. 1. Additional atmospheric and wave measurements on the tower included shortwave and longwave radiometers (Kipp and Zonen) a stereo camera pair (IOI 5MP Victorem), two RH/T sensors (Vaisala), and a standard lower-resolution barometer (Setra).

Subsurface velocity and pressure measurements were made with an array of six Nortek acoustic Doppler velocimeters (ADVs) paired with high-resolution Keller pressure sensors sampling at 32 Hz. These measurements were primarily used in this study for assessing surface wave directions. Additional water-side measurements were made (including temperature, conductivity, currents, turbulence, acoustic backscatter, and more), which will be described in a subsequent manuscript, and will not be used in the analysis here.

The sonic anemometers and pressure sensors were sampled continuously, with sampling frequencies of 20 Hz (Gill R3), 32 Hz (RM Young, booms 1–3), and 16 Hz (Paros pressure). The lidar was nominally sampled at 20 Hz and sampled for 40 min starting at the top of each hour. Data were recorded in 20-min-long files, and time averages and spectra are estimated over these 20-min segments. Timing was synchronized such that the start and end of each instrument’s 20-min files were aligned to less than 10 ms. There is some additional unknown timing delay between instrument sampling and instrument data output that depends on the manufacturer specifications which are not always published. As an example, the Gill R3-50 manual reports that sampling the three pairs of transducers in sequence over 7 ms with a 3-ms delay before sending data. We estimate the resulting time alignment between instruments as being roughly one instrument sample (50 ms) to account for variable instrument delays and timing variance from the datalogger. For waves with frequencies of 0.1, 0.2, 0.4, and 0.8 Hz, a 50-ms timing offset would result in 1.8°, 3.6°, 7.2°, and 14.4° error in phase, respectively. Therefore, more variance is expected in higher-frequency estimates of spectral phase due to the sub-ping timing alignment uncertainty.

### b. Spectral and cross-spectral analysis

Instruments were linearly interpolated to a 20-Hz time grid for spectral and cospectral processing. Power spectra and cross-spectra were estimated using the overlapping-segmented averaging method as implemented by MATLAB’s “pwelch” and “cpsd” functions. Each time series was linearly detrended and processed with 2048-sample windows (1.69 min) tapered with a Hamming window with 50% overlap between segments. The interpolation resulted in some high-frequency deviation from spectra made from the original (not interpolated) time series, although this was confined to frequencies larger than 1 Hz, above the wave band frequencies of interest to this study. For example, interpolated sonic anemometer spectra were observed to be roughly 10%–20% lower than noninterpolated spectra at 5 Hz, with no noticeable deviation at 1 Hz. Confidence intervals for power spectra follow the standard *χ*^{2} form, with normalized errors estimated as

Separation of the pressure work cospectrum into wave-coherent and wave-incoherent terms for the atmospheric measurements was done following Eq. (8), which is described in more detail in appendix B. This separation between wave-coherent and wave-incoherent components generally follows from previous studies and textbooks (e.g., Bendat and Piersol 2011; Veron et al. 2008; Grare et al. 2013b). The total wave-coherent pressure work at a measurement height can be found by integrating Eq. (8) in frequency, *f* < 1 Hz). Although this is somewhat arbitrary, it was found that decreasing the low-frequency limit added noise in the integrated *pw* estimates. Increasing the high-frequency limit resulted in minor changes but was set to where there were minimal differences between interpolated and noninterpolated spectra. Visual inspection of the MSC (Fig. 3) shows the most coherence at frequencies 0.2 Hz < *f* < 0.8 Hz, well inside the chosen integration bounds.

(a) Boom-4 wind speed. (b) Wave elevation spectra. (c) Magnitude squared coherence between the sea surface and atmospheric pressure from boom-1. (d) Magnitude squared coherence between the sea surface and atmospheric vertical velocity from boom-1.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

(a) Boom-4 wind speed. (b) Wave elevation spectra. (c) Magnitude squared coherence between the sea surface and atmospheric pressure from boom-1. (d) Magnitude squared coherence between the sea surface and atmospheric vertical velocity from boom-1.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

(a) Boom-4 wind speed. (b) Wave elevation spectra. (c) Magnitude squared coherence between the sea surface and atmospheric pressure from boom-1. (d) Magnitude squared coherence between the sea surface and atmospheric vertical velocity from boom-1.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

### c. Estimation of S_{in} and β, and Janssen’s (1999) pressure work

Wind-wave input *S*_{in} was estimated using the measured energy spectrum *E*(*f*), and the measured wind friction velocity *β* using the procedure described in Komen et al. (1996). Wave directions were measured using velocity and pressure measurements from an acoustic Doppler velocimeter (ADV) roughly 2 m below the water surface (Herbers et al. 2012; Thomson et al. 2018). Wavenumber *k*(*f*) was estimated using the finite-depth linear dispersion relation, (2*πf*)^{2} = *gk* tanh(*kd*), where *d* is the local water depth, and *g* is the acceleration due to gravity. Due to the relatively short fetch-limited waves measured at this location, most wave frequencies were unaffected by water depth via the dispersion relation. For example, at 0.2 Hz, the finite-depth wavelength is 97% of its deep-water equivalent. Surface currents were typically less than 20 cm s^{−1}, and dispersion corrections were not considered since they are relatively small for waves at frequency *f* < 1 Hz.

*β*is used. The total wave-coherent pressure work can be estimated by integrating the above equation in frequency.

## 3. Results

### a. Wave coherent pressure and velocity

Atmospheric pressure and velocity were found to be coherent with the waves, particularly during strong winds. Figure 3 shows a time series of the wave elevation spectrum along with the MSC of *ηp* and *ηw*. The strongest coherence for *p* and *w* occurs during the 3-day high-wind event on 8–10 May. Periods of high pressure coherence also align with decaying waves; however, these periods often have weaker coherence in velocity, whereas high-wind events are associated with high coherence in both *p* and *w*.

During high-wind events, the atmospheric pressure was found to be shifted from the potential flow 180° phase with the sea surface elevations, consistent with wind-wave growth theories (Miles 1957; Janssen 1991) and past measurements of wave-coherent pressure (Snyder et al. 1981; Hare et al. 1997; Donelan et al. 2005). During these events, low atmospheric pressure perturbations were seen slightly downwind of wave crests, and high atmospheric pressure perturbations were measured over wave troughs. An example is shown in Fig. 4, during a high-wind event that took place in early May where the lowest boomed instruments were particularly close to the wave crests (for the shown example, the middle of the instrument sampling volume was just under 30 cm above the highest wave crests). Wave-induced pressure perturbations are visible from all three booms with pressure perturbations as large as *p*′ ∼ 50–100*τ _{a}* over the wave crests. Pressure perturbations were generally larger at the booms closer to the surface. Although the boom-2 pressure shown in Fig. 4 occasionally exceeds the boom-1 pressure, on average the perturbations were measured to decrease with height. Perturbations were not seen over every wave crest/trough, but were seen sporadically over groups of 5–10 sequential wave crests, particularly during high-wind events.

(a) A time series of normalized pressure perturbations measured from the three Paros pressure sensors. (b) A time series of sea surface elevation measured by the single point lidar (solid blue) and the boom-1 wind and pressure sample height (dashed orange) that is closest to the water surface. Heights in (b) are referenced to the 20-min mean water level; boom-1 is closer to the surface here than referenced in Fig. 1 due to the combined tidal stage and wind-driven setup. Pressure in (a) is normalized by the wind stress, estimated using a 20-min averaging window. The pressure perturbation is the deviation from a 30-s moving average filter. Large negative atmospheric-pressure perturbations are aligned with the passage of wave crests. Waves and wind can be imagined as moving right to left in the time series, such that lidar measures the leeward (downwind) face of the wave first.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

(a) A time series of normalized pressure perturbations measured from the three Paros pressure sensors. (b) A time series of sea surface elevation measured by the single point lidar (solid blue) and the boom-1 wind and pressure sample height (dashed orange) that is closest to the water surface. Heights in (b) are referenced to the 20-min mean water level; boom-1 is closer to the surface here than referenced in Fig. 1 due to the combined tidal stage and wind-driven setup. Pressure in (a) is normalized by the wind stress, estimated using a 20-min averaging window. The pressure perturbation is the deviation from a 30-s moving average filter. Large negative atmospheric-pressure perturbations are aligned with the passage of wave crests. Waves and wind can be imagined as moving right to left in the time series, such that lidar measures the leeward (downwind) face of the wave first.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

(a) A time series of normalized pressure perturbations measured from the three Paros pressure sensors. (b) A time series of sea surface elevation measured by the single point lidar (solid blue) and the boom-1 wind and pressure sample height (dashed orange) that is closest to the water surface. Heights in (b) are referenced to the 20-min mean water level; boom-1 is closer to the surface here than referenced in Fig. 1 due to the combined tidal stage and wind-driven setup. Pressure in (a) is normalized by the wind stress, estimated using a 20-min averaging window. The pressure perturbation is the deviation from a 30-s moving average filter. Large negative atmospheric-pressure perturbations are aligned with the passage of wave crests. Waves and wind can be imagined as moving right to left in the time series, such that lidar measures the leeward (downwind) face of the wave first.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

A time series showing the spectral phase of pressure and vertical velocity for the full deployment is shown in Fig. 5. Both the pressure and the vertical velocity phase (Figs. 5a,b) exhibit a phase transition across a frequency defined as *f*_{max} = max{*f _{p}*,

*f*}, where

_{c}*f*is the peak wave frequency and

_{p}*f*is the deep-water critical layer frequency [the frequency where

_{c}*U*(

*z*) =

*c*]. Recognizing that relative wind and wave directions can result in modified or multiple critical layers, we opt for the unidirectional deep-water critical layer frequency for its simple representation. Pressure phases shift from Φ

*∼ 180° near*

_{ηp}*f*

_{max}to Φ

*< 180° for frequencies higher than*

_{ηp}*f*

_{max}. This phase shift holds for frequencies just above

*f*

_{max}during the high winds in May, which ranged between 165° < Φ

*< 180°, despite the visually apparent red coloring at high frequency in Fig. 5a. The high-frequency shift to Φ*

_{ηp}*> 180° during this event is discussed further in appendix C, and is believed to be an effective phase bias due to a horizontal offset that primarily impacts high frequencies, and not the phase difference Φ*

_{ηp}*− Φ*

_{ηp}*.*

_{ηw}Phases of (a) pressure and (b) vertical velocity with respect to sea surface elevation, with (c) the cosine of their difference. The color axis for (a) is zoomed to better see slight differences near 180° of phase, and the color axis of (c) is chosen such that blues align with a downward flux [see Eq. (8)]. The bounded white lines in (a)–(c) show max{*f _{p}*,

*f*}, where

_{c}*f*is the peak frequency and

_{p}*f*=

_{c}*g*(2

*π*〈

*u*〉)

^{−1}is frequency of the critical layer in deep water. Estimates in (c) with associated magnitude squared coherence less than 0.1 are excluded. Data have been smoothed using a 4-h moving average filter for visibility.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

Phases of (a) pressure and (b) vertical velocity with respect to sea surface elevation, with (c) the cosine of their difference. The color axis for (a) is zoomed to better see slight differences near 180° of phase, and the color axis of (c) is chosen such that blues align with a downward flux [see Eq. (8)]. The bounded white lines in (a)–(c) show max{*f _{p}*,

*f*}, where

_{c}*f*is the peak frequency and

_{p}*f*=

_{c}*g*(2

*π*〈

*u*〉)

^{−1}is frequency of the critical layer in deep water. Estimates in (c) with associated magnitude squared coherence less than 0.1 are excluded. Data have been smoothed using a 4-h moving average filter for visibility.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

Phases of (a) pressure and (b) vertical velocity with respect to sea surface elevation, with (c) the cosine of their difference. The color axis for (a) is zoomed to better see slight differences near 180° of phase, and the color axis of (c) is chosen such that blues align with a downward flux [see Eq. (8)]. The bounded white lines in (a)–(c) show max{*f _{p}*,

*f*}, where

_{c}*f*is the peak frequency and

_{p}*f*=

_{c}*g*(2

*π*〈

*u*〉)

^{−1}is frequency of the critical layer in deep water. Estimates in (c) with associated magnitude squared coherence less than 0.1 are excluded. Data have been smoothed using a 4-h moving average filter for visibility.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

Similar to previous work, vertical velocity phases consistently shift from Φ* _{ηw}* ∼ 90° at frequencies above

*f*

_{max}, to Φ

*∼ 270° below*

_{ηw}*f*

_{max}. Since both pressure and vertical velocity phases shift similarly, the difference in frequency Φ

*− Φ*

_{ηp}*varies more weakly with frequency than pressure or vertical velocity phase alone. Still, there is an important change in sign of cos(Φ*

_{ηw}*− Φ*

_{ηp}*) that occurs across*

_{ηw}*f*

_{max}, with negative values (blue in Fig. 5c) for frequencies above

*f*

_{max}and generally positive values (red in Fig. 5c) for frequencies below

*f*

_{max}. This change in sign reflects the phase difference that is consistently equal to Φ

*− Φ*

_{ηp}*≈ 110° for*

_{ηw}*f*>

*f*

_{max}. It is this small phase shift that deviates roughly 20° from quadrature that allows for the downward (negative) wave-coherent pressure work fluxes.

### b. Similarity functions and decay scales

Wave-coherent vertical velocity and pressure are consistent with a vertical decay of the form exp(−*αkz*), and are consistent with the decay rate *α* = 1 suggested by potential flow solutions (Young 1999). Figure 6 shows measurements scaled as suggested by the Hare et al. (1997) similarity functions. Both *N _{ηw}* and

*N*approach 1 at small

_{ηp}*kz*, and binned values from all three booms are in close agreement with exp(−

*kz*). Although only measurements from boom-1 extend below roughly

*kz*= 0.5, data from all three booms agree for

*kz*> 0.5. These measurements are within the range of decay rates found in previous studies

*α*= 0.63 (Grare et al. 2018) and 1 <

_{w}*α*< 1.8 (Donelan et al. 2006). However, the decay rates from this study appear much closer to the potential flow solution on average (i.e.,

_{p}*α*= 1).

Spectral similarity functions for wave-coherent (a) vertical velocity and (b) pressure plotted against vertical scale *kz*. Small colored dots show 20-min spectral data for booms-1–3 (mean boom elevations shown in Fig. 1). Estimates with low MSC have been excluded. Bin averages are made by averaging 125 sequential descending points in *kz*, and are shown with 95% confidence intervals defined as twice the standard error. Bin averages agree well with the exp(−*kz*) decay rate shown by the dashed red line over 0.2 < *kz* < 3.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

Spectral similarity functions for wave-coherent (a) vertical velocity and (b) pressure plotted against vertical scale *kz*. Small colored dots show 20-min spectral data for booms-1–3 (mean boom elevations shown in Fig. 1). Estimates with low MSC have been excluded. Bin averages are made by averaging 125 sequential descending points in *kz*, and are shown with 95% confidence intervals defined as twice the standard error. Bin averages agree well with the exp(−*kz*) decay rate shown by the dashed red line over 0.2 < *kz* < 3.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

Spectral similarity functions for wave-coherent (a) vertical velocity and (b) pressure plotted against vertical scale *kz*. Small colored dots show 20-min spectral data for booms-1–3 (mean boom elevations shown in Fig. 1). Estimates with low MSC have been excluded. Bin averages are made by averaging 125 sequential descending points in *kz*, and are shown with 95% confidence intervals defined as twice the standard error. Bin averages agree well with the exp(−*kz*) decay rate shown by the dashed red line over 0.2 < *kz* < 3.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

### c. Wave-coherent pressure work

Combining measurements of the wave-coherent pressure and vertical velocity allow for estimation of the wave-coherent pressure work through Eq. (8). These estimates can be compared against the directly measured cospectra of velocity and pressure, which include both wave-coherent and wave-incoherent information (Fig. 7a). Cospectra of pressure work show clear negative *pw* in the wave band, counter to the background wave-incoherent fluctuations which are positive (Fig. 7a). Both *p* and *w* show strong coherence with waves, *η*, in the wind wave frequency band (0.05 Hz < *f* < 1 Hz, Fig. 7b), suggesting the negative *pw* is due to surface waves. A wave-turbulence decomposition (described in appendix B) shows that the *pw* cospectrum is similar in magnitude and shape to the Janssen (1999) model [Eq. (11)]. The spectral decay term, exp(−2*kz*) is essential in this fit, and is consistent with the decays measured in Fig. 6. Still, Eq. (11) over predicts *pw* at roughly *f* < 0.25 Hz and under predicts *pw* for roughly *f* > 0.35 Hz, suggesting some possible deficiencies in the modeled vertical decay and/or growth rate related to frequency.

(a) Six 20-min *pw* cospectra from 9 May 2022 are averaged (blue) and shown alongside the decomposed turbulent [black, also shown in (c)] and wave-coherent [orange, also shown in (d)] components. (b) The magnitude squared coherence *γ*^{2} between *η* and *p*, as well as between *η* and *w* are shown. There is strong coherence for pressure and vertical velocity in the wave band (0.1 < *f* <1 Hz), which allows the (c),(d) decompositions. The Janssen (1999) model for atmospheric *pw* is shown in purple in (d) [Eq. (11)], which aligns well in both magnitude and shape to the measured cospectrum.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

(a) Six 20-min *pw* cospectra from 9 May 2022 are averaged (blue) and shown alongside the decomposed turbulent [black, also shown in (c)] and wave-coherent [orange, also shown in (d)] components. (b) The magnitude squared coherence *γ*^{2} between *η* and *p*, as well as between *η* and *w* are shown. There is strong coherence for pressure and vertical velocity in the wave band (0.1 < *f* <1 Hz), which allows the (c),(d) decompositions. The Janssen (1999) model for atmospheric *pw* is shown in purple in (d) [Eq. (11)], which aligns well in both magnitude and shape to the measured cospectrum.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

(a) Six 20-min *pw* cospectra from 9 May 2022 are averaged (blue) and shown alongside the decomposed turbulent [black, also shown in (c)] and wave-coherent [orange, also shown in (d)] components. (b) The magnitude squared coherence *γ*^{2} between *η* and *p*, as well as between *η* and *w* are shown. There is strong coherence for pressure and vertical velocity in the wave band (0.1 < *f* <1 Hz), which allows the (c),(d) decompositions. The Janssen (1999) model for atmospheric *pw* is shown in purple in (d) [Eq. (11)], which aligns well in both magnitude and shape to the measured cospectrum.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

### d. Downward pressure work fluxes during high winds

The time series of the frequency-integrated wave-coherent *pw* cospectrum (hereafter called measured pressure work) is consistent with the Janssen (1999) model at the lowest boomed level over the entire high-wind event, which lasted roughly 7–11 May 2022. Figure 8 shows the measured pressure work against the energy fluxes estimated from the Janssen (1999) model. There is good agreement in both magnitude and shape between the measurements and model, with the model slightly overpredicting energy fluxes. These results are fairly consistent if the wave growth rate *β* is taken from either Plant (1982) or Janssen (1991). Moderate agreement was seen at booms 2 and 3 (not shown), at the expected but much lower energy levels (0.06 and 0.02 W m^{−2}, respectively, during the highest winds on 9 May).

A comparison of the time series of wave-coherent energy fluxes estimated from measurements made from the lowest-elevation boom (blue) and from the Janssen (1999) model using the Plant (1982) growth rate [orange with shaded error bars, Eq. (6)], and the Janssen (1991) growth rate [purple, Eq. (7)] over the 3-day high-wind event in early May 2022. Error bars are derived from the reported coefficient uncertainty in Plant (1982).

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

A comparison of the time series of wave-coherent energy fluxes estimated from measurements made from the lowest-elevation boom (blue) and from the Janssen (1999) model using the Plant (1982) growth rate [orange with shaded error bars, Eq. (6)], and the Janssen (1991) growth rate [purple, Eq. (7)] over the 3-day high-wind event in early May 2022. Error bars are derived from the reported coefficient uncertainty in Plant (1982).

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

A comparison of the time series of wave-coherent energy fluxes estimated from measurements made from the lowest-elevation boom (blue) and from the Janssen (1999) model using the Plant (1982) growth rate [orange with shaded error bars, Eq. (6)], and the Janssen (1991) growth rate [purple, Eq. (7)] over the 3-day high-wind event in early May 2022. Error bars are derived from the reported coefficient uncertainty in Plant (1982).

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

### e. Upward pressure work fluxes during relaxing winds

The measured pressure work fluxes occasionally switch sign, such that there is an upward flux (waves giving energy to the wind). This occurs when the phase difference Φ* _{ηp}* − Φ

*is slightly less than 90°, and is most often observed for decaying waves at frequencies*

_{ηw}*f*<

*f*

_{max}. Many low-wind events correspond to low MSC, which makes estimation of the upward wave-coherent pressure work more challenging. However, during quickly decreasing winds there is enough remnant wave energy to observe high MSC and upward pressure work. For example, strong pressure-wave coherence and upward (positive) pressure work was observed on 29 May after a rapid decrease in wind speed (as shown in Fig. 9). As the winds drop, the frequency of the spectral critical layer increases. These upward fluxes from the waves are not considered in the Janssen (1999) model which may result in an overestimate of the wave energy sink in the TKE budget.

An example of upward pressure work during a decaying wind-wave event in late May. (a) 20-min mean wind speed measured from boom-4 (height roughly 10 m). (b) Significant wave height. (c) Magnitude squared coherence between the sea surface elevation and boom-1 pressure and the deep-water critical layer frequency (white line). (d) Wave-coherent pressure work spectrum (blue) shown with a scaled wave elevation spectrum (gray) for reference. Spectra in (d) were averaged over the 2-h period from 2100 to 2300 local time for visibility.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

An example of upward pressure work during a decaying wind-wave event in late May. (a) 20-min mean wind speed measured from boom-4 (height roughly 10 m). (b) Significant wave height. (c) Magnitude squared coherence between the sea surface elevation and boom-1 pressure and the deep-water critical layer frequency (white line). (d) Wave-coherent pressure work spectrum (blue) shown with a scaled wave elevation spectrum (gray) for reference. Spectra in (d) were averaged over the 2-h period from 2100 to 2300 local time for visibility.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

An example of upward pressure work during a decaying wind-wave event in late May. (a) 20-min mean wind speed measured from boom-4 (height roughly 10 m). (b) Significant wave height. (c) Magnitude squared coherence between the sea surface elevation and boom-1 pressure and the deep-water critical layer frequency (white line). (d) Wave-coherent pressure work spectrum (blue) shown with a scaled wave elevation spectrum (gray) for reference. Spectra in (d) were averaged over the 2-h period from 2100 to 2300 local time for visibility.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

Previous studies have measured waves driving mean winds in the laboratory (Harris 1966), and shown upward momentum fluxes from swell moving faster than winds (Grachev and Fairall 2001); however, treatment of these wave-to-wind fluxes in both atmospheric boundary layer schemes and community wave models remains poorly constrained by field data. Commonly implemented source terms in community wave models (i.e., the ST4 physics package; Ardhuin et al. 2010) compute swell dissipation rates largely based on the analysis of Ardhuin et al. (2009), which used altimeter data to track swell energy loss along great circles. Although there was not sufficient data from the example shown in Fig. 9 for an extensive analysis of wave-to-wind fluxes, the pressure work method presented here could prove useful for future field studies to refine understanding of processes by which swell loses energy to the atmosphere (Hanley and Belcher 2008).

### f. Overall data/model agreement

The Janssen (1999) model provides reasonable agreement with the measurements, particularly at high winds. Figure 10 shows a direct comparison for measured and modeled wave-coherent pressure work. For energy fluxes above 3 × 10^{−2} W m^{−2}, the data and model are in good agreement. For lower wind speeds (below approximately 10–12 m s^{−1}), the data and model are correlated, but with much smaller measured fluxes than predicted by the model. This may be due to the reversal of the direction of the pressure work flux at frequencies lower than the critical layer frequency, which is not considered in the model as it is calculated here. Conversely, there may be fluxes that were not measured well due to low coherence at lower frequencies, as can be seen by the data excluded in Fig. 5c. Although integrated fluxes were still calculated, there may be bias introduced from the frequencies with very low MSC. These tended to skew toward lower frequencies, which often exhibited erroneously large upward fluxes. This was found to be due to the shape of the pressure spectra, which exhibited spectral slopes between *f*^{−1} and *f*^{−2} outside of the wave band (i.e., increasing pressure variance at decreasing frequency).

A scatterplot shows a comparison of the measured and modeled frequency-integrated wave-coherent pressure work above. Data–model agreement is good during high-energy conditions. At lower energy levels, the model does not capture the reversal of sign at the critical frequency and overpredicts the energy draw down from the atmospheric boundary layer. Open gray circles show integrated 20-min cospectral values, and blue diamonds show binned averages with 95% confidence intervals. Bin averages are made by averaging 25 sequential ascending points, and confidence intervals defined as twice the standard error.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

A scatterplot shows a comparison of the measured and modeled frequency-integrated wave-coherent pressure work above. Data–model agreement is good during high-energy conditions. At lower energy levels, the model does not capture the reversal of sign at the critical frequency and overpredicts the energy draw down from the atmospheric boundary layer. Open gray circles show integrated 20-min cospectral values, and blue diamonds show binned averages with 95% confidence intervals. Bin averages are made by averaging 25 sequential ascending points, and confidence intervals defined as twice the standard error.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

A scatterplot shows a comparison of the measured and modeled frequency-integrated wave-coherent pressure work above. Data–model agreement is good during high-energy conditions. At lower energy levels, the model does not capture the reversal of sign at the critical frequency and overpredicts the energy draw down from the atmospheric boundary layer. Open gray circles show integrated 20-min cospectral values, and blue diamonds show binned averages with 95% confidence intervals. Bin averages are made by averaging 25 sequential ascending points, and confidence intervals defined as twice the standard error.

Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0097.1

## 4. Discussion

### a. Prediction of pressure-work fluxes

While there is compelling consistency between the Janssen (1999) *pw* model and the pressure work measurements during this event, it is perhaps not generalizable without more events. As noted by several authors, there is still considerable spread in existing data for growth rate *β*, resulting in 50% uncertainty in the coefficient reported by Plant (1982) [as seen in Eq. (6)]. Including this uncertainty would put the majority of measured *pw* within the uncertainty bounds in wind-wave energy flux. The spread in existing growth rate data is somewhat expected. Under the critical layer mechanism, the growth rate is governed by the ratio of wind speed curvature to shear at the critical layer height, *χ* = *U*″(*z _{c}*)/

*U*′(

*z*). The value of

_{c}*χ*is expected to be inherently variable due to the tendency of derivatives to enhance variability. Therefore, small deviations from an expected logarithmic profile (used to inform parameterized growth rates) would cause much larger variability in growth rate (Hristov and Ruiz-Plancarte 2014).

Still, close inspection of Fig. 7d shows the largest difference between the theory and measured wave-coherent pressure work occurs near 0.25 Hz, just before the measured cospectrum falls off toward zero. This effect is fairly consistent across the May high-wind event, suggesting that there may be some deficiencies of the model. However, it is unclear if the mismatch is due to the growth rate, the vertical decay function, the assumption of unidirectional waves, or another unidentified mechanism. The vertical decay function is particularly suspect, as the potential flow exponential decay has been shown to fall short when compared to more detailed numerical models (Kudryavtsev et al. 2001; Donelan et al. 2006).

Several recent studies have also suggested variability in the wave growth rate with wave steepness as well as wave age (e.g., Buckley et al. 2020; Wu et al. 2022). Therefore, it could be that the spectral growth rate decreases near the peak in relation to a roll-off in spectral steepness. Last, this roll-off could be due to unaccounted-for directional spreading effects. Plant (1982) integrated a directional spreading function, which modified the unidirectional growth rate by roughly 20% [Plant 1982, Eqs. (10)–(12) therein]. Finite depth effects that modify the wave phase speed and shape of the wave orbital motions were not expected to be large, since at 5 s (the frequency where wave-coherent pressure work approaches zero in Figs. 7a,d) the finite depth phase speed is still 97% that of the deep water limit.

As noted in Ayet and Chapron (2022), there is some disagreement over the expected shape of the wave-induced transport term Π* _{w}*, which depends heavily on wave-coherent pressure work and which could impact mean wind profiles and turbulent statistics in the atmospheric boundary layer (Fig. 5 therein). Very recent work from Janssen and Bidlot (2023) investigated the feedback between wave-supported energy fluxes and changes to the mean wind profile. Under a Miles critical layer growth mechanism, the curvature of the mean wind profile sets the wave growth rate

*β*. The discussed feedback is that the wave-coherent energy fluxes can modify the shear profile, which would then result in a smaller growth rate. Janssen and Bidlot (2023) found this nonlinear effect was visible but relatively small at 〈

*u*〉 ∼ 15 m s

^{−1}, with a large reduction of growth rate at 〈

*u*〉 = 50 m s

^{−1}(see Fig. 1 therein). Continued work to understand the magnitude and decay rate of

### b. Flux measurements from low-height sensors at sea

There is a growing interest in measuring fluxes from autonomous platforms at sea which frequently measure wind speed from a low mean height (e.g., ∼1 m for wave gliders). The approach is often to use bulk flux algorithms (Edson et al. 2013) and/or the inertial dissipation method (Edson et al. 1991; Yelland et al. 1994). The extent to which the wave boundary layer influences estimates of surface fluxes, particularly at low measurement heights, has yet to be fully determined. The results here support the arguments from Janssen (1999) that inertial dissipation estimates are likely biased due to a deficit of dissipation related to the wave growth mechanism. However, we emphasize here that this bias is small at wind speeds less than ∼10 m s^{−1}, even for low measurement heights. At moderate wind speeds, the Janssen (1999) model does a reasonable job predicting the pressure work fluxes at wind speeds 10–15 m s^{−1}, suggesting a wave correction could be applied to inertial dissipation estimates of stress. Since wave spectra are commonly measured from autonomous sea-going platforms, these corrections would not pose onerous additional measurement constraints. The reference to specific wind speeds is not intended to be universal, since the data used in this study were in a limited fetch environment and will have a different wave age than similar wind speeds in open ocean conditions.

Whether or not the mean wind speed measured from such platforms exhibits bias due to wave-affected boundary layer influences is also of debate. A discussion in Hristov and Ruiz-Plancarte (2014) suggests that the wind profile wave bias is controlled by the wave age and wave steepness, concluding that at the commonly observed wave steepness the wind profile wave bias is small (order 2%). However, this study did not consider feedbacks between sea state and the roughness length *z*_{0} and used the Charnock coefficient as an experimental parameter. These feedbacks through the roughness length were a central part of the findings of Janssen and Bidlot (2023). The results presented here focus on the energy fluxes and not the mean wind speed profiles. However, given that the pressure work fluxes are small until higher wind speeds, it is unlikely wave-driven fluxes would have a significant impact on mean wind profiles for wind speeds less than 10–12 m s^{−1}. For storm and high-wind conditions (above roughly 12–15 m s^{−1}), the wave-driven pressure work fluxes become a larger fraction of the TKE equation and the impact to Monin–Obukhov (MO) similarity functions central to bulk flux formulas (Edson et al. 2013) has yet to be determined. More work will be needed to justify the use of bulk formulas from these low-level wave-following platforms at high wind speed.

## 5. Conclusions

We have presented observations of wave-coherent pressure work in the wave-affected atmospheric boundary layer. Observations are qualitatively consistent with existing wind-wave growth rate parameterizations, and a vertical decay that depends on height and surface wavenumber. The vertical decay is roughly consistent with exp(−2*kz*), although we recognize that this form of the vertical decay function is an imperfect approximation. The agreement between measured pressure work and existing models is seen in cospectra and in a frequency-integrated time series. The simple model for atmospheric work tends to overpredict the measured pressure work by a factor of 20%–50%; however, this is generally within the reported error bounds of a coefficient for the wave growth parameterizations, and this disagreement was smaller at higher wind speeds. At rapidly decreasing winds, upward pressure work fluxes were observed from waves traveling faster than winds.

Future work is needed to understand the impact of wave coherent pressure work on MO similarity and bulk flux formulae at high winds. For the specific instrument heights and wave states in this study, measurements indicate the TKE budget (and therefore MO similarity functions by proxy) start to be modified by waves at winds above 10–12 m s^{−1}. We feel that this problem has a particular relevance given the growing interest in using low-lying anemometers on autonomous platforms at sea for measurements in high winds.

We note there is some difference in how the triple velocity product is treated in previous work. Since the focus of this work is the pressure term, not the triple velocity product, we opt for the definition used in Ayet and Chapron (2022) and note the differences here.

The statement in Janssen (1999) that potential flow leads to a decay scale of *e*^{−2}* ^{kz}* has been disputed by a reviewer of this manuscript. The reviewer argues that a constant wind speed profile is also needed to arrive at vertical decay of

*e*

^{−2}

*, and that solutions to the Rayleigh and Taylor–Goldstein equations that impose a logarithmic wind profile result in different vertical decays. In this work, we recognize the*

^{kz}*e*

^{−2}

*decay is an imperfect approximation, the validity of which will be tested against field data.*

^{kz}## Acknowledgments.

This work was funded by NSF Award 2023020. Additional support was provided by Grants NSF 2316818, NASA 80NSSC21K0832, and ONR N00014-22-1-2405. Jay Sisson and Steve Faluotico helped design, deploy, maintain, and recover the tower and associated measurement systems. Thanks to Jonah Mikutowicz and the AGM and 41° North crews for invaluable help with deployment and recovery of the tower. Jim Wilczak provided the static pressure heads used in this experiment. Two anonymous reviewers provided feedback on an initial draft, and Al Plueddemann, Tim Duda, Chris Zappa, Alejandro Cifuentes-Lorenzen, and Roger Samelson provided helpful comments and discussions.

## Data availability statement.

Data used in this work (Zippel et al. 2022) is available through the WHOI Open Access Server (WHOAS) at https://doi.org/10.26025/1912/29583.

## APPENDIX A

### Estimation of the Miles’ Parameter for Determining the Wind/Wave Energy Input

*B*. We follow the description in Komen et al. [1996, Eqs. (3.38)–(3.47) therein]. Parameter

*B*is defined as

*B*= 1.2 is a constant and

_{m}*μ*is defined as

*z*

_{0}is defined with a Charnock relation,

*τ*being the stress from surface gravity waves. This stress is defined using the initial growth rate,

_{w}*B*then requires the full growth rate, and must be done iteratively. We used an initial estimate of growth rate following Eq. (6) to solve for the wave stress, which was used to recompute the growth rate. We used a fixed number of iterations, 3, which served as a balance between computational speed and accuracy.

## APPENDIX B

### Estimation of Wave-Coherent and -Incoherent Spectra

*pw*cospectra are decomposed using the assumption of a linear spectral model which is described here. This method closely follows Bendat and Piersol (2011, section 6.2.2 “Single-Input/Multiple-Output Model”), as well as Veron et al. (2008) and Grare et al. (2013a), in which the authors use a similar decomposition. Here, we assume that the time series of

*p*′ and

*w*′ each have a wave-coherent component,

*η*, such that the Fourier transform

*H*and

_{p}*H*are complex transfer functions that depend on frequency. Here we use notation consistent with the measured time series, but note that the notation of Bendat and Piersol (2011) would equate to

_{w}*η*=

*x*(

*t*),

*p*′ =

*y*

_{1}(

*t*), and

*w*′ =

*y*

_{2}(

*t*), and use

*G*to denote the power spectrum of

_{xx}*x*, and

*x*and

*y*

_{1}. Following Bendat and Piersol’s (2011) Eq. (6.77), the complex transfer functions are defined using cross-spectra and autospectra,

*G*(

_{pη}*f*) is the complex cross-spectrum of

*p*′ and

*η*,

*G*(

_{wη}*f*) is the complex cross-spectrum of

*w*′ and

*η*, and

*G*(

_{ηη}*f*) is the real-valued power spectrum of

*η*.

*p*′ and

*w*′, assuming no wave-turbulent correlations (e.g.,

*pw*correlations contribute to the TKE equation, the wave-coherent

*pw*spectrum is estimated combining Eqs. (B3) and (B4),

*= arctan(Im{*

_{pw}*G*}/Re{

_{pw}*G*}) is the spectral phase. The wave-incoherent part

_{pw}*β*from out-of-phase pressure Im{

*G*} (Hare et al. 1997; Donelan et al. 2006). Assuming a boundary condition where the atmospheric velocity equals the wave orbital motion at

_{pη}*z*= 0, we have |

*G*(

_{wη}*z*= 0)| =

*ωG*and Φ

_{ηη}*(*

_{wη}*z*= 0) = 90. Combining with Eq. (B5),

*G*|sin(Φ) = Im{

*G*}, such that,

## APPENDIX C

### Errors in Estimates of Spectral Phase

Time and space offsets in measurements can cause spectral phase estimates to vary in frequency. For phases of the form Φ = **k** ⋅ **x** − *ωt* + *ϕ*, time offsets result in linear change in phase with frequency, ΔΦ = −*ω*Δ*t*, and space offsets result in a change of phase, ΔΦ = **k** ⋅ Δ**x**, with time and space offsets Δ*t* and Δ**x**, respectively. In this dataset, time delays were determined to be small on average resulting in minimal functional bias in phase estimates. Spatial offsets between atmospheric pressure and velocity were also determined to be relatively small at the frequencies of interest, typically resulting in directional phase shifts less ΔΦ < 5°. However, there was some uncertainty regarding the spatial location of the lidar spot on the surface. While the instrument’s dot was aligned prior to deployment of the tower, small rotations and offsets resulting during the deployment yield some uncertainty in the final horizontal position of the *η* measurement. Therefore, phases Φ* _{ηp}* and Φ

*are expected to have larger uncertainty due to unknown space offsets Δ*

_{ηw}**x**

*and Δ*

_{ηp}**x**

*. Assuming deep water dispersion, the resulting phase bias would depend on frequency squared and wave direction. In practice, this bias is small at frequencies near*

_{ηw}*f*

_{max}where both wave energy and pressure work fluxes tend to be largest. No realistic time/space offsets could explain the measured Φ

*near and just above*

_{ηp}*f*

_{max}. However, the steep increase in bias with frequency resulting from spatial offsets could plausibly explain the high-frequency phase Φ

*> 180° in Fig. 5a during the high-wind event in May. It is important to note that this high-frequency bias is not reflected in the final wave-coherent pressure work estimate since the phase difference Φ*

_{ηp}*− Φ*

_{ηp}*subtracts this unknown offset, since Δ*

_{ηw}**x**

*− Δ*

_{ηp}**x**

*= Δ*

_{ηw}**x**

*. This latter offset, Δ*

_{pw}**x**

*, is known and is small.*

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