1. Introduction
The transfer of heat, gases, momentum, and energy between the atmosphere and the ocean are known to be connected with surface wave breaking, bubble generation, and mixing by wave- and shear-driven turbulence in the upper-ocean boundary layer (Anis and Moum 1995; Melville 1996). In general, surface wave breaking increases turbulent kinetic energy (TKE) and dissipation in the near-surface layer to a depth on the order of the wave height (Gerbi et al. 2009). The existence of a turbulent diffusion sublayer beneath the wave breaking layer has been suggested, where the vertical divergence of turbulent energy fluxes is in balance with the TKE dissipation and the shear production is small due to the weak background mean flow shear (Benilov and Ly 2002; Gerbi et al. 2009). Both observations (Thorpe et al. 2003a; Stips et al. 2005; Wijesekera et al. 2013; D’Asaro et al. 2014) and model simulations (McWilliams et al. 1997; Teixeira and Belcher 2010) show that the mixing associated with Langmuir circulation (Lc) is responsible for deepening of the mixed layer and increasing the eddy viscosity. It is likely that Lc can persist much longer and reach much deeper depths than those by the direct injection of breaking wave motions (Thorpe et al. 2003a). The turbulent motions in this sublayer play a key role by transporting energy downward (Feddersen et al. 2007). The quantification of the generation, transport, and dissipation of TKE is critically important for advancing our understanding and the development of mixing parameterizations in the oceanic boundary layer. However, because of the logistical difficulties of making near-surface measurements especially during high wind conditions, only a few measurements are available (Gemmrich and Farmer 1999, 2004; Soloviev and Lukas 2003; Thorpe et al. 2003b; Bakhoday Paskyabi and Fer 2014).
Field observations at strong winds show an increasing intensity and occurrence of breaking waves on the surface as whitecaps and bubble plumes in the water column. During the wave breaking process, bubbles are injected into the water column. Within about one wave period, larger fragmented bubbles quickly rise to the surface, contributing to the formation of foam (Farmer et al. 1999; Thorpe et al. 1999). The smaller bubbles (less than 100 μm radii) left beneath the surface persist much longer in the water column and are subject to turbulent motions (Thorpe et al. 2003b). Wave-induced turbulence is responsible for the breakup of bubbles and transport of fragmented bubbles into the water column (Vagle et al. 2012). The effects of dissolution and pressure in changing bubble radii are generally negligible during turbulent injection by breaking waves, which could last about 100 s and is comparable to the bubble life time (Thorpe and Hall 1983). Slow-rising small bubbles injected by breaking waves could be treated as passive tracers for depicting dynamic processes in the near-surface mixed layer (Thorpe 1992). Thorpe (1984, 1986) studied the vertical turbulent transport in the ocean boundary layer under weak to moderate winds by analyzing the eddy diffusion coefficient derived from bubble cloud acoustic backscatter profiles.
The objective of this study is to understand and characterize wave-affected turbulence dynamics in the mixed layer under high winds based on ocean and meteorological measurements collected off the coast of the northern Gulf of Alaska near Kayak Island in December 2012. Cyclonic storms dominate the atmospheric forcing in the Gulf of Alaska (Stabeno et al. 2004). These storms can interact with the steep coastal terrain to form strong Alaskan coastal wind jets, which are also referred to as “barrier jets” (Loescher et al. 2006; Olson et al. 2007). Barrier jets follow the coastal terrains and can persist for several days because the storms tend to linger as they spin down (Wilson and Overland 1986).
We focus on the analysis of vertical eddy diffusion coefficients estimated from time-averaged bubble acoustic backscatter profiles over a selected 12-day period when the maximum surface winds and significant wave heights reached 22 m s−1 and 9 m, respectively, and the background ocean and atmospheric conditions remained nearly uniform. Section 2 describes the experiment platforms and measurements. Section 3 describes the observed marine environmental conditions including surface winds, waves, background currents, and calculated net fluxes. Analyses of acoustic backscatter profile data on a surface-following coordinate are presented in section 4. Discussion is given in section 5, and a summary of major findings is provided in section 6.
2. Experiment platforms, sensors, and measurements
Hydrographic and velocity fields, profiles of acoustic backscatter, surface waves, and surface meteorology were collected in the coastal waters off Kayak Island, Alaska, over a 6-month period (October 2012 to March 2013) for the Naval Research Laboratory’s Breaking-Wave Effects under High Winds (BWE) program (Fig. 1). Instruments were deployed in a rectangular mooring network approximately parallel to the coast between the isobaths of 60 and 90 m (Fig. 1). The network’s alongshore and cross-shore distances are about 14 and 9 km, respectively. Instrument locations, data types, and water depths are given in Table 1. At the center of the network, a BioSonics DT-X echosounder (BioSonics 2004; Depew et al. 2009; Stevens et al. 2008) was deployed on an instrument platform (hereinafter referred to as the Lander) at about 1.5 m off the bottom. This echosounder has three vertically oriented upward-looking transducers that operate at 123, 208, and 430 kHz to provide measurements of echo intensity from breaking wave–induced bubbles within the water column directly above the Lander. The beamwidth is 6.5° for all three transducers. In this study, we chose to analyze the backscatter data collected at 208 kHz, similar to the operating frequencies used by Thorpe (1984; 1986), Dahl and Jessup (1995), Trevorrow (2003), and Vagle et al. (2010). The bubble radius that is resonant and particularly effective in scattering for the operating frequency of 208 kHz is about 16 μm at the water surface and approximately 27 μm at 20-m depth (Thorpe 1992; Vagle and Farmer 1992).
The location map of the BWE field experiment; Barny (B1–B4) and string (S1–S4) moorings are marked by solid circles. The land-based weather station (NRL MET) and moored buoy (NDBC 46082) are indicated by an open triangle and circle, respectively. Depth contour lines (gray lines) are in 10-m intervals starting at 10 m. The darker contour lines are for depths of 50, 100, 150, 200, 250, and 300 m.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
The location map of the BWE field experiment; Barny (B1–B4) and string (S1–S4) moorings are marked by solid circles. The land-based weather station (NRL MET) and moored buoy (NDBC 46082) are indicated by an open triangle and circle, respectively. Depth contour lines (gray lines) are in 10-m intervals starting at 10 m. The darker contour lines are for depths of 50, 100, 150, 200, 250, and 300 m.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
The location map of the BWE field experiment; Barny (B1–B4) and string (S1–S4) moorings are marked by solid circles. The land-based weather station (NRL MET) and moored buoy (NDBC 46082) are indicated by an open triangle and circle, respectively. Depth contour lines (gray lines) are in 10-m intervals starting at 10 m. The darker contour lines are for depths of 50, 100, 150, 200, 250, and 300 m.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Location of BWE mooring stations, water depth, sensors, and data types.
The depth profile of backscatter intensity was sampled at 2 Hz with a 0.017-m vertical resolution. During the experiment, the backscatter intensity profiles were collected for 10 min starting at 0000, 0400, 0800, 1200, 1600, and 2000 UTC and for 60 min at 1100 UTC. The measured intensity was converted to volumetric backscatter strength, Mυ (m−1), by the use of the sonar equation (BioSonics 2004); Mυ is often expressed in decibels (10 log10Mυ). Figure 2a shows 10-min samples of acoustic backscatter data; here, the depth axis zd represents the vertical distance measured upward from the bottom-mounted echosounder. The instantaneous wave surface η is identified from the profile’s high backscatter intensity anomalies caused by the air–sea interface (Vagle and Farmer 1992; Gemmrich 2010). The measured backscatter data Mυ(zd) from the fixed bottom-mounted echosounder is remapped to a wave-following coordinate system Mυ(zη), where zη is the distance measured downward from the wave surface η (Fig. 2b). The remapping of bubble backscatter profiles reduces the aliasing effect on averaging due to the surface wave orbital motions. Measurements based on the wave-following coordinate are more suitable for studying near-surface turbulence (Soloviev and Lukas 2014).
Time–depth section of acoustic backscatter profiles in (a) a fixed coordinate zd distance measured upward from the echosounder and (b) wave-following coordinate zη distance measured downward from the wave surface η, shown as the solid line in (a). The color shading is the backscatter intensity 10 log10(Mυ) (dB). The wind speed and wave height are u5 = 21.1 m s−1 and Hs = 6.6 m, respectively (case 69; Table 2).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time–depth section of acoustic backscatter profiles in (a) a fixed coordinate zd distance measured upward from the echosounder and (b) wave-following coordinate zη distance measured downward from the wave surface η, shown as the solid line in (a). The color shading is the backscatter intensity 10 log10(Mυ) (dB). The wind speed and wave height are u5 = 21.1 m s−1 and Hs = 6.6 m, respectively (case 69; Table 2).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time–depth section of acoustic backscatter profiles in (a) a fixed coordinate zd distance measured upward from the echosounder and (b) wave-following coordinate zη distance measured downward from the wave surface η, shown as the solid line in (a). The color shading is the backscatter intensity 10 log10(Mυ) (dB). The wind speed and wave height are u5 = 21.1 m s−1 and Hs = 6.6 m, respectively (case 69; Table 2).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Background turbulent motions are likely to modify spatial distributions of rising bubbles in the near surface. In general, larger bubbles rise rapidly to the surface, while microbubble populations are formed into diffused bubble plumes by advective–diffusive processes controlled by turbulent motions, background currents, buoyancy forcing, and gas exchange. The plumes can persist over time during the active wave breaking period (Monahan and Lu 1990; Vagle et al. 2012). In this study, the analysis focuses on the 10-min acoustic backscatter data acquired during a 12-day period. A total of 81 10-min records were analyzed, and environmental conditions for these records are given in Table 2.
Summary of ocean surface meteorological and wave conditions and variables derived from the 81 records: buoy wind speed at 5 m height u5, buoy wind direction θu, significant wave height Hs, peak wave period Tp, surface wind stress τu, surface air temperature from the NRL weather station Tair, SST from S1, net heat flux Qnet, bubble depth Db, e-folding length λe, eddy diffusion coefficient based on Db [(4)] KυD, eddy diffusion coefficient from λe [(2)] Kυe, eddy diffusion coefficient from friction velocity [(7)], Kυu*, and λe and Kυe are not available when Db < Hs (marked as *).
The surface wave conditions are described in terms of the significant wave height Hs and the peak wave period Tp, where 
Time history of (a) wind speed u5 at 5-m height, (b) wind direction θu at NDBC buoy 46082, (c) significant wave height Hs at the Lander (solid circles) and at the buoy (dashed line), (d) peak wave period Tp at the Lander (solid triangles) and at the buoy (dashed line), and (e) surface Stokes drift Us0 at the Lander.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time history of (a) wind speed u5 at 5-m height, (b) wind direction θu at NDBC buoy 46082, (c) significant wave height Hs at the Lander (solid circles) and at the buoy (dashed line), (d) peak wave period Tp at the Lander (solid triangles) and at the buoy (dashed line), and (e) surface Stokes drift Us0 at the Lander.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time history of (a) wind speed u5 at 5-m height, (b) wind direction θu at NDBC buoy 46082, (c) significant wave height Hs at the Lander (solid circles) and at the buoy (dashed line), (d) peak wave period Tp at the Lander (solid triangles) and at the buoy (dashed line), and (e) surface Stokes drift Us0 at the Lander.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
The surface Stokes drift is computed from Sη as 
Ocean current data were collected by five upward-looking 300-kHz Teledyne RD Instruments (RDI) Workhorse ADCPs deployed at the Lander and at the four corners of the rectangular network (B1–B4; Fig. 1). The ADCPs were positioned off the bottom in trawl-resistant bottom mounts called Barnys (Perkins et al. 2000; Teague et al. 2013). The ADCPs consisted of four transducers each with a 20° beam angle to the vertical and sampled full water column profiles of zonal U, meridional V, and vertical W velocity components at 2-m vertical resolution. The deepest velocity bins were off the bottom by 4.7 m at B1, B2, B3, and B4 and by 5.7 m at the Lander. ADCP velocity data in the upper 10 m were discarded due to side-lobe interference in the ADCP beams. Ping intervals of the ADCPs were 10 s at B1, B2, and B4, 11.25 s at B3, and 2.5 s at the Lander. Current velocity component data are averaged every 15 min at B1, B2, B3, and B4 and every 2 min at the Lander. All of the ADCPs returned good quality data except for the ADCP at B4. Four subsurface string temperature–conductivity (TC) moorings (S1–S4) were deployed near B1–B4 (distances less than 170 m). The TC string moorings were equipped with either 11 or 15 instruments to measure pressure, temperature, and conductivity. The instruments consisted of Sea-Bird Electronics (SBE) 37 MicroCATs (MC; 8 or 12 on a line) and In-Situ Aqua TROLLs (AT; 3 on each line). The MC and AT instruments were evenly spaced on the lines at depth intervals of about 5 to 7 m and recorded data at 5- and 10-min intervals, respectively. A wave tide gauge (WTG; Sea-Bird Electronics 26 with an attached Sea-Bird Electronics 4 conductivity cell) was also installed on the Lander to record bottom pressure, temperature, and conductivity data every 30 min. Detailed discussions of instrumentation, data collection, sampling methods, and data processing for the long-term current and hydrographic observations are in Jarosz et al. (2016, manuscript submitted to J. Geophys. Res.). Here, we describe the subset of the data relevant to this study, that is, measurements made during late December storm events.
3. The December storm events
a. Surface meteorology
In late December 2012, a series of high wind events occurred over the BWE experiment site. Figure 3 shows the time series of winds from the nearby NDBC buoy and wave data from the buoy and the Lander during the high wind period between 23 December 2012 and 3 January 2013. The buoy wind speeds exceeded 15 m s−1 with maximum winds reaching 22 m s−1 (Fig. 3a). The strong easterly winds generated large surface waves with a maximum significant wave height of about 9 m (Fig. 3c). The averaged significant wave height and peak wave period were 4.5 m and 10.9 s, respectively (Figs. 3c,d; Table 2). The wave height and peak wave period during 28 and 29 December measured from the Lander at 73-m water depth were smaller than those from the deep-water NDBC buoy due to shoaling effects.
The surface wind stress τw, net surface heat flux Qnet (Wm−2), and Monin–Obukhov length Lmo (m) were estimated by following Fairall et al. (1996; Fig. 4). Here, we used wind speed from the NDBC buoy and sea surface temperature (SST) from S1 and air temperature, solar radiation, and relative humidity from the NRL weather station at Kayak Island. During the 12-day period, SST remained steady at about 6°C, and air temperature increased from −7° to 4°C as warmer air moved into the BWE area. Water temperatures closely followed the air temperatures after 24 December (Fig. 4b). The estimated heat flux remained mostly negative (upward) and less than 50 W m−2 on most occasions. The Monin–Obukhov length remained positive from near zero to about 3500 m, indicating surface buoyancy forcing was not a factor during high winds. The turbulent Langmuir number Lat = (u*w/Us0)0.5 was also estimated, where the friction velocity u*w is (τw/ρw)0.5 and ρw is the density of water. The estimated Lat varied between 0.16 and 0.45 with a mean value of 0.3. This is well within the Langmuir regime (Li et al. 2005; Wijesekera et al. 2013).
Time history of (a) surface wind stress τw, (b) surface air temperature (solid line) at 4-m height (Kayak Island station) and water temperature (dashed line) at 5-m depth from S1, (c) net surface heat flux Qnet, where negative Qnet represents upward heat flux, (d) Monin–Obukhov length Lmo, and (e) turbulent Langmuir number Lat at the Lander.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time history of (a) surface wind stress τw, (b) surface air temperature (solid line) at 4-m height (Kayak Island station) and water temperature (dashed line) at 5-m depth from S1, (c) net surface heat flux Qnet, where negative Qnet represents upward heat flux, (d) Monin–Obukhov length Lmo, and (e) turbulent Langmuir number Lat at the Lander.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time history of (a) surface wind stress τw, (b) surface air temperature (solid line) at 4-m height (Kayak Island station) and water temperature (dashed line) at 5-m depth from S1, (c) net surface heat flux Qnet, where negative Qnet represents upward heat flux, (d) Monin–Obukhov length Lmo, and (e) turbulent Langmuir number Lat at the Lander.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
b. Background currents and high-frequency energy
Figure 5 shows the depth–time sections of ADCP velocity components (U, V, and W) at the Lander and water temperatures, salinities, and densities within 40 m of the mean water line at the nearby S3. Westward flows were dominant and the vertical shears were weak (Fig. 5a). Data from the string mooring show a well-mixed water column (Figs. 5d,e).
Depth–time current velocity components (cm s−1) at the Lander: (a) zonal U, (b) meridional V, and (c) vertical W. (d) Potential temperature Tθ (°C), (e) salinity S (psu), and (f) potential density anomaly σt (kg m−3) at the S3 subsurface mooring. Depth is the distance measured downward from the mean water line.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Depth–time current velocity components (cm s−1) at the Lander: (a) zonal U, (b) meridional V, and (c) vertical W. (d) Potential temperature Tθ (°C), (e) salinity S (psu), and (f) potential density anomaly σt (kg m−3) at the S3 subsurface mooring. Depth is the distance measured downward from the mean water line.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Depth–time current velocity components (cm s−1) at the Lander: (a) zonal U, (b) meridional V, and (c) vertical W. (d) Potential temperature Tθ (°C), (e) salinity S (psu), and (f) potential density anomaly σt (kg m−3) at the S3 subsurface mooring. Depth is the distance measured downward from the mean water line.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Figure 6 shows the variance-preserving spectra of current velocity components at 15-m depth for the 12-day period. Horizontal velocity spectra (U and V) show most of the energy is at frequencies less than 0.5 cycles per hour (cph). The variance of the vertical velocity spectrum has most of the energy at higher frequencies between 0.5 cph and the Nyquist frequency of 15 cph (Fig. 6c). The high-frequency energy levels were largest near the Nyquist frequency, indicating that the 2-min sampling rate could only partially resolve higher-frequency vertical motions. By applying a 2-h high-pass filter to the velocity components, we separated high- and low-frequency components, expressed as U = UL + u′, V = VL + υ′, and W = WL + w′, where UL, VL, and WL are low-frequency components and u′, υ′, and w′ are high-frequency components (e.g., Wijesekera et al. 2013). Velocity fluctuations at frequencies higher than the Nyquist frequency were not included in the high-frequency components due to the averaging scheme used in ADCP sampling, and therefore we underestimate true strength of turbulent velocity components. The time averaging of squared high-pass velocities (u′, υ′, and w′) was used to approximate TKE, q2/2, where q2 = 〈u′2〉 + 〈υ′2〉 + 〈w′2〉 and the angle brackets denote 10-min averaging. The TKE was dominated by horizontal components (Fig. 6). The time–depth variation of TKE is shown in Fig. 7b. The TKE was as large as 102 cm2 s−2 at a depth of 10 m, especially when winds were strong (Fig. 7a), and decayed rapidly with depth. The time variability of TKE is strongly correlated with that of wind stress (Fig. 7). Wave orbital velocities can alias the measurement of turbulent velocity fluctuations due to the sampling rate of the ADCP. However, in this study, aliasing from wave orbital velocities associated with the dominant wave motion is likely insignificant since the dominant wave periods are 3 to 5 times that of our 2.5-s sampling rate (Table 2). The effect of aliasing on variance is estimated to be less than 1 cm2 s−2 at a depth of 10 m, and the time-averaged vertical profile of TKE is about two orders of magnitude larger than a plausible contamination of velocity variance by wave orbital motions (Fig. 8). More detailed discussion of estimating wave orbital contamination is presented in the appendix.
Variance-preserving frequency spectra of velocity components at the Lander: (a) U, (b) V, and (c) W at 15-m water depth. The semidiurnal tide (period: 12.42 h) and the diurnal tide (period: 24.48 h) are marked as circles and crosses, respectively.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Variance-preserving frequency spectra of velocity components at the Lander: (a) U, (b) V, and (c) W at 15-m water depth. The semidiurnal tide (period: 12.42 h) and the diurnal tide (period: 24.48 h) are marked as circles and crosses, respectively.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Variance-preserving frequency spectra of velocity components at the Lander: (a) U, (b) V, and (c) W at 15-m water depth. The semidiurnal tide (period: 12.42 h) and the diurnal tide (period: 24.48 h) are marked as circles and crosses, respectively.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time series of (a) friction velocity u*w (m s−1) and (b) depth profile of TKE. The color shading is log10(TKE) (cm2 s−2).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time series of (a) friction velocity u*w (m s−1) and (b) depth profile of TKE. The color shading is log10(TKE) (cm2 s−2).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time series of (a) friction velocity u*w (m s−1) and (b) depth profile of TKE. The color shading is log10(TKE) (cm2 s−2).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Mean depth profile of TKE for the 81 cases (circles). The solid line represents the exponential function [(10)]. The crosses represent the mean aliasing variance profile of simulated wave orbital motions Δwave for the 81 cases. The dotted lines show 95% confidence intervals; z is the distance measured downward from the mean water line z = 0.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Mean depth profile of TKE for the 81 cases (circles). The solid line represents the exponential function [(10)]. The crosses represent the mean aliasing variance profile of simulated wave orbital motions Δwave for the 81 cases. The dotted lines show 95% confidence intervals; z is the distance measured downward from the mean water line z = 0.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Mean depth profile of TKE for the 81 cases (circles). The solid line represents the exponential function [(10)]. The crosses represent the mean aliasing variance profile of simulated wave orbital motions Δwave for the 81 cases. The dotted lines show 95% confidence intervals; z is the distance measured downward from the mean water line z = 0.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
4. The e-folding length and bubble depth
The occurrence, depth, and intensity of bubble clouds are closely related to the surface wind and wave conditions and turbulent motions including Langmuir circulation in the surface layer. The temporal and spatial evolutions of entrained bubbles arise from a combination of turbulence and advection associated with breaking waves and Langmuir circulation (Gemmrich and Farmer 1999; Thorpe et al. 2003a). To illustrate the bubble cloud temporal and spatial variations under moderate to high winds, time–depth acoustic backscatter intensity profiles for four selected cases with surface winds u5 of 8.9, 13.7, 16, and 19.6 m s−1 are shown in Fig. 9. The corresponding significant wave heights are 4.1, 3.6, 6.0, and 8.7 m, respectively. As winds increased, the bubble cloud coverage expanded and increased intensity and penetration depth (Fig. 9). The intensity of backscatter decays rapidly with depth as illustrated in the time-averaged profiles 〈Mυ〉 in Figs. 10a and 10b. The bubble statistics (such as e-folding length and bubble depth as discussed below) depend on the averaging time. We examined the impact of averaging time scales on the bubble statistics of 〈Mυ〉 by using the averaging time intervals from 5 to 900 s from the 60-min backscatter profile data. We noted large variabilities for the time scales less than 30 s. However, the statistics become independent of the averaging scale when the averaging scale is larger than 5 min. In the following analysis, the 10-min averaging scale was used, since it provides quasi-steady estimates of bubble statistics.
The 10-min time history of acoustic backscatter depth profiles at winds u5 of (a) 8.9, (b) 13.7, (c) 16, and (d) 19.6 m s−1. Corresponding wave heights Hs are 4.1, 3.6, 6.0, and 8.7 m, respectively. They are for cases 26, 29, 16, and 70, respectively (Table 2). The color shading is 10 log10(Mυ) (dB), and zη is the distance measured downward from the free wave surface η.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
The 10-min time history of acoustic backscatter depth profiles at winds u5 of (a) 8.9, (b) 13.7, (c) 16, and (d) 19.6 m s−1. Corresponding wave heights Hs are 4.1, 3.6, 6.0, and 8.7 m, respectively. They are for cases 26, 29, 16, and 70, respectively (Table 2). The color shading is 10 log10(Mυ) (dB), and zη is the distance measured downward from the free wave surface η.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
The 10-min time history of acoustic backscatter depth profiles at winds u5 of (a) 8.9, (b) 13.7, (c) 16, and (d) 19.6 m s−1. Corresponding wave heights Hs are 4.1, 3.6, 6.0, and 8.7 m, respectively. They are for cases 26, 29, 16, and 70, respectively (Table 2). The color shading is 10 log10(Mυ) (dB), and zη is the distance measured downward from the free wave surface η.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Profiles of 10-min-averaged backscatter 〈Mυ〉 at the four wind speeds (see Fig. 9) in (a) a wave-following coordinate zη and (b) zη/Hs, scaled by significant wave height. The e-folding length is calculated for the depth profile 〈Mυ〉 between zη/Hs = 1 and 2 as marked by the two horizontal lines in (b). The vertical line in (b) represents the empirical backscatter threshold (−50 dB) for identifying the bubble depth.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Profiles of 10-min-averaged backscatter 〈Mυ〉 at the four wind speeds (see Fig. 9) in (a) a wave-following coordinate zη and (b) zη/Hs, scaled by significant wave height. The e-folding length is calculated for the depth profile 〈Mυ〉 between zη/Hs = 1 and 2 as marked by the two horizontal lines in (b). The vertical line in (b) represents the empirical backscatter threshold (−50 dB) for identifying the bubble depth.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Profiles of 10-min-averaged backscatter 〈Mυ〉 at the four wind speeds (see Fig. 9) in (a) a wave-following coordinate zη and (b) zη/Hs, scaled by significant wave height. The e-folding length is calculated for the depth profile 〈Mυ〉 between zη/Hs = 1 and 2 as marked by the two horizontal lines in (b). The vertical line in (b) represents the empirical backscatter threshold (−50 dB) for identifying the bubble depth.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Vertical distributions of breaking wave bubbles decay exponentially with depth (e.g., Thorpe 1984, 1986), and the resulting vertical distribution of acoustic backscatter is approximated by an exponentially decaying function 
Time series of (a) buoy wind speed u5 and (b) bubble depth Db (solid circles) and e-folding length λe (triangles).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time series of (a) buoy wind speed u5 and (b) bubble depth Db (solid circles) and e-folding length λe (triangles).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time series of (a) buoy wind speed u5 and (b) bubble depth Db (solid circles) and e-folding length λe (triangles).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Wind speed u5 vs (a) bubble depth Db and (b) e-folding length λe. The thick dashed lines represent the empirical linear relationships by Vagle et al. (2010). The solid circles are u5 bin-averaged Db and λe. The length of the error bars represents two standard deviations.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Wind speed u5 vs (a) bubble depth Db and (b) e-folding length λe. The thick dashed lines represent the empirical linear relationships by Vagle et al. (2010). The solid circles are u5 bin-averaged Db and λe. The length of the error bars represents two standard deviations.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Wind speed u5 vs (a) bubble depth Db and (b) e-folding length λe. The thick dashed lines represent the empirical linear relationships by Vagle et al. (2010). The solid circles are u5 bin-averaged Db and λe. The length of the error bars represents two standard deviations.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Profiles of 10-min-averaged backscatter 〈Mυ〉 for wind speed u5 ~ 14 m s−1. Case numbers are listed in Table 2, and the ratios of bubble depth to significant wave height Db/Hs are 2.9, 2.4, 2.7, 2.5, 4.3, and 2.1, respectively. The vertical dashed line represents the empirical backscatter threshold (−50 dB) used for identifying bubble depth.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Profiles of 10-min-averaged backscatter 〈Mυ〉 for wind speed u5 ~ 14 m s−1. Case numbers are listed in Table 2, and the ratios of bubble depth to significant wave height Db/Hs are 2.9, 2.4, 2.7, 2.5, 4.3, and 2.1, respectively. The vertical dashed line represents the empirical backscatter threshold (−50 dB) used for identifying bubble depth.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Profiles of 10-min-averaged backscatter 〈Mυ〉 for wind speed u5 ~ 14 m s−1. Case numbers are listed in Table 2, and the ratios of bubble depth to significant wave height Db/Hs are 2.9, 2.4, 2.7, 2.5, 4.3, and 2.1, respectively. The vertical dashed line represents the empirical backscatter threshold (−50 dB) used for identifying bubble depth.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
(a) Bubble depth scaled by wave height Db/Hs vs wave age Cp/u*a. The circles are data from the BWE experiment. The crosses are from Thorpe (1986), and u10 = 28u*a is used for replotting Thorpe’s data. (b) Bubble depth scaled by wavelength Db/Lp vs wave age Cp/u*a. Here, Lp is computed from the peak wave period Tp, using the linear dispersion relation. Bubble depth data with wind speeds less than 6 m s−1 are excluded. The solid circles are Cp/u*a bin-averaged Db/Hs and Db/Lp. The length of the error bars represents two standard deviations.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
(a) Bubble depth scaled by wave height Db/Hs vs wave age Cp/u*a. The circles are data from the BWE experiment. The crosses are from Thorpe (1986), and u10 = 28u*a is used for replotting Thorpe’s data. (b) Bubble depth scaled by wavelength Db/Lp vs wave age Cp/u*a. Here, Lp is computed from the peak wave period Tp, using the linear dispersion relation. Bubble depth data with wind speeds less than 6 m s−1 are excluded. The solid circles are Cp/u*a bin-averaged Db/Hs and Db/Lp. The length of the error bars represents two standard deviations.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
(a) Bubble depth scaled by wave height Db/Hs vs wave age Cp/u*a. The circles are data from the BWE experiment. The crosses are from Thorpe (1986), and u10 = 28u*a is used for replotting Thorpe’s data. (b) Bubble depth scaled by wavelength Db/Lp vs wave age Cp/u*a. Here, Lp is computed from the peak wave period Tp, using the linear dispersion relation. Bubble depth data with wind speeds less than 6 m s−1 are excluded. The solid circles are Cp/u*a bin-averaged Db/Hs and Db/Lp. The length of the error bars represents two standard deviations.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Downward movement of bubble clouds is driven by the convergence of Langmuir cells (e.g., Zedel and Farmer 1991; Plueddemann et al. 1996). These convergence zones have narrow spatial scales, which in turn require high spatial and temporal resolution velocity data to identify Langmuir cells. The 2-min-averaged ADCP velocities are not sufficient to determine Langmuir cell structure. Since bubble plumes are driven by Langmuir cells, their effects on backscattering distribution are explicitly included in the estimated e-folding and bubble depths from 10-min-averaged profiles of 〈Mυ〉.
5. Estimating turbulent diffusivity coefficient
Both estimates of Kυe [(2)] and KυD [(4)] computed during the December storm events were comparable and closely followed the wind speed (Fig. 15). The Kυe and KυD varied between 0.01 and 0.5 m2 s−1 and are highly correlated (Fig. 16). Figure 17 shows the relationship between eddy diffusion coefficients and surface wind speeds. For comparison, eddy diffusion coefficients estimated by Thorpe (1984) from e-folding lengths and by Dahl and Jessup (1995) from bubble depths were also included in Fig. 17. Their observations were limited to wind speeds less than 10 m s−1. The general trend of increasing eddy diffusion coefficients with increasing winds can be found from all data sources for wind speeds up to 22 m s−1. As shown in Fig. 17, our eddy diffusivities are higher than those of Thorpe (1984) and lower than those of Dahl and Jessup (1995) for winds ranging from 6 to 10 m s−1. Our results are similar to the empirical relationship of eddy viscosity and wind speed (Neumann 1952; Neumann and Pierson 1966).
Time series of (a) buoy wind speed u5 (b) eddy diffusion coefficients: Kυe [(2)] (triangles), KυD [(4)] (circles), and Kυu* (dashed line) based on wind speed and wave age [(7)].
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time series of (a) buoy wind speed u5 (b) eddy diffusion coefficients: Kυe [(2)] (triangles), KυD [(4)] (circles), and Kυu* (dashed line) based on wind speed and wave age [(7)].
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time series of (a) buoy wind speed u5 (b) eddy diffusion coefficients: Kυe [(2)] (triangles), KυD [(4)] (circles), and Kυu* (dashed line) based on wind speed and wave age [(7)].
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Eddy diffusion coefficients Kυe vs KυD. The thick dashed line represents the ratio 1:1.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Eddy diffusion coefficients Kυe vs KυD. The thick dashed line represents the ratio 1:1.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Eddy diffusion coefficients Kυe vs KυD. The thick dashed line represents the ratio 1:1.
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Wind speed u5 vs bin-averaged eddy diffusion coefficients KυD (solid circles) and Kυe (triangles). The length of the error bars represents two standard deviations of 〈KυD〉 (solid line) and 〈Kυe〉 (dashed line). Eddy diffusion data from Thorpe (1984) and Dahl and Jessup (1995) are represented, respectively, by crosses and asterisks. The gray line represents the empirical relationship of the eddy viscosity and wind speed (Neumann 1952).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Wind speed u5 vs bin-averaged eddy diffusion coefficients KυD (solid circles) and Kυe (triangles). The length of the error bars represents two standard deviations of 〈KυD〉 (solid line) and 〈Kυe〉 (dashed line). Eddy diffusion data from Thorpe (1984) and Dahl and Jessup (1995) are represented, respectively, by crosses and asterisks. The gray line represents the empirical relationship of the eddy viscosity and wind speed (Neumann 1952).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Wind speed u5 vs bin-averaged eddy diffusion coefficients KυD (solid circles) and Kυe (triangles). The length of the error bars represents two standard deviations of 〈KυD〉 (solid line) and 〈Kυe〉 (dashed line). Eddy diffusion data from Thorpe (1984) and Dahl and Jessup (1995) are represented, respectively, by crosses and asterisks. The gray line represents the empirical relationship of the eddy viscosity and wind speed (Neumann 1952).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
6. Discussion
a. Parameterizing the eddy diffusion coefficient
Cubic of waterside friction velocity (u*w)3 vs bin-averaged eddy diffusion coefficients 〈KυD〉 (solid circles) and 〈Kυe〉 (triangles). The solid and dashed lines represent the eddy diffusivity coefficient predicted by (7) for young (δ = 15) and developed (δ = 70) sea states, respectively. The length of the error bars represents two standard deviations of 〈KυD〉 (solid line) and 〈Kυe〉 (dashed line).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Cubic of waterside friction velocity (u*w)3 vs bin-averaged eddy diffusion coefficients 〈KυD〉 (solid circles) and 〈Kυe〉 (triangles). The solid and dashed lines represent the eddy diffusivity coefficient predicted by (7) for young (δ = 15) and developed (δ = 70) sea states, respectively. The length of the error bars represents two standard deviations of 〈KυD〉 (solid line) and 〈Kυe〉 (dashed line).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Cubic of waterside friction velocity (u*w)3 vs bin-averaged eddy diffusion coefficients 〈KυD〉 (solid circles) and 〈Kυe〉 (triangles). The solid and dashed lines represent the eddy diffusivity coefficient predicted by (7) for young (δ = 15) and developed (δ = 70) sea states, respectively. The length of the error bars represents two standard deviations of 〈KυD〉 (solid line) and 〈Kυe〉 (dashed line).
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
b. Estimation of turbulence diffusion flux and shear production
Time history of (a) u*w waterside friction velocity and (b) depth-averaged vertical energy diffusion 
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time history of (a) u*w waterside friction velocity and (b) depth-averaged vertical energy diffusion
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
Time history of (a) u*w waterside friction velocity and (b) depth-averaged vertical energy diffusion
Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1
The mean values and 95% confidence interval (CI) of turbulent diffusion, shear production, and energy dissipation rates averaged over 10- to 30-m depth during the December storms (W kg−1).
7. Summary and conclusions
Acoustic backscatter intensity, hydrographic and velocity fields, surface wave statistics, and meteorological measurements were used to quantify the magnitude and temporal variability of near-surface turbulent diffusivities in the coastal waters off Kayak Island, Alaska, as part of the NRL’s Breaking-Wave Effects under High Winds (BWE) program. The analysis was focused on estimating vertical eddy diffusivity from time-averaged bubble acoustic backscatter profiles in a wave-following coordinate. We used a subset of acoustic data collected with an echosounder at 208 kHz for a period of 12 days in late December 2012. During the 12-day observational period, the strong and persistent easterly winds reached 22 m s−1 (Figs. 3a,b), with an average speed of about 12 m s−1, and generated large surface waves with a maximum significant wave height of about 9 m and an average height of 4.5 m (Fig. 3c). The sea surface temperature remained steady at about 6°C, and the air temperature increased from −7° to 4°C. The sea surface was experiencing cooling, and, on average, the net surface cooling was about 50 W m−2 (Fig. 4). The 81 10-min segments of acoustic backscatter profiles were analyzed. We used the observed vertical distribution of acoustic backscattering intensity from an upward-looking high-frequency echosounder to estimate near-surface turbulent diffusivity during high-wind conditions. The major observational findings and inferences of this study are as follows:
Bubble backscattering strength decayed exponentially with depth (Fig. 10). The e-folding length λe, estimated between one and two wave heights below the wave surface, varied from about 0.6 to 6 m and was highly correlated with wind speed (Fig. 8). Similar wind speed dependence was found in the bubble penetration depth or the bubble depth Db, where Db was defined as the depth at which the backscatter intensity drops to an empirical threshold of −50 dB (Dahl and Jessup 1995; Trevorrow 2003). The bubble depth varied from about 3 to 30 m. The bubble depth scaled by the wave height (Db/Hs) varied between 0.5 and 5 (Fig. 14a).
The turbulent diffusivity in a bubble cloud layer was estimated from acoustic backscatter data by following Thorpe (1984). Here, microscale bubble clouds were treated as a scalar field and the eddy diffusion coefficient Kυe was obtained by solving the one-dimensional, steady-state, advection–diffusion equation [(2)]. The turbulent diffusivity KυD estimated from the bubble depth based on scaling arguments [(4)] is consistent with the diffusivity Kυe and varied between 0.01 and 0.5 m2 s−1 (Fig. 15). Both Kυe and KυD are closely correlated with surface wind speeds between 5 and 22 m s−1 (Fig. 17).
In the bubble layer, the shear production of TKE at a depth below one significant wave height was a similar order of magnitude as the dissipation rate predicted by the wall boundary layer theory.
The turbulent diffusivity in the bubble layer (beyond about a depth of one significant wave height from the surface) can be parameterized as a function of the cube of the wind friction velocity with a proportionality coefficient that depends weakly on wave age (Fig. 18).
Acknowledgments
This work was sponsored by the Office of Naval Research in a Naval Research Laboratory (NRL) project referred to as Breaking-Wave Effects under High Winds (BWE). We thank the assistance provided by U. S. Coast Guard. We thank Mark Hulbert, Steve Sova, Andrew Quaid, and Justin Brodersen for their technical support. We also thank the captain, crew, and marine technicians of the R/V Oceanus and the crew of Sound Pacer for their assistance. We thank the two anonymous reviewers for their thorough and careful review of the manuscript and useful comments.
APPENDIX
Estimating Wave Orbital Velocity Aliasing
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