## 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 *μ*m 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 *μ*m at the water surface and approximately 27 *μ*m at 20-m depth (Thorpe 1992; Vagle and Farmer 1992).

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, *M*_{υ} (m^{−1}), by the use of the sonar equation (BioSonics 2004); *M*_{υ} is often expressed in decibels (10 log_{10}*M*_{υ}). Figure 2a shows 10-min samples of acoustic backscatter data; here, the depth axis *z*_{d} 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 *M*_{υ}(*z*_{d}) from the fixed bottom-mounted echosounder is remapped to a wave-following coordinate system *M*_{υ}(*z*_{η}), where *z*_{η} 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 *z*_{d} distance measured upward from the echosounder and (b) wave-following coordinate *z*_{η} distance measured downward from the wave surface *η*, shown as the solid line in (a). The color shading is the backscatter intensity 10 log_{10}(*M*_{υ}) (dB). The wind speed and wave height are *u*_{5} = 21.1 m s^{−1} and *H*_{s} = 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 *z*_{d} distance measured upward from the echosounder and (b) wave-following coordinate *z*_{η} distance measured downward from the wave surface *η*, shown as the solid line in (a). The color shading is the backscatter intensity 10 log_{10}(*M*_{υ}) (dB). The wind speed and wave height are *u*_{5} = 21.1 m s^{−1} and *H*_{s} = 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 *z*_{d} distance measured upward from the echosounder and (b) wave-following coordinate *z*_{η} distance measured downward from the wave surface *η*, shown as the solid line in (a). The color shading is the backscatter intensity 10 log_{10}(*M*_{υ}) (dB). The wind speed and wave height are *u*_{5} = 21.1 m s^{−1} and *H*_{s} = 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 *u*_{5}, buoy wind direction *θ*_{u}, significant wave height *H*_{s}, peak wave period *T*_{p}, surface wind stress *τ*_{u}, surface air temperature from the NRL weather station *T*_{air}, SST from S1, net heat flux *Q*_{net}, bubble depth *D*_{b}, *e*-folding length *λ*_{e}, eddy diffusion coefficient based on *D*_{b} [(4)] *K*_{υD}, eddy diffusion coefficient from *λ*_{e} [(2)] *K*_{υe}, eddy diffusion coefficient from friction velocity [(7)], *K*_{υu*}, and *λ*_{e} and *K*_{υe} are not available when *D*_{b} < *H*_{s} (marked as *).

The surface wave conditions are described in terms of the significant wave height *H*_{s} and the peak wave period *T*_{p}, where *S*_{η}(*f*) is the wave frequency spectrum, and *f*_{1} = 0.05 Hz and *f*_{2} = 0.5 Hz are the lower and upper cutoff–frequency limits for the integration, respectively. The wave frequency spectrum *S*_{η} is computed from 10-min time series of surface displacement *η* extracted from the backscatter profile data (Fig. 2a), and *T*_{p} (peak frequency *f*_{p}) is the period of the spectral peak (Table 2). Ocean surface wave conditions were also measured at the Lander by a RDI acoustic Doppler current profiler (ADCP) equipped with a wave array system (Strong et al. 2000) and at a nearby NDBC buoy (ID 46082). The wave data from the Lander’s ADCP were collected every 2 h, and the buoy wave data were collected hourly. Measurements of wave heights and periods from the Lander mounted echosounder and ADCP (not shown here) and the NDBC buoy are similar (Figs. 3c,d).

Time history of (a) wind speed *u*_{5} at 5-m height, (b) wind direction *θ*_{u} at NDBC buoy 46082, (c) significant wave height *H*_{s} at the Lander (solid circles) and at the buoy (dashed line), (d) peak wave period *T*_{p} at the Lander (solid triangles) and at the buoy (dashed line), and (e) surface Stokes drift *U*_{s0} at the Lander.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1

Time history of (a) wind speed *u*_{5} at 5-m height, (b) wind direction *θ*_{u} at NDBC buoy 46082, (c) significant wave height *H*_{s} at the Lander (solid circles) and at the buoy (dashed line), (d) peak wave period *T*_{p} at the Lander (solid triangles) and at the buoy (dashed line), and (e) surface Stokes drift *U*_{s0} at the Lander.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1

Time history of (a) wind speed *u*_{5} at 5-m height, (b) wind direction *θ*_{u} at NDBC buoy 46082, (c) significant wave height *H*_{s} at the Lander (solid circles) and at the buoy (dashed line), (d) peak wave period *T*_{p} at the Lander (solid triangles) and at the buoy (dashed line), and (e) surface Stokes drift *U*_{s0} at the Lander.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1

The surface Stokes drift is computed from *S*_{η} as *ω* = 2*πf*, *k* is the wavenumber, *ω*^{2} = *gk* tanh(*kh*), *g* is the gravitational acceleration, *h* is the water depth, and *G*(*kh*) = cosh(2*kh*)/sinh^{2}(*kh*) is a dimensionless factor accounting for the wave shoaling effect at *h*. Here, *ω*_{1} and *ω*_{2} are the lower and upper cutoff–frequency limits for the integration, respectively. The estimated surface Stokes drifts (Fig. 3e) vary between 0.05 and 0.5 m s^{−1} and are closely related to the variation of surface winds. Surface meteorological conditions were measured at the NDBC buoy and at a land-based NRL weather station installed for the experiment on Kayak Island. The buoy winds were measured by a R. M. Young anemometer at 5 m above the sea surface. More details about the NDBC wave and meteorological sensors can be found in Steele and Mettlach (1993). The NRL land station measured wind speed and direction, air temperature, relative humidity, solar irradiance (downwelling shortwave radiation), and barometric pressure. The anemometer was approximately 8 m above the ground while the rest of the sensors were about 4 m above the ground with the exception of the barometer, which was installed in the control panel mounted 2 m above the ground. Measurements from the buoy and the NRL land station are used to estimate surface wind stress and net heat flux *Q*_{net} from the bulk flux algorithms (Fairall et al. 1996).

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 *τ*_{w}, net surface heat flux *Q*_{net} (Wm^{−2}), and Monin–Obukhov length *L*_{mo} (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 La_{t} = (*u*_{*w}/*U*_{s0})^{0.5} was also estimated, where the friction velocity *u*_{*w} is (*τ*_{w}/*ρ*_{w})^{0.5} and *ρ*_{w} is the density of water. The estimated La_{t} 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 *τ*_{w}, (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 *Q*_{net}, where negative *Q*_{net} represents upward heat flux, (d) Monin–Obukhov length *L*_{mo}, and (e) turbulent Langmuir number La_{t} at the Lander.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1

Time history of (a) surface wind stress *τ*_{w}, (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 *Q*_{net}, where negative *Q*_{net} represents upward heat flux, (d) Monin–Obukhov length *L*_{mo}, and (e) turbulent Langmuir number La_{t} at the Lander.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1

Time history of (a) surface wind stress *τ*_{w}, (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 *Q*_{net}, where negative *Q*_{net} represents upward heat flux, (d) Monin–Obukhov length *L*_{mo}, and (e) turbulent Langmuir number La_{t} 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 *T*_{θ} (°C), (e) salinity *S* (psu), and (f) potential density anomaly *σ*_{t} (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 *T*_{θ} (°C), (e) salinity *S* (psu), and (f) potential density anomaly *σ*_{t} (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 *T*_{θ} (°C), (e) salinity *S* (psu), and (f) potential density anomaly *σ*_{t} (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* = *U*_{L} + *u*′, *V* = *V*_{L} + *υ*′, and *W* = *W*_{L} + *w*′, where *U*_{L}, *V*_{L}, and *W*_{L} 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, *q*^{2}/2, where *q*^{2} = 〈*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 10^{2} cm^{2} 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 cm^{2} 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 log_{10}(TKE) (cm^{2} 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 log_{10}(TKE) (cm^{2} 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 log_{10}(TKE) (cm^{2} 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 Δ_{wave} 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 Δ_{wave} 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 Δ_{wave} 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 *u*_{5} 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 〈*M*_{υ}〉 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 〈*M*_{υ}〉 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 *u*_{5} of (a) 8.9, (b) 13.7, (c) 16, and (d) 19.6 m s^{−1}. Corresponding wave heights *H*_{s} 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 log_{10}(*M*_{υ}) (dB), and *z*_{η} 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 *u*_{5} of (a) 8.9, (b) 13.7, (c) 16, and (d) 19.6 m s^{−1}. Corresponding wave heights *H*_{s} 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 log_{10}(*M*_{υ}) (dB), and *z*_{η} 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 *u*_{5} of (a) 8.9, (b) 13.7, (c) 16, and (d) 19.6 m s^{−1}. Corresponding wave heights *H*_{s} 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 log_{10}(*M*_{υ}) (dB), and *z*_{η} 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 〈*M*_{υ}〉 at the four wind speeds (see Fig. 9) in (a) a wave-following coordinate *z*_{η} and (b) *z*_{η}/*H*_{s}, scaled by significant wave height. The *e*-folding length is calculated for the depth profile 〈*M*_{υ}〉 between *z*_{η}/*H*_{s} = 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 〈*M*_{υ}〉 at the four wind speeds (see Fig. 9) in (a) a wave-following coordinate *z*_{η} and (b) *z*_{η}/*H*_{s}, scaled by significant wave height. The *e*-folding length is calculated for the depth profile 〈*M*_{υ}〉 between *z*_{η}/*H*_{s} = 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 〈*M*_{υ}〉 at the four wind speeds (see Fig. 9) in (a) a wave-following coordinate *z*_{η} and (b) *z*_{η}/*H*_{s}, scaled by significant wave height. The *e*-folding length is calculated for the depth profile 〈*M*_{υ}〉 between *z*_{η}/*H*_{s} = 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 *M*_{υ}(0) is the backscatter at the surface and *λ*_{e} is the *e*-folding length that describes the decay rate; *z*_{η} is the distance measured downward from the wave surface *z*_{η} = 0. Here, we estimated the *e*-folding length by fitting an exponential profile to 〈*M*_{υ}〉 between *z*_{η} = *H*_{s} and *z*_{η} = 2*H*_{s} (indicated by the horizontal dotted lines in Fig. 10b). In addition to the *e*-folding length, we estimated the bubble penetration depth or the bubble depth *D*_{b}, defined as the depth where 〈*M*_{υ}〉 decreases to an empirically determined threshold value of −50 dB. Similar techniques have been used to empirically determine bubble cloud depths with threshold levels of −50 (Dahl and Jessup 1995; Trevorrow 2003) and −60 dB (Thorpe 1986). We noted that the noise levels of some profiles were close to −60 dB (Fig. 10a), but the noise levels of many profiles were closer to −50 dB. As indicated in Fig. 10b, the bubble depth is about 2 to 4 times the wave height. Figure 11b shows the time variation of estimated bubble depths and *e*-folding lengths over the 12-day period. The close relationships between wind speed and bubble depth and *e*-folding length are shown in Fig. 12. For winds less than 10 m s^{−1}, the relationships between wind and bubble depth and *e*-folding length are approximately linear and are similar to those by Vagle et al. (2010; Fig. 12). For winds higher than 10 m s^{−1}, the relationships between wind speed and bubble depth and *e*-folding length became nonlinear and are much higher than those by Vagle et al. (2010). It is noted that bubble depth and *e*-folding length in Vagle et al. (2010) were derived from acoustic bubble backscatter data averaged over 2.25 h, while the current observations were based on 10-min-averaged backscatter profiles.

Time series of (a) buoy wind speed *u*_{5} and (b) bubble depth *D*_{b} (solid circles) and *e*-folding length *λ*_{e} (triangles).

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1

Time series of (a) buoy wind speed *u*_{5} and (b) bubble depth *D*_{b} (solid circles) and *e*-folding length *λ*_{e} (triangles).

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1

Time series of (a) buoy wind speed *u*_{5} and (b) bubble depth *D*_{b} (solid circles) and *e*-folding length *λ*_{e} (triangles).

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1

Wind speed *u*_{5} vs (a) bubble depth *D*_{b} and (b) *e*-folding length *λ*_{e}. The thick dashed lines represent the empirical linear relationships by Vagle et al. (2010). The solid circles are *u*_{5} bin-averaged *D*_{b} and *λ*_{e}. 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 *u*_{5} vs (a) bubble depth *D*_{b} and (b) *e*-folding length *λ*_{e}. The thick dashed lines represent the empirical linear relationships by Vagle et al. (2010). The solid circles are *u*_{5} bin-averaged *D*_{b} and *λ*_{e}. 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 *u*_{5} vs (a) bubble depth *D*_{b} and (b) *e*-folding length *λ*_{e}. The thick dashed lines represent the empirical linear relationships by Vagle et al. (2010). The solid circles are *u*_{5} bin-averaged *D*_{b} and *λ*_{e}. The length of the error bars represents two standard deviations.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0164.1

*D*

_{b}is a function of wind speed, but there is a noticeable variation in

*D*

_{b}for a given wind speed band. Therefore, the variability in

*D*

_{b}is further examined. Figure 13 shows six backscatter profiles for wind speeds

*u*

_{5}that only slightly varied between 13.9 and 14.1 m s

^{−1}, while

*D*

_{b}/

*H*

_{s}varied from 2.1 to 4.3 (Table 2). Thorpe (1992) showed that during the early stages of sea state development surface wave breaking is dominated by plunging breakers with larger

*D*

_{b}/

*H*

_{s}. As waves continue to develop, spilling breakers dominate the breaking processes with small

*D*

_{b}/

*H*

_{s}. The wave development stage is represented by wave age

*δ*=

*C*

_{p}/

*u*

_{*a}, where

*C*

_{p}is the wave phase speed associated with peak wave period and

*u*

_{*a}is the friction velocity in the atmospheric boundary layer. The relationship between bubble depth

*D*

_{b}/

*H*

_{s}and wave age

*δ*for wind speeds greater than 6 m s

^{−1}is shown in Fig. 14a and is consistent with observations reported by Thorpe (1986, his Fig. 5). Both Thorpe (1986) and present observations show a similar trend of decreasing

*D*

_{b}/

*H*

_{s}with increasing wave age (Fig. 14), and the relationship between

*D*

_{b}/

*H*

_{s}and

*δ*can be approximated as

*β*≈ 70 was determined empirically from the wave age bin-averaged data between 15 ≤

*δ*≤ 70. A similar trend was observed between wave age and bubble depth scaled by wavelength (

*D*

_{b}/

*L*

_{p}; Fig. 14b), where

*L*

_{p}is the wavelength of the peak wave period

*T*

_{p}based on the dispersion relation for surface waves. This similarity is expected because wave heights and lengths of wind waves are closely related (Toba 1978).

Profiles of 10-min-averaged backscatter 〈*M*_{υ}〉 for wind speed *u*_{5} ~ 14 m s^{−1}. Case numbers are listed in Table 2, and the ratios of bubble depth to significant wave height *D*_{b}/*H*_{s} 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 〈*M*_{υ}〉 for wind speed *u*_{5} ~ 14 m s^{−1}. Case numbers are listed in Table 2, and the ratios of bubble depth to significant wave height *D*_{b}/*H*_{s} 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 〈*M*_{υ}〉 for wind speed *u*_{5} ~ 14 m s^{−1}. Case numbers are listed in Table 2, and the ratios of bubble depth to significant wave height *D*_{b}/*H*_{s} 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 *D*_{b}/*H*_{s} vs wave age *C*_{p}/*u*_{*a}. The circles are data from the BWE experiment. The crosses are from Thorpe (1986), and *u*_{10} = 28*u*_{*a} is used for replotting Thorpe’s data. (b) Bubble depth scaled by wavelength *D*_{b}/*L*_{p} vs wave age *C*_{p}/*u*_{*a}. Here, *L*_{p} is computed from the peak wave period *T*_{p}, using the linear dispersion relation. Bubble depth data with wind speeds less than 6 m s^{−1} are excluded. The solid circles are *C*_{p}/*u*_{*a} bin-averaged *D*_{b}/*H*_{s} and *D*_{b}/*L*_{p}. 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 *D*_{b}/*H*_{s} vs wave age *C*_{p}/*u*_{*a}. The circles are data from the BWE experiment. The crosses are from Thorpe (1986), and *u*_{10} = 28*u*_{*a} is used for replotting Thorpe’s data. (b) Bubble depth scaled by wavelength *D*_{b}/*L*_{p} vs wave age *C*_{p}/*u*_{*a}. Here, *L*_{p} is computed from the peak wave period *T*_{p}, using the linear dispersion relation. Bubble depth data with wind speeds less than 6 m s^{−1} are excluded. The solid circles are *C*_{p}/*u*_{*a} bin-averaged *D*_{b}/*H*_{s} and *D*_{b}/*L*_{p}. 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 *D*_{b}/*H*_{s} vs wave age *C*_{p}/*u*_{*a}. The circles are data from the BWE experiment. The crosses are from Thorpe (1986), and *u*_{10} = 28*u*_{*a} is used for replotting Thorpe’s data. (b) Bubble depth scaled by wavelength *D*_{b}/*L*_{p} vs wave age *C*_{p}/*u*_{*a}. Here, *L*_{p} is computed from the peak wave period *T*_{p}, using the linear dispersion relation. Bubble depth data with wind speeds less than 6 m s^{−1} are excluded. The solid circles are *C*_{p}/*u*_{*a} bin-averaged *D*_{b}/*H*_{s} and *D*_{b}/*L*_{p}. 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 〈*M*_{υ}〉.

## 5. Estimating turbulent diffusivity coefficient

*K*

_{υ}by solving the one-dimensional, steady-state, advection–diffusion equation representing the following processes: (i) turbulent transport of bubbles from the near-surface layer, (ii) buoyant bubbles rise at a constant speed, and (iii) bubble radius changes due to gas diffusion and compression from hydrostatic pressure. Thorpe (1984) proposed several formulations for estimating

*K*

_{υ}based on assumptions related to the vertical structure of the diffusivity coefficient, the rate of change of bubble radius, and the nature of the turbulent motions in the mixed layer. Here, for simplicity, the eddy diffusion coefficient is treated as a constant within the bubble layer, bubbles rise at a steady speed with a constant rate of change in radius, and the vertical distribution of bubbles is assumed to follow an exponentially decaying profile with depth. The resulting eddy diffusivity (which is also referred as turbulent diffusivity) is estimated from the

*e*-folding length

*λ*

_{e}of the bubble vertical acoustic profile (Thorpe 1984):

*λ*

_{e}is the

*e*-folding length of the bubble distribution,

*w*

_{b}is the rising speed of bubbles, and

*σ*is the rate of change of bubble radius due to diffusion of gases and compression under hydrostatic pressure. The determination of

*w*

_{b}and

*σ*requires the bubble size distribution function and the lifetime of bubbles. However, such measurements were not made during BWE observations. By following Thorpe (1984), we used

*w*

_{b}= 0.54 cm s

^{−1}, which corresponds to a bubble radius of 50

*μ*m at the peak of bubble size distributions as observed by Johnson and Cooke (1979). We used

*σ*= 0.018 s

^{−1}, which corresponds to the observed life time bubbles (~1 min) as observed in Thorpe and Hall (1983).

*V*

_{t}and a length scale

*l*

_{t}(e.g., Thorpe 1984):

*κ*is 0.41. Depth-dependent and depth-independent eddy diffusion coefficients can be derived from (3). The depth-dependent eddy diffusivity

*V*

_{t}is set to the friction velocity

*u*

_{*w}and

*l*

_{t}=

*z*, the vertical distance measured downward from the mean surface

*z*= 0. By choosing the velocity scale as

*u*

_{*w}and the vertical turbulent length scale as the bubble depth

*D*

_{b}, the depth-independent eddy diffusivity can be expressed as

Both estimates of *K*_{υe} [(2)] and *K*_{υD} [(4)] computed during the December storm events were comparable and closely followed the wind speed (Fig. 15). The *K*_{υe} and *K*_{υD} varied between 0.01 and 0.5 m^{2} 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 *u*_{5} (b) eddy diffusion coefficients: *K*_{υe} [(2)] (triangles), *K*_{υD} [(4)] (circles), and *K*_{υu*} (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 *u*_{5} (b) eddy diffusion coefficients: *K*_{υe} [(2)] (triangles), *K*_{υD} [(4)] (circles), and *K*_{υu*} (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 *u*_{5} (b) eddy diffusion coefficients: *K*_{υe} [(2)] (triangles), *K*_{υD} [(4)] (circles), and *K*_{υu*} (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 *K*_{υe} vs *K*_{υD}. 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 *K*_{υe} vs *K*_{υD}. 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 *K*_{υe} vs *K*_{υD}. 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 *u*_{5} vs bin-averaged eddy diffusion coefficients *K*_{υD} (solid circles) and *K*_{υe} (triangles). The length of the error bars represents two standard deviations of 〈*K*_{υD}〉 (solid line) and 〈*K*_{υe}〉 (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 *u*_{5} vs bin-averaged eddy diffusion coefficients *K*_{υD} (solid circles) and *K*_{υe} (triangles). The length of the error bars represents two standard deviations of 〈*K*_{υD}〉 (solid line) and 〈*K*_{υe}〉 (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 *u*_{5} vs bin-averaged eddy diffusion coefficients *K*_{υD} (solid circles) and *K*_{υe} (triangles). The length of the error bars represents two standard deviations of 〈*K*_{υD}〉 (solid line) and 〈*K*_{υe}〉 (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

*E*is a function of air friction velocity

*u*

_{*a}and wave age

*δ*, and the resulting functional form is

*E*=

*H*

_{s}

^{2}/16, the gravitational acceleration

*g*= 9.81 m s

^{−2}, and

*B*= 0.051 (Toba 1978). The air friction velocity is computed from wind stress

*τ*

_{w}=

*ρ*

_{a}(

*u*

_{*a})

^{2}=

*ρ*

_{w}(

*u*

_{*w})

^{2}, where

*ρ*

_{w}and

*ρ*

_{a}are water and air densities, respectively. By combining (5) and (6), we can express the vertical eddy diffusivity as a function of friction velocity and wave age by

*α*= 4

*kβB*

^{0.5}(

*ρ*

_{w}/

*ρ*

_{a}). For

*β*= 70 [(1)],

*B*= 0.051 [(6)], and

*ρ*

_{w}/

*ρ*

_{a}≈ 800,

*α*has a value of about 2 × 10

^{4}. Equation (7) shows that the eddy diffusivity is proportional to the cubic power of the friction wind velocity and square root of wave age. Given that

*δ*varies between 15 and 70 (Fig. 14), the impact of wave age on the variation of eddy diffusivity for the range between 0.01 and 0.4 m

^{2}s

^{−1}is within about a factor of 2 (Fig. 18). Therefore, 〈

*K*

_{υe}〉 and 〈

*K*

_{υD}〉 are well predicted by (7) at moderate to high winds, and, on average, eddy diffusivities are proportional to

^{4})

*δ*

^{0.5}] weakly dependent on the wave age.

Cubic of waterside friction velocity (*u*_{*w})^{3} vs bin-averaged eddy diffusion coefficients 〈*K*_{υD}〉 (solid circles) and 〈*K*_{υe}〉 (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 〈*K*_{υD}〉 (solid line) and 〈*K*_{υe}〉 (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 〈*K*_{υD}〉 (solid circles) and 〈*K*_{υe}〉 (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 〈*K*_{υD}〉 (solid line) and 〈*K*_{υe}〉 (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 〈*K*_{υD}〉 (solid circles) and 〈*K*_{υe}〉 (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 〈*K*_{υD}〉 (solid line) and 〈*K*_{υe}〉 (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

*q*

^{2}/2)/∂

*t*is the time derivative of TKE, Adv is the advection of TKE by mean currents,

*P*

_{w}is the pressure–velocity correlation term,

*F*

_{t}is the vertical transport of TKE by turbulent motions,

*P*

_{s}is the shear production term, and

*ε*is the energy dissipation term. We did not have data to evaluate Adv and

*P*

_{w}. However, we suspect that

*P*

_{w}may not be small in the wave dominating boundary layer. The vertical transport of TKE by turbulent motions

*F*

_{t}is typically expressed in terms of the downgradient transport of diffusion energy flux (e.g., Mellor and Yamada 1982):

*K*

_{q}is the vertical eddy diffusivity of TKE (

*q*

^{2}/2). Here, the TKE is approximated from ADCP high-frequency velocity components (Figs. 7, 8), and

*K*

_{q}is estimated from bubble eddy diffusion coefficients (Fig. 15). As shown in Fig. 8, the mean profile of TKE below 10-m depth decays exponentially with depth and can be approximated as

*α*

_{t}(≈16) and

*D*

_{t}(≈30 m) are empirically determined to match the mean TKE profiles of the 81 records. By treating

*K*

_{q}as a constant in the bubble layer and by evaluating the second derivative of

*q*

^{2}from (10), we approximate the vertical diffusion of TKE:

*P*

_{s}is

*K*

_{m}is the eddy viscosity coefficient and the vertical shear of

*U*

_{L}and

*V*

_{L}are computed from low-pass velocity components from the ADCP data. Here,

*K*

_{υD}(5) is used for

*K*

_{m}and

*K*

_{q}= 0.41

*K*

_{m}(Wijesekera et al. 2003). In addition, the TKE dissipation rate predicted by the law of the wall scaling (Soloviev and Lukas 2014) is

*C*

_{e}= 0.5

*C*

_{p}, and

*C*

_{p}is wave phase speed of the dominant wave period

*T*

_{p}. The quantities

*F*

_{t},

*P*

_{s},

*ε*

_{L}, and

*ε*

_{T}were computed from (11) to (14) based on 10-min-averaged currents, winds, and eddy diffusivity estimates. Depth-averaged values

*z*= 10 m and

*z*= 30 m and are plotted as a function of time in Fig. 19b. The shear production term

*P*

_{s}is the same order magnitude as the dissipation rate

*ε*

_{L}based on the wall boundary layer scaling. The turbulent diffusion of TKE

*F*

_{t}is about one order of magnitude smaller than the shear production, and the dissipation rate predicted from Terray et al. (1996) is one order of magnitude larger than the shear production term. The time-mean values of

^{−7}, 2 × 10

^{−6}, 1.2 × 10

^{−6}, and 1.5 × 10

^{−5}W kg

^{−1}, respectively, for the 81 records (Table 3). Our results suggest that in the bubble layer, that at least between one and two wave heights below the wave surface, the shear production is similar to the turbulent dissipation represented by the wall boundary layer theory [(13)]. Our estimates are consistent with observations reported by Thorpe et al. (2003b

**)**, who noted that waves and Lc have insignificant impact on the turbulence beyond a depth of 1.2

*H*

_{s}.

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*λ*_{e}, 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*D*_{b}, where*D*_{b}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 (*D*_{b}/*H*_{s}) 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

*K*_{υe}was obtained by solving the one-dimensional, steady-state, advection–diffusion equation [(2)]. The turbulent diffusivity*K*_{υD}estimated from the bubble depth based on scaling arguments [(4)] is consistent with the diffusivity*K*_{υe}and varied between 0.01 and 0.5 m^{2}s^{−1}(Fig. 15). Both*K*_{υe}and*K*_{υD}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

*U*

_{wave}and

*W*

_{wave}, were simulated using wave height and wave period,

*H*

_{s}and

*T*

_{p}. They are expressed as (Dean and Dalrymple 1991)

*z*is the depth measured downward from the mean water line

*z*= 0,

*h*is the water depth,

*θ*is a random phase between −

*π*and

*π*,

*ω*

_{p}= 2

*π*/

*T*

_{p}, and the wavenumber

*k*

_{p}is determined from

*T*

_{p}using the dispersion relationship (Dean and Dalrymple 1991). The mean particle velocity components

*U*

_{wave}and

*W*

_{wave}sampled at 2.5-s intervals for 2 min, the same as that used for ADCP current components. The aliasing variance of wave orbital velocity due to random phase averaging is then computed as

*Δ*

_{wave}, averaged for the 81 cases, is shown in Fig. 8. The aliasing is insignificant, and the aliasing variance in general is at least one order of magnitude smaller than the TKE.

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