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    Environmental conditions at the observation site on 14–15 Nov 2006. (top) Wind speed at 10-m height, u10 (mean wind speed, solid line; minimum and maximum during 5-min intervals, dashed lines). (middle) Wind direction dd; (bottom) significant wave height Hs (line) and dominant wave period τp (dots).

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    Example of surface tracking (1047 PST 15 Nov 2006). (a) Acoustic backscatter strength of the vertical sonar, (b) binary image of the backscatter field, and (c) surface elevation record. Range given as distance from the sonar head.

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    Structure function D(s) at (a) 0.03, (b) 0.11, and (c) 0.37 m below the surface: raw data (open circles) and least squares fit (dashed line).

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    Mean dissipation 〈ϵ〉 of a 500-s record as a function of the number Nav of velocity differences (i.e., the averaging period) utilized in the structure function [Eq. (5)].

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    Turbulence field and acoustic backscatter in the wave-surface layer (1144 PST 15 Nov 2006). (a) TKE dissipation rate ε in the upper 0.5 m beneath waves. (b) Filtered TKE dissipation field as in (a) but subject to a five-point running average. Black triangles indicate actively breaking events closely upstream of the sonar, as observed in the video recordings. (c) Average dissipation rate in the top 0.1 m. (d) Average acoustic backscatter strength γ (relative units) in the top 0.1 m (solid line) and assumed threshold for breaking waves (dashed line). Data in (a)–(d) are based on a surface-following coordinate system; that is, z = 0 represents the instantaneous free surface. (e) Corresponding surface elevation η. The dashed line indicates the extent of the dissipation profile. (See section 3.)

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    Average TKE dissipation rate in the upper 0.1 m from the free surface as a function of wave scale, expressed as the phase speed c within (a) high bubble concentrations (γ > 2.85, indicative of breaking waves) and (b) low bubble concentrations (γ < 2.8, indicative of nonbreaking waves). The vertical bars indicate ± one standard deviation. The dashed lines represent the least squares fits with slope (a) 0.48 and (b) 0.19.

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    Surface elevation spectrum at 1700 PST 15 Nov 2006. The dashed line indicates ω−4.

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    Average dissipation within 40-min periods as function of mean wave saturation σb. (a) The total dissipation E and (b),(c) the average dissipation rate 〈ϵ〉. Black upward (gray downward) triangles are data taken beneath wave crest (trough) regions. (b) Data from the upper 0.1 m from the free surface. (c) Data from 0.4 to 0.5 m below the free surface.

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    Dissipation within a wave crest Ec and the following wave trough Et. Data are integrated values from the free surface over the vertical range of the crest amplitude: (a) distribution of the ratio Ec/Et and (b) bin averages. Error bars indicate ±1 standard deviation.

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    Average coefficient of depth dependence ε ∝ zm: black upward (gray downward) triangles are data taken beneath wave crest (wave trough) regions.

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    Normalized dissipation profile during period of (a) frequent wave breaking (1400–1700 PST 15 Nov 2006) and (b) intermittent wave breaking (0200–0500 PST 15 Nov 2006): black upward (gray downward) triangles are data taken beneath wave crest (wave trough) regions.

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    Normalized dissipation profile based on observations referenced to a flat surface, where zflat is the distance to the mean waterline, during period of (a) frequent wave breaking (1400–1700 PST 15 Nov 2006) and (b) intermittent wave breaking (0200–0500 PST 15 Nov 2006).

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    (a) Probability distribution of dissipation rates P(ϵ) in the upper 0.05 m based on a 1-h-long record and (b) bubble size distribution N(a) as function of bubble radius a [inferred from (a)]. The dashed lines depicts (right) a−10/3 and (left) a−3/2 dependences as previously observed in laboratory experiments (Deane and Stokes 2002).

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Strong Turbulence in the Wave Crest Region

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  • 1 University of Victoria, Victoria, British Columbia, Canada
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Abstract

High-resolution vertical velocity profiles in the surface layer of a lake reveal the turbulence structure beneath strongly forced waves. Dissipation rates of turbulence kinetic energy are estimated based on centered second-order structure functions at 4-Hz sampling. Dissipation rates within nonbreaking wave crests are on average 3 times larger than values found at the same distance to the free surface but within the wave trough region. This ratio increases to 18 times for periods with frequent wave breaking. The depth-integrated mean dissipation rate is a function of the wave field and correlates well with the mean wave saturation in the wave band ωpω ≤ 4ωp. It shows a clear threshold behavior in accordance with the onset of wave breaking. The initial bubble size distribution is estimated from the observed distribution of energy dissipation rates, assuming the Hinze scale being the limiting size. This model yields the slope of the size distribution, , consistent with laboratory results reported in the literature, and implies that bubble fragmentation associated with intermittent high dissipation rates is a valid mechanism for the setup of bubble size spectra.

Corresponding author address: Johannes Gemmrich, Dept. of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, BC V8W 3P6, Canada. Email: gemmrich@uvic.ca

Abstract

High-resolution vertical velocity profiles in the surface layer of a lake reveal the turbulence structure beneath strongly forced waves. Dissipation rates of turbulence kinetic energy are estimated based on centered second-order structure functions at 4-Hz sampling. Dissipation rates within nonbreaking wave crests are on average 3 times larger than values found at the same distance to the free surface but within the wave trough region. This ratio increases to 18 times for periods with frequent wave breaking. The depth-integrated mean dissipation rate is a function of the wave field and correlates well with the mean wave saturation in the wave band ωpω ≤ 4ωp. It shows a clear threshold behavior in accordance with the onset of wave breaking. The initial bubble size distribution is estimated from the observed distribution of energy dissipation rates, assuming the Hinze scale being the limiting size. This model yields the slope of the size distribution, , consistent with laboratory results reported in the literature, and implies that bubble fragmentation associated with intermittent high dissipation rates is a valid mechanism for the setup of bubble size spectra.

Corresponding author address: Johannes Gemmrich, Dept. of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, BC V8W 3P6, Canada. Email: gemmrich@uvic.ca

1. Introduction

Surface waves have a strong influence on the turbulence field in the uppermost layer of natural water bodies. Wave-induced upper ocean turbulence plays a significant role in many air–sea interaction processes. The transfer of energy, momentum, heat, and gases across the air–sea interface; the enhancement of latent heat fluxes; the generation of aerosol due to spray generation; and the dispersion of nutrients and pollutants are examples of processes affected by wave-induced turbulence (for references see, e.g., Thorpe 1995; Melville 1996; Duncan 2001; Garbe et al. 2007; Zappa et al. 2007). Furthermore, energy dissipation in breaking waves limits the wave growth, but its parameterization for spectral wave models is still a topic of great debate (e.g., Young and Babanin 2006).

A common reference level for turbulence in boundary layers is the constant stress layer scaling, also called the “law of the wall.” In that case, the dissipation rate of turbulence kinetic energy (TKE) is given as εwl = u*3(κz)−1, with friction velocity u*, von Kármán constant κ, and z the distance to the rigid interface.

At the air–water interface the energy input equals the stress τ = ρu*2 times an effective phase speed ceff of waves acquiring energy from the wind (Gemmrich et al. 1994; Terray et al. 1996) plus the input to the surface drift us,
i1520-0485-40-3-583-e1
with the second term generally being small.

In a fully developed wave field the total energy dissipation E = ∫ε dz balances the energy input EEin, and even for small values of ceff = O(1 m s−1) (Gemmrich et al. 1994; Terray et al. 1996), as inferred from early field dissipation measurements (for references see Gemmrich et al. 1994), average dissipation rates ε in the surface layer of the ocean and lakes are one to two orders of magnitude larger than predicted by εwl. Detailed velocity and bubble measurements linked enhanced TKE dissipation rates ε ≫ εwl to wave breaking events (Gemmrich and Farmer 2004).

Despite the widespread importance of the process, no reliable parameterization of dissipation rates in terms of environmental parameters exists so far. A framework for a potential parameterization was developed by Phillips (1985). He suggested that wave breaking may be characterized by the length of the breaking crest (in the along-crest direction) and its propagation speed, and introduced the breaking crest length distribution Λ(c). This distribution is defined in a way that Λ(c) dc describes the average total length of breaking wave crest, per unit area, that propagate at speeds in the range c to c + dc.

Combining the concept of breaking crest length distributions with laboratory experiments (Duncan 1981) that relate the phase speed of a breaking wave to the rate of energy loss yields
i1520-0485-40-3-583-e2
where ρ is the water density, g the gravitational acceleration, and b is a nondimensional factor. The momentum flux from waves to underlying currents follows as
i1520-0485-40-3-583-e3
This concept is very attractive as it suggests the estimation of key wave dynamics from remotely sensed properties of the wave field. However, the factor b, required to link the wave kinematics to the wave dynamics, is unknown. Based on crude conceptual models it is expected that b ≪ 1 and that b depends on the breaking strength and is not constant but likely a function of the wave field itself (Gemmrich et al. 2008). However, no theoretical framework for predicting the proportionality factor b exists and it has to be determined experimentally. So far, the field studies (Ding and Farmer 1994; Phillips et al. 2001; Melville and Matusov 2002; Gemmrich et al. 2008; Thomson et al. 2009) as well as laboratory studies (Banner and Peirson 2007) are inconclusive.

To address open issues in the Duncan–Phillips concept we conducted a field experiment that combined visual and infrared observations of the breaking crests with collocated in situ measurements of TKE dissipation rates. Here we report on the detailed structure of the turbulence and bubble field within the wave-influenced near-surface water column. The breaking crest length distributions and its link to the wave dynamics are reported in Thomson et al. (2009).

2. Observations

The objective of the overall project was to investigate the link between breaking crest distributions and subsurface dissipation rates in a natural wind-driven wave field. Wave breaking in the open ocean is often affected by a complex interplay of different wave systems, and wave tank experiments do not represent natural breaking wave fields. Therefore, to avoid the complexity of oceanic breakers and the limitations of artificially forced breaking waves, we chose a site in Lake Washington as a natural laboratory. Measurements were taken 9–16 November 2006 from the R/V Henderson of the Applied Physics Laboratory, University of Washington, which was moored at the northern end of the lake (47°44.542′N, 122°16.518′W), approximately 1.5 km from the closest shore. The water depth at this site is about 11.5 m and all observed wave scales are deep water waves. Here we concentrate on a strong wind event on 14–15 November (Fig. 1), spanning the wind speed range from calm conditions to up to 15 m s−1. The wind direction was along the major axis of the narrow northern section of the lake, resulting in ∼7 km fetch. Maximum significant wave height (Hs) reached about 0.45 m and the peak period τp about 2.5 s.

a. Instrumentation

Subsurface turbulence measurements were obtained with a set of three high-resolution pulse-coherent acoustic Doppler profilers (Dopbeam, Sontek) (see Gemmrich and Farmer 2004). The Dopbeams acquired velocity profiles of 0.72-m length, in a path starting 0.14 m ahead of the sensor head and with 6 × 10−3 m radial bin size. The diameter of the sonar transducer is 25 × 10−3 m and the beamspread half-angle is 1.2°. Near-field effects are limited to the first 0.2 m (from the sensor head) and the maximum beamspread is 35 × 10−3 m at the far range of the profile. Backscatter strength is recorded along a 2.5-m path, allowing for the tracking of bubble clouds and the sea surface. The sampling rate for velocity and backscatter is 20 Hz, implying that single pings are based on 10 individual pulses. This setup yields a velocity uncertainty better than 2 × 10−3 m s−1.

The sensors were mounted on a frame extending 8 m off the stern of the R/V Henderson, which was moored with its stern pointing into the approaching waves. Due to the catamaran-style design of the vessel and the narrow directional distribution of the wave field, no noticeable disturbance of the wave field occurred at the sensor location. Furthermore, for the short dominant wavelength encountered on the lake, the vessel is very stable and the sensors remain at a constant depth. One of the profilers was upward looking with the sensor head located 0.75 m below the still water surface, providing vertical velocity profiles and, from the backscatter signal, a record of the local surface elevation. Thus, the velocity profile extended to a maximum of 0.11 m above the mean waterline, and did not reach the free surface of larger wave crests during the highest sea states. The other two profilers were oriented horizontally at 0.2-m and 0.58-m depth, with the sensor heads 0.6 m downwind from the vertical profiler and aligned so that the horizontal and vertical beams intersected each other.

The Air–Sea Interaction and Remote Sensing group at the Applied Physics Laboratory (Seattle) deployed a suite of surface imaging sensors. This component of the experiment is reported in Thomson and Jessup (2009) and Thomson et al. (2009). In addition, they provided a set of standard meteorological measurements including heat and momentum fluxes.

3. Analysis methods

The instrumentation records high-resolution velocity profiles and acoustic backscatter strength within a fixed reference frame. From these data, profiles of TKE dissipation rates, relative to the free surface, are extracted.

a. Surface tracking

The instantaneous surface yields a very strong signal in the backscatter profile of the upward-looking sonar. At higher sea states, the presence of bubble clouds may obscure the surface return signal by broadening the region of strong backscatter. To extract the instantaneous surface from the vertical backscatter profile, I developed the following scheme, based on image processing methods. The range–time evolution of a 500-s backscatter record (104 profiles) is treated as an image, with individual range bins of a single ping being equivalent to image pixels. Thus, the total image is of size 104 × 416 (Fig. 2a). In an intermediate step, a filter mask is generated. For each ping (i.e., a given x value of the image) a binary threshold is applied at the 98th percentile (Fig. 2b). The entire binary image is then dilated with a 3 × 3 structure element; that is, the maximum value of its adjacent eight pixels is assigned to each pixel. This ensures that the mask of the surface return is a continuous, interconnected image object. All objects with area less than 500 pixels are removed, leaving a mask outlining a swath slightly broader than the surface return, removing subsurface bubble cloud returns and noise. Multiplying this mask with the nondilated binary backscatter image removes most of the subsurface bubble cloud returns and all the noise at ranges larger than the surface range. For each ping, the surface is the location of the strongest backscatter signal within this mask (Fig. 2c). In the absence of significant bubble clouds this scheme yields the surface elevation with a one bin accuracy (6 × 10−3 m). Within bubble clouds, the backscatter signal may saturate and the surface return may be up to 10 pixels wide. In these cases the surface location is defined as the median value of the surface return ranges, yielding up to 3 × 10−2 m uncertainty in the surface location. Note that the sonars operate at 2 MHz and the velocity measurements are practically unaffected by the bubble clouds.

b. TKE dissipation rate

The turbulence field will be referenced to the instantaneous free surface, which is only known at the location of the upward-looking sonar but not along the paths of the horizontal beams. Therefore, the following analysis is restricted to the upward-looking sonar. The observed velocities υ(z, t) = υorb(z, t) + υ′(z, t) are the sum of the vertical component of the wave orbital motion υorb(z, t) and turbulent fluctuations υ′(z, t). The orbital component υorb, which is usually the dominant term, can be estimated from the surface elevation time series
i1520-0485-40-3-583-e4
where the instantaneous amplitude a, frequency ω, and phase θ are obtained via the Hilbert–Huang transform (see the appendix) and the wavenumber k is calculated using the linear dispersion relation.
The basis of the dissipation rate estimate is the centered second-order turbulence structure function:
i1520-0485-40-3-583-e5
where the length scale s falls within the range lKslo, with Kolmogorov scale lK and turbulence outer scale lo. The angle brackets represent time averaging over Nav data points, and the structure function is evaluated at a subsampled time stamp, t. For slK, it follows (Kolmogorov 1991; Lohse 1994) that
i1520-0485-40-3-583-e6
with C = 2.0 (Frehlich and Cornman 2002). In steep waves the linear approximation for the wave orbital motion (4) is less accurate. However, deviations between the true and the approximated orbital motion with spatial scales larger than s do not affect the structure function, and therefore, in this framework, (4) is a suitable approximation.
The structure function and thus the dissipation rate estimate are evaluated at each depth bin z. The separation distances have to fall within the inertial subrange, and here I chose 6 × 10−3 m < s/2 < smax, where smax is the minimum of (z, dυz, 0.12 m); that is, eddy sizes up to 0.24 m are considered, consistent with the inertial subranges found in Gemmrich and Farmer (2004). Squared velocity differences are averaged over five pings (see below), yielding 4-Hz sampling rate, and a least squares fit
i1520-0485-40-3-583-e7
is calculated (Wiles et al. 2006). An example of the fitted structure function is given in Fig. 3. The offset N is mainly due to noise inherent in the velocity measurement and is assumed to be independent of the location along the acoustic beam. Thus, for Nas2/3 a lower bound of the dissipation estimate is given as
i1520-0485-40-3-583-e8
which depends only on the slope A of the fit. Generally, the noise offset N in (7) as well as the slope A increases toward the boundary, and the approximation (8) is valid throughout this dataset (Fig. 3).

The estimation of dissipation rates from structure functions is a statistical concept, relying on some time averaging of squared velocity differences. Here we aim at a high temporal resolution without jeopardizing the validity of the underlying assumptions. To test the dependence of the dissipation estimate on the averaging interval in (5) I processed a record of 104 velocity profiles with varying length Nav of the averaging period. This resulted in fields of dissipation rates with varying temporal, but fixed spatial, resolution. To compare the results the mean dissipation rate 〈ϵ〉 for the entire 500-s record, and taken over a range of 0.3 m in the center of the profile, was calculated. No clear trend or asymptotic behavior of 〈ϵ〉 versus Nav exists for the range of Nav between 5 and 2000, corresponding to averaging times Tav of 0.25–100 s (Fig. 4), but fluctuations of up to 25% occur. In the wave zone, short pulses of high dissipation are observed (Gemmrich and Farmer 2004), which will be better resolved by shorter averaging periods. Therefore, an increase of 〈ϵ〉 toward smaller Nav is expected. The lower bound of Tav is given by the time scale associated with the evolution of the turbulent eddies τt = (ν/ϵ)1/2, where ν is the kinematic viscosity: Tavτt. In the wave zone we find τt < 0.1 s. Based on these constraints, I set Nav = 5 (i.e., Tav = 0.25 s) throughout the following analysis.

The spatial resolution of the dissipation estimates is the same as of the velocity measurement, that is, 6 × 10−3 m. However, each dissipation data point is based on a range of neighboring velocity measurements [Eq. (5), see Fig. 3] representing the local eddy cascade. Compared to the spectral inertial method (Gemmrich and Farmer 2004), this structure function method yields a clear improvement in spatial resolution.

In higher sea states the surface of some wave crests is beyond the range of the velocity profile, and dissipation profiles are truncated. The total dissipation is dominated by the near-surface region, and the fraction of unresolved data bins within this region depends on the significant wave height Hs. At Hs = 0.25 m, less than 1% of the data bins within the uppermost 0.1 m are unresolved. This fraction increases to 4%, 6.5%, and 9% for 0.35 m, 0.4 m, and 0.45 m, respectively. For the entire dataset the fraction is 2.3% (see also Fig. 5).

The estimation of ε from structure functions requires the existence of an inertial subrange (Wiles et al. 2006) in the eddy cascade as well as local isotropy. There is evidence that the latter is a reasonable assumption even beneath breaking waves (Gemmrich and Farmer 2004). Immediately following a breaking event the eddy cascade may not be developed yet (Gemmrich and Farmer 2004), so no inertial subrange exists. Thus, in some cases dissipation estimates may not be available for up to 1 s. Furthermore, all data at z ≤ 0.012 m are ignored.

4. Results

a. Turbulence field

Figure 5 shows an example of the turbulence and bubble field observed in the surface layer. The high-resolution dissipation rates reveal fast fluctuations in the surface layer of more than four orders of magnitude (Fig. 5a). Generally, highest values are close to the surface and are rapidly decaying with depth, consistent with TKE injection by waves (Stewart and Grant 1962). In this very energetic environment high dissipation rates are strongly localized and strong temporal fluctuations are observed within the narrow sonar beam. Applying a five-point running average filter to the dissipation time series reveals blobs of high turbulence (Fig. 5b). In many cases the high surface region dissipation rates are associated with active whitecaps upstream of the sonar beam (Figs. 5a–c), as observed in video recordings (kindly provided by J. Thomson). However, there are also frequent cases of high dissipation in the top 0.1 m without any visible breaking signal in the vicinity. These high dissipation events may be generated by steep nonbreaking waves (Gemmrich and Farmer 2004), or breakers without visible air entrainment, or are the remainder of strong turbulence generated farther upstream. Individual breaking waves generate small bubble clouds and their downward spreading is the result of turbulent motion being stronger than the bubble rise speed due to buoyancy. Subsurface bubble clouds are more persistent than high dissipation rates and are also advected by the mean flow and larger-scale structures, resulting in persistent intermediate to high backscatter regions. Nevertheless, the backscatter signal may be utilized for identifying wave breaking events. In deep water, breaking waves are mainly spilling waves with the highest air fraction close to the surface. Therefore, the association to the breaking status is based on the mean acoustic backscatter γ in a relatively shallow surface layer (0.03 m < z ≤ 0.13 m). Wave breaking is defined as periods with γ larger than 95% of the maximum observed values, that is, γ ≥ 2.85: nonbreaking waves and older decaying wave breaking events as γ ≤ 2.5. (Note that the acoustic backscatter strength is not calibrated and ranges from 0 to 3.) A cross-check with the video recordings confirmed that this scheme captured most of the breaking events (Fig. 5).

The combination of surface elevation record, wave breaking indication, and TKE dissipation rates (Figs. 5d–a) allows us to stratify energy dissipation rates by wave scale. The Duncan–Phillips concept (Duncan 1981; Phillips 1985) predicts the energy dissipation rate due to wave breaking as a function of the wave scale, represented as phase speed c (2). In the equilibrium subrange the wave height spectrum scales as S(ω) ∝ ω−4. Phillips (1985) postulated for the local equilibrium that energy input Sin, nonlinear wave–wave interactions Snl, and energy dissipation Sdis are proportional, which implies the same scale dependence. Therefore, the breaking crest length distribution follows as Λ(c) ∝ c−6 (Melville and Matusov 2002; Gemmrich et al. 2008). If the proportionality factor b were scale independent, although not necessarily constant, it would imply ε ∝ c−1 (i.e., smaller waves dissipating more wave energy than larger waves).

Figure 6 shows the observed mean energy dissipation in the upper 0.1 m as function of the wave speed, for the period 1400–1900 PST 15 November, separated into cases of wave breaking (upper panel) and excluding breaking waves (lower panel). Individual waves are defined as the period between two successive zero-up crossings of the surface elevation record, and the mean dissipation within this period is calculated. For each wave crest the local frequency, based on the Hilbert–Huang transform (see the appendix), is estimated and the phase speed c is calculated from the linear dispersion relation. This method is an efficient scheme for correctly extracting wave scales from time series, even if frequent riding waves are present. All waves are then sorted into phase speed bins and mean dissipation rates, and their standard deviations are estimated by the bootstrap method.

Breaking waves show a distinct increase of dissipation rates with increasing wave speed, whereas excluding the breaking waves results in a much weaker wave-scale dependence (Fig. 6). A least squares fit ε ∝ cm yields m = 0.48 and m = 0.19 for breaking and nonbreaking waves, respectively. Allowing error bars of one standard deviation (Fig. 6) widens the slope estimate to 0.05 ≤ m ≤ 0.67 for breaking waves and 0.06 ≤ m ≤ 0.23 if breaking waves are excluded. The wave spectrum during this period is consistent with the equilibrium range scaling S(ω) ∝ ω−4 (Fig. 7), suggesting that in this short-fetched wave field the energy terms Sin, Snl, and Sdis are not proportional. It also implies a weaker scale dependence of the product of the scaling factor b and breaking crest density bΛ(c) ∝ cp, p ≈ −4.5, than p = −6 predicted by the equilibrium scaling (Melville and Matusov 2002).

The rate of wave breaking has been linked to the saturation of the wave spectrum (Banner et al. 2002):
i1520-0485-40-3-583-e9
where D(ω) is the directional spreading of the wave height spectrum (Banner et al. 2002; Gemmrich et al. 2008). Independent of the wave scale, the onset of wave breaking occurs at a threshold σ = 4.5 × 10−3, and the breaking rate increases rapidly with increasing saturation (Banner et al. 2002). Here I characterize the wave field using the weighted, band-averaged saturation
i1520-0485-40-3-583-e10
that is, the saturation in the wave spectral band expected to include the scales of whitecapping. The ω−1 weighting emphasizes larger-scale breakers and, more importantly, makes σb nearly independent of the high frequency cutoff.
The mean total dissipation,
i1520-0485-40-3-583-e11
with T = 40 min and zmax = 0.5 m, follows a similar threshold behavior as found for the breaking probability (Fig. 8a). For the given choice of bandwidth in (10), the threshold is σbt = 10−2. At low saturation levels σb < σbt, the total dissipation remains constant at a background level, E ≈ 0.7 W m−2. At higher saturation levels, σbσbt wave breaking sets in and the dissipation increases rapidly—in our dataset by a factor 4.
Thomson et al. (2009) estimated the total dissipation from the distribution of breaking crest length and found that 67% of the total dissipation can be associated with the dominant wave band ω/ωp = [0.7, 1.3]. Here I calculate the total dissipation during a single wave period
i1520-0485-40-3-583-e12
where tc is the time of the wave crest passage and τ is the local wave period, and find that waves in the dominant wave band make up 56% of the total dissipation. However, wave-induced turbulence persists for several wave periods, although decaying rapidly (Gemmrich and Farmer 2004), and a single profile measurement cannot resolve completely the dissipation associated with a single wave. Therefore, this in situ dissipation estimate provides only a coarse lower bound of the total dissipation associated with a single wave. Nevertheless, it supports the notion that infrequent large-scale breakers dominate the total wave dissipation.

The enhanced dissipation is entirely confined to the top of the wave crest zone (Fig. 8b). Below the wave zone, at 0.4-m depth, dissipation rates are independent of the wave saturation (Fig. 8c), whereas in the uppermost 0.1 m a tenfold increase is observed (Fig. 8b). The transition occurs somewhere around 0.2 m but no direct correlation with the significant wave height is found. Furthermore, the data clearly stratify with wave phase. At low wave saturation levels, dissipation values beneath wave crests are about three times higher than beneath neighboring trough regions (Fig. 8b). Near-surface dissipation rates beneath troughs do not vary with wave saturation and for (σb > σbt) the crest–trough dissipation ratio increases to up to 30 times.

To further evaluate the contribution of the wave crest to the total dissipation, I calculate the integral dissipation in the wave crest region,
i1520-0485-40-3-583-e13
and the corresponding dissipation in the following trough region,
i1520-0485-40-3-583-e14
where η is the local crest elevation and tc and tt are the time of the wave crest and the following wave trough, respectively.

The distribution of the ratio of crest to trough dissipation is clearly skewed toward larger dissipation in the crest region (Fig. 9a), with the average value of Ec/Et = 3.0. However, for the most energetic events, this ratio increase to Ec/Et = 51 (Fig. 9b).

Stewart and Grant (1962) argued in their pioneering work on wave turbulence that wave-induced turbulence is injected at the breaking crest, and therefore more than one-half of the energy is dissipated above the mean waterline. Terray et al. (1996) introduced a three-layer concept for breaking wave-induced turbulence, with a constant high dissipation layer of thickness zb, a wave turbulence layer at zb < zzt, and shear-induced turbulence at z < zt. Although their measurement did not reach into the near-surface constant dissipation layer, a balance between energy input and dissipation suggested that approximately half of the total dissipation occurs between the surface and zb ≈ 0.6Hs. Wave tank measurements of spilling waves on a sloping beach revealed that 80% of the depth-integrated dissipation occurred above the still waterline (Govender et al. 2004). Here I show the first direct field observations to demonstrate this turbulence enhancement beneath deep water wave crests. Previous field measurements of wave-zone dissipation (for reference see Burchard et al. 2008) were obtained below the trough line, and thus the resolved direct measurements could have underestimated the total dissipation by more than 80%. Assuming a certain depth dependence, the resolved measurements could be extrapolated to the surface, thus obtaining an estimate of total dissipation in the water column (Terray et al. 1996).

The depth dependence of dissipation rates is ε(z) ∝ zm, with m = −1 for flows along a solid wall. Field observations of profiles of wave-induced turbulence, taken as the average of several tens of minutes and referenced to a flat surface located at the mean waterline, yielded m = −2 to m = −4 (for references see Burchard et al. 2008).

Here I calculate the power law fit for every individual profile, referenced to the instantaneous free surface, and calculate the mean of the exponent for a 40-min period. This average exponent m depends on wave saturation and wave phase (Fig. 10). Beneath wave troughs the profiles are consistent with wall-layer scaling, m = −1, and independent of wave saturation. In contrast, the shape of the dissipation profile beneath wave crests is a threshold function of wave saturation. In the absence of wave breaking the exponent is approximately m = −1.1. At higher wave saturation levels, when wave breaking is frequent, the exponent decreases with wave saturation, reaching m = −1.6 at the highest saturation levels.

It should be noted that these depth dependences are all referenced to the instantaneous free surface, whereas published values of the coefficient m are all based on a flat, mean water level surface (in the following denoted by a subscript f ). A general conversion of the coefficient mf to a corresponding coefficient ms in a surface-following coordinate system is not possible. For example, assuming mf = −2.5 converts to ms ≈ −1.9 beneath a wave crest, but ms ≈ −5.4 beneath a wave trough, if measured at a depth range Hs/2 ≤ zflat ≤ 5Hs. For nonsymmetric wave forms and dissipation time series, the resulting time-average value of ms would not be a known function of the trough and crest values of ms. Therefore, the depth dependence of the dissipation profile in the wave-effected near-surface layer strongly depends on the choice of surface reference.

Turbulence enhancement, relative to a z−1 dependence, is mainly restricted to the crest region (Fig. 11). The reference speed uref was chosen so that the mean value of the normalized dissipation profiles equals 1. Commonly, the normalization is based on wall-layer scaling; that is, uref = u*. However, the friction velocity is not a universal scaling parameter that collapses our datasets, yielding an apparent mean enhancement ε/εwl of 5 during high wind speeds, compared to an enhancement factor of 20 during lower wind speeds. This difference in turbulence enhancement is puzzling and seems to be counterintuitive. Furthermore, adjusting the reference speed to achieve a common mean normalized dissipation value highlights the shape of the profiles. The scaling factors are uref = 1.58u* in the high wind case and uref = 2.7u* for the lower wind speed subset. The deviation from the z−1 dependence is almost entirely confined to the wave crest and is most prominent above the mean waterline. In the trough region, wall-layer depth dependence seems to be valid, even close to the surface. In a surface-following coordinate system, the constant dissipation layer, implied by the Terray et al. (1996) scaling, would include almost the entire crest region profile 0 ≤ z/Hs ≤ 1.2, but only the top bin in the trough region profile. There is no evidence of a constant dissipation layer in this dataset.

Most previous observations of dissipation profiles were referenced to a fixed coordinate system with the origin at the mean water level (for reference see Burchard et al. 2008). However, close to the surface, the shape of the dissipation profile and the magnitude of the wall-layer dissipation is severely affected by the choice of the reference system. Figure 12 shows the same data as in Fig. 11 but now analyzed in a flat-surface coordinate system, where zflat is the distance to the mean water level, and no differentiation regarding the wave phase is made. The average dissipation values are very similar in both representations. However, in a fixed-frame reference, the normalized dissipation peaks at zflat ≃ 0.3Hs. Above this level, dissipation falls off rapidly. The wave-affected layer extends to a depth of approximately one significant wave height. Below this layer the profile is consistent with wall-layer scaling. Previous fixed-frame dissipation profiles did not report data above zflat ≃ 0.3Hs and the apparent dropoff in dissipation, which is an artifact of the choice of the coordinate system, was not resolved.

To account for the unresolved near-surface sections, previous dissipation measurements were commonly extrapolated to the mean water level rather than the free surface. The extrapolation assumes a constant dissipation layer 0 < zflat < αHs (Terray et al. 1996) with α = (0.1 to 1) (for reference see Burchard et al. 2008) Reanalysis of our data, using a very conservative α = 0.1, shows that on average this extrapolation scheme misses 45% of the total dissipation for sea states with active wave breaking, σb > σbt, and 24% during low sea states (σb < σbt). The underestimation reaches 58% at the largest saturation levels. This severe underestimation of the total dissipation in this extrapolation scheme is primarily due to the cutoff at the mean water level. On the other hand, truncating the dissipation profiles at 0.15 m below the mean waterline, and subsequent extrapolation to a flat surface based on local exponents m, would overestimate the total dissipation by up to 90%. Particularly in deep troughs, the power law extrapolation can generate large values near the assumed flat surface. Therefore, the extrapolated dissipation values in the trough region, above the true surface, are on average larger than the true dissipation rates in the lower crest regions. This imbalance yields a significant, positive difference of the extrapolated trough values and the rejected crest values and, thus, an overestimation of the total dissipation.

Here the observed dissipation profiles include the wave crest regions, and the integrated total dissipation E is close to the total dissipation in the water column. Achieving an approximate balance of energy input and dissipation EEin (neglecting wave energy flux divergence) implies ceffu10; that is, in this young wave field the energy input occurs predominately at the peak of the spectrum. Furthermore, this analysis indicates that E values were underestimated in previous work, suggesting that ceff might be larger than given by Gemmrich et al. (1994) or Terray et al. (1996).

b. Bubble size distribution

Air bubbles are broken up by turbulence pressure fluctuations beneath breaking waves, and thus there exists a relation between bubble sizes and dissipation rates. The maximum bubble radius sustained by surface tension, with larger bubbles being disrupted by the turbulence pressure forces, is the Hinze scale (Garrett et al. 2000):
i1520-0485-40-3-583-e15
where CH = 0.5 and γw and ρ are surface tension and density of the water, respectively. Garrett et al. (2000) hypothesize that the initial bubble size spectrum in a breaking wave is a measure of the probability of the intermittent dissipation rates:
i1520-0485-40-3-583-e16
This yields a scaling of the bubble size distribution:
i1520-0485-40-3-583-e17

In a spilling breaker bubble generation is concentrated within a shallow surface layer only, and we take the observed dissipation rates within the upper 0.05 m to evaluate (17). The distribution of dissipation rates during 1600–1700 PST 15 November, when wave breaking was frequent, and the inferred bubble size distribution (17) are given in Fig. 13. The absolute number of bubbles depends on the air fraction of the breaking wave and only the shape of the distribution can be obtained from (17). The average dissipation rate = ∫εP(ε) dε/∫P(ε) dε is ≃ 0.2 m2 s−3, resulting in a bubble radius of aH ≃ 2700 μm. However, the high temporal and spatial resolutions of the dissipation estimates also reveal a tail of the dissipation distribution with values as much as three orders of magnitude higher than the average value. These intermittent high dissipation rates correspond to Hinze scales O(100 μm)!

The Hinze scale is a measure of the largest bubbles sustained in a turbulent flow. A measure of the average bubble size is given by the Sauter scale, a32, with a32 = AsaH. Although the proportionality constant As depends on the turbulence levels of the flow (Pacek et al. 1998), it is generally assumed to be 0.6 ≤ As ≤ 0.7 (Angle and Hamza 2006), and the average bubble size is about ⅔ of the Hinze scale.

Our data show that the model of bubble breakup by turbulent pressure forces is able to explain the presence of the reported small bubble sizes O(100 μm) (Vagle and Farmer 1998). The slope of N(a) ∝ aβ is consistent with observed in the laboratory (Deane and Stokes 2002). Thus, a slope can be the result of bubble fragmentation by intermittent high turbulence levels and does not necessarily imply a bubble cascade. The latter concept had been proposed by Garrett et al. (2000) as an alternative method for the generation of the bubble size spectrum and had been favored as an explanation for the Deane and Stokes (2002) observations. Unlike the laboratory experiments where dissipation values are likely much smaller than observed in this field study, the entire size spectrum can be attributed to bubble fragmentation. The rolloff to N(a) ∝ aα, for small bubbles seen in the laboratory experiment at a ≈ 2 mm is not supported by these field observations.

5. Conclusions

This work has shown that acoustic remote sensing techniques are capable of measuring turbulence within an active wave zone, including the wave crests. These unique high-resolution dissipation rate data provide new insight into several areas relevant to air–sea interaction. Turbulence levels are highest within the wave crest region and are generally much larger than previous observations, all of them excluding the crest regions, suggested. Near-surface turbulence governs air–sea heat and gas exchange, as well as sea spray generation, which is of importance to hurricane dynamics, and affects the distribution of zooplankton and phytoplankton. Furthermore, dissipation is arguably the most uncertain term in spectral wave models. The threshold behavior and scale dependence of the energy dissipation found in this study provides a basis for a physically based parameterization of this term, as well as for calibrating ocean mixing models. Our observations support the concept of bubble fragmentation by turbulence as the major mechanism for establishing observed bubble size distributions. Thus, the parameterization of processes and properties that depend on bubble populations, like air–sea gas exchange, sound transmission, or optical sensing of near-surface layers, can be improved by including a turbulence dependence, which in turn depends on the wave field saturation. Similarly, the fragmentation model may be extended to predictions of the breakup of oil spills and other pollutants.

Acknowledgments

I thank A. Jessup and J. Thomson (APL-UW) for their help with the experimental work and for providing auxiliary data, and C. Garrett, J. Thomson, A. Babanin, and M. Banner for many helpful discussions. Comments from two reviewers greatly improved this manuscript. The Dopbeam system was developed and is being maintained by the Ocean Acoustics/Instrument Development Group of the Institute of Ocean Sciences (Sidney, British Columbia, Canada). Funding was provided by NSF and ONR.

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APPENDIX

Instantaneous Frequency

The following derivation of the local amplitude and frequency of a surface elevation record is based on the Hilbert analysis and was first presented in Gemmrich (2005).

The Hilbert transform of a signal X(t) is defined as
i1520-0485-40-3-583-ea1
taken as a Cauchy principal value P. The local amplitude is then given as
i1520-0485-40-3-583-ea2
and the local frequency is
i1520-0485-40-3-583-ea3
where Φ = tan−1(H(t)/X(t)). [Note that if X(t) = cos(ωt), it follows that H(t) = sin(ωt) and Φ(t) = ωt.] However, the estimation of a local amplitude and frequency by the classical Hilbert analysis is only meaningful for narrow-banded signals. Broad-banded signals, like surface elevation records, have to be divided into a finite number of intrinsic mode functions (IMFs), by way of the empirical mode decomposition (EMD), before the Hilbert analysis can be applied (Huang et al. 1998). An IMF is a function in which the number of zero crossings and the number of local maxima differ by not more than one. The IMFs are generated in an iterative way as the difference between the signal and a slowly varying background. The background signal is the mean of two spline fittings: one through all local maxima and the second through all local minima. This sifting process is repeated until the IMF condition is met. The so-obtained IMF is subtracted from the time series and the iteration continues with the residual signal until only a monotonic signal remains. Typically, surface height records yield decomposition into N = O(10) IMFs. The surface elevation time series may be represented as a Hilbert expansion:
i1520-0485-40-3-583-ea4
where aj, Φj, and ωj = dΦj/dt are amplitude, phase, and frequency of the IMFs, respectively. The instantaneous amplitude of the signal is
i1520-0485-40-3-583-ea5
and the instantaneous frequency is
i1520-0485-40-3-583-ea6
where k corresponds to the dominant IMF at time t: ak(t) = max(aj(t)), j = 1, … , N.

Fig. 1.
Fig. 1.

Environmental conditions at the observation site on 14–15 Nov 2006. (top) Wind speed at 10-m height, u10 (mean wind speed, solid line; minimum and maximum during 5-min intervals, dashed lines). (middle) Wind direction dd; (bottom) significant wave height Hs (line) and dominant wave period τp (dots).

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 2.
Fig. 2.

Example of surface tracking (1047 PST 15 Nov 2006). (a) Acoustic backscatter strength of the vertical sonar, (b) binary image of the backscatter field, and (c) surface elevation record. Range given as distance from the sonar head.

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 3.
Fig. 3.

Structure function D(s) at (a) 0.03, (b) 0.11, and (c) 0.37 m below the surface: raw data (open circles) and least squares fit (dashed line).

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 4.
Fig. 4.

Mean dissipation 〈ϵ〉 of a 500-s record as a function of the number Nav of velocity differences (i.e., the averaging period) utilized in the structure function [Eq. (5)].

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 5.
Fig. 5.

Turbulence field and acoustic backscatter in the wave-surface layer (1144 PST 15 Nov 2006). (a) TKE dissipation rate ε in the upper 0.5 m beneath waves. (b) Filtered TKE dissipation field as in (a) but subject to a five-point running average. Black triangles indicate actively breaking events closely upstream of the sonar, as observed in the video recordings. (c) Average dissipation rate in the top 0.1 m. (d) Average acoustic backscatter strength γ (relative units) in the top 0.1 m (solid line) and assumed threshold for breaking waves (dashed line). Data in (a)–(d) are based on a surface-following coordinate system; that is, z = 0 represents the instantaneous free surface. (e) Corresponding surface elevation η. The dashed line indicates the extent of the dissipation profile. (See section 3.)

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 6.
Fig. 6.

Average TKE dissipation rate in the upper 0.1 m from the free surface as a function of wave scale, expressed as the phase speed c within (a) high bubble concentrations (γ > 2.85, indicative of breaking waves) and (b) low bubble concentrations (γ < 2.8, indicative of nonbreaking waves). The vertical bars indicate ± one standard deviation. The dashed lines represent the least squares fits with slope (a) 0.48 and (b) 0.19.

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 7.
Fig. 7.

Surface elevation spectrum at 1700 PST 15 Nov 2006. The dashed line indicates ω−4.

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 8.
Fig. 8.

Average dissipation within 40-min periods as function of mean wave saturation σb. (a) The total dissipation E and (b),(c) the average dissipation rate 〈ϵ〉. Black upward (gray downward) triangles are data taken beneath wave crest (trough) regions. (b) Data from the upper 0.1 m from the free surface. (c) Data from 0.4 to 0.5 m below the free surface.

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 9.
Fig. 9.

Dissipation within a wave crest Ec and the following wave trough Et. Data are integrated values from the free surface over the vertical range of the crest amplitude: (a) distribution of the ratio Ec/Et and (b) bin averages. Error bars indicate ±1 standard deviation.

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 10.
Fig. 10.

Average coefficient of depth dependence ε ∝ zm: black upward (gray downward) triangles are data taken beneath wave crest (wave trough) regions.

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 11.
Fig. 11.

Normalized dissipation profile during period of (a) frequent wave breaking (1400–1700 PST 15 Nov 2006) and (b) intermittent wave breaking (0200–0500 PST 15 Nov 2006): black upward (gray downward) triangles are data taken beneath wave crest (wave trough) regions.

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 12.
Fig. 12.

Normalized dissipation profile based on observations referenced to a flat surface, where zflat is the distance to the mean waterline, during period of (a) frequent wave breaking (1400–1700 PST 15 Nov 2006) and (b) intermittent wave breaking (0200–0500 PST 15 Nov 2006).

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

Fig. 13.
Fig. 13.

(a) Probability distribution of dissipation rates P(ϵ) in the upper 0.05 m based on a 1-h-long record and (b) bubble size distribution N(a) as function of bubble radius a [inferred from (a)]. The dashed lines depicts (right) a−10/3 and (left) a−3/2 dependences as previously observed in laboratory experiments (Deane and Stokes 2002).

Citation: Journal of Physical Oceanography 40, 3; 10.1175/2009JPO4179.1

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