Active and total whitecap fractions quantify the spatial extent of oceanic whitecaps in different lifetime stages. Total whitecap fraction W includes both the dynamic foam patches of the initial breaking and the static foam patches during whitecap decay. Dynamic air–sea processes in the upper ocean are best parameterized in terms of active whitecap fraction WA associated with actively breaking crests. The conventional intensity threshold approach used to extract WA from photographs is subjective, which contributes to the wide spread of WA data. A novel approach of obtaining WA from energy dissipation rate ε is proposed. An expression for WA is derived in terms of energy dissipation rate WA(ε) on the basis of the Phillips concept of breaking crest length distribution. This approach allows more objective determination of WA using the breaker kinematic and dynamic properties yet avoids the use of measuring breaking crest distribution from photographs. The feasibility of using WA(ε) is demonstrated with one possible implementation using buoy data and a parametric model for the energy dissipation rate. Results from WA(ε) are compared to WA from photographic data. Sensitivity analysis quantifies variations in WA estimates caused by different parameter choices in the WA(ε) expression. The breaking strength parameter b has the greatest influence on the WA(ε) estimates, followed by the breaker minimal speed and bubble persistence time. The merits and caveats of the novel approach, possible improvements, and implications for using the WA(ε) expression to extract WA from satellite-based radiometric measurements of W are discussed.
Oceanic whitecaps are the most direct surface expression of breaking wind waves in the ocean. The whitecap fraction W quantifies the breaking events and is thus a suitable forcing variable for parameterizing and predicting air–sea interaction processes enhanced by breaking waves. Whitecaps are dynamic features on the ocean surface that evolve quickly (Monahan et al. 1982; Anguelova and Huq 2012; Callaghan et al. 2012) and thus have markedly different properties in different lifetime stages (Monahan and Woolf 1989). At the moment of active breaking, the whitecaps are bright with high albedo, comprise a wide range of bubble sizes, are fast changing, move along with the wave crest, and cover less surface area. We refer to this actively generated sea foam as active whitecaps or stage A (young) whitecaps (Monahan and Woolf 1989). In contrast, the decaying foam after the breaking event and the froth formed by the bubbles rising from below are dimmer with lower albedo, comprise mostly smaller bubbles, linger almost motionless behind the wave that has created them, and spread over a larger area. We refer to these decaying foam patches as residual whitecaps or stage B (mature) whitecaps.
Both active and residual whitecaps contribute to the total value W, with their relative contributions depending on the wind speed and other meteorological and environmental factors. Total whitecap fraction W is associated with bubble-mediated sea spray production and heat exchange (Andreas et al. 2008; de Leeuw et al. 2011). Many other more dynamic air–sea processes must be represented in terms of active whitecap fraction WA, such as production of spume droplets (important for the intensification of tropical storms), momentum flux, turbulent mixing, gas exchange, and generation of ambient noise in the ocean (Pumphrey 1989; Melville 1996; Medwin and Clay 1998; Asher et al. 1998; Andreas et al. 2008).
Photographic measurements of whitecap fraction often quantify W without a clear measure of the relative contributions of active and residual whitecaps (Monahan et al. 1983; Stramska and Petelski 2003; Callaghan et al. 2008b). Deliberate efforts in processing and analyzing photographic images of the sea state are required to separate the active part of the whitecaps and obtain WA. Usually a suitable intensity threshold separating the dynamic and static foam patches is sought (Monahan and Woolf 1989; Asher et al. 2002). This thresholding technique could be enhanced by additional information for whitecap shapes and motions (Mironov and Dulov 2008; Kleiss and Melville 2010; Scanlon and Ward 2013).
This study describes a new approach of employing the concept of breaking crest statistics introduced by Phillips (1985) to obtain active whitecap fraction WA. Wind waves in the ocean break over a wide range of scales (Gemmrich and Farmer 1999). Describing and predicting the upper ocean dynamics require characterization of the breaking wave scales (Thorpe 1995). Phillips (1985) realized that the length of the breaking crest Λ and its propagation velocity c (bold symbol denotes a vector quantity) provide a good representation of the scale of a breaking wave. Thus Phillips defined a statistical variable called breaking crest length distribution Λ(c)dc, which quantifies the total length of breaking crests per unit area moving with a velocity in the range (c, c + dc).
Phillips (1985) uses moments of the breaking crest length distribution to define various statistics for breaking waves; hereafter we refer to this idea as the Phillips concept. Combining the first moment of Λ(c)dc with the persistence time of the bubbles T gives the active whitecap fraction:
The fifth moment of the breaking crest length distributions determines the energy dissipation rate:
where g is the acceleration due to gravity, and b is the so-called breaking strength parameter (Gemmrich et al. 2008; Drazen et al. 2008). Because previous experience has shown that extracting WA from photographs is subjective (Callaghan and White 2009) and it is difficult to measure ε (Gemmrich and Farmer 2004), the usefulness of Λ(c)dc for obtaining WA and ε (and other) breaking wave characteristics has prompted numerous experimental studies in the last decade aiming to measure Λ(c)dc instead (Phillips et al. 2001; Melville and Matusov 2002; Gemmrich et al. 2008; Thomson et al. 2009; Kleiss and Melville 2010; Zappa et al. 2012). However, the method of measuring Λ(c)dc is somewhat similar to measuring W because it does involve a choice of intensity threshold to extract breaking crest lengths from photographic images (Gemmrich et al. 2008; Kleiss and Melville 2010, 2011). Thus some of the uncertainty associated with the intensity threshold approach remains and is transferred to WA and ε determined with (1) and (2).
Our approach follows the Phillips concept in reverse: instead of trying to measure Λ(c)dc in order to obtain WA from (1) or ε from (2), we use the Phillips concept to derive an expression for WA in terms of ε. An expression WA(ε) can provide estimates of the active whitecap fraction using measured or modeled energy dissipation rate ε. This is feasible because, although difficult, measurements and models of ε are increasingly more manageable and accurate (Gemmrich and Farmer 2004; Hwang and Sletten 2008; Gerbi et al. 2009; Thomson et al. 2009; Rogers et al. 2012; Gemmrich et al. 2013; Schwendeman et al. 2014). The merit of this approach is that by relying on the kinematics and dynamics of the breaking waves, one can more objectively quantify active whitecap fraction WA while avoiding the pitfalls of the intensity threshold approach.
This paper describes the physical basis of this new approach (section 2b) and one possible implementation (section 3). Our results demonstrate the feasibility of obtaining WA using the Phillips concept (section 4a), while our sensitivity analysis ranks the tuning parameters for improved results (section 4b). Merits and caveats, possible expansions and improvements, and implications for remote sensing of whitecaps are also discussed (section 5).
2. Approaches to separate active whitecap fraction
a. Conventional approach: Photographs and intensity threshold
Extracting WA from images of the sea surface requires a step beyond a general discerning of any white (foamy) water from the surrounding seawater. Table 1 summarizes 13 datasets for which various techniques were used to ensure extraction of WA specifically.
Most datasets listed in Table 1 have been processed by setting intensity threshold high enough to discard dim, residual foam and extract only the bright active crests. The lower-intensity resolution of the video images that Monahan et al. (1985, 1988) processed had been successfully used to discriminate not only spurious features such as sun glints but also the less bright residual foam patches (Monahan and Woolf 1989). Pursuing a parameterization of gas transfer velocity in terms of whitecap fraction, Asher et al. (2002) had consistently set manually chosen intensity threshold that extract only stage A whitecaps. Similarly, in a study on energy dissipation, Hanson and Phillips (1999) used an image processing technique that aimed specifically at quantifying stage A whitecaps. For four datasets in Table 1 (Nordberg et al. 1971; Ross and Cardone 1974; Bondur and Sharkov 1982; Bortkovskii 1987), the choice of the intensity threshold was aided by categorizing the white water areas in photographs as crests and streaks and specifically delineating their shapes. Scanlon and Ward (2013) optimized the separation procedure by defining a secondary intensity threshold on the basis of the shape of the whitecaps and their location relative to the crests.
This conventional intensity threshold approach has been strengthened recently by adding more information to the processing procedure. Mironov and Dulov (2008) extended the intensity thresholding with an algorithm that takes into consideration the different kinematics of stage A, stage B, and mixed whitecaps. Mironov and Dulov identified the deviation of a whitecap group (consisting of crests and residual patches) motion from the mean velocity of an “ideal breaker” as the most effective measure. For further refinement, Mironov and Dulov considered the temporal evolution of the whitecap area by assigning whitecaps with an increasing area as active and with a decreasing area as residual foam. Further filtering of active whitecaps from whitecaps with mixed stages was aided by considering the direction of the movement: active whitecaps moved in the direction of the wave. Kleiss and Melville (2010, 2011) also extracted WA by tracking the whitecap motion; the significant difference from other processing procedures was that they used measured Λ values for this tracking.
While viable WA data points were extracted using intensity threshold, aided occasionally by shape delineating, the approach is always somewhat subjective and this contributes to a relatively large scatter of the extracted values. This is demonstrated in Fig. 1a, which shows a plot of WA as a function of wind speed at 10-m reference height U10 for all datasets in Table 1 (black symbols). These WA values are compared to values of total whitecap fraction W obtained from photographs (gray symbols) from datasets summarized by Anguelova and Webster (2006, their Table 2). The power-law parameterization of the W(U10) relationship with wind speed exponent of 3.41 based on data for the total whitecap fraction (Monahan and Muircheartaigh 1980) is also plotted for reference (purple line denoted MOM80 in the legend). Besides the wide spread of both W and WA values, we draw attention to two more observations in Fig. 1a: 1) as expected, WA values are indeed an order or two lower than W so that almost all WA values are below the MOM80 line; and 2) the wind speed dependence WA(U10) is stronger than that of the W(U10) relationship.
Values of active whitecap fraction WA obtained with additional kinematic considerations (Mironov and Dulov 2008; Kleiss and Melville 2010) are distinguished from the intensity-based WA values in Fig. 1b (magenta symbols). A noteworthy observation here is the smaller spread among these “kinematic” WA values. While the reason for this could partially be the fact that WA values obtained with kinematic considerations are fewer, it also points to the potential for improved WA extraction using information in addition to intensity threshold.
b. Novel approach: The Phillips concept and energy dissipation rate
Kleiss and Melville (2010) are the first to use the Phillips concept of breaking crest length distribution Λ(c)dc to estimate active whitecap fraction WA. They use sea surface imagery of breaking waves and whitecaps to obtain Λ(c)dc and then these measured Λ values assist the extraction of WA from the images. Here we describe our approach to obtaining active whitecap fraction WA by using the Phillips concept of Λ(c)dc in a different way.
The physical basis for our approach is the connection between active whitecap fraction WA and the energy dissipation rate ε provided by the moments of Λ(c)dc (section 1). Solving for the first moment cΛ(c)dc in (2) and substituting it in (1), we obtain
where the persistence time of the bubbles T is assumed a constant. In (3), the term ε(c)dc can be presented with an expression for the spectral rate of energy loss from the wave components in the equilibrium range of the wave spectrum [Phillips 1985, Eq. (6.6) therein]. Using such an expression, we integrate (3) over a suitable range of breaking front speeds c = (cmin, cmax). In the result of this integration, one can recognize an expression that can be obtained from the total (i.e., integrated) dissipation rate . Then for the active whitecap fraction we have
Having this expression, one needs values for εt, T, b, cmin, and cmax to obtain WA. What is important in representing the active whitecap fraction with (4) is that via the breaker speed c and the total energy dissipation rate εt the kinematics and dynamics of the breaking waves determine the values of WA. That is, the active, turbulent part of the whitecaps that moves along with the breaker is objectively separated from the residual, more static part of the whitecaps. In this, our approach is similar to those of Mironov and Dulov (2008) and Kleiss and Melville (2010). Our approach differs from theirs in that that it avoids (or minimizes) the use of photographs and video images; rather it relies primarily on measured or modeled values for εt. Further discussion on the merits and caveats of this approach is given in section 5.
The challenge in obtaining reliable estimates of active whitecap fraction WA with (4) is to obtain and choose well-constrained values for the total energy dissipation rate εt and the various parameters in (4). In this section we present one possible implementation of this approach. We describe existing methodologies to obtain εt [section 3a(1)], the parametric model we use [section 3a(2)], and the buoy data used to calculate εt with this parametric model [section 3a(3)]. We also justify choices for parameters T, b, cmin, and cmax in (4) (section 3b), and variations of these choices used to investigate the sensitivity of the WA estimates to the parameter values (section 3c).
a. Total energy dissipation rate
While kinematic and dynamic considerations ensure that (4) can determine the active whitecap fraction, the more general premise of using (4) is that the total energy dissipation rate εt is a suitable measure for wave breaking with air entrainment. The empirical expression W(ε) that Hanson and Phillips (1999) derived using field data attests to the validity of this general premise by showing improved correlation coefficient of 0.86 for W(ε) compared to a correlation of 0.72 for the W(U10) relationship. Hanson and Phillips (1999) expected such an outcome on the basis of two laboratory observations: 1) the energy transfer from breaking waves to the upper ocean layer may be comparable to that transferred from the wind (Melville and Rapp 1985) and 2) the air volume entrained by a breaking wave is proportional to the energy dissipated by the wave (Lamarre and Melville 1991). Field observations also clearly link enhanced energy dissipation rates in the upper ocean to wave breaking (Agrawal et al. 1992; Drennan et al. 1996; Terray et al. 1996; Gemmrich and Farmer 2004; D’Asaro et al. 2014).
1) Existing methodologies
Measurements of εt usually record turbulent velocity profiles with different sensors mounted on different platforms (Lueck et al. 2002) including a current meter (Drennan et al. 1996) and acoustic Doppler profiler (Veron and Melville 1999). To effectively separate the turbulence properties from the surface wave signal, the sensors need to be optimized for noise level, response time, and spatial resolution (Soloviev et al. 1999) so that small velocity fluctuations are obtained at time scales of breaking events and length scales within the turbulent inertial subrange (defined as the intermediate range of turbulent scales or wavelengths that is smaller than the energy-containing eddies but larger than viscous eddies).
Different processing techniques are applied to obtain energy dissipation rates from the measured velocity profiles. Dissipation rates obtained using turbulence structure function are directly linked to turbulent eddy scales corresponding to different depths below the surface (Thomson et al. 2009; Gemmrich 2010; Schwendeman et al. 2014). A turbulence structure function provides the highest spatial resolution compared to other processing techniques (Gemmrich 2010). Reported εt values obtained with this technique range from less than 0.1 to ≅3 W m−2 (Thomson et al. 2009, Fig. 1b therein) and from ≅0.5 up to 6 W m−2 (Gemmrich 2010, Fig. 8a therein). Figure 2 (magenta symbols) shows εt measured in the Strait of Juan de Fuca in fetch-limited (up to 15 km) stormy conditions with wind speeds from 9 to 18 m s−1 (Schwendeman et al. 2014).
The spectral inertial method relates dissipation rates to the wavenumber spectrum on the basis of the Kolmogorov hypothesis that in the inertial subrange the wavenumber spectrum has a universal form depending only on the energy dissipation (Soloviev and Lukas 2003; Gemmrich and Farmer 2004; Gerbi et al. 2009). In this method, the measured velocity profiles are converted to wavenumber spectra using spectral analysis; for example, Gemmrich and Farmer (2004) use Hilbert transform. The reported spectral energy dissipation rates range from 10−6 to 10−2 m2 s−3. Note that energy dissipation rate can be expressed in different units; in citing different works, we adhere to the units used by individual authors. See appendix B for details on the ε units.
One can obtain the total energy dissipation rate εt from measured or modeled wavenumber k or frequency ω spectra, S(k) or S(ω). This approach is based on the equilibrium range theory of Phillips (1985) developed with the assumption that in the equilibrium range the energy source and sink terms that drive the evolution and propagation of the wave energy spectral density (Komen et al. 1994) are in balance (Hanson and Phillips 1999):
In (5), Sin is the energy input from the wind, Snl is the nonlinear wave–wave interaction that transfers the input energy from large to small scales, and Sds is the spectral energy dissipation. Because Snl merely redistributes the energy between different wave scales, it vanishes in the total integrated energy balance over k (or ω) and direction θ so that the spectrally averaged wind input 〈Sin〉 and dissipation 〈Sds〉 balance each other. Assuming small to negligible local and advective wave growth (e.g., ≅6%; Hasselmann et al. 1973; Donelan 1998; Csanady 2001; Hwang and Sletten 2008), one may consider the total wind energy input primarily dissipated by breaking waves. One can therefore use Sds or Sin to obtain total energy dissipation rate εt. Phillips (1985) suggested that in the equilibrium range, where the energy conservation law given with (5) holds, each process modeled in (5) can be expressed in terms of the wave spectrum in this equilibrium range.
Phillips (1985) approach was tested with measured wave frequency spectra. Felizardo and Melville (1995) summarized spectral dissipation models cast in terms of frequency spectrum and used them to obtain time series of total energy dissipation rate from measurements of the wave spectrum in deep water (3000 m) in the Pacific Ocean (130 km west of Reedsport, Oregon). The estimated εt showed wide dynamic range from less than 1 × 10−4 to ~1 kg s−3 (which is equivalent to W m−2, as shown in appendix B) for wind speeds from 1 to 12 m s−1. Starting with ε(k) = ωSds(k), converting to ε(ω), and using Phillips (1985) equilibrium range wave spectrum S(ω), Hanson and Phillips (1999) developed expressions for 〈Sds〉 and εt [their Eq. (26)]:
Using (6), Hanson and Phillips (1999) obtained total dissipation rate εt for open ocean conditions from measurements of directional wave spectra in the Gulf of Alaska using the time series of heave, pitch, and roll buoy. When obtaining εt from measured wave spectra, it is important to use the wind sea components of the wave spectra and remove the swell. Hanson and Phillips (1999) remove the swell using a topographic minima method (Hanson and Phillips 2001). The resulting εt ranged from 1 × 10−3 to 2 kg s−3 (≡W m−2) for wind speeds from 5 to more than 15 m s−1. These values are shown in Fig. 2 (green symbols). Use of (6) or other functional forms for 〈Sds〉 with a modeled wave spectrum S(ω) comes naturally as another possibility to obtain εt (e.g., Rogers et al. 2012).
As mentioned earlier (section 1), only recently (2) has been used to obtain spectral ε(c) and total εt dissipation rates using measurements of the breaking crest length distribution Λ(c)dc. The latest development of this method is the use of a stereo pair of longwave infrared (IR) cameras to reconstruct the surface elevation and velocity fields (Sutherland and Melville 2013). The sensitivity of the IR cameras to the disruption of the cool skin layer allows microscale breakers, without air entrainment, to be detected. This extends the range of energy dissipation estimates to breaker velocities pertinent for short gravity to capillary waves. While overall it is useful to expand measurements of energy dissipation to short scales, estimates of εt with this technique would overestimate air entrainment and the active whitecap fraction WA considered here.
2) Parametric model used in this study
In this study, we use a parameterization of εt developed by Hwang and Sletten (2008) based on the energy balance of a fetch- or duration-limited wind wave system. It is expressed in terms of the neutral reference wind speed U10 and wave characteristics in the form
Here, ρa is the air density and α is a wave parameter determined using nondimensional values of the peak frequency of the wave spectrum ωp, , and the root-mean-square value of the (wind sea) surface displacement ηrms, . The model builds on assumption (5) and uses wind input parameterizations. Appendix B gives a brief account of the main steps in deriving the parametric model shown in (7).
Hwang and Sletten (2008, their Fig. 4) show that for the conditions frequently encountered in the field (ω* between about 0.8 and 2, but computed to 10), the values of α are within a narrow range (3.7, …, 5.7) × 10−4. The εt values computed with this parametric model compare reasonably well with field measurements of εt (Hanson and Phillips 1999; Felizardo and Melville 1995; Terray et al. 1996). Figure 2 compares εt estimates from this parametric model (red lines) to the field measurements of Hanson and Phillips (1999) and Schwendeman et al. (2014).
3) Data used with the parametric model
We obtain total energy dissipation rate εt with (7) using data from the National Data Buoy Center (NDBC) for buoy 46001 moored in the western Gulf of Alaska at (56.3°N, 147.9°W) in depth of 4206 m (ndbc.noaa.gov/station_page.php?station=46001). The buoy data are archived as 1-h time series of wind vector (speed and direction) and wave field characteristics (significant wave height and peak wave period) for 2006. Because these wave characteristics are for mixed sea (swell and wind waves), while wave parameter α in (7) should be calculated for wind seas, we also use the buoy archived data for spectral wave density to separate sea and swell with the method described by Hwang et al. (2012).
The wind speed and direction are collected by an anemometer located on the buoy’s mast 5 m above the sea level. We converted the wind speed values to winds at 10-m reference height using a multiplication factor of 1.06 (Wentz 1992; Hwang and Sletten 2008). The wind speed U10 and wind direction ϕ are the average values over an 8-min acquisition period starting at 42 minutes after the hour; the wind center time is thus at 46 minutes of each hour.
NDBC-reported wave measurements are based on the heave acceleration or the vertical displacement of the buoy hull during the wave acquisition time. A fast Fourier transform is applied to derive nondirectional spectral wave measurements such as wave energies with their associated frequencies (ndbc.noaa.gov/wave.shtml). The significant wave height and peak wave period are then derived from the spectral energies. Buoy heave motion spectrum is measured over a 20-min acquisition period starting at 30 minutes after the hour (i.e., the wave center time is at 40 minutes of each hour). With the swell removed from the spectral energies, we determined the significant wave height Hs and peak wave period for the wind seas Tpw. Phase speed for the wind sea was obtained as cpw = gTpw/(2π). Wind sea wave period Tpw = 2π/ωpw and significant wave height Hs = 4ηrms from the wind sea portion of the buoy data were used to calculate the nondimensional frequency ω* and nondimensional displacement variance η* needed for the wave parameter α in (7).
b. Parameter choices
There are four parameters in (4) whose values need to be chosen to obtain active whitecap fraction—the breaking strength parameter b, the bubble persistence time T, the threshold breaker speed cmin, and the upper limit of the range of the breaker speeds cmax. We made a literature review for each parameter to find suitable measured and/or suggested values.
The breaking strength parameter b has been initially considered to be a constant (Phillips 1985). Laboratory and field experiments, as well as modeling studies, have shown that parameter b varies widely. Table 2 expands the table with b values compiled by Drazen et al. (2008) with newly published values. Figure 3 visualizes the wide range of variation of b, from less than 0.1 × 10−3 to around 50 × 10−3. The value of 0.06 (≡60 × 10−3) suggested by Phillips (1985) on the basis of Duncan (1981) laboratory experiments is on the large end of values (see data points in Fig. 3 and entries in Table 2 at reference numbers 1 and 2). Gemmrich et al. (2013) comment that a value of 70 × 10−3 would be too high to give realistic breaking rates. Still, several experimental b values are of this order (reference numbers 10–12 in Fig. 3). A group of experimental and modeling values are of order 0.01 (≡10 × 10−3, reference numbers 3, 4, 6, 7, 13, 14). Babanin et al. [2010, their Eqs. (23)–(25)] combine modeled dissipation values Sds with the Λ(c) parameterization of Melville and Matusov (2002) to predict b. Babanin et al. (2010) obtain b = O(10−2) only in the range of the wind forcing conditions 1.5 < U10/cp < 2, where U10/cp (=ω*) is the inverse wave age with cp being the phase speed of the spectral-peak waves. This model prediction for b (reference number 14 in Fig. 3) is comparable to the value of 8.5 × 10−3 that Melville and Matusov (2002) used to obtain Λ(c) moments. Gemmrich et al. (2008, 2013) report the lowest b values (reference numbers 9 and 16 in Fig. 3). Breaking parameters obtained from other recent field campaigns (Romero et al. 2012; Schwendeman et al. 2014) revise the high b values previously reported by the these same groups—compare in Fig. 3 the b values at reference numbers 15 and 17 to those at reference numbers 10 and 11, respectively. The wide range of observed values for b and the physical understanding that a parameter describing the strength of breaking waves having various scales could be expected to vary with breaking wave scale as well both justify the expectation that b is not a constant. This has led to efforts to measure spectrally resolved breaking parameter b(c) or b(k); results have been reported only recently (Romero et al. 2012; Gemmrich et al. 2013). Experimental and modeling approaches to estimate b are currently an active area of research, so presently there is not a good physical justification to consider some values as more suitable than others. Fully aware that this argument needs to be revised in time, for now we use the average of all reported values b = 12.997 × 10−3 (≅0.013, red symbol in Fig. 3).
For the bubble persistence time, we use a constant value T = 2 s. This value is based on the field measurements of Callaghan et al. (2012) and is the time for rapid growth of the whitecap area, a process associated with the active phase of the whitecaps (e.g., Mironov and Dulov 2008; Anguelova and Huq 2012). Two comparisons help to evaluate this choice. First, T = 2 s is roughly twice shorter than the peak of the probability density functions of whitecap decay times reported by Callaghan et al. (2012) for field data and the whitecap decay time of 3.53 s reported by Monahan et al. (1982) for laboratory data. Second, this value is an upper limit to the mean breaking event durations reported by Ding and Farmer (1994), from 1.4 to 1.8 s for winds from 6 to 12 m s−1, and obtained from acoustic measurements that are closely related to active breaking crests. Mueller and Veron (2009) justified the use of T = βTTp, where Tp is the dominant wave period at the peak of the wave spectrum. Coefficient βT was chosen to be 0.2 so that their values of WA match closely the WA data of Monahan (1993). Proportionality between T and Tp assumes noticeable influence of the wave field on the bubble persistence. We opine, however, that the wave field has weak, if any, effect on bubble persistence time, and factors like salinity, water temperature, and surfactants affect T much more. Because current knowledge of the effect of these quantities on T is limited (Zheng et al. 1983), we opted for the constant value.
A host of values can be found in the literature for the threshold breaker speed cmin (denoted cT by Phillips (1985) or cb or cbr elsewhere in the literature). It becomes increasingly clear that the breaking crest speeds are lower than the phase speed cp of the underlying dominant wave (Frasier et al. 1998; Hwang et al. 2008; Hwang 2009; Babanin et al. 2010; Banner et al. 2014a) so that cmin = αccp The factor αc is not known precisely (Banner et al. 2014b). Values αc ≥ 0.8 have been suggested and used (Melville and Matusov 2002; Banner and Peirson 2007; Romero et al. 2012). Laboratory measurements of breaking crests velocities with two different techniques have shown that, depending on the experimental conditions and the measuring technique, this factor varies over a wide range, 0.22 ≤ αc ≤ 0.85 (Jessup and Phadnis 2005). Feature tracking and Doppler analysis of radar sea spikes (closely associated with wave breaking) at low grazing angle also show that the dominant velocity of breaking features is much smaller than cp (Frasier et al. 1998; Hwang et al. 2008). Using field experiment, Gemmrich et al. (2008) found that 0.1 ≤ αc ≤ 1 for developing sea conditions, but αc values were restricted to 0.1 ≤ αc ≤ 0.4 for fully developed conditions. Following Gemmrich et al. (2008), we choose to work with αc = 0.3.
For the maximum value cmax, a choice of the ratio cmax/cmin = 10 seems well justified from available data for breaking statics and Λ(c) distributions. Ding and Farmer (1994) found nonnegligible breaking event speeds up to 10 m s−1 for winds from 6 to 12 m s−1. This minimum to maximum breaker speed change is comparable to that measured by Gemmrich et al. (2008, 2013) and Zappa et al. (2012). Kleiss and Melville (2011) present Λ(c) values from observed breaking events over the same speed range of up to 10 m s−1. Meanwhile, Phillips et al. (2001) obtained from field radar data much lower intrinsic event speeds, from 3 m s−1 to a maximum of 6 m s−1. Similarly, Jessup and Phadnis (2005) observed in laboratory experiment a factor of 2 to 3 difference between minimum and maximum breaker speeds.
c. Sensitivity to parameter choices
Equation (4) with the expression of active whitecap fraction in terms of energy dissipation rate WA(ε) shows how each of the chosen parameters (b, T, αc, and cmax/cmin) can influence the WA estimates. It is clear that WA will increase if we select longer T and lower b, and restrict the breakers to slower speeds with a lower ratio cmax/cmin. Conversely, shorter T, higher b, and faster breakers (determined by higher αc so that cmin approaches cp) will result in lower WA values. Of interest, however, is to evaluate the degree of influence that each of these parameters has on the WA(ε) estimates.
We evaluate the effect of parameter choices with a sensitivity analysis in which we obtained WA(ε) with (4) for different sets of parameters (b, T, αc, and cmax/cmin). Table 3 lists the chosen set of parameters as set 0. Four more sets of parameters are formed by changing only one parameter and keeping the remaining three parameters equal to those in set 0. The parameter changes that we made are informed by the possible range of values available from the literature review (section 3b). Set 1 changes αc from 0.3 to 0.8 in order to evaluate the effect of using higher threshold speed for the breakers. Set 2 evaluates the effect of lower upper limit of the breaker speeds by changing cmax/cmin from 10 to 3. The effect that the bubble persistence time has on WA is evaluated with set 3 in which T is changed from 2 to 4 s. Set 4 aims to show the influence of the breaking parameter b. Because the chosen value of 0.013 (≡13 × 10−3) represents mostly the higher b values (Table 2 and Fig. 3), we decreased it by an order of magnitude to 0.001 (≡1 × 10−3) to evaluate the effect of the reported low b values.
We use three different metrics to quantify the sensitivity analysis results. The average and standard deviation of all WA(ε) estimates obtained with (4) for each set of parameters in Table 3, denoted hereafter with 〈WA〉 and , track changes in the obtained WA values due to different parameter choices. We assess differences between the WA values from the energy dissipation approach WA(ε) and the WA values from the intensity threshold approach with the percent difference PD. Designating the average of all WA values from photographs (those plotted with black symbols in Fig. 1) as a reference, hereafter denoted as 〈WA〉r, we determine the percent difference as . Note that PD gives relative differences between two quantities in percent, which is different from the unit of the absolute whitecap fraction values reported throughout the text in percent, not as fractions. Finally, we assess differences in wind speed trends between the new WA(ε) estimates and the conventional WA from photographs by considering the bias ΔWA between them. For this purpose we binned the new WA(ε) estimates and the available photographic WA values by wind speed in bins of 1 m s−1. The bias is formulated as ΔWA = WA − WA(ε) for each bin. Only bins containing more than 10 data points are used.
Figure 4 shows wind speed U10, wind direction ϕ, significant wave height Hs, and wind sea period Tpw measured at buoy 46001 for the period of 24 May–16 November (days 140–320 of the year). For this 6-month period, wind speeds up to ~23 m s−1 were recorded (Table 4) with most probable values of 5–12 m s−1 determined from data histogram. Wind direction was in most of the cases from northeast to southwest and from east to west (ϕ in the range from ~225° to 270°). It could be expected that such wind directions exclude the presence of swell coming from the south in most instances. This expectation is consistent with the fact that the most probable values of the mixed sea periods Tp obtained from NDBC (not shown) are below 10 s. The majority of the wind sea periods Tpw obtained after swell removal are also predominantly below 10 s (Fig. 4d and Table 4).
a. Active whitecap fraction
Figure 5 shows the wave parameter α as a function of dimensionless frequency ω*. The effect that the swell has on α is illustrated by comparing the wind sea α values to those of the mixed seas (blue symbols). It is seen that excluding swell, many α values at low ω* are removed (Fig. 5a). The trends of the same data binned by ω* in bins of 0.125 (Fig. 5b) show that, in absence of swell, α generally decreases (black symbols with error bars). The α values calculated with the buoy data for ω* ∈ (0.007–2.5) span the range (0.013–6) × 10−3. This range is much wider than the narrow range of α values for ω* ∈ (0.8–2) that Hwang and Sletten (2008) considered [recall section 3a(2)].
Figure 6 shows the total dissipation rate εt for wind seas as a function of wind speed U10. Shown for comparison is εt from (7) with the mean value of the wave parameter (Fig. 5) α = 5.7 × 10−4 (red line). Compared to our εt values are εt from the field campaigns in the Gulf of Alaska (Hanson and Phillips 1999) and the Strait of Juan de Fuca (Schwendeman et al. 2014) plotted in Fig. 2. For wind speeds above 3 m s−1, our total energy dissipation rates obtained from buoy data are up to 8.6 W m−2 (Table 4). Note that the spread of the εt values at a given wind speed is about one order of magnitude or less.
The active whitecap fraction is obtained with (4) using εt (Fig. 6 and Table 4) and the set of chosen parameters (set 0 in Table 3): b = 0.013, T = 2 s, αc = 0.3 for cmin, and cmax/cmin = 10. With the data for cpw, we obtain cmin values in the range from 1.16 to 8.14 m s−1 (Table 4). This range of breaker speeds seems reasonable when compared to the speed range of 1.5 to 15 m s−1 that Gemmrich et al. (2008) used to evaluate momentum flux and energy dissipation rate for waves generating visible whitecaps. Figure 7a shows the results for active whitecap fraction WA for wind speeds above 3 m s−1. The WA(ε) estimates span a wide range of values, from ~10−4% to about 7% (Table 4). Although the εt values show relatively small spread (Fig. 6), the resulting WA values show a spread of about two orders of magnitude; this is further discussed in section 5.
For the set of chosen parameters, the WA(ε) estimates are comparable in magnitude to WA values from previous photographic measurements (Fig. 7a, magenta symbols). The percent difference PD between the two approaches is +80% (Table 3), meaning that on average the photographic WA data are 80% higher relative to the WA(ε) estimates with the chosen parameters. Figure 7b shows the same data binned by wind speed. The wind speed dependences for the two approaches seem similar. However, the bias ΔWA (the difference between the binned trends as defined in section 3c) shows that the wind speed dependences diverge noticeably at high winds (Fig. 7c, solid black symbols). We verified the statistical significance of the biases ΔWA by performing a Student’s t statistic for the photographic WA values and the dissipation WA(ε) values at the chosen set of parameters. That is, we evaluated the probability that the means of the sampled photographic and dissipation WA values in each wind speed bin in Fig. 7b are significantly different. We found significant difference (with a p value below 0.05) between the means of the two samples for the wind speed bins from 7 to 19 m s−1. The bias ΔWA is as little as 5 × 10−3% and up to 2% (in absolute units % not relative units as in PD above); we discuss this further in section 4b.
b. Sensitivity analysis
Figure 8, Table 3, and Fig. 7c (open symbols) show results for the sensitivity of the WA(ε) estimates to the parameter choices. As expected from (4), we obtain lower WA(ε) values when the threshold breaker speed cmin increases from 0.3cpw to 0.8cpw (set 1 in Table 3 and Fig. 8a). This is physically plausible because choosing a higher breaker speed we place more emphasis on faster thus longer waves that may break less frequently (Phillips et al. 2001). The percent difference between the resulting WA(ε) values and the photographic WA data increases to about 100%. The bias ΔWA (Fig. 7c, squares) ranges from ~10−2% to 2.27%.
Figure 8b shows the effect of the upper integration limit in (3). If we conform to observations of narrow range of breaker speeds (Phillips et al. 2001; Jessup and Phadnis 2005) and choose lower limit cmax/cmin = 3, the resulting WA(ε) values are only 58% lower relatively to the photographic WA data (set 2 in Table 3). This is expected because the logarithmic value of cmax/cmin appears in (4), making this a less sensitive variable. The bias ΔWA (Fig. 7c, triangles) ranges from negligible ~10−4% to 1.72%. However, the small differences compared to the photographic data do not necessarily mean that the WA(ε) values are closer to the true values. Rather, small differences remind us of the difficulties that the intensity threshold approach has in discriminating between active crests and decaying whitecaps. Because a lower limit of integration suggests slower breakers, this might reflect the presence of decaying whitecaps in our WA(ε) estimates. This would therefore bring the reported smaller difference with photographic WA data, which could plausibly contain estimates for decaying whitecaps as well.
Figure 8c shows that doubling the persistence of bubbles T from 2 to 4 s leads to higher coverage by active whitecaps with 〈WA〉 increasing by a factor of 2 compared to 〈WA〉 for the set of chosen parameters (set 3 in Table 3). The increase of the WA(ε) values could be expected because larger T values suggest long-lived and thus decaying foam, which covers a larger area. The PD for these WA(ε) values and the photographic data is about 60%. The bias ΔWA (Fig. 7c, diamonds) is comparable to that of set 2 with a range from ~10−3% to 1.74%. That the effect of set 3 on WA(ε) values is comparable to that of the upper limit integration (set 2) supports our reasoning above regarding the impact of set 2. That is, the smaller PD in WA(ε) values with set 3 is not necessarily a sign of improved WA estimates.
Figure 8d shows that the impact of the breaking parameter b on the WA(ε) values is by far the most influential. The choice of b = 0.001 (set 4 in Table 3) is an upper limit to the low values obtained by Gemmrich et al. (2008, 2013) and a lower limit of the recent b evaluations of Romero et al. (2012) and Schwendeman et al. (2014). Lower b values increase the WA(ε) estimates relative to the photographic WA data by about 161% (see Table 3). Figure 7c (red circles) shows the largest bias ΔWA, up to about 3%. As expected, larger b values, say b = 0.06 that Phillips (1985) initially proposed, leads to lower WA(ε) estimates that differ relative to the photographic data by 96% (not shown).
The biases ΔWA for all sets of parameters (Fig. 7c) show statistically significant (section 4a) divergence of the WA(ε) estimates from photographic WA values as the wind speed increases, especially above 15 m s−1. This suggest leveling off in the wind speed dependence of the WA(ε) values at high winds, while the photographic WA values keep increasing with U10. Note that such a leveling off might be different (or absent) for WA(ε) values obtained with data from other buoys. While we do not attempt to identify the physical reasons for this leveling off, we note that Holthuijsen et al. (2012, their Fig. 6c) show almost flat wind speed dependence for the active whitecap fraction and strong wind speed dependence for the streak coverage (i.e., residual whitecap fraction). In light of Holthuijsen et al.’s (2012) observation, the leveling off in the wind speed dependence of the WA(ε) estimates in Fig. 7c is an indirect support of the notion that these estimates are primarily associated with the active breaking crests and more reliably exclude the residual whitecaps than the photographic WA values.
a. Merits and limitations of the novel approach
The results in Fig. 7 show that energy dissipation rate can be used successfully to obtain whitecap fraction as an alternative to using photographs of the sea state. Together with the possibility to obtain whitecap fraction from radiometric observations of the sea surface emissivity (Anguelova and Webster 2006; Salisbury et al. 2013), there are now three different and independent methods to quantify the whitecapping in the ocean. While it is not yet easy to pinpoint differences caused by different principles of measurement, in time, as data amass, the availability of whitecap fraction estimates obtained with different methods can help to better constrain them and decrease their spread. Understanding and minimizing variability in whitecap fraction values due to measuring methodology will help to better evaluate the natural variability of whitecap formation, spatial extent, and persistence.
Another merit of the novel approach of obtaining whitecap fraction is that the energy dissipation rate can be obtained in different ways. Here we demonstrate that the parametric model of Hwang and Sletten (2008) gives quite good results. However, one can use any of the methodologies, experimental or modeling, listed in section 3a(1) to provide εt in (4). Usage of different measurements for ε (e.g., measurements of the velocity fields or directional wavenumber spectra) would expand experimental observations of whitecaps to day and night, avoiding the daylight need for video records and widening the range of environmental conditions that affect whitecaps.
Considering the motivation of this study (section 1), we see the most important merit of the novel approach in the ability to objectively obtain the active whitecap fraction WA on the basis of the kinematic and dynamic characteristics of breaking waves. A solid basis for this objectivity is the derivation of the WA(ε) expression from the Phillips (1985) concept for breaking wave statistics. Furthermore, using the WA(ε) expression, we circumvent the need to measure the breaking crest length distribution Λ(c) in order to obtain WA, and rely on measurements and/or models for energy dissipation rate ε which are increasingly available and steadily improving.
One caveat for the novel approach of (4) is that it still could be somewhat dependent on photographic data and intensity threshold processing. Indirect influence comes about in our results in two ways: via comparison (calibration) of our WA(ε) values to photographic WA observations and via parameter b in the WA(ε) expression. Figure 1b shows narrower spread of the “kinematic” WA data points (section 2a) as compared to the conventional ones. We envision that the increasing availability of such kinematic-improved conventional WA data for calibration of WA(ε) results would minimize this influence of the intensity threshold.
Indirect intensity threshold influence on the WA(ε) values comes via the breaking parameter because b is usually obtained from measurements of Λ(c), which, in turn, is obtained from photographs. The image processing used to obtain Λ may involve the conventional intensity thresholding (e.g., Kleiss and Melville 2011, their section 3) or application of intensity thresholding to the difference of consecutive images (Gemmrich et al. 2008; Kleiss and Melville 2011, their section 4). The use of image differences effectively removes the static (residual) foam and defines well only the moving breaking crest. This ensures more objective determination of Λ (thus b and ultimately WA) compared to the conventional method. Parameter b can also be determined from measurements of wave steepness (e.g., Drazen et al. 2008) and avoid the use of Λ measurements from photographs altogether. In short, novel and improved measurements of parameter b have the potential to decrease the influence of photographic intensity threshold on WA(ε) values.
Another limitation of the novel WA(ε) approach may come from the underlying assumptions (section 3a) and the reliance on the equilibrium theory to fulfill (5). While experiments have shown that energy dissipation rates in the upper ocean are primarily related to wave breaking, there could be conditions for which substantial portion of the wave energy is lost to turbulence production without air entrainment and work against buoyancy forces. New laboratory and field experiments on momentum and energy transfer at the air–sea interface can help to more accurately partition the energy loss via different pathways (breaking long waves, short waves, and viscous shear) (Csanady 2001). Continuous calibration to improved photographic WA observations will also help to better constrain the assumptions for WA(ε).
b. Improvements and implications
Figure 7a shows two order of magnitude spread of the WA(ε) estimates. Considering that εt in Fig. 6 shows relatively small spread, the parameter choices in (4) are the most probable reason for the uncertainty. Of course, natural variability of whitecaps can also contribute to the scatter of WA(ε) values because bubble size distributions (and hence whitecap extent and persistence) at a given dissipation rate also depend on breaking severity, water temperature, and other factors (Anguelova and Webster 2006; Callaghan et al. 2008a). For example, Stramska and Petelski (2003) found that whitecap coverage for the same wind conditions in polar regions is about a quarter of what is observed at other latitudes. However, using radiometric observations of whitecap fraction at 10 GHz—a proxy for active whitecap fraction—Salisbury et al. (2013) showed that secondary factors have smaller influence on the variability of WA than on W. Therefore, currently the major path for improving WA(ε) values is by tuning the parameters used in (4), such as the bubble persistence time T and the breaker speed cmin. As anticipated (section 3b), the choice of the breaking strength parameter b has the largest effect on the calculated WA(ε) values for the simple reason that b has the largest uncertainty (Table 2). Thus to better constrain WA(ε) values, we must start with a refined choice of the b value.
We now present some possible improvements. First, note that whereas the set of parameters in (4) introduces a spread in the WA(ε) estimates, the sensitivity analysis shows that variations of these parameters do not yield variations in the spread of WA(ε) values (Fig. 8). Indeed, using 〈WA〉 and in Table 3, one can estimate that the relative variations remain the same for all sets of parameters. One possibility to diminish the spread of the WA(ε) estimates is to use parameters varying with the different breaking scales. We demonstrate this by evaluating after changing T and/or b from constants to some suitable representations T(c) and b(c). Let us use Mueller and Veron’s (2009) suggestion for the bubble persistence time T = βTTp with βT = 0.2 (section 3b). With the wind sea periods Tpw available from buoy 46001 [section 3a(3) and Table 4], the possible range of T to be used is from ~0.5 to ~3.5 s; note that our choice of T = 2 s, justified with different reasoning (section 3b), is close to the mean T value of 1.54 s in this range. We express T in terms of breaker speed c and modify (3) to
where b is still kept constant. Using derivation similar to that shown in appendix A, we modify (4) to a new expression that accounts for varying T values. With these changes, the spread of the WA(ε) estimates decreased by 30% compared to the WA(ε) scatter seen in Fig. 7a. On one hand, this is a somewhat small decrease, which supports our opinion that the influence of the wave field on T via the T–Tp relationship is relatively weak (section 3b). On the other, the diminished scatter in the WA(ε) estimates shows clearly that one way to reduce the uncertainty of the novel WA(ε) approach is to account for varying T values in different environmental conditions.
Next, a scale-dependent breaking parameter can be obtained by following Drazen et al. (2008) to express b in terms of the wave slope S, b ∝ S5/2. Using the wave spectral characteristics from our buoy data [section 3a(3)] to estimate the spectrally averaged wave slope as , we obtained a different b value for each data point available from (8). The slopes thus obtained are from 0.013 to 0.2, a range comparable to the slopes measured by Banner and Peirson (2007, their Fig. 7). Using an ad hoc coefficient of proportionality of 10, these slopes yielded b values from 0.1 × 10−3 to 178 × 10−3 with a mean of 17 × 10−3. With these b values, the new expression from (8) resulted in a much closer clustering of the WA(ε) estimates. Overall, using T(c) and scale-dependent b values in (8), the spread of the WA(ε) data points decreased by a factor of 3.8. This substantial decrease in the WA(ε) uncertainty came mainly from the use of varying breaking parameter. As new and improved b(S) representations become available (e.g., Romero et al. 2012), there is an opportunity to improve WA(ε) estimates as well. The order-of-magnitude evaluation here should be replaced with more rigorous approach that uses b(c) and T(c) relationships and numerically integrate the expression
The significant decrease of the WA(ε) uncertainty that resulted from the use of scale-dependent b can help to better understand the nature of the breaking strength parameter. Phillips (1985) postulated on dimensional grounds that b is a constant, yet published work (Table 2) and our analysis show how widely it may vary. The fact that b is changing so much implies (according to one of the reviewers of this paper) that the dimensional scaling of the energy dissipation rate is not all inclusive and an important scaling is missing (i.e., b should be presented by a true constant and additional quantity). The representation b(S) of Drazen et al. (2008) improved our estimates significantly, strongly suggesting that b should not be treated as a constant. Further work on b(S) and other scaling of b, including a relationship to the breaker speed b(c), are thus well justified.
A useful implication of objectively obtaining WA from ε is to use it in combination with radiometric measurements of whitecap fraction from satellites. The basic principle of passive microwave measurements determines the outcome of radiometric observations to be the total whitecap fraction W (Anguelova and Webster 2006; Anguelova and Gaiser 2011). It has been shown that an extensive database of W from satellite-based radiometric observations is useful for studying and parameterizing the variability of whitecap fraction (Anguelova et al. 2010; Salisbury et al. 2013, 2014). However, to make such a database useful for dynamic processes in the upper ocean, it is necessary to find a way to extract WA from W. Matching in time and space satellite-based W data with in situ data for WA affords estimates of a scaling factor R = WA/W. Having such a scaling factor parameterized in terms of readily measured variables (e.g., U10 or latitude), we can provide both W and WA from satellite radiometers. Preliminary results on combining W from satellite and WA(ε) from buoys have been presented (Anguelova and Hwang 2012); further work on this topic will be reported in a forthcoming paper.
Enhancement of air–sea interaction processes by wave breaking with air entrainment can be parameterized either in terms of total whitecap fraction W, which includes foam generated during active breaking of wind-driven waves and residual foam left behind by these breaking waves, or in terms of active whitecap fraction WA associated solely with the actively breaking crests. Predictions of sea spray aerosol production and heat exchange use values of W, while WA is needed for dynamic air–sea processes in the upper ocean such as turbulent mixing, gas exchange, ocean ambient noise, and spray-mediated intensification of tropical storms. To parameterize such processes, objective evaluations of WA separate from W are needed.
In this study we propose an approach to obtain active whitecap fraction WA using data for energy dissipation rate ε as an alternative to extracting WA from photographs using intensity threshold approach. We derive the WA(ε) expression (4) using the Phillips (1985) concept of breaking crest length distribution (section 2b). We calculate total (integrated) dissipation rate εt using a parametric model [section 3a(2)] and data from buoy 46001 for 2006 [section 3a(3)]. We justify suitable values for four parameters in the WA(ε) expression [(4)] (section 3b), namely breaking strength parameter b = 0.013, bubble persistence time T = 2 s, threshold breaker speed cmin = 0.3cp, and upper limit of the range of the breaker speeds with cmax/cmin = 10. We perform a sensitivity analysis to quantify the degree to which parameter choices influence the resulting WA(ε) estimates (section 3c).
Our results demonstrate the feasibility of obtaining WA from data for ε (Figs. 6 and 7). With the chosen parameters, the active whitecap fraction from the WA(ε) approach ranges from ~10−4% to about 7% for wind speeds up to 23 m s−1 (Table 4). The spread of the WA(ε) is about two orders of magnitudes (Fig. 7a), most probably because of using b and T as constants. The WA(ε) values are comparable to photographic WA values with relative differences no more than 80% (Table 3). The absolute bias between the WA(ε) and the photographic WA increases for higher winds (Fig. 7c, solid symbols) and is up to 2%. The sensitivity analysis (Table 3; Figs. 8 and 7c, open symbols) shows that the breaking parameter b has the greatest influence on the WA(ε) estimates, followed by the breaker minimal speed and bubble persistence time.
The main merit of the novel approach is the ability to objectively obtain the active whitecap fraction WA on the basis of the kinematic and dynamic characteristics of breaking waves. As anticipated, the use of parameters commensurate with different scales of breaking waves (e.g., T and b) in the WA(ε) expression [(8)] diminishes the spread of the WA(ε) estimates by a factor of 3.8. A useful implication of the WA(ε) expression is the possibility to extract WA from satellite-based radiometric measurements of W.
This work was sponsored by the Office of Naval Research, NRL Program element 61153N WU 4500. We thank Brian Ward and Brian Scanlon for sharing whitecap data from the Knorr field campaign. We also thank the two anonymous reviewers for their careful reviews and valuable suggestions, which strengthened the paper.
Derivation of WA(ε)
Here is the surface wave spreading function [Felizardo and Melville 1995, their Eq. (4.5)]. Using this notation, we obtain
Here the approximation is used because for the fastest breaker speeds becomes very small number.
To further simplify (A4), we express the quantity in (A4) with the total energy input per unit volume into the surface layer turbulence by breaking waves in terms of the breaker velocity c [similarly to Eq. (4.5) in P85]:
The multiplication by the seawater density ρw gives the energy dissipation per unit volume and the correct units for εt [P85, Eq. (4.6) therein; Felizardo and Melville 1995, their Eq. (4.9); Hanson and Phillips 1999, their Eq. (21)]. From (A.5), we recognize that
Derivation of the Parametric Model for εt
The basis for the model is the balance equation for wave energy spectral density involving terms for wind input, nonlinear wave–wave interaction, and dissipation [section 3a(1)]. As reasoned in section 3a(1), the spectrally integrated (represented with angle brackets) wave–wave interaction terms is 〈Snl〉 = 0; note that HS08 use Q instead of S for the terms in (5). Using field data, HS08 prove the expected balance 〈Sin〉 + 〈Sds〉 = 〈Snet〉 ≅ 0 so that 〈Sin〉 ≅ 〈Sds〉 is justified.
To obtain an expression for the total energy dissipation rate εt, HS08 start with the energy E(ω, t) of a time-varying field comprising a range of spectral wave components with a spectrum S(ω, t):
where ρw is the seawater density and g is acceleration due to gravity. On the basis of (5) and 〈Sin〉 ≅ 〈Sds〉, the spectrally averaged nonstationary wave field can be represented with any of these terms 〈Sx〉, where x = in or ds, as follows:
The total energy of the wave field then is
When 〈Sx〉 ≡ 〈Sds〉, we get the total energy dissipation rate:
Because the functional formulations for wind input term 〈Sin〉 are considerably more advanced than those of breaking dissipation term 〈Sds〉, HS08 represent the dissipation term 〈Sds〉 with a function derived for 〈Sin〉 on the basis of 〈Sin〉 ≅ 〈Sds〉. Term 〈Sin〉 (or 〈Sds〉) can be represented as a function of wave field characteristics such as spectral peak frequency ωp and the variance of the surface elevation using the fetch- or duration-limited wave development relationship [section 2.1 in HS08]:
Here 〈γ〉 is the nondimensional integrated wind input and s = ρa/ρw is the ratio of air and seawater densities. For deep water waves, HS08 (their section 2.2) approximate 〈γ〉 as
where ω* is nondimensional frequency. HS08 nondimensionalize ωp and using g and neutral wind speed at 10-m reference height U10 as follows:
Grouping the wave field characteristics into a wave field parameter , we obtain the parametric model for total energy dissipation (7) as .
To find expression (B.3) for 〈γ〉, HS08 define the integrated wind input 〈γ〉 as
where γ is the growth rate for individual wave components and χ(ω) is the wave frequency spectral density. Because the wind input is dominated in the short wave section of the wave spectrum, the integration is carried out from the peak frequency ωp to Nωp, where N is a positive number. HS08 conducted sensitivity tests to determine N and the number 10 is recommended as the best match to the growth rate measurements (see their appendix A and the corrigendum). In practice, it is difficult to measure the intrinsic frequency spectrum beyond a couple of ωp because of the Doppler frequency shift problem.
For the range of integration in (B6), the spectral function is assumed to be a power-law function χ(ω) = Aω−a. HS08 use the Joint North Sea Wave Project (JONSWAP) and Donelan spectra (Hasselmann et al. 1973; Donelan et al. 1985) to determine coefficients A and a for χ(ω).
The growth rate γ is parameterized in terms of the ratio of wind speed U10 and wave phase speed cp, U10/cp or (U10/cp − 1). The parameterizations are in power-law form [Eqs. (8) and (9) in HS08] and were obtained with least squares fitting to field data. The field data used are those of Snyder et al. (1981), Hsiao and Shemdin (1983), and Donelan et al. (2006), who obtained γ from direct, independent, and simultaneous measurements of surface elevation, air pressure, and wind flow profile.
Because ε represents dissipation of turbulent kinetic energy (e.g., Gemmrich and Farmer 2004), the units of ε are determined by expressing the change of the kinetic energy Ek in time t per unit mass m in a flow with velocity V, as
The equivalency between the three unit expressions is obtained using the SI units N = kg m s−2, J = N m, and W = J s−1. Expression εd = ρε gives dissipation density (i.e., energy dissipation per unit volume) in units of watts per cubic meter. Most often energy dissipation per unit area A is used, as εt in the main text of this paper, so
where the unit equivalency is obtained with the same SI units listed above.