Breaking waves in deep water: measurements and modeling of energy dissipation
In the presence of strong winds, ocean surface waves dissipate significant amounts of energy by breaking. Here, breaking rates and wave-following turbulent dissipation rate measurements are compared with numerical WAVEWATCH III estimates of bulk energy dissipation rate. At high winds, the measurements suggest that turbulent dissipation becomes saturated; however, the modeled bulk dissipation continues to increase as a cubic function of wind speed. Similarly, the mean square slope (i.e., the steepness) of the measured waves becomes saturated, while the modeled mean squared slope grows linearly with wind speed. Only a weak relation is observed between breaker fraction and wind speed, possibly because these metrics do not capture the scale (e.g., crest length) of the breakers. Finally, the model skill for basic parameters such as significant wave height is shown to be sensitive to the dissipation rate, indicating that the model skill may be compromised under energetic conditions.
KeywordsWave breaking Energy dissipation Turbulence Prediction of wave dissipation Spectral wave model
Dissipation due to turbulent kinetic energy (DTKE) and work against buoyancy (Dbuoy) are typically the most important terms contributing to the total wave-breaking dissipation (Sds). DTKE values are commonly obtained by depth-integrating the profiles of turbulent kinetic energy rates, ϵ(z), measured using Doppler sonars near the sea surface. The buoyancy work done on bubbles, however, is difficult to measure and cannot be resolved by Doppler sonars. A simplistic approach is sometimes applied to estimate wave-breaking dissipation by assuming it is equal to TKE dissipation only and that all the other dissipation terms are small (e.g., Carini et al. 2015; Thomson et al. 2016). This simplified relation has been shown to be accurate under moderate conditions, but neglecting the additional dissipation terms may lead to a significant underprediction of wave dissipation in the presence of strong winds (i.e., U10 > 15 m/s).
The importance of DTKE and Dbuoy relative to Sds has been the subject of several studies. For example, in a laboratory experiment, Lamarre and Melville (1991) found that for deep-water breaking waves, the contribution of buoyancy to the energy dissipation budget is 30–50%, while a laboratory experiment by Blenkinsopp and Chaplin (2007) and numerical simulations by Deike et al. (2016) find this ratio to be between 5 and 15%. More recently, laboratory work by Deane et al. (2016) suggests that bubble clouds may lead to an upper limit of turbulent dissipation rates of ϵ = O(102) m2 s−3. In the field, Thomson et al. (2016) observed a saturation of TKE dissipation rates for high wind conditions up to 20 m/s, though maximum values were only ϵ = 10−2 m2 s−3. Similarly, the TKE dissipation rates reported in Sutherland and Melville (2015), which were measured in winds up to 18 m/s, do not exceed ϵ = 10−2 m2 s−3. Setting aside the issue of the vertical distribution of turbulent dissipation for the moment, both of these field studies indicate a similar upper bound in the integrated turbulence DTKE. Yet, commonly used theories suggest that the bulk dissipation Sds should increase with increasing wind speeds, following the logic that wind input and wave energy dissipation are in quasi-equilibrium (Phillips 1985), and wind input increases with wind speed (Donelan et al. 2006). This suggests that the relative role of DTKE and other processes (such as Dbuoy) changes significantly as wind speed increases. The discrepancy between the in situ measurements and models can be partially explained by the occlusion of turbulent velocity measurements in bubble plumes, as well as limitations in the depths of the measurements; recent work has estimated that these effects bias the in situ dissipation estimates low by up to a factor of four in high winds (Derakhti et al., in prep).
Improving the performance of wave forecasts likely requires improving our physical understanding of energy dissipation during wave breaking. Forecasting wave models solve the spectral action balance equation and, simply put, estimate wave energy dissipation as the energy that balances the input energy from wind forcing. In addition, some formulations assume pre-defined spectral shape constraints (such as an f−5 diagnostic tail at the high-frequency end of the spectrum), while others incorporate bulk estimates of breaking and swell dissipation. These bulk parameterizations of energy dissipation do not explicitly resolve the physical processes involved in dissipating wave energy given in Eq. (1), and as such may limit the accuracy of spectral predictions.
Here, we analyze independent estimates of bulk wave-breaking dissipation and dissipation due to turbulence, not only against each other but also as they relate to wind speed, wave slopes, individual breakers, and overall predictability. The focus of this work is on high wind conditions, though not extreme conditions such as hurricanes.
To obtain estimates of turbulent dissipation rate DTKE, a Surface Wave Instrument Float with Tracking (SWIFT) is utilized. A SWIFT estimates turbulent dissipation rates in the upper layer of the water column using acoustic Doppler profilers. It also measures the wave spectrum and collects surface images for visual confirmation of wave breaking. The SWIFT was deployed twice in the winter of 2017, in the presence of strong winds up to 23 m/s.
To estimate bulk dissipation Sds in the wave field, we employ the third-generation numerical wave forecasting model WAVEWATCH III (henceforth WW3). This model is generally skilled in predicting wave height (e.g., Bi et al. 2015; Yang and Neary 2017; Ellenson and Özkan-Haller 2018), lending some confidence to the wave energy dissipation estimates produced by the model. Different physics packages can be applied in WW3 to model the relationship between wind input energy and bulk wave energy dissipation. These packages differ from each other in the way they consider the details of breaking dissipation and the ramifications on spectral shape. The packages have evolved over the course of several decades to include increasing levels of understanding of the dissipation mechanisms by wave breaking, including the effects of dissipation of short waves by long breaking waves, or the effects of favorable or adverse winds.
Details of field collection activities and hindcast simulation setup are described in the “Observations and simulations of interest” section. In the “Comparison of measured and modeled wave parameters” section, we assess the skill of the model in estimating bulk wave parameters as compared with the measurements. In the “Energy dissipation” section, we explore the relationship between measured turbulent dissipation rates and modeled wave dissipation rates. Energy spectra and breaking metrics, in connection with energy dissipation, are discussed in the “Discussion” section. The influence of energy dissipation on the estimation of bulk parameters is also discussed in this section. Finally, we outline our conclusions in the “Conclusions” section.
2 Observations and simulations of interest
2.1 SWIFT measurements
A Surface Wave Instrument Float with Tracking was deployed from a helicopter offshore from the Oregon coast on two occasions in the winter of 2017. SWIFTs are free-drifting buoys equipped with GPS and IMU sensors to measure wave-following motion, Doppler profilers (Nortek Aquadopp HR) to measure turbulent velocity profiles, and cameras to take pictures of the sea surface. Data are recorded in 9-min bursts, sampled at 25 Hz every 12 min. The additional 3 min is used to average, process, and telemeter the data collected during the previous 9 min. Images are captured every 4–6 s. An in-depth description of how SWIFTs operate is found in Thomson (2012).
To keep our scope on whitecapping—as opposed to nearshore, depth-limited, wave breaking—we only utilize observations made at depths greater than 50 m in our analysis. In both deployments, this cut-off point roughly corresponds to a non-dimensional (kh) depth of 2.5 and allows us to focus on wave breaking in deep water and in upper intermediate-water. In addition, we observed that during the April 2017 event, when the SWIFT first reached a 50-m water depth, the wind speed had dropped to roughly 15% of that recorded at the beginning of the deployment (from 22.3 down to 3.7 m/s), giving us confidence that this threshold captures the observations of whitecapping driven by strong winds.
2.2 WAVEWATCH III hindcasts
The most commonly used physics packages, implemented to describe the interrelation of Sin and Sds, are ST2, ST4, and ST6. The widely used ST2 package, developed by Tolman and Chalikov (1996), uses different criteria for dissipating energy at low and high spectral frequencies; at low frequencies, the dissipation parameterization is akin to turbulent dissipation, and at high frequencies, an f−5 spectral roll-off is prescribed. The formulation in the ST4 physics package, developed by Ardhuin et al. (2010), forgoes a pre-defined spectral shape and rather parameterizes energy dissipation based on steepness for swell conditions, and exceedance thresholds that capture directionality and dissipation of short waves by long breaking waves. The most recent physics package, ST6, was developed by Zieger et al. (2015) with a special focus on whitecapping dissipation. ST6 accounts for negative wind input (i.e., both favorable and adverse winds) and full air-flow separation and, after exceeding a steepness threshold, implements a dissipation formulation that depends on (1) an inherent breaking dissipation term as a function of the spectral density and (2) a forced term that becomes active at frequencies greater than the spectral peak, both proportional to the threshold exceedance. ST6 also enforces a spectral shape with an f−5 spectral roll-off. For a complete description, the reader is referred to the references cited above and the WW3 user manual (The WAVEWATCH III Development Group 2016).
In this study, the ST6 physics was chosen because of the emphasis on whitecapping and wave turbulence in its formulation. The non-linear wave-wave interaction term Snl was parameterized using the discrete interaction approximation (DIA) package described in Hasselmann et al. (1985).
The WW3 hindcasts were produced using three nested grids of increasing resolution as developed in a previous study of the Oregon coast completed by Ellenson and Özkan-Haller (2018). The largest of the grids covered the entire Pacific Ocean at a 30 arc-minute resolution. The second grid spanned the eastern North Pacific at a 7.5 arc-minute resolution. The third grid extended over the continental shelf (40.3 ° − 49.5 ° N, 233 ° − 236.25 ° W) at a 90 arc-seconds resolution.
The bathymetry for all grids was obtained from the National Geophysical Data Center’s ETOPO1 (Amante and Eakins 2009), and the third grid was also integrated with bathymetry from NOAA’s Gridded Tsunami Bathymetry. Wind input at 10-m elevation and air-sea temperature differences were obtained from NCEP’s Global Forecast System (GFS), at a 0.5 ° × 0.5° spatial resolution and 3-h time resolution. Wave frequencies are resolved between 0.003 and 1.23 Hz with a resolution ranging from 0.003 Hz for low frequencies up to 0.11 Hz for high frequencies. The wave directional resolution is 10°.
The output resolution of the hindcasts is 1 h. The output of the directional wave spectra was obtained at pre-determined spatial points chosen to match the mean location of the SWIFT at each hour of each deployment. In addition, several other parameters were computed over a grid spanning the finest resolution grid described earlier. Standard mean wave parameters that are available on a grid include significant wave height, mean wave period, and peak wave direction. Gridded parameters of the atmosphere-waves layer include wind-to-wave energy flux and friction velocity. The gridded output of wave to sea energy flux (i.e., depth-averaged wave dissipation rate) was also obtained.
2.3 Observed and modeled conditions
At the first hour of the February event, waves were predominantly coming from the SW, with a significant wave height of 6.3 m and a mean period of 9.8 s. Later during the deployment (not shown), the largest area of high significant wave height and dissipation shifted northward and away from the buoy. The smaller energetic area south of the SWIFT remained roughly in place and faded over time. At the first hour of the April event, mean waves were coming from the SSW, with a significant wave height of 7.3 m and a mean period of 9.5 s. In later hours of this deployment (not shown), the storm-associated area on the grid plots shifted northward along the shelf as the SWIFT drifted towards the coast. Overall, as both deployments progressed, the wave conditions became less energetic and fewer waves were captured breaking by the SWIFT camera.
3.1 Comparison of measured and modeled wave parameters
All WW3 gridded parameters were spatially interpolated to obtain SWIFT-following time series to enable model/data comparisons. The interpolation was done bilinearly by first finding the grid quadrant that contained the location of the point of interest that is the mean SWIFT position at a given hour. WW3 output values at the SWIFT locations were then found by linearly interpolating in one direction (west-east) and then the other (north-south).
3.2 Energy dissipation
Doppler sonars on board the SWIFT were used to measure turbulent velocity fluctuations and estimate turbulent kinetic energy rate profiles, ϵ(z), over the top 0.62 m of the ocean immediately below the instantaneous sea surface. The methods apply the second-order structure function, with details described in Thomson (2012). Prior to calculating the structure function, the raw velocity data are screened to remove points with low correlations. These are known to be associated with high void fractions and bubble clouds in the water column. Thus, the observed dissipation rates preferentially represent the persistent turbulence related to wave breaking outside of episodic bubble injection events. The recent work of Derakhti et al. (in prep) suggests that this screening of the raw data results in dissipation estimates that are biased low by up to a factor of 3. Combined with the finite depth of the measurements, Derakhti et al. (in prep) indicate that the overall bias in high sea states is a factor of 4.
Here, ϵ(z) is the TKE dissipation rate measured at depth z, Hs is the significant wave height, F is the wind-to-wave input energy rate, and λ is a tuning parameter. For the combined data from both deployments, this dependence is best fit by a tuning parameter of λcombined = 1.3, matching previously reported fits using analogous wave-following data. In Gemmrich (2010), the tuning parameter reaches upward values of λG10 = 1.6 when breaking is confirmed (λG10 = 1.1 when wave conditions are calm), Sutherland and Melville (2015) reported values of λSM15 = 1 close to the sea surface and λSM15 = 2 at depths greater than one significant wave height, and in Thomson et al. (2016), the depth dependence is characterized by λT16 = 1.4 for wind observations up to 20 m/s.
The color information shown in Fig. 5 illustrates the correlation between high values of breaker fraction and most profiles of high dissipation rates. The light gray lines are profiles that have a breaker fraction equal to or lower than 0.02. All the lines in color have dissipation values greater than the background signal at all, or at least most, depths. Near the surface, the value of the greatest dissipation rate was recorded during the April 2017 deployment (ϵz = − 0.02 m = 0.012 m2/s3) with a corresponding breaker fraction of 0.04. The highest breaker fraction estimated was also for the April 2017 deployment, with a value of 0.07 and a corresponding dissipation rate of 0.005 m2/s3 near the surface.
In this section, we analyze wave spectra and breaking metrics, to explore the linkage between wave breaking, energy dissipation, and wind speed. We evaluate plausible mechanisms for the observed saturation of turbulent dissipation rates at high winds. We also asses the influence of energy dissipation on model performance.
4.1 Influence of spectral shape
In this study, the mean squared slope is calculated from measured and modeled spectra over a frequency range of 0.045 − 0.49 Hz, which is the range of frequencies resolved by the SWIFT instrumentation. This slope metric is effectively the fourth moment of the scalar energy spectrum, and thus, the higher frequencies (which show the significant model-data difference in Fig. 8) are most important.
To ensure these mss findings are not an artifact of our data or specific to the hydrodynamic response of the SWIFT buoy itself, we briefly reference a dataset of wind and wave measurements in the Pacific Northwest presented in Schwendeman and Thomson (2015) and in Thomson et al. (2016). Data acquisition took place in January of 2015 during a voyage aboard the R/V Thomas G. Thompson near Ocean Weather Station P (OWS-P, located at 50 ° N, 145 ° W). Wind measurements were collected with a three-axis sonic anemometer mounted on the bow of the ship. Wave spectral data were obtained simultaneously using free-drifting SWIFTs and a moored 0.9-m diameter Datawell directional Waverider MKIII (WR). These data are utilized here to analyze the relationship between mss and wind speed for this independent data set.
The corroboration of the saturation of mss from the Waverider measurements suggests there are real physical mechanisms at work limiting wave growth with additional wind input. This in turn may limit the amount of wave-breaking dissipation due to turbulence at high winds, even though total wave-breaking dissipation may still increase with increasing wind speed. For example, it is possible that the “clipping” of wave crests and the generation of sea spray, spindrift, and spume at high winds are sufficient mechanisms to prevent waves from increasing steepness beyond the suggested limiting range (mss ≈ 0.014 − 0.018). Such mechanisms would dissipate energy above, or right at, the sea surface, and this dissipation would not be captured in the SWIFT turbulence measurements just below the surface (hence the saturation in those values as well). In this mechanism, the wind input and wave dissipation could both continue to increase with wind speed, but it would not manifest as steeper waves or as more turbulence below the surface. Alternatively, increasing total wave-breaking dissipation in the presence of a limiting turbulent energy dissipation may point towards breaking events that are associated with deeper penetration of bubble clouds (which would also not be captured by the SWIFT turbulence measurements).
4.2 Consideration of individual breakers
In Eq. (8), |ai| is the complex amplitude of the STFT at frequencies f ≥ 2 Hz, and the subscript i represents each direction of the NWU reference frame. Frequencies lower than 2 Hz were filtered out to limit the influence of the SWIFT’s resonant response. For this dataset, waves are said to be breaking during windows with α values greater than 3.25 m/s2. This empirical threshold was chosen such that for times the algorithm indicated breakers, the corresponding available images confirmed whitecapping.
where Nb is the number of observed breakers during a period of observation τ, and T is the corresponding wave period. In Brown et al. (2018), values of Qb were computed for each data burst (τ = 12 min). In this study, breaker fraction bursts were averaged over each hour of each deployment.
Breaking waves were also identified visually from video imaging captured by the SWIFT camera at a rate of 635 frames per hour. To reduce the size of the datasets, images captured from sunset to sunrise, and those taken once the buoy reached the coastline, were removed. Images were then categorized based on a three-level criterion in which waves could be breaking, possibly breaking, or not breaking. Waves were classified as breaking when they were photographed just as their crests were breaking near the buoy. Images were also flagged as breaking when the presence of a whitecap indicated a recently broken wave. The images shown in Fig. 3 are examples of breaking waves. Possibly breaking waves were generally those where the presence of glare in the image, or the long-lasting prevalence of drifting foam over seemingly calm water, made it difficult to determine if a wave was breaking or had recently broken. Waves that were thought to be breaking at a distance, but it was uncertain if their impact would have been recorded by the other SWIFT instrumentation, were also flagged as possibly breaking. All remaining images were identified as not breaking.
A breaking parameter that has been better correlated with wind speed is whitecap coverage; it is most commonly estimated from video imagery and used to approximate wave-breaking dissipation. We are unable to calculate whitecap coverage because of the limited field of view for these images, as well as the inherent complexity in normalizing the camera orientation in each frame. It is worth mentioning, however, that some studies that have delved into determining the best fit to the wind speed-whitecap coverage relationship (typically a power law or cubic function) have observed a leveling off in whitecap coverage with high winds (e.g., Callaghan et al. 2008; Schwendeman and Thomson 2015.). The latest WW3 user manual (The WAVEWATCH III Development Group 2016, corresponding to version 5.16), indicates that whitecap coverage will be made available as a gridded output in a future update. Care should be taken in the model characterization of this parameter since, based on this information, whitecap coverage may be overly estimated at high wind speeds by a power law or cubic fit.
4.3 Fidelity of wave parameters relative to dissipation
We describe highly energetic sea states, driven by winds up to 23 m/s, during two SWIFT deployments made in the winter of 2017. Field measurements of waves and wave breaking collected with the SWIFT are compared with coincident model results from WAVEWATCH III simulations using the ST6 physics package. ST6 wave dissipation is dependent on wave steepness so that the model is internally consistent in continuing to increase both steepness and dissipation with increasing winds.
WW3 estimates of significant wave height are found to have high skill, while the mean wave period is generally underpredicted by the model. Estimates of peak wave direction have moderate skill for the first deployment but a high skill for the second. The unidirectional wave energy spectra are generally well predicted by the model, particularly when wind conditions were moderate.
A direct comparison of turbulent and bulk dissipation rates shows that measured DTKE only amounts to a fraction of modeled Sds in the presence of high winds. Occasionally, the opposite is observed, but to a lesser degree; at such times, the model may not be dissipating enough energy. It is worth remembering at this point that turbulent dissipation is only used as a proxy of total wave dissipation and that differences between the two do not necessarily invalidate either. A comparison of these two parameters to wind speed reveals a magnitude saturation of DTKE at high wind speeds, in contrast to an increasing trend, best fit by a cubic function, for Sds. Similar trends in the relation between mean squared slope and wind speed are found—this time with modeled mss having a more linear relation with wind speed while measured mss saturates at a value of 0.014 for strong winds. The saturation of mean squared slope observed by the SWIFT is consistent with similar measurements collected with a Datawell Waverider buoy near Ocean Weather Station P, with mss measurements saturating at about 0.018.
The saturation in the measured turbulence and mean square slopes, in combination with increasing total dissipation in the wavefield with increasing wind, suggests the existence of a different dynamic regime for wave breaking for winds above 15 m/s, where dissipation due to processes such as spindrift or bubble clouds become increasingly important and contribute to the increasing dissipation with increasing wind speed despite saturation in turbulence. A better understanding of the relative importance of the various whitecapping dissipation mechanisms is crucial before attempting to improve the estimation of total deep-water energy dissipation in WW3. Further direct observations of these processes are required as well as additional studies that involve higher resolution modeling. The saturation of turbulence and mean squared slope should also be explored at greater length not only as they pertain to moderately large wind events but also considering swell conditions.
Video imaging data confirms that waves were breaking particularly at the beginning of each deployment. For a more qualitative analysis of whitecaps, we obtained breaker fraction estimates from a companion publication: Brown et al. (2018). The time series of breaker fraction estimates were found to correlate well with the image counts of breakers and possible-breakers identified visually, but more so for the April 2017 deployment. On average, an increasing linear trend is observed between breaker fraction and wind speed, in contrast to the saturation of whitecap coverage reported in other studies.
The skill of the model in simulating the encountered field conditions was further analyzed in the context of energy dissipation. When compared with SWIFT measures, significant wave height has lower skill when the difference between modeled Sds and measured DTKE is high. No conclusive arguments could be made about the accuracy of wave period and direction estimates relative to the difference of dissipation metrics.
Alex de Klerk and Joe Talbert prepared the SWIFT for deployment. Brimm Aviation carried out the helicopter deployments.
We are thankful to the US Department of Energy whose support funds the Advanced Laboratory and Field Arrays (ALFA) for Marine Energy project (task 3).
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