Analysis of radar sea return for breaking wave investigation

Transcription

1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007jc004319, 2008 Analysis of radar sea return for breaking wave investigation Paul A. Hwang, 1 Mark A. Sletten, 1 and Jakov V. Toporkov 1 Received 4 May 2007; revised 24 September 2007; accepted 9 November 2007; published 6 February [1] Low-grazing angle backscattering data collected by a coherent dual-polarized radar installed on a fixed tower in the ocean are analyzed to investigate the properties of sea spikes attributable to wave breaking. The distribution of breaking wave speed is narrow-banded with an average speed between 2.0 and 2.6 m/s in mixed seas with wind speeds between 7 and 14.5 m/s. The corresponding breaking wavelength is between 2.5 and 4.3 m. The length or velocity scale of wave breaking is not proportional to the length or velocity scale of the dominant wave. This observation reflects the localized nature of the breaking process and may have significant implications on quantifying various breaking properties such as the energy dissipation or area of turnover by breaking waves. The fraction of sea spike coverage generally increases with wind speed but the trend of increase is modified by the intensity and relative direction of background swell. Parameterizations of sea spike coverage needs to take into consideration both wind and wave factors. Similarities and differences between sea spikes and whitecaps are discussed. In particular, while both quantities show a similar power law dependence on wind speed, the fraction of sea spike coverage is considerably higher than that of whitecap coverage. This result reflects the prevalence of steep features that produce quasi-specular facets and short-scale waves bounded to intermediate waves during breaking. These quasi-specular facets and bound waves contribute significantly to enhancing the radar sea return but may not entrain air to produce whitecap signature. Citation: Hwang, P. A., M. A. Sletten, and J. V. Toporkov (2008), Analysis of radar sea return for breaking wave investigation, J. Geophys. Res., 113,, doi: /2007jc Introduction [2] The prevalence of radar sea spikes is a well-known characteristic of low-grazing angle radar backscatter from the ocean surface. Their presence has been linked to surface wave breaking based on comprehensive comparisons of simultaneous and collocated radar and video measurements in the field and the laboratory. Using a focused phased-array imaging radar (FOPAIR), Frasier et al. [1998] and Liu et al. [1998] acquired continuous sequences of dual-polarized radar backscatter from the ocean surface covering an area of approximately 60 m by 90 m. Simultaneous and collocated video images were analyzed. They found that for young and developed seas, radar sea spikes are associated with about 30% of video images with total or partial whitecapping and approximately 60% with steep wave features. For decaying sea, the fractions with whitecaps and steep waves are 3% and 92%, respectively. These observations suggest that microscale breaking is likely taking place on these steep features. Sletten et al. [2003] conducted laboratory experiment using a dual-polarized radar with very high range-resolution (0.04 m) on spilling and plunging breakers of about 0.8-m wavelength produced 1 Remote Sensing Division, Naval Research Laboratory, Washington, DC, USA. Copyright 2008 by the American Geophysical Union /08/2007JC004319$09.00 by dispersive focusing [Duncan et al., 1999]. Comparing the radar measurements with simultaneous high-speed optical images of the breaking waves, they conclude that for the spilling breaker, over 90% of the horizontally (HH) polarized radar backscatter is generated during the initial stage of breaking by the small bulge near the wave crest. For vertical (VV) polarization, the crest bulge produces about 60% of the total backscattered energy. For both polarizations, the Doppler velocity associated with the enhanced scattering is very close to the phase speed of the dominant wave in the wave packet. For the plunging breaker, the initial feature on the crest of an overturning jet generates a lower percentage of the total backscattered energy. The connection between sea spikes and breaking waves offers the possibility for radar remote sensing to serve as a powerful tool to study the very difficult subject of ocean wave breaking. Properties of wave breaking that can be deduced from sea spike analysis include its wind speed dependence, frequency of occurrence, propagation speed, event duration, length scale, momentum flux and energy dissipation [e.g., Phillips, 1988; Frasier et al., 1998; Phillips et al., 2001; Melief et al., 2006]. [3] In this paper, we present an analysis of the 1D and 2D probability density functions (pdf) of radar backscattering cross section and the Doppler frequency of the scattering elements. On the basis of the analysis, we apply the threshold condition of polarization ratio exceeding one (0 db) for detecting sea spikes associated with breaking 1of16

2 waves. The probability of breaking is obtained for a range of wind speed from 7 to 15 m/s. The Doppler frequency of sea spike events is used to derive the phase speed of breaking wavelets. The result is in good agreement with that obtained from feature tracking method [Liu et al., 1998; Frasier et al., 1998] and combinations of multiple thresholds based on the polarization ratio, the cross section magnitude and the Doppler velocity magnitude [Melief et al., 2006]. Using the geometric similarity property of breaking patches derived from earlier laboratory experiments [e.g., Duncan, 1981; Hwang et al., 1989], the area of breaking patch is estimated by the square of the breaking wavelength calculated from the Doppler frequency. The fraction of the surface area covered by breaking waves can then be quantified by the product of breaking probability and the area of breaking patches. The result from this computation is in good agreement with that derived from direct measurements of Frasier et al. [1998] applying the feature tracking method on their 3D (horizontal space and time) mapping of radar scatter from the ocean surface acquired by the FOPAIR. The proportionality coefficient relating the length scale of breaking patch to the length scale of the breaking surface wave is determined to be about 0.1. In mixed seas, parameterizations with dimensionless wave phase speed or dimensionless wave height (normalized by wind speed) collapse the observed results in a more organized manner than parameterization with wind speed alone. The physical interpretation is suggested. We also illustrate the similarity and difference between the wind speed dependence of radar sea spikes and whitecaps on the ocean surface, the latter measurement represents one of the oldest ways (and probably still the most convenient way) of observing and quantifying surface wave breaking in the ocean. [4] In the following, section 2 describes the radar measurements, including the instrumentation, the environmental conditions and a brief description of data analysis. Section 3 presents the results on the pdf of radar backscattering cross section and Doppler frequency. The latter is converted to the breaking wave phase speed, from which one can estimate the average length of breaking fronts per unit area, L(c), as discussed by Phillips [1985] and Phillips et al. [2001]. Section 4 discusses the breaking probability, velocity, length scale, and area of coverage, as well as their dependence on wind and wave parameters. A comparison of sea spike and whitecap properties is also presented in this section. Finally, section 5 is a summary. 2. Radar Backscatter Measurements 2.1. Background and Radar System [5] In 2006, NRL deployed a modified marine radar, an acoustic array, and a video camera on an offshore tower off the Georgia coast (Station SPAG1 at N, W, local water depth 25 m; see station_page.php?station=spag1) as part of a research program designed to develop a model for the acoustic noise generated by a spatially varying breaking wavefield. One of the objectives of this program is to exploit the relationship between sea spikes and breaking waves by developing a low-grazing angle, dual-polarized, coherent radar as a breaking wave detector, using the imagery from the visible camera as well as published reports as a source of ground truth. [6] The radar consists of a Raytheon Pathfinder magnetron transmitter with a center frequency of 9.3 GHz, two marine-radar-type fan beam antennas (one vertically polarized, the other horizontal), and a custom, coherent receiver. Pulse-to-pulse switching between the two antennas is used to collect horizontal-transmit-horizontal-receive (HH) and vertical-transmit-vertical-receive (VV) backscatter on alternate pulses at a per-polarization pulse repetition frequency (PRF) of up to 1200 Hz. All the data discussed in this paper were collected with PRF of 300 per polarization channel. Coherency is achieved by sampling the transmit waveform at the start of each pulse and determining the random transmit phase from it during post-processing. This random phase is then removed from the recorded IF backscatter generated by the sea surface. The magnetron pulse width is 50 ns, resulting in a range resolution of about 8 m. The data acquisition is programmed to over-sample and produces 1.5 m ground range resolution. Peak transmitted power is 1 kw. The range of grazing angles, q g, in the data set presented here is 0.5 to 6.3. The radar antennas are placed at the northeastern corner of the tower on a stairway railing 12 m above the mean water level and positioned to look horizontally to the north. The azimuthal beam width is 1.2 and the elevation beam width 22. Figure 1 shows an example of the acquired data of the relative normalized cross section, s VV and s HH, and the Doppler frequency, w DV and w DH. Modulation effects of surface waves on radar sea return can be clearly visualized. Spikiness in the scattered signal is quite appreciable especially in the horizontal return. These range-time mappings of the sea surface by radar represent a valuable data source to study ocean surface waves in great detail Environmental Conditions [7] The Station SPAG1 is instrumented with basic wind and wave sensors. Data available on the website mentioned above are the hourly wind direction, wind speed, wind gust, significant wave height, and air and water temperatures. Additional information of the peak wave period and surface wave spectrum is available from a nearby buoy (at N, W, 18 m depth) maintained by the National Data Buoy Center (NDBC), about 28.5 km to the west of SPAG1. Radar measurements were collected over three days in March and April These data are referred to as 22Mar06, 10Apr06 and 11Apr06 from here on. The wind and wave measurements from these two stations are shown in Figure 2 for 22Mar06 and Figure 3 for 10Apr06 and 11Apr06. For each day, the radar data acquisition was manually started at approximately hourly intervals. The data length of each collection episode is about ten minutes. The starting time of radar measurement is marked with a square symbol in the figures. The reference wind speed, U, significant wave height, H s, and peak wave period, T p,atthe time of radar operation are interpolated from the in situ measurements (SPAG1 for U and H s and for T p ) and listed in Table 1. Also shown in the table are the breaking probability, phase speed and wavelength derived from sea spike analysis, which will be described in more detail in section 4. 2of16

3 Figure 1. Range-time mappings of backscattering cross section (left column) and Doppler frequency (right column); the upper row is for VV, and the lower row for HH. [8] The wave conditions of the three days are all mixed seas. For 22Mar06, a southerly wind event of up to about 19 m/s occurred a day before and built up waves reaching about 2 m height and 7 s period. The wind died down then reversed direction and a northerly wind of about 13 m/s prevailed at the beginning of experiment. A new wind wave system developed over the old adverse swell as can be inferred from the time series depicted in Figure 2. The air temperature is about 4 C colder than the water temperature so a mildly unstable condition can be expected for this data set. [9] For 10Apr06 and 11Apr06, the wind direction remained steady from NNE the whole time. Wind speed varied about diurnally. This long and quasi-steady wind episode started more than one day prior to radar data acquisition. The wave height continued to increase in the first half and then decreased in the second half of 10Apr06 while wave period remained almost unchanged. For 11Apr06, the wave period displayed a general increasing trend the whole time while wave height was mostly constant at the first half and then slowly decreased in the second half. Water temperature on 10Apr06 was 2 to 3 C warmer than Figure 2. Wind and wave conditions relevant to data set 22 March 2006: (a) wind speed, (b) wind direction, (c) significant wave height, (d) peak wave period, and (e) air and water temperatures. Measurements from both stations SPAG1 and are shown. 3of16

4 Figure 3. Same as Figure 2 but for data sets 10 April 2006 and 11 April the air temperature. For 11Apr06, the temperature difference gradually became smaller and by the second half, the air was actually slightly warmer than water. The wind, wave and temperature conditions on these three days are quite complicated. Such complexity seems to reflect in the sea spike features, which will be described in section Data Analysis [10] We consider the complex envelope of the received radar signal, V(t, r), which is formed using the Hilbert transform [e.g., Papoulis, 1991] of each radar pulse. It is written as if t;r Vðt; rþ ¼ at; ð rþe ð Þ ; ð1þ Table 1. Sea Spike Statistics and Relevant Environmental Conditions a U (m/s) H s (m) T p (s) P s (%) c s (m/s) l s (m) # a #1-9: 22 March 2006, #10-18: 10 April 2006, #19-29: 11 April of16

5 where t is time, r ground range, a amplitude and f phase. The relative normalized cross section is derived from the square of the amplitude taking into account the cubic range falloff, sðt; rþ ¼ Cr ðþ a2 ðt; rþ r 3 ; ð2þ where the factor C includes the antenna pattern provided by the manufacturer and an unknown calibration reference. The instantaneous Doppler frequency of surface scattering element is calculated by the temporal derivative of the unwrapped phase of the complex signal, w D ðt; rþ ð t; r Þ : [11] Thompson and Jansen [1993] showed that the Doppler frequency derived from this approach is equivalent to the mean Doppler frequency calculated from the first moment of the Doppler spectrum (their equations (1) (6)) for a small differential time. For the data presented here, the differential time is 1/300 s. To reduce data noise, a running average with a window of 50 temporal pixels is performed for s and w D, yielding an equivalent integration time of 1/6 s. [12] The Doppler frequency is caused by the motion of the scattering element, so the radial component of the advection velocity (referred to as the Doppler velocity from here on) of the scattering element can be calculated by u D ðt; r Þ ¼ w Dðt; rþ ; 2k r cos q g where k r is the radar wave number and q g the grazing angle. For the radar frequency used in this experiment, k r = 197 rad/m, and the denominator on the right hand side of (4) is 394 rad/m at q g =1 and 392 rad/m at q g =6. [13] Plant [1997] presents a model of microwave Doppler sea return of Bragg scattering from bounded, tilted waves. He postulates that ocean surface waves of the order of a few meters long are frequently steep enough to generate bound centimetric waves in the Bragg resonance scale. The bound centimetric waves have a nonzero mean tilt and move at the speed of the intermediate waves. Applying composite surface scattering theory to this sea surface model, he shows that much of the apparently anomalous behavior of microwave sea return measured at incidence angles between 50 and 80 can be explained. The picture of the sea surface features responsible for microwave scattering as described by Plant [1997] seems to be in very good agreement with the comparison of sea spike measurements with simultaneous video recording that indicates the majority of sea spike events occurs with steep wave features, as reviewed in section 1. Plant [2003a] advanced this picture of the ocean surface roughness by showing that the combination of free waves and bound waves can provide an alternative explanation of the asymmetric sea surface slope pdf observed by Cox and Munk [1954]. We therefore equate the advective ð4þ speed calculated by (4) to the intrinsic phase speed of the facets carrying the scattering surface roughness, c, c ¼ u D : [14] For facets that produce sea spikes, c is the phase speed of the breaking wavelet, referred to as the breaking wave speed from here on and is denoted by c s. The subscript s may be dropped in the subsequent discussions for simplicity unless clarification is needed. [15] It is pointed out by one of the reviewers that while the denominator of the Doppler frequency equation (4) is of the same expression as that of the Bragg wave number, the equation is equally valid for a point target without declaring any scattering mechanism. As a result, justification of equation (5) does not require the Bragg scattering of bound waves. The investigation of the scattering mechanisms requires more extensive analysis of the radar return signals and is beyond the scope of this paper. 3. Statistical Properties of Radar Return at Low Grazing Angle [16] This section presents the analysis of the basic statistics of radar returns as well as the 1D and 2D pdf of the backscattering cross section and Doppler velocity. In the subsequent discussions, the angle brackets represent time average. Figure 4 shows an example of the grazing angle dependence of the average radar cross section. The wind speed is about 13 m/s. The rate of decrease of hs VV i is much larger than hs HH i. On average, hs VV i decreases about 10 db for q g between 6 and 1 while the variation of hs HH i remains within 2 db for the same grazing angle range. At q g = 6, hs VV i is about 8 db higher than hs HH i, the difference decreases steadily toward lower grazing angle and at q g = 1, it is only about 1 db. The average polarization ratio, hr s i = hs HH /s VV i, is approximately 8 to 5 dbforq g from 6 to 3, and about 1 to 2 db for q g <2 ; there is an apparent change in the grazing angle dependence of hr s i near q g =3 (Figure 4c). The average Doppler velocity, hu D i, of backscatter is consistently higher in HH than in VV (Figure 4b) as first noticed by Pidgeon [1968]. The magnitude of hu D i in both VV and HH increases toward decreasing q g but the difference remains almost constant for q g >2. The HH scatterers move about 0.5 m/s faster than the VV scatterers (about 2 m/s vs. 1.5 m/s average for q g >3 ). [17] The significant increase of the instantaneous polarization ratio has been used to associate radar sea spikes with surface wave breaking at low grazing angle [e.g., Trizna, 1991; Lee et al., 1996; Frasier et al., 1998; Plant, 2003b; Forget et al., 2006; Melief et al., 2006]. The joint pdf (jpdf) constructed from pairs of R s and s or u D provides further information on using sea spikes in the radar sea return for breaking wave detection (Figure 5). In particular, for both VV and HH, the magnitude of s or u D in the subpopulation with R s > 0 db spreads over a wide range. The jpdf analysis illustrates convincingly that events with R s > 0 db occurs over a broad range of s and u D, which is not unexpected as it is well known that wave breaking occurs at various length scales and intensity levels. It is judged that for wave breaking detection, setting other thresholds in addition to ð5þ 5of16

6 Figure 4. Grazing angle dependence of (a) mean relative normalized cross section, (b) mean Doppler velocity, (c) mean polarization ratio (HH/VV), and (d) mean Doppler velocity difference (HH-VV). the polarization ratio is not necessary or is even undesirable. The results presented in the following are derived from a single criterion of R s > 0 db. Frasier et al. [1998] applied different threshold levels for sea spike detection, ranging from 1 (0 db) to 8 (9 db). As expected, varying the threshold level changes the absolute magnitude of breaking probability but does not change markedly its dependence on wind or wave parameters (their Figure 14). 4. Sea Spike Analysis 4.1. Sea Spike Distribution and Dynamic Properties [18] Because of the limitation of computer resources, during post processing the 10-min data are divided into 20-s segments and 17 segments of each data collection episode are averaged together. The results of basic sea spike statistics and relevant wind and wave properties are tabulated in Table 1. These results will be further discussed in more detail in the later part of this section and in section 4.2. [19] Examples of the pdf of s and u D for both VV and HH polarizations of the full population and subpopulation with R s > 0 db are shown in Figures 6 and 7. The data are divided into subsets of grazing angles in 1 bins. Broadening of the pdf toward lower grazing angle for both s and u D is apparent. Quite interestingly, the pdf of s gradually develops a bimodal feature as the radar scatter returns from near horizon, probably as a result of increased geometrical Figure 5. Jpdf of (a) (R s, s VV ), (b) (R s, u DV ), (c) (R s, s HH ), and (d) (R s, u DH ). 6of16

7 Figure 6. Pdf of relative normalized cross section in 1 grazing angle bins. Upper row is VV, low row is HH; left column is for all population, right column is for sea spikes only (R s >0 db). sheltering. For the whole population, the Doppler velocity from such near-horizon returns increases considerably in both VV and HH polarizations (Figures 7a and 7c). In the subpopulation with R s > 0 db, the pdf of u DH is almost invariant with respect to q g and only relatively small variation with q g is found in the pdf of u DV. These results reconfirm that the Doppler velocity of horizontal polarization radar return is less ambiguous in identifying sea spikes of breaking wave origin because the contribution of direct Bragg scattering mechanism is less in HH than in VV. In the following sea spike analysis, the HH Doppler data are used. [20] Figure 8a shows an example of the pdf of breaking wave speed, p s (c), with c calculated by the Doppler frequency of sea spike events, (4) and (5). The result is obtained from the data with q g between 3 and 6 for better signal-to-noise ratio. The fraction of sea spike pixels in the three q g bins are 15%, 14% and 12% of the total number of pixels. Noticeably, the breaking wave speed is distributed in a narrow range between 1 and 4 m/s with a peak near 2.5 m/s. This result is similar to the sea spike analyses reported by Frasier et al. [1998, Figure 10] using feature tracking and Melief et al. [2006, Figure 5] using multiple thresholds for sea spike classification. (Melief et al. [2006] assume that the Bragg resonant waves are free propagating. Their reported breaking wave phase velocity is thus about 0.4 m/s smaller than the present analysis that do not assume any particular scattering mechanisms. The difference includes the phase speed of free Bragg waves, about 0.25 m/s, and wind Figure 7. Same as Figure 6 but for the Doppler velocity. 7of16

8 Figure 8. (a) Pdf of breaking wave speed from sea spike analysis with data in three different grazing angle bins, (b) the third moment of the pdf, (c) the seventh moment of the pdf, and (d) the eighth moment of the pdf. Figures 8b, 8c, and 8d are expected to be proportional to the average length of breaking wavefront, the momentum flux due to wave braking, and the energy dissipation due to wave braking, respectively [Phillips et al., 2001]. surface drift, about 3% of wind speed. The subsequently derived breaking wavelength is also reduced somewhat.) [21] Phillips et al. [2001] apply feature tracking to analyze the evolution of sea spikes measured by an X-band radar with very high spatial and temporal resolution (0.3 m in range and 2000-Hz pulse repetition rate). The result yields not only the number of breaking events per unit area per unit time but also the distribution of the length of breaking front per unit area of ocean surface as a function of breaker phase velocity, L(c). The L(c) is of fundamental importance and can be used to derive the fraction of surface area turned over by breaking waves as well as the momentum flux and energy dissipation of wave breaking [Phillips, 1985]. With the event duration, d, and event speed, c, obtained from feature tracking of sea spikes, Phillips et al. [2001] explain that L(c) can be calculated by LðÞ¼ c ac ð P d 2 Þ ; ð6þ ATDc where a is a constant (about 0.35), A the swath area, T the total observation time and Dc the speed interval. In the analysis presented here, event duration is not available but both feature tracking analyses by Frasier et al. [1998] and Phillips et al. [2001] show a fairly clear linear proportionality between the average event duration and the event speed. It is therefore reasonable to assume that L(c) c 3 p s (c). According to Phillips [1985] and Phillips et al. [2001], the area of turnover, the momentum flux and wave energy dissipation due to wave breaking are related to L(c) by cl(c), c 4 L(c), and c 5 L(c), respectively, thus they are also proportional to c 4 p s (c), c 7 p s (c), and c 8 p s (c), respectively (Figures 8b, 8c, and 8d). All these functions remain narrowly distributed with a distribution peak near 3 m/s. The peak speed is about 0.3 of the dominant phase speed of the wavefield, which is about 10 m/s for this case (first data collection period of 10Apr06). Table 2 lists the integrated velocity scale of the first eight moments of the breaking velocity distribution averaged over q g between 3 and 6. The integrated velocity scale of the n-th moment is calculated by Z 1=n c n ¼ c n p s ðþdc c : ð7þ [22] The corresponding peak velocities of the distributions of the eight moments are listed in Table 3. These characteristic velocities of the eight moments vary only slightly in all the data analyzed. This result reflects, again, the narrowness of the distribution of breaking wave speeds. The standard deviation of the distribution can be calculated from the integrated velocity scales of the first two moments, it ranges between 0.43 and 0.52 m/s. The integrated velocity scale of the first moment of the distribution ranges between 2.03 and 2.63 m/s. The mean velocity computed from the average of the sea spike population is about 0.05 m/s smaller than the integrated velocity scale of the first moment of the breaking velocity distribution. [23] The result on the phase speed distribution of sea spike events reported by Phillips et al. [2001] also display a local peak near c = 4.3 m/s but they consider that the low-velocity events from feature tracking may be less accurate and the local peak is probably caused by the data processing artifact. Melville and Matusov [2002] applied a technique similar to that used in particle imaging velocimeter to obtain the length of breaking front from video recording of whitecaps from a low-flying aircraft. The result from their analysis is more 8of16

9 Table 2. Integrated Velocity Scales (m/s) of the First Eight Moments of the Breaking Velocity Distribution a c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 # a #1-9: 22 March 2006, #10-18: 10 April 2006, #19-29: 11 April broad-banded and the peak of the dissipation distribution for the three cases in their paper is near c = 9 m/s for U = 7.2 and 9.8 m/s, and about c = 7 m/s for U = 13.6 m/s. On the other hand, Hwang and Wang [2004] present an analysis of source function balance of wind-generated surface waves with data acquired by wave gauge arrays mounted on a free-drifting buoy to alleviate the complication of Doppler frequency shift on the data processing of short and intermediate scale waves. An interesting result derived from that analysis is that the dissipation function of wind-generated waves displays a Table 3. Peak Velocity (m/s) of the First Eight Moments of the Breaking Velocity Distribution a c p1 c p2 c p3 c p4 c p5 c p6 c p7 c p8 # a #1-9: 22 March 2006, #10-18: 10 April 2006, #19-29: 11 April of16

10 Figure 9. Wind speed dependence of (a) probability of sea spike occurrence, and comparison with the sea spike density obtained by Frasier et al. [1998], (b) breaking wavelength deduced from the Doppler velocity of sea spike facets, and (c) the sea spike coverage. singularity behavior in the wave number range between 3 and 30 rad/m with a sharp peak near 4 rad/m (wavelength about 1.6 m), the corresponding phase speed of the distribution peak component is 1.6 m/s. The results of breaking length scale from source function analysis by Hwang and Wang [2004] and from the present sea spike analysis are based on very different approaches but they reach very similar conclusion that the breaking dissipation function is narrowly distributed. The peaks of the phase speed distribution derived from the two approaches differ only slightly (within a factor of two). Plant [1997] also found that the primary breaking waves have a mean length of 2.5 m (phase speed 2 m/s) at wind speed of 7 m/s and 4 m (phase speed 2.5 m/s) at 16 m/s. Gemmrich et al. [1994] present a model relating the energy dissipation in the ocean mixed layer to the energy input into the surface wavefield. Using measurements of turbulent kinetic energy dissipation, they determine that the average phase speed of waves acquiring maximum energy input from the wind is between 0.55 and 0.72 m/s. Thorpe [1993] estimates the ratio of breaking wave speed to the peak wave phase speed, c b /c p, to be about This result is based on observations of the frequency of wind wave breaking and laboratory estimates of the energy loss of a steady breaking wave to infer the net rate of energy transfer to the mixed layer from breaking waves as a function of wind speed. Melville [1994] factors in the unsteadiness of spilling breaking in surface waves in the ocean in comparison to the steady breaking in laboratory simulation and the effects of the mixed layer thickness, and reached the conclusion that c b /c p 0.4 to Because the wavelength ratio scales with the square of phase speed ratio, such difference in the estimate of breaking phase speed can result in an order of magnitude difference in the breaking length scale. Additional discussions on the breaking wavelength scale based on measurements of acoustic noise, whitecaps and sea spikes are presented by Hwang [2007]. Because of the difficulty of measurements, the issue of the spectral distribution of wave dissipation function requires more theoretical and experimental research. In section 4.3, we present the breaking velocity and length scales as well as the speed ratio estimated from the present sea spike analysis. [24] The sea spike probability, P s, is computed from integrating the pdf of R s for R s > 0 db. Figure 9a shows the result of P s as a function of wind speed. The quantity P s is proportional to the sea spike density (in m 2 s 1 ) reported by Frasier et al. [1998, Figure 14]. Their result is also shown in the same plot for comparison. Phillips et al. [2001] also reported sea spike event density of /m 2 /s at U = 9.3 m/s in their feature tracking analysis. The number is about 50 times smaller than that of Frasier et al. [1998] at a comparable wind speed. As pointed out by one of the reviewers, possible reasons for the discrepancy in the numerical values of the sea spike density and the breaking probability reported from different sources may be attributed to the threshold setting in data processing and the radar systems used. The resolution cell sizes for all three systems are quite different, and since sea spikes tend to be point-like targets rather than distributed targets, the issue of thresholds, radar sensitivity and area averaging all come into play. [25] The length scale of sea spikes, l s =2p/k s, can be derived from the mean Doppler frequency of the sea spike population using (4) and (5) and the dispersion relation, k s = g/c s 2. The result from our data shows that the average l s is distributed in a narrow range between 2.4 and 4.5 m (Figure 9b, Table 1). These numbers are in good agreement with those reported by Frasier et al. [1998], Plant [1997], and also with those of Melief et al. [2006] if the surface drift and Bragg wave phase speed are taken into account. Applying the geometric similarity property of breaking waves observed from laboratory experiments [e.g., Duncan, 10 of 16

11 and O Muircheartaigh, 1986; Wu, 1988; Zhao and Toba, 2001; Melville and Matusov, 2002; Lafon et al., 2004, 2007; Anguelova and Webster, 2006; Sugihara et al., 2007; and the references therein]. The research of the whitecap properties is also important in ocean remote sensing because the electromagnetic properties of foams are considerably different from those of the seawater. Since radar sea spikes and whitecaps share the common generation mechanism of wave breaking, we expect that these two phenomena would exhibit some similar properties. [28] Figure 10 replots the result of sea spike coverage from the present analysis and those of Frasier et al. [1998] (Figure 9c) together with the whitecap measurements reproduced from the tabulated data of Monahan [1971], Toba and Chaen [1973], Ross and Cardone [1974], Xu et al. [2000], Lafon et al. [2004, 2007] and Sugihara et al. [2007]. These whitecap measurements are collectively referred to as the MTRXLS data set. Monahan [1971] suggests that P w = U 3.4 forms an upper envelop of his whitecap measurements. The envelop function is also applicable to the assembled data here. The mean data trend follows very well the semi-analytical function P w = U 3.75 suggested by Wu [1988]. Interestingly, when a threshold wind speed is introduced, a cubic wind speed relation, Figure 10. Comparison of sea spike and whitecap coverage. 1981; Hwang et al., 1989], the fraction of the ocean surface covered by breaking waves identified by sea spikes, P sc, can be estimated by P s l s 2 (Figure 9c). Using the continuous time series of the horizontal spatial images of radar backscatter acquired from FOPAIR, Frasier et al. [1998] perform detail kinematic and geometric analysis of sea spike evolution. Their direct measurements of the fraction of sea spike area coverage of the ocean surface is used here to estimate the proportionality constant connecting P sc and P s l s 2, which is found to be P sc ¼ P sð0:1l s Þ 2 ¼ 0:055P sl 2 s ; ð8þ A 1 A p where A p is the average pixel area (1.5 m 3.6 m), and A 1 =1m 2, introduced in the equation to emphasize that P sc is dimensionless. This result also suggests that the length scale of the breaking patch is about 10% of the length scale of the surface wavelet undergoing breaking process. [26] The result of sea spike coverage reported by Frasier et al. [1998] and from the present analysis is shown in Figure 9c. The agreement on the wind speed dependence of our estimation using P s l 2 s and their direct measurements is reasonable, considering the difference in the two approaches and the fact that wind speed scaling is probably insufficient in mixed sea conditions. Further discussion on the swell effect is given in section Sea Spikes and Whitecaps [27] Whitecaps have been used as surrogate of breaking waves in the ocean for some time [e.g., Monahan, 1971; Toba and Chaen, 1973; Ross and Cardone, 1974; Monahan P w ¼ 1: ðu 2Þ 3 ¼ 1: U 3 1: ; ð9þ fits the data equally well or better, especially for the measurements in the lower wind conditions. Indeed, a cubic wind speed relationship is expected from the point-of-view of energy dissipation [e.g., Phillips, 1985, 1988; Thorpe, 1993; Melville, 1994; Phillips et al., 2001; Melville and Matusov, 2002; Hwang and Sletten, 2008]. Similar cubic wind speed relationship of whitecap coverage has also been proposed by several researchers [e.g., Bondur and Sharkov, 1982; Monahan, 1993; Asher and Wanninkhof, 1998; Asher et al., 2002; Reising et al., 2002; Stramska and Petelski, 2003; see Table 1 of Anguelova and Webster, 2006]. [29] Somewhat surprisingly, the data of sea spike coverage lie near the upper bound of whitecap results. Conceptually, sea spikes are caused by the active portion of breaking waves while whitecaps include the active white water near the wave crest and passive foams left over following the active phase of wave breaking and from previous breaking events. Ross and Cardone [1974] report both active and passive fractions of their airborne whitecap measurements and the result shows that the passive fraction is of the same magnitude as the active fraction. If sea spikes represent only the active portion of wave breaking, one would expect a much smaller fraction of coverage than that of whitecaps. On the other hand, as reviewed in section 1, comparisons of simultaneous radar and video measurements suggest that most sea spikes occur in steep wave features that may be undergoing microscale breaking undetectable by the whitecap analysis. From this perspective, the sea spike fraction would actually be higher than the whitecap fraction under the same wind and wave conditions. Plant [2003a], in his new interpretation of the pdf of the sea surface slopes, points out that the probability of finding bound waves (presumably produced by steep surfaces during breaking) on the ocean surface is much larger than 11 of 16

12 Figure 11. The hourly surface wave spectra covering the radar measurement periods, (a) 22 March 2006, (b) 10 April 2006, and (c) 11 April (d) The peak wave period and (e) the significant wave height of the swell component. the probability of whitecaps. The property of bound waves plays an important role in his scattering model, especially in the interpretation of Doppler shifts in microwave backscatter from the sea surface at high incident angle. We hope that as more sea spike and other breaking wave data become available, issues such as the surface roughness composition and electromagnetic scattering mechanisms of breaking waves can be further clarified eventually. [30] It is also noticeable that the data scatter in sea spike measurements seems to be much less than that of whitecaps. Frasier et al. [1998] also performed whitecap analysis using their video images. They found that while the range of sea spike coverage is slightly less than one order of magnitude in their total data set, the range of whitecap coverage spreads almost three orders of magnitude (their Figure 17). It is possible that the larger data scatter in the whitecap measurements is caused by the passive nature of the technique, thus the whitecap data are more influenced by environmental conditions such as sun angle, camera angle, cloud reflection or shadow, exposure setting, as well as subjective criteria placed during the processing phase, such as the contrast level to separate whitecaps from ambient water. Figure 4 of Sugihara et al. [2007] presents a graphic example showing that the processed fraction of whitecap coverage may vary by a factor of four when the threshold setting was varied by a factor of two. The sea spike analysis method presented here is relatively straightforward and can be applied to any radar measurements operated in coherent and dual-polarization mode. These results complement whitecap observations and can be quite helpful in enhancing our breaking wave database from the open ocean Swell Influence [31] As illustrated in section 4.1, when our measurements from three separate days are viewed together, there is an apparent increasing trend of the sea spike probability or sea spike coverage as a function of wind speed (Figure 9a). Closer inspection shows delicate differences in the subsets of the data. In particular, the breaking probability in 10Apr06 is about 30 to 50 percent lower than that of 22Mar06 for the same wind speed, and the data in 11Apr06 (with wind speeds mostly near 10 m/s) seem to be more scattered than those of the other two days (Figure 9a). These anomalies are also present in the sea spike coverage result (Figure 9c). As commented in section 2.2, from examining the measured wind and wave conditions, it is found that on 22Mar06, the wind reversed direction several hours before data collection and the wavefield is characterized by fresh wind-generated waves superimposed on mild adverse background swell (Figure 2). On 10Apr06 and 11Apr06, the wind direction had been steady for almost 24 h prior to data acquisition. It remained steady during the two-day period and only the wind speed showed some diurnal fluctuations. The wavefields for these two days were much more complicated with alternating growth in wave height and wave period as described in section 2.2 (Figure 3). Figures 11a to 11c show the wave spectra measured by buoy covering the duration of radar data acquisition. The wave period and wave height of the swell component are obtained from the spectra (Figures 11d and 11e). The swell intensity in terms of wave height is mostly less than 0.2 m for 22Mar06 and it is much larger in 10Apr06 and 11Apr06. As discussed in section 2.2, examination of the time history of wind events suggests that the wind and swell were in opposite directions in 22Mar06 while in 10Apr06 and 11Apr06 they were probably in the same direction. [32] As shown in Figure 12, parameterizations of the sea spike coverage incorporating both wind and wave properties (Figures 12c 12d) appear to yield a better correlation with the observed sea spike coverage than those with wind speed (Figure 12a, reproducing Figure 9a and placed here for easier comparison) or (dominant) wave properties alone 12 of 16

13 Figure 12. Sea spike coverage as a function of (a) wind speed, (b) dominant wave steepness, (c) dimensionless wave velocity, and (d) dimensionless wave height. (Figure 12b). A plausible interpretation is that the predominant scale of breakers is on the order of a few meters. Such short wind-generated waves do not contribute significantly to the global properties of the wave spectrum of ocean waves, typically represented by the peak wave frequency and significant wave height. Therefore parameterization with dominant wave properties alone, such as K p H s /2 shown in Figure 12b, may miss the predominantly short-scale breaking features. On the other hand, in mixed seas, short waves are modified by the properties of the background long-scale waves, which are swells propagated from outside of the area influenced by the local wind field. Parameterization with wind speed alone would miss the influence of non-locally generated swell. [33] The ratio c b /c p discussed in section 4.1 can be estimated from the sea spike result of c s /c p (Figure 13). Under a given wind speed, the result shows that the relative length scale of breaking is much larger in 22Mar06, with c s / c p ranging mostly between 0.3 and 0.4, than that for the other two days (10Apr06 and 11Apr06), with c s /c p mostly between 0.2 and 0.3. Again, parameterizations combining both wind and wave factors seem to give some order to the data points but the separation of 22Mar06 from 10Apr06 and 11Apr06 remains consistent in all parameterizations Figure 13. Same as Figure 12 but for c s /c p. 13 of 16

14 Figure 14. Breaking wave speed as a function of (a) dimensionless wave velocity, and (b) dimensionless wave height; breaking wavelength as a function of (c) dimensionless wave velocity, and (d) dimensionless wave height. applied here. Quite interestingly, while the absolute breaking length scale (proportional to the square of the velocity scale) differs only slightly between 22Mar06, 10Apr06 and 11Apr06 data sets (Figure 9b), when normalized by the peak wavelength, 22Mar06 data stand out from the other two data sets. As noted earlier, the most outstanding feature in the wind and wave conditions of 22Mar06 is its wind sea in mild counter-swell condition, while the wind sea and swell are in the same direction in 10Apr06 and 11Apr06. When wind and swell are counter to each other, the sea becomes choppier and wave breaking occurs much earlier than when wind and swell are in the same direction. Thus one can expect the dominant scale of wind waves to be longer in the latter case as the wind sea can continue to grow over a much longer period before interrupted by intensive breaking. The large difference in c s /c p reflects the change of c p rather than the change of c s. The observation that the length scale of breaking waves seems to be independent of the length scale of dominant waves is noteworthy. Furthermore, the distribution of breaking length scale is quite narrow for a broad range of wind seas or mixed seas, unlike the distribution of dominant wavelength scale. Normalizing c s with c p in fact separates rather than brings together data groups from different sea state conditions. This decoupling of c s and c p may be an important clue for advancing our understanding of the wave breaking process. A possible explanation of this result is that even in the case of dominant wave breaking, the wave profile is distorted severely and deviates from the gentle sinusoidal waveform. The length scale of the steepening segment that leads to instability and eventual breaking is quite localized and it is not strongly dependent on the wavelength defining the dominant wave. The dissipation rate and other related breaking properties are determined by the local breaking wavelength rather than that of the dominant wave that spawned the breaking wavelets. Additional discussions of breaking length scale and implications on breaking process are given by Hwang [2007]. [34] Comparing the results shown in Figures 12c and 13c versus Figures 12d and 13d, it is evident that parameters U 2 / gh s and U/c p are more preferred than U or K p H s /2 for bringing different data sets together although the number of data points is small and the scatter large. The physical meaning of U 2 /gh s is dimensionless energy and U/c p dimensionless momentum or inverse wave age. On the basis of the limited data available here, the following empirical functions from least squares fitting are proposed: P sc ¼ 3: U 1:27 ¼ 4: U 2 1:17 ; ð10þ c p gh s c s ¼ 2: U 0:68 ¼ 9: U 2 0:56 : ð11þ c p c p gh s [35] These empirical relations are also shown in Figures 12 and 13. As discussed in the last paragraph, the normalization of length or velocity scale by that of the dominant waves yields more data scatter rather than collapse data sets from different conditions. The least squares fitting equations for the breaking wave speed and wavelength (Figure 14) are: c s ¼ 2:3 U c p l s ¼ 3:4 U c p 0:17 ¼ 1:8 U 2 0:16 ; ð12þ gh s 0:34 ¼ 2:0 U 2 0:33 ; ð13þ gh s 14 of 16