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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, C02003, doi:10.1029/2007JC004319, 2008
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Analysis of radar sea return for breaking wave investigation
Paul A. Hwang,1 Mark A. Sletten,1 and Jakov V. Toporkov1
Received 4 May 2007; revised 24 September 2007; accepted 9 November 2007; published 6 February 2008.
[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, C02003, doi:10.1029/2007JC004319.
1. 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.
0148-0227/08/2007JC004319$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
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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 31.38°N, 80.57°W,
local water depth 25 m; see http://www.ndbc.noaa.gov/
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
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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, qg, 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,
sVV and sHH, and the Doppler frequency, wDV and wDH.
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.
2.2. 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 41008 (at
31.40°N, 80.87°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 2006. 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, Hs, and peak wave period, Tp, at the
time of radar operation are interpolated from the in situ
measurements (SPAG1 for U and Hs and 41008 for Tp) 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.
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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 41008 are shown.
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Figure 3. Same as Figure 2 but for data sets 10 April 2006 and 11 April 2006.
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 4.
2.3. 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
V ðt; rÞ ¼ aðt; rÞeifðt;rÞ ;
ð1Þ
Table 1. Sea Spike Statistics and Relevant Environmental Conditionsa
U (m/s)
Hs (m)
Tp(s)
Ps (%)
cs (m/s)
ls (m)
#
12.9
12.9
12.9
12.1
10.5
9.8
8.9
8.2
6.9
13.0
14.5
14.4
13.7
12.9
12.7
11.3
10.7
10.5
13.0
12.0
12.0
10.7
10.0
10.0
10.0
10.0
10.0
10.0
9.2
1.4
1.8
1.8
1.5
1.4
1.3
1.3
1.3
1.2
2.1
2.3
2.4
2.7
2.7
2.6
2.1
2.0
2.0
2.3
2.2
2.5
2.3
2.3
2.4
2.3
2.3
2.3
2.1
2.1
3.6
4.1
4.2
4.7
4.6
4.5
4.5
4.8
4.6
5.9
5.7
5.8
6.1
5.7
5.9
5.7
5.5
5.4
5.5
5.9
6.7
6.6
6.5
7.1
6.7
6.5
6.1
6.0
7.5
7.86
8.61
8.21
6.28
7.53
7.29
5.79
3.43
4.40
6.93
6.39
5.28
5.51
5.88
5.72
6.14
4.78
5.05
7.94
7.40
6.27
4.82
3.73
5.08
5.63
5.21
3.36
6.53
6.73
2.44
2.55
2.50
2.59
2.48
2.48
2.45
2.40
1.97
2.58
2.51
2.60
2.57
2.44
2.37
2.24
2.24
2.32
2.42
2.41
2.54
2.56
2.53
2.41
2.28
2.27
2.08
2.18
2.01
3.83
4.16
4.00
4.29
3.95
3.95
3.85
3.71
2.48
4.27
4.04
4.33
4.22
3.83
3.59
3.22
3.22
3.45
3.77
3.72
4.13
4.22
4.11
3.73
3.34
3.30
2.78
3.05
2.60
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
a
#1-9: 22 March 2006, #10-18: 10 April 2006, #19-29: 11 April 2006.
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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 C ðrÞ
a2 ðt; rÞ
;
r3
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,
wD ðt; rÞ ¼
@fðt; rÞ
:
@t
ð3Þ
[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 wD, 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
uD ðt; rÞ ¼
wD ðt; rÞ
;
2kr cos qg
ð4Þ
where kr is the radar wave number and qg the grazing angle. For
the radar frequency used in this experiment, kr = 197 rad/m,
and the denominator on the right hand side of (4) is 394 rad/m
at qg = 1° and 392 rad/m at qg = 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
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speed calculated by (4) to the intrinsic phase speed of the
facets carrying the scattering surface roughness, c,
c ¼ uD :
ð5Þ
[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 cs. 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 hsVVi is much
larger than hsHHi. On average, hsVVi decreases about 10 dB
for qg between 6 and 1° while the variation of hsHHi
remains within 2 dB for the same grazing angle range. At
qg = 6°, hsVVi is about 8 dB higher than hsHHi, the
difference decreases steadily toward lower grazing angle
and at qg = 1°, it is only about 1 dB. The average
polarization ratio, hRsi = hsHH/sVVi, is approximately
8 to 5 dB for qg from 6 to 3°, and about 1 to 2 dB
for qg < 2°; there is an apparent change in the grazing angle
dependence of hRsi near qg = 3° (Figure 4c). The average
Doppler velocity, huDi, of backscatter is consistently higher
in HH than in VV (Figure 4b) as first noticed by Pidgeon
[1968]. The magnitude of huDi in both VV and HH
increases toward decreasing qg but the difference remains
almost constant for qg > 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 qg > 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 Rs and s or uD 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 uD in the subpopulation
with Rs > 0 dB spreads over a wide range. The jpdf analysis
illustrates convincingly that events with Rs > 0 dB occurs
over a broad range of s and uD, 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
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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 Rs > 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 uD for both VV and HH
polarizations of the full population and subpopulation with
Rs > 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 uD
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) (Rs, sVV), (b) (Rs, uDV), (c) (Rs, sHH), and (d) (Rs, uDH).
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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 (Rs>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 Rs > 0 dB, the pdf of uDH is almost
invariant with respect to qg and only relatively small
variation with qg is found in the pdf of uDV. 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, ps(c), with c calculated by the Doppler frequency of sea spike events, (4) and (5). The result is
obtained from the data with qg between 3 and 6° for better
signal-to-noise ratio. The fraction of sea spike pixels in the
three qg 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.
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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
LðcÞ ¼
P
acð d 2 Þ
;
AT Dc
ð6Þ
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) c3ps(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), c4L(c), and c5L(c), respectively, thus they are also
proportional to c4ps(c), c7ps(c), and c8ps(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 qg between 3 and 6°.
The integrated velocity scale of the n-th moment is
calculated by
Z
cn ¼
cn ps ðcÞdc
1=n
:
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
8 of 16
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Table 2. Integrated Velocity Scales (m/s) of the First Eight Moments of the Breaking Velocity Distributiona
c1
c2
c3
c4
c5
c6
c7
c8
#
2.51
2.58
2.51
2.59
2.52
2.53
2.49
2.42
2.04
2.57
2.57
2.63
2.57
2.50
2.44
2.32
2.34
2.35
2.44
2.46
2.58
2.58
2.59
2.48
2.37
2.32
2.18
2.20
2.08
2.55
2.62
2.55
2.63
2.56
2.57
2.54
2.46
2.08
2.61
2.61
2.67
2.62
2.54
2.49
2.37
2.39
2.40
2.49
2.51
2.62
2.63
2.63
2.53
2.43
2.37
2.24
2.26
2.14
2.59
2.66
2.59
2.67
2.60
2.61
2.58
2.50
2.12
2.65
2.65
2.71
2.66
2.59
2.54
2.42
2.44
2.45
2.53
2.55
2.65
2.67
2.68
2.58
2.48
2.42
2.29
2.30
2.20
2.63
2.70
2.62
2.70
2.63
2.65
2.61
2.54
2.16
2.68
2.68
2.74
2.70
2.62
2.58
2.47
2.48
2.49
2.57
2.58
2.69
2.70
2.71
2.62
2.52
2.46
2.34
2.35
2.25
2.66
2.73
2.65
2.74
2.67
2.68
2.65
2.57
2.20
2.71
2.71
2.78
2.74
2.66
2.62
2.51
2.52
2.53
2.61
2.62
2.72
2.74
2.75
2.66
2.56
2.50
2.39
2.39
2.30
2.69
2.76
2.68
2.77
2.70
2.71
2.68
2.61
2.24
2.74
2.74
2.81
2.78
2.69
2.66
2.55
2.56
2.57
2.65
2.65
2.75
2.77
2.78
2.69
2.60
2.53
2.43
2.43
2.35
2.72
2.79
2.71
2.80
2.73
2.74
2.71
2.64
2.27
2.77
2.77
2.83
2.81
2.72
2.69
2.58
2.60
2.60
2.68
2.68
2.78
2.80
2.81
2.73
2.64
2.57
2.47
2.46
2.39
2.75
2.82
2.73
2.82
2.75
2.77
2.74
2.67
2.31
2.80
2.80
2.86
2.84
2.75
2.72
2.62
2.63
2.63
2.72
2.70
2.81
2.83
2.84
2.76
2.68
2.60
2.51
2.50
2.43
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
a
#1-9: 22 March 2006, #10-18: 10 April 2006, #19-29: 11 April 2006.
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 Distributiona
cp1
cp2
cp3
cp4
cp5
cp6
cp7
cp8
#
2.63
2.73
2.76
2.84
2.68
2.68
2.60
2.60
2.17
2.81
2.84
2.87
2.84
2.68
2.87
2.46
2.46
2.54
2.66
2.70
2.81
2.73
2.89
2.51
2.51
2.46
2.47
2.41
2.33
2.78
2.79
2.82
2.84
2.76
2.74
2.89
2.60
2.17
2.81
2.84
2.92
2.89
2.86
2.90
2.52
2.46
2.66
2.66
2.70
2.89
2.87
2.95
2.82
2.51
2.46
2.62
2.49
2.38
2.86
2.87
2.82
3.02
2.86
2.81
2.94
2.70
2.17
2.92
2.92
3.11
2.98
2.89
2.90
2.73
2.74
2.73
2.86
2.70
2.92
2.87
2.95
3.02
2.82
2.68
2.68
2.63
2.60
2.92
2.89
2.89
3.02
2.87
2.86
2.97
2.87
2.31
3.02
2.97
3.11
3.11
3.02
2.97
2.73
2.74
2.86
2.89
2.94
3.06
3.14
2.97
3.02
2.87
2.68
2.74
2.65
2.60
3.03
3.00
2.95
3.03
3.03
2.97
3.05
2.89
2.41
3.03
2.98
3.13
3.11
3.06
2.98
2.73
2.84
2.97
2.89
2.97
3.06
3.14
3.02
3.05
2.97
2.90
2.81
2.66
2.70
3.03
3.02
2.97
3.03
3.14
3.10
3.06
2.97
2.66
3.06
3.11
3.13
3.19
3.13
2.98
3.05
2.95
3.06
2.95
3.00
3.10
3.14
3.02
3.05
3.13
2.90
2.90
2.86
2.73
3.08
3.16
2.97
3.06
3.14
3.16
3.10
3.00
2.66
3.06
3.13
3.13
3.22
3.13
2.98
3.05
2.95
3.06
3.02
3.00
3.10
3.14
3.11
3.06
3.13
2.90
2.90
2.97
2.87
3.11
3.16
3.03
3.11
3.14
3.18
3.14
3.00
2.81
3.06
3.13
3.14
3.22
3.13
2.98
3.05
2.97
3.06
3.02
3.00
3.10
3.14
3.11
3.11
3.16
2.95
2.90
2.97
2.87
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
a
#1-9: 22 March 2006, #10-18: 10 April 2006, #19-29: 11 April 2006.
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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, cb/cp, to be about
0.25. 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 cb/cp 0.4 to 0.63. 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, Ps, is computed from
integrating the pdf of Rs for Rs > 0 dB. Figure 9a shows
the result of Ps as a function of wind speed. The quantity Ps
is proportional to the sea spike density (in m2 s1) 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 1.2 104/m2/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, ls = 2p/ks, can be
derived from the mean Doppler frequency of the sea spike
population using (4) and (5) and the dispersion relation, ks =
g/c2s . The result from our data shows that the average ls 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,
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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 Pw =
1.35 105U3.4 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 Pw = 1.7 106U3.75
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, Psc, can
be estimated by Psl2s (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 Psc and Psl2s , which is
found to be
Psc ¼
Ps ð0:1ls Þ2 0:055Ps l2s
¼
;
A1
Ap
ð8Þ
where Ap is the average pixel area (1.5 m 3.6 m), and
A1 = 1 m2, introduced in the equation to emphasize that Psc
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 Psl2s 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 4.3.
4.2. 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
Pw ¼ 1:5 105 ðU 2Þ3 ¼ 1:5 105 U 3 1:2 104 ;
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
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Figure 11. The hourly surface wave spectra covering the radar measurement periods, (a) 22 March
2006, (b) 10 April 2006, and (c) 11 April 2006. (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.
4.3. 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 41008 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
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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 KpHs/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. Parameteri-
zation with wind speed alone would miss the influence of
non-locally generated swell.
[33] The ratio cb/cp discussed in section 4.1 can be
estimated from the sea spike result of cs/cp (Figure 13).
Under a given wind speed, the result shows that the relative
length scale of breaking is much larger in 22Mar06, with cs/
cp ranging mostly between 0.3 and 0.4, than that for the
other two days (10Apr06 and 11Apr06), with cs/cp 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 cs/cp.
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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 cs/cp reflects
the change of cp rather than the change of cs. 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 cs with cp in fact separates rather than
brings together data groups from different sea state conditions. This decoupling of cs and cp 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 U2/
gHs and U/cp are more preferred than U or KpHs/2 for
bringing different data sets together although the number of
data points is small and the scatter large. The physical
meaning of U2/gHs is dimensionless energy and U/cp
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:
1:27
2 1:17
U
U
¼ 4:9 103
;
gHs
cp
ð10Þ
0:68
2 0:56
cs
U
U
¼ 2:3 101
¼ 9:7 102
:
cp
gHs
cp
ð11Þ
Psc ¼ 3:1 102
[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:
14 of 16
0:17
2 0:16
U
U
cs ¼ 2:3
¼ 1:8
;
gHs
cp
ð12Þ
0:34
2 0:33
U
U
ls ¼ 3:4
¼ 2:0
;
gHs
cp
ð13Þ
HWANG ET AL.: RADAR SEA SPIKES AND BREAKING WAVES
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Table 4. Fitting Coefficient, A, Exponent, a, for the Velocity
Scales of the First Eight Moments of the Breaking Velocity
Distributiona
n
A1
a1
A2
a2
1
2
3
4
5
6
7
8
2.343
2.394
2.441
2.484
2.524
2.561
2.596
2.629
0.151
0.142
0.134
0.127
0.121
0.115
0.109
0.104
1.873
1.938
1.999
2.055
2.108
2.157
2.204
2.248
0.142
0.134
0.127
0.120
0.114
0.109
0.104
0.099
a
y = Axa, where y is velocity scale, x is U/cp or U2/gHs. Subscript 1 is
for parameterization with U/cp, subscript 2 for parameterization with U2/
gHs.
The coefficients and exponents of least squares fitting
applied to the velocity scales of the first eight moments of
breaking velocity distribution are listed in Table 4.
5. Summary
[36] We have presented the statistical analysis of low
grazing angle radar return from the sea surface obtained
by a dual-polarized coherent radar. The dependence of
scattering cross section and Doppler velocity on the grazing
angle is outlined in section 3. The distribution of breaking
wave velocity and wavelength is quite narrow and the
average scales are much smaller than those of the dominant
component of the surface wave spectrum. The length or
velocity scale of breaking waves is uncoupled from the
length or velocity scale of dominant waves, reflecting the
spatially localized nature of breaking process. The fraction
of sea spike coverage is considerably higher than the
whitecap coverage at a given wind speed, reflecting the
prevalence of steep features during wave breaking that
produce significant enhancement in radar backscatter but
undetectable by whitecap analysis (section 4). Clarification
of the length scale of breaking waves is important to
accurate quantification of properties such as rates of surface
turn over and energy dissipation of wave breaking, which in
turn are important to a broad range of subjects such as airsea gas, momentum and energy transfers, passive and active
ocean remote sensing, ocean engineering and wave dynamics. The breaking probability and sea spike coverage show a
general increase with wind speed but the trend is complicated by the background wave conditions. Parameterizations of breaking probability and sea spike coverage
combining both wind and wave factors show some apparent
advantage than parameterizations with wind or wave factors
alone (Figures 12– 14, equations (10) – (13)). However, with
only a small quantity of sea spike data available the data
scatter is large. The analysis method presented here for sea
spike processing is relatively simple and easy to implement.
With radar remote sensing of the ocean becomes more
popular, we hope that more results on the analyses of sea
spikes and breaking waves will become available to clarify
many of the difficult issues concerning breaking waves,
such as the generation process, length scale, probability
distribution, turn-over area, momentum flux and energy
dissipation.
C02003
[37] Acknowledgments. This work is sponsored by the Office of
Naval Research (NRL work unit 6863 and 8953, and program element
62435N and 61153N; NRL contribution number NRL/JA/7260—07-0199).
We are grateful to the insightful comments from two anonymous reviewers.
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