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1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 3. DATES COVERED (From -To)
12-07-2007 Journal Article _
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Spectral Signature Wave Breaking in Surface Wave Components of
Intermediate Length Scale 5b. GRANT NUMBER
5c. PROGRAM ELEMENT NUMBER
0602435N
6. AUTHOR(S) 5d. PROJECT NUMBER
Paul A. Hwang
5e. TASK NUMBER
5f. WORK UNIT NUMBER
73-6628-85-5
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
REPORT NUMBER
Naval Research Laboratory
NRL/JA/7330-05-5262
Oceanography Division
Stennis Space Center, MS 39529-5004
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Approved for public release, distribution is unlimited.
13. SUPPLEMENTARY NOTES
14. ABSTRACT
This paper investigates the length scale of ocean surface breaking waves in the spectral range of intermediate wavelength components a few
centimeters to a few meters long. The spectral properties of wave breaking are examined first with the dissipation function of the wave action density
conservation equation. The analysis reveals a strong breaking signature in wave components between 0.15 and 1.5 m long in the form of a quasi-
singular behavior of the dissipation function using the present formulation of the wind-generation and breaking dissipation functions. Independent
studies of more-direct breaking observations of radar tracking of sea spikes in the past have shown close correlation between sea spiked and
scatterers traveling at the speed of surface waves a few meters long and much shorter than the dominant wavelength. The intermediate-scale waves
are the primary contributor of the ocean surface mean-square slope, the close correlation between the gas transfer rate and the mean-square slope
has been demonstrated repeatedly. A better understanding of the wave dynamics of intermediate-scale waves is important for clarification of various
gas transfer mechanisms.
15. SUBJECT TERMS
wave breaking, surface roughness, intermediate-length scale, dissipation, sea spikes
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE ABSTRACT OF Paul A. Hwang
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Available online at www.sciencedirect.com
JOURNAL OF
- ScienceDirect M A R I N E
SYSTEMS
ELSEVIER
Journal of Marine Systems 66 (2007) 28- 37
www.clsevier.com'Iocatewiiiarsys
Spectral signature of wave breaking in surface wave
components of intermediate-length scale
Paul A. Hwang *
Bldg, 2. Roon 244E. Naval Researuh Laboratoir 4555 Overlook Avenue SW Ifitshington. DC 20375. U.S,4.
Received 12 September 2005; accepted I I November 2005
AMailable online 7 September 2006
Abstract
This paper investigates the length scale of ocean surface breaking waves in the spectral range of intermediate wavelength
components a few centimeters to a few meters long. The spectral properties of wave breaking are examined first with the
dissipation function of the wave action density conservation equation. The analysis reveals a strong breaking signature in wave
components between 0. 15 and 1.5 m long in the form of a quasi-singular beha, ior of the dissipation function using the present
fornnulation of the wind-generation and breaking dissipation functions. Independent studies of more-direct breaking observations of
radar tracking of sea spikes in the past have shown close correlation between sea spikes and scatterers traveling at the speed of
surface waves a few meters long and much shorter than the dominant wavelength. This feature of sea-spike properties is consistent
with the breaking signature of the dissipation function in similar wavelengths. The intermediate-scale waves are the primary
contributor of the ocean surface mean-square slope. The close correlation between the gas transfer rate and the mean-square slope
has been demonstrated repeatedly. A better understanding of the wave dynamics of intermediate-scale waves is important for
clarification of various gas transfer mechanisms.
,,c 2006 Else\,ier B.V. All rights reserved.
Keywords: Wave breaking; Surface roughness; Intermediate-length scale: Dissipation; Sea spikes
1. Introduction 1994; Monahan, 2002; Asher et al., 2002). Clarification of
the spectral properties of breaking surface waves may
Wave breaking has profound impacts on the upper- contribute to a better understanding of the gas transfer
ocean turbulence properties, the disruption of the ocean process in the ocean. Wave breaking is also an important
surface cool skin, the generation and entrainment of air subject in many different ocean remote sensing applica-
bubbles into the water column, and the composition of the tions, for example, sea spikes in radar scatter from the
ocean surtfce roughness. All factors cited above arc ocean surface (e.g., Lee et al., 1996; Frasier et al., 1998;
known to produce significant modifications of the gas Liuetal., 1998; Phillips et al., 2001) and bubble effects on
transfer rate across the air-sea intertfce (e.g., Phillips, underwater acoustics (e.g., Ding and Farmer, 1994; Deane
1985; Wallace and Wirick, 1992; Thorpe, 1993; Melville, and Stokes, 2002; Melville and Matusov, 2002).
In this paper, experimental results on the length scale
of breaking waves are examined. Section 2 describes the
• Iel.: 1I 202 767 0800W fax: 1 202 767 5599. indirect approach of breaking wave analysis through the
E-mciladd.vs.: phxNangeaccs.nrl.navy.mil. investigation of the dissipation function of the wave
0924-7963/S -scc front matter i(ý 2006 Iseier 1V. All rights reserved,
do i:1 0. l0 16,'j.j.m ars s.2005.l .015 U; TUT B; ON 3T AT 7 iT 7N T A
Appi Oved 0or Public Pelease
U161ribution Unlimited
PA, Iti ang / Journalo ! Marine S~stems 66 (2007) 28 37 29
action density conservation equation (e.g., Phillips, source functions due to nonlinear wave wave interac-
1984. 1985). The method has been applied to the spectra tion, wind input and breaking dissipation. N is related to
of intermediate-scale waves (from 0.02 to 6.3 m long) the 2D wave displacement spectral density, z(k), by N
collected in the ocean using a free-drifting measurement (k) =gX(k)i/r, g the gravitational acceleration, and a the
technique to mitigate the problem of Doppler frequency intrinsic frequency of the wave component. In the
shift in converting the measured encounter-frequency following, the result applicable to the omni-directional
spectrum to the wavcnurnber domain (Hwang and spectrum,X(k)=iX(k)kdO and N(k)-gZ(k)/1r, is
Wang, 2004). Interesting features of the spectral discussed.
properties of intermcdiate-scalc surface waves in the Theoretical and experimental studies of the energy
ocean derived from the analysis are discussed. In transfer from wind to waves have led to the following
Section 3, the wind-speed dependence of the wave parameterization function for the wind input source
spectral density is presented. The topic is of great function (Plant, 1982; Phillips, 1984)
interest in many areas of research including wave
dynamics, air sea interaction and microwave remote Sw(k) nmu(-.N(k). (2)
sensing of the ocean. Results from earlier studies on
surface waves of length scales from long gravity waves where n 0.04. For gravity waves several times shorter
to short capillary wvaves are found to be in good than the spectral peak component. the nonlinear wave
agreement with the intermediate-scale wave data. In
SSeeccttiioonn n44t, ,ththtei leddi issssiippaattioionn fnfu ncticotionn ddeerirvvede dffarmotma . tt he wttee armvems in((PPteh riailllc itpipos, n , 1t1e 9r7m77 . i,s 1m998u84 c)h). sTmhhae lldedryy n thaaman i nct h ce b bao ltahane crce et wooof
action density conservation equation is described. The intermediate- and short-scale gravity waves, therefore, is
most interesting result is that the dissipation function mainly determined by the wind-input and breaking
displays a quasi-singular property in the middle range dissipation terms. Phillips (1984, 1985) suggested the
(about 0.15 to 1.5 in) of the intennediate-length scale. following expression for the omni-directional dissipa-
Such a feature is interpreted as a distinct signature of tion sink term
wave breaking in wave components of intennediate-
length scale. Results on the length scale of wave D = gkf1(B). (3)
breaking obtained from more-direct wave breaking
measurement techniques, including tracking of bubble- where B=k3X=k3aN/g is the dimensionless wave
generated acoustic noise (Ding and Fanner, 1994), radar spectrum representing the degree of saturation (Phillips,
sea spikes (Lee et al., 1996; Liu et al., 1998; Frasier 1985). Eq. (2) can be expressed in terms of B,
et al., 1998; Phillips et al., 2001), whitecaps (Melville,,,\2
and Matusov, 2002), and high-speed photographs of Sw(k) = i(n gk 3B(k). (4)
bubble plume formation (Deane and Stokes, 2002) are
investigated in Section 5. It is shown that the length At equilibrium dNidt=0. Equating (3) and (4), Phillips
scales of wave breaking derived from direct and indirect (1984) concluded that the solution to the unknown
methods are in very good agreement. Additional dissipation functionfi(B) relies on the knowledge of the
discussions on issues of wave dynamics are given in dependence of B on the wind speed. Experimental data
Section 6 and a summary is presented in Section 7. show that the dependence of the spectral density on
wind speed can be represented by a power-law function
2. Analysis (Hwang and Wang. 2004)
Wave breaking is the dominant dissipation mecha- B(-) ((5)
nismn in the source function balance of the action or C /
energy conservation equation of surface waves. The
conservation equation of wave action, N(k), can be Assuming a power-law relation for the unknown
expressed as (e.g., Phillips, 1984, 1985) function in the dissipation term (Phillips, 1984, 1985)
f(B) = AdB"" (6)
dN e'I S
dt (cid:127)I the expressions for ad and Ad become simply
where k is the wavenumber vector with modulus k. The ad I J 2 Ad - 1. (7a. b)
three terms on the right-hand side of (I) represent the a0
30 PA. Hiwang "Inournal t lMarine SYitewns 66 (2007) 28 37
3. Spectral dependence on wind speed by two separate sensor systems. The first system is a two-
dimensional scanning laser slope sensor that generates
The approach described above is most suitable for directly the wavenumber spectrum from the spatial
applications in the region of intermediate- and short- images of the surface slopes over an area 0.1 m x 0.1 m.
scale waves where the magnitude of the nonlinear The second system constitutes of thin-wire wave gauges
wave-wave interaction term is much smaller than that at fixed stations that provide the frequency spectrum of
of the wind input or the breaking dissipation term. surface waves measured at essentially the same wind fttch
Obtaining reliable measurements of intermediate- and as that of the scanning slope sensor system. The labo-
short-scale waves in the ocean is a complicated task. ratory measurements further confirn the robust power-
One of the most difficult issues is the Doppler frequency law relationship of B(u,!c) as illustrated by the examples
shift that renders the interpretation of the length scale of shown in Fig. lb. From these field and laboratory data, .,4o
measured encounter frequency spectrum an uncertain and ao can be obtained by least-square fitting of the
task (Hwang, 2006). This problem can be mitigated by spectral measurements for each wavenumber component.
free-drifting operation that provides measurements in a The results arc given in Fig. Ic and d, respectively. An
frame moving with the advecting currents that caused interesting feature in the results of At and au is their
the Doppler frequency shift. Phillips (1984) method of nomnonotonic dependence on k, as clearly illustrated in
analysis described in the last section is applied to the the field data. The results from laboratory experiment (all
spectra of intermediate-scale waves (wavelengths be- wind-sea conditions) show a similar dependence of A,
tween 0.02 and 6.3 m) measured in the ocean using and an on k in the wavenumber range comparable to those
wave gauges mounted on a free-drifting platform of the field data, but because the length scale of laboratory
(Htwang and Wang, 2004). The results show that B(u,! wind-generated waves is much shorter than that in the
c) can be represented by a power-law function (5). An field, the interesting nonmonotonic feature revealed by
example is shown in Fig. Ia . the field data cannot be detected in the laboratory results.
Recently, laboratory experiments were conducted in Also, there is an apparent wavenumber downshift in the
the large wind- wave facility at the Institut de Recherche cluster of laboratory data derived from temporal measure-
sur les Phenomenes Hors Equilibre (IRPHE), Marseille, menits by stationary sensors when compared to the result
France. The wind-generated surface waves are measured derived from spatial measurements. This is likely due to
(a) (c) -_ __ __ __-_..._II
Field drifling (k= 16) 0 Field (0Aind ,wad
"I0 Field (mixed sa,)
"- Fquilihnuin aymptole
" Lahoraory iemporal
10-20-
10o1 full MIoit 2
(b) (d) k( r.udm
a tLah spalial (k=452) 1.5 E
* Lah temporal (k=264)
10-2 A
(cid:127) +
a (1.5
100
10.1 Io0 i00 I01 102
(t,/C ) k (rad/rn)
Fig. I. Examples of the power-law dependence ofBltinrj: (a) field measurements by wave gauges mounted on a free-driffing platform, (b) laboratory
measurements by sLalionary wave gauges (temporal) and scanning laser slope sensor (spatial). The wavenumber dependence of(c) the proportionality
coetficient. A,, and (d) the exponent, a(. of the power-law function derised from both field and laboratory experiments.
P.A. HIwang Journalo l Marine Systerns 66 (200') 28 37 31
the Doppler frequency shift induced by the orbital summarized by many researchers (e.g., Toba, 1973;
velocity of longer waves. Because the Jacobian connect- Phillips, 1985; Forristall, 1981; Donelan ct al., 1985:
ing the wavenumber-to-frequency conversion is nonlinear Hwang et al., 2000). Based on the 2D wavenumber
with respect to the advecting current, the Doppler fre- spectrum derived from the spatial 3D surface wave
qucncy shift due to the background orbital velocity does topography of a steady and equilibrium wind-generated
not vanish on averaging over integral cycles of long wave field obtained by an airborne scanning lidar system,
waves. Instead, the advection by orbital velocity produces Hwang et al. (2000) reported b z5.26 x 10 2. The
a net downshifl in the resulting encounter frequency for a analysis of intenrediate-scale wave spectra also yields
given wavcnumber component (Hwang. 2006). similar values of the proportionality coefficient and
The nonmonotonic behavior of the spectral dcpen- exponent (A1. and a() in the lower limit of the resolved
dence on wind speed as quantified by au is noteworthy. wavenumber range (I!< k_< 316 rad/m) (Fig. Ic -d).
Discussions of this feature have been given by Hwang and Banner ct al. (1989) reported wavenumber spectra of
Wang (2004) and they are briefly summarized in the surface waves derived from 3D surface topography
following. As illustrated in Fig. Id , tbr both conditions of measured by stereo photography. The resolved wavc-
wind sea and mixed sea, the wind speed dependence of the lengths are between 0.2 and 1.6 m (31 Ž k Ž4 rad/m). The
wave spectral density approaches linear toward the long range of wind speeds is between 5.5 and 13.3 mns. Their
gravity wave region. This is in agreement with many results show a very weak wind-speed dependence of the
earlier investigations on the wind-speed dependence of wave spectrum, with B(k) -u,1 (indicated by a short
the surface wave spectrum in the equilibrium range, line-segment in Fig. 2). The analysis of intermediate-scale
wave spectra expands the resolved wavelengths to a range
-2. (8) from 0.02 to 6.3 m (I _k<_< 316 rad/m). For wind sea, the
C wind-speed exponent drops sharply from approaching 1.0
The corresponding dimensionless spectrum is (linear) for long gravity waves to about 0.22 neark= 5 rad/
m; at, remains to be less than 0.5 through k=40 rad/m. In
b (91) mixed sea, the minimum value of ao is also about 0.22,
'
B(cid:127),(k) h " (9) and a,, < 0.3 fork between 8 and 25 rad/m, and ao- :0.5
tbr k between 3 and 80 radim (Fig. 2).
The range of h varies approximately within a factor-of- For short gravity waves and capillary waves, field data
two range based on field observations, as has been are usually obtained by radar backscatter measurements
3 1.... 1.1 . . ... I . . .I .I .. .I 1 1 1 11 11
- Slerco Phodo (B89) 0
- Driffing gauge, wind sca (W04)
2.5 --- " " ixed % -a
0 Radar. )>O.@6in (fom Fig. 5, H97) 0
"O.A>X,>0.2
in
* 0.2A>>0.05 in
2 o 0.05>X.>O.03 inI *0 0
0 0.03>".02 il
0.02A.r>0.001 in o~o
j1.5 - Fquilibrium asymptote + 00
0 #a
0 , ,, I 1 II 1
I0W 10I 102 103
k Iraid/tn)
Fig. 2. The wvind-specd exponent of a given wase spectral component obtained by different methods including radar return (Jones and Schroeder,
1978, Stewart, 1985; Masuko et al.. 1986; Phillips, 1989; Weissinan ctal., 1994; Colton ci al., 1995; summnarized in Fig. 5 of4lwang. 1997), stereo
photography (Banner ct al.. 1989), and ffre-drifiing wVave gauge (I hvang and Wang. 2004); ., is the radar wavelength.
32 PA. Hwang /,Journal of Marine Systens 66 (2007) 28 37
using diftcrent radar frequencies and incident angles (e.g., short waves. The results suggest that the spectral
Jones and Schroeder, 1978; Stewart, 1985: Masuko et al., signature of wave breaking is localized in the wave-
1986; Phillips, 1988; Weissman et al., 1994; Colton et al., number space in the intermediate-length scale. [Detailed
1995). Fig. 5 of Hwang (1997) summarized the radar data breaking wave observations also show the obvious
with incidence angles between 250 and 600, in which signature at the dominant-wave length scale (e.g.,
range the dominant scattering mechanism is the Bragg Banner et al., 2000, 2002). The results presented here
resonance so the radar cross section is proportional to the do not resolve wavelengths near the spectral peak
spectral density of the resonant roughness spectral component. The focus of the present research is on
component. The radar data are shown in Fig. 2 together surface waves in the intermediate-length scale. These
with the result derived from the analysis of intermediate- intennediatc-scale waves contribute more than 78% to
scale wave spectra described above. Despite the large the total mean-square slope of gravity waves generated
scatter in the radar data, there are apparent similarities in by winds up to 20 rn/s (Hwang, 2005).] From the point-
the results obtained by these fundamentally different of-view of the spectral response to wind input, as shown
approaches of obtaining the wind-speed exponent of the in Fig. I d or Fig. 2, the presence of a localized region in
wave spectral component. the wavenumber space where the spectral density is only
weakly dependent on the forcing wind condition
4. Dissipation function suggests a localized region where the wave growth is
quenched by strong dissipation. Further discussions of
Fig. 3 shows the coefficients (7) of the dissipation this point will be described in Section 6. The range of
function (6) obtained from field and laboratory wavelengths with at<0.3 is about 0.15 _<), _ 1.5 m.
measurements. The breaking dissipation function dis- These wave components are sufficiently long such that
plays a quasi-singular behavior in the neighborhood of the energy loss due to viscous dissipation (Lamb, 1945)
meter-long wave components. In mixed sea, the or generation of parasitic capillary waves (Longuet-
breaking scale shifts toward smaller breaker size and Higgins, 1992, 1995) may be neglected. It is reasonable
the size range expands in comparison with those of the to assume that wave breaking is the major dissipation
wind sea, reflecting the modification by long waves and mechanism for the observed peculiarity in the dissipa-
interaction between background orbital velocity and tion function of intermediate-scalc waves.
(a) (b)
-- Field (MS)
aLab temp.
op, Lah spatial
IoEqui.asymp.
< I
* A' it
6a* OF
F~ig _i.T rhe wavcnumber dependence of (a) the proportionality coefficient. Ad, and (b) the exponent, u,j, of the power-law represe2ntation of"the
dissipation l'unction, I(B). Results from free-driffing measurements in the field in wind sea and mixed sea as well as laboratory data by stationary
temporal and spatial measurements are presented.
PA. Htwanig Journal o/ Marine Systems 66 (2007) 28 37 33
Microwave observations of the ocean surface over breaking events as a function of the breaker phase speed.
the last fe~w decades indicate that Doppler shifts at HH The length scales of breaking waves can be calculated
(horizontal transmitting, horizontal receiving) polariza- from the reported statistics, they range from about 0. I to
tion and low grazing angles are due to scatterers 3 in at 4 mis wind speed and from 3 to 20 m at 15 m/s
traveling at the speed of surface waves of a few meters wind speed (Fig. 4a). Two major factors contribute to
long (e.g., Lee et al., 1996; Frasier et al., 1998; and the the observed wide range of the breaking length scales at
references therein). This feature is consistent with the a given wind speed. The first is the dynamic range of the
strong signature of wave breaking in the intermediate- sensors. In particular, the passive acoustic tracking of
length wave components. In the next section, some breaking events cannot detect small-scale breakings due
recent breaking-wave observations made by more-dircct to the problem of ambient noise (Ding and Fanner,
measurement techniques are summarized to compare 1994). The second factor is the stage of wave
with the result using the indirect approach of dissipa- development. The experiments reported by Lee et al.
tion-function analysis. (1996) were conducted in a lake and a protected bay
with limited wind tiztch so the wave field is relatively
5. Breaking length scale young compared to the wave conditions of the others
obtained in the open ocean environment. The second
In the last decade or so, many more-direct measure- factor can be compensated somewhat by normalizing
ments of breaking waves have been reported, using the breaking length scale, h.a,b y the dominant
techniques such as passive acoustic tracking of the wavelength of the wave field, .P.W ith this normaliza-
ambient noise produced by bubbles generated by tion, the majority of radar data are clustered in the -bi!X
1
breaking waves (Ding and Farmer, 1994), radar tracking range between 0.04 and 0.2 over the full range of wind
of sea spikes (Lee et al., 1996: Frasier et al., 1998; Liu speeds encountered in the experiments (Fig. 4b). The
et al., 1998, Phillips et al., 2001), and video tracking of mean value with a standard deviation is 2ýi;.-o=0.092 1
whitecap evolution (Melville and Matusov, 2002). 0.051, thus the representative breaking length scale is
These remote sensing techniques usually process a about one order of magnitude shorter than the dominant
very large population of breaking events, on the order of wavelength of the wave field based on sea-spike
tens of thousands. From these analyses, the authors observations. [Although the normalized breaking length
report statistics of the frequency of occurrences of scale seems to bring together data sets from very
(a) (b)
o Video whitecaps MM, 2
o tydrol.hone. 1)F94
*Radar PO I 0 o
0 Radar irisingswellh. F98 ( 00°o
. Radar (sicad ) / o 0 0 0
10 2 . Radarideca)ing) / 00
* Radar 1[96 00 0
Wire (vwind sea)1I)I 4o
- - Wire onixed seas)/,+
-- Well dkwehoc(cid:127)d' O °i
0 0 01 00
101 s 0-
--
4
10 E• o I
g 4 h r n li0gbtih scale , obt.in. y d nict m.h. . . i b i
o 0 - 0 -- .. ÷
)"(cid:127)............... .. ........".:.
I "I '
i0-I II, ,I
alyiisgahdaboe1.0h1I0D mniols brakn legtI 0 scale, 1(,10 (cid:127).I01
[tl ot( IS) ( 11 i0 nit(cid:127)s
Fig. 4. (a) Theli representative breaking length scale, )-b, obtained by different more-direct measurement techniques of wave breaking including
tiacking (if acoustic noisc' , whitecaps and sea spike~s. Estimiation of thc upper and lower hounds of the breaking length scales froin the dissipation
analysis is graphed as boxes. (b) Dimensionless breaking length scale, A+)P.
34 PA. HIwang .Journalo l Marine Sysntms 66 (2007) 28 37
different wave development stages, it should be pointed acceleration and the surface tension divided by the water
out that for open ocean measurements the data scatter density, and is almost constant in sea water without
with breaking length given in dimensional form is much surface slicks), the reduced scatter in the dimensional
smaller than that given in dimensionless forn (Fig. 4). representation of the breaking length scale would be
The cause and significance of this interesting result is sensible. However, as discussed earlier, the range of the
not obvious at this stage but it seems to suggest that the length scales of nonmonotonic behavior is between 0.15
peak wavelength is not necessarily the proper scaling and 1.5 m, and far away from the capillary influence. A
factor for intermediate-length wave components several better explanation remains elusive.]
times shortcr than the spectral peak wavelength. If the The breaking length scale can be estimated from the
scaling length is the wavelength at minimum phase indirect breaking analysis using the dissipation function
speed (determined by the ratio of the gravitational by setting a threshold in the result of Ad(k) or a (k)
1
Fig. 5. Pictures from high-speed photography of hubbtc plume linnalion under breaking waves (Fig. 2 of' Dearie and Stokes, 2002).
PA. /lhang Journal ol Marime Svstemns 66 (2007) 28 37 35
shown in Fig. 3. For example, applying a threshold of models. In most applications, the action density
106 times less than the peak value of Ad, one can derive conservation equation fornulated as Eq. (1) is sufficient.
the breaking length scales from the lower and upper In coastal areas, the bottom dissipation function needs to
bounds of the wave numbers that Ad, exceeds the be included on the right-hand side of the equation. For
threshold. The result is shown in Fig. 4. The breaking intermediate-scale waves, the situation is not as clear. A
length, Lb,i s between 0.6 and 1.5 m for wind sea, and key question is whether wind forcing is the sole source
between 0.15 and 1.2 ril for mixed sea. The ratio A8//v0 of wave generation. The results from radar sea-spike
ranges between 0.013 and 0.025 for wind sea, and from analyses are very instructive. Many of those experi-
0.008 to 0.05 for mixed sea. Similar results can be ments are accompanied with detailed video recording of
derived by using a threshold (of about 7) in ad. These the wave field in the radar field-of-view (e.g., Frasier
values of breaking length scales are smaller than those et al., 1998; Liu et al., 1998). The close association of
derived from sea-spike measurements. It is likely that sea spikes with (large-scale, or dominant) wave
sea-spike tracking overestimates the breaking length breaking or steep waves is convincingly established,
scale because the tracking procedure requires a finite yet the phase velocity of the breakers derived from
dwell time of the breaking event in the field of tracking sea spikes is much slower than the phase speed
measurements. For example, Frasier et al. (1998) set a of the dominant wave component. The logical explana-
threshold of I-s minimal dwell time for the sea-spike tion of this result is in the distinction of wave breaking
event to be included in the breaking analysis, thus events (as in dominant wave breaking) vs. breaking
excluding shorter breaking events in the final statistics. patches (as in sea spikes or the length scale of the
This problem is similar to the ambient-noise limitation dissipation function). The breaking patches are obvi-
on the length-scale resolution in the acoustic tracking of ously generated by breaking of dominant waves. For a
breaking-induced acoustic signature encountered by very short moment, they retain a phase speed close to or
Ding and Farmer (1994). Taking this bias into exceeding that of the dominant waves (as a bound-wave
consideration, it is concluded that the results from direct component). Judging from the sea-spike tracking
and indirect measurements ofthe breaking length scales analysis, this duration is very short and for the
are comparable. remaining part of the sea-spike (breaking) lifetime, the
Deane and Stokes (2002) conducted laboratory study breaking patches propagate as free waves with phase
on the generation mechanism of bubble plumes under velocities according to their sizes (in the intermediate-
breaking waves. Using high-speed photography, they length scale). In other words, breaking plays double
captured many interesting hydrodynamic processes in the roles in the intermediate-scale wave components: it is a
active phase of wave breaking. Relevant to the present dissipation function of wave energy and an important
study is the sequence of pictures depicting the develop- source function of intermediate-scale wave generation.
ment of bubble plume in the first two seconds after Although it is reasonable to state that there is an
formation of the air cavity trapped by the breaking jet additional source function due to breaking-wave
(their Fig. 2, reproduced as Fig. 5 here). The dominant generation in the intermediate-scale spectral compo-
frequency of the wave field is 0.73 liz and the nents, it remains a difficult task explaining the lack of
corresponding wavelength is 2.3 m. As illustrated by response to wind-forcing in the mid-range components
the sequence of pictures, after more than one wave period of intermediate-scale waves, observed by both in situ
tollowing the formation of the air cavity produced by the wave measurements and remote-sensing radar scatter
jet of wave breaking, the length scale of the active analyses (Fig. 2). It is generally accepted that breaking
breaking region with intensive dissipation (the vortex probability increases with the cube of wind speed, so
regions in the photographs) remains on the order of 0.1 m. incorporation of breaking as a generation term for
This gives an estimate of the ratio )4/;,Pzz0 .025, which is intermediate-scale waves should increase the wind-
in very good agreement with the results derived from the speed exponent for the wave components affected,
analysis of the dissipation function shown in Fig. 4b. contradicting to the observations of decreased wind-
speed exponent in the middle range of the intermediate-
6. Discussions length wave components. It remains a tough challenge
to explain the observed peculiar nonmonotonic behavior
Investigation of the breaking dissipation function is of the spectral properties of intermediate-scale wave
of great interest to many areas of research ranging from components shown in Fig. 2.
ocean and coastal engineering, wave dynamics, air-sea Eq. (1) is obviously inadequate for describing the
interaction, remote sensing, and numerical wave dynamics of intermediate-scale waves. While it worked
36 P/A. twang '.Journal o(cid:127) Marine Systems 66 (2007) 28 37
nicely for identifying the breaking length scales, it is breaking in intermediate-scale spectral components
emphasized that the proposed local balance of breaking can lead to a better understanding of the gas transfer
dissipation and wind generation functions is a hypoth- processes.
esis that leads to the prediction of enhanced dissipation
in the intennediatc-length wave components. Further- Acknowledgements
more, in side-by-side comparison of radar images of sea
spikes and video images of ocean surface waves, it was This work is supported by the Office of Naval
found that the majority of sea-spike events are Research (Naval Research Laboratory PE61153N and
associated with steep waves without visible breaking PE62435N). NRL Contribution JA-7330-05-5262.
whitecaps - approximately 60% 'for the young and the
developed sea and 92%/, for decaying sea (Liu et al., References
1998). It is not clear whether these sea-spike-associated
steep waves were undergoing microscale breaking, so Asher, W., Fdson, J.. McGillis, W., Wanninkhof, R., Jo, D.T,
an unequivocal causal link between sea spikes and Litchcndorf, T.. 2002. Fractional area whitecap coverage and air
sea gas transfer velocities measured during GasEx-98. In: Donclan,
breaking waves is yet to be established. The observa- MA., Drennan, W.M., Saltzman, E.S., Wanninkhof. R.( Eds.). Gas
tions described above are supportive but not conclusive Transfer at Water Surfaces. AGU Press. pp. 199-203.
and further observations of enhanced breaking at the Banner, M.L., Jones, I.S.F., Trinder, J.C., 1989. Wavenumber spectra
intermediate- and short-scale waves are needed to of short gravity waves. J. Fluid Mech. 198. 321 344.
substantiate this prediction conclusively. Banner, MIL.. Babanin, A.V.. Young, IR.. 2000. Breaking probability
for dominant waves on the sea surface. J. Phys. Oceanogr. 30.
3145 3160.
7. Summary Banner, M.L.. Genimrich, J.R.. Farmer, D).R., 21tX2. Multiscale
measurements of ocean wave breaking probability. J. Phys.
Properties of wave breaking can be investigated Oceanogr. 32, 3364 3375.
through the dissipation function of the wave action Colton, M.C., Plant, WI. Keller, W.C.,(jeemaert, G.L. 1995. Tower-
based measurements of normalized radar cross section from Lake
Ontario(cid:127) evidence of wind stress dependence. J. Gcophys. Rcs.
scribed by Phillips (1984), the functional dependence 100, 8791 8813.
of B(u*!c) is tbund to f6ollow closely the power-law Deane, GB., Stokes. M.D., 2002. Scale dependence of bubble creation
relationship based on data obtained from both field and mechanisms in breaking waves. Nature 418. 839 844.
laboratory environments. The exponent of the power- Ding, L., Fanner, D.M.. 1994. Observations of breaking surface wave
law, ao, represents the wind-speed dependence of the statistics. J. Phys. Oceanogr. 24, 1368 1387.
Donelan. M.A.. Hamilton, J.. lfir, W.I I., 1985. Directional spectra of%m ind-
spectral wave components. Results from field data that generated waves. Philos. Trins. R. Soc. Lond., A A315, 509 562.
resolve wavelengths between 0.02 and 6.3 m reveal a Forristall, G.Z., 198l. Measurements of a saturated range in ocean
nonmonotonic behavior of at as a function of k. The wave spectra. J.G eophys. Res. 86, 8075-8084.
spectral densities of wave components between 0.15 and Frasier, S.J., Liu, Y., McIntosh, R.E., 1998. Space time properties of
1.5 m long depend only weakly on the forcing windJ.eohsRe.r1a3da1r7 s4ea8 5s.pikes and their relation to wind and wave conditions.
J. Geophys. Res. 103, 18745--18757.
condition. The dissipation function displays a quasi- llwang, P.A., 1997. A study of the %,avenumbesrp ectra of short isater
singular behavior in the corresponding wavelength waves in the ocean: Part 2. Spectral model and mean square slope.
range, suggesting a localized region in the wavenumber J.A tnos. Ocean. Technol. 14. 1174 1186.
domain with a strong breaking signature. This result is llwang, P.A.. 2006. Doppler frequency shift in ocean \vave
consistent with extensive observations of radar sea measurements: frequency downshifl of a wavenumber component
conkesistent wthee xstfensive obseatoss hof darc sea by adsection of background wave orbital velocity. J. Geophys.
spikes during the last few decades showing a close Res. III. C06033,
correlation between sea spikes and ocean surface llwang. P.A.. 2005. Wavenumber spectrum and mean-square slope of
scatterers traveling at the speed of surface waves a few intermediate-scale ocean surface waves. J. (ieophys. Res. I10.
meters long. The strong signature of breaking waves in C(10029. doi: I0.1029/2005JC003002.
the decimeter- to meter-long waves reflects the I twang, P.A.. Wang. D.W_. 2004. An empirical investigation of source
term balance of small scale surface waves. Geophys. Res. Loet. 31,
important impact of wave breaking on the dynamics of L15301. doi: 10.1029,'2004iL20080.
intermediate-scale ocean surface waves that are the lItsang, P.A., Wang. D.W., Walsh. E.J., Krabill, W.B., Swilt. RN., 20)O.
major contributor to the ocean surface roughness Airborne measurements of the directional wa enumber spectra of
(Hwang, 2005). The close con-elation between the gas ocean surface waves: Part I. Spectral slope and dimensionless
spectral coefficient. J. Phys. Oceanogr. 30, 2753 2767.
transfer velocity and the ocean surface roughness Jones. W.L., Schroeder. L.C.. 1978. Radar backscatter from the ocean:
(mean-square slope) has been demonstrated repeatedly. dependence on surface friction %elocity. Boundary - Layer
A better understanding of the properties of wave Meteorol. 13, 133 149.
