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On discriminating swell and wind-driven seas in Voluntary
Observing Ship data
S. I. Badulin
1,2,3
and V. G. Grigorieva
1,4
Received 30 January 2012; revised 27 June 2012; accepted 3 July 2012; published 14 August 2012.
[1]The global visual wave observations are reanalyzed within the theoretical concept
of self-similar wind-driven seas. The core of the analysis is one-parametric dependencies of
wave height on wave period. Theoretically, wind-driven seas are governed by power-like
laws with exponents close to Toba’s one 3/2 while the corresponding swell exponent
(1/2) has an opposite signature. This simple criterion was used and appeared to be
adequate to the problem of swell and wind-driven waves discrimination. This theoretically
based discrimination does not follow exactly the Voluntary Observing Ship (VOS) data.
This important issue is considered both in the context of methodology of obtaining VOS
data and within the physics of wind waves. The results are detailed for global estimates
and for analysis of particular areas of the Pacific Ocean. Prospects of further studies are
discussed. In particular, satellite data are seen to be promising for tracking ocean swell
and for studies of physical mechanisms of its evolution.
Citation: Badulin, S. I., and V. G. Grigorieva (2012), On discriminating swell and wind-driven seas in Voluntary Observing
Ship data, J. Geophys. Res.,117, C00J29, doi:10.1029/2012JC007937.
1. Introduction
[2] The understanding physics of wind-driven waves and
wind-wave coupling is extremely important both for funda-
mental science and numerous practical applications. This is
why experimental efforts are targeted at getting reliable
information on wind waves in a wide range of spatiotemporal
scales: from campaign measurements in areas of special
interest to global monitoring wind seas using sophisticated
satellite methods. Being collected in global databases (like
ICOADS and others) these experimental data form a basis of
climate studies, wave forecasting and maritime safety.
[3] At the same time, there is a lack of experimental data
which are suitable for relating observations and measure-
ments to wave theory. Precise wave measurements in special
field experiments are very few, extremely expensive and
their correspondence to theoretical concepts and models is,
in many cases, quite questionable. On the other hand, the
most abundant sources (e.g., satellite data) quite often pro-
vide incomplete, inaccurate or irregular (in space and time)
wave information.
[4] An example of such a rich data source is Voluntary
Observing Ship (VOS) data that cover all the World Ocean
since 1870. Last 50 years of the data collection are charac-
terized by more high and homogeneous density of observa-
tions and the errors in the well-sampled regions are estimated
as less than 10% of monthly mean values [Gulev and
Grigorieva, 2003]. Thus, the VOS collection can be regar-
ded as a self-consistent source of wave data of limited (not
high) accuracy. The relatively low accuracy of these data is
balanced by their abundance and, to an extent, by the well-
elaborated methods of the data quality control.
[5] In addition to the longest continuity these data contain
an important supplement, separate estimates of wind wave
and swell parameters. These estimates are made visually
and, evidently, suffer from subjectivity. But they represent,
in a sense, two dynamical extremes of sea state. A concep-
tual difference of these extremes is in their tie with a wind:
wind waves are generally considered as affected heavily by
wind while the swell is seen as a wind-independent phe-
nomenon that evolves mainly due to its inherent dynamics
[Komen et al., 1995]. This conceptual difference causes
trouble for conventional analysis of wave data: wind speed is
considered as a useful physical scale for wind-driven waves
but is not relevant to the swell case.
[6] In this paper we are trying to fit VOS data [Gulev and
Grigorieva, 2003] to a theoretical concept of self-similar
wind driven seas presented as split balance model [Badulin
et al., 2005, 2007]. The core of the concept is an assump-
tion of dominating inherently nonlinear wave dynamics as
compared to wind input and wave dissipation. This does not
mean a disregard of wind-wave coupling but just putting
each physical mechanism in its proper place when describ-
ing evolution of spectra of wind-driven waves. As a result,
1
P. P. Shirshov Institute of Oceanology, Russian Academy of Science,
Moscow, Russia.
2
Also at Laboratory of Nonlinear Wave Processes, Novosibirsk State
University, Novosibirsk, Russia.
3
Also at Satellite Oceanography Laboratory, Russian State
Hydrometeorological University, Saint Petersburg, Russia.
4
Also at Natural Risk Assessment Laboratory, Moscow State University,
Moscow, Russia.
Corresponding author: S. I. Badulin, P. P. Shirshov Institute of
Oceanology of the Russian Academy of Science, 36 Nakhimovsky Pr.,
117997 Moscow, Russia. (badulin@ioran.ru)
©2012. American Geophysical Union. All Rights Reserved.
0148-0227/12/2012JC007937
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, C00J29, doi:10.1029/2012JC007937, 2012
C00J29 1of13
wave dynamics due to the dominating four-wave resonant
interactions causes extremely fast relaxation of wave spectra
to an inherent state while total external forcing (wind gen-
eration plus dissipation) affects parameters of this inherent
state at relatively slow spatiotemporal scales. This model
(when the theory key assumptions are true and it is valid)
leads to conceptual gains.
[7] First, within the model the strong nonlinearity pro-
vides a strong tendency of wave spectra to universal shapes.
It is in line with conventional idea of quasi-universality of
wind wave spectra that found their implementation in the
widely used parameterizations of wave spectra like one by
Pierson and Moskowitz [1964] or JONSWAP spectrum
[Hasselmann et al., 1973] when wave spectra evolution can
be described reasonably well in terms of small number of
parameters, first of all, mean wave height H
m
(or significant
H
s
) and mean period T
m
(or significant one T
s
or spectral
peak one T
p
).
[8] Secondly, wave input and dissipation contribute at
relatively slower scales as integral quantities and transforms
the quasi-universal wave spectra “as a whole.”This integral
effect of external forcing (input and dissipation) makes the
wave evolution to be robust: particular mechanisms of wave
generation or dissipation and their distributions in wave
scales become unimportant, in a sense. Their net integral
values only affect the wave growth. The recent attempts to
treat experimental [Badulin et al., 2007] and numerical
results [Gagnaire-Renou et al., 2011] in terms of the integral
quantities and key spectral parameters H
m
(H
s
), T
m
(T
s
)
showed “the right to life”of the concept of self-similar wind-
driven seas.
[9] This paper analysis, at the first glance, is based on very
particular result of the above theory: for standard setups of
duration- and fetch-limited growth (spatially homogeneous
and stationary problems, correspondingly) it predicts power
law dependence
~
H¼B
~
TRð1Þ
for nondimensional wave heights
~
Hand periods
~
Twhich
scaling will be specified later. The relationship (1) is widely
used in wind-wave studies. For exponent R= 3/2 it gives the
well-known law of Toba [1972] with scaling of friction
velocity u
*
and significant heights and periods H
s
,T
s
. The
preexponent Bwithin the theory becomes a universal con-
stant B= 0.062 under assumption that “the work done by
wind stress to wind waves, or the time rate of the average
wave energy”is constant [Toba, 1972, p. 112].
[10] Similar one-parametric dependence for R= 5/3 has
been obtained by Hasselmann et al. [1976] as a particular
solution of the wave balance equation with a net wave input
corresponding to constant rate of wave momentum. This
special case has been detailed by Resio and Perrie [1989]
and justified later by thorough analysis of equilibria ranges
of wave spectra [Resio et al., 2004].
[11]Zakharov and Zaslavsky [1983] were the first who
associated the case R= 4/3 with the weakly turbulent theory
of wind waves: they related it with spectral flux cascading
rather than with a particular model of wind-wave coupling
and parametrization of the coupling in terms of wind speed.
Their solution provides a constant time rate of wave action
growth.
[12] From our “weakly turbulent viewpoint”the family of
one-parametric dependencies (1) with arbitrary Rvarying in
a wide range describes different regimes of wind-wave
coupling. The particular values of R(4/3, 3/2 or 5/3) can be
considered as reference cases when this coupling keeps rates
of one of the basic physical quantities, momentum, energy or
action, to be constant.
[13] In our analysis of the VOS data we use exponent Rin
(1) as an indicator of wind wave dynamics. There is a critical
point of such analysis, the coefficient Bthat varies in a wide
range and depends on a number of parameters of wind-sea
interaction: wind speed, stratification of air flow, gustiness,
etc. High dispersion of Bdoes not allow for reliable dis-
criminating particular cases of wave growth basing on (1).
Fortunately, the solution of the kinetic equation for swell
gives the power law dependence (1) as well [see Zaslavskii,
2000; Badulin et al., 2005]. In this case R=1/2 has an
opposite signature as compared to the case of growing wind
sea. We use this simple fact as a yardstick of our analysis of
VOS data for discriminating swell and growing seas.
[14] We start with a brief overview of the physical back-
ground of our approach to the data analysis in section 2. The
VOS data and features of their processing are given in
section 3. Relatively low quality of these data requires spe-
cial procedures of data selection and discriminating wave
dynamics (wind sea and swell). Results of the data analysis
are presented in section 4. The paper is finalized by con-
clusions and discussion in section 5.
2. Split Balance Model and Reference Cases
of Sea Wave Growth
[15] The today research and wind-wave prediction models
start with the basic equation for statistical description of
random field of weakly nonlinear water waves, the kinetic
equation, widely known as the Hasselmann [1962, 1963a,
1963b] equation
∂Nk
∂tþr
kwkrrNk¼Snl þSin þSdiss:ð2Þ
Subscripts k,rfor rare used for gradients in wave vector k
and coordinate rspaces correspondingly. For N
k
(r,t), wave
action spectral density and linear wave frequency wk¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g∣k∣tanh ∣k∣d
p(dis water depth) the subscript kmeans
dependence on wave vector. Further we discuss the deep
water case only, i.e., the power law dependence wk¼ffiffiffiffiffiffiffiffiffi
g∣k∣
p.
[16] Strictly speaking, Klauss Hasselmann derived the
conservative kinetic equation for potential water waves with
the only term S
nl
that describes four-wave resonant interac-
tions. This term given by explicit cumbersome formulas is
extremely inconvenient for simulation (time-consuming,
requires special accuracy control, etc.). Its accurate and
effective calculation in research and operational models
remains a burning problem so far [e.g., Cavaleri et al., 2007,
Figure 7]. At the same time, homogeneity properties of S
nl
in
the deep water limit allows to advance in theoretical studies
of the conservative Hasselmann equation. Exact solutions of
the equation [Zakharov and Filonenko, 1966; Zakharov and
Zaslavsky, 1982] that correspond to constant spectral fluxes
of wave energy and action (the so-called direct and inverse
BADULIN AND GRIGORIEVA: ON DISCRIMINATING SWELL AND WIND WAVES C00J29C00J29
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cascades) laid the foundation of the today theory of weak
turbulence of wind waves.
[17] These basic results can be generalized for cases when
S
in
and S
diss
in (2) are not plain zeroes. The key assumption
of the split balance model [Badulin et al., 2005, 2007] is
dominating of wave-wave interactions over the effects of
wave generation and dissipation. A naive treatment of the
assumption as inequalities
Snl ≫Sin;Snl ≫Sdiss
does not reflect the physical roots and consequences of the
model. As it has been noted by Young and van Vledder
[1993] the significance of the wave-wave interaction term
is in its ability to provide very fast relaxation of a wave
spectrum to an inherent quasi-universal shape. This effect
has been detailed in an extensive numerical study [Badulin
et al., 2005] and in recent analysis of the nonlinear transfer
term S
nl
as a relaxation term [Zakharov and Badulin, 2011].
The S
nl
can be presented as a sum of two terms, nonlinear
forcing F
k
and nonlinear damping G
k
N
k
which absolute
magnitudes are much greater than one of S
nl
itself. Thus,
terms S
in
,S
diss
should be compared with F
k
,G
k
N
k
but not
with S
nl
. Evidently, this assumption of dominating wave-
wave interaction is valid for certain range of physical con-
ditions only. These conditions are likely satisfied quite often
for wind sea as the cited works showed [e.g., Badulin et al.,
2005, 2007; Gagnaire-Renou et al., 2011]. Somewhat indi-
rect but important support of the assumption can be found in
the well-known fact of quasi-universal wind-wave shaping.
This fact is widely used in parameterizing wave spectra and
features of wind-wave growth [e.g., Pierson and Moskowitz,
1964; Hasselmann et al., 1973; Babanin and Soloviev,
1998b].
[18] Accepting this assumption one can propose an
asymptotic model of wind-wave growth described by the
following system of two equations:
dNk
dt ¼Snl;ð3Þ
dN
k
hi
dt ¼Sin þSdiss
hi
:ð4Þ
The conservative kinetic equation (3) represents the lowest-
order approximation of the asymptotic theory while the second
equation (4) can be considered as a closure condition of the
theory for formally small terms S
in
and S
diss
. The model (3) and
(4) is physically transparent: (3) gives a family of solutions
that are determined by nonlinear transfer only and does not
depend explicitly on wave input or dissipation while (4) con-
trols an integral balance of the wave spectra.
[19] The good prospects of the model (3) and (4) become
apparent when analyzing its self-similar solutions for partic-
ular cases of duration- and fetch-limited wave spectra evo-
lution [Badulin et al., 2005]. These solutions correspond to
power law dependencies of wave energy (action, momen-
tum) and a characteristic wave scale (wave frequency, wave
number) on duration (time) or fetch, i.e.,
E¼E0tpt;w¼w0tqt;ð5Þ
E¼E0xpc;w¼w0xqc:ð6Þ
Equations (5) and (6) can be related to widely used forms of
experimental laws of wave growth [e.g., Babanin and
Soloviev, 1998b]. Total wave action and wave momentum,
evidently, can be expressed as power law dependencies in a
similar way.
[20] Cases of linear in time growth of wave energy (5), action
and momentum are of special interest because they correspond
to constant rates of production of these quantities. They
provide a physical ground for speculating about mechanisms
of wave growth. For instance, wave momentum can be
associated naturally with turbulent wind stress and, hence,
the corresponding case takes its self-consistent physical
treatment: waves acquire a permanent fraction of the turbu-
lent wind stress at permanent wind conditions.
[21] It is useful to pass from expressions (5) and (6) to time-
, fetch-independent one-parametric dependencies of wave
height on wave period for wind-driven waves
HTR:
In such form exponents Rappear to be the same for duration-
and fetch-limited dependencies (5) and (6) in the special
cases of constant production of wave energy, action or
momentum. These special exponents and the corresponding
references are given in Table 1.
[22] This list of Table 1 can be extended by the swell case.
The corresponding self-similar solution is one of the con-
servative kinetic equation (3) with an additional condition of
conservation of wave action
ZZ∞
∞
Nk;tðÞdk¼Z∞
0Zp
p
Nw;q;tðÞdwdq¼const:ð7Þ
The total energy of the solution E=RE(k,t)dkdecays
slowly with time
Etot t1=11 ð8Þ
or fetch
Etot x1=12 ð9Þ
that is very difficult to observe in experimental studies.
[23] The key property of the swell solution, nonconserva-
tion of total energy in absence of wind input and dissipation
is well known [Zakharov, 2010; V. E. Zakharov, Direct
and inverse cascades in the wind-driven sea, unpublished
paper, 2005, available at http://math.arizona.edu/zakharov/
1Articles/Cascades.pdf] but quite often is not taken into
Table 1. Reference Cases of Self-Similar Evolution of Sea Waves
Case
Regime of Wave
Production RReference
AdM
hi
dt ¼const 5/3 Hasselmann et al. [1976]
BdEhi
dt ¼const 3/2 Toba [1972]
CdN
hi
dt ¼const 4/3 Zakharov and Zaslavsky [1983]
DdNhi
dt ¼01/2 Zaslavskii [2000]
BADULIN AND GRIGORIEVA: ON DISCRIMINATING SWELL AND WIND WAVES C00J29C00J29
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account when treating swell observations or its simulations.
The observed swell decay is usually considered as a result of
quasi-linear dissipation due to various dissipation mechan-
isms while the inherent nonlinear wave dynamics is not dis-
cussed as one responsible for this decay. These quasi-linear
models predict rather strong attenuation of sea swell [see
Kudryavtsev and Makin, 2004] that is in contradiction with
available observations [e.g., Ardhuin et al., 2009; Soloviev
and Kudryavtsev, 2010].
[24] The essentially nonlocal effect of nonconservation of
energy is quite difficult to capture within simulations in a
finite domain of wave scales and within the Discrete Inter-
action Approximation [Hasselmann et al., 1985] widely
used for calculation S
nl
. Dissipation terms are usually intro-
duced to make such calculations stable. Nowadays, exact
calculation of the wave-wave interaction term S
nl
allows for
reproducing theoretical features of swell in their full
[Badulin et al., 2005; Benoit and Gagnaire-Renou, 2007]
without an additional dissipation.
[25] Two features of the swell solutions (8) and (9) should
be stressed. First, this is a power law decay in contrast to
exponential one as usually discussed for quasi-linear dissi-
pation mechanisms. Second, exponents of the swell decay
are quite low (1/11 and 1/12) and are difficult for observing
this phenomenon in the sea. The one-parametric height-to-
period relationship for the swell looks like a good luck in the
context of the problem of quantifying the swell evolution.
One gets the law
HT1=2ð10Þ
with “observable”exponent 1/2.
[26] Figure 1 gives a graphical summary of four reference
cases of Table 1 where these cases are shown as different R,
tangents of one-parametric dependencies height-to-period
in logarithmic axes. Reference cases of growing wind sea
are shown as the young sea growth at permanent wave
momentum production (exponent R= 5/3 by Hasselmann
et al. [1976]), growing Toba’s sea (R= 3/2) and old pre-
mature sea by Zakharov and Zaslavsky [1983] with R= 4/3.
The stage ranges in terms of wave age
a¼Cph
Uwind ð11Þ
estimated for spectral peak phase speed C
p
(deep water case)
Cp¼gTp
2pð12Þ
have been estimated recently in numerical study [see
Gagnaire-Renou et al., 2011, Figure 10]. Wind speed in def-
inition (11)is usually takenfor neutrally stable atmosphere at a
reference height of observations (as a rule, at h=10m)inthe
dominant wave direction
Uwind ¼U10 cosQð13Þ
(Qis angle between wind and dominant wave directions).
Note, that definitions (11)–(13) are conventional rather than
physically based.
[27] The exponents 1/2 < T< 1/2 fall into range where
our asymptotic approach of split balance is not formally
valid. This range is shaded in Figure 1 in order to show a gap
between cases of swell and growing wind sea. This gap
contains cases of slowly growing waves which theoretical
study is extremely difficult and observations are quite rare.
[28] Theoretically, the presented scheme gives a hope that
different physical regimes of growing wind sea and swell
can be delineated in terms of one-parametric dependencies
(1), i.e., exponents R. In fact, the values of Rfor three cases
A,B,Cof growing wind sea are quite close and in a thor-
ough laboratory studies only this hope has a little chance to
be realized [Badulin and Caulliez, 2009]. On the other hand,
the swell case with opposite signature of Rlooks to be more
promising for relating the above theoretical scheme with
wave measurements. Unfortunately, the exponent Rcannot
be used straightforwardly without the effect of preexponent
Btaken into account and proper scaling of wave heights and
periods in (1). We need a background of facts, theoretical
estimates and additional hypothesis to say: “The effect of
preexponent Bis not critical in the problem of delineating of
two extremes of sea waves, wind waves and swell in terms
of one-parametric dependencies (1)”.
[29] This background will be presented below in section 4.
3. Voluntary Observing Ship Data as a Source
of Wave Data
[30] Besides model hindcasts, satellite and buoy measure-
ments Voluntary Observing Ship (VOS) data represent now
an important source of global wave information. We used the
latest update of the global archive of visual wind wave data
based on the ICOADS (International Comprehensive Ocean-
Atmosphere Data Set) [Woodruff et al., 2011] collection of
marine meteorological observations. Visual wave observa-
tions were extensively used for the description of climato-
logical characteristics of wind waves [Gulev and Grigorieva,
2003], for the assessment of the long-term tendencies in
wave parameters [Gulev et al., 2004] and for the study of
extreme waves [Gulev and Grigorieva, 2006; Grigorieva
Figure 1. Reference cases of wave growth as one-paramet-
ric dependencies H
s
(T
s
) (see Table 1). Cases of Toba [1972]
R= 3/2 law and swell with R=1/2 are shown by bold
lines. Domain where the asymptotic scheme is formally
invalid [Badulin et al., 2007] is shaded.
BADULIN AND GRIGORIEVA: ON DISCRIMINATING SWELL AND WIND WAVES C00J29C00J29
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and Gulev, 2008, 2011]. However, they have not been yet
employed for regular testing any wave theories. We used
these data to check consistency of the theoretical criterium of
discriminating wind-driven and swell seas in terms of expo-
nents Rin (1) with visual results of observers. Visual data are
not as accurate as other sources of wave information, but
their strong points are (1) great number of observations (that
is crucial for statistical evaluation) and (2) separate esti-
mates of wind sea and swell.
3.1. General Quality Check
[31] First visual wave observations in ICOADS date back to
1870. Coding systems have been changed several times, but the
most meaningful change was in 1950s, when officers started to
report sea and swell parameters separately but not the signifi-
cant wave height (SWH) only. Additionally, the upper code
limit for the highest waves was extended from 16 to 25 m.
[32] The data since 1970 have been taken for our study in
order to provide a representative statistical analysis of the
global wave parameters within the proposed theoretical
paradigm. In the last 40 years the number of observations
became stable and exceeded 100,000 per month. The dis-
tribution of reports of different quality is shown in Figure 2.
All the records have been reduced to a uniform format and
passed thorough quality check. Totally, more than 35 billion
records underlie the experimental basis of the study even
after the very strict quality control.
[33] Preliminary data control and data preprocessing are
regarded as important preconditions of substantial analysis
of visual wave observations. Note the most important points
of the procedure: (1) Only reports containing all the basic
wave parameters, sea and swell heights, directions, periods
and wind observations have been taken for further analysis,
i.e., almost 80% of total number of records have been
withdrawn from use. (2) Screening for observational arti-
facts, such as reports of unrealistic dates or nonzero wave
heights with zero periods eliminated about 3% of total
number of records. (3) Wave steepness control (values well
above the theoretical limit of wave breaking 2pH
s
/(gT
s
2
)>
1/7 [e.g., Longuet-Higgins, 1988, 1996]) has eliminated up
to 10% of all reports for some months. (4) Wave age control
a< 1.6 (see for details section 3.3) was applied to remove
unrealistically long waves from reports of wind-driven
waves. The number of the records can reach 15% of data left
after checkpoints (1)–(3). All the above checkpoints can be
considered as quite general and self-evident. They do not
involve wave physics explicitly and after all leave no more
than 10% of total data collection for further analysis.
3.2. Discriminating Swell and Wind-Driven Seas
in Visual Estimates
[34] The next step of preprocessing follows a way of dis-
criminating swell and wind sea based on a “traditional”
vision of sea wave dynamics. It does not eliminate any sig-
nificant portion of selected data and could be missed in the
paper as an optional comment. Nevertheless, we give some
details of the procedure in order to emphasize key points of
our approach once more.
[35] The discriminating wind and swell sea is an important
and quite delicate problem that comes from actual needs of
sea state forecasting. The most advanced methods operate
with wave spectra [e.g., Tracy et al., 2007]. They are aimed
at delineation of wind sea and swell for their separate use in
numerical models of sea state. Such approach implies con-
ceptually different physics of the sea wave constituents:
wind waves are considered as locally generated and closely
Figure 2. Number of VOS reports for the period 1950–2007 and quality of the data (in legend).
BADULIN AND GRIGORIEVA: ON DISCRIMINATING SWELL AND WIND WAVES C00J29C00J29
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linked to wind conditions while the swell is associated with
wave generation at distant sites and its directional and scale
characteristics are not linked definitely with local wind.
[36] Similar ideas allow for reducing uncertainty in dis-
criminating swell and wind seas in visual estimates. The
problem appears when the well-developed wind sea is
reported as swell and when low swell is treated as wind
waves. Gulev and Hasse [1999] computed joint probability
distributions of wave height and wind speed for both wind
sea and swell and overplotted these distributions by the
JONSWAP curves, representing wave height as a function
of wind speed and duration [Carter, 1982]. Dependencies of
wave height on wind speed at durations 6 and 18 h for
JONSWAP parameterizations of wave growth have been
used as low and upper bounds of wind sea state. Gulev and
Hasse [1999, Figure 4] showed that the main portion of
observations reported as wind sea falls well between the two
bounds while 80% of waves observed as swell lie above the
upper curve. The wind sea criteria appear to be compatible
with estimates of timescales of wind wave growth by
Golitsyn’s [2010] table (see line “Fetch time, h”). At the
same time, critical points of the analysis by Gulev and Hasse
[1999] should be fixed definitely. Experimental para-
meterizations of wind wave growth show very high disper-
sion [Babanin and Soloviev, 1998a] that leads to high
uncertainty of the corresponding bound criteria. Thus, this
approach should be considered as heuristical rather than
physically or even phenomenologically based.
[37]“More physical”critics of the approach concerns the
use of particular dependencies by Carter [1982]. In fact,
simple power law dependencies [Carter, 1982, equa-
tions 15–17] rely upon particular case of wave growth found
by Hasselmann et al. [1976], i.e., upon reference case A of
our Table 1 and Figure 1 with exponent R= 5/3 in (1). As we
mentioned above, this particular case corresponds to rela-
tively fast growth of young wind waves governed by a
quasi-constant wind stress. Thus, strictly speaking, the cri-
teria by Gulev and Hasse [1999] of discrimination wind and
swell seas are seen too restrictive. They, evidently, do not
take into account the most frequently observed regime by
Toba [1972] (case B in Table 1) and case of premature sea
by Zakharov and Zaslavsky [1983] (case C in Table 1).
3.3. Visual Observations and Wind Speed Scaling
[38] An alternative approach through wave age estimate
[e.g., see Smith, 1991, definitions (11–13)] can also be used
to assess sea and swell separation. The simplest scheme of
wind-wave coupling predicts wave growth for waves mov-
ing slower than wind (a< 1) and arrest of the growth or even
wave decay when waves are faster than wind (a> 1). Gulev
and Hasse [1998] followed this scheme to derive the North
Atlantic climatology of wave age of VOS data. Less than
0.5% of the total number of wind sea observations have been
qualified as swell in accordance with criterium a> 1.2. Our
study found approximately 3% of wind sea data that should
be considered as swell with the above limitation. Note, that
this criterium is close to the wave age limit of fully devel-
oped sea by Pierson and Moskowitz [1964] a
PM
= 1.22.
[39] VOS data provides an abundant source of all wave
characteristics with an important supplement, separate esti-
mates of swell and wind sea. Global climatology based on
these data demonstrates quite consistent spatial distributions
of wave heights and periods and partitioning of sea waves
into wind waves and swell (see Global Wave Atlas on http://
www.sail.msk.ru\atlas\index.html). Deeper analysis of the
data, generally, follows a conventional vision of sea wave
physics where wind-sea coupling predetermines wave
dynamics completely. The corresponding wind speed-based
criteria of JONSWAP growth curves (section 3.2) and wave
age (section 3.3) allow to delineate wind sea and swell and,
to an extent, to strengthen our confidence to visual obser-
vations. At the same time, these approaches impoverish
heavily the physics itself of sea waves, leaving no room for
inherent wave dynamics, different scenarios of wave growth
and wave-swell interactions. Wind speed data themselves
appear to be an additional source of uncertainties and errors.
Therefore, the wind speed scaling must be avoided both by
“conceptual”and this purely “technical”reasonings.
[40] Within our split balance approach the data should be
scaled by spectral flux (total net input). At the first glance,
such scaling is advantageous theoretically only as far as the
spectral flux is not easy to specify basing on available data
(in contrast to inaccurate but easy-to-measure wind speed).
[41] The gain of the spectral flux scaling becomes appar-
ent when considering the swell. In this case the spectral flux
is provided by leakage of wave energy at constant total wave
action. For the one-parametric dependence (10) it immedi-
ately dictates a link of observed swell height and period with
a distant swell state (“initial”height H
0
and period T
0
)
HT1=2¼H0T1=2
0:ð14Þ
An important question how to specify the initial state H
0
,T
0
requires a special thorough study. For the problem discussed
we propose a simplified solution basing on our preliminary
study of statistical features of the quantity H
0
T
0
1/2
. Estimates
for the whole World Ocean and for some particular regions
where strong swell is generated (Roaring Forties) showed
that H
0
T
0
1/2
has a sharp distribution and decays rapidly both
with H
0
and T
0
. It allows to fix the scales H
0
and T
0
in our
study and, hence, to fix preexponent B= const in (1). Within
such scaling the exponents Rwill be slightly underestimated
(exponents will be shifted to higher negative values). In
other words, keeping B= const in (1) at scaling Hand Ton
arbitrary but fixed values H
*
,T
*
(say, H
*
= 1 m and T
*
= 1s)
makes the gap between domains of swell reference case D
and growing wind waves (cases A, B, C) in Figure 1 to be
wider.
[42] As far as we follow the fixed scaling H
*
=
const
1
,T
*
= const
2
for estimation exponent Rfor swell,
continue to use the same approach for wind waves. Similar
statistical argumentation can be proposed basing on dis-
tributions of wind speed (see below). First, as a rule, these
distributions appear to be relatively narrow, i.e., strongly
localized near a certain value of wind speed. Secondly, at
fixed scales H
*
,T
*
preexponent Bin (1) is estimated as
growing function of wind speed (e.g., the Toba law
H
s
= 0.062(gu
*
)
1/2
T
s
3/2
). The latter means overestimating of
the exponent Rin (1), again, the fixed scaling makes a gap
between reference cases of swell and wind sea to be wider
(see Figure 1).
[43] In our argumentation of the fixed H
*
,T
*
and Bwe
deviated, in a sense, from our spectral flux line when refer to
dependence of Bon wind speed. In fact, there is no apostasy
BADULIN AND GRIGORIEVA: ON DISCRIMINATING SWELL AND WIND WAVES C00J29C00J29
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in the reference to the well-known Toba law in its explicit
dimensional form. Similar qualitative arguments on Bas
growing function of wave input can be given basing on
recent works [e.g., Gagnaire-Renou et al., 2011]. The
arguments presented above can be accepted as assumptions
as well. Thorough analysis of these assumptions is a subject
of special studies. More arguments related with the VOS
data features that justify legality of our approach will be
given below.
4. Swell and Wind-Driven Sea in Terms
of Reference Cases of Wave Growth
[44] In this section we use results of section 3.3 for
quantifying one-parametric dependencies. This consider-
ation is based essentially on the fixed data scaling and on the
assumption of small effect of variations of preexponent Bon
estimates of exponent R. As it is shown above, the
assumption B= const is enlarging a gap of estimates of Rfor
wind waves and swell, thus, making their delineation more
definite. Minor variations of Rfor wind waves can be
associated with different cases of Table 1 tentatively only.
[45] The “red line”of further data processing and analysis
can be presented as “homogenization”of data subsets in
order to minimize the effect of uncertainty of the pre-
exponent Bin (1). Abundance of the VOS data gives a good
chance when following this “red line”. We consider two
ways of the “homogenization”in this study. First, we make
estimates for particular regions of the World Ocean where
sea state conditions are “more or less homogeneous”. Sec-
ondly, we specify subranges of wave periods for wind sea
and swell in order to minimize poorly known conditions of
wind wave and swell generation. Note, that the VOS data
itself are inherently “more or less homogeneous”: generally,
marine officers prefer regular routes and avoid severe sea.
4.1. Swell and Wind Sea Discriminating:
A General Look
[46] As the very first step of our analysis look at one-
parametric dependencies for swell and wind sea derived
from the whole data collection (after preprocessing proce-
dures described in sections 3.1–3.3). Average values of
wave heights are given for each value of wave period for
wind and swell components separately in Figure 3.
[47] Coarse sampling is a well-known problem of the VOS
data: 1 s for periods and 0.5 m for wave heights. Thus,
average or interpolated values should be used for tracking
one-parametric dependencies H(T) and correct estimates of
their power law fits. In this study we operated with subsets
of wave heights at fixed wave periods. A weighted mean
values of Hat the given periods Twere calculated for the
whole World Ocean or coordinate boxes 2020and
monthly or all data of the full duration of observations 1970–
2007 years. Typically, more than 1000 values of Hwere
averaged for each period Tin coordinate boxes 2020
and more than 50,000 for the whole World Ocean. The
resulting dispersions are ranged from 1% for the most
abundant subsets up to 10% for some coordinate boxes and
periods. Generally, dispersions are high for rarely observed
big periods and heights. Subsets shorter than 10 values were
eliminated from further consideration. Different weight
function were tried to get the H(T) dependencies. An
important but not surprising outcome of the attempts is that
choice of a weight function does not affect significantly the
resulting dependencies. Such robustness is, evidently, deal-
ing with great number of observations and thorough data
preprocessing. A simple average values of Hhave been used
for further analysis that eased verification and treatment of
the results.
[48] The resulting dependencies and their power law fits
show nothing but clear difference of wind sea and swell
components: slope of H(T) curve for wind sea (Figure 3, top)
is much steeper than one for swell. At the same time,
exponent R= 0.96 for wind waves is significantly lower than
ones of reference cases A, B, C of Table 1. Similarly, R=
0.54 of swell observations is definitely higher than the case
D reference value R=1/2. The presented general look is
not a failure of our theoretical approach. It reflects nothing
but a wide range of conditions of wave generation or, in
other words, high dispersion of preexponents Bin (1).
[49] Figure 4 presents histograms of exponents Rcalcu-
lated separately for different months and coordinate
boxes 2020(totally 1481 dependencies containing data
subsets valid for our analysis). The distributions are rather
Figure 3. Dependencies H(T) and their power law fits (1)
for the whole World Ocean, 1970–2007 for (top) wind
waves and (bottom) swell. Lines marked as A,B,C,Dshow
reference power laws of Table 1. The exponents of the
experimental fits R= 0.96 for wind sea and R= 0.54 for
swell are found to be quite far from the reference cases.
Totally, 36,356,695 reports have been used for wind waves
and 31,041,169 for swell observations.
BADULIN AND GRIGORIEVA: ON DISCRIMINATING SWELL AND WIND WAVES C00J29C00J29
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broad both for wind sea (Figure 4, top) and swell (Figure 4,
bottom) and overlapped by about 1/3. At the same time, the
corresponding peaks are well separated: R
max
= 1 for wind
waves and R
max
= 0.6 for swell. Both patterns resemble
those of wind speed in Figure 5. For the histograms of wind
speeds we used the same reports as for constructing H(T)
dependencies. Annual observations (Figure 5, top) and
subsets for particular months (Figure 5, bottom, January
observations shown) give rather broad distributions that
explains why the effect of wind speed variability on expo-
nent R can be so strong. Thus, global estimates of the power
law fits and exponent Rcan be treated tentatively only as
ones reflecting the most pronounced effects and tendencies.
Such pronounced effect, difference of Rfor swell and wind-
driven sea is clearly seen when comparing power law fits for
swell and wind waves in Figure 3.
[50] Figure 5 shows a critical problem of wind speed data
of the VOS collection. For the abundant data set containing
millions records one can see pronounced peaks in the dis-
tribution at bins 13, 16, 18 m/s that looks quite strange for
the huge data set. In our opinion, an explanation of the
strange fact can be quite trivial: low accuracy and inaccurate
conversion of the measurements when observations in knots
are recasted in m/s. Speed 13 m/s gives 25 knots with high
accuracy. Similar strange peaks correspond to 16 m/s ≈
30 knots and 18 m/s ≈35 knots. It is extremely difficult to
set off the effect of the flaw of the VOS collection. This is
one of a number of reasons why we avoid the use of wind
speed scaling in this study.
4.2. Swell and Wind Sea: Wave-Scale Selection
[51] Generalized wave growth curves H(T) in Figure 3 show
clearly an important feature: they are markedly steeper for
short waves. The curve slope is milder for long wind waves
and manifests a sort of saturation for long swell. The pro-
nounced break for swell curve at wave period T=10sec
corresponds to phase speed C
ph
≈16 m/s. The latter is con-
sistent with global wind speed distributions in Figure 5 where
more than 90% of records give wind speeds below 16 m/s. In
other words, generally, waves with periods T>10saretrav-
eling faster than wind and, hence, are affecting by wind
slightly. It can explain a sort of saturation for T=10–20 s. For
longer swell one can see a sudden change and rather high
dispersion. Partially it can be explained by low number of
observations of swell with periods T=20–30 and extreme
wavelengths exceeding 600 m.
[52] The features of the generalized curves H(T) in Figure 3
give an idea to focus on special ranges of swell and wind
waves. We took the range T=5–10 s for such “true”wind-
driven waves and 10–20 s for “true”swell basing on Figure 5
hints. The growth curves for the corresponding data subsets
of wind waves and the whole World Ocean in Figure 6 (top)
Figure 4. Histograms of exponents Rof power law fits of
H(T) dependencies calculated monthly for 2020boxes
of the World Ocean (1481 dependencies of total 8 lati-
tudes 18 longitudes 12 months = 1782 subsets): (top)
wind sea and (bottom) swell.
Figure 5. Histograms of wind speed observations in the
World Ocean (top) during 1970–2007 (36,356,695 observa-
tions) and (bottom) for January, 1970–2007 (3,124,678
observations).
BADULIN AND GRIGORIEVA: ON DISCRIMINATING SWELL AND WIND WAVES C00J29C00J29
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illustrate this trivial idea perfectly well: the dependence
shows very good coincidence with the theoretical scheme
recapitulated in Table 1. Exponent R= 1.37 is very close to
the case of premature sea by Zakharov and Zaslavsky [1983]
(Table 1). Thus, in wind wave records one can see manifes-
tations of asymptotic laws of Table 1.
[53] Distributions of Rillustrates the effect of our selection
of wave periods fairly well. For wind waves (cf. Figures 4,
top and 7, bottom) this selection just moves the distribution
to higher values Rwith no essential change of the distribu-
tion shape. The swell selection changes the distribution
dramatically (cf. Figures 4, bottom and 7, top). The initial
(without the period selection) well-localized distribution is
converted by this selection to very broad one with more than
twice lower peak magnitude, probability exceeds 1% in a
wide range 1<R< 1. This dramatic changes of the swell
distribution deserves special discussion.
[54] More than half of cases with R>0(≈52%) can be
treated as swell energy growth and, hence, the swell pump-
ing due to a number of mechanisms, wind effect, coupling
with wind waves, etc. (we do not discuss here the very
complex question of accuracy of our estimates of R). In fact,
the range 1/2 < R< 0 that is associated with energy loss
can also be treated as swell pumping. The energy loss within
the statistical description of wind waves is dealing with
nonconservation of energy within the kinetic equation (2)
where wave input and dissipation are plain zeroes. This
loss is supported by inherent wave evolution, an irreversible
cascading to infinitely short scales where the energy dis-
sipates (Zakharov, unpublished paper, 2005) but not by
external forcing. The dissipation of “true conservative”
quantity, wave action, implies R<1/2. Thus, the cases
with R>1/2 should be qualified as swell pumping but not
as swell decay. We stress once again this point to fix the
conceptual difference of our scheme with conventional
vision of swell dissipation [e.g., Ardhuin et al., 2009].
[55] Less than one quarter only of the derived dependen-
cies fall into range R<1/2, that is, show true swell decay
both for the conventional criterium of energy decay R<0
and for our more restrictive criterium of wave action leakage
R<1/2 (a long tail R<1 is not shown in Figure 7, top).
In fact, all the speculations around exponents Rcannot led to
decisive conclusions at the present state of our knowledge.
Independence or weak coupling of swell with local wind
appears to be a bad luck in this case. Swell evolution
depends on physical parameters at distant sites (initial con-
ditions) or/and on the swell “history,”a number of processes
that affect the propagating swell (dissipation, swell-current,
swell-wave interactions, etc.). Spatial tracking of the swell
Figure 6. Dependencies H(T) and their power law fits (1) for
the whole World Ocean, 1970–2007 for (top) wind wave per-
iods 5–10 s and (bottom) swell range 10–20 s. Lines marked as
B,Dshow reference power laws of Toba and swell of Table 1.
The fits R= 1.37 for wind sea and R= 0.06 for swell are found
to be quite close to the reference cases. Totally, 15,172,871
reports have been used for wind waves and 6,158,859 for swell
observations.
Figure 7. Histograms of exponents Rof power law fits of
H(T) dependencies calculated monthly for 2020boxes
of the World Ocean (1481 dependencies of total 1782 = 8
18 12 coordinate boxes) for special ranges of wave
periods: (bottom) wind sea with T=5–10 s and (top) swell
with periods T=10–20 s.
BADULIN AND GRIGORIEVA: ON DISCRIMINATING SWELL AND WIND WAVES C00J29C00J29
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using satellite data could be an adequate tool for correct
assessment of exponents Rin power law fits (1). Neverthe-
less, even for rather rough estimates in Figure 7 we see a
ground for an important observation. Within our approach at
least three quarters of the available dependencies H(T) show
swell pumping (R>1/2). The wind effect as a possible
mechanism of the pumping looks attractive as an explana-
tion. In fact, we would like to draw attention to an important
alternative to the wind-sea coupling, interaction of swell and
wind-driven sea. As experimental data [Kahma and
Pettersson, 1994; Young, 2006] and theoretical models
[Badulin et al., 2008] show such interaction can be anoma-
lously strong: the swell can play a role of an effective
absorber of locally generated wind waves.
4.3. Swell and Wind Sea Discriminating:
Regional Variations
[56] The selection in wave periods is found to be the most
effective way of homogenization of initial data collection for
the problem of discriminating swell and wind waves. It
appears to be very robust: different subsets of the full data
collection show consistent picture of coexisting swell and
wind sea and their possible coupling. Other ways of selec-
tion in direction, wave age, wind speeds do not provide such
robustness. In this section we apply our approach to analysis
of regional features of sea waves. The regional features
provide a specific type of selection. A number of physical
parameters can be affected by this selection simultaneously:
including wind features, proximity to swell sources, etc.
[57] Results for four coordinate boxes 2020are pre-
sented in this study. The wave characteristics we introduced
in this paper being averaged in areas of about 1000 square
nautical miles or even more give a consistent physical pic-
ture of sea wave dynamics on the meridional cross section in
the Pacific Ocean from 60Nto40
S. Coordinates of the
boxes are given in Table 2. Again we construct dependen-
cies H(T) for data subsets in all coordinate boxes. H(T)
dependencies in Figures 8 and 9 cover shorter range of wave
periods as compared to their counterparts in global estimates
of Figure 4. At the same time, they allow for estimating
exponents Rboth for the whole period range and for sub-
ranges we introduced for wind waves (T=5–10 s) and swell
(T=10–20 s). Estimates of Rin Figures 8 and 9 are close to
the mean values over the World Ocean (cf. Figure 4). The
estimates for the subranges are given in R
ww
and R
sw
col-
umns of Table 2. Estimates for wind waves (R
ww
column)
are quite homogeneous and look reasonable for monotonic
dependencies in Figure 8. On the contrary, high dispersions
Table 2. Coordinate Boxes of a Meridional Cross Section
a
Box Latitude Longitude R
ww
R
sw
211 140W–160W40
N–60N 1.02 0.34
411 140W–160W0
N–20N 1.02 0.20
511 140W–160W0
S–20S 1.15 0.28
614 80W–100W20
S–40S 1.13 0.67
a
Exponents of power fits for wind sea R
ww
and swell R
sw
are estimated for
ranges of periods 5–10 and 10–20 s, respectively.
Figure 8. Growth curves H(T) of wind waves for coordinate boxes of Table 2 for January, 1970–2007.
Lines marked as B,Dshow reference power laws of Toba and swell of Table 1. Exponent of power law fit
Ris given for the whole range of observed wave periods T.
BADULIN AND GRIGORIEVA: ON DISCRIMINATING SWELL AND WIND WAVES C00J29C00J29
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of experimental points of swell dependencies H(T)in
Figure 9 make the estimates (the swell exponent R
sw
in
Table 2) quite questionable.
[58] Wind speed distributions for all the boxes looks sur-
prisingly close to each other. For three last boxes differences
in bins do not exceed, generally, a couple of percents. Sim-
ilarly, exponents Rfor wind waves are remarkably close to
each other (see Table 2, R
ww
column). At the same time, all
these distributions differ dramatically from the mean over
ocean pattern (cf. Figures 5 and 10) and show visible dif-
ference in H(T) dependencies where wave amplitudes vary
significantly, say, two times for boxes 211 and 511. Thus,
we see once again inadequacy of wind speed as a key
physical scale in the problem of wave growth.
[59] The dramatic difference of mean over ocean and
regional wind speed distributions is twofold. First, mean
values are essentially different. Secondly, the distribution
shapes are also different: in coordinate boxes low winds are
absent almost completely and the patterns themselves are
better localized. Accordingly, the effect of wind dispersion
in the regional distributions is essentially lower than in
global estimates. This note is very important in view of our
consideration in section 4: in the regional subsets physical
parameters that affect wave growth (wind speed) can be more
homogeneous. Thus, the estimates of wave growth exponents
Rare more reliable and can be used as characteristics of
wave dynamics. Very close values of Rfor wind waves in
Table 2 are in agreement with this conclusion.
[60] One more remark can be made on wind speed dis-
tributions in Figure 10. Again, quite similarly to the whole
ocean distributions (cf. Figure 5) the pronounced peaks are
clearly seen at “round bins”11 m/s ≈20 knots, 13 m/s ≈
25 knots, 16 m/s ≈30 knots, etc.
5. Conclusions and Discussion
[61] We apply the theoretical scheme of discriminating
wind waves and swell to the VOS data. Within the approach
the exponents of power law fits of dependencies H(T) can be
considered as indicators of wave growth. Two, in a sense,
extremes of sea state, growing wind waves and swell, cor-
respond to opposite signatures of the exponents. This simple
fact is suggested for discriminating wind seas and swell.
[62] With the exponent Ras indicator of sea wave
dynamics we make a conceptual step: we study a link of
wave heights Hand periods Trather than features of the
independent data sets. The separate analysis basing on VOS
[Gulev et al., 2004] or satellite data [e.g., Zieger, 2010] gives
valuable information on ranges of wave parameters and their
geographical variability but propose quite primitive vision of
wave dynamics. Recent attempts to combine satellite altim-
eter observations of wave heights and mathematical model-
ing of wave dynamics [Laugel et al., 2012] propose
reconstructions of full spatiotemporal structure of wind
wavefield. This study is based on extensive simulations and
requires thorough theoretical analysis. The interpretation of
Figure 9. Growth curves H(T) of swell for coordinate boxes of Table 2 for January, 1970–2007. Lines
marked as B,Dshow reference power laws of Toba and swell of Table 1. Exponent of power law fit Ris
given for the whole range of observed wave periods T.
BADULIN AND GRIGORIEVA: ON DISCRIMINATING SWELL AND WIND WAVES C00J29C00J29
11 of 13
its results in the context of burning problems of sea wave
physics shows a good prospect for further study.
[63] The proposed theoretical scheme relates H(T) depen-
dencies with different cases of wind-wave coupling. This
scheme cannot be realized in its full due to features of the
VOS data: relatively low data accuracy, coarse data sam-
pling, etc., but is shown to be useful for delineating two
extremes, wind waves and swell. A wide gap between values
of Rfor wind waves and ones of swell allows to realize this
delineation basing on wave heights and periods data only,
without wind speed data also available in the VOS
collection.
[64] We avoided to use wind speed data by a number of
reasons. Our theoretical scheme implies but does not dictate
the wind speed to be a key physical scale. Say, in swell case,
such scaling is, evidently, misleading. Additionally, quality
of wind data collected by ship observers is rather low, the
data flaws are seen clearly in statistical distributions of wind
speeds (see Figures 5 and 10).
[65] We accepted an assumption that uncertainty of
knowledge of wind wave input (wind speed) and parameters
of swell generation (see scaling condition 10) does not affect
critically our estimates of exponent Rin (1). Fortunately, the
proposed approach gave interesting results and showed its
promising prospects.
[66] Even for global estimates of exponents Rswell and
wind wave data showed definite difference in mean values
and in distribution patterns (see Figures 4 and 7). First of all,
this result can be considered as a justification of good quality
of discriminating swell and wind-driven sea by observers.
Secondly, it shows robustness of the proposed criterium for
delineating wind waves and swell: a gap between values R
for wind waves and swell is sufficiently wide.
[67] Selection in wave periods (T=5–10 s for wind waves
and 10–20 s for swell) makes exponent Rto be more definite
indicator of wave growth. This selection works quite well for
interpretation of wind-wave growth in the spirit of the pre-
sented simple theory. In contrast, swell dynamics in terms of
Rlooks more complex. We explain this fact by complexity
of swell dynamics itself when its coupling, first of all, with
locally generated wind waves can be extremely important.
[68] Example of section 4.3 showed that Ris quite good as
indicator of growing wind waves. At the same time, it fails
to explain strong regional variability of wind wave and swell
magnitudes. Surprisingly (at the very first glance), but wind
speed is failed to explain these variations as well: all the
wind speed distributions are quite close to each other.
[69] The presented results show good prospects of further
study, in particular, in constructing a sort of climatology in
terms of exponent R. Irregular data sampling of VOS data is
usually seen as a bad luck when constructing global dis-
tributions. In fact, this point of concern can be used in a
positive sense. VOS data are associated mostly with regular
ship routes, mariners avoid bad weather conditions, etc. All
these factors, in fact, homogenize conditions of wave gen-
eration and propagation. The latter makes the physical situ-
ation closer to the physical model considered in this paper.
Figure 10. Wind speed histograms for coordinate boxes in Table 2 for January, 1970–2007. Box ID and
number of observations are shown in legends.
BADULIN AND GRIGORIEVA: ON DISCRIMINATING SWELL AND WIND WAVES C00J29C00J29
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[70] Our main hopes are satellite data that allow for
tracking spatial wave evolution. For relatively long swell
this task is feasible with today technologies and methods of
remote measurements.
[71]Acknowledgments. The work was sponsored by the Russian gov-
ernment contracts 11.G34.31.0035 and 11.G34.31.0078, Russian Foundation
for Basic Research grant 11-05-01114-a and ONR grant N000141010991.
Authors gratefully acknowledge continuing support of these foundations.
Authors appreciate the critics of the reviewers that helped in the paper revision.
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