### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Split Balance Model and Reference Cases of Sea Wave Growth
- 3. Voluntary Observing Ship Data as a Source of Wave Data
- 4. Swell and Wind-Driven Sea in Terms of Reference Cases of Wave Growth
- 5. Conclusions and Discussion
- Acknowledgments
- References
- Supporting Information

[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.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Split Balance Model and Reference Cases of Sea Wave Growth
- 3. Voluntary Observing Ship Data as a Source of Wave Data
- 4. Swell and Wind-Driven Sea in Terms of Reference Cases of Wave Growth
- 5. Conclusions and Discussion
- Acknowledgments
- References
- Supporting Information

[2] The understanding physics of wind-driven waves and wind-wave coupling is extremely important both for fundamental 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 measurements 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 provide 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 characterized by more high and homogeneous density of observations 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 regarded 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 conceptual 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 phenomenon 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 assumption 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 describing evolution of spectra of wind-driven waves. As a result, 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 generation 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 provides 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

for nondimensional wave heights and periods which 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 *B* within the theory becomes a universal constant *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 *R* varying 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 *R* in (1) as an indicator of wind wave dynamics. There is a critical point of such analysis, the coefficient *B* that 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 *B* does not allow for reliable discriminating 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 background 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 special 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 conclusions and discussion in section 5.

### 2. Split Balance Model and Reference Cases of Sea Wave Growth

- Top of page
- Abstract
- 1. Introduction
- 2. Split Balance Model and Reference Cases of Sea Wave Growth
- 3. Voluntary Observing Ship Data as a Source of Wave Data
- 4. Swell and Wind-Driven Sea in Terms of Reference Cases of Wave Growth
- 5. Conclusions and Discussion
- Acknowledgments
- References
- Supporting Information

[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

Subscripts **k**, **r** for ∇ are used for gradients in wave vector **k** and coordinate **r** spaces correspondingly. For *N*_{k}(**r**, *t*), wave action spectral density and linear wave frequency (*d* is water depth) the subscript **k** means dependence on wave vector. Further we discuss the deep water case only, i.e., the power law dependence .

[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 interactions. 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 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

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 Γ_{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}, Γ_{k}*N*_{k} but not with *S*_{nl}. Evidently, this assumption of dominating wave-wave interaction is valid for certain range of physical conditions 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 indirect 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:

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) controls 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 particular cases of duration- and fetch-limited wave spectra evolution [*Badulin et al.*, 2005]. These solutions correspond to power law dependencies of wave energy (action, momentum) and a characteristic wave scale (wave frequency, wave number) on duration (time) or fetch, i.e.,

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 turbulent 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

In such form exponents *R* appear 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 conservative kinetic equation (3) with an additional condition of conservation of wave action

The total energy of the solution *E* = ∫ *E*(**k**, *t*)*d***k** decays slowly with time

or fetch

that is very difficult to observe in experimental studies.

[23] The key property of the swell solution, nonconservation 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 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 mechanisms while the inherent nonlinear wave dynamics is not discussed 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 Interaction Approximation [*Hasselmann et al.*, 1985] widely used for calculation *S*_{nl}. Dissipation terms are usually introduced 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 dissipation 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

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 premature sea by *Zakharov and Zaslavsky* [1983] with *R* = 4/3. The stage ranges in terms of wave age

estimated for spectral peak phase speed *C*_{p} (deep water case)

have been estimated recently in numerical study [see *Gagnaire-Renou et al.*, 2011, Figure 10]. Wind speed in definition (11) is usually taken for neutrally stable atmosphere at a reference height of observations (as a rule, at *h* = 10 m) in the dominant wave direction

(Θ is 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 *R* for three cases *A*, *B*, *C* of growing wind sea are quite close and in a thorough 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 *R* looks to be more promising for relating the above theoretical scheme with wave measurements. Unfortunately, the exponent *R* cannot be used straightforwardly without the effect of preexponent *B* taken 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 *B* is 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.

### 5. Conclusions and Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Split Balance Model and Reference Cases of Sea Wave Growth
- 3. Voluntary Observing Ship Data as a Source of Wave Data
- 4. Swell and Wind-Driven Sea in Terms of Reference Cases of Wave Growth
- 5. Conclusions and Discussion
- Acknowledgments
- References
- Supporting Information

[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, correspond to opposite signatures of the exponents. This simple fact is suggested for discriminating wind seas and swell.

[62] With the exponent *R* as indicator of sea wave dynamics we make a conceptual step: we study a link of wave heights *H* and periods *T* rather 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 altimeter observations of wave heights and mathematical modeling 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 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*) dependencies 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 sampling, etc., but is shown to be useful for delineating two extremes, wind waves and swell. A wide gap between values of *R* for 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 *R* in (1). Fortunately, the proposed approach gave interesting results and showed its promising prospects.

[66] Even for global estimates of exponents *R* swell 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 *R* to be more definite indicator of wave growth. This selection works quite well for interpretation of wind-wave growth in the spirit of the presented simple theory. In contrast, swell dynamics in terms of *R* looks 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 *R* is 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 distributions. 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 generation and propagation. The latter makes the physical situation closer to the physical model considered in this paper.

[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.