2.1. The physical model of self-similar wind-driven seas

We follow a statistical description of a random field of weakly nonlinear wind-driven waves under the effect of wind forcing and wave dissipation. The spectral density of the wave action $N(\boldsymbol{k},\boldsymbol{x},t)$ as a function of wavenumber $\boldsymbol{k}$ , spatial coordinate $\boldsymbol{x}=(x,y)$ and time $t$ can be described by the kinetic equation (Hasselmann Reference Hasselmann1962) as follows:

The idea of a balance between wind input $S_{in}$ , wave dissipation $S_{diss}$ and wave–wave interactions $S_{nl}$ has been circulating since long before World War II (e.g. Sverdrup & Munk Reference Sverdrup and Munk1947; Lavrenov Reference Lavrenov2003b ). The start of the modern concept of the spectral balance of a wind-wave field is usually attributed to the paper by Gelci, Cazalé & Vassal (Reference Gelci, Cazalé and Vassal1957) where all the terms in (2.1) have been treated as wave-scale dependent.

The milestone papers of the early 1960s by Klauss Hasselmann (Reference Hasselmann1962, Reference Hasselmann1963a ,Reference Hasselmann b ) provided a consistent physical description of the term for four-wave resonant interactions $S_{nl}$ . The role of these interactions in the evolution of wind-driven waves has been recognized but has not been realized in full. The basic results of the theory of weak turbulence of water waves (Zakharov & Filonenko Reference Zakharov and Filonenko1966; Katz & Kontorovich Reference Katz and Kontorovich1971; Katz et al. Reference Katz, Kontorovich, Moiseev and Novikov1975; Zakharov & Zaslavsky Reference Zakharov and Zaslavsky1983; Zakharov et al. Reference Zakharov, Lvov and Falkovich1992) including those for the anisotropic Kolmogorov–Zakharov spectra (Katz & Kontorovich Reference Katz and Kontorovich1974) remained outside the chief topics studied by the wind-wave community. The efforts were focused mostly on numerical aspects of accounting for the effect of wave–wave interactions in operational and research models.

The knowledge today of the terms $S_{in}$ and $S_{diss}$ on the right-hand side of (2.1) is based mostly on empirical parameterizations. This represents an additional problem for wind-wave studies when correct modelling of wave evolution requires tuning to the features of a particular environment. Below we present results that do not depend on these features and do not contain any parameters of wave generation or dissipation explicitly. The physical roots of this surprising result lie in the leading role of the wave–wave interaction term $S_{nl}$ (e.g. Hasselmann et al. Reference Hasselmann, Barnett, Bouws, Carlson, Cartwright, Enke, Ewing, Gienapp, Hasselmann, Kruseman, Meerburg, Muller, Olbers, Richter, Sell and Walden1973; Young & van Vledder Reference Young and van Vledder1993; Badulin et al. Reference Badulin, Pushkarev, Resio and Zakharov2005; Zakharov & Badulin Reference Zakharov and Badulin2011): the effects of external forcing appear directly embedded in the intrinsic parameters of the nonlinear wave field.

Following the previous works (e.g. Badulin et al. Reference Badulin, Pushkarev, Resio and Zakharov2005, Reference Badulin, Babanin, Resio and Zakharov2007a ; Zakharov Reference Zakharov2005) we consider an asymptotic model describing the wind-driven seas. Assuming wave–wave interactions to be dominant compared to wind forcing and wave dissipation one can split (2.1) into two equations. In terms of the spectral energy density $E(\boldsymbol{k},\boldsymbol{x},t)$ the model takes the form

The angular brackets in (2.2) denote integration over wave scales (i.e. in wave vector space). Equation (2.2a ) describes the effect of resonant wave–wave interactions only. Equation (2.2b ) imposes closure conditions corresponding to the balance of the total energy: the net input (input and dissipation) is equal to the growth rate of the total wave energy. Hwang & Sletten (Reference Hwang and Sletten2008) used (2.2b ) to address the dissipation issue of wind generated waves in field experiments.

A breakthrough can be made for deep water waves when the wave dispersion relation and the wave–wave interaction term $S_{nl}$ are homogeneous functions of the spectral energy density $E(\boldsymbol{k},\boldsymbol{x},t)$ and the wave vector $\boldsymbol{k}$ , i.e.

with ${\it\upsilon}$ and ${\it\varrho}$ being arbitrary positive coefficients (e.g. Zakharov Reference Zakharov1999). This important property allows one to look for self-similar solutions as function of time (fetch) and wave frequency (wavenumber).

2.2. Power-law dependence of the self-similar solutions

Now we briefly outline features of self-similar solutions for the system (2.2), details being given in appendix A. Let us introduce dimensionless variables for the model (2.2) as follows (Badulin et al. Reference Badulin, Pushkarev, Resio and Zakharov2005, Reference Badulin, Babanin, Resio and Zakharov2007a ; Zakharov Reference Zakharov2005):

Note that the time and length scales $t_{0}$ and $l_{0}$ can be chosen arbitrarily in the deep water case, say, by accepting wind speed scaling (1.1), (1.2) as an option (e.g. Hwang Reference Hwang2006).

For the duration-limited setup one has in (2.2)

and the solution in the form of the so-called incomplete self-similarity can be written as

where ${\bf\xi}=b_{{\it\tau}}\boldsymbol{k}t^{2q_{{\it\tau}}}$ . Substitution of (2.6) into (2.2) leads to two constraints on parameters $p_{{\it\tau}},q_{{\it\tau}}$ and $a_{{\it\tau}},b_{{\it\tau}}$ that are of key importance for further analysis. Thus, energy growth and frequency downshift are governed by the linear relationship

while the parameters of the solution amplitude $a_{{\it\tau}}$ and its width in wavenumber space $b_{{\it\tau}}$ obey the equation

These useful relationships confirm empirical power-like laws (1.3a,b ) with

Here $I_{{\it\tau}},J_{{\it\tau}}$ (see (A 3), (A 5) in appendix A) are integral expressions of the shape function ${\it\Phi}_{p_{{\it\tau}}}({\bf\xi})$ in (2.6) that do not depend on exponent $p_{{\it\tau}}$ explicitly. After combining relationships (1.3a,b ), (2.7), (2.9) in the form of the invariant (1.4) we observe a remarkable outcome: the result does not depend on time and initial state (i.e. on pre-exponent $a_{{\it\tau}}$ ). Moreover, its implicit dependence on exponent $p_{{\it\tau}}$ is expressed by integrals of the shape function ${\it\Phi}_{p_{{\it\tau}}}({\bf\xi})$ . Assuming spectral shape invariance, i.e. integrals $I_{{\it\tau}},J_{{\it\tau}}$ to be constants, we get immediately that ${\it\alpha}_{0}$ in (1.4) is constant. This is what we call the universality of wind-driven seas.

Note that the assumption of spectral shape invariance is introduced here for the integral quantities $I_{{\it\tau}}$ and $J_{{\it\tau}}$ and is not equivalent to a point-by-point matching of the shape functions ${\it\Phi}_{p_{{\it\tau}}}({\bf\xi})$ for different exponents $p_{{\it\tau}}$ . This assumption has been carefully checked in previous extensive numerical studies (Badulin et al. Reference Badulin, Pushkarev, Resio and Zakharov2005, Reference Badulin, Babanin, Resio and Zakharov2007a , Reference Badulin, Babanin, Resio, Zakharov, Borisov, Kozlov, Mamaev and Sokolovskiy2008). A similar assumption has been exploited by Hasselmann et al. (Reference Hasselmann, Ross, Müller and Sell1976, see § 2).

The same universal behaviour holds in the fetch-limited setup. Now one has

and the self-similar solution is given by the expression

with ${\bf\zeta}=b_{{\it\chi}}\tilde{\boldsymbol{k}}\tilde{x}^{2q_{{\it\chi}}}$ . Again, substitution into (2.2) gives links between exponents

and pre-exponents

Similarly to the duration-limited case the pre-exponents of wave growth in (1.3c,d ) are

It is easy to check that the invariant (1.4) keeps the same form as the one in the duration-limited case with the number of waves ${\it\nu}$ defined in terms of spatial wave period. Again, the invariant does not depend on time and initial state (i.e. on pre-exponent $a_{{\it\chi}}$ ). The implicit dependence of the integrals $I_{{\it\chi}}$ and $J_{{\it\chi}}$ in (A 10), (A 13) on exponent $p_{{\it\chi}}$ is weak and the assumption of spectral shape invariance can be accepted to fix the right-hand-side value as constant ${\it\alpha}_{0}$ .

2.3. Universal constant ${\it\alpha}_{0}$ for duration- and fetch-limited setups

The importance of links between parameters of self-similar solutions (2.7), (2.8), (2.12), (2.13) was first realized by Badulin et al. (Reference Badulin, Babanin, Resio and Zakharov2007a , see (1.9)) in the form of the so-called weakly turbulent law of wind-wave growth

Here ${\it\alpha}_{ss}$ is a coefficient that depends on the exponent of wave energy growth $p_{{\it\tau}}$ ( $p_{{\it\chi}}$ ) only. Assuming spectral shape invariance (quasi-universality of spectral shape in the words of Badulin et al. Reference Badulin, Babanin, Resio and Zakharov2007a ) one obtains ${\it\alpha}_{ss}\sim p_{{\it\tau}}^{-1/3}$ ( ${\it\alpha}_{ss}\sim p_{{\it\chi}}^{-1/3}$ ). For a given parameter $p_{{\it\tau}}(p_{{\it\chi}})$ , i.e. for a power-law growth of wave energy, the relationship (2.15) allows for the conversion of the instant wave energy and frequency into the instant wave input or vice versa (see § 5 in Badulin et al. Reference Badulin, Babanin, Resio and Zakharov2007a ).

In contrast to ${\it\alpha}_{ss}$ in (2.15), ${\it\alpha}_{0}$ in the invariant (1.4) is constant (with the assumption of spectral shape invariance) and the invariant itself does not contain time or space derivatives but physical quantities that we are interested in only (wave height and period). Thus, the conservation law (1.4) can be treated as an adiabatic invariant for (2.15) and for the corresponding families of self-similar solutions of the model (2.2) with $p_{{\it\tau}}(p_{{\it\chi}})$ being formally a slowly varying parameter. The features of this adiabatic invariant are, first, independence of the parameter of adiabaticity $p_{{\it\tau}}(p_{{\it\chi}})$ and, secondly, independence of the initial state of the system. The first one implies an arbitrary dependence of wave growth on time or fetch. The second feature looks strange amongst examples of classic mechanics when a conservative quantity, say the wave action of an oscillator, is determined by its initial energy and frequency. In fact, our special case reflects an asymptotic nature of the self-similar solutions of the kinetic equation and their inherent nonlinearity that forces the system to forget the initial state.

It is useful to specify different values of the constant ${\it\alpha}_{0}$ in (1.4) for duration- and fetch-limited setups. The correspondence of these cases comes directly from the relationships between the partial derivative in time and the convective operator in the model (2.2), i.e.

The velocity $V$ is associated with averaging the wave energy flux over the wave-scale range, i.e.

Generally, $V$ differs from the group velocity of the spectral peak given by

a quantity we are exploiting in our analysis. This case allows a simple definition of the number of waves ${\it\nu}$ in the fetch-limited case, that is

which will be used below in this paper for the fetch-limited case. Furthermore, to check the law (1.4) we take the definitions (1.6) and (2.19) for duration- and fetch-limited cases respectively as follows:

The notation ${\it\alpha}_{0}$ without the extended subscript is used below provided this does not lead to confusion. Following Badulin et al. (Reference Badulin, Babanin, Resio and Zakharov2007a ) and Gagnaire-Renou et al. (Reference Gagnaire-Renou, Benoit and Badulin2011) one can obtain the two estimates

The difference in magnitude of ${\it\alpha}_{0(d)}$ and ${\it\alpha}_{0(f)}$ can be partially related to the difference in shape of the wave spectra of duration- and fetch-limited seas. This is unlikely to be the major effect if we accept spectral shape invariance. It is more natural to treat the small (about 10 %) difference as one between the characteristic velocity $V$ and the group velocity of the spectral peak. If we are looking for a ‘perfect universality’ of our law, i.e. equivalence between ${\it\alpha}_{0(f)}$ and ${\it\alpha}_{0(d)}$ , we have to take into account this difference between $V$ and $C_{g}({\it\omega}_{p})$ in the definition of the number of waves ${\it\nu}$ in (2.19), i.e.

The characteristic velocity $V$ for wind-wave spectra appears to be approximately 30 % smaller than that of the spectral peak $C_{g}({\it\omega}_{p})$ which matches quite well with previous experimental results (e.g. Yefimov & Babanin Reference Yefimov and Babanin1991; Hwang & Wang Reference Hwang and Wang2004; Hwang Reference Hwang2006).

3.1. Physical scaling of self-similar wave growth

The invariant (1.4) of self-similar solutions for wind-driven seas can be written in the form of a dependence of wave height on wave period. An example of such a dependence is the famous Toba (Reference Toba1972) law of $3/2$ (the dimensionless wave height is proportional to the power $3/2$ of non-dimensional wave period). The key difference of the new dependence is in the physical scaling: the conventional scaling is based on wind speed (1.1), (1.2) while the new one implied by invariant (1.4) is wind-speed free.

Let us introduce the dimensionless wave height and period for the fetch-limited case as follows:

for fetch $x$ , the spectral peak period $T$ and the significant wave height $H_{s}=4\sqrt{E}$ . For the duration-limited case similar quantities can be introduced as

The dimensionless periods defined by (3.1), (3.2) have a simple physical meaning as they express wave lifetime in terms of the number of instantaneous temporal or spatial wave periods. For the duration-limited case (3.2) it follows that

and for the fetch-limited case (3.1)

Definitions (3.3), (3.4) represent a kinematic treatment of invariant (1.4): the instantaneous wave steepness is thus determined by the time (distance) of wave evolution expressed in dimensionless instantaneous wave periods.

One can propose a dynamical interpretation of (1.4). The ${\it\mu}^{4}$ defined by (1.5) gives a scale for the nonlinear relaxation of a deep-water wave field (see (22), (23) in Zakharov & Badulin Reference Zakharov and Badulin2011). Thus, one can treat dimensionless time ${\it\nu}$ as a dynamical lifetime:

where ${\it\tau}_{nl}$ , given by

is a characteristic time of the nonlinear relaxation of a deep-water wave field. $B$ is a large coefficient as shown by Zakharov & Badulin (Reference Zakharov and Badulin2011) ( $B=36{\rm\pi}$ in the limit of a narrow angular wave spectrum, $B=22.5{\rm\pi}$ for an isotropic wave field). Thus, (1.4) states that the wave age $t{\it\tau}_{nl}^{-1}$ measured on the nonlinear relaxation scale ${\it\tau}_{nl}$ remains constant for growing wind waves. In accordance with (3.5) and estimates by Zakharov & Badulin (Reference Zakharov and Badulin2011) time $t$ does not exceed 100 relaxation times ${\it\tau}_{nl}$ .

Within the new wind-free scaling (3.2) one gets a law of $9/4$ for the duration-limited setup, namely

The fetch-limited dependence differs from (3.7) by a factor of $2$ , but, what is more significant is the exponent. It gives a $5/2$ law:

The difference between the exponents in (3.7) and (3.8) provides a quantitative criterion for discriminating between spatial and temporal scenarios of wave growth.

3.2. A parametric model by Hasselmann et al. (Reference Hasselmann, Ross, Müller and Sell1976) and universality of wave growth

The exponents $9/4$ and $5/2$ in (3.7), (3.8) look confusing in view of their counterparts of dimensionless single-parameter dependence of wave height on period scaled by wind speed (e.g. Toba Reference Toba1972; Hasselmann et al. Reference Hasselmann, Ross, Müller and Sell1976; Zakharov & Zaslavsky Reference Zakharov and Zaslavsky1983). The latter set of exponents conforms with a specific ABC of wind-wave growth (see Badulin Reference Badulin2010). These well-known exponents of $5/3$ (Hasselmann et al. Reference Hasselmann, Ross, Müller and Sell1976), $3/2$ (Toba Reference Toba1972) and $4/3$ (Zakharov & Zaslavsky Reference Zakharov and Zaslavsky1983) correspond to different reference regimes of wind–wave coupling associated with permanent fluxes of momentum, energy and wave action (Gagnaire-Renou et al. Reference Gagnaire-Renou, Benoit and Badulin2011; Badulin & Grigorieva Reference Badulin and Grigorieva2012). The laws of $9/4$ and $5/2$ presented above do not allow one to discriminate between these reference dynamical regimes of wave growth. Instead, they describe a continuous evolution from one regime to the other in a universal way. We will show in the following analysis of the wave prediction model by Hasselmann et al. (Reference Hasselmann, Ross, Müller and Sell1976) that these laws are fully consistent with the previous studies.

4.1. Duration-limited growth

The results of simulations (e.g. Badulin et al. Reference Badulin, Pushkarev, Resio and Zakharov2005) are used here for verification of the law (1.4) for duration-limited wave growth. Figure 2 presents somewhat eclectic dependence of the wind-free invariant ${\it\alpha}_{0}=({\it\mu}^{4}{\it\nu})^{1/3}$ on non-dimensional duration ${\it\tau}=tg/U_{10}$ (figure 2 a) and inverse wave age ${\it\sigma}={\it\omega}_{p}U_{10}/g$ (figure 2 b). Wave input functions and wind speed values are shown in the legends. The straight line ${\it\alpha}_{0(d)}=0.7$ is shown as a reference. All the simulations except the last one (our reproduction of the case by Komatsu & Masuda Reference Komatsu and Masuda1996) have been carried out with a primitive dissipation function associated with a hyper-dissipation at high frequencies (see Badulin et al. Reference Badulin, Pushkarev, Resio and Zakharov2005). In these cases there is no saturated (fully developed or mature) wind-sea state and all the dependences are tending to ${\it\alpha}_{0(d)}\approx 0.7$ in full agreement with the results of § 2. The inverse wave age ${\it\omega}_{p}U_{10}/g$ can be considerably less than unity, i.e. nonlinear interactions support the growth of waves that can propagate significantly faster than wind (Glazman Reference Glazman1994).

To describe wave dissipation Komatsu & Masuda (Reference Komatsu and Masuda1996) used the more sophisticated ‘white-capping’ function by Hasselmann (Reference Hasselmann1974) (see also Komen, Hasselmann & Hasselmann Reference Komen, Hasselmann and Hasselmann1984). In this case, wave height and period tend to their limits at large time. This feature is evidenced by a break of the dependence from the general tendency at inverse wave age ${\it\omega}_{p}U_{10}/g\lesssim 1$ in figure 2. Wave steepness ${\it\mu}$ and frequency ${\it\omega}_{p}$ are then approaching the limiting values while time $t$ and, evidently, number of waves ${\it\nu}$ continue to grow.

4.2. Simulations of fetch-limited growth

The fetch-limited growth has been simulated by Zakharov et al. (Reference Zakharov, Resio and Pushkarev2012) starting from an initial white-noise spatially homogeneous wave field in a fetch interval 0–41 km and for times up to 385 000 s (approximately 107 h). Strictly speaking, there is no classic fetch-limited regime as a stationary state in these simulations. The evolution looks like a sequence of stages where reference dependences (3.7), (3.8) describe intermediate asymptotics. Figure 3 presents these results in terms of dependence on time at fixed fetches. Curves for log-spaced fetches $1,2,4,8,16$ and 32 km are shown.

Figure 3.

Wave growth curves in simulations of the fetch-limited setup (Zakharov et al. Reference Zakharov, Resio and Pushkarev2012) with: (a) fetch scaling (3.1) and (b) duration scaling (3.2). Curves are given for fixed fetches 1, 2, 4, 8, 16, 32 km (see legends). Theoretical dependences (3.7) and (3.8) are shown by dotted lines.

Figure 3(a) shows the evolution of wave parameters for wind-free scaling from the lower left to upper right corner and demonstrates an impressive coincidence with the $5/2$ power law (3.8) over a wide range. Deviations from this dependence are seen at small dimensionless periods $\tilde{T}$ (short times, lower left in figure 3 a) when wave spectra are far from self-similarity. For longer periods $\tilde{T}$ (longer times, upper right in figure 3 a) a saturation of the wave field is reflected in the vertical asymptotes of the curves: wave periods (both dimensional and dimensionless) tend to their finite limits while wave heights continue to grow. This effect is probably associated with a simulation setup where the waves are modelled in a finite spatial box. At long times when the quasi-stationary state is reaching the upper bound of the box, the quasi-stationary fetch-limited growth then gives way to a duration-limited scenario.

The time-scaled representation (3.2) in figure 3(b) traces the wave evolution in an opposite sense when compared with figure 3(a): both $\tilde{H}$ and $\tilde{T}$ are decreasing with time. Curves plotted for different fetches appear to be remarkably close to each other and fit the theoretical tangent $9/4$ in a range of more than one decade of $\tilde{T}$ . For small dimensionless periods $\tilde{T}<2\times 10^{-5}$ and, correspondingly, long times and a large number of waves ( $\tilde{T}\sim {\it\nu}^{-1}$ ), the saturation of wave field growth is evident as curves deviating well above the reference dependence (3.7), counterparts of the saturation stage in figure 3(a). Thus, the wind-free relationships $\tilde{H}(\tilde{T})$ (3.7), (3.8) provide a good reference for the physical analysis of wave evolution.

5.1. Classic experiments on duration- and fetch-limited growth

As discussed in Hwang & Wang (Reference Hwang and Wang2004), two classes of ocean wave measurements are of great importance for the study of the generation of ocean waves by wind. The first class is the fetch-limited growth condition, under which the wave development is limited by the available spatial coverage upwind of the measurement location. Over the years, there have been several successful field experiments reported. Extensive reviews and analyses of these datasets have been reported previously (e.g. Kahma & Calkoen Reference Kahma and Calkoen1992, Reference Kahma, Calkoen, Komen, Cavaleri, Donelan, Hasselmann, Hasselmann and Janssen1994; Young Reference Young1999; Hwang & Wang Reference Hwang and Wang2004; Hwang Reference Hwang2006; Badulin et al. Reference Badulin, Babanin, Resio and Zakharov2007a ; Hwang et al. Reference Hwang, Garciá-Nava and Ocampo-Torres2011).

The second class is the duration-limited growth condition, under which the wave development is limited by the temporal duration of the steady wind event acting on the water surface. The ideal initial and boundary conditions satisfying the duration-limited wave growth rarely occur in nature and it is no wonder that reports of such data are very scarce. Young (Reference Young1999) presents an extensive review of fetch- and duration-limited wave growth studies. The only duration-limited datasets cited are Sverdrup & Munk (Reference Sverdrup and Munk1947), Bretschneider (Reference Bretschneider1952a ,Reference Bretschneider b ) and Darbyshire (Reference Darbyshire1959), as compiled by Wiegel (Reference Wiegel1961). DeLeonibus & Simpson (Reference DeLeonibus and Simpson1972) report field data that contain duration growth information. Liu (Reference Liu1985) describes an interesting episode of almost 60 hours of quasi-steady wind forcing of wave growth measured by an NDBC wave buoy in Lake Superior. All these data are obtained at later stages of wave development with dimensionless time, ${\it\tau}=gt/U_{10}$ , greater than about $7000$ . By chance, Hwang & Wang (Reference Hwang and Wang2004) obtained one data set describing the duration growth in the first two hours of wind-wave generation and extended the data coverage by more than one order of magnitude ( ${\it\tau}$ between $498$ and $8801$ ).

For our quantitative analysis, the fetch or duration data sets require information on the complete triplets of ( ${\it\varepsilon},{\it\sigma}$ and ${\it\tau}$ ) or ( ${\it\varepsilon},{\it\sigma}$ and ${\it\chi}$ ). We have been able to assemble in figure 4 nine fetch-limited data sets (Burling Reference Burling1959; Hasselmann et al. Reference Hasselmann, Barnett, Bouws, Carlson, Cartwright, Enke, Ewing, Gienapp, Hasselmann, Kruseman, Meerburg, Muller, Olbers, Richter, Sell and Walden1973; Dobson, Perrie & Toulany Reference Dobson, Perrie and Toulany1989; Babanin & Soloviev Reference Babanin and Soloviev1998; Donelan Reference Donelan1979; Merzi & Graf Reference Merzi and Graf1985; Garciá-Nava et al. Reference Garciá-Nava, Ocampo-Torres, Osuna and Donelan2009; Romero & Melville Reference Romero and Melville2010, the first five data sets are called collectively BHDDB, and the others D79, M85, G09 and R10 respectively), and three duration-limited data sets (DeLeonibus & Simpson Reference DeLeonibus and Simpson1972; Liu Reference Liu1985; Hwang & Wang Reference Hwang and Wang2004, referred to as D72, L85 and H04).

In keeping with the conventional observation that the wave growth can be fitted by power-law functions (1.3) but allowing the exponents to vary with the range of dimensionless fetch or duration in different experiments, Hwang & Wang (Reference Hwang and Wang2004) developed a second-order fitting of the power-law functions to the BHDDB fetch-limited data and obtained (cf. (1.3), notation is slightly modified compared with Hwang & Wang Reference Hwang and Wang2004)

where the exponents are slowly varying functions of dimensionless fetch ${\it\chi}$ given by with ${\it\varepsilon}_{0{\it\chi}}=\exp (-17.6158)$ , ${\it\alpha}_{1}=1.7645$ , ${\it\alpha}_{2}=-0.0647$ , ${\it\sigma}_{0{\it\chi}}=\exp (3.0377)$ , ${\it\beta}_{1}=0.3990$ , and ${\it\beta}_{2}=-0.0110$ . A sufficient range of the dimensionless fetch ${\it\chi}$ for computation is $10^{0}{-}10^{6}$ .

Excluding fetch ${\it\chi}$ in (5.2a,b ) one has a counterpart of the linear link (2.12) for self-similar solutions of the model (2.2)

where

are remarkably close to the theoretical value of unity. The duration-limited growth functions can be derived from a similar approach using a formal conversion of fetch $x$ to time $t$ (see Hwang & Wang Reference Hwang and Wang2004, for details). Trivial algebra leads to a similar relationship between $p_{{\it\tau}}$ and $q_{{\it\tau}}$ (cf. (2.7) and (5.3)), i.e.

where the values of coefficients $P_{{\it\tau}},S_{{\it\tau}}$

again, appear to be quite close to the theoretical value of unity. The empirical fits (5.3), (5.5) and their theoretical counterparts (2.7), (2.12) are shown in figure 1.

One can examine the correspondence between the theory and the empirical knowledge by following the notes in the Introduction. After excluding wind speed from (1.3) one can easily obtain ‘empirical’ invariants (1.7):

in which exponents $r_{{\it\tau}},r_{{\it\chi}}$ depend on growth rates $p_{{\it\tau}}$ ( $p_{{\it\chi}}$ ). Generally, these exponents are close to unity as predicted by our theory (see (1.4)) but can vary in a wide range, even being negative for high exponents $p_{{\it\tau}},p_{{\it\chi}}$ .

Figure 4.

Dependence of experimental estimate of self-similarity parameter ${\it\alpha}_{0}$ : (a) on non-dimensional fetch ${\it\chi}=xg/U_{10}^{2}$ and duration ${\it\tau}=tg/U_{10}$ ; (b) on non-dimensional frequency of spectral peak ${\it\sigma}={\it\omega}_{p}U_{10}/g$ , for a combined collection of duration- and fetch-limited datasets. The dashed line shows the theoretical value ${\it\alpha}_{0(f)}=0.62$ , the solid line is the empirical power-law fit (5.8) by Hwang & Wang (Reference Hwang and Wang2004) for the collection.

An alternative formulation of the results allows tracing of the effect of fetch as follows:

The additional dependence on dimensionless fetch ${\it\chi}$ yields a variation of ${\it\alpha}_{0}$ in (5.8) by a factor $2.3833$ when dimensionless fetch varies over a range of five orders of magnitude, ${\it\chi}=10{-}10^{6}$ . Figure 4 show the results of these field measurements similarly to figure 2 for simulations. The theoretical reference value of ${\it\alpha}_{0}$ is reasonably consistent with experimental data and with the empirical fitting (5.8) by Hwang & Wang (Reference Hwang and Wang2004). Typically, the duration-limited data show larger scatter, reflecting the departure from ideal duration-limited wave generation conditions as discussed at the beginning of this section.

Figure 5.

Data by Sverdrup & Munk (Reference Sverdrup and Munk1947): (a,b) wave height–period dependence with the conventional wind speed scaling; (c,d)  $\tilde{H}(\tilde{T})$ scaled by fetch (c) or time (d); (e,f) estimates of invariants ${\it\alpha}_{0(f)}$ and ${\it\alpha}_{0(d)}$ . The dotted lines in (a–d) show the Toba (Reference Toba1972) law, and the dashed lines in (e,f) our theoretical values of ${\it\alpha}_{0(f)}$ and ${\it\alpha}_{0(d)}$ . ID in (e,f) corresponds to records’ ID in tables of Sverdrup & Munk (Reference Sverdrup and Munk1947). (a,c,e) Data of table II for fetch-limited cases; (b,d,f) data of table III for duration-limited cases. Names of observation ship and authors are given in the legends of (a,b).

5.2. Data by Sverdrup & Munk (Reference Sverdrup and Munk1947)

The special attention we pay to the work by Sverdrup & Munk (Reference Sverdrup and Munk1947) is not limited to the historical aspects of wind-wave studies. The theoretical construction of their paper is quite close to our physical approach. Ten years before the formulation of the spectral approach for wind waves Sverdrup & Munk (Reference Sverdrup and Munk1947) showed that ‘the concept of ‘significant waves’ is essential for purpose of forecasting’. They started their theory from the equation for the integral balance of energy (see (47) in Sverdrup & Munk Reference Sverdrup and Munk1947), i.e. the counterpart of the second equation of our model (2.2). The effect of nonlinear interactions has been described by a semi-empirical relationship between two dimensionless parameters: wave steepness and wave age. The experimental data available at that time have been used to specify parameters of their model.

Figure 5 presents the data of tables II and III of appendix II by Sverdrup & Munk (Reference Sverdrup and Munk1947). Only measurements containing full triads of wave height, period and fetch (duration) were taken into account. The upper row shows fetch- (figure 5 a) and duration-limited (figure 5 b) data with the conventional wind speed scaling. There is a rather large scatter of data around Toba’s $3/2$ law (dotted line). Strong deviations from Toba’s law are associated with four particular data groups: US Eng. and Cornish for fetch- and Krümmel, Dover for duration-limited observations.

The alternative wind-free scaling plots in figure 5(c,d) also show large deviations from the reference dependences (3.7), (3.8) but the ranges of the corresponding dimensionless values are wider, especially, for the duration-limited case (figure 5 d). Again, we can notice a clear disparity of points associated with the source of data and the method of estimating wave parameters. The data of USS Augusta and Gibson (fetch-limited) and of USS Augusta and Berkeley (duration-limited) show better agreement with theoretical dependence (dotted lines). The wave period in these observations was estimated directly whereas those of Krümmel and Paris have been computed from measurements of wavelength.

The data in figure 5(c,d) cover the same range of dimensionless $\tilde{T}$ and $\tilde{H}$ as the simulations by Zakharov et al. (Reference Zakharov, Resio and Pushkarev2012) in figure 3. One point of duration-limited data (figure 5 d, Paris) appears below the lower limit of $\tilde{T}$ in figure 3(b), i.e. in the range we treated as a saturated wave field (see comments in § 4.2). As the opposite extreme, note the data of Gibson that fit both approaches (Toba’s and the new one) fairly well (cf. figure 5 a,c). Two points of Gibson correspond to very young waves in terms of wind speed scaling (wave age $C/U_{10}=0.22$ and $C/U_{10}=0.28$ , see figure 5 a). Short fetches ( $1.3$ and 3.5 km) put these points into the upper right corner of figure 5(c). In terms of the ‘life distance’ $x/{\it\lambda}$ , these points correspond to approximately $152$ and $267$ wave periods.

The bottom row, figure 5(e,f) demonstrates a surprising correspondence of the experimental estimates of the invariant ${\it\alpha}_{0}$ to our theoretical values. Again, data of USS Augusta, Gibson and Berkeley give the best fit to ${\it\alpha}_{0(d)}=0.7$ and ${\it\alpha}_{0(f)}=0.62$ . The agreement is considered quite good given the relatively poor quality of these early data sets.

5.3. Data of the field research facility wave riders

The data of wave riders in a near-shore area look very attractive for illustrating the law (1.4) in terms of the dependence of wave height on wave period (3.8). This simple $5/2$ power-law dependence is verified with data available at the website of the Field Research Facility of the US Army Corps of Engineers http://www.frf.usace.army.mil/. A summary of the data is given in table 1. The wave riders collected data for many years; for example, Buoy 630 has been operational since 1997 and Buoy 430 since 2008. The most recent data downloaded for this analysis are dated September 2013. The total number of measurements and the number of data points relevant to the fetch-limited setup are listed in table 1. We have selected records when offshore wind direction was $\pm 30^{\circ }$ from the coast normal. Buoy 200 is sheltered by a cape from the north; this is the closest coastline but there is almost no wind from that direction. Direction $280^{\circ }$ has been taken as an alternative reference and the corresponding fetch has been set as three times longer than the one from the closest shoreline.

Table 1.

Summary of wave rider data of the Field Research Facility of the US Army Corps of Engineers. Wave direction is reported as the direction from which the waves originate, e.g. waves coming from the west have direction $270^{\circ }$ .

Wind from the ocean is dominating in this area and only 0.31 % of total number of data points (938 of 305 261) qualifies as consistent with a fetch-limited setup of wave growth. When presented in dimensionless variables (3.1) these data match the theoretical dependence $\tilde{H}(\tilde{T})$ (3.8) fairly well as seen in figure 6. The data of the buoys cover a wide range of dimensionless periods (‘life distances’), similar to figures 2–5 for simulations of wave growth, more recent wave growth data reported in Hwang & Wang (Reference Hwang and Wang2004) and the historical data by Sverdrup & Munk (Reference Sverdrup and Munk1947). A slight overshoot relative to the theoretical dependence can be explained by a systematic underestimating of wave periods. Underestimating of fetch can also contribute to this overshoot when the wave development is slowed down by the coast sheltering effect. The scaling gives a good approximation for the problem of wave growth at relatively small slanting fetches, for our present knowledge up to $40^{\circ }$ from the coastline normal. At larger angles the wave growth off-shore is accompanied by a complex system of along-shore modes as shown in recent simulations of fetch-limited growth in which the effect of nonlinear wave–wave interactions has been accounted for in full (e.g. Gagnaire-Renou Reference Gagnaire-Renou2009; Zakharov et al. Reference Zakharov, Resio and Pushkarev2012).

Figure 6.

Data of Field Research Facility wave riders versus theoretical dependence (3.8) shown as dotted line. Different buoy data are shown by different symbols (see legend and table 1).

6.1. Wind speed scaling in experiments by Toba (Reference Toba1961, Reference Toba1972) and Caulliez et al. (Reference Caulliez, Makin and Kudryavtsev2008) and Caulliez (Reference Caulliez2013)

Figure 7(a) shows the classical representation of wind-wave growth in terms of the dimensionless wind-dependent wave parameters $H^{\ast }$ and $T^{\ast }$ , using the definitions based on wind friction velocity as given by Toba (Reference Toba1972). The whole sets of data obtained respectively by Toba (Reference Toba1972, table 1 therein, filled symbols) and by Caulliez (Reference Caulliez2013, partially given in table 1, open symbols) are displayed. In particular, the Toba and Caulliez data were collected for a number of similar conditions, namely for reference wind speeds between $5$ and $12~\text{m}~\text{s}^{-1}$ and fetches between $6$ and 14 m but in facilities of quite different sizes, the water tanks being respectively $21$ and 40 m long.

In addition, the air layer above Toba’s water tank was just 50 cm in height $\times ~75~\text{cm}$ in width and, clearly, the air flow above waves might be affected by the walls. The wind friction velocity was determined from the mean velocity profiles measured at three fetches by means of small cup anemometers while the significant wave height was derived from the mean wave height estimates. When compared to the Marseille data as given in Caulliez et al. (Reference Caulliez, Makin and Kudryavtsev2008) and Caulliez (Reference Caulliez2013), the original data reported by Toba (Reference Toba1972) in figure 7(a) show laboratory waves ‘unrealistically young’ exhibiting abnormally small $T^{\ast }$ associated with inverse wave age in the range 20–45, i.e. propagating 20–45 times slower than the wind speed estimated at the standard 10 m level (note that for short gravity waves, when neglecting drift current effects on wave propagation, $T^{\ast }$ corresponds approximately to $2{\rm\pi}$ times the wave age $C/u_{\ast }$ ). A thorough examination of the data set has shown such very small values of wave age are related to abnormally high values of friction velocity.

The observations reported by Caulliez (Reference Caulliez2013) in the large wind-wave tank in Marseille were made for a wider range of wind speeds and fetches, from 2 m up to 26 m, with an air layer above the water surface of 1.5 m in height and 3.2 m in width. The friction velocity was determined from careful hot-X-wire measurements of the vertical profiles of the turbulent momentum flux in air, a quantity found constant within the whole water surface boundary layer (Caulliez et al. Reference Caulliez, Makin and Kudryavtsev2008). The significant wave height was estimated as four times the root-mean-square value of the water surface displacements. In this experiment, the inverse wave age varied between $4$ and  $18$ .

In figure 7(a), data sets by both Toba and Caulliez follow quite well the Toba $3/2$ power law. However, the Toba data are strictly separated from the Caulliez ones in the range of wave age $T^{\ast }$ observed. Additionally, all the Toba data are quite close to the $3/2$ power law while the Caulliez data observed at the shortest fetches, in a range depending on wind speed (up to 26 m at $2.5~\text{m}~\text{s}^{-1}$ to less than 6 m at $10~\text{m}~\text{s}^{-1}$ ) show a transition to a ‘saturated wave field state’, in the words of Toba (Reference Toba1972). These transitional points of the Caulliez data appear as outliers from the ones of Toba collected in the same range of wave age and fetches (e.g. between $6$ and 14 m).

Therefore, Toba’s data have been re-analyzed by using more realistic values of wind and wave parameters based on the detailed measurements made in the large Marseille wind-wave tank. In brief, new friction velocity values have been derived from drag coefficient estimates obtained in Marseille at the same fetches and reference wind speeds as Toba’s data. Note that these new values are in very good agreement with the $u_{\ast }$ values measured by Kawai (Reference Kawai1979) in a wind-wave tank of size comparable to the tank used by Toba (Reference Toba1961) in the experiments analyzed here. In addition, significant wave height has been evaluated from the mean wave height given by Toba (Reference Toba1972) on the basis of the wave height probability distribution observed in Marseille and not from the Rayleigh distribution generally assumed for in situ conditions (see for instance figure 9 in Caulliez et al. Reference Caulliez, Makin and Kudryavtsev2008). When these new parameters are used for estimating $T^{\ast }$ and $H^{\ast }$ , it is striking to see in figure 7(b) that both data sets again agree very well but now, in a more realistic way, fall into the same range of wave age $T^{\ast }$ . As previously mentioned, they also follow remarkably well the $3/2$ power law except for the non-saturated wave fields observed at the shortest fetches.

Figure 7.

Data of wind-wave tank studies of wave growth by Toba (Reference Toba1972) (filled symbols) and by Caulliez (Reference Caulliez2013) (open symbols) made dimensionless by using the gravity parameter $g$ and the conventional friction velocity $u^{\ast }=\langle U^{\prime }W^{\prime }\rangle ^{1/2}$ ( $T^{\ast }=gT/u^{\ast }$ and $H^{\ast }=gH_{s}/{u^{\ast }}^{2}$ ). (a) Toba’s data derived from table 1 in Toba (Reference Toba1972); (b) friction velocity values $u^{\ast }$ given in Toba (Reference Toba1972) are replaced by estimates at the same reference wind speed and fetch made on the basis of the Caulliez measurements in the large Marseille wind wave tank. Solid line is the Toba (Reference Toba1972)  $3/2$ law.

Figure 8.

Data derived from wind-wave tank studies of wave growth by Toba (Reference Toba1972) (filled symbols) and by Caulliez (Reference Caulliez2013) (open symbols) plotted in terms of the new scaling (3.1). Solid line is the Toba (Reference Toba1972)  $3/2$ law.

6.2. Wind-free scaling in experiments by Toba (Reference Toba1961, Reference Toba1972) and Caulliez (Reference Caulliez2013)

Figure 8 shows the same data sets obtained by Caulliez and Toba but plotted with the new scaling representation (3.1) based on the law of universality (1.4). For Toba’s data in figure 8, the significant wave height has been estimated in the same way as in figure 7(b). The correspondence between the two experimental data sets and the theoretical $5/2$ power-law dependence (3.8) looks very good for the well-developed wave fields observed at the largest fetches, but the experimental points of Toba (Reference Toba1972) are located about 1.5 times higher than the theoretical curve given by (3.8). This overshoot may be explained by the use of different methods for estimating the spatial wave period $\tilde{T}$ as given by (2.19) and (3.4).

For fetch-limited conditions, as seen from (2.19), the dimensionless spatial wave period is a measure of the number of waves propagating over the fetch distance $x$ , i.e. $\tilde{T}=(2|\boldsymbol{k}_{p}|x)^{-1/2}$ . In the Toba (Reference Toba1972) experiments based on one-point wave gauge measurements, wave period was first derived from the mean-over-spectrum or dominant peak time scale and the related spatial quantity $\tilde{T}$ was estimated by using the linear dispersion relation for gravity waves. Caulliez (Reference Caulliez2013) however measured directly the phase speed $C_{p}$ of dominant waves by means of a cross-correlation method between two wave signals recorded by a pair of wave gauges (Dudis Reference Dudis1981). This method enables direct measurement of the velocity of wave propagation accounting for the effect of surface drift current and, for the shortest waves, the effect of surface tension. In the observations reported here, the magnitude of these combined effects varies between 70 % of the linear phase velocity value for small fetches and wind speeds, to 10 % for large fetches and winds. This clearly indicates that the dynamics of wind-wave fields observed in the laboratory are governed not only by the gravity force but also by surface drift and, at the shortest scales, by capillary forces.

The good agreement observed in figure 8 between the Caulliez (Reference Caulliez2013) data at large fetches and the theoretical curve given by (3.8), in contrast to the Toba (Reference Toba1972) data, validates the use of this experimental method for estimating $\tilde{T}$ for wind-wave tank observations. This noticeable fit also supports the appropriateness of the new scaling approach for describing well-developed wind-wave fields observed in laboratory at large fetches.

Finally, figures 7 and 8 enable us to consider two issues thoroughly: the success of the new approach for wind-wave tank observations and the related relevance of the conventional wind speed scaling under Toba’s law. To support this viewpoint, one can also notice that, in figure 8, the range of dimensionless periods $\tilde{T}$ observed at the smallest wind speeds overlaps the data of the Field Research Facility for buoys 190 and 630 located at the minimal distances from the coast, i.e. at 3 and 6 km respectively (see figure 6). The equivalent dimensionless fetch in terms of wavelength – the number of waves ${\it\nu}$ – for the wind-wave tank data can be estimated as $50$ to $300$ , i.e. values quite close to those obtained by Gibson ( $\tilde{T}$ is $152$ and $266$ at fetches $1.3$ and 3.5 km and wind speeds $16.5$ and $16~\text{m}~\text{s}^{-1}$ respectively) described in our analysis of the Sverdrup & Munk (Reference Sverdrup and Munk1947) results. Thus, with the new approach one can see that wind waves of lengths higher than roughly 15 cm observed in large water tanks are developed enough to be considered for modelling the nonlinear interactions and statistical properties of the resulting wave field.

6.3. Toba’s law and the wind-free approach: similarities and dissimilarities

The data presented here and in the previous sections (see figure 5 for data by Sverdrup & Munk Reference Sverdrup and Munk1947) show quite good correspondence of the observations both with our theoretical dependences and with Toba’s power law, especially for wind-wave tank experiments. It reflects, in a sense, the similarities between the basic physical assumptions of both the purely theoretical model described here and the theoretical–empirical model by Toba (Reference Toba1972). Both models consider physical mechanisms providing a local balance (as emphasized in the title of the paper by Toba Reference Toba1972).

Toba (Reference Toba1972) develops his model using a dimensional analysis, associating the wave energy growth rate with the work of the wave-induced wind stress in the field of wave orbital motions. Assuming that this work is directly controlled by the local air flow (and not by wave parameters), and then is proportional to $u^{\ast 3}$ , he derived the so-called $3/2$ Toba relationship between the dimensionless wave height and wave period. Note that this law is equivalent to proportionality of the total drift current including the Stokes one (i.e. the averaged value of the orbital velocity) and the friction velocity $u^{\ast }$ . This relation that links the wave steepness to the root-mean-square of the inverse wave age expresses the local equilibrium of the dominant wind-wave field but here the treatment is heuristical rather than theoretically well-grounded.

One of the reviewers pointed out that the Stokes drift current is an essentially nonlinear effect of second order in the wave steepness and, thus, unrelated to viscosity and to friction velocity: it is present in a nonlinear wave field in the absence of wind as well. The Stokes drift current therefore can be easily observed in laboratory flumes where waves are generated mechanically by a wavemaker. In the present use of the terminology (of Stokes drift), we mean a somewhat different contribution to the total drift current that results from the shear stress on the air–water interface due to the presence of wind. Interestingly, in laboratory wind-wave facilities these two different contributions to the drift current at the water surface are often of comparable magnitude.

On the contrary, in the present theoretical model we assume the nonlinear interactions to be dominant compared to the external forcing terms $S_{in}$ and $S_{diss}$ and exploit the key features of nonlinear transfer among the random water wave field components explicitly. The remarkable fact is that one can express the corresponding balance of the wave field energy (or momentum, or action) in terms of wave parameters only, without any reference to wind forcing and/or dissipation. However, when examining this model in terms of the dependence of wave height on wave period, this balance gives the Toba $3/2$ law but as an important special case of local wave field equilibrium for which the flux of energy to waves (or the ‘rate of work done by the wind stress to wind waves’ in the words of Toba Reference Toba1972) remains constant. Furthermore, as shown in Gagnaire-Renou et al. (Reference Gagnaire-Renou, Benoit and Badulin2011), the wave energy growth rate can be expressed as a function of $u^{\ast 3}$ . The present model thus leads to expressing the strong assumption made by Toba (Reference Toba1972) as the result of the predominance of nonlinear interactions among the other processes governing the wind-wave field evolution. It also means that, in this case, both models represent very similar physics, but here our model reveals more precisely the governing physical mechanisms.

The mathematically consistent analysis of the model (2.2) allows the effect of dominating nonlinearity to be accounted for in a quite ‘flexible’ way as two-parametric families of self-similar solutions (2.6), (2.11) and, thus, a description of energy flux that depends on time or fetch. The ‘freedom’ of these solutions comes from homogeneity properties (2.3), i.e. from arbitrariness of physical scaling of deep-water wave lengths and heights (coefficients ${\it\upsilon}$ and ${\it\varrho}$ in (2.3)). It allows the balancing of the dominating nonlinear wave transfer and the total net input (see (2.2b )) in a wide range of physically relevant wind-wave growth scenarios when energy income depends on time or fetch. On the contrary, the restrictiveness of Toba’s model is inherently associated with the definite physical scale of wind speed. Toba fixes this scale by postulating that ‘the growth of wind waves is predicted by an integration with respect to the fetch and duration’ (Toba Reference Toba1972, p. 18) and, hence, in fact, by freezing a particular scenario of wave development at constant energy growth rate.

A divergence of the two approaches is then observed beyond the quite restrictive Toba’s wave growth scenario, i.e. when the energy flux to waves varies with time or fetch. The cases by Hasselmann et al. (Reference Hasselmann, Ross, Müller and Sell1976) and Zakharov & Zaslavsky (Reference Zakharov and Zaslavsky1983) considered above (see also Gagnaire-Renou et al. Reference Gagnaire-Renou, Benoit and Badulin2011) provide other exponents in the dependence of wave height on frequency. In the context of this problem, the consistency between the ‘wind-based’ model by Toba and our ‘wind-free’ model, especially for wind-wave tank observations (cf. figures 7 and 8) might be the most striking manifestation of the ‘universal’ behaviour of wind–wave coupling occurring in the laboratory when external air and water flow disturbances are minimized. These conditions make the friction velocity $u^{\ast }$ a fully representative physical parameter.

7.1. Theory

The key result of this paper is the invariant (1.4) of wind-wave field growth. The basic assumption of the dominating role of nonlinear wave transfer leads to the property of self-similarity of wind-wave spectra for the reference cases of duration- and fetch-limited wave growth and the power-law dependence of dimensionless wave parameters on fetch and time duration (1.3). In contrast to the traditional power-law fits with four free parameters, the theoretical analysis demonstrates the existence of rigid links between exponents and pre-exponents of (1.3). These universal links then give us the universal form of the invariant (1.4), (cf. (1.7)) with the right-hand-side ${\it\alpha}_{0}$ being expressed by cumbersome integral dependences of spectral shape functions (e.g. Hasselmann et al. Reference Hasselmann, Ross, Müller and Sell1976). We assume ${\it\alpha}_{0}$ to be constant based on previous experimental and numerical studies that showed spectral shape invariance of growing wind seas. The ultimate result of these theoretical considerations implies a surprising conclusion that appears to contradict the present understanding of wind-wave growth: the invariant (1.4) does not contain wind parameters explicitly. In other words:

This wind-free paradigm implies a new scaling of dimensionless wave height and period (3.7), (3.8): the simple time duration or fetch becomes a key physical scale irrespective of the conditions of wind forcing. We show that the new theoretical dependences agree fairly well with conventional theoretical–empirical ones (e.g. Hasselmann et al. Reference Hasselmann, Ross, Müller and Sell1976; Carter Reference Carter1982). Exclusion of wind speed from these dependences gives exactly exponents $9/4,5/2$ and, what is more surprising, provides consistent estimates of basic constants ${\it\alpha}_{0(d)},{\it\alpha}_{0(f)}$ (see (3.11)). In particular, the reference to the theoretical–empirical model by Hasselmann et al. (Reference Hasselmann, Ross, Müller and Sell1976) is quite representative. The purely theoretical approach shows that the empirical estimations of the parameters in the JONSWAP spectrum and its specific spectral shape are, in a sense, exhaustive for the analysis: the results are substantiated by much more general physical principles.

7.2. Simulations of wind-wave growth

We stress that all the numerical experiments were accomplished almost ten years ago with no reference to our findings in this paper. A series of runs for a duration-limited setup from Badulin et al. (Reference Badulin, Pushkarev, Resio and Zakharov2005, Reference Badulin, Babanin, Resio and Zakharov2007a , Reference Badulin, Babanin, Resio, Zakharov, Borisov, Kozlov, Mamaev and Sokolovskiy2008) taken arbitrarily showed a very good agreement with the results of this new theory.

The recent numerical study of fetch-limited growth by Zakharov et al. (Reference Zakharov, Resio and Pushkarev2012) gives us a broader view of the classic problem of wind-wave growth. We found that the fetch-limited regime is just an intermediate asymptotic stage when wave growth is limited in space. When waves from the coast arrive at the opposite side of the simulation domain the waves continue to grow but according to a duration-limited scenario.

7.3. Field experiments on wind-wave growth

Field studies of wave growth provide the dominant support for the new theory. First, we show that the parameters of the power-law experimental fitting (1.3a−d ) are linked. The theoretical links (2.7), (2.12) reproduce the parameterizations by Hwang & Wang (Reference Hwang and Wang2004) reasonably well: a minor quantitative difference makes the invariant (1.4) weakly dependent on dimensionless fetch (or wave age).

The outcome of our historical review of the brilliant paper by Sverdrup & Munk (Reference Sverdrup and Munk1947) is two-fold. First, we show that the relatively coarse wave observations made during World War II are consistent with our theory. Secondly, we emphasize the parallel between our self-similarity approach and the concept of significant wave height. Both approaches describe the wave field with a minimal number of parameters: significant wave height and peak period, in contrast to the more detailed but much more expensive representation of wave field as an ensemble of a great number of spectral components.

Data from wave riders of the US East Coast were analyzed as an additional justification of these new theoretical outcomes. Wave heights and periods scaled by the new wind-speed-free parameterization show remarkable closeness to the $5/2$ law for the fetch-limited case. We expect that more experimental proof of the validity of such an approach can be found in more wave data.

7.4. Wind-wave tank experiments: beyond the formal validity of the statistical description?

The results obtained from wind-wave tank experiments are found to be well representative. Even if the wind-wave tank experiments are generally considered as too far from wind-sea reality, careful data treatment enables one to show that it is probably not the case. In terms of the new wind-free scaling the results of experiments in the Large Air-Sea Interaction Facility (LASIF) in Marseille fit the theoretical dependences remarkably well. They correspond to high values of dimensionless $\tilde{H},\tilde{T}$ , i.e. those obtained at relatively short fetches. As expected, the statistical description used in this work may not apply in this case. At the same time, the range of wave age observed in the wind-wave tank experiments overlaps those observed by the wave riders. This finding affords promising perspectives for modelling sea waves in large wind-wave facilities (Zavadsky, Liberzon & Shemer Reference Zavadsky, Liberzon and Shemer2013).

7.5. A final remark

Evidently, the invariant (1.4) does not provide a full description of wind-wave field evolution for the classic setups of duration- and fetch-limited growth. Essentially, it shows that the problem can be split into two steps and, in the first step, gives an essential physical constraint that does not contain any parameters of wind forcing. The effect of wind can be accounted for in the next step to get the full description of wave growth in terms of wave height and period as functions of time, fetch and parameters of wind–sea coupling.

Thus, the concise expression (1.4) shows the validity and prospects of an analytical theory of wind-driven seas. To a certain extent, this result breaks a long-lived belief that the description of wind-driven seas must be the subject of extensive experimental studies and costly simulations rather than the result of an elegant physical theory.