Abstract

It has long been known that nonlinear wave–wave interactions produce stationary solutions related to constant energy flux through the equilibrium range when a deep-water spectrum follows an f−4 form, as has been verified in numerical studies in which spectra follow a constant angular spreading distribution. This paper shows that, although energy fluxes through such spectra remain essentially constant, momentum fluxes do not. On the other hand, if the angular distribution of a spectrum is allowed to behave in a manner consistent with observations, both the energy flux and the momentum flux tend to remain constant through a major portion of the spectrum. Thus, it appears that directional distributions of energy within wind-wave spectra adjust to a form consistent with nondivergent nonlinear fluxes, suggesting that these fluxes likely play a very prominent role in the evolution of directional spectra during wave generation.

1. Introduction

Over the last 40 yr, two classical spectral forms for ocean wave spectra have evolved within the scientific literature. Essentially, both forms are expressed in terms of power laws of frequency: that is, , where n is an integer constant and is a parameter with appropriate dimensions. The earliest of these, the f−5 spectral form, was initially based on dimensional scaling that used only frequency and gravity (Phillips 1958) but subsequently evolved into forms that included wind speed (Kitaigorodskii 1962; Pierson and Moskowitz 1964; Hasselmann et al. 1973). The second form, involving an f−4 spectral dependence, originated in the work of Zakharov and Filonenko (1966) and Toba (1972, 1973). Theoretical arguments have suggested that the f−5 form is created when wave breaking is the dominant source (sink) term affecting the spectral energies (Phillips 1958; Kitaigorodskii 1962; Pierson and Moskowitz 1964; Hasselmann et al. 1973), whereas the f−4 form has been theoretically linked to nonlinear energy fluxes from low- to high-frequency regions of the spectrum (Zakharov and Filonenko 1966; Kitaigorodskii 1983). Because different portions of a spectrum can have different source term balances affecting them, it is possible that spectral shapes will change their form within different frequency ranges. In fact, Forristall (1981) and Long and Resio (2007) have presented observational evidence suggesting that spectra shift from an f−4 form in the equilibrium range to an f−5 form at high frequencies.

Two schools of thought relative to the role of nonlinear fluxes during wave generation have also emerged during the latter part of the last century. The primary difference in these approaches centers on the relative magnitude of the nonlinear transfer source term compared to other source terms in the energy-balance equation, written here as

 
formula

where E(f, θ) is the spectral energy density at frequency f and propagation angle θ, cg is the vector group velocity, is the spatial gradient operator, and Sk(f, θ) is the contribution of the kth source. In deep water, it is commonly believed that the source term balance involves three primary terms: wind input Sin, dissipation Sds, and nonlinear wave–wave interactions Snl. Kitaigorodskii (1983), following the theoretical works of Zakharov and Filonenko (1966) and Zakharov and Zaslavskii (1982, 1983), advanced the concept that the effect of Snl dominates the source term balance in the spectral equilibrium range. Alternately, Phillips (1985) hypothesized that all three source terms could be of comparable magnitudes; hence, all three terms could conceivably be of equal importance within the equilibrium range.

In addition to the influence of nonlinear fluxes on spectral shape, it has long been recognized that these fluxes play an important role in the net balances of action, energy, and momentum within the coupled wind-wave system. However, most theoretical formulations have focused only on the role of energy fluxes during spectral development and have not spent much time considering the role of momentum fluxes. Assuming that energy fluxes alone dictate spectral shape has led to some elegant mathematics in terms of derived stationary solutions [often termed the Kolmogorov–Zakharov (K–Z) solutions]. The K–Z solutions tacitly assume that directional characteristics of wave spectra are of secondary importance in terms of their influence on fluxes within a spectrum.

This paper will investigate the influence of directional energy distributions on nonlinear momentum fluxes through wave spectra as well as the potential role of combined energy and momentum fluxes on spectral shape and evolution. Previous studies have certainly dealt with the influence of nonlinear fluxes on spectral shape. For example, Hasselmann et al. (1973) touched on this topic when they discussed the relative partitioning of total momentum entering and retained within wave spectra during their pioneering fetch-limited wave generation experiment [the Joint North Sea Wave Project (JONSWAP)]; Resio and Perrie (1991) also considered this topic when they showed that variations in the frequency power-law exponent n had a very large feedback on the evolution of local energy levels within a spectrum. Young and Van Vledder (1993) qualitatively examined the “central role” of nonlinear interactions on the total energy balance during wave generation; recently, Badulin et al. (2005, 2007) have offered substantial evidence that the entire wave generation process and pattern of spectral evolution is consistent with self-similar spectral shapes created by nonlinear fluxes.

Here, we shall build upon these earlier works, beginning with examinations of energy fluxes through idealized spectra with directional spreading that is independent of frequency. In spectra that maintain an f−4 form to very high frequencies, it will be seen that the net energy flux is directed toward higher frequencies through the equilibrium range for broad directional spreading but decreases as the directional spread narrows, eventually becoming negative throughout the equilibrium range. Although energy fluxes through these spectra are quite constant through the equilibrium range, momentum fluxes in the same spectra will be shown to be strongly divergent in this range.

Because observational evidence supports a shift toward an f−5 form at high frequencies, we will also investigate how such a transition affects fluxes through the equilibrium range. It will be shown that this change in the characteristic power law for spectral energy leads to divergence in the energy fluxes within the range in which they were formerly constant and exacerbates the momentum flux divergence in this same range.

Essentially all studies of directional spreading since the 1970s (e.g., Mitsuyasu et al. 1975; Hasselmann et al. 1980; Donelan et al. 1985; Young et al. 1995; Ewans 1998; Wang and Hwang 2001; Long and Resio 2007) have supported the prevalence of frequency-dependent directional spreading in wind-wave spectra. In addition to our analysis of fluxes through spectra with constant angular spreading, we will examine energy and momentum fluxes through spectra with directional spreading consistent with frequency-dependent characteristics similar to observations. It will be shown that both energy and momentum fluxes remain approximately constant through the equilibrium range in such spectra. These results suggest that both energy fluxes and momentum fluxes must be considered in any attempt to determine a quasi-stationary directional spectrum at frequencies higher than the spectral peak. Furthermore, by investigating both energy fluxes and momentum fluxes in comparisons with observed spectral characteristics, a framework can be established for quantifying constraints on various source terms within the wave generation process. Although action fluxes will likely be very important in the low-frequency regions of wind-generated spectra, they will not be treated in this paper.

2. Theoretical perspective and observational evidence

The general form for the Boltzmann integral as derived by Hasselmann (1962) is well known. However, Hasselmann’s derivation depends on asymptotic relations that are strictly valid only for spectral evolution on time scales longer then about O[Tp/(akp)2], where Tp is the spectral peak wave period, a is wave amplitude, and kp is the wavenumber of the spectral peak. Furthermore, some investigators have offered theoretical arguments that simulations over thousands of wave periods would be required to cancel the effects of nonresonant interactions (Annenkov and Shira 2006). Thus, the strict applicability of these interactions to the wave generation process might seem questionable; however, recent direct numerical simulations of the deterministic primitive equations for surface gravity waves in deep water by Tanaka (2001a,b, 2007), Korotkevich et al. (2007), and Toffoli et al. (2010), among others, have clearly demonstrated that the rate of energy change predicted by the Hasselmann equation agrees very well with that of the direct numerical simulations for time scales as small as 10 times the period of the spectral peak.

The numerical form of the Boltzmann integral used here has been discussed previously (Resio and Perrie 1991; Resio et al. 2001) and will not be revisited in this paper. Directionally integrated fluxes of action , energy , and momentum components and through a spectrum can be directly calculated from integral forms as shown by Resio and Perrie (1991). Here, we shall adopt the convention that positive fluxes represent fluxes from lower frequencies to higher frequencies and that negative fluxes are in the opposite direction.

Here, we consider self-similar spectra of the form

 
formula

where fp is the spectral peak frequency, θ0 is the mean propagation energy over the entire spectrum, and φE is a function of relative frequency and relative angle describing the distribution of energy with respect to frequency and direction. Theoretical analyses and numerical experiments show that, for such spectra, the nonlinear source term Snl can be scaled exactly as

 
formula

where φnl is another function of relative frequency and relative angle. A consequence of this is that for an f−4 spectrum and that for an f−5 spectrum. Thus, for self-similar spectra, fp and can both be scaled out of estimates of Snl with no loss of accuracy.

Most recent observations tend to support the existence of an f−4 form over a large range of basin sizes (Donelan et al. 1985; Resio et al. 2004; Long and Resio 2007). Although some studies have suggested a transition to an f−5 form at high frequencies (Forristall 1981; Long and Resio 2007), to the authors’ knowledge, no recent papers have argued that the f−5 form is characteristic of the equilibrium range in deep-water wind waves. Although Hara and Belcher (2002) base their spectral shape on an f−5 principle related to dissipation, they include additional terms within their spectral shape function that create a significant deviation from the f−5 shape in the equilibrium range. In this paper, we shall focus our attention on the f−4 class of self-similar spectra in our investigations of spectral fluxes and inherent energy-balance constraints.

A good way to visualize spectral shapes for a given class of self-similar spectra is to introduce a “compensated” spectrum , which is defined here as . Figure 1 shows characteristic compensated spectral shapes for sets of observations taken in six different areas around the world from Resio et al. (2004) and Long and Resio (2007). As can be seen in this figure, the f−4 form appears to fit these spectra reasonably well in all of the areas examined within a band of frequencies taken to be representative of an equilibrium range. Furthermore, these studies show that the equilibrium range coefficient can be better expressed as a function of a velocity term ua, which combines wind speed and spectral peak phase speed , than in terms of wind velocity alone. Here, cp is the phase speed of the spectral peak, u10 is the wind measured at a 10-m reference level, u* is the friction velocity, and uλ is the wind speed at the reference height proposed by Miles (1993) for a particular peak wavenumber.

Fig. 1.

Compensated spectra from Resio et al. (2004) for (top to bottom) different locations showing tendency for spectra to follow an f−4 form in the spectral region above the spectral peak, followed by a transition to an f−5 form at higher frequencies.

Fig. 1.

Compensated spectra from Resio et al. (2004) for (top to bottom) different locations showing tendency for spectra to follow an f−4 form in the spectral region above the spectral peak, followed by a transition to an f−5 form at higher frequencies.

The prevalence of spectra with approximately f−4 forms from so many basins of different sizes around the world suggests that the mechanism that creates this balance is fairly universal in nature. Such shapes are very close to their theoretical “constant energy flux” forms predicted for exact K–Z solutions to the stationary kinetic equation, which is consistent with the arguments of Badulin et al. (2005, 2007). A number of investigators (e.g., Phillips 1985; Hara and Belcher 2002; Alves and Banner 2003) have argued that the equilibrium range must also consider the effects of wind input and wave breaking on the local energy balance within this frequency range. Hara and Belcher (2002) assume that Snl is negligible in this range, which would be true for directionally integrated energy fluxes if the spectrum followed an exact f−4 form and the directional spreading was constant throughout the spectrum. If the sum of the wind input and wave breaking terms were nonzero over the range of frequencies within the equilibrium range, Snl would not remain zero. Instead, the form of the spectrum would shift toward a balance such that

 
formula

where are the net energy fluxes at the high- and low-frequency limits of the equilibrium range, respectively. Because we expect the net energy flux to flow from the peak of the spectrum toward high frequencies, any deviation in the slope of the energy densities from an f−4 form can be interpreted in terms of the integral of the net external source term (Sin + Sds), with slopes lower than f−4 indicating a dominance of the wind input over wave dissipation and slopes higher than f−4 indicating a dominance of dissipation over wind input. For the equilibrium range to remain close to an f−4 form, as seen in many observations, the sum of the wind input and spectral dissipation terms in the equilibrium range must also be close to zero.

3. Estimates of energy fluxes through the equilibrium range in spectra with constant directional spreading

In this section, we begin with a treatment of the directionally integrated spectrum. Perhaps the best known spectral parameterization is the JONSWAP spectrum developed by Hasselmann et al. (1973), which was an extension of the Pierson–Moskowitz spectral form (Pierson and Moskowitz 1964). This parameterization is typically written in terms of an f−5 power law,

 
formula

where E(f) is the spectral energy density at f,

 
formula
 
formula

In Eq. (5), α5 is a dimensionless constant; γ5 is the peakedness parameter; and the standard values of the spectral peak width parameters σa and σb are taken to be 0.07 and 0.09, respectively. In this type of spectrum, given that the values for σa and σb are set to constants and that both fp and αs can be scaled out of the Boltzmann integral, the peakedness parameter γ5 becomes the primary variable utilized in sensitivity studies of the nonlinear source terms.

Because the f−5 form is multiplied by an exponential term in the standard JONSWAP spectrum, it is difficult to represent energy fluxes through the equilibrium range in terms of a straightforward power-law spectrum. On the other hand, spectra following an f−4 power law have a clear interpretation in terms of energy fluxes through the equilibrium range; consequently, we will use a simplified version of the spectrum introduced by Resio and Perrie (1989) for our energy flux estimates. This spectrum can be expressed as

 
formula

where

  • β is the equilibrium range constant as defined by Resio et al. (2004);

  • H[·] is the Heaviside function = 1 for [·] ≥ 0; = 0 for [·] < 0;

  • γr is the relative peakedness as defined by Long and Resio (2007); and

  • Θ4 is a peakedness factor given by 
    formula
    with σ values the same as for the JONSWAP spectrum.

One feature of this spectral form is that the peakedness value can be viewed simply as the ratio of the compensated energy density at the spectral peak to the compensated energy density in the equilibrium range. Long and Resio (2007) have shown that peakedness defined in this manner appears to have a relatively clear dependence on wave age, with values of γr in the range of 2–3 for very young waves and values of γr approaching 1 as waves become fully developed. Figure 2a shows idealized compensated spectra for three different values of relative peakedness (γr = 1.0, 1.75, and 2.50), with no transition to an f−5 form at high frequency. Figure 2b shows the same idealized compensated spectra with a transition to an f−5 form at 4fp. As can be seen in this figure, the energy enhancement in the vicinity of the spectral peak extends to about 1.75fp in these spectra.

Fig. 2.

Idealized f−4 form for spectrum, with peakedness values γr = to 1.0, 1.75, and 2.5 with (a) representing a spectrum with no transition to an f−5 form at high frequencies and (b) representing a spectrum with transition to an f−5 form at 4fp. The compensated energy densities here are normalized by their value in the equilibrium range at 3fp [i.e. α4 = 2βg3/2/(2π)3].

Fig. 2.

Idealized f−4 form for spectrum, with peakedness values γr = to 1.0, 1.75, and 2.5 with (a) representing a spectrum with no transition to an f−5 form at high frequencies and (b) representing a spectrum with transition to an f−5 form at 4fp. The compensated energy densities here are normalized by their value in the equilibrium range at 3fp [i.e. α4 = 2βg3/2/(2π)3].

Longuet-Higgins et al. (1963) introduced a simple form for the directional spreading of energy in wave spectra, in terms of a power law for the cosine function,

 
formula

where Λ is a normalization coefficient such that

 
formula

This form for directional spreading or variations of it has been used in most of the idealized numerical simulations of energy fluxes in spectra to date. For this reason, we will examine fluxes through spectra with this type of angular spreading as our baseline case.

Figure 3 shows the net nonlinear energy fluxes through spectra with varying peakednesses (γr = 1.0, 1.75, and 2.50) and different directional spreading exponents, which do not vary with frequency (n = 1, 2, 4, and 8) for the spectra shown in Fig. 2a, where the f−4 spectral tail extends past the upper limit of integration. In our numerical representation of these spectra, the relative frequency spacing is set such that fi+1 = ɛfi, with ɛ = 1.045. The spectral peak in these calculations is located consistently in the 14th integration ring, and the integral covers a frequency range from 0.45fp to 12.30fp. All tests were executed for the case of a spectral peak frequency of 0.1 Hz in a depth = 10 000 m, which ensures that all frequencies in the spectrum are in deep water. Figure 3 shows that a near-constant energy flux is only achieved at about 2.5fp; beyond this point, the spectral fluxes for the case of constant directional spreading (independent of frequency) become quite constant. The net fluxes become slightly negative (i.e. directed toward lower frequencies) through the equilibrium range for the two narrowest directional distributions (n = 4 and n = 8) but become positive for broader angular distributions (n = 1 and n = 2).

Fig. 3.

Net energy fluxes through directionally integrated spectra shown in Fig. 2a for varying values of n in cos2n directional form: n = (a) 1, (b) 2, (c) 4, and (d) 8.

Fig. 3.

Net energy fluxes through directionally integrated spectra shown in Fig. 2a for varying values of n in cos2n directional form: n = (a) 1, (b) 2, (c) 4, and (d) 8.

Figure 4 shows energy fluxes through spectra with the same set of directional distributions as used for Fig. 3 but with a transition from an f−4 form to an f−5 form at 4fp, as shown in Fig. 2b. This transition markedly affects the fluxes past about 2.5fp, the location where they tended to become constant in spectra with no transition to an f−5 form (Fig. 3); however, the transition does not appear to affect fluxes markedly near the spectral peak. Thus, such a transition should not play a dominant role in the nonlinear source term in the spectral peak region. As can be seen in Fig. 4, the positive fluxes increase significantly over the range from 3fp to slightly above 4fp, where the shape transition occurs. This increase in fluxes creates a flux divergence over this range of frequencies, producing a net energy loss in this area. A somewhat smaller flux convergence occurs in frequencies slightly higher than the shape transition, because the fluxes diminish somewhat with increasing frequency, producing a net energy gain in this region. A consequence of this sink-source pattern is to push the spectral shape back toward a local f−4 form. Even for the case of the narrowest directional spreading tested in Fig. 4, the fluxes through the equilibrium ranges in spectra that transition to an f−5 form remain positive, unlike the situation in spectra with no transition to an f−5 form.

Fig. 4.

As in Fig. 3, but for spectra shown in Fig. 2b.

Fig. 4.

As in Fig. 3, but for spectra shown in Fig. 2b.

4. Estimates of momentum fluxes through the equilibrium range in spectra with constant directional spreading

Although many papers have discussed energy fluxes through the equilibrium range (e.g., Zakharov and Filonenko 1966; Kitaigorodskii 1983; Resio 1987; Resio and Perrie 1991; Young and Van Vledder 1993), few studies have quantified the role of momentum fluxes through a spectrum due to nonlinear interactions. Hasselmann et al. (1973) stated that the incoming atmospheric momentum is partitioned at the sea surface in a consistent manner, such that about 20% of the total air–sea momentum flux enters the wave spectrum near the spectral peak; about 5% is transferred to lower frequencies and is therefore retained within the wave field; and about 15% is transferred to high frequencies, where it is presumably lost. Resio et al. (2004) argue that their results for energy levels within the equilibrium range appear reasonably consistent with such a momentum partitioning.

Momentum fluxes were calculated during the same computer runs in which the energy fluxes shown in Figs. 3 and 4 were generated, assuming that the mean propagation direction is along the positive x axis. Figure 5 shows the net nonlinear fluxes of x momentum through spectra for the cases of directionally integrated spectra that maintain an f−4 form to very high frequencies, as shown in Fig. 3. Because the spectra were constructed to be symmetric about the x axis, the net y-momentum fluxes are zero in these tests. Figure 5 shows that spectra with a constant directional spreading, although producing essentially constant energy fluxes through the equilibrium range, produce highly divergent momentum fluxes through this range. Resulting divergence in x-momentum fluxes would produce an adjustment in the directional distribution such that an increasing fraction of the total momentum flux is diverted to y-momentum fluxes (symmetrically in these cases to maintain zero net y-momentum flux), leading to a directional distribution with spreading that should increase with frequency. This result suggests that a spectrum with an initially constant directional spreading, allowed to change under the action of nonlinear interactions, would evolve into a spectrum with a directional spreading that would increase with increasing frequency. Figure 6 shows the resulting momentum fluxes for the cases of spectra with constant directional spreading that transition to an f−5 form at 4fp. It is apparent that this transition exacerbates the momentum divergence in the equilibrium range rather than diminishing it.

Fig. 5.

As in Fig. 3, but for momentum fluxes.

Fig. 5.

As in Fig. 3, but for momentum fluxes.

Fig. 6.

As in Fig. 5, but spectra shown in Fig. 2b.

Fig. 6.

As in Fig. 5, but spectra shown in Fig. 2b.

5. Energy and momentum fluxes through the equilibrium range in spectra with variable directional spreading

Mitsuyasu et al. (1975) and Hasselmann et al. (1980) both examined large sets of observations and developed unimodal frequency-dependent functions for directional spreading. Such a single-peaked distribution was also adopted by Donelan et al. (1985). However, in recent years, it has been widely observed that the directional distribution of energy in the equilibrium range tends to be markedly bimodal in fetch-limited waves (Young et al. 1995; Ewans 1998; Wang and Hwang 2001; Long and Resio 2007). To provide some idea of the comparability of these bimodal spectra fit to the unimodal directional distributions of Mitsuyasu et al. (1975) and Hasselmann et al. (1980), we computed a best fit power-law to the Currituck Sound dataset discussed in Long and Resio (2007), using the same cosine functions as used in the unimodal studies (Fig. 7). The results show that the observed spreading in all of these datasets seems quite similar, about equally narrow near fp, and progressively broadening at almost identical rates for both the equilibrium range and the spectral forward face region.

Fig. 7.

Comparison of directional spreading characteristics, represented in terms of a cos2n form, from Currituck Sound, Hasselmann et al. (1980), and Mitsuyasu et al. (1975) for different ranges of inverse wave age (u10/cp).

Fig. 7.

Comparison of directional spreading characteristics, represented in terms of a cos2n form, from Currituck Sound, Hasselmann et al. (1980), and Mitsuyasu et al. (1975) for different ranges of inverse wave age (u10/cp).

Using Ewans’s (1998) parameterization of observed energy densities in fetch-limited waves, included here in the  appendix, combined with the directionally integrated form given in Eq. (6), we can construct a directional spectrum which is consistent with most recent observations. Figure 8 shows a three-dimensional projection of the energy densities for such a spectrum for the specific case of a peakedness equal to 1.75.

Fig. 8.

Three-dimensional projection of energy densities in a spectrum of the type used in numerical tests in this paper, compensated by multiplying by (2π)3f4/2βg3/2.

Fig. 8.

Three-dimensional projection of energy densities in a spectrum of the type used in numerical tests in this paper, compensated by multiplying by (2π)3f4/2βg3/2.

Figures 9a,b show calculated energy fluxes through spectra with Ewans’s (1998) directional spreading and varying peakedness. Figure 9a shows fluxes for the cases having a bimodal angular distribution with no transition to an f−5 form. Figure 9b shows the fluxes for the cases having a bimodal angular distribution with a transition to an f−5 form at 4fp. The effect of the transition to an f−5 form is not nearly so pronounced in the bimodal angular distributions as in spectra with constant directional spreading. Fluxes through the spectra for the cases with no transition show a slight decrease in flux rates with increasing frequency and essentially no dependence on peakedness in the equilibrium range. Fluxes for the cases with a transition at 4fp show little or no variation in the flux rates through the equilibrium range.

Fig. 9.

Net energy fluxes for spectra with Ewans (1998) class of bimodal spreading combined with (a) a directionally integrated form that maintains an f−4 form to very high frequencies and (b) one that transitions to an f−5 form at 4fp.

Fig. 9.

Net energy fluxes for spectra with Ewans (1998) class of bimodal spreading combined with (a) a directionally integrated form that maintains an f−4 form to very high frequencies and (b) one that transitions to an f−5 form at 4fp.

Figure 10 shows the calculated momentum fluxes for the same spectra used in Fig. 9, with Fig. 10a representing the cases with no transition to an f−5 form and Fig. 10b representing the cases with a transition at 4fp, both compared to momentum fluxes through spectra with the same directionally integrated shape, but with constant angular spreading of cos8θ for reference. Unlike net momentum fluxes through spectra with constant directional spreading, momentum fluxes for the bimodal angular distribution remain reasonably constant throughout the equilibrium range. Figure 10c shows net momentum fluxes through the same bimodal spectrum used in Fig. 10a compared to fluxes through the same directionally integrated spectrum with a frequency-dependent angular spreading as specified by Mitsuyasu et al. (1975). In this case, we see that the difference between fluxes for bimodal and unimodal cases is quite small. Figure 10d shows the net momentum fluxes through a bimodal spectrum used in Fig. 10b compared to fluxes through the same directionally integrated spectrum with a frequency-dependent angular spreading as specified by Mitsuyasu et al. (1975). In this case, the net fluxes through the bimodal spectrum are substantially less divergent than the fluxes for the unimodal case. The gist of these results is that wave spectra that are consistent with observed spreading appear to approach a condition in which both the energy fluxes and the momentum fluxes are nondivergent.

Fig. 10.

Net momentum fluxes for spectra having Ewans (1998) class of bimodal spreading combined (a) with a directionally integrated form that maintains an f−4 form to very high frequencies compared to net momentum fluxes for a constant cos8θ spreading with the same directionally integrated form; (b) with a directionally integrated form that transitions to an f−5 form at 4fp compared to net momentum fluxes for a constant cos8θ spreading with the same directionally integrated form; (c) with a directionally integrated form that maintains an f−4 form to very high frequencies compared to net momentum fluxes for unimodal frequency-dependent spreading with the same directionally integrated form; and (d) with a directionally integrated form that transitions to an f−5 form at 4fp compared to net momentum fluxes for unimodal frequency-dependent spreading with the same directionally integrated form.

Fig. 10.

Net momentum fluxes for spectra having Ewans (1998) class of bimodal spreading combined (a) with a directionally integrated form that maintains an f−4 form to very high frequencies compared to net momentum fluxes for a constant cos8θ spreading with the same directionally integrated form; (b) with a directionally integrated form that transitions to an f−5 form at 4fp compared to net momentum fluxes for a constant cos8θ spreading with the same directionally integrated form; (c) with a directionally integrated form that maintains an f−4 form to very high frequencies compared to net momentum fluxes for unimodal frequency-dependent spreading with the same directionally integrated form; and (d) with a directionally integrated form that transitions to an f−5 form at 4fp compared to net momentum fluxes for unimodal frequency-dependent spreading with the same directionally integrated form.

6. Some implications of nonlinear momentum fluxes related to generation and its associated source term balance

Many laboratory and field studies have represented the growth of dimensionless wave energy in terms of a simple power law with respect to dimensionless fetch,

 
formula

where

 
formula

and

 
formula

The value of m2 from several field experiments depends somewhat on which wind scaling term is used, on the atmospheric stability in the fetch area, and on wind variations along the fetch within each site; however, in general, its value lies in the vicinity of 1 [e.g., m2 = 1 in Mitsuyasu (1968); m2 = 1 in Hasselmann et al. (1973); m2 = 1 (as reported in Komen et al. 1994); 0.84 < m2 < 1.04 in Perrie and Toulany (1990); m2 = 0.9 in Kahma and Calkoen (1992); and m2 = 0.9 in Breugem and Holthuijsen (2007)]. As noted by Resio and Perrie (1989), a value of m2 close to 1 implies that the wave field retains approximately a constant fraction of the total momentum transferred from the air into the water in the early stage of wave growth along a fetch, consistent with Hasselmann et al.’s (1973) claim that a constant fraction of about 5% of the total momentum crossing from the atmosphere into the water is retained in the wave field.

Hasselmann et al.’s (1973) numerical calculations suggested that about 3 times more momentum is directed toward high frequencies (positive flux) than toward low frequencies (negative flux); however, this 3:1 momentum flux ratio is specific to a JONSWAP spectrum with constant directional spreading. Here, we obtain a ratio of about 2:1 using our numerical calculations for these fluxes in the f−4 spectra with constant directional spreading examined here. This is probably in reasonable agreement with these earlier estimates, given both the numerical difficulties in some of the earliest integration methods used for the Hasselmann equation and differences in the spectral forms used in these numerical studies. However, if we modify our spectral shape to include the Ewans directional distribution along with our f−4 form, our estimates of the proportion of the x-momentum flux suggest that the positive and negative fluxes out of the spectral peak region remain comparable for the entire peakedness range considered in our simulations (γr values from 1 to 2.5). Thus, our results suggest that momentum fluxes to lower frequencies, which should approximately equal the retained momentum, yields the following estimate for the momentum flux entering the equilibrium range from the spectral peak region:

 
formula

where is the positive momentum flux into the equilibrium range due to nonlinear interactions, is the negative momentum flux moving to frequencies lower than the frequency of the spectral peak via nonlinear interactions, λret is the fraction of the total air-to-sea momentum flux retained by the wave field, ρa is the density of air, and ρw is the density of water. Hristov et al. (2003) point out the importance of separating the wave-coherent signal from the total atmosphere–water momentum transfers to quantify the amount of momentum entering the wave field. Their results indicate that about 10% of the total transfer into the water is in the form of wave-coherent velocities, which presumably serves as input into the waves. Assuming that equal amounts are transported to higher and lower frequencies, the Hasselmann et al. (1973) and Hristov et al. (2003) results suggest that about 5%–10% of the total momentum should be transported into high frequencies from the spectral peak region.

The early stages of wave growth are well represented by the Currituck Sound dataset (Long and Resio 2007). Using these data along with other datasets, Resio et al. (2004) developed regression relationships between the velocity parameters introduced earlier and energy levels within the equilibrium range. Using the Resio et al. (2004) regression equations, we can produce a specific spectrum for any given wind condition essentially identical to those described by Long and Resio (2007). For example, if we take a 13 m s−1 wind speed, which is a typical value for many of the wave generation sequences in the Currituck Sound dataset, we can develop a wave spectrum with a bimodal directional distribution in the tail of the spectrum and thus calculate the momentum flux through this spectrum for comparison to Eq. (9). Such a spectrum can be characterized by an equilibrium range coefficient determined by the regression equation for u* given in Resio et al. (2004), a peakedness value γr equal to 2.5, and a transition from an f−4 form to an f−5 form at about 3fp.

Figure 11 shows the calculated net x-momentum flux through the spectrum defined in the previous paragraph. The nonlinear x-momentum flux in the range where the fluxes become quasi constant is calculated to be approximately 1.1 × 10−7 m. Assuming a logarithmic boundary layer and a Charnock (1955) surface roughness, the coefficient of drag for a 13 m s−1 wind speed is estimated to be 1.6 × 10−3. Combining this with the wind speed provides an estimate of u* and an estimate of the x momentum into the wave field from Eq. (9), which for the case described here is 10% of 1.3 × 10−6 m, or 1.3 × 10−7 m. Although this value is slightly higher than the net nonlinear flux value, the agreement is indeed remarkable because these two terms have been estimated independently and because there are no “tunable” coefficients in the fluxes estimated from the Hasselmann equation.

Fig. 11.

Calculated momentum fluxes through a typical Currituck Sound directional spectrum, showing the tendency toward a constant value of about 1.1 × 10−7 m at high frequencies.

Fig. 11.

Calculated momentum fluxes through a typical Currituck Sound directional spectrum, showing the tendency toward a constant value of about 1.1 × 10−7 m at high frequencies.

To convert our estimated momentum flux to an estimated energy flux out of the spectral peak region, which could then drive energy fluxes through the equilibrium range, we have to add another velocity term to Eq. (9). Lighthill (1962) provides an excellent discussion of the physical interpretation of the Miles (1957) mechanism for wave generation by the wind. Both show that the flux of momentum from the atmosphere into the waves can be written as

 
formula

where L is the wavelength of the waves receiving the momentum, u(z) is the wind profile as a function of height z above the mean surface, zc is the “critical layer” at which the wind speed is equal to the phase speed of the waves, and w0 is the vertical displacement of the perturbed flow. The energy flux from the atmosphere into the waves can be expressed as

 
formula

where c is the phase speed of the waves. A comparison of Eqs. (10) and (11) shows that the ratio of net energy flux to net momentum flux from the atmosphere into the wave field, in terms of a wind source term of the “Miles” type, should be equal to the phase speed of the waves c. The underlying reason for this result is that the wind velocity at the critical layer is equal to the phase speed. In fact, it would seem that any pattern of momentum–energy transfers from the atmosphere into the water must entail a work rate involving correlations among wave phases and overlying perturbations, whether this would be due to sheltering, a resonant feedback, or some other mechanism. If the ratio of energy transfer to momentum transfer does not equal the phase speed, such transfers would require a compensatory momentum sink–source somewhere within the wind-wave system to ensure that all constants of motion remain conserved.

Equation (9), combined with the relationship between wind energy and momentum input rates shown in Eq. (10), suggests that the net energy flux moving through the equilibrium range can be estimated as

 
formula

where is the net energy flux passing through the equilibrium range. Resio et al. (2004) found that the scaling for energy levels (β values) in the equilibrium range fit their composite dataset quite well, whereas attempts to scale the energy levels with velocity parameters, which did not include phase speed tended to create large discrepancies among the datasets. If we substitute this relationship for β into estimates of energy fluxes through the equilibrium range, we obtain a relationship between (i) the nonlinear energy flux from the spectral peak region transmitted to higher frequencies and (ii) the energy entering the spectrum near the spectral peak,

 
formula

where Λnl is an empirical constant which depends on the type of angular spreading used. Thus, if one adopts a Miles-type wind input, it follows that a balance with the nonlinear fluxes is possible only if β is proportional to , as found in Resio et al. (2004).

7. Discussion

A key result emerging from this study is the surprisingly strong dependence of nonlinear energy and momentum fluxes on the directional distribution of wind-wave spectra. Previously, it had been hypothesized that nonlinear energy fluxes were only weakly dependent on directional spreading; however, these observations were based on studies conducted with relatively broad directional spreading functions (typically cos2θ or cos4θ), which did not vary with frequency. Energy and momentum fluxes through spectra with frequency-dependent angular spreading can deviate significantly from fluxes through spectra with constant angular spreading. In particular, energy fluxes in spectra with variable directional spreading can be substantially reduced compared to energy fluxes through spectra with constant directional spreading, which leads to reductions in the overall values of the nonlinear source terms Snl. For example if we examine the nonlinear source term for a spectrum with constant angular spreading, chosen to be consistent with angular spreading in the primary energy containing region near the spectral peak (taken here as cos16θ), and a spectrum with the Ewans (1998) directional spreading for the same directionally integrated spectrum utilized in Fig. 11, we see that the directionally integrated nonlinear source terms differ by almost a factor of 2 in their peak values (Fig. 12).

Fig. 12.

Comparison of nonlinear source terms for identical directionally integrated spectra combined with constant (cos16θ) angular spreading (solid line) and a frequency-dependent spreading due to Ewans (1998) (dashed line).

Fig. 12.

Comparison of nonlinear source terms for identical directionally integrated spectra combined with constant (cos16θ) angular spreading (solid line) and a frequency-dependent spreading due to Ewans (1998) (dashed line).

Furthermore, as can be seen in a comparison of Figs. 5 and 10, the momentum fluxes, which strongly diverge for the case of constant angular spreading in f−4 spectra, become relatively constant when the same directionally integrated spectra are combined with frequency-dependent directional spreading that is consistent with observations. Thus, it seems that the directional distributions of wind waves in nature have adjusted to a shape in which the both energy and momentum fluxes are nondivergent, suggesting that the role of nonlinear interactions is quite critical to spectral evolution during wave generation.

To examine the potential role of nonlinear interactions in wave generation, let us follow a particular frequency through time during the wave generation process. When this frequency is located at 0.8fp, the directional distribution is quite broad, equivalent to about a cos4(θ/2) spreading. At a later time, if waves continue to grow and the peak frequency continues to decrease, this frequency will become colocated with the spectral peak. At this time, the directional distribution has become quite narrow [equivalent to about a cos20(θ/2) spreading]. This narrowing of the directional spread cannot be produced by nonlinear interactions, as can be verified by experiments with external source terms set to zero, so it must be produced by some external source term acting on the spectrum in the frequency range 0.8fp–1.0fp. Because the sum of the source terms is strongly positive in this region of the spectrum, it is logical to assume that wind input is the dominant source term here. Furthermore, to counterbalance the natural tendency of nonlinear interactions in this spectral range to produce increased directional scattering, it would seem that the wind input must be quite narrow directionally to achieve this directional concentration of energy.

At a later time, as the spectral peak continues to propagate into still lower frequencies, let us examine the situation when this frequency is located at 1.8fp. Here, the spreading becomes much broader again, equivalent to about a cos6(θ/2) spreading. In contrast to the period of time during which the frequency moved from 0.8fp to 1.0fp, the change in directional spreading is now quite consistent with what would be expected in a situation where nonlinear interactions are very dominant. If the wind is strongly transferring energy into this part of the spectrum, why are its effects on the directional distribution so weak, particularly in comparison to its apparent dominant influence on the low-frequency side of the spectral peak? Because wind input in most theories tends to be scaled as a function of (u/c)n, where n is typically taken to equal either 1 or 2, one might expect a stronger influence of wind input in this region of the spectrum than in the vicinity of the spectral peak.

If the wind input and dissipation within the equilibrium range were much larger than the nonlinear source term, as assumed in some studies, the tendency would be to suppress the development of directional bimodality. Furthermore, given the dependence on (u/c)n noted above, along with the fixed dependence of wind input on wave angle in existing wind input formulations, there would also be little reason for angular spreading to increase in this range of frequencies. Hence, both the existence of pronounced bimodality in early stages of wave growth and the increasing angular spreading as a function of frequency in older waves seem inconsistent with the interpretation that the spectrum in this region is controlled by a balance of wind input and wave breaking. On the other hand, because the angular spreading appears to increase at almost precisely the rate needed for the nonlinear momentum fluxes to remain constant with increasing frequency, it seems logical to believe that these fluxes are the dominant source terms within this spectral range.

Our discussion shows that, by simultaneously examining directionally integrated spectral characteristics and directional distributions within a spectrum, one can potentially gain important insights into the wave generation process. Phillips (1985) questioned the efficacy of theories based on dominant nonlinear fluxes to represent the effects of the three primary source terms acting on the spectrum during wave generation, and rebuttals of Phillips’s arguments based solely on directionally integrated concepts of spectral evolution have not proven very definitive. This has left a situation in wave prediction in which many wind input and dissipation source terms have been hypothesized and judged only with respect to performance in model testing, primarily in an operational forecast mode. It would seem extremely important for all of these source term combinations to be tested with respect to what is now emerging as a good understanding of the evolution of directional spectra. To the authors’ knowledge, Alves and Banner (2003) and Banner and Morison (2010) have provided the only strong foray into this area.

8. Conclusions

This paper has examined nonlinear fluxes of energy and momentum through wave spectra using the WRT integration method. Conclusions based on our results are listed below.

  1. Although nonlinear energy fluxes through a spectrum with constant angular spreading are quite constant through the equilibrium range, momentum fluxes for such a spectrum are highly divergent. Therefore, such a situation cannot persist very long, even if it were somehow generated as an initial condition.

  2. The bimodal structure observed in many studies of the directional distributions of wind-wave spectra (Young et al. 1995; Ewans 1998; Wang and Hwang 2001; Long and Resio 2007; Toffoli et al. 2010) can be shown to be relatively consistent with the cos2n angular distributions, at least in a bulk sense, as derived in earlier field studies (Mitsuyasu et al. 1975; Hasselmann et al. 1980).

  3. Spectra with bimodal directional distributions of energy consistent with recent observations produce relatively constant fluxes of both energy and momentum through the equilibrium range, which suggests that the role of nonlinear interactions is quite critical to the directional evolution of wave spectral.

  4. Nonlinear momentum fluxes from the spectra peak region through the equilibrium range, obtained from a numerical solution to the Boltzmann integral, are in good agreement with the expected momentum balance within the spectrum.

  5. The very peaked directional distribution of energy near the spectral peak suggests that wind input into the wave field should have a very narrow angular distribution in this region of the spectrum.

  6. Consideration of both energy and momentum fluxes due to nonlinear interactions appears to provide an improved basis for understanding the wave generation process compared to the consideration of only scalar quantities (action and energy). Such a perspective should yield valuable constraints on the external source terms (wind input and dissipation) within the overall source term balance during wave generation.

Acknowledgments

Support for this research comes from the U.S. Army Corps of Engineers, the Canadian Panel on Energy Research and Development (PERD—Offshore Environmental Factors Program), and the U.S. Office of Naval Research (ONR) through the National Oceanographic Partnership Program.

APPENDIX

Characterization of Directional Distribution Function

This appendix is included to provide sufficient information to interested researchers to allow them to reproduce results obtained in this paper. Here, the directional function φE in Eq. (2) is characterized to allow a frequency-dependent directional spread with two modal peaks or a single modal peak, using the symmetric form of the double-Gaussian distribution suggested by Ewans (1998), which, in terms of dimensionless frequency , is expressed as

 
formula

where θ0 is overall mean wave direction, θm is the frequency-dependent azimuth of each modal peak on either side of the mean wave direction, and σ is the frequency-dependent parameter that governs the directional spread of each mode. All direction parameters are in units of radians. For θm > 0, the parameterization yields a bimodal directional distribution, whereas, for θm = 0, it gives a directional distribution with a single mode. Equation (A1) satisfies the conventional constraint

 
formula

We characterize functional forms for θm and σ in terms of and inverse wave age (u10/cp) using θm data published by Ewans (1998, his Fig. 12, assuming the modal separation data therein represents 2θm) and a subset of the Currituck Sound directional distribution data described by Long and Resio (2007) that are reasonably symmetric, which we averaged in three wave-age classes that are younger than the cases shown by Ewans (1998). As noted by Ewans (1998) and Long and Resio (2007), wave-age dependence of directional distributions is not particularly pronounced in these datasets, but there does seem to be a trend and our regressions were improved when we included wave age as a discriminator.

Using θm data from Ewans (1998) and fitting Eq. (A1) to the Currituck Sound data by varying θm and σ to minimize the variance in the region (where modal peaks are clearly indentified) and then using simple functions to model , we characterized the modal displacement as

 
formula

where

 
formula
 
formula
 
formula
 
formula

Figure A1 shows the data we used and curves based on Eqs. (A3) and (A4a)(A4d), with data and curves for each wave-age grouping offset along the ordinate for clarity. The curves suggest that the two modes merge (θm → 0) for , but this is somewhat subjective because it becomes difficult to distinguish modal spread and modal separation when the two modes are in close proximity. In our analysis, we use the fitted curves for θm in Eq. (A1) and then find a σ that best fits our observations. The curves in Fig. A1 suggest that the wave-age dependence is rather subtle, with θm ranging from 56° for the oldest waves (u10/cp ≈ 0.7) to about 62° for the youngest waves (u10/cp ≈ 2.75) at .

Fig. A1.

Modal displacement θm data and fitted curves as functions of f/fp in classes of inverse wave age. Solid symbols are from Ewans (1998). Open symbols are from a subset of Currituck Sound observations (Long and Resio 2007) that are reasonably symmetric about the mean direction. The bottom curve and data are to scale. Each subsequent set (increasing u10/cp) is offset by 10° on the ordinate for clarity. Curves are based on Eqs. (A2) and (A4a)(A4d).

Fig. A1.

Modal displacement θm data and fitted curves as functions of f/fp in classes of inverse wave age. Solid symbols are from Ewans (1998). Open symbols are from a subset of Currituck Sound observations (Long and Resio 2007) that are reasonably symmetric about the mean direction. The bottom curve and data are to scale. Each subsequent set (increasing u10/cp) is offset by 10° on the ordinate for clarity. Curves are based on Eqs. (A2) and (A4a)(A4d).

To characterize the behavior of , we use Eqs. (A3) and (A4a)(A4d) in Eq. (A1) and vary σ to minimize the variance with the three wave-age classed directional distribution observations from Currituck Sound. Figure A2 shows as symbols the resulting estimates of σ in terms of for the three age classes. The lowest set of points is to scale. The other sets are offset on the ordinate by 20° each for clarity. Solid lines in Fig. A2 are curves that we fitted to the data points and are described by

 
formula

where

 
formula
 
formula
 
formula
 
formula
 
formula
 
formula
 
formula
 
formula
 
formula
 
formula

The algebraic complexity of this algorithm owes to the disparate behaviors of various parts of the spectra, particularly in the region near the spectral peak where the modal spread parameters reach minima. Notably, the minimum in σ (in combination with the variation in θm) causes the maxima of the directional distributions for our two youngest cases to occur at frequencies slightly less than the spectral peak, qualitatively similar to observations made by Donelan et al. (1985). The directional spread increases rapidly at frequencies lower than the spectral peak region, in consonance with results from other investigators; however, we limit σ to 120° at the lowest because we do not resolve it very well in our observations. At frequencies above the spectral peak region, the individual modal spreads appear to increase monotonically with . Our observational range of does not extend to high values, so, for computational purposes, we assume that σ continues to increase in accordance with Eq. (A5).

Fig. A2.

Modal spread σ data and fitted curves as functions of f/fp in classes of inverse wave age. Data are from a subset of Currituck Sound observations (Long and Resio 2007) that are reasonably symmetric about the mean direction. The bottom curve is to scale. Subsequent curves (increasing u10/cp) are offset along the ordinate by 20° each for clarity. Curves are based on Eqs. (A5) and (A6a)(A6j).

Fig. A2.

Modal spread σ data and fitted curves as functions of f/fp in classes of inverse wave age. Data are from a subset of Currituck Sound observations (Long and Resio 2007) that are reasonably symmetric about the mean direction. The bottom curve is to scale. Subsequent curves (increasing u10/cp) are offset along the ordinate by 20° each for clarity. Curves are based on Eqs. (A5) and (A6a)(A6j).

As a check on our algorithms for representing directional spread, Fig. A3 shows the observed Currituck Sound directional distributions compared with the directional distributions resulting from our curve-fitting exercise. The gross features of the fitted shapes vary only slightly, consistent with weak wave-age dependence, and appear to be reasonably consistent with our observations. Because projections such as those shown in Fig. A3 are sometimes difficult to interpret quantitatively, in Fig. A4 we include contour plots of the same information provided in Fig. A3.

Fig. A3.

Comparison of observed Currituck Sound directional distributions φE averaged in inverse-wave-age classes (u10/cp) = (a) 1.5–2.0, (b) 2.0–2.5, and (c) 2.5–3.0 with the Ewans (1998) model based on the modal separation and modal spread parameters specified at (u10/cp) = (d) 1.75, (e) 2.25, (f) 2.75, showing the ability of the equations in the  appendix to characterize these spectra.

Fig. A3.

Comparison of observed Currituck Sound directional distributions φE averaged in inverse-wave-age classes (u10/cp) = (a) 1.5–2.0, (b) 2.0–2.5, and (c) 2.5–3.0 with the Ewans (1998) model based on the modal separation and modal spread parameters specified at (u10/cp) = (d) 1.75, (e) 2.25, (f) 2.75, showing the ability of the equations in the  appendix to characterize these spectra.

Fig. A4.

Contour plots of, and in the same order as, the φE shown in Fig. A3. Contour interval is 0.1 rad−1.

Fig. A4.

Contour plots of, and in the same order as, the φE shown in Fig. A3. Contour interval is 0.1 rad−1.

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