1. Introduction

The detailed physical mechanisms by which wind waves grow and evolve remain an intriguing unsolved mystery of fluid mechanics. Most field observations of wind wave growth characteristics have been made in well-defined geometrical scenarios with steady prevailing winds, well-understood fetch, and no swell. An analysis of fully developed wind seas by Pierson and Moskowitz (1964) verified earlier theoretical notions that there is a universal similarity in spectral shape (Kitaigorodskii 1962) with an asymptotic upper bound (Phillips 1958). However, a peak enhancement effect observed during the Joint North Sea Wave Project (JONSWAP) was found to have no correlation with various fetch relationships (Hasselmann et al. 1973). In a study of fetch-limited waves on Lake Ontario, Donelan et al. (1985) noted that the peak enhancement depends on inverse wave age U_c/c_p, where U_c is the component of the 10-m-elevation wind speed (U₁₀) in the mean direction of the waves at the spectral peak, and c_p is the wave phase speed. To further refine the wave fetch relationships, Donelan et al. (1992) made extensive wind and wave measurements on Lake St. Clair out to a fetch of approximately 21 km. They note a very high degree of correlation of nondimensional wind sea energy e′ = eg²/U⁴_c with U_c/c_p, where the total energy (variance) is defined by integrating the directional wave spectrum S(ω, θ) over frequency (ω) and direction (θ):

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The effect of swell on wind wave growth has been a topic of active research for many years with inconsistent results. Selected findings are summarized in Table 1. Although numerous well-documented experiments show unequivocally that short laboratory wind waves exhibit reduced amplitudes when in the presence of longer paddle-generated waves, the details are often contradictory among investigations. Furthermore, there remain a variety of competing theories to explain these phenomena. Most surprising, perhaps, is that modification of wind-driven gravity waves by swell has yet to be demonstrated in the real ocean. This is partially due to the difficulties of separating naturally observed wind sea from swell. Note that only a single study from Table 1 was conducted in oceanic conditions (Dobson et al. 1988). In a fetch-limited coastal environment, Dobson and colleagues do not report any significant growth effects resulting from the presence of an opposing swell. They present results that are statistically identical to the Lake Ontario results of Donelan et al. (1985). They attribute this similarity to the short fetches involved with both studies.

Breaking surface waves are observed on any open body of water over which the wind exceeds approximately 3 m s⁻¹. In a recent review, Melville (1996) notes that breaking waves contribute to wave evolution by effectively dissipating excess energy originally input by the wind. The importance of this role is evidenced in the radiative transfer equation, which describes the conservation of wave action spectral density N(k) = gS(k)/ω of waves interacting with a current of velocity U (Phillips 1977; Komen et al. 1994) in wave number (k) space:

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where c_g is the wave group velocity; −∇_k · T(k) is the action spectral flux due to nonlinear wave–wave interactions; S_w represents action input by the wind; and D is the dissipation, primarily due to wave breaking. Since an explicit definition of D is not yet available, wave evolution models such as WAM rely on a tunable dissipation source term. As assessed by Banner and Young (1994), shortcomings in the dissipation source term lead to significant differences between predicted and observed spectral properties of fetch-limited waves.

In addition to the role in wave evolution, wave breaking has important dynamical roles in facilitating air–sea transfers. The breaking process acts to transfer momentum and energy from the waves to the upper ocean; Melville and Rapp (1985) showed that the momentum flux by breaking may be comparable to that transferred directly from the wind. As revealed in numerous observations of turbulent dissipation beneath surface waves, collated and presented by Agrawal et al. (1992) and further analyzed by Terray et al. (1996), the momentum flux from breaking waves provides an energy source for enhanced turbulence and mixing in the upper ocean with turbulent kinetic energy (TKE) dissipation rates that are one or two orders of magnitude above levels expected from a wall-bounded shear flow. Using laboratory data (Loewen and Melville 1991; Rapp and Melville 1990) and theoretical arguments (Phillips 1985), Melville (1994) concluded that the surface layer should be well mixed to a depth comparable to the breaking wave height with TKE dissipation rates in the range observed by Agrawal et al. (1992).

The entrainment of air is another important attribute of surface wave breaking. Using electrical conductivity sensors to map the air content, or void fraction distribution, under breaking laboratory waves of different amplitudes, Lamarre and Melville (1991) observed that void fractions of >20% were sustained for up to one-half wave period after breaking. Furthermore, the initial volume of entrained air was found to scale with the total energy dissipated by each breaking event; this led them to conclude that wave parameters obtained by remote sensing techniques may be used to quantify air entrainment and gas transfer in the field.

Wave breaking and the bubble structures associated with breaking are now known to influence a variety of processes of global and operational significance (Banner and Donelan 1992; Melville 1996). For example, air–sea gas flux is enhanced by a factor of 2 or more above molecular diffusion in the open ocean when waves are breaking (Wallace and Wirick 1992; Farmer et al. 1993). In addition, Jessup et al. (1990) reported strong microwave returns from breaking waves in the field. They note that the contribution of breaking waves to the radar cross section increases approximately as u³*, in agreement with theoretical arguments by Phillips (1988). Furthermore, the bubbles entrained by breaking waves are found to be the dominant natural source of underwater ambient noise when winds are above the 3 m s⁻¹ breaking threshold (Kerman 1988, 1993; Buckingham and Potter 1995). It is thought that ambient noise could be an indicator of air–sea gas transfer (Melville 1996). Finally, near-surface bubble populations are now known to be the dominant contributor to low-frequency (≤1000 Hz) acoustic scatter and reverberation near the ocean surface (Ogden and Erskine 1994; McDaniel 1993).

In place of a detailed physical representation of breaking waves and the subsequent development of bubble clouds, empirical process models for air–sea gas flux, sonar and radar performance, etc., typically employ wind speed expressions to describe the wave-induced effects. Examples include the thin-film model for air–water gas transfer (Jähne 1990), which includes a wind-speed-dependent diffusion coefficient; the Wenz curves for ambient noise (Wenz 1962), which yield a power-law dependence of ambient noise on wind speed; and the Ogden–Erskine model for low-frequency acoustic backscatter (Ogden and Erskine 1994), which employs wind speed as the only environmental variable in a multiparameter fit to experimental data. The use of short-term averaged wind speed to describe wave-related processes will lead to prediction inaccuracies as the state of wind wave development at any given time represents an integrated effect of both the previous wind history and its spatial variability (Huang 1986).

Attempts have been made to augment or replace wind speed with wave parameters in air–sea process modeling. These have usually failed when bulk wave statistics were chosen; average wave parameters tend to smear the information from distinct wind sea and swell wave systems. Wallace and Wirick (1992), however, note a better correlation of mixed-layer oxygen saturation levels with wave height than with wind speed. Their results do not match the standard thin-film gas diffusion model but are more comparable to predictions from a model by Thorpe (1984) that include enhanced transfer by bubbles. The correlation between ambient noise A_N and various surface wave parameters was investigated by Felizardo and Melville (1995). They isolate wind sea spectra from full energy–frequency spectra by locating a spectral minimum between wind sea and swell peaks and then filtering the lower frequencies from the data. They note a high correlation between A_N and wind sea rms amplitude that was comparable to that between wind speed U and A_N. The correlation of A_N to various estimates of wave dissipation rate, obtained from the wind and wave observations using the separate ideas of Phillips (1985) and Hasselmann (1974), were also quite successful. These results, along with those obtained by Wallace and Wirick, suggest that properly chosen wave parameters can be used to augment or replace wind forcing parameters to improve the predictions of air–sea process models.

The primary objectives of the work reported here were to investigate wind sea growth and dissipation in the open ocean and to determine if wave parameters are more appropriate than wind speed parameters for estimation of wave breaking phenomena. These were accomplished using a spectral partitioning method for the isolation of distinct wind sea and swell wave components from directional wave spectra obtained from a heave, pitch, and roll buoy. Statistics generated from the wind sea partitions were employed to explore the influence of wind history and background swell on wave growth in the open ocean environment.

Furthermore, the equilibrium range theory of Phillips (1985) was employed to estimate the total rate of wave dissipation by breaking and to explore the relationships among wave dissipation, related air–sea parameters, and oceanic whitecap coverage. We believe the results have important implications for the use of surface wave parameters in air–sea process models.

2. Observations

Open ocean wind and wave observations were obtained during the Gulf of Alaska Surface Scatter and Air–Sea Interaction Experiment, conducted from 24 February through 1 March 1992, as part of the Critical Sea Test Program (Tyler 1992; Hanson and Erskine 1992; Hanson 1996). Located in the central Gulf of Alaska at 48°45′N, 150°00′W, the experiment was specifically designed to study the influence of the air–sea boundary zone on underwater acoustic scatter and reverberation at frequencies below 1000 Hz. Here we focus on the wind, wave, and whitecap observations made during this experiment. The collection and processing of these data, described elsewhere (Hanson 1996; Monahan and Wilson 1993), are briefly summarized below for completeness.

3. Wind sea growth in the open ocean

From comprehensive observations of wind wave growth on Lake St. Clair, Donelan et al. (1992) report a high correlation of nondimensional wave energy e′ = eg²/U⁴_c with inverse wave age U_c/c_p. For our analysis, U₁₀ has been used in place of U_c since wave-induced motions of floating meteorological stations have been found to induce error into the directional resolution of attached wind sensors (Dugan 1991). Indeed, attempts to use U_c in these analyses resulted in poorer correlations. Furthermore, observations made with U₁₀ < 5.0 m s⁻¹ were dropped from the record because of the greater uncertainty in buoy measurements of small, high-frequency waves.

The Gulf of Alaska results, appearing in Fig. 3, depict a high correlation of wind sea energy to inverse wave age. The 90% confidence interval for these observations, estimated using the approach of Skafel and Donelan (1983), appears on the plot. A least squares regression curve on the plot is given by

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with a correlation of r = 0.94. Also included on Fig. 3 is a regression from the Lake St. Clair observations of Donelan et al. (1992):

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where the L subscript is used to denote lake data. The open ocean observations are statistically identical with the Lake St. Clair results. This agreement is remarkable considering the drastic differences between the two sites: one a swell-dominated open ocean environment and the other a sheltered lake setting with no swell. From these results there is no directly discernible effect of swell on wind sea growth. There is, however, sufficient scatter beyond the 90% confidence limits to warrant additional investigation of these data. In the following sections we examine, first, the effect of wind history variability on wind sea growth and then look in greater detail at wind sea modification by swell.

a. Wind history effects

Since the response of the ocean surface to wind speed and direction changes is not instantaneous, it is likely that some of the remaining scatter in Fig. 3 is attributable to the recent wind history prior to each observation. Specifically, the mean rate at which wind speeds are rising or falling prior to each wave observation ought to determine, in addition to the current wind speed, the state of sea development. Indeed, it was noted by Toba et al. (1988) that the equilibrium range spectral level α becomes smaller for accelerating wind conditions and vice versa. To quantify the wind history conditions for each Gulf of Alaska wave observation, a 3-h averaging window was used to compute the mean wind speed U₁₀. The mean rate of change or acceleration of the wind was then computed for each observation:

a_UU₁₀t,(7)

where Δt represents a 15-min interval. A comparison of the U₁₀, U₁₀, and a_U records from the Gulf of Alaska appears in Fig. 4.

The nondimensional wind sea energies observed during rising (a_U > 0.0002 m s⁻²) and falling (a_U < −0.0002 m s⁻²) wind conditions appear in Fig. 5 with separate symbols and regression lines for each. Again, observations made with U₁₀ < 5 m s⁻¹ are excluded from the analysis. A distinct grouping of the two observation sets indicates that wind seas under falling wind conditions are more developed both in energy content and location of the spectral peak. The latter is evident by the shift of falling wind observations to larger inverse wave ages. Although not shown (to preserve clarity), a regression through a steady wind observation subset (−5 × 10⁻⁵ < a_U < 5 × 10⁻⁵) is given by

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which falls directly between the rising and falling wind regressions shown in Fig. 5. These results depict a trend of increasing energy with decreasing wind acceleration for observations at a given wave age and are in agreement with the findings of Toba et al. (1988). This important effect is ignored by air–sea process models that employ a short-term average wind speed forcing.

b. Interaction with swell

As discussed in the introduction, to date there has been little or no evidence that swells influence wind sea growth, over the range of frequencies observed by wave buoys, in the open ocean. It is clear that, at least for laboratory waves, the presence of long waves results in the attenuation of aligned wind waves. The mechanism for this process is as yet unresolved; the results from numerous laboratory and theoretical treatments are inconclusive and at times contradictory (Table 1). The agreement shown here between observations in the swell-dominated Gulf of Alaska and those from Lake St. Clair suggests that the relationship between eg²/U⁴ and U/c_p of natural wind waves is not affected by the presence of swell.

To more fully address the interaction between swell and wind sea in the Gulf of Alaska, the information assimilated through the wave spectral partitioning process was employed. To isolate specific swell conditions, the wave observations were sorted based on ratios between wind sea significant slope Ψ_ws, dominant swell partition significant slope Ψ_s1, the significant slope of the second largest swell partition Ψ_s2, and the angle θ_s−ws between the wind sea and dominant swell directions. Here the significant slope (Huang 1986)

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is computed using the peak frequency (ω_p) and H_s statistics from the appropriate spectral partition. Four wavecondition categories were formed as identified in Table 2. Most of the observations fall under the confused sea category, indicating the simultaneous existence of two or more swell systems with comparable steepness.

Nondimensional wind sea energy as a function of inverse wave age for the opposed swell and aligned swell categories (Table 2) appears in Fig. 6. Also appearing on the plot is the best-fit regression of the no-swell observations, given by

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with a correlation of r = 0.94. The results suggest that the relationship of nondimensional wind sea energy to inverse wave age is independent of the presence of swell, regardless of the direction of swell propagation relative to the wind sea direction.

These results do not necessarily contradict the laboratory results from previous investigations. Chu et al. (1992) observed the interactions between statistically steady wind-generated waves and groups of longer waves of various slopes. During the interaction phase, they noted both a reduction of wind wave amplitudes by breaking near the long wave crests and a shortening of the wind waves due to straining by the wave orbital velocities of the group. The important point is that these two interaction effects are compensating, such that wind wave slope was essentially unchanged from its original value before the interaction. In terms of the nondimensionalization employed herein, swell interactions would be expected to decrease e′ because of enhanced breaking while simultaneously decreasing the wavelength λ because of orbital straining. Hence, assuming that the deep water simplification

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holds for this scenario, a corresponding increase in U/c_p would occur. This simultaneous decrease of e′ and increase of U/c_p suggests that the occurrence of swell would not change the dependence of e′ on U/c_p.

4. Wave dissipation

a. Equilibrium range model

The equilibrium range theory of Phillips (1985) provides the necessary framework for obtaining estimates, from actual wave observations, of the total wave energy dissipation rate ε_t. By assuming a dependence of each term on the developing spectrum, Phillips employed theoretical and empirical arguments to derive the equilibrium range wave number and frequency spectral forms:

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where I(p) is a spreading function given by

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and β is related to Toba’s constant (α) and I(p) by

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The equilibrium range concepts allow for the expression of each source term as a function of the equilibrium range spectrum. The spectral rate of energy loss in the equilibrium range is given by

kωDkγβ³δ^3pu³*k⁻²(15)

This expression indicates that breaking waves provide energy for near-surface turbulence over the full range of scales in the equilibrium range. Equating the dissipation rate to the rate of energy input by the wind reveals

γβ²m(16)

and combining Eqs. (14) and (16),

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The equilibrium range ideas can be used to develop an expression for the total rate of wave dissipation ε_t in terms of a measured directional wave spectrum S(ω, θ). First it is necessary to express the spectral dissipation rate [Eq. (15)] in terms of frequency. This transformation is given by (Phillips 1977, 1985)

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Substituting Eq. (15) into the above and solving yields

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for k = ω²/g.

The spectral dissipation rate can be cast in terms of the equilibrium range spectrum by first solving equation (12) for u∗ and then inserting this result into Eq. (19):

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Assuming that the equilibrium range extends from ω_p to higher frequencies, integrating over this range and putting in the water density factor yields the dimensionally correct total rate of wave energy dissipation

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Felizardo and Melville (1995) use this expression to compare wave dissipation rate estimates with ambient sound signatures of breaking waves. They use wind sea frequency spectra, obtained by filtering out what was considered to be swell energy from observed wave frequency spectra, in place of S(ω) in Eq. (21), along with constant I(p) spreading terms. Felizardo and Melville compare results of using Eq. (21) with those obtained using a dissipation expression proposed by Komen et al. (1984). This alternate approach follows theoretical arguments of Hasselmann (1974) that spectral dissipation should approach a linear proportionality with S(ω) and exhibit a damping coefficient proportional to ω². In Felizardo and Melville’s correlations of spectral dissipation rate with ambient noise levels, Eq. (21) performed at a level comparable to, if not slightly better than, the Komen et al. approach (see Fig. 19 in Felizardo and Melville 1995). Thus we will proceed with the development of Eq. (21), however noting the existence of an alternate method that has been shown to produce statistically similar results.

Perhaps the most significant limitation of the equilibrium range theory is the I(p) directional spread factor defined by Eq. (13). This factor has its roots in the Plant (1982) wind input equation, which reasonably uses cos δ to obtain the component of wind stress that is in the direction of wave propagation. However, the use of this method to calculate I(p) yields a spreading function that assumes a symmetrical wave spectrum developed through a hypothetical u∗ stationarity. As directional distributions of wave spectra are now routinely measured, it is conceivable that the I(p) terms of Eq. (21) could also be replaced with observational data.

A convenient directional spread indicator that has the same limits as cos δ, yet is derived from observations, is the normalized direction spectrum

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where the wave direction spectrum in the equilibrium range is defined by

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and S(θ_p) represents the equilibrium range peak. New spreading functions are defined as

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Substitution of I₁ and I₃ in place of I(p) and I(3p), respectively, in Eq. (21) yields an estimate of the total rate of wave energy dissipation based on attributes of the directional wave spectrum:

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with

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5. Discussion

a. Wave dissipation estimates

Estimates of the total rate of wave energy dissipation by breaking surface waves in the Gulf of Alaska have been conveniently obtained from wave buoy observations. On average, the Gulf of Alaska dissipation rates fell below those estimated from wave measurements off the coast of Oregon by FM95 (Fig. 8). Several factors could contribute to the increase in FM95 wave dissipation estimates over those obtained from the Gulf of Alaska, including 1) exclusion of dissipation in the spectral tail region, 2) the method employed to isolate the equilibrium range, and 3) geographic and seasonal trends in atmospheric forcing. Each of these will be discussed in turn.

1) Dissipation in the spectral tail region

FM95 used wire wave gauges to estimate wave frequency spectra; they present these estimates out to a cutoff frequency f_c = 2.0 Hz. If they estimated equilibrium range dissipation out to 2.0 Hz, then their dissipation estimates include more of the spectral tail region than those presented for the Gulf of Alaska (f_c = 0.5 Hz) and, hence, would be expected to be higher. But how much higher? To determine the importance of dissipation in the spectral tail, a simple relationship can be established assuming an ω⁻⁵ spectral falloff in this region as observed by Donelan et al. (1985). The total dissipation rate given in Eq. (26) can be expressed as

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where P simply refers to the term in front of the integral. In line with Donelan et al. (1985), assume

Sωω⁻⁵Cω⁻⁵(33)

Substituting this into Eq. (32) and solving over the spectral tail region,

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where the integral has been evaluated from the radian cutoff frequency ω₀ = 2πf_c out to ∞. Since C = S(ω)ω⁵, the total dissipation rate in the spectral tail can be expressed as a function of S(ω₀):

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Application of the tail dissipation correction factor to the Gulf of Alaska dissipation estimates results in just a few percent increase in log(ε_t). As the FM95 estimates are some 30%–40% higher than the Gulf of Alaska estimates, spectral dissipation in the tail region is not the primary cause of this offset.

2) Isolation of the equilibrium range

The theoretical concepts discussed earlier specify the equilibrium range to be a spectral region where the action flux due to wind input, nonlinear transfer, and wave dissipation are all in balance and proportional to each other. This refers to that portion of the spectrum where active exchanges are occurring and, hence, specifies the range over which the waves are breaking. Observations by Farmer and Vagle (1988) indicate a wide range of breaking scales, supporting the concept here that the equilibrium range extends from near the wind sea spectral peak out through the observed spectral tail.

FM95 isolated wind sea frequency spectra by locating a minimum between wind sea and swell peaks and filtering out spectral components at frequencies below this minimum. It appears that all spectral components above the minimum were considered to be wind sea. They defined the equilibrium range, for estimation of dissipation rate, as extending from the peak of the filtered wind seas to the end of the spectral tail at f_c. To this they applied a constant directional spreading of I(p) = 2.4.

As described, the Gulf of Alaska directional wind sea spectra were isolated using an automated spectral partitioning approach. An α-threshold test was employed to identify and remove young swell components that were found to exist within the wind-forced spectral domain. The premise here is that rapidly varying wind conditions can lead to a close spectral mix of wind sea and swell components and that these swell components, moving at speeds comparable to or slightly greater than the wind speed, are not actively involved in the equilibrium range exchanges. The removal of young swell components from isolated wind sea spectra would likely result in dissipation rate estimates that are lower than those obtained by the FM95 method. Although the constant I(p) spectral spreading employed by FM95 would not contribute to a bias of the mean, it could result in an increased level of data scatter about the mean due to wind direction variability.

3) Geographic and seasonal effects

As was demonstrated by Fig. 9, wind history trends factor prominently in the distribution of ε_t with U₁₀. The FM95 data were obtained during September at a location approximately 130 km off the Oregon coast. The atmospheric forcing trends, and the responses of the ocean to these trends, are expected to be different than those observed in the Gulf of Alaska during winter. Indeed, inspection of the FM95 wind records indicates that wind speeds varied on much slower timescales than those experienced in the Gulf of Alaska. Furthermore, the FM95 wind directions were reasonably steady during most of their experiment. As a result, the FM95 wave field, at a given wind speed, tended more toward full development and, hence, would be expected to exhibit higher dissipation rates than were observed in the highly variable Gulf of Alaska environment.

6. Conclusions

A wave spectral partitioning approach has allowed an investigation of wind sea growth and dissipation in a swell-dominated open ocean environment. When scaled by eg²/U⁴, Gulf of Alaska wind seas exhibit the same functional dependence on inverse wave age U/c_p that was observed in a relatively benign lake setting by Donelan et al. (1992). It is concluded that the chosen scaling compensates for wind sea modification by swell;a decrease of eg²/U⁴ through swell interactions is accompanied by a compensating increase of U/c_p. This finding is consistent with the laboratory results of Chu et al. (1992) that show wind sea slope is preserved during interactions between wind sea and swell.

Using the 3-h average wind acceleration as a crude indicator of wind trend, it was found that the dependence of eg²/U⁴ on U/c_p is significantly influenced by wind history. At a given wave age, wind seas are more developed during falling winds due to the recent history of high winds, and they are less developed during rising winds for the opposite reason. This effect is not accounted for in empirical process models for air–sea gas flux (Jähne 1990), acoustic reverberation (Ogden and Erskine 1994), and ambient noise (Wenz 1962; Cato et al. 1995). It is likely that accounting for wind history effects would improve the accuracy of these models.

The total rate of wave energy dissipation (ε_t) was estimated from the Gulf of Alaska wind sea partitions using concepts from the Phillips (1985) equilibrium range theory [Eq. (24)–(27)]. Equilibrium ranges of directional wave spectra were isolated using a wave-age-dependent partitioning approach with an α-threshold criterion for the removal of young swell components near the wind sea peak. Furthermore, the I(p) spreading functions in the Phillips theory have been replaced by directional spread indicators obtained from the wave observations.

Remote sensing techniques, wave buoy arrays, and regional wave modeling efforts are continually being refined so that soon high-quality directional wave spectra will be routinely available on a global scale (Beal 1991). It is conceivable that air–sea process models for such mechanisms as gas flux, radar backscatter, underwater ambient noise, and near-surface acoustic reverberation will soon experience significant improvements due to the use of surface wave spectral parameters in place of external forcing parameters. Since wave-related processes are dynamically linked to intrinsic properties of the wave field (not necessarily to properties of the external forcing on the waves from winds, currents, bathymetry, etc.), use of wave parameters should improve model performance.