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    Wind wave development results of IntOA experiment (García-Nava et al. 2009; Ocampo-Torres et al. 2011) and comparison with the empirical fetch-limited wave growth functions scaled by (a) U10 and (b) u*. The empirical functions for U10 scaling (H04) and conversion to u* scaling (Hwang 2006) are derived from the BHDDB dataset (Burling 1959; Hasselmann et al. 1973; Donelan et al. 1985; Dobson et al. 1989; Babanin and Soloviev 1998). The BHDDB data are shown in the background. The range of x# in the IntOA data is between 523 and 4751.

  • View in gallery

    As in Fig. 1, but for the field data of GOTEX (Romero and Melville 2010a). The range of x# in GOTEX is between 208 and 38315. Corrections applied to spatial measurements are discussed in section 4b.

  • View in gallery

    Wind wave development results expressed in terms of (a) σ#(ω#) and (b) σ**(ω**). In these representations, the wind fetch is not needed and many more datasets can be assembled for comparison. The datasets illustrated here include BHDDB, IntOA, and GOTEX described in Figs. 1 and 2, as well as DMAJ (Donelan 1979; Merzi and Graf 1985; Anctil and Donelan 1996; Janssen 1997), T96, and H04.

  • View in gallery

    Wind and wave parameters in the IntOA experiment relevant to this study: (a) wind speed, (b) friction velocity, (c) peak wave periods of wind sea and swell, (d) significant wave heights of wind sea and swell, and (e) fetch. Five wind events are highlighted for detailed discussion in section 4a.

  • View in gallery

    The temporal variation of the wind speed dependence of (a) σ** (x**), (b) ω** (x**), (c) σ** (ω**), (d) Hsw(U10), (e) Tpw(U10), and (f) u*(U10) for the quasi-steady case (event 2). The starting, peak, and end of the event are shown with a star, triangle, and circle, respectively.

  • View in gallery

    As in Fig. 5, but for an unsteady case (event 5).

  • View in gallery

    Wind wave development results, (a) σ#(x#) and (b) σ** (x**), for the four groups of IntOA data: growing (GS) and decaying (DS) phases of quasi-steady case (event 2), and accelerating (AU) and decelerating (DU) phases of unsteady cases (events 1, 3, 4, and 5).

  • View in gallery

    As in Fig. 7, but for (a) ω#(x#) and (b) ω** (x**).

  • View in gallery

    As in Fig. 7, but for (a) σ#(ω#) and (b) σ** (ω**).

  • View in gallery

    The probability density functions of (a) , (b) , (c) , (d) , (e) , and (f) for the four groups of IntOA data: growing (GS) and decaying (DS) phases of the quasi-steady case (event 2) and accelerating (AU) and decelerating (DU) phases of the unsteady cases (events 1, 3, 4, and 5).

  • View in gallery

    (a) A comparison of the spectral peak wavenumber calculated through the dispersion relation using the peak component of a frequency spectrum from temporal measurement F(f) and the peak component of the corresponding wavenumber spectrum from spatial measurement S(k). The computations are based on the DP87 spectrum model and include both considerations with and without Doppler frequency shift. (b) As in (a), but showing the IntOA wave array data processed to obtain both frequency and wavenumber spectra.

  • View in gallery

    Comparison of (a) C10 and (b) u**, of IntOA and GOTEX measurements. The sensor height is 6.5 m in IntOA and 40 or 50 m in GOTEX. Data points with peak wavelength less than 40 and 80 m in GOTEX are highlighted. A decreasing trend of GOTEX wind stress measurements toward shorter wavelengths is detectable. Corrected results for GOTEX using the similarity relation of friction coefficient (A6) are also illustrated.

  • View in gallery

    Comparison of (a) σ** (x**) and ω** (x**) and (b) σ** (ω**) of the GOTEX measurements using corrected and uncorrected u**.

  • View in gallery

    The probability density functions of (a) , (b) , (c) , (d) , (e) , and (f) for the IntOA and GOTEX measurements.

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    Field measurements showing a decreased wind stress for a give wind speed in mixed sea in comparison to that in wind sea. The results are shown in term of (a) u*(U10) and (b) C10(U10), which is the normalized wind stress in the present context. More detailed discussion of the four groups of IntOA data is given in section 4a.

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Observations of Wind Wave Development in Mixed Seas and Unsteady Wind Forcing

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  • 1 Remote Sensing Division, Naval Research Laboratory, Washington, D.C.
  • | 2 Departamento de Oceanografía Física, Centro de Investigación Científica y de Educación Superior de Ensenada, Ensenada, Mexico
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Abstract

Theoretical study and experimental verification of wind wave generation and evolution focus generally on ideal conditions of steady state and quiescent initial background, of which the ideal fetch-limited wind wave growth is an important benchmark. In nature, unsteady winds and swell presence are more common. Here, the observations of wind wave development in mixed seas under unsteady and quasi-steady wind forcing are presented. With reference to the ideal fetch-limited growth functions established under steady wind forcing in the absence of swell, the analysis shows that the wind-steadiness factor impacts wave growth. The wind wave variance in mixed sea is enhanced in both accelerating and decelerating phases of an unsteady wind event, with a larger enhancement in the accelerating phase than in the decelerating phase. Spatial and temporal wind wave measurements under similar environmental conditions are also compared; the quantifiable differences in the wave development are attributable to the wind-steadiness factor. Coupled with the empirical observation that the average wind stress is decreased in mixed sea, these results suggest that wind wave generation and development are more efficient in mixed sea than in wind sea. Possible causes include (i) oscillatory modulation of surface roughness increases air–sea exchanges, (ii) background surface motion reduces energy waste for cold start of wind wave generation from a quiescent state, and (iii) breaking of short waves redistributes wind input and allows more of the available wind power to be directed to the longer waves for their continuous growth.

Naval Research Laboratory Contribution Number JA/7260-11-0061.

Corresponding author address: Dr. Paul A. Hwang, Remote Sensing Division, Naval Research Laboratory, 4555 Overlook Avenue SW, Washington, DC 20375. E-mail: paul.hwang@nrl.navy.mil

Abstract

Theoretical study and experimental verification of wind wave generation and evolution focus generally on ideal conditions of steady state and quiescent initial background, of which the ideal fetch-limited wind wave growth is an important benchmark. In nature, unsteady winds and swell presence are more common. Here, the observations of wind wave development in mixed seas under unsteady and quasi-steady wind forcing are presented. With reference to the ideal fetch-limited growth functions established under steady wind forcing in the absence of swell, the analysis shows that the wind-steadiness factor impacts wave growth. The wind wave variance in mixed sea is enhanced in both accelerating and decelerating phases of an unsteady wind event, with a larger enhancement in the accelerating phase than in the decelerating phase. Spatial and temporal wind wave measurements under similar environmental conditions are also compared; the quantifiable differences in the wave development are attributable to the wind-steadiness factor. Coupled with the empirical observation that the average wind stress is decreased in mixed sea, these results suggest that wind wave generation and development are more efficient in mixed sea than in wind sea. Possible causes include (i) oscillatory modulation of surface roughness increases air–sea exchanges, (ii) background surface motion reduces energy waste for cold start of wind wave generation from a quiescent state, and (iii) breaking of short waves redistributes wind input and allows more of the available wind power to be directed to the longer waves for their continuous growth.

Naval Research Laboratory Contribution Number JA/7260-11-0061.

Corresponding author address: Dr. Paul A. Hwang, Remote Sensing Division, Naval Research Laboratory, 4555 Overlook Avenue SW, Washington, DC 20375. E-mail: paul.hwang@nrl.navy.mil

1. Introduction

In the ocean environment, the presence of surface gravity waves is one of the most conspicuous phenomena. The waves grow from absorbing the energy and momentum of the forcing wind. Nonlinear wave–wave interaction transfers energy from the spectral peak region to both lower and higher frequencies and broadens the wave spectrum. During the growth process, waves also become steeper and eventually break with accompanying energy dissipation. The wind input, breaking dissipation, and nonlinear wave–wave interaction are the three most important source terms governing the dynamics of the surface gravity wave evolution (e.g., Hasselmann et al. 1973; Phillips 1977, 1985; Komen et al. 1994; Janssen 2004). The details of the source functions are still active research subjects, and one of the methods to gauge the performance of numerical or theoretical wave models is to compare the model results with fetch- or duration-limited wave growth functions (e.g., Komen et al. 1984, 1994; Janssen et al. 1994; Janssen 2004; Ardhuin et al. 2007; Romero and Melville 2010b). Because the duration-limited condition occurs rarely in the natural environment, by far the fetch-limited growth research contributes the most to the benchmark database. Over the last few decades, many fetch growth functions have been proposed (see, e.g., reviews by Hasselmann et al. 1973; Donelan et al. 1985; Kahma and Calkoen 1994; Young 1999). The experimental data suggest that the fetch-limited growth can be expressed in power-law functions, but the coefficients of the power-law functions vary with the atmospheric stability condition and the range of wind fetch coverage in the data. Hwang (2006) presents an analysis of the wind wave growth functions with three different scaling wind velocities for both fetch- and duration-limited conditions under near-neutral atmospheric stability. The growth functions are expressed as power-law functions with proportionality coefficients and exponents dependent on the dimensionless fetch or duration. A brief summary of the analysis is given in section 2, with more details provided in the appendix.

In principle, fetch- or duration-limited condition implies stationarity in the wind field and quiescent surface condition prior to the beginning of wind events. Thus, the majority of fetch- or duration-limited experiments have been carried out in nearshore environments or enclosed water bodies such as reservoirs, lakes, or bays. In the open ocean, wind fields are rarely stationary and background surface motions induced by swell are the norm rather than the exception. Although wind wave development in the general open ocean conditions appears to follow similar wave growth functions as observed in the more ideal fetch- or duration-limited configurations, it is uncertain whether the impact of mixed sea and unsteady wind is to enhance or impede wind-generated wave development. (For brevity, we use the term “wind sea” or “simple wind sea” for the wind-generated waves in the absence of swell, in contrast to the term “mixed sea.” The wind-generated waves represent the wind-sea portion of the wave spectrum in both wind-sea and mixed-sea conditions.)

In this paper, we present an analysis of open ocean wind and wave observations in the presence of background swell and under unsteady and quasi-steady wind forcing. Data from two recent field experiments conducted in the Gulf of Tehuantepec are used for the analysis, the Gulf of Tehuantepec Air–Sea Interaction Experiment (IntOA) (García-Nava et al. 2009; Ocampo-Torres et al. 2011) and the Gulf of Tehuantepec Experiment (GOTEX) (Romero and Melville 2010a). The regional coastline runs approximately in the east–west orientation (see, e.g., García-Nava et al. 2009, Fig. 1; Romero and Melville 2010a, Fig. 2; Ocampo-Torres et al. 2011, Fig. 1), and in the winter strong mountain gap winds from the north (locally known as the Tehuanos) are very common, so the geometry is conducive to fetch-limited wave growth, although the wind field may not be ideally uniform or steady. The IntOA measurements are from an air–sea interaction spar (ASIS) buoy moored at 22 km offshore at 60-m water depth over a period of about 2 months with about 1 month’s worth of simultaneous wind and wave data. The GOTEX wind and wave measurements are obtained from sensors on an aircraft that can provide more than 500-km fetch coverage within less than 2 h (the nominal aircraft speed is 100 m s−1); a total of six flight tracks of various fetch distances are reported. The south side of Gulf of Tehuantepec is open to the Pacific Ocean and swell is a constant presence, so, during the Tehuano events, the surface wave condition can be represented by an almost ideal bimodal frequency spectrum with locally generated sea propagating against swell, although multiple swell components are also observed sometimes. The complete set of the IntOA wave spectra used in this study have been presented in Fig. 4 of García-Nava et al. (2009); most of the spectra are distinctly bimodal.

The temporal history of a typical Tehuano event is characterized by a sharp increase of wind speed, reaching to more than 20 m s−1 in some cases, and then followed by a similarly sharp drop of the wind speed. The event may last from about one day to several days. The wind spreads out from the mountain gap like a point source into the gulf and creates a rather inhomogeneous wind field both vertically and horizontally: for example, see the numerical simulation presented in Romero and Melville (2010b, Fig. 3). Despite the obvious deviation from the ideal conditions for fetch-limited wave growth (quiescent background and steady wind forcing), the IntOA and GOTEX measurements show similar power-law dimensionless fetch dependence of wind wave variance and peak frequency as those obtained in more ideal wind seas in the absence of swell, but with some measurable differences (section 3). There are also discernible differences between the spatial and temporal measurements. Our analysis indicates that the differences may be connected to the steadiness factor of the measured wind condition. Overall, the investigation leads to the conclusion that wind wave development is more efficient in mixed sea than in wind sea. The details of the analysis are discussed in section 4. Finally, a summary is given in section 5.

2. A brief summary of the empirical fetch-limited growth functions

The fetch-limited wind wave development can be expressed with the following two power-law functions relating the dimensionless wave variance and spectral peak frequency to wind fetch (appendix),
e1
where , , , σ is the root-mean-square (rms) surface elevation of the wind-sea portion of the wave spectrum, g is the gravitational acceleration, U10 is the neutral wind speed at 10-m elevation, ωp is the peak frequency of the wind-sea portion of the wave spectrum, and x is the wind fetch. (Because the properties of swell cannot be related to the local wind condition, in the studies of wind wave generation and development, it is understood that σ2 and ωp are derived from the wind-sea portion of the wave spectrum. A subscript w may be added for wind sea if distinction is necessary. The corresponding quantities of the swell components are identified with a subscript s.) The coefficients and exponents of the power-law functions are fetch dependent,
e1b
with α0 = −17.6158, α1 = 1.7645, α2 = −0.0647, β0 = 3.0377, β1 = −0.3990, and β2 = 0.0110, based on the analysis of an assembly of five field datasets obtained in near-neutral atmospheric stability and steady wind conditions [Burling 1959, Hasselmann et al. 1973, Donelan et al. 1985, Dobson et al. 1989, and Babanin and Soloviev 1998 (BHDDB)]. Further details are given in the appendix.
The growth function can also be expressed as power-law functions without the fetch factor by combining and , which yields
e2a
with the coefficients
e2b
The coefficients R and r can also be obtained from direct fitting of , which gives (Hwang 2005b, appendix A)
e2c
with γ0 = −6.1384, γ1 = −2.4019, and γ2 = −0.6102. Computed results using (2b) and (2c) are very similar, and either set of the R and r expressions can be chosen for convenience. The results presented in this paper are based on (2b). Similar expressions of the wave growth functions with u* as the scaling velocity can be derived from taking into account the similarity relation of the surface friction coefficient from wavelength scaling (Hwang 2006) as outlined in the appendix. In what follows, a prime (′) is used to denote dimensionless parameter with either U10 or u* scaling, and subscripts # and ** distinguish scalings with U10 and u*, respectively.

Physically, σ′ represents the surface wave energy in dimensionless form, and ω′ can be regarded as the surface wind forcing in dimensionless form, so (2) can be interpreted as a similarity relation of air–sea energy and momentum exchange through surface waves. In the next section, all three forms of the wave growth functions, , , and , are used to investigate the effect of mixed sea and unsteady wind on the development of wind-generated waves in the open ocean.

3. Field observations of wave development in the open ocean

Here, we study the wind wave development functions in the open ocean conditions (unsteady winds and mixed seas) using the more ideal fetch-limited growth functions as reference. As mentioned in section 1, data from two field experiments conducted in the Gulf of Tehuantepec are used in this analysis. In one experiment (IntOA), the wind and wave parameters are measured with in situ sensors mounted on a spar buoy stationed at about 22 km from the coast. The duration of the experiment is from 22 February to 24 April 2005, and several high wind events are encountered (García-Nava et al. 2009; Ocampo-Torres et al. 2011). The atmospheric stability is near-neutral based on the bulk Richardson number, , where z is sensor height (6.5 m) and Ta and Tw are air and water temperatures (in Kelvins). In the IntOA dataset, Ta > Tw; the maximum Ri is 0.024; and 362 out of the total 396 data have |Ri| < 0.01, which is classified as neutral condition according to Donelan (1990). In the other experiment (GOTEX) (Romero and Melville 2010a), wind and wave measurements are obtained with sensors carried on an airplane traveling at a nominal speed of 100 m s−1; the range of the covered fetch is from 6 to 509 km. The experiment is conducted from 17 to 28 February 2004 during two high wind events. The atmospheric stability during GOTEX is expected to be also near neutral.

Figure 1 shows the IntOA results of and . The minimum and maximum fetch of the dataset are 21.4 and 27.6 km, and, with U10 between 7.1 and 20.3 m s−1, the range of x# is between 523 and 4751. For comparison, the BHDDB results are superimposed in the background. The distribution and scatter in the IntOA and BHDDB datasets are very similar. In Fig. 1a, the dimensionless parameters are scaled with U10. The dashed curve represents the best-fit function (1) of the BHDDB data (Hwang and Wang 2004, hereafter H04). In Fig. 1b, the dimensionless parameters are scaled with u*. The smooth curves represent the corresponding conversion of the U10 scaling functions (1) to u* scaling functions (A12) using the friction coefficient similarity relation (A6) for mixed seas, with the factor (see appendix) averaged over the wind speed range of 7–20 m s−1.

Fig. 1.
Fig. 1.

Wind wave development results of IntOA experiment (García-Nava et al. 2009; Ocampo-Torres et al. 2011) and comparison with the empirical fetch-limited wave growth functions scaled by (a) U10 and (b) u*. The empirical functions for U10 scaling (H04) and conversion to u* scaling (Hwang 2006) are derived from the BHDDB dataset (Burling 1959; Hasselmann et al. 1973; Donelan et al. 1985; Dobson et al. 1989; Babanin and Soloviev 1998). The BHDDB data are shown in the background. The range of x# in the IntOA data is between 523 and 4751.

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

In general, with the u* scaling, the dimensionless variance is mostly higher in the IntOA data than the reference BHDDB average, whereas the dimensionless frequency of the IntOA data is lower than the BHDDB reference. These results signify that, in the mixed sea (wind against swell), higher wave energy and longer wave period are observed (in comparison to the more ideal fetch-limited condition). More significantly, the enhanced wave development in the mixed-sea data (IntOA) is achieved with a decreased wind stress: comparing to the simple wind-sea condition represented by the Donelan (1979), Merzi and Graf (1985), Anctil and Donelan (1996), and Janssen (1997) (DMAJ) dataset, an overall decrease of about 20% in the friction coefficient (and wind stress) for the same wind speed is found in the mixed-sea data (Fig. A1; see also García-Nava et al. 2009, section 5.2). Further discussion of the enhanced wave development of IntOA data is presented in sections 4a and 4c.

The GOTEX is conducted in the same region of IntOA under similar Tehuano events but with airborne sensors covering a much broader range of the wind fetch; the duration of the experiment is between 17 and 28 February 2004 (Romero and Melville 2010a). The wind fetch in the airborne data is from 6 to 509 km, the wind speed is from 11.4 to 20.4 m s−1, and the range of x# is between 208 and 38315. The average aircraft speed is about 100 m s−1, so a full flight track is typically completed in less than 2 h. The results of fetch development are shown in Fig. 2. The rate of growth is fetch dependent as in the BHDDB data, but the slowing down of wave growth in large fetches seems to be less prominent in the GOTEX data, especially when expressed in u* scaling. Overall, the wave growth in GOTEX is very similar to that in the BHDDB data as well as the derived empirical growth functions.

Fig. 2.
Fig. 2.

As in Fig. 1, but for the field data of GOTEX (Romero and Melville 2010a). The range of x# in GOTEX is between 208 and 38315. Corrections applied to spatial measurements are discussed in section 4b.

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

The airborne sensing produces spatial measurements yielding the wavenumber spectrum rather than the frequency spectrum; special care is needed to make use of the empirical growth functions obtained from frequency spectrum processing, because the spectral peak components of wavenumber and frequency spectra are not identical (Plant 2009). Also, the wind stress data in the airborne experiment are measured at an elevation of 40 or 50 m; some underestimation of the wind stress is noticed. The frequency–wavenumber spectrum peak conversion and wind stress correction, which have been applied to the results in Fig. 2, are discussed in more detail in section 4b.

Figure 3 shows the wave development in terms of the dimensionless wave variance as a function of dimensionless peak frequency . Because this representation of the wave development function is independent of the wind fetch, many more datasets can be compared. Illustrated in the figure include the mixed-sea results of IntOA and GOTEX as well as measurements from 11 wind-sea-dominant field experiments, BHDDB, DMAJ, Terray et al. (1996, hereafter T96), and H04. All data show very good agreement with the empirical growth curves developed with the BHDDB data, although the experimental conditions of IntOA and GOTEX deviate considerably from the ideal fetch- or duration-limited growth conditions. Interestingly, there are measurable differences between the results of IntOA and GOTEX. Given the significant similarities in the environmental wind and wave conditions between the two experiments—particularly the unsteady northerly wind events (Tehuanos) spreading out into the Gulf of Tehuantepec that opens to the Pacific for swell propagating from south to the region—the results may reflect factors such as temporal versus spatial measurements and the wind variations within the sampling periods as obtained from the two modes of sensing the environment (spatial and temporal, which represent the typical difference between remote sensing and in situ observations). These issues are further discussed in section 4b.

Fig. 3.
Fig. 3.

Wind wave development results expressed in terms of (a) σ#(ω#) and (b) σ**(ω**). In these representations, the wind fetch is not needed and many more datasets can be assembled for comparison. The datasets illustrated here include BHDDB, IntOA, and GOTEX described in Figs. 1 and 2, as well as DMAJ (Donelan 1979; Merzi and Graf 1985; Anctil and Donelan 1996; Janssen 1997), T96, and H04.

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

4. Discussion

a. Effects of mixed sea and unsteady wind

As mentioned in section 1, several high wind events are encountered during the IntOA experiment. Five of the prominent events are identified in Fig. 4, which shows the time series of wind speed, friction velocity, the peak wave period and significant wave height computed from the wind-sea and swell portions of the wave spectrum, and fetch. Event 2 is quasi steady with wind speed remaining mostly within 12 ±2 m s−1 over a duration of about 36 h. Events 1, 3, 4, and 5 are typical Tehuanos: the local mountain gap winds from north lasting from just over one day to several days. The wind speed during the first part of each Tehuano event is generally fast accelerating to a peak wind speed that may exceed 20 m s−1. During the second part of each event, the wind decelerates at a somewhat slower rate. The swell conditions of the five events differ slightly. The average period Tps, significant wave height Hss, and steepness KAs of the swell are listed in Table 1. As shown in Figs. 4c,d, the wave period of the swell component display a slowly decreasing trend during each Tehuano event, but the swell height may have the trend of increase (event 1), steady (event 2), or decrease (events 3–5).

Fig. 4.
Fig. 4.

Wind and wave parameters in the IntOA experiment relevant to this study: (a) wind speed, (b) friction velocity, (c) peak wave periods of wind sea and swell, (d) significant wave heights of wind sea and swell, and (e) fetch. Five wind events are highlighted for detailed discussion in section 4a.

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

Table 1.

Swell condition of the five events identified in Fig. 4 and number of data in the accelerating and decelerating phases of unsteady events (events 1, 3, 4, and 5) and growing and decaying phases of the quasi-steady event (event 2).

Table 1.

To derive a better understanding of the wave development in the open ocean, we examine the temporal variation of dimensional and dimensionless variables. The analysis of two events is described here. In the top panels of Fig. 5, σ**(x**), ω** (x**), and σ** (ω**) of the growing and decaying phases of event 2 (quasi-steady condition) are shown with connected curves to illustrate the temporal variation. A star indicates the starting condition, a triangle represents the condition at the maximal wind speed that separates the growing and decaying phases of the event, and a circle denotes the end of the event. Because of the mixed-sea condition, the starting state (indicated by the star) may deviate considerably from the more ideal fetch-limited growth curves. As the wind event begins, the wind sea quickly adjusts to the condition similar to that of the ideal fetch-limited wave growth functions. In the bottom panels of Fig. 5, the wind speed dependence of Hsw, Tpw, and u** are depicted. Because the wind fetch is essentially constant, from the ideal fetch-limited wave growth relation, the wind dependence of Hsw, Tpw, and u** should be deterministic. In reality, considerable differences are observed in the rates of change with U10 for the three variables. In particular, Hsw and Tpw show the hysteresis characteristic with a much slower rate of change with respect to wind speed in the decaying phase than in the growing phase. In contrast to the response of Hsw and Tpw to the changing U10, the correlation between u** and U10 is much closer to instantaneous adjustment although some hysteresis effect remains. This fast response between u* and U10 indicates that the length scale of the surface roughness relevant to wind drag is much shorter than the energetic portion of the wind wave spectrum, of which the significant wave height Hsw is evaluated, and Hsw has been shown to have significant lags in response to wind change. Because of the unsteady and inhomogeneous nature of the wind field produced by the point source of the Tehuano mountain gap wind, propagation lag may be an important factor contributing to this observation. Estimating from the group velocity of the peak wind wave component, a propagation lag of about 2–4 h can be expected for the 22-km wind fetch.

Fig. 5.
Fig. 5.

The temporal variation of the wind speed dependence of (a) σ** (x**), (b) ω** (x**), (c) σ** (ω**), (d) Hsw(U10), (e) Tpw(U10), and (f) u*(U10) for the quasi-steady case (event 2). The starting, peak, and end of the event are shown with a star, triangle, and circle, respectively.

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

Figure 6 displays the variations of σ**(x**), ω** (x**), σ** (ω**), Hsw(U10), Tpw(U10), and u*(U10) for an unsteady case (event 5). Compared to event 2, the swell period is shorter and the wave height is smaller, but the steepness is higher (Table 1). Similar to the quasi-steady case, the general behavior of wave development follows a trend close to the more ideal fetch-limited wind wave growth functions. The hysteresis property of wave height, wave period, and wind stress in response to wind speed change between accelerating and decelerating phases is also similar to the quasi-steady case. However, the magnitude of the dimensionless wave variance is generally larger in the unsteady case than that in the quasi-steady case. In other words, the field data shows that, given the same dimensionless wind forcing condition, which is represented by or , the wave height in unsteady wind fields is greater than that in steady wind fields.

Fig. 6.
Fig. 6.

As in Fig. 5, but for an unsteady case (event 5).

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

Figures 79 show the expanded views of the wave development functions, , , and , for the results of four groups of data: growing and decaying phases of event 2 (quasi-steady case) and accelerating and decelerating phases of events 1, 3, 4, and 5 (unsteady cases). For brevity, the four groups are identified as growing steady (GS), decaying steady (DS), accelerating unsteady (AU), and decelerating unsteady (DU), respectively. Using the BHDDB fitted curves as a reference, the values of and are generally larger in the AU, DU, and DS groups, with the enhancement factor as high as about 4 but mostly between 1 and 2. The average magnitude of the GS group is very close to the reference BHDDB result, but the scatter seems to be larger than the other three groups. This result is probably related to the much larger swell height of event 2 (Fig. 4d and Table 1). For , the average magnitude of all four groups is about the same as the wind sea (BHDDB) results, but the magnitude of at larger is smaller in mixed sea for decelerating or decaying phase, reflecting the slower decreasing trend of wave period in the decaying or decelerating phase of a wind event (Figs. 5e, 6e).

Fig. 7.
Fig. 7.

Wind wave development results, (a) σ#(x#) and (b) σ** (x**), for the four groups of IntOA data: growing (GS) and decaying (DS) phases of quasi-steady case (event 2), and accelerating (AU) and decelerating (DU) phases of unsteady cases (events 1, 3, 4, and 5).

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

Fig. 8.
Fig. 8.

As in Fig. 7, but for (a) ω#(x#) and (b) ω** (x**).

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

Fig. 9.
Fig. 9.

As in Fig. 7, but for (a) σ#(ω#) and (b) σ** (ω**).

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

The difference of wave development in mixed sea and wind sea can be quantified by the ratios , which are , , and normalized by the corresponding values of the more ideal fetch-limited growth functions established on BHDDB. Note that these ratios of the dimensionless parameters are identical to the equivalent ratios of the dimensional quantities of , , , , , and with respect to the BHDDB references, because the computation of both measurements and references are using identical wind velocity or friction velocity and wind fetch for each datum. The ratios of the dimensionless parameters, , are the same as the ratios between the measured dimensional peak frequency and wave variance with reference to the wave growth functions derived from the more ideal fetch-limited conditions: (1) and (2) for U10 scaling and (A11) and (A12) for u** scaling. In fact, this similarity of dimensional and dimensionless analyses is also applicable to the discussions of Figs. 13 and 79 in regard to the comparison of the dimensionless and dimensional variance and peak frequency with respect to the reference growth functions.

Figure 10 shows the probability density functions (pdfs) of for the four groups of data: AU, DU, GS, and DS. As illustrated in Figs. 10a,d, the result indicates that the evolution of the spectral peak wave period follows closely the fetch-limited wave growth function of and can be predicted accurately; the mean value and standard deviation of and the corresponding values in U10 scaling for the four groups (AU, DU, GS, and DS) are listed in Table 2. The ratios for the four groups are all within 6% of unity (1 ±0.06), and most data points are distributed within a narrow region. The wave height, on the other hand, shows significant variations in response to the wind fluctuation; the mean and standard deviation of , , , and for the four groups are also listed in Table 2. The mean ratio of the dimensionless or dimensional variance in unsteady wind (AU and DU groups) ranges between 1.24 and 1.44. In quasi-steady wind, the mean ratio ranges from 0.91 to 1.05 for the growing phase and from 1.09 to 1.37 for the decaying phase. There is a conspicuous bimodal feature of the dimensionless wave variance in the pdfs of and in all four groups of data. The feature is especially prominent in the growing phase of the quasi-steady (GS) group (Figs. 10b,c,e,f) and may suggest the overshoot–undershoot characteristic of wave development on its way to the equilibrium condition. The seesaw behavior may be enhanced by small fluctuations of the quasi-steady wind field and swell modulation over the observation duration of each datum in the mixed-sea condition. Overall, for the same dimensionless fetch or inverse wave age, the wave variance in the unsteady wind field is larger than that in the quasi-steady wind condition. Further quantification of various contributing physical mechanisms such as propagation lag, wind field inhomogeneity, wind-steadiness factor, and swell parameters would require comprehensive modeling of air–sea interaction and wind wave evolution.

Fig. 10.
Fig. 10.

The probability density functions of (a) , (b) , (c) , (d) , (e) , and (f) for the four groups of IntOA data: growing (GS) and decaying (DS) phases of the quasi-steady case (event 2) and accelerating (AU) and decelerating (DU) phases of the unsteady cases (events 1, 3, 4, and 5).

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

Table 2.

Mean and standard deviation of the ratios for the four groups of IntOA data: growing (GS) and decaying (DS) phases of quasi-steady case (event 2) and accelerating (AU) and decelerating (DU) phases of unsteady cases (events 1, 3, 4, and 5).

Table 2.

b. Spatial measurements

1) Spectral peak component

The wave data of Romero and Melville (2010a) are based on wavenumber spectrum obtained from spatial measurements using an airborne scanning lidar system. Plant (2009) emphasizes that the spectral peak component of a wave field obtained from its wavenumber spectrum is different from the one derived from its frequency spectrum. Because spatial measurements of surface waves are becoming more prevalent as imaging radars are now frequently used in coastal and spaceborne ocean sensing, the topic of spatial versus temporal surface wave measurements deserves some attention.

In Plant’s (2009) discussion, the wavenumber spectrum is given by F(k) and the frequency spectrum is F(f), where f = ω/2π and is the wave variance. More commonly, kp of a wavenumber spectrum obtained from 3D spatial measurements is evaluated from the directionally integrated 1D spectrum, (Hwang et al. 2000; Romero and Melville 2010a). A more consistent comparison is between S(k) and F(f). For deep-water waves, F(f) ~ fS(k); thus, the spectral peak component in the frequency spectrum is expected to shift to a shorter-scale wave in comparison to the spectral peak component of the wavenumber spectrum. We can evaluate the difference of the frequency and wavenumber spectral peak components using analytical spectrum models. For example, based on the wind-sea spectrum model of Donelan et al. (1985) and Donelan and Pierson (1987, hereafter DP87), the results with and without considering Doppler frequency shift caused by surface drift current (assumed to be 3% of wind speed) are shown in Fig. 11a. An empirical factor of is obtained for adjusting the dispersion calculation when the spectrum peak is obtained with the spatial measurements (wavenumber spectrum) in order to make use of the growth functions established from frequency spectrum measurements.

Fig. 11.
Fig. 11.

(a) A comparison of the spectral peak wavenumber calculated through the dispersion relation using the peak component of a frequency spectrum from temporal measurement F(f) and the peak component of the corresponding wavenumber spectrum from spatial measurement S(k). The computations are based on the DP87 spectrum model and include both considerations with and without Doppler frequency shift. (b) As in (a), but showing the IntOA wave array data processed to obtain both frequency and wavenumber spectra.

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

The wave gauge array on the ASIS allows the derivation of both the frequency spectrum using the conventional time series spectral analysis of single-gauge data and the wavenumber spectrum using the wavelet analysis combining multiple-gauge measurements (Donelan et al. 1996). Figure 11b shows the comparison of spectral peaks derived from the frequency and wavenumber spectral analyses of the IntOA data. The empirical factor of obtained from the wind-sea spectrum model yields some agreement, although field data of wavenumber spectra obtained with the wavelet analysis show considerable scatter.

In this paper, the multiplication factor of 1.25 has been applied to kp of GOTEX wavenumber spectrum measurements for the calculation of ωp using the dispersion relation. The factor 1.25 is based on the numerical simulations of wavenumber and frequency spectra using the analytical wave spectrum model as shown in Fig. 11a. We are not as confident on the fidelity of the wavenumber spectrum in the field data, which is derived from wavelet method using measurements from a small number of wave gauges rather than from the true spatial mapping of the wave field.

2) Wind stress measurement

When the wind stress data of IntOA and GOTEX are compared (Fig. 12), obvious differences between the two datasets are noticed although the wind stresses in both experiments have been derived using the eddy correlation method. Particularly, the GOTEX wind stress is generally smaller than that of the IntOA for a given wind speed: the difference is about 20%–30% in C10 or 10%–15% in u*. Several factors can contribute to the observed differences. Romero and Melville (2010a) obtain the wind stress data from their lowest flights, mostly at 40-m altitude, except for one flight at 50 m. Although the high sensor elevation may not impact the accuracy of the mean wind speed [see their Fig. 3, showing excellent agreement between the airborne measurements and Quick Scatterometer (QuikScat) U10], the influence on wind stress measurements is more difficult to assess.

Fig. 12.
Fig. 12.

Comparison of (a) C10 and (b) u**, of IntOA and GOTEX measurements. The sensor height is 6.5 m in IntOA and 40 or 50 m in GOTEX. Data points with peak wavelength less than 40 and 80 m in GOTEX are highlighted. A decreasing trend of GOTEX wind stress measurements toward shorter wavelengths is detectable. Corrected results for GOTEX using the similarity relation of friction coefficient (A6) are also illustrated.

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

The sample rate of the wind sensor is 25 Hz, and each wind stress measurement is an average of 50-s data. With a nominal aircraft velocity of 100 m s−1, the sample spacing is 4 m. From the frozen eddy assumption, the effective sample rate would be fe = U10/4 Hz. With the wind speed ranging between 11.4 and 20.4 m s−1 in the dataset, the effective sample rate is between 2.9 and 5.1 Hz. We have resampled the IntOA data from 20 to 10, 5, and 2 Hz, and reprocessed u* using the resampled data. There is only up to about 5% underestimation of u* because of a lower sample rate, which is not enough to account for the observed differences between the two datasets. On the other hand, the 50-s averaging time imposes a severe limitation on resolving the low-frequency components of the wind stress spectrum. Based on the IntOA spectrum of uw′, where u is the longitudinal velocity and w is the vertical velocity, 20%–25% of stress is contained in frequencies less than 0.02 Hz. However, in aircraft measurements, 50 s represents 5-km spatial coverage, and the comparison of low-frequency resolution may not be straightforward. Using the frozen eddy assumption again, the 5-km length would correspond to 0.002 and 0.004 Hz for U10 = 10 and 20 m s−1, respectively. Based on the IntOA turbulence spectrum, the spectral energy below 0.003 Hz is negligible.

Finally, the wavelength scaling similarity relation of the ocean surface friction coefficient emphasizes the importance of the characteristic wavelength λp in the air–sea boundary layer properties (Hwang 2004; Hwang et al. 2011). Because the vertical extent of the dynamic influence of waves is approximately λp/2, we can expect that the underestimation of u* determined at 40-m level would be more severe for shorter-wavelength conditions. In Fig. 12, the GOTEX data points with λp less than 40 and 80 m are highlighted. The result shows the general trend of progressively lower estimation of u* for shorter λp at a given wind speed. It seems that the underestimation of wind stress in the GOTEX data is caused by a combination of greater sensor height and shorter integration period.

In their discussion of the wind stress data, Romero and Melville (2010a, section 3) state that

…the momentum budget on the GOTEX measurements reported by Friehe et al. (2006) showed that the flux divergence was significant near shore (nearly balanced by the pressure gradient) but small at long fetches. However, in this study, the stress divergence will be neglected, because extrapolation of wind stress between 30 to 10 m above mean sea level, based on the stress divergence estimates between 30 and 800 m reported by Friehe et al. (2006), would only increase the stress by about 10%, which is small compared to the 35% rms error of the measured stress divergence.

The analysis presented in the last paragraph suggests that, in the wave-modulated boundary layer, the vertical extent of no vertical stress divergence is further restricted by the characteristic wavelength of the wave field. This is consistent with their discussion regarding significant flux divergence near shore, where the waves are shorter.

We have used the similarity relation of wind friction coefficient (A6) established from the IntOA data to recalculate the surface wind stress using the wind speed and spectral peak frequency of the GOTEX measurements. The results are shown as stars in Fig. 12. The wave development results of σ**(x**), ω** (x**), and σ** (ω**), calculated with corrected and uncorrected u*, are displayed in Fig. 13. The main difference occurs in σ** (x**) and the small ω** portion of σ** (ω**). The GOTEX data are in very good agreement with the BHDDB reference.

Fig. 13.
Fig. 13.

Comparison of (a) σ** (x**) and ω** (x**) and (b) σ** (ω**) of the GOTEX measurements using corrected and uncorrected u**.

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

3) Comparison with temporal measurements

As shown in Figs. 2 and 3, the wave development results, , , and , obtained from airborne spatial measurements are in close agreement with the more ideal fetch-limited wave growth functions derived from the BHDDB data. It is reasonable to assume that the experimental conditions of BHDDB are under the steady wind condition because the researchers of fetch-limited wind wave growth experiments usually use the wind steadiness as one of the primary screening factors for excluding unfavorable conditions during data processing. The environmental conditions between IntOA and GOTEX, conducted in the same region under Tehuano events, are expected to be very similar. The results from IntOA show clear enhancement of wave development except in the growing phase of the rare occasion of a quasi-steady condition (Fig. 10). Figure 14 shows the comparison of the pdf of for the full datasets of IntOA and GOTEX. The mean and standard deviation are listed in Table 3. In general, the dimensionless frequencies derived from both experiments are very similar to the BHDDB data, but a much larger magnitude of the dimensionless variance is found in the IntOA data. For example, the mean values of are (0.99, 1.25, 1.20) for IntOA and (1.02, 0.94, 0.98) for GOTEX (Table 3).

Fig. 14.
Fig. 14.

The probability density functions of (a) , (b) , (c) , (d) , (e) , and (f) for the IntOA and GOTEX measurements.

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

Table 3.

Mean and standard deviation of the ratios for IntOA and GOTEX.

Table 3.

The accuracy of wave measurements using the airborne scanning lidar system has been examined in great detail and the comparison studies show close agreement between the measured wavenumber spectra and wind wave equilibrium spectrum models or in situ buoy measurements (e.g., Hwang et al. 2000; Romero and Melville 2010a). In the airborne spatial measurements, the wave field is essentially a frozen version of the environment. For example, the data are collected within about 50 s for each wave field in GOTEX, and the survey of one full track of several hundred kilometers is completed within a couple of hours. The corresponding driving wind field in the airborne measurements can be expected to have much less fluctuations in comparison to the moored IntOA measurements, of which 30-min segments are used to process the data that stress the long-term (order of days) evolution of the wind and wave fields. Judging from the analysis of the unsteady and quasi-steady cases presented in section 4a, the observed difference of 20%–30% (on average) in wave variance found between IntOA and GOTEX probably reflects the wind field fluctuations in the wave generation process within the data segments (30 min versus 50 s) of the two sensing techniques.

c. Implications on wind wave development

It has been noticed for some time that there are considerable differences in the wind wave properties between mixed sea and simple wind sea. In a study of the velocity scale of breaking waves, Hwang (2009) discusses the observational evidence showing that cpw in mixed seas is in general larger than that in the absence of swell and suggests the following:

Several possible explanations for this observation include: (a) The effective surface wind stress under oscillatory modulation is higher than that in the absence of modulation, resulting in the development of higher and longer dominant waves. (b) Wave generation process becomes more “efficient” in the presence of background oscillation in a similar sense that a machine runs more efficiently after warming up; in other words, the presence of swell reduces the energy waste for cold start of the wind wave generation process from a quiescent state. (c) The premature breaking of short waves blocks the wind input to those wave components and allows more of the available wind power to be directed to the longer waves for continuous growth.

Hwang’s factor (a) is inferred from numerical simulations of airflow above water waves. For example, Gent and Taylor (1976) and Gent (1977) illustrate through numerical models that when the dynamic roughness is allowed to vary with position along the wave, the fractional rate of energy input from wind to waves (active wave generation case, with wind faster than waves) or waves to wind (waves faster than or against wind) can be significantly increased in comparison to the condition of isotropic roughness distribution. Given the empirical observations that the average wind stress is decreased in mixed sea in comparison to the wind stress in simple wind sea for a given wind speed (e.g., Fig. A1), Hwang’s factor (a) should be rephrased as follows: oscillatory modulation of surface roughness increases air–sea exchanges for a given average wind stress. To our knowledge, Hwang’s factors (b) and (c) have not been quantitatively investigated either numerically or theoretically, and we are not aware of any quantitative measurements in field or in laboratory.

With regard to the data processing of fetch-limited wind wave growth measurements, there have been suggestions that the average wind speed over the fetch should be used as the scaling velocity instead of the local wind velocity (e.g., Dobson et al. 1989). The reason is that, with the land–sea transition, the wind field is rarely uniform in space along the fetch in the region where fetch-limited experiments are conducted. Although the concept is sound, such exercise has not produced convincing improvement in reducing the data scatter (e.g., Dobson et al. 1989, Figs. 5, 6). In our analysis, we have investigated both temporal measurements of IntOA and spatial measurements of GOTEX conducted in the same geographical location and under similar wind forcing conditions. Processed identically with the local wind speed (instead of the along-fetch average) as the velocity scale, the GOTEX results show excellent agreement with the reference ideal fetch-limited growth functions (Table 3 and Fig. 14), whereas the degree of agreement between the IntOA data and the BHDDB reference varies with the wind-steadiness factor: in good agreement for quasi-steady wind forcing and with an average of about 24%–44% enhancement of wave variance in unsteady wind fields (Table 2 and Fig. 10), the overall average of the full IntOA dataset is 18%–25% variance enhancement (Table 3 and Fig. 14).

One plausible explanation of the observed discrepancy between spatial and temporal measurements is that, when the wind field is unsteady, the surface wave field is under a combination of fetch-limited and duration-limited growth. Because of the asymmetric rates of increase with wind speed of wave height and wave period in the accelerating and decelerating phases of the (short term) wind fluctuations similar to those displayed in Figs. 5 and 6, the final product of wave growth is dependent on the wind-steadiness factor. In the airborne spatial measurements, the quasi-frozen snapshots (50 s in the case of GOTEX) minimize fluctuations of both the environmental forcing conditions and the resulting wave fields, whereas in the temporal measurements the complete influence of the fluctuation effects over the duration of each data processing segment is absorbed (30 min in the case of IntOA).

In wave modeling, fetch- and duration-limited growths are not always represented equally well. Researchers frequently attribute this to errors in spectral source term balances. The present data comparison may suggest that fetch- and duration-limited growth are not as compatible as we think in realistic field environments with typically unsteady, nonuniform wind fields and complex mixed-sea conditions.

Finally, despite the delicate differences of the wave properties under steady and unsteady wind forcings in mixed seas as described in this paper, it is quite amazing that, given the nonuniformity and unsteadiness of the wind fields and prominent swell components, the IntOA and GOTEX measurements are in such a good agreement with the ideal fetch-limited growth functions (the agreement is well within a factor of 2, as shown in Figs. 10, 14 and Tables 2, 3). The results presented here are useful in a practical sense as they show from observations why wave models tuned for idealized conditions work well in a confused reality.

5. Summary

In this paper, we present an analysis of wind wave development in mixed seas under unsteady and quasi-steady wind forcing. Two recent datasets of wind and wave measurements in the Gulf of Tehuantepec under strong mountain gap winds are used for the study. The first dataset (IntOA) is based on in situ sensors moored at a fixed location and provides long-term monitoring to cover a wide range of environmental conditions. The second dataset (GOTEX) is from an airborne platform that yields rapid surveys at a speed of about 100 m s−1 that can cover an impressive fetch of 506 km in less than 1½ h. The spatial measurements represent a nearly frozen version of the environmental conditions and minimize the fluctuations of the driving winds or the resulting wave fields.

Compared with the ideal growth functions, the IntOA long-term observations show some deviation, whereas the GOTEX snapshots are in very good agreement. Several factors can contribute to the observed differences. The prominent ones include the following: (i) the Tehuano event is similar to a point source spreading out from the mountain gap into the Gulf of Tehuantepec, producing an inhomogeneous wind field with distinctively unsteady accelerating and decelerating phases, and (ii) under the Tehuano events the wave field is young wind sea propagating against swell. Quantitatively, with reference to the fetch-limited wave growth functions, for unsteady wind events the dimensionless wave variance is about 1.5–2 times higher in the accelerating phase and about 1–1.5 times higher in the decelerating phase. For a quasi-steady event, the average wave variance in the growing phase is about the same as the reference condition. As the quasi-steady starts a decreasing trend, the decay of the wave field lags that of the wind field and the dimensionless wave variance is again about 1–1.5 times the reference condition (Figs. 7, 9). For all four groups of data (AU, DU, GS, and DS), the pdf of the normalized wave variance (with respect to the reference fetch-limited conditions) displays a bimodal distribution, which reflects the overshoot–undershoot behavior as the wave system struggles to reach an equilibrium condition (Fig. 10), subjecting to the different temporal response scales (relaxation rates) from the various source terms (e.g., Hwang and Shemdin 1990). Additional contributing factors of the seesaw behavior may include swell properties and small-scale wind fluctuations.

Although the apparent wind condition is unsteady (Tehuanos), the results from the airborne spatial measurements (GOTEX) fall into the quasi-steady wind forcing category because each wave scene for data processing is acquired within about 50 s; the forcing wind field can be considered essentially frozen. The dimensionless variance in the airborne measurements is very close to the reference condition (Figs. 2, 14).

Because the effective wind stress in mixed sea is reduced by about 20% in comparison to that in the simple wind-sea conditions (Fig. A1), the results of wind wave development in quasi-steady and unsteady wind fields in both the IntOA and GOTEX experiments indicate more efficient wind wave generation in mixed seas than in simple wind seas (section 4c). Possible contributing factors include the following: (i) oscillatory modulation of surface roughness increases air–sea exchanges, (ii) background surface motion reduces energy waste for cold start of wind wave generation from a quiescent state, and (iii) breaking of short waves redistributes wind input and allows more of the available wind power to be directed to the longer waves for their continuous growth.

The wave conditions in both IntOA and GOTEX are typical of young wind waves propagating against swell. It would be of interest to apply the same analysis to other directional configurations between wind waves and swell.

Acknowledgments

This work is sponsored by the Office of Naval Research (NRL Program Element 61153N) and CONACYT (Project 62520, DirocIOA). The IntOA field experiment was supported by CONACYT (SEP-2003-C02-44718). We are grateful for the comments, suggestions, and thought-provoking questions from three anonymous reviewers.

APPENDIX

Empirical Fetch-Limited Growth Functions

The material in this appendix is provided for the sake of completeness and for the convenience of readers because it encompasses a diverse range of topics, including fetch-limited wave growth for a broad range of the dimensionless fetch, its scaling with U10 and u*, the scaling of ocean surface drag coefficient in wind sea and mixed sea, and conversion of drag coefficient from similarity relation (through wavelength scaling referenced to Uλ/2, the neutral wind speed at an elevation equal to one-half of the spectral peak wavelength) to practical application of using U10 as the reference wind speed.

From experimental observations and theoretical considerations, ocean surface waves grow longer and higher with the wind fetch under steady wind forcing. The experimental results can be organized in similarity relations when presented in dimensionless parameters
ea1
where σ2 is the variance of surface displacement, ωp is the peak wave frequency, U is a reference wind speed, x is fetch, g is gravitational acceleration, and a prime (′) on a variable denotes its dimensionless representation. When it is necessary to distinguish between normalizations with U10 and u*, the dimensionless variables are differentiated by subscripts # and **, respectively: that is, , and so on. Because the properties of swell cannot be related to the local wind condition, in the studies of wind wave generation and development, it is understood that σ2 and ωp are derived from the wind-sea portion of the wave spectrum.

Hwang (2006) presents a discussion of the fetch- or duration-limited wave growth functions scaled with three different reference wind speeds: U10, Uλ/2, and u*. A summary is given here for the fetch-limited growth functions scaled with U10 and u*. With regard to Uλ/2, although it is an important quantity serving as the free-stream velocity of wave-modulated boundary layer flows and the application of which leads to the establishment of the similarity relations of the ocean surface friction coefficient (Hwang 2004; Hwang et al. 2011), its role as a scaling velocity can be assumed by u*. In terms of convenience and practicality of application, Uλ/2 is surpassed by U10 by a long shot: Uλ/2 cannot be measured directly, because the computation of which needs wind speed measurement at a fixed elevation, wind friction velocity, and spectral peak wavenumber or wave frequency; the last item is frequently not well documented in many experiments. H04 (Fig. C2) show a comparison of duration-limited growth functions processed with U10 and Uλ/2 scalings. The data scatter of the processed results is not distinguishable between the two velocity scales, so the Uλ/2 scaling is not pursued further.

The fetch-limited growth of wind-generated waves can be expressed by
ea2
Extensive efforts have been devoted to the establishment of the fetch-limited growth functions f1 and f2 (e.g., Burling 1959; Hasselmann et al. 1973; Donelan et al. 1985; Dobson et al. 1989; Kahma and Calkoen 1992, 1994; Babanin and Soloviev 1998; Young 1999; H04). These efforts lead to the conclusion that f1 and f2 can be represented by power-law functions,
ea3a
The coefficients and exponents of the power-law functions vary in different reports, even for neutral stability conditions, mainly caused by the different range of the dimensionless fetch in the data used for analyses (see, e.g., Hwang 2006, appendix A). The wave growth can be represented in the conventional power-law functions (A3a) but with the coefficients and exponents given as functions of the dimensionless fetch, duration, or wave frequency (H04). Applying the analysis to an assembly of five field datasets (BHDDB) obtained in near-neutral and steady wind conditions, the fetch-dependent coefficients for the growth functions (A3a) with U10 scaling are
ea3b
with α0 = −17.6158, α1 = 1.7645, α2 = −0.0647, β0 = 3.0377, β1 = −0.3990, and β2 = 0.0110.
When the reference wind speed is changed from U10 to u*, the dimensionless parameters are related to each other by the friction coefficient C10,
ea4
Substituting (A4) to (A3), the growth functions in terms of u* are
ea5
It is difficult to obtain a consistent parameterization of the ocean surface friction coefficient in terms of C10. As described in Hwang (2004) and Hwang et al. (2011), a similarity relation of the ocean surface friction coefficient exists when Uλ/2 is taken as the reference wind speed. The corresponding friction coefficient is Cλ/2, and the similarity relation of the ocean surface friction coefficient can be expressed as
ea6
For wind sea, the coefficients are obtained from processing the DMAJ data: Acw = 1.22 × 10−2, acw = 0.704, A10w = 1.29 × 10−3, and a10w = 0.815. For mixed sea, the coefficients are obtained from processing the IntOA data: Acm = 4.43 × 10−3, acm = 0.380, A10m = 1.33 × 10−3, and a10m = 0.398.

Figure A1 shows the field measurements of wind stress expressed in both the wind friction velocity and the friction coefficient, which is the normalized wind stress in the present context. The DMAJ dataset represents the wind-sea conditions. The average is taken over the same range of the wave age in the IntOA data (U10/cpw between 1.3 and 3.3, where cpw is the phase speed of the wind-sea spectral peak component). The difference of wind stress between the four groups of IntOA data (described in section 4a) is rather small. Comparing to the simple wind-sea condition (represented by the DMAJ data), an overall decrease of about 20% in the friction coefficient (and wind stress) for the same wind speed is found in the mixed-sea data.

Fig. A1.
Fig. A1.

Field measurements showing a decreased wind stress for a give wind speed in mixed sea in comparison to that in wind sea. The results are shown in term of (a) u*(U10) and (b) C10(U10), which is the normalized wind stress in the present context. More detailed discussion of the four groups of IntOA data is given in section 4a.

Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-044.1

The relation between Cλ/2 and C10 can be obtained with reference to the logarithmic wind speed profile,
ea7
where z is the sensor height, z0 is the dynamic roughness, and κ = 0.4 is the von Kármán constant. Rearranging terms and setting z = λp/2, then
ea8
For simplicity, the deep-water dispersion relation, ω2 = gk, is used throughout this paper. Full expressions with general depth conditions are given in Hwang (2004, 2005a). From the logarithmic wind speed profile (A7),
ea9
With the definition of wind stress, ,
ea10
Substituting (A10) in (A5), then
ea11
The growth function can also be expressed as power-law functions without the fetch factor by combining and ,
ea12a
From (A3) and (A11),
ea12b
One advantage of using (A12) is that many more field datasets can be assembled for comparison because the fetch information is no longer needed. Furthermore, one source (fetch) of measurement errors is eliminated in the data processing of the growth function (A12).

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