Abstract
 Top of page
 Abstract
 1. Introduction
 2. Measurement Scheme
 3. Results
 4. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[1] We examine a set of 1626 highresolution frequencydirection wind wave spectra and collocated winds collected during a 7month period at a site in Currituck Sound, North Carolina, in terms of onedimensional spectral structure and directional distribution functions. The data set includes cases of shorenormal winds in broadfetch conditions as well as winds oblique to the basin geometry, with all fetches of order 10 km or less. Using equilibriumrange scaling, all onedimensional spectra have a spectral peak region, an equilibrium range of finite bandwidth following an f^{−4} slope at slightly higher frequencies, and a highfrequency tail that falls off more rapidly than f^{−4}. For shorenormal winds, spectral peakedness appears to be high and approximately constant for young waves, low and approximately constant for old waves, and steeply graded for intermediate inverse wave ages in the range 1.0 < u_{10}/c_{p} < 1.7. Equilibriumrange bandwidth seems to be narrow for young waves and increases with increasing wave age. Directional distribution functions in shorenormal winds are symmetric about the wind direction, narrow at spectral peaks, and broad at high frequencies with distinct directionally bimodal peaks, consistent with other observations. In obliquewind cases, directional distribution functions are asymmetric and directionally sheared in spectral peak regions, with peak directions aligned with longer fetch directions. At high frequencies, directional distributions are more nearly symmetric about the wind direction. Onedimensional spectra tend to have reduced spectral peakedness and highly variable equilibriumrange bandwidths in obliquewind conditions, clearly indicating a more complex balance of source terms in these cases than in the more elementary situation of shorenormal winds. These complications are not without consequence in wave modeling, as any bounded or semibounded lake or estuary will be subject to oblique winds, and current operational models do not deal well with conditions like those we find here.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Measurement Scheme
 3. Results
 4. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[2] The frequency and directional distributions of surface gravity wave spectra resulting from wind forcing are of great theoretical interest and practical importance, yet the governing processes are incompletely understood. As part of the work reported by Resio et al. [2004], a substantial set of observations of frequencydirection spectra E(f, θ) was obtained in Currituck Sound, a semienclosed water body of limited fetch on the landward side of the northern Outer Banks of North Carolina. Motivation for these measurements initially came from an attempt to resolve apparent scaling differences between spectra observed in large basins and those obtained in smaller basins, such as those found in Lake George, Australia [Young et al., 1995; Babanin et al., 2001]. For that purpose, Resio et al. [2004] used only a small part of the Currituck Sound data in their work. This paper will examine the entire data set in detail in order to determine differences and commonality among the observed spectra in this estuarine area and those reported elsewhere [Donelan et al., 1985; Birch and Ewing, 1986; Pettersson, 2004]. It is expected that these observations will also be of considerable specific value for formulating and testing hypotheses related to spectral characteristics in estuarine areas.
[3] Equilibriumrange scaling is useful in analysis of onedimensional frequency spectra in a frequency basis, E(f), or in a corresponding wave number basis F(k). Here, f is cyclic frequency, k is wave number modulus and, using linear theory and neglecting mean currents, the spectra are related by F(k) = (c_{g}/2π) E(f), where c_{g} is group velocity. Linear dispersion (2πf)^{2} = gk tanh (kd) relates f to k, given water depth d and gravitational acceleration g. Phillips [1958] introduced the original concept of a deepwater spectral equilibrium range where energy density would saturate to scales dominated by gravitational acceleration at frequencies higher than the spectral peak frequency f_{p}, and having one of the forms
where α_{5} is a dimensionless universal constant. Subsequent field studies, for example, the Joint North Sea Wave Project (JONSWAP) experiment [Hasselmann et al., 1973], failed to support the hypothesis of a constant α_{5}, and attention was focused on the possible importance of nonlinear fourwave interactions in governing energy distributions over a significant part of Phillips' equilibrium range. Scaling arguments by Toba [1972], Kitaigorodskii [1983], and Resio [1987], observations by Toba [1973], Mitsuyasu et al. [1980], Forristall [1981], and Donelan et al. [1985], and numerical studies by Resio and Perrie [1991] and Resio et al. [2001] using the nonlinear interaction theories of Hasselmann [1962], Zakharov and Filonenko [1966], Webb [1978], and Herterich and Hasselmann [1980] provide compelling arguments for a deepwater equilibriumrange spectrum spanning the approximate frequency band 1.5f_{p} < f < 3f_{p} of the form
where u is a scale velocity and α_{4} is a dimensionless universal constant. Spectra at higher frequencies may take other forms. Phillips [1985], in arguing for an E_{4} spectrum, notes that highfrequency parts of spectra used in his prior work [Phillips, 1977] still appear to have an E_{5} form. Forristall [1981] finds a transition from an E_{4} spectrum to an E_{5} spectrum such that α_{5} is approximately constant at high frequencies. As will be shown, we, too, find a break in spectral slope, but with a structure that varies considerably within the young waves and basin geometry of our observations.
[4] Resio [1987] shows that equation (2) is the deepwater asymptotic form of the more general wave number expression for the equilibriumrange spectrum, which is valid into water of finite depth. We write this general form as
where
Various forms for the scale velocity u have been proposed in the literature, including the wind speed at 10m elevation u_{10} and the windstressbased friction velocity u_{*}. Resio et al. [2004] use data from several diverse data sets to intercompare a number of possible forms for u. Among their various candidate scales, they find that a combination of the equilibriumrange scaling concept expressed by Resio and Perrie [1989] and the windspeed scale u_{λ} (defined below) derived by Resio et al. [1999] yields a good data characterization consistent with a constant α_{4}. The resulting expression for β is
where α_{4} = 0.00553 and u_{0} = 1.93 m/s. In equation (5), c_{p} is spectral peak phase speed and u_{λ} is wind speed at an elevation equal to a fixed fraction λ (= 0.065) of the spectral peak wavelength 2π/k_{p}, where k_{p} is spectral peak wave number. Resio et al. [2004] assume all winds to follow a neutrally stratified logarithmic profile having a von Karman coefficient κ = 0.41 and subject to a Charnock [1955] surface roughness z_{0} = α_{C}u_{*}^{2}/g with α_{C} = 0.015. Thus
and we note that this wind scale differs only slightly from the “critical layer” velocity derived by Miles [1993], for which λ = 0.045, given the same boundary layer structure.
[5] Figure 1, adapted from Resio et al. [2004], shows the correlation of β, estimated from data by the average 〈k^{5/2}F(k)〉 over a suitable range of equilibriumrange frequencies, with (u_{λ}^{2}c_{p})^{1/3}g^{−1/2} determined from a number of samples from a diverse set of sources. The regression line, equation (5), seems a reasonable characterization of all observations. Data used in Figure 1 were selected as representative of conditions of steady or rising wind speeds in excess of 5 m/s with reasonably steady wind direction and minimal swell. Sources span a wide range of scales from small, enclosed water bodies (Lake George, cited previously, and Currituck Sound, discussed here), to two finitedepth sites (a Waverider buoy 6 km offshore in 20 m of water and a Baylor gauge 0.5 km offshore in 8 m of water, both at the U. S. Army Engineer Field Research Facility (FRF) at Duck, North Carolina, Figure 2), to two National Data Buoy Center deep ocean sites (a 6m NOMAD buoy at their Station 41001, off the southeast U. S. coast, and a 12m discus buoy at their Station 46035 in the Bering Sea). We chose the National Data Buoy Center (NDBC) data from 15–20 storm events at each site that occurred during October, November, and December of 1999 and 2000. In this paper, we use data from these two deep ocean sites to help put the Currituck Sound observations in a broader context.
2. Measurement Scheme
 Top of page
 Abstract
 1. Introduction
 2. Measurement Scheme
 3. Results
 4. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[6] Instruments on a sled platform provided observations of directional wave spectra and local winds at a fixed location on the east side of Currituck Sound in the vicinity of the Field Research Facility (Figure 2). The sound is a body of water elongated in the north–south direction, residing between the northern North Carolina Outer Banks and the mainland to the west. The nearest ocean inlet is about 40 km to the south. As can be seen in Figure 2, the southern end of Currituck Sound opens into Albemarle Sound. Aside from this opening, Currituck Sound is bounded by land running approximately from northwest to southeast along both sides. From the sled location, the shortest fetch to the east is 0.9 km. To the west, the shortest fetch is 5.5 km. Longer fetches exist to the northwest and southwest.
[7] In the vicinity of the sled site, there is a broad channel that follows the long axis of the sound with shoal areas on the east and west sides; however, information on which this information is based is somewhat dated. A recent bathymetric survey over a rectangular area 0.5 km by 3 km with the long dimension in the east–west direction across the sound helped to improve our understanding of the local bathymetry. This survey confirmed the presence of a relatively flat area in the vicinity of the sled, with depths in the range of 1.8 to 2.4 meters. About 0.5 km east of the sled, a steep bank was found where depths rapidly shoaled to about 0.6 m (NGVD). Outside the surveyed area, a NOAA chart and local fishing lore indicate numerous shoal areas to the north and south of the sled site.
[8] The sled provided hourly observations of winds and waves from 24 October 2001 to 19 April 2002. From the great number of samples it was possible to apply rather strict editing to ensure reasonable results. For winds from the east, air passes over the rather large roughness elements of the barrier island and then over a rather short fetch to the sled location, which allows the possibility that local easterly winds may retain significant landinduced effects and not be representative of a pure airwater dynamic balance in a marine boundary layer. The bathymetric information suggests that water of nominally uniform depth exists over an arc of directions generally from the west, where, in addition, the fetch is large enough for the airwater boundary layer to be better established. Consequently, data used in Figure 1 were selected from situations in which winds arrived at the sled from within a 140° arc of directions centered on magnetic west (which is approximately normal to the ocean shoreline in Figure 2).
[9] Figure 3 shows the sled platform in a view looking west from its deployed position. A threelegged pipeframe tower with a working stage near its top holds a solar panel and battery box to provide electrical power, a weatherproof box housing multiplexing circuitry and a radio transmitter to send data to the Field Research Facility (FRF) main building via an antenna (on the short mast above the handrail), a lightning rod (on the tall mast), and an anemometer (R. M. Young model 09101 digital wind speed and direction) on the third mast. The anemometer is at a nominal elevation of 5 m above the water surface. On the far side of the pipe tower in Figure 3 is a framed structure that penetrates the water surface and holds a directional wave gauge. That gauge consists of a spatial array of nine capacitancetype wave rods of 0.6mm diameter and 3m length (Ocean Sensor Systems, Coral Spring, Florida) with internal digitization of water levels into 4,096 steps along each rod length. Static calibration of the rods showed a very linear response (r^{2} > 0.999) over the central 2 m of their lengths. The gauge support structure consists of two horizontal cruciform frames separated in the vertical by pipes at the frame ends. The wave rods pass through holes in the cruciform frames, which thus hold the tops and bottoms of the wave rods in a fixed geometry. The wave rods are not rigid but were constrained via horizontal stays (made of fishing line and attached to the vertical members of the gauge support structure) at intermediate levels along the wave rods.
[10] Two editing constraints arise from this sled design. First, the discrete steps associated with waverod sampling result in a finite noise floor in spectral estimates that must be avoided to retain valid wave data. To reduce noise contamination from this source, we eliminated from consideration any spectral density estimate that was less than ten times the digitization noise floor. This filter caused a truncation of some spectra and the complete elimination of verylowenergy spectra. The second constraint arises in cases where waves pass through one of the horizontal wavegauge stays in such a way that the stay is partially submerged. The effect is to modify the electrical path employed by the wave rods in such a way that detected wave shapes can be substantially distorted. Such cases exhibit obvious signatures in results and have been removed from our analyses.
[11] Figure 4 shows the geometry of the directional array in plan view. It has the form of two orthogonal, fiveelement linear arrays with one shared gauge. The twodimensional character of this array allows full 360° directional resolution. The design of each lineararray arm follows the guidance given by Davis and Regier [1977]. Minimum gauge spacing is 0.1 m, which corresponds to a spatial Nyquist wavelength of 0.2 m or, in deep water, an upper frequency limit of about 2.8 Hz. Maximum gauge spacing is 1.6 m. Reasonably detailed directional resolution is expected for waves several times this length, the constraint being the ability of the array to determine the crossspectral phase pattern of directionally distributed waves of a given length. Array dimensions are thus compatible with energetic wind wave frequencies and wavelengths expected to exist in Currituck Sound. The directional algorithm for this array is the iterative maximum likelihood estimator (IMLE) of Pawka [1983] adapted to two dimensions. Numerous tests by Pawka [1982], Long [1995], and others have indicated that the estimator is quite reliable in determining major directional distribution features, especially when crossspectral estimation degrees of freedom are high.
[12] Data collection consisted of synchronous, 12,288point sampling of wave rod outputs at 0.2048s intervals (about 4.88 Hz), resulting in records of about 41min duration originating at the top of every hour. The 2.44Hz Nyquist frequency is compatible with the minimum gauge spacing of the directional array. Fourier analysis with segment and band averaging yielded auto and crossspectral estimates having 120 degrees of freedom in bands of about 0.024Hz width. We averaged autospectral estimates from the nine wave rods to obtain an estimate of total variance density E(f) at each frequency. The IMLE provided estimates of the directional distribution function D(f, θ), so that the frequencydirection spectrum is
where θ is relative to sled coordinates and, in the rectangular (rather than polar) coordinate system used here, the distribution function satisfies
[13] We determined mean wind speed u_{z} and direction θ_{w} at anemometer elevation z above mean water level by vector averaging 1Hz speed and direction output over the same duration as the wave records. Mean water level varied in the sound owing to wind effects, so z was the difference of the fixed anemometer elevation above the bottom and water depth determined from the wave rod signal means. An estimate of friction velocity u_{*} was the solution of
under the assumption of a neutrally stratified, logarithmic boundary layer. Other elevationrelated wind parameters were found by replacing z in the righthand side of equation (9) with, for example, z_{10} = 10 m for the 10m wind speed u_{10} or z_{λ} = 2πλ/k_{p} for reference wind speed u_{λ}, as in equation (6).
3. Results
 Top of page
 Abstract
 1. Introduction
 2. Measurement Scheme
 3. Results
 4. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[14] There were 1626 sample spectra remaining after our editing and filtering procedures. Table 1 shows the ranges of some common spectral parameters represented by this data set. For the indicated range of u_{10}, inverse wave ages u_{10}/c_{p} are relatively large, indicating young, fetch or durationlimited waves, consistent with small characteristic wave heights H_{mo} and high peak frequencies and wave numbers. The longest spectral peak wavelength is about seven times maximum gauge spacing in the directional array, so we expect reasonable directional resolution for waves at and above this wavelength, but a degradation of resolution for k < 0.4 rad/m (wavelengths greater than ten times maximum gauge spacing), which is just below the lowest k_{p} in these data. Water depths are small, but wave scales are such that equilibrium ranges of all spectra are in deep water, as usually indicated by kd > π. Though our range of k_{p}d extends down to 1.2, waves at k ≅ 2k_{p} (for which f ≅ 1.4f_{p}, roughly the lowfrequency bound of the equilibrium range) have 2k_{p}d = 2.8 ≈ π, and so are sufficiently short still to be considered deepwater waves. However, c_{p} at these low k_{p}d are slower by about 5% than deepwater values, so there are some minor finitedepth influences in the spectral peak regions of some of our spectra. Bulk steepness H_{mo}k_{p} and relative wave height H_{mo}/d are small enough for all cases that conventional wave breaking limits are not exceeded and that fourwave interaction scales are appropriate for the equilibrium ranges of these spectra, as discussed by Resio et al. [2004].
Table 1. Parameter Ranges Associated With Currituck Sound and NDBC Buoy Data^{a}  Currituck Sound  NDBC 41001  NDBC 46035 


Number of cases  1626  212  144 
u_{10}, m/s  5.0–16.2  9.7–24.4  7.9–23.9 
H_{mo}, m  0.08–0.59  2.6–8.1  3.3–10.2 
f_{p}, Hz  0.32–0.78  0.08–0.15  0.07–0.14 
k_{p}, rad/m  0.44–2.29  0.02–0.09  0.02–0.08 
d, m  2.24–2.93  >4300  >3600 
k_{p}d  1.17–5.68   
H_{mo}k_{p}  0.10–0.38  0.12–0.28  0.10–0.28 
H_{mo}/d  0.03–0.22   
u_{10}/c_{p}  1.43–4.06  0.63–1.45  0.46–1.58 
[15] Figure 5 shows a typical directional spectrum, E(f, θ), in sled directional coordinates (for which direction increases counterclockwise from magnetic east), with waves from the southwest. The spectrum has a directionally unimodal peak, finite directional spread at all frequencies and some evidence of a directionally multimodal highfrequency tail. Aside from locally windgenerated waves, there is no other signal, which means that observations appear free of background swell. It is expected that spectral structures and parametric relationships found from the small, young waves in this venue might thus help clarify the early stages of pure windwave growth. Because the sound is nearly completely bounded, there exist a large number of slanting fetch observations, and because the sound is elongated in one direction, there exist cases of broad and narrowfetch spectra as discussed by Pettersson [2004] in her studies in the Gulf of Finland. Our observation site is to one side of the experiment basin (Figure 2), somewhat like the arrangement used by Donelan et al. [1985], and so is more suited to broadfetch observations for winds from the west than for narrowfetch situations for winds from the north or south.
3.1. Fetch and Mean Direction Considerations
[16] To investigate fetch compatibility of wind and wave scales for winds from various directions, we compare apparent fetch with maximum fetch for these observations to see if the evident origins of our wave data appear consistent with the geometry of the basin boundaries. We do not know enough about the shoal areas on the edges of Currituck Sound to specify effective fetches from actual bathymetry, but a rough estimate can be obtained from an inverse analysis based on fetchgrowth models. This procedure yields estimated apparent fetches from sledderived parameters that can be compared to maximum possible fetches derived from shoreline location. Here, we use an adaptation of the JONSWAP energyfetch relationship in the form given by Resio and Perrie [1989] as
where H_{mo} is variancebased characteristic wave height, and the JONSWAP peakfrequencyfetch relationship in the form
where, in this expression, we assume deepwater peak phase speed c_{p} = g/2πf_{p}.
[17] Figure 6a shows as symbols estimated fetch based on equation (10) as a function of mean wind direction θ_{w} for our 1626 observations. The solid line in Figure 6a is an estimate of absolute limiting fetch based on distance from the sled position to solid land along various possible wind azimuths. For winds generally from the north, east and west, waves evidently originate within the body of the sound, suggesting fetch or durationlimited growth. A finite offset from an upwind boundary is sensible for fetchlimited growth if one considers that it takes a distance of order 100 times the height of an upwind roughness element to form a new internal boundary layer governed by local roughness. The sound is lined with trees, dunes, and buildings with heights of order 10 m, so it may require about 1 km of open water for waves to begin following a fetchgrowth model. If significant shoals exist, evident fetches may be considerably smaller than distance to upwind solid land. In this context, our observed energies are not terribly out of line with the JONSWAP relationship of equation (10).
[18] Figure 6b shows estimated fetch based on equation (11) as a function of θ_{w}. In contrast with Figure 6a, the energy and frequency fetch estimates are in nominal agreement only for winds from the westerly direction. Otherwise, frequencybased fetch estimates exceed energybased fetch estimates, and, notably, the frequencybased fetch estimates consistently exceed available fetch for winds from the northeast and southeast, which indicates that peak frequency has evolved to lower values in these slanting fetch situations than would occur in the normal fetch geometry of the JONSWAP experiment. This behavior is consistent with observations by Donelan et al. [1985], who noted that peak wave directions in slanting fetches tend to align with the direction of longer fetch.
[19] To pursue this idea, we compared spectral peak mean wave direction with wind direction for our observations. Following Kuik et al. [1988], a general expression for mean wave direction within a range of frequencies f_{1} < f < f_{2}, is
To obtain a reasonably consistent definition for a mean direction of waves within the spectral peak region, we defined a peak mean wave direction _{pk} for frequencies satisfying 0.9 < f/f_{p} < 1.5. Figure 7a shows _{pk} as a function of θ_{w} for our observations. The pattern is consistent and quite clear. Winds and peak waves tend to align only for waves coming across the sound from the westerly direction. In all other cases, peak waves tend to align with directions of longer fetch. For example, winds from the northeast, θ_{w} ≈ 45°, yield waves roughly from the north, _{pk} ≈ 90°, which is approximately alongshore and a direction of greater fetch (Figure 2). Figure 7a is very much like results reported by Donelan et al. [1985].
[20] To see if wind and wave directions differed in other parts of our spectra, we defined an equilibriumrange mean wave direction _{er} using equation (12) with frequencies satisfying 1.5 < f/f_{p} < 2.1, a highfrequency mean wave direction _{hf} for frequencies f/f_{p} > 2.1, and averaged the mean wave directions in 10° bins of mean wind direction. Figure 7b shows the three binaveraged mean wave directions in terms of mean wind direction. The pattern of the peak wave direction of Figure 7a is duplicated in Figure 7b, but, notably, the highfrequency mean wave directions tend to align much more nearly with the wind, even in cases where peak wave directions deviate strongly from wind direction. This result suggests there is considerable directional shear in our obliquewind data, which we examine in more detail in section 3.3.
3.2. OneDimensional Spectra
[21] Given the significance of inverse wave age u_{10}/c_{p} in past analyses of spectral shape and the evident importance of wind direction over a given basin geometry as noted by Donelan et al. [1985] and Pettersson [2004], we initially grouped our onedimensional spectra in five classes of inverse wave age ranging from 1.5 to 4.0 with bin widths of 0.5 and 18 classes of wind direction of 20° bin widths, which collectively span a full circle. We analyzed our spectra in terms of equilibriumrange scales. The E(f) obtained from wave gauge measurements were first converted to wave number (modulus) spectra F(k) using the method outlined in the introduction, and then normalized by the equilibriumrange model of equation (3) to form
where β is given by equation (5). Each k has a corresponding f so that we averaged (k) in discrete bins of f/f_{p} to examine the distribution relative to peak frequency.
[22] Figure 8, which shows classaverage means as symbols and standard deviations as error bars in each f/f_{p} bin, indicates that there appears to be less variation in resulting class mean spectra of different inverse wave ages within a winddirection class than between direction classes. Figures 8a–8c show three different waveage class means for winds crossing the sound from the west, ∣θ_{w}∣ > 170°. Figure 8d shows the mean of all spectra, irrespective of wave age, for that winddirection class. Figures 8e–8g show waveage class means of the same inverse wave ages as those in Figures 8a–8c, respectively, for winds roughly alongshore from the south, −110° < θ_{w} < −90°. Figure 8h shows the mean of all spectra for that winddirection class. Though there appear to be some variations with wave age in each direction class, the error bars suggest the waveage distinction is relatively minor.
[23] Comparison of the two simple, directionclassaveraged spectra, Figures 8d and 8h, is of interest. The deviation of the normalized spectral value at the spectral peak, f/f_{p} = 1, above the equilibriumrange reference line, = 1, is greater in Figure 8d than in Figure 8h. This suggests a simple parameter with which to characterize the spectral peak in equilibriumrange scaling as used here and, for example, by Donelan et al. [1985] and Pettersson [2004]. Calling it relative peakedness γ_{r}, it is defined simply in our case by
[24] In Figure 8d, γ_{r} ≈ 2.5. In Figure 8h it is somewhat less, with γ_{r} ≈ 1.2. The flattened part of the spectrum that indicates the equilibrium range in Figure 8h is somewhat less than the equilibriumrange reference line, suggesting that a model for β that is based on wide fetch conditions out of the west tends to overestimate its value for this class of spectra. As we show later, there is an indication that γ_{r} might be an indication of inverse wave age in conditions not involving severely oblique winds, with smaller values indicating older spectra, somewhat like the JONSWAP peakedness parameter γ.
[25] In both Figures 8d and 8h, the spectra tend to flatten for frequencies greater than about 1.5f_{p}, indicating the existence of equilibrium ranges in each. In Figure 8h, for our shoreparallel winds, the apparent equilibrium range extends to about 3.5f_{p} with no strong evidence of a change in slope, a condition rather like the cases illustrated by Donelan et al. [1985]. On the other hand, the spectrum in Figure 8d, for winds crossing the sound roughly normal to shore, shows a distinct deviation from the equilibrium range at frequencies greater than about 2.1f_{p}. This condition suggests a shift in the balance of wave evolution source terms, possibly to a condition of increased dissipation at the high frequencies in this case.
[26] We note that Donelan et al. [1985] classified their normalized spectra in terms of an inverse wave age based on the wind component aligned with the mean peak wave direction. In our data the wind and peak wave direction are nearly the same for the shorenormal conditions in Figure 8d and differ at most by about 20° for the shoreparallel conditions of Figure 8h. Modifying the scale wind speed by the cosine of 20° does not alter the inverse wave age classifications of Figures 8e–8g to make them distinct from the classifications of Figures 8a–8c, so it does not appear that the classification scheme of Donelan et al. [1985] applies to our data. Pettersson [2004] reached a similar conclusion in regard to her observations.
[27] Because of the small variation in normalized spectra among waveage classes within a winddirection classification, we redid our analysis and simply averaged spectra in winddirection classes of the same 20° width as we used previously. Of the 18 classification bins, two had too few cases to provide a reliable average. These two classes involved our shortest fetches involving waves from the east that were too small to measure with our gauging system. Class averages for the remaining 16 classes are illustrated in Figures 9a–9p with reference to winddirectionclass bin boundaries superimposed on the map of Figure 2.
[28] Figure 9 shows that mean spectra have rather common characteristics for winds along an arc from the north counterclockwise to just south of west (Figures 9e–9j), with nominally constant relative peakedness near 2.5 and an equilibrium range that terminates with a break in slope at frequencies near 2.1f_{p}. For winds along an arc from the southwest counterclockwise to the southeast (Figures 9k–9o), the relative peakedness drops systematically from about 2.0 to about 1.0 and the break in slope appears to migrate to higher frequencies, with Figure 9o having a form very much like the normalized spectra shown by Mitsuyasu et al. [1980, Figures 9 and 10], which had much greater wave ages. A similar reduction in relative peakedness appears to occur for winds along an arc from north–northeast clockwise to northeast (Figures 9d, 9c, and 9b). In Figures 9b and 9p the equilibrium range appears to be either very poorly formed or very narrow, suggesting that the balance of dynamic processes produces a somewhat different structure for this situation (short fetches with wind directions roughly ±45° to shore). Figure 9a, representing very short fetches in which the winds are blowing almost normal to the shore (essentially out of the east), returns to somewhat the same pattern observed for winds over longer fetches out of the west.
[29] The pattern observed in our data is undoubtedly dependent on our basin geometry, the wind field over that geometry and our point of observation within that geometry. Further observations are required to demonstrate any generality in our results, but it is quite likely that observations in other bounded water bodies (e.g., Lake Ontario or Lake George) may be subject to similar variability related to the proximity of “side” boundaries within the basins.
[30] The more traditional approach to fetchlimited wave growth has focused on situations in which it has been assumed that the effects of side boundaries are negligible. Our data can be used to contribute to the broader picture of simple waveagerelated wave growth in this situation if we consider only observations from an arc of directions essentially out of the west. To test this idea, we averaged all observations that contribute to Figures 9h–9j, for winds from the westerly directions and which include data representing inverse wave ages in the range 1.6 < u_{10}/c_{p} < 3.4, to generate a youngwave spectrum that might be characteristic of the broadfetch class of observations described by Pettersson [2004]. This spectrum is shown as the top curve in Figure 10. The other spectra in Figure 10 arise from waveageclass averages of data from the NDBC buoy data described in the introduction and for which the parameter ranges are listed in Table 1. These spectra are all older than our Currituck Sound data and are from sites at a great distance from any boundary and so should be purely agedependent. There are about 200 observations in our Currituck Sound spectrum and 20 to 40 observations in each of the NDBC age classes. Figure 10 suggests a pattern of spectral development where relative peakedness γ_{r} is roughly constant with a value near 2.5 for all of our young waves, then goes through a transition to a value near unity at an inverse wave age near unity, and settles to a value just less than unity for yet older waves. The pattern also suggests that as relative peakedness decreases, the relative frequency f/f_{p} at which a break in slope from equilibrium range to some other highfrequency range migrates to higher values. The break in slope occurs near 2.1f_{p} for our young Currituck Sound data, evolves to about 3f_{p} for u_{10}/c_{p} ≈ 1, and appears to approach 4f_{p} for the oldest waves in Figure 10.
[31] The waveage behavior of γ_{r} suggested by Figure 10 is supported by observations from other sources. Figure 11 shows as symbols the relation of γ_{r} to u_{10}/c_{p} from the data shown in Figure 10. To this is added data from an analysis by Birch and Ewing [1986] of two young spectra derived from observations in a water reservoir near Hersham, England, where the observation point was near (but not at) the longest fetch of the water body. We also include points derived from figures published by Pettersson [2004] relating to observations in the Gulf of Finland and the Golfe du Lion in the Mediterranean Sea. She distinguishes her data among broad, narrow, and slantingfetch cases, but her observations were nearer the center axis than the edge of the Gulf of Finland and near the center of the Golfe du Lion so her results may be less influenced by the proximity of land as influenced our observations and possibly those of Donelan et al. [1985]. Data from the latter reference indicated substantially smaller γ_{r} for young waves than is illustrated in Figure 11, perhaps because of conditions like those we found in Figure 8. The pattern we see in Figure 11 suggests a weak variation of γ_{r} for u_{10}/c_{p} > 2, a weak variation for u_{10}/c_{p} < 1 and a sharp gradient for inverse wave ages between 1 and 2.
3.3. Directional Distributions
[32] To characterize our directional distribution observations, we initially classified our D(f, θ) in groups of inverse wave age and wind direction, as we did with the onedimensional spectra. As with the onedimensional spectra, we found less variation in directional structure with wave age within a winddirection classification than between direction classes, so we simply grouped data by wind direction (18 classes of 20° width, as we used previously, with two of the 18 classes having too few observations to average). To average the distributions in a common context, we shifted the direction axis of each observation so that the directional distribution was relative to wind direction rather than absolute direction, then converted frequency f to wave number k and normalized k with peak wave number k_{p} so that each D(f, θ) was transformed to D(k/k_{p}, θ − θ_{w}). We then formed mean distributions by averaging elements of D(k/k_{p}, θ − θ_{w}) in discrete bins of k/k_{p} (25 bins of width 0.5, with bin centers ranging from 0 to 12) and θ − θ_{w} (90 bins of 4° width). The highest relative wave number bin k/k_{p} = 12 corresponds approximately to f/f_{p} = 3.5 and so corresponds to the higher relative frequency bins typical of the onedimensional spectra of Figure 9.
[33] Figures 12 and 13 illustrate the 16 resulting mean directional distributions, referenced to the basin boundaries of Figure 2 for each of the winddirection classes. Figure 12 shows the distributions as threedimensional surfaces and provides a qualitative characterization of the directional distribution shapes. Figure 13 shows contours of the distributions, with contour lines at tenths of the directional distribution maximum in each plot, and enables a somewhat more quantitative characterization of directional structure relative to mean wind direction. We note that the sinuous directional distributions in the lowest wave number bins in Figure 12 are artifacts of poor directional estimation owing to the finite spatial dimension of our directional gauge array. Except for directional distributions that are very peaked, these lowwavenumber observations should not be interpreted as wellresolved directional distributions.
[34] The estimates of mean directional distributions in Figures 12 and 13 suggest a lot of structural variability, some of which possibly could be a consequence of averaging, but there are gross features in common with many of the class averages. In all cases, there are narrow directional distributions near spectral peak wave numbers and broad distributions at higher wave numbers, features that are commonly reported in the literature. In Figures 12c–12n, the broad directional distributions at high wave numbers are characterized by two modal peaks, with the arc Δθ separating the modal peaks typically near 100° at the higher wave numbers, qualitatively consistent with observations by Young et al. [1995] in Lake George and by Ewans [1998] in the Cook Strait region of New Zealand.
[35] At slightly greater detail, Figures 12c–12n and Figures 13c–13n suggest that the directional distributions are nominally symmetric about the mean wind direction for k/k_{p} > 4.4, which corresponds approximately to the f/f_{p} > 2.1 we used to define _{hf} in Figure 7, where we found that the mean wave direction tends to align with the wind direction. There is somewhat less symmetry in some of the directional distributions for 2.3 < k/k_{p} < 4.4, which corresponds to the equilibriumrange mean direction _{er} in Figure 7. A much more dramatic variation occurs at the directional peaks in Figures 12 and 13, where some directional peaks are aligned with the wind (e.g., Figures 13h and 13m) and some directional peaks deviate from the wind direction by 50°–60° (Figures 13b and 13p), consistent with the mean peak wave _{pk} directional variation in Figure 7. The sequence of Figures 13h–13o suggests a strong influence by available fetch in these results. In Figure 13m, the wind is coming from the longfetch opening at the south end of the sound (Figure 2), the directional distribution peak is very near the wind direction and the directional distribution function is reasonably symmetric about the wind direction. Figures 13l and 13n show wind direction classes that are 20° on either side of the class in Figure 13m. In Figure 13n, the directional peak is approximately 20° less than the wind direction and thus tends to align with the longfetch direction. In Figure 13l, the directional peak is about 20° greater than the wind direction and also tends to align with the longfetch direction. Figures 13k and 13o, in direction classes that are 40° away from the class in Figure 13m, have directional peaks shifted roughly 40° from the wind direction and toward the longfetch direction. In the extreme case of Figure 13p, a wind direction class 60° from the longfetch direction, the directional peak is not only offset from the wind by about 60°, but, as shown in Figure 12p, there appear to be two directional functions, one associated with the directional peak and another at higher frequencies with directions on the positive side of the wind direction.
[36] A similar but slightly more complicated behavior occurs for wind classes with north components. The directional distributions of Figures 13b–13g suggest the influence of two longfetch directions, one being near 100° and the other near 140° in our directional coordinate system. These directions do not line up with the maximum possible fetch direction at 120° shown in Figure 6, but, from the inset map of Figure 13, align with the alongshore direction to the north and some point to the west of the islands and shoals to the north–northwest of our observation site. Figures 13b–13d in particular show primary and secondary directional peaks associated with the two apparent longfetch directions, with primary peaks associated with the 100° fetch and secondary peaks at lower frequencies associated with the slightly longer 140° fetch. The secondary peaks can just be seen in Figures 12b–12d, but are nearly blocked from view by the primary peaks owing to the perspective of those figures.
[37] The effect of longfetch oblique waves mixing with shortfetch windaligned waves must have a consequent effect on the resulting onedimensional spectra and we speculate on this effect in section 4. If any of our observations are to be useful in the narrower perspective of simple wave evolution, they must be relatively free of slanting or narrowfetch effects. As a characterization of such conditions, we group and average directional distributions from the three winddirection classes representing winds crossing the sound from the west, that is, those shown in Figures 12h–12j and 13h–13j, where directional distribution peaks are nearly aligned with the wind and conditions are like the broadfetch case of Pettersson [2004].
[38] Figure 14 is the result of this average and shows the mean of all directional distributions for which ∣θ_{w}∣ > 150°, with a threedimensional perspective plot in Figure 14a and a contour plot in Figure 14b. The distribution is reasonably smooth and retains the gross structural characteristics described previously: a unimodal peak, directionally bimodal highwavenumber tail and symmetry about the wind direction. The smoothness of the distribution allows a simple detection of the maxima of the directional modal peaks. The directional locations of these maxima are shown by the bold lines in Figure 14b. The arc Δθ(k/k_{p}) between the two modal peaks at each relative wave number can then be computed and compared to results reported elsewhere. Here, we refer to the paper by Ewans [1998] (his Figure 12, with his f/f_{p} squared to compare with our k/k_{p}). He determined directional modal separations Δθ for observations in four u_{10}/c_{p} classes ranging from 0.8 to 1.6 at a site having a nominal fetch of 194 km. Our Figure 14 represents observations from a nominal 5.5 km fetch and includes cases with inverse wave ages ranging from 1.6 to 3.4. Figure 15 shows results from the two sources. In this comparison, our results are somewhat narrower than those of Ewans [1998] in the spectral peak region, k/k_{p} < 2.3, very nearly equal to his through part of the equilibrium range we see in our Figure 10, 2.3 < k/k_{p} < 4.4, and broader than his, but following a similar trend, at high wave numbers. Our results are certainly in general agreement with his observations. A possible interpretation of Figure 15 is that there is a waveagedependent variation in directional modal separation, as Ewans' oldest curve (Figure 15, solid squares) is most different from our very young observations (Figure 15, solid circles), with his intermediate ages falling between these two extremes. If so, Figure 15 suggests that directional distributions tend to narrow with increasing wave age, and it might be useful to seek the verity of this suggested behavior in future work.
4. Discussion and Conclusions
 Top of page
 Abstract
 1. Introduction
 2. Measurement Scheme
 3. Results
 4. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[39] Several months of hourly wind wave observations from a site on the east side of Currituck Sound, North Carolina (Figure 2), indicate a pattern where spectral structure varies considerably with both wind direction and wave age, with the effects of wind direction becoming dominant for wind angles oblique to shore. Inverse wave ages of these waves fall in a range 1.5 < u_{10}/c_{p} < 4 associated with very young seas appropriate to the order 1 to 10km fetches from our observation site to the boundaries of the sound. Except for an arc of angles approximately 60° wide centered on winds directly out of the west, spectra tend to be characterized by directional shear and misalignment with local winds near spectral peaks, variable spectral peakedness and differences in equilibriumrange bandwidth that apparently owe to narrow and slantingfetch geometries within the basin.
[40] Waves from the westerly direction appear to be consistent with fetchlimited wave generation situations in which effects of side boundaries are minimal. Directionally integrated spectra from the west, normalized by equilibriumrange scales, equations (5) and (13), and averaged in bins of f/f_{p}, form part of a pattern that, with likewise normalized spectra from the deep ocean, suggest relative peakedness, equation (14), decreases and equilibriumrange bandwidth increases with increasing wave age (Figure 10). For this class of wave spectra, the highfrequency portion of the equilibrium range consistently deviates from an f^{−4} form and toward an f^{−5} form, suggestive of increased dissipation processes at these high frequencies.
[41] For young waves from the west in Currituck Sound, this characteristic break in slope occurs near f/f_{p} ≈ 2.1 so that the highfrequency spectra tend to fall off more rapidly than k^{−5/2} (or f^{−4} in deep water) and might more nearly approximate a slope of k^{−3} (or f^{−5} in deep water) as suggested by Phillips [1985], though Figure 10 suggests the break in slope is agedependent and not curtailed by an absolute saturation limit. As regards relative peakedness, these results, in combination with observations by Pettersson [2004] and Birch and Ewing [1986], indicate the main variation in γ_{r} occurs within a small range of inverse wave age 1 < u_{10}/c_{p} < 2, with γ_{r} ≈ 0.6–0.8 for u_{10}/c_{p} < 1 and γ_{r} ≈ 2.5 for u_{10}/c_{p} > 2 (Figure 11). This is somewhat different than the behavior suggested by Donelan et al. [1985], who suggest that peakedness varies roughly monotonically over a broad range of wave ages.
[42] The average directional distribution of waves from the west in Currituck Sound is characterized by a unimodal, narrow directional distribution near the spectral peak, a broadening of the directional distribution at higher frequencies and a tendency for energy to concentrate along two directional modes at frequencies in and above those of the equilibrium range (Figure 14). These results are qualitatively consistent with observations by Young et al. [1995], and the modal locations are quantitatively consistent with observations by Ewans [1998], even suggesting a weak tendency for modal separations to decrease with increasing wave age (Figure 15).
[43] Our results show that substantial variation in the structure of directional spectra in Currituck Sound is related to the direction of the wind within the basin. In particular, for obliquewind situations, spectra exhibit marked differences in the directional shear near their peaks, differences in spectral peakedness, and changes in the bandwidth of the equilibriumrange, as a function of both wave age and wind direction. As an example of this, consider the winddirection class that includes the opening at the south end of the sound in Figure 2. The mean, normalized directionally integrated spectrum for this class of spectra is shown in Figure 9m, and the mean directional distribution function is illustrated in Figures 12m and 13m. The onedimensional spectrum for this class has γ_{r} ≈ 1.2 and little evidence of a break in slope at high frequencies, and so differs from the characteristic spectrum associated with westerly winds (upper curve in Figure 10) discussed above. The directional distribution has a peak direction very nearly aligned with the wind and with the center of the opening to the south, a slightly narrower spread in the equilibrium range than the characteristic westerly case of Figure 14 and the distribution is reasonably symmetric about the wind direction. We expect waves in this class generated within a distance of about 6 km from the sled site to have spectral characteristics similar to the westerly waves of Figures 10 and 14, generated over a similar fetch. However, these waves have a somewhat greater JONSWAPtype f_{p} fetch (Figure 6b) and shape characteristics of older waves, as deduced from Figure 10, suggesting an influence by waves entering the sound from the south. If such waves have the broad directional distributions we observe in the westerly waves, then only part of their energy can enter the sound from the south and reach our sled position owing to the narrowness of the southern opening. Consequently, our observed waves in this class are a mix of waves generated within the sound and directionally filtered, longer fetch waves from the south.
[44] The characteristic evolution of spectra within Currituck Sound documented here suggests that the physical balance between source terms and propagation effects in areas of complex geometry (and possible in most cases of winds very oblique to a coast) varies strongly as a function of wind direction. It would be difficult, if not impossible, for secondgeneration spectral wave models properly to capture these effects, as they do not model each source term within a detailedbalance framework. However, existing thirdgeneration wave models, which do attempt to represent detailedbalance effects within the wave generationpropagation process, also do not appear to be able to capture the observed patterns of change within the directional spectrum [Pettersson, 2004]. Instead, these models appear to produce results that are roughly consistent with observed openwater or simple fetchlimited conditions with negligible influence of side boundaries, situations that can also be modeled relatively well by secondgeneration models. In the context of observations presented here, it appears that there remains a considerable need to quantify the detailed balance of source terms and propagation effects in complex situations. Lacking this capability, results of existing wave models used to establish wave climates and estimate nearshore processes in semienclosed and enclosed water bodies could be seriously degraded.
[45] Our primary conclusions regarding wave generation situations for which the effects of side boundaries are negligible are as follows:
[46] 1. In equilibriumrange scaling, spectral peakedness appears to be approximately constant at γ_{r} ≈ 2.5 for u_{10}/c_{p} > 1.7, decreases rapidly for 1.7 > u_{10}/c_{p} > 1.0 to a value of about 0.7 to 0.8, and remains fairly constant at lower inverse wave ages.
[48] 3. In conjunction with results from Ewans [1998], there is an indication that the angular separation of the highfrequency directional modal peaks might be waveagedependent.
[49] 4. There is a consistent break in spectral slope from an f^{−4} form to an f^{−5} form, similar to that observed by Forristall [1981]; however, the location of the transition point does not appear to occur at a fixed value of dimensionless frequency as suggested by Forristall [1981].
[50] Our primary findings for situations influenced by side boundaries are as follows:
[51] 1. Directional shear in the spectral peak region is similar to that observed by Donelan et al. [1985]; however, the directional bifurcation, albeit occasionally very asymmetric, persists throughout spectra generated under these conditions, suggesting that this process is central to the wave generation process even in complex situations.
[52] 2. At high frequencies, mean wave direction tends to align with the wind, suggesting that wind input for shorter waves might be less directly affected by side boundaries; net spectral evolution, including directional asymmetry, thus tends to be a consequence of directionally sheared nonlinear coupling between these windcentered highfrequency waves and longfetchdirected spectral peak waves.
[53] 3. The effects of side boundaries appear to diminish spectral peakedness in a consistent fashion, making spectra generated under such situations appear older in terms of wave age than occurs under conditions of broadfetch, shorenormal winds.
[54] 4. The effects of side boundaries influence the transition of the spectrum from a characteristic f^{−4} form toward an f^{−5} form at high frequencies. In some cases the transition is shifted toward higher frequencies relative to f_{p} than found for waves driven by shorenormal winds and, in other cases, the transition appears to disappear entirely, with little evidence of a welldefined equilibrium range.
[55] 5. Except for cases of very shortfetch winds where the effective wind scale is less clear, spectral equilibrium ranges in the inclusion set examined here are consistent with the coefficient deduced by Resio et al. [2004].