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Keywords:

  • wind waves;
  • wave generation;
  • wave spectra

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Measurement Scheme
  5. 3. Results
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References
  9. Supporting Information

[1] We examine a set of 1626 high-resolution frequency-direction wind wave spectra and collocated winds collected during a 7-month period at a site in Currituck Sound, North Carolina, in terms of one-dimensional spectral structure and directional distribution functions. The data set includes cases of shore-normal winds in broad-fetch conditions as well as winds oblique to the basin geometry, with all fetches of order 10 km or less. Using equilibrium-range scaling, all one-dimensional spectra have a spectral peak region, an equilibrium range of finite bandwidth following an f−4 slope at slightly higher frequencies, and a high-frequency tail that falls off more rapidly than f−4. For shore-normal 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 < u10/cp < 1.7. Equilibrium-range bandwidth seems to be narrow for young waves and increases with increasing wave age. Directional distribution functions in shore-normal 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 oblique-wind 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. One-dimensional spectra tend to have reduced spectral peakedness and highly variable equilibrium-range bandwidths in oblique-wind conditions, clearly indicating a more complex balance of source terms in these cases than in the more elementary situation of shore-normal 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Measurement Scheme
  5. 3. Results
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References
  9. 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 frequency-direction spectra E(f, θ) was obtained in Currituck Sound, a semi-enclosed 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] Equilibrium-range scaling is useful in analysis of one-dimensional 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) = (cg/2π) E(f), where cg 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 fp, and having one of the forms

  • equation image

where α5 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 α5, and attention was focused on the possible importance of nonlinear four-wave 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 equilibrium-range spectrum spanning the approximate frequency band 1.5fp < f < 3fp of the form

  • equation image

where u is a scale velocity and α4 is a dimensionless universal constant. Spectra at higher frequencies may take other forms. Phillips [1985], in arguing for an E4 spectrum, notes that high-frequency parts of spectra used in his prior work [Phillips, 1977] still appear to have an E5 form. Forristall [1981] finds a transition from an E4 spectrum to an E5 spectrum such that α5 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 equilibrium-range spectrum, which is valid into water of finite depth. We write this general form as

  • equation image

where

  • equation image

Various forms for the scale velocity u have been proposed in the literature, including the wind speed at 10-m elevation u10 and the wind-stress-based 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 equilibrium-range scaling concept expressed by Resio and Perrie [1989] and the wind-speed scale uλ (defined below) derived by Resio et al. [1999] yields a good data characterization consistent with a constant α4. The resulting expression for β is

  • equation image

where α4 = 0.00553 and u0 = 1.93 m/s. In equation (5), cp is spectral peak phase speed and uλ is wind speed at an elevation equal to a fixed fraction λ (= 0.065) of the spectral peak wavelength 2π/kp, where kp 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 z0 = αCu*2/g with αC = 0.015. Thus

  • equation image

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 〈k5/2F(k)〉 over a suitable range of equilibrium-range frequencies, with (uλ2cp)1/3g−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 finite-depth 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 6-m NOMAD buoy at their Station 41001, off the southeast U. S. coast, and a 12-m 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.

image

Figure 1. Correlation (coefficient r2 indicated) of equilibrium-range coefficient β with (uλ2cp)1/3/g1/2 based on data from six disparate sources. The (bold) regression line forms the basis of equation (5), which, with equation (4), provides a model for the equilibrium range of one-dimensional wave number spectra. Equation (6) defines velocity scale uλ. Only a subset of Currituck Sound data are used in this correlation, but we use the model defined by equations (4), (5), and (6) to normalize all spectra in the entire Currituck Sound data set presented in this paper. NDBC, National Data Buoy Center; FRF, Field Research Facility.

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image

Figure 2. Site map showing the boundaries of Currituck Sound nearby the instrument platform (solid circle labeled “sled”). The sound is isolated from the Atlantic Ocean to the east by the barrier island of North Carolina's Outer Banks. At the nearest points, the sled is about 1 km from the east side of the sound and about 6 km from the west side. A broad channel of about 2.5-m depth runs along the long axis of the sound in the vicinity of the sled. This channel narrows considerably to the north. The opening to the south, representing an azimuthal arc of about 20° from the sled location, leads to the larger, deeper Albemarle Sound. Solid circles marked #625 and #650 show locations of Baylor gauge and Waverider buoy, respectively, that provided data used in Figure 1.

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2. Measurement Scheme

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Measurement Scheme
  5. 3. Results
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References
  9. 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 land-induced effects and not be representative of a pure air-water 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 air-water 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 three-legged pipe-frame 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 capacitance-type wave rods of 0.6-mm diameter and 3-m 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 (r2 > 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.

image

Figure 3. Photograph of the sled platform, looking west from its deployed position, illustrating the directional array of wave rods in the cage-like structure on the far side of the sled and the impellor-vane wind sensor at a nominal elevation of 5 m above the water surface. Other fixtures are for power, communication, and lightning protection.

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[10] Two editing constraints arise from this sled design. First, the discrete steps associated with wave-rod 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 very-low-energy spectra. The second constraint arises in cases where waves pass through one of the horizontal wave-gauge 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, five-element linear arrays with one shared gauge. The two-dimensional character of this array allows full 360° directional resolution. The design of each linear-array 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 cross-spectral 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 cross-spectral estimation degrees of freedom are high.

image

Figure 4. Pattern and dimensions of gauge array used for wave directional estimation. Gauge spacing along each arm follows the guidance of Davis and Regier [1977]. Orthogonal arms allow 360° directional resolution.

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[12] Data collection consisted of synchronous, 12,288-point sampling of wave rod outputs at 0.2048-s intervals (about 4.88 Hz), resulting in records of about 41-min duration originating at the top of every hour. The 2.44-Hz Nyquist frequency is compatible with the minimum gauge spacing of the directional array. Fourier analysis with segment and band averaging yielded auto- and cross-spectral estimates having 120 degrees of freedom in bands of about 0.024-Hz width. We averaged auto-spectral 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 frequency-direction spectrum is

  • equation image

where θ is relative to sled coordinates and, in the rectangular (rather than polar) coordinate system used here, the distribution function satisfies

  • equation image

[13] We determined mean wind speed uz and direction θw at anemometer elevation z above mean water level by vector averaging 1-Hz 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

  • equation image

under the assumption of a neutrally stratified, logarithmic boundary layer. Other elevation-related wind parameters were found by replacing z in the right-hand side of equation (9) with, for example, z10 = 10 m for the 10-m wind speed u10 or zλ = 2πλ/kp for reference wind speed uλ, as in equation (6).

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Measurement Scheme
  5. 3. Results
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References
  9. 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 u10, inverse wave ages u10/cp are relatively large, indicating young, fetch- or duration-limited waves, consistent with small characteristic wave heights Hmo 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 kp 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 kpd extends down to 1.2, waves at k ≅ 2kp (for which f ≅ 1.4fp, roughly the low-frequency bound of the equilibrium range) have 2kpd = 2.8 ≈ π, and so are sufficiently short still to be considered deepwater waves. However, cp at these low kpd are slower by about 5% than deepwater values, so there are some minor finite-depth influences in the spectral peak regions of some of our spectra. Bulk steepness Hmokp and relative wave height Hmo/d are small enough for all cases that conventional wave breaking limits are not exceeded and that four-wave 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 Dataa
 Currituck SoundNDBC 41001NDBC 46035
  • a

    NDBC, National Data Buoy Center.

Number of cases1626212144
   u10, m/s5.0–16.29.7–24.47.9–23.9
   Hmo, m0.08–0.592.6–8.13.3–10.2
   fp, Hz0.32–0.780.08–0.150.07–0.14
   kp, rad/m0.44–2.290.02–0.090.02–0.08
   d, m2.24–2.93>4300>3600
   kpd1.17–5.68  
   Hmokp0.10–0.380.12–0.280.10–0.28
   Hmo/d0.03–0.22  
   u10/cp1.43–4.060.63–1.450.46–1.58

[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 high-frequency tail. Aside from locally wind-generated 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 wind-wave 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 narrow-fetch 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 broad-fetch observations for winds from the west than for narrow-fetch situations for winds from the north or south.

image

Figure 5. Example of a frequency-direction spectrum measured in Currituck Sound. Here, Hmo = 0.35 m, fp = 0.42 Hz, and mean waves are from −137° and subject to 9.3 m/s winds at 5-m elevation from −145°. The direction axis originates at magnetic east and increases counterclockwise. The directional distribution is narrow at the spectral peak, broadens at somewhat higher frequencies, and indicates directional bimodality at yet higher frequencies.

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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 fetch-growth models. This procedure yields estimated apparent fetches from sled-derived parameters that can be compared to maximum possible fetches derived from shoreline location. Here, we use an adaptation of the JONSWAP energy-fetch relationship in the form given by Resio and Perrie [1989] as

  • equation image

where Hmo is variance-based characteristic wave height, and the JONSWAP peak-frequency-fetch relationship in the form

  • equation image

where, in this expression, we assume deepwater peak phase speed cp = g/2πfp.

[17] Figure 6a shows as symbols estimated fetch based on equation (10) as a function of mean wind direction θw 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 duration-limited growth. A finite offset from an upwind boundary is sensible for fetch-limited 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 fetch-growth 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).

image

Figure 6. Estimates of fetch (symbols) from observed wave parameters based on inversions of the Joint North Sea Wave Project (JONSWAP) dimensionless fetch relationships for (a) energy (in terms of Hmo) and (b) peak frequency (in terms of cp), which are plotted in terms of mean wind direction and compared to estimates of maximum possible fetch (bold lines) along azimuthal lines from the sled location to solid land. The two fetch estimates agree approximately only for a narrow range of westerly directions, which correspond to broad-fetch, shore-normal wind conditions described by Pettersson [2004]. In all other cases, there is substantial disagreement, suggesting that obliquity of the wind to the basin geometry has a strong effect on wave energy distributions.

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[18] Figure 6b shows estimated fetch based on equation (11) as a function of θw. In contrast with Figure 6a, the energy and frequency fetch estimates are in nominal agreement only for winds from the westerly direction. Otherwise, frequency-based fetch estimates exceed energy-based fetch estimates, and, notably, the frequency-based 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 f1 < f < f2, is

  • equation image

To obtain a reasonably consistent definition for a mean direction of waves within the spectral peak region, we defined a peak mean wave direction equation imagepk for frequencies satisfying 0.9 < f/fp < 1.5. Figure 7a shows equation imagepk as a function of θw 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, θw ≈ 45°, yield waves roughly from the north, equation imagepk ≈ 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].

image

Figure 7. Correlations of mean wave directions in various frequency bands with wind direction: (a) mean wave direction equation imagepk in spectral peak region 0.9 < f/fp < 1.5 versus θw for all observations (symbols), with the solid line representing perfect correlation and (b) averages in 10° bins of θw of equation imagepk (triangles, pointing up), equilibrium-range mean wave direction equation imageer for 1.5 < f/fp < 2.1 (squares) and high-frequency mean wave direction equation imagehf for 2.1 < f/fp (triangles, pointing down). Deviations of equation imagepk from θw are toward directions of longer fetch, consistent with observations of Donelan et al. [1985]. Nearer agreement of equation imagehf with θw suggests a strong direct wind influence at high frequencies and, in oblique-wind conditions, considerable directional shear in directional spectra.

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[20] To see if wind and wave directions differed in other parts of our spectra, we defined an equilibrium-range mean wave direction equation imageer using equation (12) with frequencies satisfying 1.5 < f/fp < 2.1, a high-frequency mean wave direction equation imagehf for frequencies f/fp > 2.1, and averaged the mean wave directions in 10° bins of mean wind direction. Figure 7b shows the three bin-averaged 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 high-frequency 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 oblique-wind data, which we examine in more detail in section 3.3.

3.2. One-Dimensional Spectra

[21] Given the significance of inverse wave age u10/cp 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 one-dimensional 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 equilibrium-range 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 equilibrium-range model of equation (3) to form

  • equation image

where β is given by equation (5). Each k has a corresponding f so that we averaged equation image(k) in discrete bins of f/fp to examine the distribution relative to peak frequency.

[22] Figure 8, which shows class-average means as symbols and standard deviations as error bars in each f/fp bin, indicates that there appears to be less variation in resulting class mean spectra of different inverse wave ages within a wind-direction class than between direction classes. Figures 8a–8c show three different wave-age class means for winds crossing the sound from the west, ∣θw∣ > 170°. Figure 8d shows the mean of all spectra, irrespective of wave age, for that wind-direction class. Figures 8e–8g show wave-age class means of the same inverse wave ages as those in Figures 8a–8c, respectively, for winds roughly alongshore from the south, −110° < θw < −90°. Figure 8h shows the mean of all spectra for that wind-direction class. Though there appear to be some variations with wave age in each direction class, the error bars suggest the wave-age distinction is relatively minor.

image

Figure 8. Representative wave-age and wind-direction class means (symbols) and standard deviations (error bars) of normalized one-dimensional wave number spectra averaged in discrete bins of f/fp. The left panels show spectra in broad-fetch, shore-normal wind conditions ∣θw∣ > 170° for inverse wave ages such that (a) 1.5 < u10/cp < 2.0, (b) 2.0 < u10/cp < 2.5, (c) 2.5 < u10/cp < 3.0, and (d) all u10/cp. The right panels show spectra in roughly shore parallel wind conditions −110° < θw < −90° for inverse wave age ranges the same as (e) Figure 8a, (f) Figure 8b, (g) Figure 8c, and (h) all u10/cp. Clearly, spectral structure varies more between wind-direction classes than with inverse wave age within a wind-direction class.

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[23] Comparison of the two simple, direction-class-averaged spectra, Figures 8d and 8h, is of interest. The deviation of the normalized spectral value at the spectral peak, f/fp = 1, above the equilibrium-range reference line, equation image = 1, is greater in Figure 8d than in Figure 8h. This suggests a simple parameter with which to characterize the spectral peak in equilibrium-range scaling as used here and, for example, by Donelan et al. [1985] and Pettersson [2004]. Calling it relative peakedness γr, it is defined simply in our case by

  • equation image

[24] In Figure 8d, γr ≈ 2.5. In Figure 8h it is somewhat less, with γr ≈ 1.2. The flattened part of the spectrum that indicates the equilibrium range in Figure 8h is somewhat less than the equilibrium-range 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 γr 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 1.5fp, indicating the existence of equilibrium ranges in each. In Figure 8h, for our shore-parallel winds, the apparent equilibrium range extends to about 3.5fp 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 2.1fp. 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 shore-normal conditions in Figure 8d and differ at most by about 20° for the shore-parallel 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 wave-age classes within a wind-direction classification, we redid our analysis and simply averaged spectra in wind-direction 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 wind-direction-class bin boundaries superimposed on the map of Figure 2.

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Figure 9. (a–p) Wind-direction class (20° bins of θw) means (symbols) and standard deviations (error bars) of equation image(k) averaged in discrete bins of f/fp, representing all data in the Currituck Sound data set, with reference to basin boundary geometry (central graph). Axis labels of all spectral plots are the same as those in Figure 9n. Sparse data precludes samples in classes near 0°. Figures 9h–9j represent broad-fetch, nearly shore-normal conditions for winds from the west. Relatively smooth gradation of spectral shapes in the sequence of Figures 9k–9o shows apparent influence of long-fetch waves from the south–southwest. A similar, but less sharply graded, pattern in the sequence of Figures 9c–9g suggests an influence of the more restricted long fetch to the northwest. Figures 9b and 9p, representing direction classes for short-fetch winds near 45° to the shoreline, suggest less well defined equilibrium ranges. Figure 9a, approximating broad-fetch, nearly shore-normal conditions for winds from the east, has a shape rather like corresponding westerly classes. Overestimation of equilibrium-range scale β for this class might be due to local winds somewhat misrepresenting wind conditions over this very short fetch.

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Figure 9. (continued)

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[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 9e9j), with nominally constant relative peakedness near 2.5 and an equilibrium range that terminates with a break in slope at frequencies near 2.1fp. For winds along an arc from the southwest counterclockwise to the southeast (Figures 9k9o), 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 fetch-limited 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 wave-age-related 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 9h9j, for winds from the westerly directions and which include data representing inverse wave ages in the range 1.6 < u10/cp < 3.4, to generate a young-wave spectrum that might be characteristic of the broad-fetch 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 wave-age-class 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 age-dependent. 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 γr 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/fp at which a break in slope from equilibrium range to some other high-frequency range migrates to higher values. The break in slope occurs near 2.1fp for our young Currituck Sound data, evolves to about 3fp for u10/cp ≈ 1, and appears to approach 4fp for the oldest waves in Figure 10.

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Figure 10. Mean equation image(k) averaged in discrete bins of f/fp for broad-fetch, nearly shore-normal conditions ∣θw∣ > 150° from Currituck Sound (young waves, top curve) and (all other curves, older waves) inverse-wave-age classes of deep-ocean data from NDBC buoys 41001 and 46035, for which fetch is less influential. The pattern suggests that the relative peakedness γr = equation image(kp) (i.e., at f/fp = 1) is larger than unity for young waves, near unity for wave ages near unity, and less than unity for old waves. The pattern also suggests wave-age, or perhaps γr, dependence of high-frequency end of equilibrium range, where spectral slope steepens from k−5/2 (f−4 in deep water) to something more like k−3 (f−5 in deep water), with the break in the slope increasing with increasing wave age.

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[31] The wave-age behavior of γr suggested by Figure 10 is supported by observations from other sources. Figure 11 shows as symbols the relation of γr to u10/cp 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 slanting-fetch 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 γr 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 γr for u10/cp > 2, a weak variation for u10/cp < 1 and a sharp gradient for inverse wave ages between 1 and 2.

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Figure 11. Inverse-wave-age dependence of relative spectral peakedness suggested by broad-fetch, nearly shore-normal Currituck Sound observations (open diamonds with the data grouped into three age classes for illustrative purposes and with standard deviations indicated by error bars), NDBC buoy data (crosses with standard deviations as error bars), fetch-orientation-grouped observations from Pettersson [2004], and reservoir data from Birch and Ewing [1986]. Data from Birch and Ewing [1986] are based on single observations. Data from Pettersson [2004] are based on multiple observations with scatter (not shown) about mean values of an order similar to that for Currituck Sound data. The pattern suggests γr ≈ 2.5 for u10/cp > 1.7, γr ≈ 0.7–0.8 for u10/cp < 1.0, and a steep gradient of γr for 1.0 < u10/cp < 1.7.

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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 one-dimensional spectra. As with the one-dimensional spectra, we found less variation in directional structure with wave age within a wind-direction 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 kp so that each D(f, θ) was transformed to D(k/kp, θ − θw). We then formed mean distributions by averaging elements of D(k/kp, θ − θw) in discrete bins of k/kp (25 bins of width 0.5, with bin centers ranging from 0 to 12) and θ − θw (90 bins of 4° width). The highest relative wave number bin k/kp = 12 corresponds approximately to f/fp = 3.5 and so corresponds to the higher relative frequency bins typical of the one-dimensional 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 wind-direction classes. Figure 12 shows the distributions as three-dimensional 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 low-wave-number observations should not be interpreted as well-resolved directional distributions.

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Figure 12. (a–p) Wind-direction class (20° bins of θw, as in Figure 9) mean directional distribution functions in direction coordinates relative to wind direction, averaged in discrete bins of k/kp (extending to 12, or f/fp ≈ 3.5), as three-dimensional surfaces, with each plot referenced to basin boundary geometry (central graph). Axis labels for all directional distribution plots are the same as those in Figure 12n. Amid considerable finer structure, persistent gross features of these distributions are directionally narrow peaks, increasing spread at higher wave numbers, and a tendency for energy to concentrate along two directional modes at the highest wave numbers. Unrealistic sinuous distributions at very low wave numbers are an artifact of poor directional resolution of long waves (for which energy is relatively small). Truncation of some class averages is a consequence of the noise floor constraint on acceptable data described in section 2. Directional shear suggested by Figure 7 is present subtly in these plots (e.g., compare Figure 12k with Figure 12p), owing to their perspective, but is more obvious in Figure 13.

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Figure 12. (continued)

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Figure 13. (a–p) Contour plots of the directional distribution functions in Figure 12. Plot layout organization is the same as that in Figure 12. Contours are at tenths of the directional distribution maximum in each plot. Directional narrowness of spectral peaks, increasing directional spread at higher wave numbers, and persistent tendency for high wave number distributions to be directionally bimodal are quite obvious in these plots. Skewing of directional peaks away from oblique-wind directions and toward long-fetch directions is also apparent, especially in wind-direction classes near Figure 13m, where winds and peak wave directions are aligned with the south opening of the sound. Figure 13n represents cases with winds 20° on the positive side of Figure 13m, yet its peak direction is about 20° on the negative side of the wind, and so aligns with the long fetch of Figure 13m. Figure 13o, with winds 40° positive from those of Figure 13m, has a peak direction about 40° negative to the wind, which, again, aligns with the long fetch of Figure 13m. Figures 13l and 13k, at 20° and 40° on the negative side of Figure 13m, respectively, have peak directions about 20° and 40°, respectively, positive to the wind, evidently strongly influenced by the long fetch of Figure 13m. A somewhat more complex pattern is evident in Figures 13b–13e, where there are primary and secondary peaks offset from the wind, owing possibly to more complicated fetch boundaries to the north and northwest. Directional shear near spectral peaks is less pronounced in Figures 13h–13j, which appear more characteristic of cases with broad-fetch, shore-normal winds.

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Figure 13. (continued)

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[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 12c12n, 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 12c12n and Figures 13c13n suggest that the directional distributions are nominally symmetric about the mean wind direction for k/kp > 4.4, which corresponds approximately to the f/fp > 2.1 we used to define equation imagehf 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/kp < 4.4, which corresponds to the equilibrium-range mean direction equation imageer 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 equation imagepk directional variation in Figure 7. The sequence of Figures 13h13o suggests a strong influence by available fetch in these results. In Figure 13m, the wind is coming from the long-fetch 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 long-fetch direction. In Figure 13l, the directional peak is about 20° greater than the wind direction and also tends to align with the long-fetch 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 long-fetch direction. In the extreme case of Figure 13p, a wind direction class 60° from the long-fetch 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 13b13g suggest the influence of two long-fetch 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 13b13d in particular show primary and secondary directional peaks associated with the two apparent long-fetch 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 12b12d, but are nearly blocked from view by the primary peaks owing to the perspective of those figures.

[37] The effect of long-fetch oblique waves mixing with short-fetch wind-aligned waves must have a consequent effect on the resulting one-dimensional 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 narrow-fetch effects. As a characterization of such conditions, we group and average directional distributions from the three wind-direction classes representing winds crossing the sound from the west, that is, those shown in Figures 12h12j and 13h13j, where directional distribution peaks are nearly aligned with the wind and conditions are like the broad-fetch case of Pettersson [2004].

[38] Figure 14 is the result of this average and shows the mean of all directional distributions for which ∣θw∣ > 150°, with a three-dimensional 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 high-wave-number 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/kp) 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/fp squared to compare with our k/kp). He determined directional modal separations Δθ for observations in four u10/cp 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/kp < 2.3, very nearly equal to his through part of the equilibrium range we see in our Figure 10, 2.3 < k/kp < 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 wave-age-dependent 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.

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Figure 14. Mean directional distribution function representing conditions of broad-fetch, shore-normal winds, based on averaged directional functions in direction coordinates relative to wind direction, accumulated in discrete bins of k/kp, for all observations with ∣θw∣ > 150°: (a) as a three-dimensional surface and (b) contoured at tenths of directional distribution maximum. Smoothing over many observations emphasizes major features, notably increasing directional spread at wave numbers above the peak, directional bimodality at high wave numbers, and reasonable symmetry of the distribution about the wind direction. The bold lines in Figure 14b follow the modal maxima evident in Figure 14a.

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Figure 15. Directional separation of modal maxima Δθ from bold lines in Figure 14 for young Currituck Sound waves under nominally shore-normal winds (solid circles) compared with modal separations for four older classes of waves reported by Ewans [1998]. Trend of Currituck Sound observations is like that of Ewans [1998] at high wave numbers. The evident gradation of Δθ curves from the oldest waves (solid squares) to young Currituck Sound observations suggests a modest age dependence of Δθ in evolving waves.

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4. Discussion and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Measurement Scheme
  5. 3. Results
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References
  9. 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 < u10/cp < 4 associated with very young seas appropriate to the order 1- to 10-km 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 equilibrium-range bandwidth that apparently owe to narrow- and slanting-fetch geometries within the basin.

[40] Waves from the westerly direction appear to be consistent with fetch-limited wave generation situations in which effects of side boundaries are minimal. Directionally integrated spectra from the west, normalized by equilibrium-range scales, equations (5) and (13), and averaged in bins of f/fp, form part of a pattern that, with likewise normalized spectra from the deep ocean, suggest relative peakedness, equation (14), decreases and equilibrium-range bandwidth increases with increasing wave age (Figure 10). For this class of wave spectra, the high-frequency 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/fp ≈ 2.1 so that the high-frequency 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 age-dependent 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 γr occurs within a small range of inverse wave age 1 < u10/cp < 2, with γr ≈ 0.6–0.8 for u10/cp < 1 and γr ≈ 2.5 for u10/cp > 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 oblique-wind situations, spectra exhibit marked differences in the directional shear near their peaks, differences in spectral peakedness, and changes in the bandwidth of the equilibrium-range, as a function of both wave age and wind direction. As an example of this, consider the wind-direction 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 one-dimensional spectrum for this class has γr ≈ 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 JONSWAP-type fp 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 second-generation spectral wave models properly to capture these effects, as they do not model each source term within a detailed-balance framework. However, existing third-generation wave models, which do attempt to represent detailed-balance effects within the wave generation-propagation 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 open-water or simple fetch-limited conditions with negligible influence of side boundaries, situations that can also be modeled relatively well by second-generation 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 semi-enclosed 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 equilibrium-range scaling, spectral peakedness appears to be approximately constant at γr ≈ 2.5 for u10/cp > 1.7, decreases rapidly for 1.7 > u10/cp > 1.0 to a value of about 0.7 to 0.8, and remains fairly constant at lower inverse wave ages.

[47] 2. The directional distribution exhibits a consistent, pronounced bifurcation above the spectral peak frequency, consistent with observations of Young et al. [1995], Ewans [1998], and Wang and Hwang [2001].

[48] 3. In conjunction with results from Ewans [1998], there is an indication that the angular separation of the high-frequency directional modal peaks might be wave-age-dependent.

[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 wind-centered high-frequency waves and long-fetch-directed 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 broad-fetch, shore-normal 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 fp than found for waves driven by shore-normal winds and, in other cases, the transition appears to disappear entirely, with little evidence of a well-defined equilibrium range.

[55] 5. Except for cases of very short-fetch 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].

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Measurement Scheme
  5. 3. Results
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References
  9. Supporting Information

[56] We are most grateful for the talent and dedication of Brian Scarborough, Kent Hathaway, Bill Grogg, Dan Freer, Ray Townsend, and Mike Leffler of the FRF staff in the execution of our field campaign. The work presented in this paper was supported by the U.S. Army Corps of Engineers RDT&E program. Permission to publish was granted by the Office, Chief of Engineers.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Measurement Scheme
  5. 3. Results
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References
  9. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Measurement Scheme
  5. 3. Results
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References
  9. Supporting Information
FilenameFormatSizeDescription
jgrc10471-sup-0001-t01.txtplain text document0KTab-delimited Table 1.

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