## 1. Introduction

[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*) = (*c*_{g}/2π) *E*(*f*), where *c*_{g} is group velocity. Linear dispersion (2*πf*)^{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 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 1.5*f*_{p} < *f* < 3*f*_{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 high-frequency 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 equilibrium-range 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 10-m elevation *u*_{10} 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 *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 2*π*/*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 equilibrium-range 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 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.