## 1. Introduction

[2] Ocean surface waves are the result of wind blowing along a fetch distance for a duration of time. The evolution of ocean surface waves is described by the wave-action equation,

in which a wave energy spectrum *E*(*f*, *θ*) of frequency *f* and directional components *θ* propagates at group velocities *c _{g}*(

*f*) and is altered by spectral source/sink terms: input from the wind

*S*, dissipation via breaking

_{wind}*S*, and nonlinear interactions between wave frequencies

_{brk}*S*. This is also called the radiative transfer equation [

_{nl}*Young*, 1999].

[3] *Phillips* [1985] postulated that a portion of the wave energy spectrum would be in equilibrium such that the source/sink terms would balance. By assuming wave growth to be slow and flux divergence to be negligible at small scales, the left-hand side of equation (1) would be zero at first order. The remaining source/sink terms on the right-hand side then participate at first order in the equilibrium range of the energy spectrum *E*. Assuming wind input *S _{wind}* scales with the wind friction velocity squared, , as empirically determined by

*Plant*[1982],

*Phillips*[1985] derived an analytic expression for the energy spectrum as a function of wave number

*k*in the equilibrium range, which can be rewritten in terms of frequency

*f*as

where *β* is a constant, *I*(*p*) is a directional spreading function, *g* is gravitational acceleration, and *u*_{*} is the wind friction velocity. The cyclic frequency *f* is used throughout; it is related to the radian frequency ω by . The *f*^{−4} spectral shape was first suggested as a universal form based on observations by *Toba* [1973], prior to the dynamic justification proposed by *Phillips* [1985]. An alternate derivation based on a wave number cascade is given by *Kitaigorodski* [1983]. The *f*^{−4} form is commonly used in determining the mean square slope of a wave spectrum, which is given by

[4] The implication of equation (2) is that, given *β* and *I*(*p*), wind friction velocity *u*_{*} (and thus wind stress) directly controls wave energy spectra levels at high frequencies. More information is required, however, to compare wave energy spectra to a measured wind speed *U _{z}* at a given height

*z*(commonly

*U*

_{10}). In a constant stress “law of the wall” boundary layer, the vertical profile of horizontal wind velocity is

where *κ* = 0.4 is the Von Karman constant and *z*_{0} is the roughness length. The roughness length is commonly estimated from the *Charnock* [1955] relation

where *α* is assumed to be 0.012. Thus, by combining the *Phillips* [1985] equilibrium formulation and the *Charnock* [1955] relation, a wind speed *U _{z}* can be estimated from wave energy spectra.

[5] There are known changes in roughness length *z*_{0} due to waves (i.e., deviations from the *Charnock* [1955] relation). These are second-order corrections and typically associated with the nondimensional wave age or , where *c _{p}* is the phase speed of the dominant sea. Although roughness length is an important quantity for the wind profile, it is independent of the wave equilibrium hypothesis (equation (2)). This is because

*u*

_{*}uniquely characterizes the surface stress, via . The roughness length

*z*

_{0}and profile

*U*(

*z*) thus are addressed here solely for the purpose of comparison with measured wind speeds. More central to the equilibrium concept is the directionality of the waves relative to the wind.

[6] The *Phillips* [1985] nondimensional directional function *I*(*p*) in equation (2) integrates over directions θ relative to the wind direction (i.e., *θ* = 0 indicates waves aligned with the wind), such that

where *p* is a directional spreading parameter (increasing for narrower directional distributions). Physically, a narrower directional spectrum is more effective at capturing the wind and thus has a higher *I*(*p*). In the equilibrium range, *Phillips* [1985] found the ratio of the downwind wave slope to the total wave slope to be . In their pioneering study of wave slopes, *Cox and Munk* [1954] found this ratio to range from 0.5 to 0.64. *Juszko et al*. [1995] found similar ratios, solving for *p* values ranging from 0.0 to 12.5, although typically less than 1, and *I*(*p*) values ranging from 1.9 to 3.1.

[7] *Juszko et al*. [1995] successfully showed equilibrium over a limited set of conditions (four storms), and obtained a mean value for the constant *β* = 0.012 from a range of 0.006 < *β* < 0.024. Specifically, *Juszko et al*. [1995] showed agreement between the equilibrium stress and the stress calculated with standard drag laws [e.g., *Smith*, 1980; *Large and Pond*, 1981] , where *C _{D}* is a drag coefficient.

*Donelan et al*. [1985] and

*Dobson et al*. [1989] suggested that

*β*depends on the wave age, however

*Juszko et al*. [1995] found negligible improvement to

*u*

_{*}estimates when incorporating a variable

*β*. The

*β*values in

*Juszko et al*. [1995] are consistent with the

*Toba*[1973] constant of empirical

*f*

^{−4}spectra and the observed values of

*α*= 0.06 from

_{T}*Kawai et al*. [1977] and

*α*= 0.13 from

_{T}*Battjes et al*. [1987].

[8] Related recent work includes *Long and Resio* [2007], who examine equilibrium range bandwidth under different fetch-limited conditions and directional cases, and *Takagaki et al*. [2012], who show a relation between the wind stress and spectral levels of both the equilibrium range and the swell range.

[9] Here, we extend the results of *Juszko et al*. [1995] to a much larger data set and include detailed observations of the equilibrium balance in equation (1), where . The primary data are process measurements with drifting buoys and shipboard instruments in the vicinity of OWS-P during a 3 week research cruise. The secondary data long-term mooring observations from Ocean Weather Station P (OWS-P), an ongoing reference site at 50°N 145°W in the North Pacific Ocean. The site has long been used to study air–sea interaction [e.g., *Large and Pond*, 1981], because of its deep location, weak currents, and large range of conditions.

[10] The data collection and processing are described in section 2. The results and sensitivities are in section 3. Errors and potential application are discussed in section 4. The conclusions are in section 5.