2840 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 A Sea State Parameterization with Nonarbitrary Wave Age Applicable to Low and Moderate Wind Speeds MARK A. BOURASSA Center for Ocean±Atmospheric Prediction Studies, The Florida State University, Tallahassee, Florida DAYTON G. VINCENT Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, Indiana W. L. W OOD* School of Civil Engineering, Purdue University, West Lafayette, Indiana (Manuscript received 8 March 1999, in ®nal form 23 January 2001) ABSTRACT A simple coupled ¯ux and sea state model is developed. It is applicable to low, moderate, and high wind speeds, with a nonarbitrary wave age. It is fully consistent with an atmospheric ¯ux parameterization. The ¯ux model includes the in¯uence of capillary waves on surface stress, which dominate surface stress on a wave- 21 perturbed surface for U10 , 7ms . The coupled model is veri®ed for conditions of local equilibrium and it is used to examine the in¯uences of surface tension and capillary waves on the equilibrium sea state at low and 21 moderate wind speeds (U10 , 7ms ). Capillary waves can directly in¯uence characteristics of the air¯ow (roughness length and friction velocity), and surface tension can directly in¯uence wave characteristics (period, phase speed, and wave age). The in¯uences of capillary waves and surface tension on wave characteristics are 21 found to be noticeable for U10 , 6ms , and are likely to be signi®cant in most applications when U10 , 5 ms21. The conditions under which relations valid for higher wind speeds break down are discussed. The new 21 sea state parameterization is an improvement over previous relations (which apply well for U10 . 7ms ) because it is also applicable for low and moderate winds. The mean wind speed over most of the world's oceans is less than7ms21, so there is considerable need for such a parameterization. The nonarbitrary wave age is particularly important because of the in¯uence of wave age on the shape and size of waves, as well as ¯uxes of momentum, heat, and moisture. Additional results include a criterion for the capillary cutoff, signi®cant wave height, and dominant wave period (as functions of wind speed and wave age). Consistency with a ¯ux parameterization is crucial because of the in¯uence of atmospheric stability on surface stress, which in¯uences wave characteristics. The modeled signi®cant wave height for local equilibrium is shown to be consistent with several parameterizations. The 21 modeled dominant period is a good match to that of the Pierson±Moskowitz spectrum when U10 . 13ms ; however, for lower wind speeds, the modeled period differs signi®cantly from that of the Pierson±Moskowitz spectrum. The new model is validated with ®eld observations. The differences are largely due to the capillary cutoff, which was not considered by Pierson and Moskowitz. 1. Introduction arbitrary sea state parameterization as a function of wind speed. Contrary ®ndings have been expressed by Yel- Recent efforts in wave modeling (Janssen 1989; GuÈn- land et al. (1998); however, the observations used in ther et al. 1992; Komen et al. 1994) and air±sea ¯ux that study were not sorted to remove the in¯uences of observations and modeling (Maat et al. 1991; Smith et swell (Bourassa et al. 1999). In practice, the sea state al. 1992; Perrie and Toulany 1990; Donelan et al. 1997; parameters required for ¯ux models are rarely known, Bourassa et al. 1999) have shown the need for a non- and arbitrary assumptions must be made regarding the sea state. For example, one of the most detailed wave * Deceased. models currently in use, WAMS (Wave Model: GuÈnther et al. 1992), requires an arbitrarily assigned wave age. This problem is signi®cant because of the in¯uence of Corresponding author address: Dr. Mark A. Bourassa, Center for Ocean±Atmospheric Prediction Studies, The Florida State University, wave age on the shape and size of waves, as well as Tallahassee, FL 32306-2840. ¯uxes of momentum, heat, and moisture. The parame- E-mail: [email protected] terization of wave age is further complicated by a wide q 2001 American Meteorological Society OCTOBER 2001 BOURASSA ET AL. 2841 variety of de®nitions for wave age [reviews include cutoff. The model veri®cation includes a check that the Donelan (1990)]: c /u (Maat et al. 1991; Smith et al. model results for high wind speeds are consistent with p * 1992), cp/Ul (Stewart 1974), cp/U10 (Perrie and Toulany the accepted relations. The widely accepted observation, 1990), cp/Ul/2 (Donelan 1990), and u ´ eà i/cpi (Bourassa that the direct in¯uence of surface tension on wave char- * 21 et al. 1999). In these de®nitions cp is the phase speed acteristics is negligible for at least U10 . 7ms ,is of the dominant waves, u* is the friction velocity, U is consistent with the coupled model. Modeled wave char- the wind speed at a height equal to the subscript (wave- acteristics are examined for low and moderate wind length of the dominant waves, half that wavelength, or speeds, to determine for which conditions the similarity 21 10 m), and eà i are basis vectors with components parallel relations that apply for U10 . 7ms break down. It (i 5 1) and perpendicular (i 5 2) to the dominant wave's will be shown that surface tension and capillary waves 21 mean direction of propagation. The forms with a wind are important to most applications for U10 , 5ms . 21 speed are often applied when an accurate measure of For U10 , 13 m s the modeled period differs signif- friction velocity is not available. A sea state parame- icantly from that of the Pierson±Moskowitz (P±M) spec- terization (modeling wave age, signi®cant wave height, trum. The model results are shown to be a close match phase speed of the dominant waves, etc.) for conditions to observations (section 4b). The differences are largely of local wind±wave equilibrium is discussed herein. due to the capillary cutoff, which was not considered Capillary waves and surface tension, which dominate by Pierson and Moskowitz. Applications that require atmospheric turbulent surface ¯uxes for low and mod- wave characteristics for low and moderate wind speed 21 erate wind speeds (U10 , 7ms ; Bourassa et al. 1999), conditions are expected to greatly bene®t from this re- are examined in the context of their in¯uence on sea sult. state. Relations that are often used to calculate cp (e.g., 2. Background Pierson and Moskowitz 1964) and u* (e.g., Smith 1980) were developed for moderate and high wind speed con- There has been a tendency to dismiss capillary waves 21 as too small (compared to gravity waves) to contribute to ditions (U10 . 8ms ), for which surface tension and capillary (surface tension) waves have been considered the surface roughness (Hay 1955; Miles 1957). For high negligible in comparison to gravity waves (Bourassa et wind speeds, this argument is probably valid; there is no al. 1999). The advantages of the model developed herein evidence that capillary waves need be directly considered are most evident at low and moderate wind speeds (U in parameterizations of roughness length (zo) for U10 . 7 10 21 , 7ms21), where similarity relations that apply well ms . The contribution to roughness length from one at greater wind speeds break down (Bourassa et al. capillary wave is certainly negligible compared to an in- 1999). Two causes of this breakdown are the in¯uence dividual gravity wave; however, the roughness length of of capillary waves on the characteristic of the air¯ow a surface is related to the sum of contributing roughness (roughness length and friction velocity) and surface ten- elements (i.e., waves). There are many capillary waves for each gravity wave; the ratio of wavelengths (and hence sion on wave characteristics (period, phase speed, and 4 wave age). Our new sea state model is fully coupled density) is typically greater than 10 :1. When superposi- tion and the cross¯ow direction are considered the ratio with a surface ¯ux model (Bourassa et al. 1999) that 7 considers atmospheric turbulent mixing due to capillary can be 10 :1. Individual capillary waves do not have to waves. This ¯ux model is brie¯y reviewed in section make much of a contribution to roughness length for cap- 2b(1). The coupling of the sea state and ¯ux model is illary waves to contribute to a large fraction of the rough- self-consistent in a manner that accounts for the three ness length (Bourassa et al. 1999). The background begins with a short review of how strongest in¯uences on ¯uxes: wind speed, atmospheric waves are in¯uenced by the vertical wind pro®le. The stability, and sea state (i.e., wave age and swell direc- rest of the background is divided into two sections: the tion). The model will be used to examine the wave age in¯uence of capillary waves on the shape of the wind corresponding to local wind±wave equilibrium, and to pro®le, and the in¯uence of surface tension on wave examine the in¯uence of capillary waves on this wave characteristics. These considerations are linked through age. the wave age parameterization, which is developed in The theoretical development of the sea state param- section 3, for use in the coupled ¯ux and sea state model. eterization (section 3), for the couple ¯ux and sea state A key to correctly coupling the ¯ux and sea state models model, avoids relations that apply only to high wind is an accurate determination of the friction velocity, speeds or gravity waves. The in¯uence of surface ten- which appears in parameterizations for the vertical pro- sion on signi®cant wave heights is not well known, and ®le of wind speed, wave age, surface stress, atmospheric the in¯uence of wave age on the capillary cutoff has stability, and signi®cant wave height.