A Sea State Parameterization with Nonarbitrary Wave Age Applicable to Low and Moderate Wind Speeds

Home , Capillary wave, Ocean observations, Sea state

2840 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

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- Ϫ1 perturbed surface for U10 Ͻ 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 Ϫ1 moderate wind speeds (U10 Ͻ 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 Ϫ1 found to be noticeable for U10 Ͻ 6ms , and are likely to be signi®cant in most applications when U10 Ͻ 5 msϪ1. The conditions under which relations valid for higher wind speeds break down are discussed. The new Ϫ1 sea state parameterization is an improvement over previous relations (which apply well for U10 Ͼ 7ms ) because it is also applicable for low and moderate winds. The mean wind speed over most of the world's oceans is less than7msϪ1, 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 Ϫ1 modeled dominant period is a good match to that of the Pierson±Moskowitz spectrum when U10 Ͼ 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

᭧ 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/U␭ (Stewart 1974), cp/U10 (Perrie and Toulany the accepted relations. The widely accepted observation,

1990), cp/U␭/2 (Donelan 1990), and u ´ eÃ i/cpi (Bourassa that the direct in¯uence of surface tension on wave char- * Ϫ1 et al. 1999). In these de®nitions cp is the phase speed acteristics is negligible for at least U10 Ͼ 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 Ϫ1 10 m), and eÃ i are basis vectors with components parallel relations that apply for U10 Ͼ 7ms break down. It (i ϭ 1) and perpendicular (i ϭ 2) to the dominant wave's will be shown that surface tension and capillary waves Ϫ1 mean direction of propagation. The forms with a wind are important to most applications for U10 Ͻ 5ms . Ϫ1 speed are often applied when an accurate measure of For U10 Ͻ 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 Ϫ1 erate wind speeds (U10 Ͻ 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 Ϫ1 as too small (compared to gravity waves) to contribute to ditions (U10 Ͼ 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 Ͼ 7 10 Ϫ1 Ͻ 7msϪ1), 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. not previously been examined. For conditions of local equilibrium between the wind and the waves, this cutoff is often described as the wind speed below which waves a. Link between the wind pro®le and waves will not be generated. In section 3c, it will be shown There have been many studies linking wave charac- how the phase speed can be used to specify a capillary teristics such as amplitude and phase speed to the wind 2842 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

TABLE 1. The physical considerations in the roughness length parameterizations of Charnock (1955), Smith et al. (1992), a variation of Smith (1988), and Bourassa et al. (1999). Charnock S92 S88 BVW Gravity waves yes yes yes yes Capillary waves no no no yes Aerodynamically smooth no no yes yes Surface wave age no yes yes yes Two horizontal dimensions no no no yes pro®le. The Miles (1957) theory is one such commonly accepted wave generation model based on form drag. Such models consider the wind speed passing imme- diately over the wave as equal to the wave's phase speed. If this condition is met, air pressure ¯uctuations are in phase (resonance) with the water surface, and the growth mechanism is as described by Jeffereys (1925). Typi- cally this theory is applied to each frequency component of the wave ®eld. This approach allows a complicated wave ®eld to be considered; however, it is computationally expensive (Janssen 1991). There have been nu- FIG. 1. Wave age as a function of U10 determined by Eq. (14) with merous attempts to apply variations of the Miles theory four roughness length parameterizations. to modeling wave ®elds (Phillips 1977; Janssen 1982, 1989, 1991; Jenkins 1992) and surface stresses (Miles 1957; Janssen 1991; Jenkins 1992, 1993). wind speed and the roughness length (z ) of the surface. An alternative approach to linking waves and a wind o For low and moderate wind speeds (U Ͻ 7msϪ1), pro®le is to solve the nonlinear hydrodynamic equations 10 capillary waves can make a large contribution to this with the application of turbulence closure schemes roughness length (Wu 1968; Bourassa et al. 1999), and (Chalikov 1978, 1986; Makin 1980, 1982, 1987; Makin largely determine the value of the friction velocity. and Panchenko 1986; Chalikov and Makin 1991). This The direct in¯uence of capillary waves on ¯ux and approach and the Miles theory require a model for the air¯ow characteristics (u and z ) is determined by the wind pro®le. The physics describing the shape of the * o relation between u and z . Given z (u ) and the mod- wind pro®le over water waves is a matter of ongoing * o o * i®ed log-wind relation U(z), it is possible to iteratively research. However, there have been recent improve- solve for u (U) and ␶(U). The modi®ed log-wind re- ments in modeling stresses for low and moderate wind * lation is speeds, which bring together in situ and wave-tank observations in a consistent manner and apply to two (hor- u* z izontal) dimensions (Bourassa et al. 1999). We build U(z) Ϫ U ϭ ln ϩ 1 ϩ ␸(z, z , L) , (2) so␬ z upon these results (summarized below) in the devel- []΂΃o opment of our model. where Us is the mean surface current, z is the mean height above the surface, ␬ is von KaÂrmaÂn's constant, b. The in¯uence of capillary waves on characteristics ␸ is the atmospheric stability term, and L is the Monin± of the air¯ow Obukhov scale length (a measure of atmospheric stability in the boundary layer). The Bourassa±Vincent± Our original interest in the sea state problem arose Wood ¯ux model (BVW: Bourassa et al. 1999) uses the from attempts to develop accurate ¯ux parameteriza- stability parameterizations of Beljaars and Holtslag tions for a wide variety of wind and sea conditions. (1991) for stable air, and Benoit (1977) for unstable air.

Consequently, one requirement was that the sea state The value of zo is dependent on the choice of frame parameterization be fully compatible with the parame- reference (Us; Bourassa et al. 1999), which (in the BVW terization of the momentum ¯ux. The downward mo- model) is the local mean surface current. mentum ¯ux (␶) can be modeled in terms of the friction velocity (u ): * 1) ROUGHNESS LENGTH PARAMETERIZATIONS ␶ ϭ ␳ u | u |, (1) ** Four dimensionally sound roughness length param- where ␳ is the density of the air. The friction velocity eterizations will be contrasted to illustrate key physical is also a key parameter in determining the wind pro®le. concepts: Charnock (1955), Smith (1988), Smith et al. The magnitude of the friction velocity depends on the (1992), Bourassa et al. (1999). These parameterizations OCTOBER 2001 BOURASSA ET AL. 2843

FIG. 3. Period of the dominant waves modeled with our sea state parameterization compared to estimates from other observationally based parameterizations. The dots are binned open ocean observations from SCOPE and SWADE, in ϳ2msϪ1 bins (seven and eight observations per bin), for wave ages between 22 and 35. The error bars indicate one standard deviation from the mean within the bin. The mean wave age in the higher wind speed bin is Ͼ28 (the equilibrium

value), consistent with an overestimation of Tp. The mean wave age in the lower wind speed bin is Ͻ28, consistent with an underesti-

mation of Tp.

several examinations of the dependence of Charnock's constant (a) on the sea state (Kusaba and Masuda 1988; Geernaert et al. (1986); Toba et al. 1990; Perrie and Toulany 1990; Maat et al. 1991; Smith et al. 1992; Yel- land et al. 1998), for which Smith et al. (1992, thereafter S92) found

2 0.48 u* zo ϭ , (4) wga FIG. 2. Signi®cant wave height modeled with our sea state param- where w ϭ c /u is wave age. This ®nding applied only a p * eterization (a) compared to estimates from other observationally in the absence of swell, when the mean wind and waves based parameterizations, and (b) demonstrating the importance of capillary waves in the roughness length (dashed and solid lines) and were moving in parallel directions. There are several con- surface tension in the phase relation (dotted and solid lines). troversies regarding this ®nding for these highly restrictive conditions (Yelland et al. 1998; Bourassa et al. 1999). Toba et al. (1990) combined ®eld and wave tank data to de-

veloped an alternative relation where zo ϰ wa. Further consider different sets of in¯uences (Table 1) in the discussion of this result can be found in Donelan et al. relation between u and z . Charnock's (1955), is the (1993). In S92 gravity waves are the only roughness el- * o simplest of these roughness length parameterizations: ements; the improvement over Charnock's relation is the consideration of nondirectional sea state (i.e., wave age). z ϭ au2 /g, (3) o * Both of the above roughness length parameterizations Ϫ1 where a ϭ 0.185 (Wu 1980) is an empirically derived are based upon observations with U10 Ͼ 6ms , and constant. Only gravity waves are considered to contrib- are not adequate at low and moderate wind speeds. ute to the roughness length, and zo is independent of Smith (1988, hereafter S88) attempted to rectify this sea state. Charnock considered that zo could be depen- shortcoming though consideration of the mixing in- dent on sea state; however, he lacked the appropriate duced by an aerodynamically smooth surface in addition data to test such a hypothesis. Recently, there have been to that induced by gravity waves (in this case there were 2844 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

Ϫ1 only three observations with U10 Ͻ 8ms ). He as- 2) SUMMARY OF THE DEPENDENCE OF FRICTION sumed that the roughness length for an aerodynamically VELOCITY ON WAVE AGE smooth surface (Nikuradse 1933; Kondo 1975; Brut- Of the four roughness length parameterizations de- saert 1982) could be added linearly to the gravity wave scribed above (and in Table 1), only Charnock's is in- roughness length: dependent of wave age; the other three parameterizations require that wave age be known through theory 0.11␯ 0.48 u2 or observation. The friction velocity can be determined z ϭϩ*, (5) o uwg given wave age, Eq. (2), and the corresponding equa- * a tions for air temperature and moisture (which are used where ␯ is the molecular viscosity. Smith (1988) used a to determine the atmospheric stability parameter L) (Liu constant value of Charnock's ``constant,'' rather than the et al. 1979; Bourassa et al. 1999), one of Eqs. (3) to wave-age-dependent form in S92 and Eq. (5) herein; S88 (6), and typical ®eld observations (wind velocity, sea refers to hybrid parameterization (5) considering both light surface temperature, air temperature, and a measure of winds and sea state. This parameterization is valid when atmospheric moisture). The BVW calculation for u* is Ϫ1 also dependent on the angle between the mean wind gravity waves dominate the ¯ow (U10 Ͼϳ7ms ) and direction and the propagation direction of the dominant when the surface is aerodynamically smooth (U10 Ͻϳ2 msϪ1). It would also be valid for the intermediate wind waves. This angle is equal to zero for conditions of local speeds if capillary waves made a negligible total contri- wind±wave equilibrium. The in¯uence of this angle on bution to the roughness length. Bourassa et al. (1999) ¯uxes has recently been discussed in terms of obser- showed that the S88 roughness length was too small (for vations (Donelan et al. 1997) and modeling (Bourassa et al. 1999). Hereafter, the discussion will be simpli®ed aerodynamically rough surfaces) at moderate wind speeds by assuming that the mean wind is parallel to the di- (ϳ2 Ͻ U Ͻ 7msϪ1) and applied wave tank observations 10 rection of wave propagation. to develop a capillary wave roughness length, which lead to improved matches to observed ¯uxes. The key aspects of the BVW model are summarized as follows. The rough- c. In¯uence of surface tension on wave ness length is modeled as the weighted root-mean-square characteristics sum of roughness lengths for gravity waves, capillary The relation between phase speed (cp) and period (Tp) waves, and an aerodynamically smooth surface (respec- of the signi®cant waves and surface tension (␴) is given tively indicated by subscripts of g, c, and s). The weighting in the phase relation for surface water waves: terms (␤Ј and ␤) consider whether or not the type of wave gT 2␲␴ is present (␤Ј), as well as shifts in the velocity frame of c ϭϩp , (7) p 2␲ T ␳ reference due to the orbital motion of gravity waves (␤c): pw

where ␳w is the density of water. For applications restricted to long periods (equivalent to long wave- 0.11␯ 22b␴ lengths), the surface tension term is often dropped; how- z ϭ ␤ ϩ ␤Ј␤ ever, the surface tension term will be retained herein. It osi |u | cc␳|u ||u ´ eÃ | [΂΃΂***ii ΃ will be important in determining our capillary cutoff

0.5 and in examining the in¯uence of surface tension for 0.48 |u ||u ´ eÃ | 2 ** i moderate wind speeds. The phase relation can be written ϩ ␤Јg . (6) wg as a second-order polynomial in Tp. The positive root ΂΃ai ] of this quadratic equation is the period of the signi®cant The unit vectors eÃ i are two perpendicular basis vectors, waves (opposed to the period of capillary waves with ␴ is surface tension, ␳w is the density of water, b ϭ 0.06 the same phase speed): is a dimensionless constant, and g is gravitational ac- ␲ celeration. Sea state enters (6) through the wave age T ϭ c ϩ c242Ϫ c /c , (8) ppppp΂΃͙ min (wa), shifts in the velocity frame of reference (␤c), and g

the capillary cutoff (through␤Јc ). For point ¯uxes over 1/4 Ϫ1 where cpmin ϭ (4g␴/␳w) ഠ 23.2 cm s is the minimum an aerodynamically smooth surface ␤g ϭ ␤c ϭ 0 and phase speed for surface waves. ␤s ϭ 1. The main difference between the BVW model Most wave applications, such as spectral wave mod- and the S88 model applicable herein is the consideration els, determine the dominant wave period from the Pier- of capillary waves. son±Moskowitz relation: The considerations in these four roughness length pa- T ϭ 7.1U /g. (9) rameterizations are summarized in Table 1. The range p 19.5

of conditions to which the models are applicable in- The relationship between wind speed and the Tp decreases from left to right. These are increasingly detailed veloped by P±M was based on Kitaigorodskii's (1961) methods for determining the friction velocity. relation between frequency (T Ϫ1) and friction velocity. OCTOBER 2001 BOURASSA ET AL. 2845

Pierson±Moskowitz assumed (seemingly reasonably at ditions on modifying drag coef®cients have been shown the time) that the wind speed at a height of 19.5 m is in Bourassa et al. (1999). directly proportional to the friction velocity, and that The new sea state parameterization has several sim- the proportionality is constant. This assumption implies ilarities with the Miles (1957) model of wave growth. that the drag coef®cient is independent of wind speed, The BVW sea state parameterization can be used to which is no longer considered to be a reasonable ap- describe departures from a condition of local wind± proximation (reviews include Donelan 1990). wave equilibrium. However, due to a lack of suf®cient The P±M relation was developed for waves in ap- observations to test the model for nonequilibrium con- proximate equilibrium with winds for the range from ditions, the focus of this study is the sea state for local Ϫ1 U10 ϭ 10.3 to 20.6 m s (20 to 40 kt), rather than wind±wave equilibrium. A wave ®eld is considered to waves associated with low and moderate winds (U10 Ͻ be in local equilibrium with the mean wind when all 7msϪ1). This limitation of (9) is rarely stated in de- probability density functions describing the wind and scriptions of wave models that utilize (9). Pierson±Mos- wave characteristics are constant with respect to time kowitz chose the best linear ®t of the observed Tp(U19.5), (provided a suf®ciently long averaging period). Most assuming that the line passed through the origin. Con- applications of the Miles theory considered the mean sequently, the period in (9) approaches zero as the wind wind speed passing over the top of each Fourier com- speed approaches zero. This approximation con¯icts ponent of the wave ®eld, and related this wind speed with observations that the capillary cutoff occurs in the to the phase speed of these components. In contrast, our Ϫ1 range of 1 Ͻ U10 Ͻ 4ms (Ursell 1956). sea state model considers the mean wind speed at the signi®cant wave height, and considers only the frequency corresponding to the dominant waves. This ap- 3. Model development proach is far less computationally demanding than considering a wide range of frequencies, as is done in the In order to determine the period as a function of wind WAMS model (GuÈnther et al. 1992; Janssen 1989, speed and sea state, the phase speed is described in terms 1991). Clearly our model cannot be applied to highly of wave age (hereafter wave age will refer to the c /u p * complicated sea states; however, it will be shown to form) and friction velocity: work well for local equilibrium conditions and to qualitatively explain physics for nonequilibrium conditions. cpaϭ uw. (10) * The sea states that can be modeled (or well approxi- This has rarely been done because most applications mated) using this model are much more diverse than determine wave age from c and u ; however, it has the p * can usually be considered with ¯ux models. For ex- advantage of allowing wave and atmospheric ¯ow char- ample, in the BVW ¯ux model, swell can be speci®ed acteristics to be parameterized in terms of wave age and from one direction while local equilibrium is speci®ed wind speed. in a perpendicular direction. The sea state is often described in terms of wave age, The log-wind pro®le does not apply within several which is a ®rst-order approximation of the equilibrium signi®cant wave heights of the surface (Large et al. sea state (Huang et al. 1990). By convention, a sea with 1995; Bourassa et al. 1999); however, the log-wind re- a small wave age is called a young sea. For strong wind lation (2) can be extrapolated to the signi®cant wave conditions, a young sea is usually a growing sea. Con- height (Hs). This mean wind speed is split into two versely, a sea with decaying waves has a large wave components: the speed needed for local equilibrium age and is called an old sea. Swell from distant storms Ueql(Hs) and a perturbation component Une(Hs): is an example of an old sea. One purpose of the sea U(H ) U (H ) U (H ). (11) state parameterization presented herein is to determine s ϭ eql s ϩ ne s local-equilibrium values of wave age and other wave This equation is transformed into a convenient nondi- characteristics (e.g., signi®cant wave height) as a func- mensional form by subtracting the (extrapolated) mean tion of wind speed. For conditions of local equilibrium wind speed at the lower boundary (i.e., the mean surface and 10-m winds in excess of 10 m sϪ1, the wave age current), and by dividing the difference by the phase determined from observations has been found to be be- speed of the dominant waves: tween 26 and 28 (Donelan 1990). Wave age is not a complete description of the wave ®eld, even as a ®rst- U(Hss) Ϫ UUeql(Hs) Ϫ UU s ne(H s) ϭ 1 ϩ . order approximation (Huang et al. 1990). Donelan's de- ccUpp΂΃eql(Hss) Ϫ U scription of wave age for U Ͼ 10 m sϪ1 required a 10 (12) statement regarding wave ®eld growth and decay.

Therefore, wave age will be modeled as a function of To at least a ®rst-order approximation, the term [Ueql(Hs) wind stress and a growth term. Regrettably, it is beyond Ϫ Us]/cp is constant (Kitaigorodskii 1970). This term the scope of this article to develop a model of wave will be referred to as the wind±wave equilibrium term evolution (i.e., the rate of change of the growth term). (G); the value of G will be determined in section 3b.

However, the effects of wave growth and decay con- The surface speed Us is retained in Eq. (12) because it 2846 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

can be signi®cant in ®eld applications (Chou 1993), and g is gravitational acceleration. The period (Tp) can be because of the added elegance of making G and the determined through (8), and friction velocity can be term in brackets invariant throughout Newtonian frames determined from (2) and (6). This signi®cant wave of reference. The convenience of a constant parameter height can then be used to iteratively solve (14). is a key reason for the choice of U(Hs), rather than the mean wind speed at a different height. Another reason for choosing this height is that wave age is a description b. The wind±wave equilibrium parameter of the dominant gravity waves and, hence, can be ex- There are few studies of wind speed as a function of pected to be related to Hs. vertical and horizontal position relative to wave crests The term in brackets on the right-hand side of (12) (Kawai 1981, 1982; Csanady 1985). Due to dif®culties is related to local equilibrium, or a lack thereof, and in making these measurements in turbulent ¯ows, there will be referred to as the wind±wave stability term (W). has been little progress in developing theoretically A value of W Ͼ 1(Une Ͼ 0) indicates a mean wind sound models of the mean velocity ®eld. Therefore, the speed greater than that needed for local wind±wave parameter G is determined indirectly through more eas- equilibrium, which is a growth condition. Conversely, ily observed phenomena. For local equilibrium and U10 a value of W Ͻ 1(U Ͻ 0) indicates insuf®cient wind Ϫ1 ne Ϫ Us Ͼ 10ms , it is assumed that wa is constant. For to maintain the wave ®eld, which is a decay condition. these conditions, the value of G is solely dependent on In summary: the functionality of the roughness length for gravity  waves. For conditions of wind±wave equilibrium, a neu- Ͼ1, a growing wave field Ϫ1  trally stable atmosphere, and U10 Ϫ Us Ͼ 10ms ,  W ϭϭ 1, local equilibrium (13) wave age has been observed to reach a plateau with a Ͻ1, a decaying wave field. value between 26 and 28 (Donelan 1990). The wave ages derived from Eq. (14) are consistent with this ob- The wind±wave stability parameter (W) can be ex- servation when ␬G is between 0.32 (wa ϭ 28) and 0.34 pressed in terms of wave age by substitution of (12) (w ϭ 26). A value of G ϭ 0.81 was chosen to be into (2), and replacement of c /u with wave age w : a p * a consistent with ␬ ϭ 0.4 and the popular approximation that the value of the saturation wave age is 28. Some 1 H w ϭ ln s ϩ 1 ϩ ␸(H , z , L) . (14) cautions should be expressed regarding this result, as it aso␬GW z []΂΃o is dependent on the Humidity Exchange Over the Sea In order to use (14) to determine wave age, it is nec- (HEXOS) result (S92), which has been questioned (Yel- essary to determine the signi®cant wave height without land et al. 1998; Bourassa et al. 1999). The HEXOS introducing new unknowns. result is used in the absence of a better evaluation of the sea state dependence, and it appears to work well for the restrictive conditions for which it applies, as well a. Wave height relations as old seas [for which the sea state is dominated by ␤Ј (Bourassa et al. 1999)]. The above results can be There are a wide variety of parameterizations for c used to determine the local equilibrium values of wa, wave height (Kitaigorodskii 1970; Toba 1972; Hsu as functions of wind speed, for the S88, S92, and BVW 1974; GuÈnther et al. 1992; reviews include Haung et al. parameterizations (Fig. 1). The consideration of wave 1990). The WAMS model (GuÈnther et al. 1992) was not age and non-gravity-wave terms (in z ) causes w to used in the new sea state parameterization because it o a increase as U increases, until wa reaches values where was too computationally expensive and explicitly did it is independent of U. not consider the effects of atmospheric stability on the wind pro®le. The WAMS model was inconsistent with the goal of a sea state parameterization that is fully c. Capillary cutoff compatible with the ¯ux parameterization. Near the sur- Both ®eld and wind tunnel observations show that, face, the wind velocity modi®cations due to atmospheric for conditions of local equilibrium, there is a low wind stability are relatively large and should not be ignored speed cutoff below which neither capillary nor gravity (Stull 1988). Toba's parameterization (15) was favored waves exist. This is usually observed in the range of 1± over Kitaigorodskii's (1970) and Hsu's (1974) relations 4msϪ1 (Ursell 1956). Observations have shown that because it is based on parameters that are usually more ultragravity waves (with both gravity and surface ten- accurately determined from observations, that is, H , s sion contributing to the restoring force) are generated u , and the period of the dominant waves (T ), rather * p prior to capillary waves. For local equilibrium condi- than z , c , and the root-mean-square surface displace- o p tions, we have assumed that neither type of surface wave ment: exists if the modeled phase speed of the dominant waves 3 0.5 Hspϭ B(gu T ), (15) is less than the minimum phase speed for water waves * (i.e., ultragravity waves): where B ϭ 0.602 is an empirically derived constant and OCTOBER 2001 BOURASSA ET AL. 2847

Ͼc , surface waves exist, u*w pmin (16) a Ͻc , surface waves do not exist. Ά pmin The actual minimum phase speed associated with the cutoff is likely to be greater than the minimum phase speed of water waves; consequently this constraint un- derestimates the phase speed and wind speed associated

-U10 k 1 near the capץ/cpץ ,with the cutoff. However illary cutoff. Consequently, the error in the wind speed associated with the cutoff is small. The capillary cutoff is used to distinguish between aerodynamically smooth and rough surfaces. This is an important distinction because the stress over a rough surface is much greater than that over a smooth surface (Byushev and Kuznetsov 1969; Bourassa et al. 1999). The low wind speed cutoff is also useful computationally because it eliminates any chance of Eq. (8) pro- ducing a complex number for the period of the dominant waves. The value of the capillary cutoff is highly dependent on the relation between roughness length and friction velocity. The wave ages determined from the four roughness length parameterizations for conditions of local equilibrium and neutral atmospheric stability are shown in Fig. 1. The greatest differences in modeled wave ages occur at low and moderate wind speeds: the C55 and S92 parameterizations indicate that the equilibrium wave age is approximately constant with wind speed, whereas the other parameterizations are consistent with observations (Donelan 1990) in predicting smaller wave ages for smaller wind speeds. However, the small wave ages modeled with S88 are smaller than observed in ®eld conditions (Ursell 1956; Donelan 1990). The low wave ages predicted with the S88 parameterization are similar to those predicted with the BVW parameterization; the major difference between these two predictions is the range of wind speeds over which the equilibrium value drops. This range is dependent on the magnitude of the roughness length component(s) other than gravity waves. For neutral atmospheric stability and local equilibrium, the capillary cutoff of S88 is 0.8 m sϪ1, which is unrealistically low.

The capillary cutoff for BVW is U10 ϭ 1.8 Ϯ 0.05 m sϪ1 (for a water temperature of 20ЊC), which is within the more commonly accepted range (Ursell 1956).

4. Results Several model results can be veri®ed in comparisons to existing, observation-based, parameterizations. The modeled signi®cant wave height and period of the dom-

←

FIG. 4. Modeled wave ®eld characteristics for various values of W: (a) wave age, (b) signi®cant height, and (c) signi®cant slope. Dotted line shows local equilibrium, solid lines indicate rising seas, and dashed lines show falling seas. 2848 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

inant waves are examined (as functions of U10) for con- solid and dashed lines) results in a large percentage drop ditions of local equilibrium (W ϭ 1) and neutral at- in Hs as U10 approaches the capillary cutoff. When stress Ϫ1 mospheric stability (L ϭ 0). The impact (on HS)of due to capillary waves is not considered (dashed line) capillary waves in the roughness length (6) and surface this reduction can exceed 70% of Hs based solely on tension in the phase relation (7) are also examined to gravity waves, and it can exceed 60% when capillary demonstrate the relative importance of these consider- waves are also considered (solid line). This problem is ations. These considerations are examined by comparing further complicated by the dependence of the capillary the output of the unmodi®ed model to cases where ␤Јc cutoff on the sea surface temperature, which is consid-

ϭ 0 and/or cpmin ϭ 0. ered in the model but will not be discussed further herein. Stress related to capillary waves, without considering the surface tension in the phase speed (dotted line), is a. Hs(U10) for local equilibrium Ϫ1 only signi®cant for U10 Ͻ 0.5ms . The resulting Observations of Hs and U10 are the basis of the Sver- Hs(U10) is incompatible with the many observations that drup±Munk, theory and the Sverdrup±Munk±Bretshnei- the capillary cutoff occurs near the 2±4 m sϪ1 range. der (SMB) nomogram. Sverdrup±Munk theory (S±M) The consideration of surface tension in the phase re- is a best ®t between observations of Hs and U10: lation is largely responsible for the differences from models based solely on gravity waves and for this mod- H ϭ 2.667 ϫ 10Ϫ42U . (17) s 10 el's reasonable capillary cutoff.

The modeled Hs(U10) is similar to the S±M curve and the SMB nomogram (Fig. 2a). The model predicts b. T (U ) for local equilibrium slightly smaller amplitude waves than the SMB no- p 10 Ϫ1 mogram for U10 Ͻ 15 m s and slightly greater am- The period of the modeled waves (Fig. 3) differs sig- plitude waves for greater winds. The differences with ni®cantly from those given by the Sverdrup±Munk± S±M theory could simply be due to forcing the S±M Bretshneider nomogram and the Pierson±Moskowitz Ϫ1 Hs(U10) to a dimensionally sound polynomial relation (1964) spectrum. For U10 Ͻ 13ms the BVW period 2 (i.e., Hs ϰ U 10/g). is approximately one second less than that predicted by Ϫ1 By de®nition, the model results also match Toba's the SMB nomogram. For U10 Ͻ 7ms , the P±M period (1972) parameterization. The wave height parameteri- is similar to that of the nomogram. The differences are zations of Kitaigorodskii (1970) and Hsu (1974) are also physically signi®cant, and of great interest because the shown (Fig. 2a). Kitaigorodskii's relation P±M spectrum is the basis of many spectral models of waves (Kitaigorodskii 1970; Hasselman et al. 1976) for ␴ ϭ 13.3z exp(␬c /u ) (18) op* conditions near local equilibrium. The P±M assumptions

(where Hs ϭ 4␴ and ␴ is the rms wave height) models that U19.5 was in constant proportion to the friction ve- wave heights that are much greater than the other pa- locity, and that the period approaches zero as the wind rameterizations. This problem with Kitaigorodskii's re- speed approaches zero, lead to errors for low and mod- lation has been previously observed. In contrast, Hsu's erate wind speeds where the validity of these assump- relation tions breaks down. If the P±M relation was adjusted to pass through (T , U ), rather than through the (T, ␴ ϭ 1.57z (c /u )2 (19) cutoff cutoff op* U) origin, the P±M periods would be reduced and be has lower signi®cant wave heights than the other pa- much more similar to the model developed herein. rameterizations. The differences in modeled Hs among The assertion that the errors are in P±M rather than Kitaigorodskii's, Toba's, and Hsu's parameterizations BVW is supported by a comparison (Fig. 3a) to ®eld are large, but they could be accounted for as systematic observations (Fairall et al. 1996; Donelan et al. 1997). errors in the calculation of zo (Bourassa et al. 1999; The San Clemente Ocean Probing Experiment (SCOPE: Bourassa 2000), as well as other systematic errors re- Fairall et al. 1996) observations were taken from the viewed by Donelan (1990). Toba's choice of parameters Scripps Institute Floating Instrument Platform, R/P greatly reduced the magnitude of systematic errors, as FLIP, in deep water off the California coast. Phase indicated by the similarity between the wave heights speeds were determined through pressure sensors, and modeled by his parameterization (in the BVW sea state friction velocity was calculated through the inertial dis- parameterization), and those in the observation-based sipation method (as well as other techniques). These Sverdrup±Munk theory. observations are used to calculate wave age and select The signi®cance of capillary waves and surface ten- near-equilibrium conditions (22 Ͻ c /u Ͻ 35). The p * sion (in the phase relation) on signi®cant wave height period of the dominant waves was calculated through is shown in Fig. 2b. Clearly, they are only signi®cant (8) and averaged within the 2 m sϪ1 wind speed bin Ϫ1 for low and moderate wind speeds (U10 Ͻ 4ms ). within which most of the observations fell. Observations Surface tension in the phase relation causes the nonzero from the Surface Wave Dynamics Experiment (SWA- minimum in phase speed, which is a key component in DE: Donelan et al. 1997) in deep water off eastern Cape the capillary cutoff. Surface tension (considered in the Henlopen and Cape Hatteras included directional wave OCTOBER 2001 BOURASSA ET AL. 2849 spectra that were used to isolate wind-driven seas. Phase sonal communication regarding observations by M. A. speeds of the dominant waves were provided by W. Donelan). This observation is often considered contra- Drennan (2000, personal communication). Friction ve- dictory to common sense; however, it can be explained locities were determined through eddy correlation and with this coupled ¯ux and sea state model. Consider a inertial dissipation methods, and the phase speeds were departure from the equilibrium value of Hs without a determined from a wave staff array. The periods (av- change in U10: Hs Ͼ Hs(Ueql) is consistent with falling eraged over the range of wave ages; 22 Ͻ c /u Ͻ 35) seas (W Ͻ 1), and H Ͻ H (U ) is consistent with rising p * s s eql from these ®eld experiments show that P±M overesti- seas (W Ͼ 1). For a speci®c wind speed, the modeled mates the period for wind speeds near 7 and9msϪ1, drag coef®cients are larger for growing seas than for and the sea state model developed herein is a much falling seas, consistent with observations in question. better match to the observations. However, this consistency does not clearly explain why a drag coef®cient decreases as signi®cant wave height increases. The answer is related to the shape of the 5. Discussion waves: signi®cant wave height is not the only charac- The dependency of several wave characteristics on teristic that changes. The wavelength of the dominant conditions of wave growth and decay will be discussed waves (␭) changes more rapidly than Hs, causing the in this section. The model results are qualitatively con- signi®cant slope (Hs/␭) to decrease as Hs increases sistent with observations. Variations in the capillary cut- (Figs. 4b,c). The drag coef®cient decreases as Hs in- off are also examined in section 5a, and variations in creases because the wave ®eld of a falling sea is smooth- wave shape and drag coef®cient are examined in section er than the wave ®eld of a growing sea. 5b. c. Fetch limited conditions a. Dependence of capillary cutoff on W In the past decade, research interests have shifted The wave age (Fig. 4a) shows a change in the cap- from the open ocean to the nearshore environment. illary cutoff as a function of W. The values of W shown Therefore, the importance of limited fetch and duration are 0.4, 0.6, 0.8, 1.0 (dotted line), 1.2, 1.5, 1.8, with limited conditions (W Ͼ 1) have increased in interest. rising sea indicated by solid lines, and falling seas in- The effect of fetch can be described in terms of W: for dicated by dashed lines. This ®nding is consistent with very short fetch U(Hs) Ͼ Ueql and W Ͼ 1, and as the → → observations that the capillary cutoff for rising seas is fetch increases U(Hs) Ueql and W 1. The relevance greater than that for decaying waves (M. A. Donelan of nonequilibrium conditions is likely to become in- and W. J. Plant 1997, personal communication). The creasingly important as the understanding of wave± BVW ¯ux model, which considers stress related to cap- wave and air±sea interaction mechanisms increases, and illary waves, has large differences in roughness length remote sensing allows calculation of short timescale associated with the transition between aerodynamically ¯uxes. Accurate simulations of wave age for duration- smooth and rough surface. The modeled changes are limited situations and swell are important in the mod- related to the differences in roughness length (which are eling of ¯uxes and upper-ocean mixing (Bourassa 2000). related to differences in gravity wave shape as discussed Without high temporal resolution satellite observa- in the following section) and to differences in wave± tions of sea state, the fetch problem is unlikely to be wave interaction related to the orbital velocity of the solved without the consideration of wave ®eld evolu- dominant waves. More steeply sloped waves have great- tion. Coupling a wave evolution model with the BVW er orbital velocities. The orbital velocity of the dominant model would allow wave height and W (and/or another waves reduces the wind shear related to the capillary growth parameter) to be determined as a function of wave riding on these waves and, hence, acts to reduce time. The advantages of such a coupled model would the stress (Bourassa et al. 1999). This mechanism causes be tremendous in that it would allow fetch to be con- a greater reduction of stress on growing (i.e., more sidered in the classic sense (the distance downwind from steeply sloped) waves and, hence, reduces the phase land), as well as modeling open-ocean evolution of wave speeds calculated with (16). However, removing this characteristics due to rising or falling winds (duration form of wave±wave interaction from the model does not limited conditions). Several of the dif®culties in cou- qualitatively change the dependence of Ucutoff on W. The pling these models have already been discussed. The dominant in¯uence is the change in roughness length evolution models are spectral models, and for low winds related to the differences in wave shape. (more precisely, for small dominant phase speeds) there is a signi®cant difference between the BVW and P±M peak periods. Another problem is that the spectral mod- b. Wave shape and roughness length els do not consider atmospheric stability, which has an 2 2 It has been observed that the drag coef®cient (u* /U ), effect on the capillary cutoff. A third problem is that and hence roughness length, tends to decrease as the the value of U(Hs)/cp for conditions of local equilibrium signi®cant wave height decreases (W. Peirson 1997, per- is assumed to equal unity in spectral models but is found 2850 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 herein to be 0.32/␬ (ഠ0.81). 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