Journal of , Vol. 57, pp. 519 to 533, 2001

Adjustment of Wind Waves to Sudden Changes of Wind Speed

1 2 3 TAKUJI WASEDA *, YOSHIAKI TOBA and MARSHALL P. T ULIN

1Frontier Research System for Global Change and International Pacific Research Center, University of Hawaii, HI 96822, U.S.A. 2Earth Observation Research Center, National Space Development Agency of Japan, Tokyo 104-6023, Japan; and Japan Marine Science and Technology Center, Yokosuka 237-0061, Japan 3Ocean Engineering Laboratory, University of California, Santa Barbara, CA 93106, U.S.A.

(Received 4 August 2000; in revised form 22 January 2001; accepted 15 March 2001)

An experiment was conducted in a small wind-wave facility at the Engineering Keywords: Laboratory, California, to address the following question: when the wind speed ⋅ Wind waves, changes rapidly, how quickly and in what manner do the short wind waves respond? ⋅ local equilibrium, ⋅ To answer this question we have produced a very rapid change in wind speed between fetch, Ð1 Ð1 ⋅ wind gust. Ulow (4.6 m s ) and Uhigh (7.1 m s ). Water surface elevation and air turbulence were monitored up to a fetch of 5.5 m. The cycle of increasing and decreasing wind speed was repeated 20 times to assure statistical accuracy in the measurement by taking an ensemble mean. In this way, we were able to study in detail the processes by which the young laboratory wind waves adjust to wind speed perturbations. We found that the wind-wave response occurs over two time scales determined by local equilibrium ad- ∆ ∆ justment and fetch adjustment, t1/T = O(10) and t2/T = O(100), respectively, in the current tank. The steady state is characterized by a constant non-dimensional (H/gT2 or equivalently, the wave steepness for linear gravity waves) depending on wind speed. This equilibrium state was found in our non-steady experiments to apply at all fetches, even during the long transition to steady state, but only after a ∆ short initial relaxation t1/T of O(10) following a sudden change in wind speed. The ∆ complete transition to the new steady state takes much longer, t2/T of O(100) at the largest fetch, during which time propagates over the entire fetch along the rays (dx/dt = cg) and grows under the influence of wind pumping. At the same time, frequency downshifts. Although the current study is limited in scale variations, we believe that the suggestion that the two adjustment time scales are related to local equilibrium adjustment and fetch adjustment is also applicable to the ocean.

1. Introduction et al. (1987)). When such formulae are applicable, Sea states are typically characterized by two param- states offshore of straight coastlines can be uniquely de- eters: , Hs, and significant wave scribed as a function of fetch and wind speed in non-di- period, Ts. For a young sea, these parameters vary in space mensional form. and time, even in the presence of uniform wind. In the In this uniform wind case, wind waves grow with uniform wind case, however, empirical evolution laws distance offshore, following the fetch law, eventually describe the variation of Hs (or total energy, ET) and Ts reaching an equilibrium state where growth ceases. How- σ (or frequency, s) in space (fetch limited) or time (dura- ever, even while growing, wind waves have been per- tion limited). Wilson’s (1965) fetch formula is an early ceived to be in a local equilibrium with the wind (Toba, example (see also Kahma and Calkoen (1992) and Battjes 1972; Masuda and Kusaba, 1987), since a single empiri- cal relationship between Hs and Ts exists everywhere: the 2 ∝ 3 so-called 3/2 power law, wherein Hs gu*Ts , where g * Corresponding author. E-mail: [email protected] is the acceleration of gravity, and u* is the air friction Copyright © The Oceanographic Society of Japan. velocity. Leaving the detailed discussion for the main text,

519 we simply state at this point that this 3/2 law does not quasi-equilibrium, even following discrete changes in hold for sufficiently short wind waves. We have found wind speed, seems to be of great significance for these here that the proportionality H ∝ gT2 represents the short short wind waves and likely is significant in the response wind-wave state better, in accordance with the earlier of short waves to perturbations other than by the wind, experimental observations made by Kunishi (1963) and such as effects. Moreover, the consequence Mitsuyasu and Rikiishi (1978) and the theoretical of this adjustment process for remote sensing seems ob- modeling of Phillips (1958). vious. The 3/2 law ceases to be observed even for energetic Section 2 describes the experimental facility and in- waves in the ocean when the wind speed changes suffi- strumentation, the gust experimental procedure and analy- ciently quickly, and the actual process by which wind sis techniques; Section 3 discusses steady state wind waves adjust to changing wind fields and stress has been waves that are relevant for understanding the adjustment previously studied using ocean data by Toba et al. (1988) processes in a short tank. The results of the gust experi- and Toba and Ebuchi (1991). The equilibrium range spec- ment from fetches 5.5 m, 4.7 m, 3.9 m and 3.1 m are pre- tral level has been observed by Toba et al. (1988) to de- sented in Section 4. Detailed analyses of the results re- crease for rising wind conditions and vice versa during a veal various interesting phenomena resulting from short time scale. Hanson and Phillips (1999) also depicted changes in wind speed, and an interpretation of the re- the trend of increasing energy with decreasing wind ac- sponses based on further analysis of the data is presented. celeration and vice versa at a given wave age, in agree- Further discussion of local equilibrium is given in Sec- ment with Toba et al. (1988). Young et al. (1987) used tion 5. wave models to study the responses of wind-wave spec- trum to sudden shifts of wind direction restricted to a 2. Experiments duration-limited growth of a homogeneous wave field. Komen et al. (1994) summarizes recent work on the modi- 2.1 Facility and instrumentation fication of the wind-wave growth due to stochastic wind Experiments were conducted in a small-scale wind- forcing. In the laboratory, Wu (1975a) studied the effect wave facility at the Ocean Engineering Laboratory of the of pulsating wind on the wind profile and the microstruc- University of California, Santa Barbara. The small wind- ture of the water surface. The growth of wind waves in wave tank is 7.0 m long, 30 cm wide, 15 cm deep, with a the duration-limited case was studied by Mitsuyasu and 15-cm air passage height; wind speeds are variable be- Rikiishi (1978) by abruptly starting the wind over a wa- tween 3 and 8 m sÐ1 (Fig. 1). Wave wires of diameter 0.1 ter surface. In order to study this wind-wave adjustment mm were located at seven test sections along the tank; process in more detail in unsteady winds, we have con- 0.4, 1.7, 2.4, 3.1, 3.9, 4.7, and 5.5 m fetches. The sensi- ducted an experiment in a small laboratory wind-wave tivity of the wave wires was about 50 µm, with a dynamic Ð1 tank, changing wind speeds from Ulow = 4.6 m s to range of 10 cm. A pitot-static tube and pressure trans- Ð1 Uhigh = 7.1 m s and vice versa. After an abrupt change ducer were located at the entrance of the tank in order to of wind speed, we observed the response of wind waves monitor the mean wind velocity, U. An X hot-film an- with time throughout the tank until temporal equilibrium emometer was used at 5.5 m fetch and 1.9 cm height to was reached everywhere, i.e., the waves were everywhere measure the Reynolds stress in the air. During the gust in their fetch-limited (steady) state. experiment, both the X hot-film anemometer signal and In an initial adjustment stage, the waves reached a the pressure transducer signal were sampled simultane- state of quasi-equilibrium or local equilibrium after a very ously with the wave wire signal at 50 Hz, using a compu- short adjustment time. The strong tendency toward this ter.

Fig. 1. Diagram of the wind-wave tank: 7.0 m long, 30 cm wide, 15 cm water depth with 15 cm air passage height. Scales are distorted.

520 T. Waseda et al. 2.2 Gust experiment and data analyses The shutter was opened and closed 40 times at an 2.2.1 Sudden change of wind speed interval of about 2 minutes, giving 20 increasing steps A manual shutter that reduces the area of the fan and 20 decreasing steps per run. The response time for outlet was used to create an abrupt change in the mean the wave spectrum at fetch 5.5 m to reach an equilibrium wind speed, from 7.1 m sÐ1 to 4.6 m sÐ1. The response state after a step-like increase/decrease of the wind speed time of the mean wind speed was short, typically around was much less than 2 minutes. 0.3 s, Fig. 2. It is obvious from this figure that the transi- 2.2.2 Analysis technique tion time for the mean wind speed is much less than a Data samples of 2 s length were sub-sampled from second and the sharp steps in the mean wind speed record the original time series; the 2 s interval began at times ∆t are considered appropriate to serve as a time reference after the steps; ∆t was varied in 1 s increments; twenty of for the further analysis described below. The signals of these samples were taken for each ∆t and these comprised the hot-film anemometer at fetch 5.5 m have the same an ensemble. The ensembles of these 2 s samples were response time, too, and there was no obvious signature used to construct the final spectral estimate of approxi- due to the propagation of pressure disturbances from up- mately 40 degrees of freedom (frequency resolution 0.5 stream in response to the opening and closing of the shut- Hz, Nyquist frequency 50 Hz, with Hanning window ap- ter. plied) and the mean properties, Reynolds stress, −ρuv′′, It is known that an abrupt change in the wind field mean wave height, H , and mean wave period, T . The will cause an oscillation of the tank water level, or a total energy ET and the high frequency energy Ehigh were . In this case, a mode 1 standing wave pattern was obtained by integration of the spectral estimate.* The observed and the period of the oscillation determined for wave height and the wave period are defined from the the tank geometry using the Merian formula is about 12 zero-up-crossing waves in the time series. s. The oscillation has a node at the center of the tank (3.1 2.2.3 Determination of the friction velocity during the m fetch) and the change of the mean water level at fetch gust experiment 1.7 m was too small (5 mm) to cause any significant air A measurement of the friction velocity prior to the flow variation within the same time scale. gust experiment indicated the existence of a constant momentum flux layer, very close to the air-water inter- face. The u* estimated by direct turbulent flux measure- ment ( −uv′′)1/2 compared well with that from the profil- ing method with Von Kàrmàn constant κ = 0.37 ± 0.03 and a small adjustment of the reference height (0.2 cm) for the high wind speed case; the range of parameters used is in accord with the earlier study by Tseng et al. (1992). In the current experiment, at fetch 5.5 m, a hot-film X probe was located at about 1.9 cm above the mean water level in order to avoid immersion of the probe in the wa- ter at the highest wind speed. Therefore, in the case of Ulow, the probe was out of the constant momentum flux layer near the air-water boundary and the measured Fig. 2. Time evolution of the mean wind speed (Ð2 to 8 s) illustrating transition time. Reynolds stress is not the same as the shear stress within the constant momentum flux layer. A simple correction for the lower wind speed case can be made by artificially adding a constant offset comparing ( −uv′′)1/2 obtained from this run to the Reynolds stress obtained earlier within the constant momentum flux layer. The estimated u* (solid line) is plotted together with the measured Reynolds stress ( −uv′′)1/2 (dotted line) in Fig. 3. When the shutter was opened/closed, the mean wind speed changed rapidly (within half a second) throughout

*The range of integration corresponds to frequencies higher than the second harmonic of the spectral peak of the σ Ulow spectrum; c is 11 Hz, 13 Hz, 15 Hz and 17 Hz 1/2 Fig. 3. Estimated u* (—) and ( −uv′′) (á á á) during gust ex- for fetch 5.5 m, 4.7 m, 3.9 m and 3.1 m, respectively (see Figs. periment. 6 and 7).

Adjustment of Wind Waves to Sudden Changes of Wind Speed 521 the tank (see Fig. 2). The air boundary layer adjusts rap- In a previous study (Waseda, 1997), the detailed spec- idly within this short time, but the Reynolds stress con- tral evolution was reported and it was shown that three tinues to change as the underlying water waves continue distinct regions exist, with different physical processes; to grow until they reach an equilibrium state. The precise similar descriptions can be found in Kunishi (1963) and estimation of the error associated with the u* requires in- Plate et al. (1969), though these aspects have not been formation from different locations in the tank as well as emphasised thus far. The evolution can be separated into an instantaneous profile measurement at a fixed location, three stages: in Stage 1, “initial wavelets” are generated, and those are not available in this study. presumably as a result of a coupled viscous shear flow instability, as originally found by Kawai (1979) in the 3. Steady State Wind Waves temporal generation of wind-waves. In Fig. 4(a), for the We introduce here steady state wind-wave evolution 4.6 m sÐ1 wind speed, at fetches 0.4Ð1.7 m, an initial wave- in a short tank, including our tank and others of compa- let is found around 17 Hz. Note that the energy level of rable size. In order to understand the transient evolution the initial wavelet is much smaller than the high frequency process that is the main focus of this paper, we need a ranges of the spectrum at larger fetches. (This demon- description of steady state wind-wave evolution, particu- strates the great sensitivity of the wave wires that were larly because the wind-wave evolution in a short tank is used in this experiment.) This spatial initial wavelet di- complicated. minishes as the new energy peak appears at much lower Wind-wave spectra, Φ(σ), were obtained from the frequencies but not necessarily at half the frequency of constant wind speed runs; examples are shown for the the initial wavelet; i.e. the downshifting does not depend Ð1 Ð1 4.6 m s (Ulow) and 7.1 m s (Uhigh) cases (Fig. 4). The on period doubling, as has earlier been speculated spectral shapes are typical of those obtained in a narrow (Janssen, 1986). The growth of this secondary peak at tank: a steep drop of the high frequency energy, at a rate lower frequency characterizes Stage 2 where the surface of fÐ(8Ð10); a small hump at about twice the spectral peak undulation acquires a three-dimensional rhombic struc- frequency corresponding to the bound waves; and the ture. The start of Stage 3 coincides roughly with the dis- leveling off of the high frequency energy, which seems appearance of this rhombic structure when the wind waves to saturate at levels that increase with wind speed. As we have a more coherent yet highly asymmetric profile, pre- discuss below, the evolution of this wind-wave spectrum ceded by parasitic capillary waves at the wave crests, in- with fetch has distinctive features related to the range of dicating the strong forcing of the wind to be parameterized Ð1 the experiment with respect to the wind-wave develop- by u*. The spectral evolutions of the steady 7.1 m s wind ment stages and, therefore, does not necessarily follow speed are shown in Fig. 4(b). As we dicuss further below, the same fetch laws of the open ocean. the wind waves observed for this wind speed case are at

Fig. 4. Wind-wave spectra for constant wind speed case; (a) 4.6 m sÐ1 (b) 7.1 m sÐ1. 170 degrees of freedom, 0.0488 Hz resolution interval. Hanning window was applied.

522 T. Waseda et al. 3.1 m fetch and above are considered representative of line Stage 3 waves that have near constant wave steepnesses 2 ν ∝ ν or H/gT . u*L/ a u*H/ a, (1) In order to illustrate better the evolution of wind waves in these three different stages, we present a new suggests a tendency toward a constant steepness defined π plotting of data obtained from Kunishi (1963). The main by H/L or (Hk)s, where k = 2 /L, H is an average wave data set comes from fetches in the range 2 m to 15 m and height, L is wavelength estimated from an average wave the mean wind speed ranged between 2 m sÐ1 and 10 period using a linear relation. This behavior m sÐ1, somewhat stronger than our experimental case. applies to the data obtained from the current study too, Figure 5(a) shows the nondimensional mean wavelength and is discussed in full in later sections which describe vs. the nondimensional mean wave height, which may be the response of wind waves to the change in wind speed. interpreted as the evolution of wavelength with fetch since It is significant that the Stage 3 data in Fig. 5 are some- the mean wave height is an increasing function of fetch. what scattered, indicating a small wind speed dependence The following parameter values indicate the transition of of the proportionality (1). ν stages: u*H/ a = 1.0 for the transition between Stages 1 In contrast to the steepness law found here for Stage ν and 2, u*H/ a = 100 for transition from Stage 2 to 3, where 3 waves, Toba (1972) found that the steepness decreased ν a is the kinematic viscosity of air. Stages 1, 2 and 3 are with increasing wave ages in the ocean denoted by the symbols, ᮀ, +, and ᭺, respectively. In the ∝ 1/2 first stage, the wavelengths or frequencies remain nearly (ak)s (u*/cp) , (2) constant whereas the wave heights increase with fetch, characterizing the growth of the initial wavelet by insta- where (ak)s is the steepness of the significant wave and bility. Stage 2 may be considered a transition to Stage 3, cp is its phase velocity. The latter relation is equivalent to ν where a universal law for the relation between u*L/ a and the 3/2 law (Toba, 1972; Kawai et al., 1977; Ebuchi et ν u*H/ a is indicated by a diagonal line in Fig. 5(a). The al., 1992),

ν ν Fig. 5. Evolution of short wind waves in a short wind-wave flume from Kunishi (1963). (a) u*L/ a plotted against u*H/ a. ᮀ ν Symbols denote different stages, where the boundary was chosen arbitrarily: Stage 1 for u*H/ a < 1.0; + Stage 2 for 1.0 < ν ᭺ ν ν ∝ ν u*H/ a < 100; Stage 3 for u*H/ a > 100. The line is an arbitrary fit indicating u*L/ a u*H/ a. (b) Same data plotted for 2 2 β 2 β gH/u* against gT/u*. Symbol nomenclature same as (a). Solid line indicates gH/u* = (gT/u*) where is from Kusaba and Masuda (1988), and the broken line indicates Toba’s 3/2 law. The dotted line box indicates the region of Fig. 12.

Adjustment of Wind Waves to Sudden Changes of Wind Speed 523 32/ In order to analyse our data, it should be noted that gH  gT  ss= B  , ()3 our experiment thus ranged from the early to the final 2   u∗ u∗ phase of Stage 3, as shown in Fig. 5(b). We show that relation (4), found for short fetch wind-waves, is satis- where B is an empirical constant, 0.062, representing con- fied during the adjustment to an abrupt wind speed change. ditions for wind waves in local equilibrium with the wind The implication of such an observation is discussed later in the ocean. Figure 5(b) shows this law (broken line) in connection with the concept of local equilibrium inter- plotted together with Kunishi’s data. As may be observed preted as a balance between wind input and energy dissi- in this figure, Stage 3 waves approach the 3/2 law, along pation. the solid line corresponding to 4. Adjustment of Wind Waves to Sudden Changes 2 β 2 gH/u* = (gT/u*) . (4) in Wind Speed

This transition situation is very similar to a tank observa- 4.1 Adjustment of spectral shape tion by Kusaba and Masuda (1988); the full line plotted We present here the spectral evolution in time after in Fig. 5(b) is for β = 0.0149 (=(5.5á10Ð2/(2π)4)1/2á2.51) sudden increases, I, and decreases, D, in the wind speed from their study. This relation is the same as (1), assum- (Figs. 6 and 7). These figures present only selected spec- ing that the waves are in the regime, L = tra with time lapse 0 s, 1 s, and 4 s and the representative g/(2π)áT2. This observation strongly suggests that Stage steady state spectra for wind speed 4.6 m sÐ1 and 7.1 3 is yet another transitional stage approaching the equi- m sÐ1. As mentioned earlier, the high frequency energy of librium state, which is satisfied for wind waves at much the steady state spectra seems saturated for either wind longer fetches (Toba, 1972), which must be considered speed and for all the fetches presented: 5.5 m, 4.7 m, 3.9 the ultimate stage. A similar relation was observed in a m and 3.1 m. lake by Donelan et al. (1992) and was confirmed by a In I, Fig. 6, the spectral energy increases rapidly more recent ocean observation by Johnson et al. (1998). throughout the whole frequency range. While the spec- The transition of the stages from (4) to (3) takes place at trum at time lapse 0 s (solid line without marker) lies on Ð1 around gT/u* = 6~8, which approximately coincides with top of the steady state spectrum at 4.6 m s (dotted line), ω ᮀ Kusaba and Masuda’s (1988) criterion ( p u*/g > 1) for the spectrum at time lapse 1 s (solid line with ) devi- waves not satisfying the 3/2 law but having a constant β. ates from it. The energy throughout the spectrum contin- In this study, the constant β showed a weak wind speed ues to increase until at time lapse 4 s (solid line with ᭹) dependency. the high frequency energy level for frequencies larger than It is interesting to consider dimensionally the transi- the second harmonic (denoted by a vertical line) ceases tion within Stage 3 from its earlier stage as expressed by to grow after reaching the energy level of the 7.1 m sÐ1 (4) to the final stage as expressed by (3). We assume that steady state spectrum (denoted by a broken line). From the statistical state of local equilibrium of wind and wind then on, the high frequency energy level remains satu- waves is determined by four parameters: u*, g, T and H. rated whereas the energy at lower frequencies continues This is a line similar to considerations adopted by Toba to grow. This is best observed in the evolution diagram (1972), Masuda and Kusaba (1987) and Kusaba and of Ehigh, Fig. 8. It should be noted that the increase of the Masuda (1988). In the earlier phase of Stage 3, u* does lower frequency tail of the spectrum around 0Ð2 Hz is not appear explicitly, and the regime is expressed by the due to energy leakage from the seiche component at 0.833 ω 4 2 following equation E p /g = const. Kusaba and Masuda Hz. reported that their experimental value of the constant was During this stage of rapid energy increase, the spec- 5.5 × 10Ð2. (This regime corresponds to the very high- tral peak frequency remains unchanged. The peak starts frequency equilibrium range of the ocean wind-wave spec- to downshift only after a short delay time, roughly in the tra, which is proportional to gωÐ5.) However, since the same time it takes for the high frequency spectral energy shear flow is driven by u*, there is a possibility that the to saturate. The gradual increase of energy and the re- β constant is a function of u*, and our data show that , duction of the peak frequency then continues for about which is equivalent to the constant, varies with u*. 20 s or so at fetch 5.5 m, and less for shorter fetches, As the wind waves become large in wave height and until the spectrum ceases to grow and finally reaches the length, a new regime begins, and u* becomes explicit and steady state; the spectrum at this time is omitted from ω 4 2 × Ð2 ω is expressed as E p /g = 4.76 10 pu*/g. This regime Fig. 6. This completes the transition between the 4.6 is expressed by the line of Fig. 5(b). It corresponds to the m sÐ1 and 7.1 m sÐ1 steady states. Such observations sug- equilibrium range of ocean wind-wave spectra, which is gest the existence of two time scales in the response of ωÐ4 proportional to gu* (e.g., Toba, 1973; Phillips, 1985). wind waves; the shorter is characterized by a change in

524 T. Waseda et al. Fig. 6. Spectral development during increasing wind speed case; á á á 4.6 m sÐ1 equilibrium spectrum; — increase the wind speed, ᮀ ᭹ ∆ Ð1 t = 0; - - t = 1 s; - - t = 4 s ( t1); - - - 7.1 m s equilibrium spectrum. 40 degrees of freedom, 0.5 Hz resolution interval. Hanning window was applied to the data after the linear trend was removed. The vertical line indicates the lower frequency limit for the computation of the high frequency energy.

wave steepness without downshifting; the longer time 4.2 Adjustment time scale scale is characterized by an energy increase together with Figure 9 presents the time evolution of H for both I continuous downshifting. and D for fetches from 2.4 to 5.5 m. Note that for clarity In D, Fig. 7, as in I, shortly after the speed transi- the curves have been shifted vertically; the offset refer- tion, within 4 s, the energy level decreases while no shift ence positions are indicated in the diagram by horizontal of the spectral peak occurs. At time lapse 4 s the high line segments. For all fetches, H starts to increase or de- frequency energy seems to drop to the 4.6 m sÐ1 steady crease immediately after the change in the wind speed. ∆ state energy level, with a similar time scale as observed The time lapses t2 are estimated from the diagram and in I. Once again, the complete transition to the 4.6 m sÐ1 are indicated by the vertical dotted lines in Fig. 9. The steady state takes much longer than the rapid energy ad- vertical positions of the circles placed at the lower end of justment stage; the former was estimated to take about these lines give the fetch and therefore the curve con- 30 s at fetch 5.5 m, and less for shorter fetches. necting these circles is the ray on which the energy propa-

Adjustment of Wind Waves to Sudden Changes of Wind Speed 525 Fig. 7. Spectral development during decreasing wind speed case; - - - 7.1 m sÐ1 equilibrium spectrum; — decrease the wind ᮀ ᭹ ∆ Ð1 speed, t = 0; - - t = 1 s; - - t = 4 s ( t1); á á á 4.6 m s equilibrium spectrum. 40 degrees of freedom, 0.5 Hz resolution interval. Hanning window was applied to the data after the linear trend was removed. The vertical line indicates the lower frequency limit for the computation of the high frequency energy.

gates. The slope gives a rough estimate of the group speed. represent the anomalous wind waves independently gen- Comparing I and D, it is clear that the group speed is erated at the entrance to the tank due to disturbances by larger for I, possibly as a result of the larger drift current. the seiche, but are incidental to the principal processes For the decreasing wind speed case there are numer- discussed here. ous peaks which propagate at the wind-wave group ve- The time evolution of T, Fig. 10, once again illus- ∆ locity (indicated by arrows in Fig. 9). The spacings be- trates the existence of t2 and fetch limited equilibrium. tween these peaks are around 12 s, indicating that these The nomenclature of the figures is the same as the H dia- peaks are somehow related to the presence of the seiche gram presented earlier, Fig. 9. Once again, the trace of in the tank. However, these peaks are neither the seiche the circles depicts the ray on which the energy propa- itself nor any propagating wave train, since they propa- gates. An important difference, however, is observed be- gate at speeds close to the group velocity of the wind tween the time evolution of T and of H. Within a few waves corrected by the drift current. These peaks likely seconds after the change of wind speed, for both I and D,

526 T. Waseda et al. Fig. 8. Response of total energy Ehigh after a sudden increase (left column) and decrease (right column) of the wind speed at fetches 5.5 m, 4.7 m, 3.9 m and 3.1 m. Standard deviation of the mean is 5Ð15% of the mean value.

Fig. 9. Response of mean wave height H after a sudden increase (left column) and decrease (right column) of the wind speed at fetches 5.5 m, 4.7 m, 3.9 m and 3.1 m. Standard deviation of the mean is 4Ð8% of the mean value. An arrow indicates the propagation of a peak at the group velocity, which was possibly generated at fetch 0 m under the influence of the seiche. The arrows are equally spaced at 12 s intervals, the seiche period.

Fig. 10. Response of mean wave period T after a sudden increase (left column) and decrease (right column) of the wind speed at fetches 5.5 m, 4.7 m, 3.9 m and 3.1 m. Standard deviation of the mean is 1.5Ð2.5% of the mean value.

Adjustment of Wind Waves to Sudden Changes of Wind Speed 527 the magnitude of T remains nearly unchanged. This slight certainties that are probably the largest among the vari- delay in the response indicates the existence of a shorter ables in this experiment. time scale. The statistical significance of the two time It was suggested in Section 3 that under steady con- scales was confirmed by the usual technique of null hy- ditions a local equilibrium exists which determines the pothesis testing. wave steepness Hk (or ak, where a = H/2). In the short gravity wind wave regime in these experiments, we have 4.3 Short time scale adjustment suggested a relation H/gT2 = const. ≡ β, or equivalently, Within the short adjustment period, the high fre- the steepness of linear gravity wave varies with the wind quency energy grew to its final equilibrium value, Fig. 8. speed. The experiments (Fig. 11, left) indicate that for Because the response time of the friction velocity was the higher wind speed, 7.1 m sÐ1, β is independent of fetch β ≈ much longer, it suggests that the aerodynamic roughness with a value ( )7.1 0.25, confirming such a relation. It is related to wind-waves of all the frequencies at this scale. is remarkable that in this case following the increase in This is a speculation, however, since both the high-fre- wind speed, the equilibrium value of β is reached within quency wave energy and the friction velocity have un- about 4Ð5 s, much early than the fetch adjustment time in

Fig. 11. Response β ≡ H/gT2 after a sudden increase (left column) and decrease (right column) of the wind speed at fetches 5.5 m, 4.7 m, 3.9 m and 3.1 m. The curves are offset vertically for clarity. The dotted horizontal lines are the corresponding ak = 0.1 lines.

Fig. 12. Trajectory of the loci of non-dimensional mean wave height and wave period. Dotted line indicates Toba’s 3/2 law for energetic ocean waves, and the dot-broken line indicates the local equilibrium law satisfied by tank wind-waves. Data from fetch 5.5 m.

528 T. Waseda et al. this case, which is about 20 sec at the 5.5 m fetch. The speeds up to three times higher. The results are incorpo- change in the magnitude of β can partly, but not entirely, rated in Fig. 13. The two solid lines bracket the Kunishi’s be explained by the change in the drift current, particu- data (open circles) and may be explained by the variation larly at shorter fetches. The rapid jump of β may indicate in wind-speeds for his data. The two broken lines are the the local equilibrium process, which is discussed further same as those plotted in Fig. 12, indicating relation H = in Section 5. A similar result is obtained in D, Fig. 11 βgT2 as obtained from the unsteady experiment. Both (right), but only at the longest fetch. This indicates that Kunishi’s data and our data lie at approximately the same for the lower wind speed the waves at the earlier fetches level of β, showing that both steady-state and time-vary- are still in Stage 2 (see Fig. 5) with a continuous transi- ing wind wave fields follow the law of local equilibrium. ν tion to Stage 3 at 5.5 m, where u*H/ a reaches a value An observation like this gives great insight into the physi- well over 100. The data at earlier fetches at this lower cal processes responsible, which we suggest in the dis- wind speed are only relevant to Stage 2 processes. They cussion involve the balance between wind pumping and do confirm Kunishi’s data (Fig. 8(b)), which show a con- dissipation. tinuously increasing wave steepness with increase in wave height (or fetch). 4.4 Long time-scale adjustment The measured time variation of the wave parameters, As described earlier in Subsection 4.2, the energy H and T, at 5.5 m fetch are shown in Fig. 12 for both I propagation speed is greater than the value one may esti- and D, showing the tendency toward local equilibrium. mate using the group velocity alone and is understood as In this figure the straight lines through the data are dis- the sum of group and drift current speeds. The wave fre- placed, indicating a change of steepness with wind speed. quencies range roughly from 3 to 7 Hz, which gives a The transition from one equilibrium curve to the other group speed of 18Ð25 cm sÐ1. In the increasing wind speed ∆ takes around 5 s, as estimated earlier in Subsection 4.2. case, t2 at 5.5 m fetch was around 20 s and therefore the Note here that the conclusion would not change if a dif- group speed alone is not sufficient to explain the time ∆ ferent scaling variable, such as mean wind speed U, were scale of t2. In Fig. 14 we have plotted the ray diagram − ′′1/2 ∆ used instead of u* (or ( uv) ), since the equilibrium using the t2 obtained both by visual inspection and by law is independent of velocity scale (H/gT2 = const.), the statistical method. The propagation speed increases except for a slight dependence of the constant value on with fetch as the waves grow. The short solid lines indi- wind speed. cate the slopes of the ray as computed by cg + ud where These results may be compared with those of Kunishi the effective convection speed due to the drift current, ≈ which were taken at longer fetches (up to 18 m) and wind ud 0.3 u* was used. The estimated ud is between the measurements of Tokuda and Toba (1982) and that of Wu Ð1 (1975b). In the increasing case, ud = 9 cm s and for the Ð1 decreasing case, ud = 6 cm s was used. Those values were estimated using the observed value of u* at the steady state.

Fig. 14. Ray diagram constructed using the estimated long time ∆ scale t2 from the unsteady experiment. Circles indicate the observed value from different fetches. The short solid line 2 Fig. 13. Plot of gH/u* versus gT/u* from Kunishi’s data and has a slope of group velocity corrected for an appropriate this study for both increasing and decreasing wind speed drift velocity. The dotted line is a polynomial curve fit to case. Symbols: ᭺ Kunishi’s data. the observed data.

Adjustment of Wind Waves to Sudden Changes of Wind Speed 529 This observation suggests that the longer time scale side has the form of dissipation of wave energy by large ∆ t2 is due to energy propagation with fetch. The time, scale wave breaking (Tulin, 1996). In other words, the ∆ t2, necessary for the wave group to propagate to the fetch 3/2 law can be understood as the proportionality between X˜ is energy input from the wind and dissipation of energy by wave breaking. This is consistent with the proposal by ˜ Phillips (1985) that the wind input, dissipation, and the ∆ = X dx () t2 ∫ 5 non-linear energy transfer are in some fixed proportion 0 c g to each other. The application of this important physical assump- ∆ and obviously the duration t2 in (5) is an increasing func- tion tion of X˜ . This is verified in the observed growth dia- grams of both H and T, Figs. 9 and 10, as well as Fig. 14. wind input ∝ energy dissipation (8)

5. Discussion is not restricted to particular forms of wind input and en- In the current experimental study of short time scales, ergy dissipation, such as (7). Variations in wind input and we have shown that there seems to be a short and a long energy dissipation will lead to different local equilibrium adjustment time scale of wind waves in response to sud- laws between Hs and Ts. For the small scale waves and den increase and decrease of wind. The short time scale small wave ages studied here, taking the wind input to be is associated with transition from one quasi-equilibrium given by Plant’s (1982) empirical formula, proportional state to the other, represented by H ∝ gT2. This relation σ 2 to ET p(u*/cp) , and assuming that the dissipation varies differs from the 3/2 law that is believed to be satisfied β ρ 2 as ( u* cp), (8) leads to the steepness relation found in for energetic wind waves in local equilibrium with wind. Stage 3, What are the mechanisms that determine the differ- ence in the laws for Stage 3 and the ultimate Stage? Toba, ∝ β 2 Hs gTs . (9) who first introduced the local equilibrium concept, as- sumed a particular relationship between wind pumping ρ 2 The dissipation term u* cp may be interpreted as and the wind. He defined non-dimensional parameters de- follows: the dissipation of energy for waves of short wave- * ≡ 2 * ≡ scribing the local : H gH/u* and T gT/u*; length and very young wave age is primarily due to the * ≡ 3 ν and for the wind condition: u* u* /g . Writing the rate quasi-steady separation “cap” as described by Okuda of acquisition of wave energy in terms of the wave pa- (1982); the “cap” must be maintained in place by wind * rameters and assuming it proportional to u* , he obtained shear at the air-water interface, and therefore the dissipa- (3). On the other hand, Masuda and Kusaba (1987) gen- tion should be proportional to the product of wind shear eralized the concept of local equilibrium of wind waves ρ 2 u* and the wave speed cp, which is the rate at which the through a description of the sea state in terms of two non- wind shear does work. σ 4 σ σ dimensional parameters: ET p /g and pu*/g, where p is The use of different wind input formulas for differ- the frequency of the spectral peak and ET is the total en- ent stages may appear somewhat manipulative but it is in ergy. When the proportionality of these parameters are accordance with the recent theoretical assessment by assumed, Belcher and Hunt (1993), where the wind-wave growth rates are given to have quadratic dependence on u*/c, and E σ 4/g ∝ σ u /g, (6) T p p * the coefficients are functions of both roughness and u*/c. As can be inferred from their paper, the dependence of (3) is recovered. the coefficient on u*/c is such that for larger roughness We propose here a new general interpretation of both the resulting growth rates approach linearity with respect Toba’s and the steepness laws based on the balance be- to u*/c. The roughness in this small wind-wave tank tween wind input and dissipation. Equation (6) is in this ν 2 (ku*/ ~ O(10 )) corresponds to an aerodynamically way given both a physical interpretation and justification. smooth to transitional flow (Wu, 1969), and therefore the σ By multiplying both sides of (6) by ET p, and rewriting application of Plant’s law in this regime is considered ap- the left-hand side in terms of steepness (ak)s, we obtain propriate. On the other hand, in the ultimate regime, where the 3/2 law applies, the application of Kahma’s or σ 2 ∝ σ ET p(ak)s ET p(u*/cp) (7) Snyder’s law is called for. A lower bound for the short adjustment time to this and the right-hand side now has the form of the rate of local equilibrium may be estimated from the conserva- change of the energy of energetic waves by wind forcing tive assumption that the wind pumping supplies the re- (Snyder et al., 1981; Kahma, 1981), while the left-hand quired changes in energy without consideration of dissi-

530 T. Waseda et al. ∆ Table 1. Estimated lower bounds of t1 for different wave ages.

Estimated Measured ∆ ∆ ∆ cp/u* t1/T t1/T t1 1/T UCSB 1Ð1 1/2 2Ð12 15~20 4 s 5Ð7 Hz Shirahama 9 ~100 210~420 7 min 0.5Ð1 Hz

pation. Then,

∆t  E  1 = ln n  ()β / f ()10 T  Eo 

where En and Eo are the at the new and old quasi- balance with the wind, respectively, T = 1/f is the wave period, and β is an empirical wind growth rate, based on Plant (1982) for the laboratory case, Snyder et al. (1981) for the field.* By using (10), for a 50% increase in the wind speed, the number of wave periods necessary for the adjustment was estimated (Table 1). ∆ The t1 for the field case is inferred from figure 3 of Toba et al. (1988). Upon taking into account in that case that approximately half of the wind pumping is lost to dissipation, the estimate begins to agree with those ob- servations of energetic waves. Uncertainties about cp (which was not directly measured) and in Plants formula create a large spread in the estimates for the laboratory measurements. Upon doubling these to take into account the loss of wind pumping effectiveness due to dissipa- tion, the estimate then includes the measured range. It ∆ thus seems likely that the initial relaxation time, t1, is accounted for by the finite rate of wind pumping. Note that this time, when measured in terms of the wave pe- riod T, is not in fact very short. Fig. 15. Time evolution of the net-source term during the short adjustment time after an abrupt increase of wind speed. Finally, a brief remark on the nonlinear energy trans- fer among wave modes should be made. The bulk formu- las such as (3) and (4) implies the downshifting of spec- tral peak as the wave energy increases. From a spectral point of view, this downshifting can be characterized by speed, the net-source term first loses its negative peaks a particular shape of the so-called net-source term that is (Fig. 15(b)), then broadens (Fig. 15(c)), and becomes dual the spectral representation of the energy growth of each peaked (Fig. 15(d)), indicating a broadening of the spec- ∆ wave modes including effects of wind pumping and dis- tral shape. Then, after the short adjustment time, t1, the sipation. The net-source term was estimated from two net-source term retains its original shape, Fig. 15(f). This adjacent wave wires at the very end of the tank, by finite rather complicated change in the shape of the net-source differencing the spectrum in space and in time (Fig. 15). term likely suggests alteration of the non-linear source During the steady state, Fig. 15(a), the net-source term term during this adjustment phase and it is caused not appears similar to that of the young wind waves (e.g. just by the change in the wind-pumping and dissipation Komen et al., 1994). After the sudden change of wind source terms. This last point needs further investigation, and may involve scale separation of each source term, such as that done for the dissipation term by Tolman and β ± 2 π Chalikov (1996). This is beyond the scope of the present *Plant’s formula: /f = (0.04 0.02)(u*/c) (2 ) and the β π study. Snyder et al.’s formula: /f = 0.0003(28(u*/c) Ð 1)(2 ).

Adjustment of Wind Waves to Sudden Changes of Wind Speed 531 ∆ 6. Concluding Remarks rium within the time t1 after the wind change, although Remarkably little has hitherto been understood about the low frequency spectrum changes continuously the basic response of wind waves to changing wind con- throughout the fetch adjustment process. This demon- ditions, despite the great importance of the subject for strates that, at least for gravity waves of this small scale, the . The reason for this lies in the relative the high frequency spectrum is controlled by the wind, underutilization of laboratory experiments. The present independent of energy levels reached near the peak. studies were motivated by a desire to better interpret gust Remote sensing has provided ample evidence of the effects in the ocean. changes in the short waves which are brought about by The experiments have led to a reasonably coherent energy transfer to the waves, not by the wind, but by sub- view of adjustment processes. The adjustment to sudden surface disturbances such as internal waves and oceanic changes in wind speed was observed to involve two dis- fronts. The evidence here is that local equilibrium is crete stages. reached rather quickly during energy transfer and the bal- In the first stage the sudden imbalance in the wind ance underlying local equilibrium may be applicable in pumping results in an adjustment in the wave energy in a the latter cases. ∆ time t1/T of O(10) in the laboratory. This initial adjust- Finally, the present work was necessarily limited in ment is completed when the waves reach “local equilib- scope and scale. The presence of lateral boundary due to rium.” In the case of energetic ocean waves, this local the tank walls acts as a wave-guide and would limit the equilibrium is described by the 3/2-power law, (3). In the directionality of the wind-wave spectrum. The breaking case of short wind waves, but in the gravity regime, the mechanism of the short waves studied here differs from present experiments, together with earlier ones, suggest that of long ocean waves since the surface tension is not another law, (4), in which the wave steepness depends negligible at this scale. It would clearly be highly desir- weakly upon the wind speed, but not specifically upon able that further controlled studies of these non-steady the wave age, as in the case of the 3/2-power law (cf. processes be carried out in extension of the present re- Bailey et al., 1991). Insight into the origin of these two sults, and particularly over a wider range of scale. laws of local equilibrium is provided here by the sugges- tion that they each describe a local balance between wind Acknowledgements pumping and dissipation of wave energy; indeed, upon We express our thanks to Mr. Leif Andersen for his using laws for the latter which may be appropriate to the important participation in the experiments. We also thank two separate regimes, energetic and short waves, it is Dr. Ian Jones for his comments on the manuscript and the demonstrated that the form of each of the two separate anonymous reviewers for their valuable comments. An local equilibrium laws follows from the pumping-dissi- essential part of this work was done at the OEL/UCSB, pation balance. as partial fulfillment of the first author’s Ph.D. work with In the case where the waves vary with fetch, as in support from the Ocean Technology Program of the Of- these laboratory experiments, or for near coastal waves fice of Naval Research, Thomas Swean, Jr., program in the presence of an offshore wind, a longer adjustment manager, and the Advanced Sensor Applications Program process is required, as the wave originating at the shore of ISSO, Donna Kulla, program manager. A part of the moves offshore continuously, acquiring energy from wind manuscript preparation was done at the International Pa- pumping which moves at the local group velocity along cific Research Center (IPRC). IPRC is partially funded rays in space time, dx/dt = cg. The group velocity itself by the Frontier Research System for Global Change. This grows concurrently with frequency downshifting, which manuscript is SOEST contribution 5433 and IPRC con- is embedded in wave prediction methods. Here this proc- tribution 83. ess was observed in the variation of H and T in time and fetch. Its consequence is that, while the wave slopes come References ∆ Bailey, R. J., I. S. F. Jones and Y. Toba (1991): The steepness into local equilibrium within the time t1, the fetch-con- trolled local energy, and peak frequency levels only reach and shape of wind waves. J. Oceanogr. Soc. Japan, 47, 249Ð an equilibrium at all fetches, following the much longer 264. fetch adjustment time, ∆t , which itself increases with Battjes, J. A., T. J. Zitman and L. H. Holthuijsen (1987): A 2 reanalysis of the spectra observed in jonswap. J. Phys. fetch. ∆ Oceanogr., 17, 1288Ð1295. During the entire fetch adjustment time, t2, it was Belcher, S. E. and J. C. R. Hunt (1993): Turbulent shear flow ∝ 2 remarkable to observe that the relation H gT was ob- over slowly moving waves. J. Fluid Mech., 251, 109Ð148. ∆ served at all times after t1 and at all fetches (for 7.1 Donelan, M. A., M. Skafel, H. Graber, P. Liu, D. Shwab and S. m/sec, wind speeds). It was also observed in spectral Venkatesh (1992): On the growth rate of wind-generated measurements that the high frequency spectrum (above waves. Atmosphere-Ocean, 30, 457Ð478. twice the peak frequency) reaches a permanent equilib- Ebuchi, N., Y. Toba and H. Kawamura (1992): Statistical study

532 T. Waseda et al. on the local equilibrium between wind and wind waves by on the generation of small water waves by wind. J. Fluid using data from ocean data buoy stations. J. Oceanogr., 48, Mech., 35(4), 625Ð656. 77Ð92. Snyder, R. L., F. W. Dobson, J. A. Elliott and R. B. Long (1981): Hanson, J. L. and O. M. Phillips (1999): Wind sea growth and Array measurement of atmospheric pressure fluctuations dissipation in the open ocean. J. Phys. Oceanogr., 29, 1633Ð above surface gravity waves. J. Fluid Mech., 102, 1Ð59. 1648. Toba, Y. (1972): Local equilibrium in the air-sea boundary proc- Janssen, P. A. E. M. (1986): The period-doubling of gravity- esses I. On the growth process of wind waves. J. Oceanogr. capillary waves. J. Fluid Mech., 172, 531Ð546. Soc. Japan, 28, 109Ð121. Johnson, H. K., J. Hojstrup, H. J. Vested and S. E. Larsen (1998): Toba, Y. (1973): Local balance in the air-sea boundary proc- On the dependence of sea surface roughness on wind waves. esses, III. J. Oceanogr. Soc. Japan, 29, 209Ð220. J. Phys. Oceanogr., 28, 1702Ð1716. Toba, Y. and N. Ebuchi (1991): Sea-surface roughness length Kahma, K. K. (1981): A study of the growth of the wave spec- fluctuating in concert with wind and waves. J. Oceanogr. trum with fetch. J. Phys. Oceanogr., 11, 1503Ð1515. Soc. Japan, 47, 63Ð79. Kahma, K. K. and C. J. Calkoen (1992): Reconsiling discrep- Toba, Y., K. Okada and I. S. F. Jones (1988): The response of ancies in the observed growth of wind-generated waves. J. wind-wave spectra to changing winds. Part I: increasing Phys. Oceanogr., 22, 1389Ð1405. winds. J. Phys. Oceanogr., 47, 1231Ð1240. Kawai, S. (1979): Generation of initial wavelets by instability Tokuda, M. and Y. Toba (1982): Statistical characteristics of of a coupled shear flow and their evolution to wind waves. individual waves in laboratory wind waves. II. Self-con- J. Fluid Mech., 93(4), 661Ð703. sistent similarity regime. J. Oceanogr. Soc. Japan, 38, 8Ð Kawai, S., K. Okada and Y. Toba (1977): Field data support of 14. σÐ4 three-seconds power law and gu* -spectral form for grow- Tolman, H. L. and D. Chalikov (1996): Source terms in a third- ing wind waves. J. Oceanogr. Soc. Japan, 33, 137Ð150. generation wind wave model. J. Phys. Oceanogr., 26, 2497Ð Komen, G. J., L. Cavaleri, M. Donelan, K. Hasselmann, S. 2518. Hasselmann and P. A. E. M. Janssen (1994): Dynamics and Tseng, R. S., Y. H. L. Hsu and J. Wu (1992): Methods of meas- Modelling of Ocean Waves. Cambridge University Press. uring wind stress over a water surface—discussions of dis- Kunishi, H. (1963): An experimental study on the generation placement height and von karman constant. Boundary-Layer and growth of wind waves. Disaster Prev. Res. Inst. Kyoto Meterorol., 58, 51Ð68. Univ. Bull., 61, 1Ð41. Tulin, M. P. (1996): Breaking of ocean waves and downshifting. Kusaba, T. and A. Masuda (1988): Wind-wave spectra based p. 177Ð190. In Waves and Nonlinear Processes in Hydro- on the hypothesis of local equilibrium. J. Oceanogr. Soc. dynamics, ed. by J. Grue, B. Gjevik and J. E. Weber, Kluwer Japan, 45, 45Ð64. Academic Publishers. Masuda, A. and T. Kusaba (1987): On the local equilibrium of Waseda, T. (1997): Laboratory study of wind- and mechani- winds and wind-waves in relation to surface drag. J. cally-generated water waves. Ph.D. Thesis, University of Oceanogr. Soc. Japan, 43, 28Ð36. California, Santa Barbara. Mitsuyasu, H. and K. Rikiishi (1978): The growth of duration- Wilson, B. W. (1965): Numerical prediction of ocean waves in limited wind waves. J. Fluid Mech., 85(4), 705Ð730. the North Atlantic for December, 1959. Deut. Hydrogr. Z., Okuda, K. (1982): Internal flow structure of short wind waves 18, 114Ð130. part I. On the internal vorticity structure. J. Oceanogr. Soc. Wu, J. (1969): Wind stress and surface roughness at air-sea in- Japan, 38, 28Ð42. terface. J. Geophy. Res., 74, 444Ð455. Phillips, O. M. (1958): The equilibrium range in the spectrum Wu, J. (1975a): Effects of pulsating wind on velocity profiles of wind-generated waves. J. Fluid Mech., 4(4), 426Ð434. and microstructures. J. Phys. Oceanogr., 5(4), 782Ð789. Phillips, O. M. (1985): Spectral and statistical properties of the Wu, J. (1975b): Wind-induced drift currents. J. Fluid Mech., equilibrium range in wind-generated gravity waves. J. Fluid 68, 49Ð70. Mech., 156, 505Ð531. Young, I. R., S. Hasselmann and K. Hasselmann (1987): Com- Plant, W. J. (1982): A relation between wind stress and wave putations of the response of a wave spectrum to a sudden slope. J. Geophys. Res., 87(C3), 1961Ð1967. change in wind direction. J. Phys. Oceanogr., 17, 1317Ð Plate, E. J., P. C. Chang and G. M. Hidy (1969): Experiments 1338.

Adjustment of Wind Waves to Sudden Changes of Wind Speed 533