Adjustment of Wind Waves to Sudden Changes of Wind Speed
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Journal of Oceanography, 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 Ocean 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, –1 –1 ⋅ 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 wave height (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 energy 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 Sea states are typically characterized by two param- states offshore of straight coastlines can be uniquely de- eters: significant wave height, 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 internal wave 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 s–1 (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 –1 tank, changing wind speeds from Ulow = 4.6 m s to range of 10 cm. A pitot-static tube and pressure trans- –1 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 s–1 to 4.6 m s–1. 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.