2015 Volume 91 Issue 4 Pages 109-130
The wind blowing over sea surface generates tiny wind waves. They develop with time and space absorbing wind energy, and become huge wind waves usually referred to ocean surface waves. The wind waves cause not only serious sea disasters but also take important roles in the local and global climate changes by affecting the fluxes of momentum, heat and gases (e.g. CO2) through the air-sea boundary. The present paper reviews the selected studies on wind waves conducted by our group in the Research Institute for Applied Mechanics (RIAM), Kyushu University. The themes discussed are interactions between water waves and winds, the energy spectrum of wind waves, nonlinear properties of wind waves, and the effects of surfactant on some air-sea interaction phenomena.
The modern studies of wind waves can be said to have started with a pioneering study by Sverdrup and Munk (1947).1) Before their study wind waves were considered to be too disordered to be treated mathematically. It is said that Lord Rayleigh remarked, “The basic law of the seaway is the apparent lack of any law”.2)
To solve the difficult problem Sverdrup and Munk (1947)1) introduced a concept of significant waves that have a kind of mean wave height and mean wave period to describe quantitatively wind waves which show random properties. They also introduced a concept of energy balance in a wave system to understand and describe the wave evolution. Furthermore they determined empirical relationships for the evolution of wind waves by using accumulated wave data.
Although these concepts are common knowledge today, it is really surprising that such ideas were proposed when the similar ideas or studies were absent except for primitive and purely empirical formulas. After that time a great many papers were written following their initiative, and fundamental properties of wind waves were gradually clarified.
On the other hand, in the beginning of 1950s a group of New York University led by W. J. Pierson and G. Neumann started to study the spectrum of wind waves and developed a wave forecasting method by using wave spectra (Pierson, Neumann and James, 1955).3) In 1950s the progress in the study of random noise greatly affected the study of wind waves to describe their statistical properties.
As for the dynamics of wind waves, Ursell (1956)4) wrote a very important review on wave generation by wind. After the critical review of the previous studies on this problem he concluded that the present state of our knowledge is profoundly unsatisfactory. The review gave a strong impact to fluid-dynamicists and physical oceanographers. Stimulated by the review two outstanding theories on the generation of wind waves by wind were presented simultaneously by Miles (1957)5) and Phillips (1957).6) Many studies followed them to test the theories experimentally or to improve their theories, and the field of studies on wind waves became very active and exciting.
In 1965 I started fundamental studies on wind waves in the Research Institute for Applied Mechanics (RIAM), Kyushu University and continued the study until my retirement from the RIAM in 1992. It was very gratifying that I could continue the studies on wind waves in such an active and rapidly developing field of the study.
In the present review the results of typical studies of our group in the RIAM are discussed. The following eight themes are selected chronologically and discussed: interactions between water waves and winds, nonlinear energy transfer in the spectrum of wind waves, the growth of the spectrum of wind waves, the directional spectrum of ocean surface waves, the dispersion relation of random water waves, wind-induced growth of water waves, the momentum transfer from winds to waves, and the effects of surfactant on some air-sea interaction phenomena.
In the discussion of each study, the motivation and background of the study are included as much as possible to explain the state of development of scientific knowledge in the study of wind waves at the time.
In 1965 I moved from the Port and Harbor Research Institute, Ministry of Transportation, to the RIAM, Kyushu University. In those days I was much interested in the experimental verification of Miles’ (1957)5) theory for wave generation by wind, but there were no experimental facilities at all in our laboratory. In such a time I had an opportunity to use a big wave tank: 70 m long, 8 m wide and 2.9 m deep. Fortunately the tank was equipped with a mechanical wave generator and a wind blower.
By using the wave tank I began to measure the wind-induced growth of mechanically-generated regular waves. Immediately after the start of the study, however, I found that the facility was not suitable for the study: the wind diverged horizontally and vertically and wind field became non-uniform along the tank. This was due to the absence of a ceiling and side walls in the wind passage of the tank. In such a condition we lost the merit of the laboratory experiment to study the phenomenon under simple and controlled conditions.
Unexpected new findings.Although the original purpose of the study encountered the difficulty, I discovered the very interesting phenomenon that wind waves in a generation area in the tank attenuate immediately after the passage of the regular waves propagating in the direction of the wind. Furthermore, the attenuation of wind waves seemed to depend only on the local steepness H/L of the regular waves, where H is the wave height and L is the wave length of the regular waves.
In order to clarify the interesting phenomenon careful measurements were repeated by changing successively wave steepness and wind speed. A typical result is shown in Fig. 1, where the relation between a normalized wind wave energy, E2/(E2)0, and the local steepness of the regular waves, H/L is shown. Here E2 is the energy of wind waves co-existing with regular waves and (E2)0 is the energy of wind waves unaffected by regular waves, i.e., the energy of wind waves before the arrival of the regular waves. As shown in Fig. 1 the normalized energy of wind waves decreases gradually with the increase of the steepness of regular waves; it becomes almost negligible near the steepness 7 × 10−2.
Normalized wind wave energy E2/(E2)0 versus steepness H/L of swell (mechanically-generated regular waves).7) E2: energy of wind waves co-existing with swell, (E2)0: energy of wind waves without swell, H: wave height of the swell, L: wave length of the swell, wave 2–6 etc.: index of the swell (cf. original paper).
This quite new finding was published in the report of the RIAM (Mitsuyasu, 1966).7) Although I expected various responses to the new finding, there were no responses at all not only from Japanese researchers but also from foreign researchers.
Phillips and Banner’s mechanism.8)In 1974, eight years after the publication of Mitsuyasu (1966),7) I received a letter from Professor Phillips of the Johns Hopkins University. His letter informed me that he conducted a study similar to ours and obtained almost the same results, and asked me about the recent progress in the studies relating to our previous one.
After the study of Mitsuyasu (1966),7) I repeated the experiment on the same phenomena under more controlled condition: the wind distribution was made uniform by improving the experimental setup. However, the results were almost the same to those of our previous study. I informed to Professor Phillips on this matter.
Immediately after the exchange of our letters, he published the paper, Phillips and Banner (1974)8) in Journal of Fluid Mechanics (JFM). Their experimental results were almost the same to Mitsuyasu (1966),7) but they proposed an important mechanism for the attenuation of the wind waves by steep long waves propagating in the direction of the wind. The point of the mechanism is that the wind-induced surface current is nonlinearly augmented at the crest of the long waves, and short wind waves tend to break and lose their energy at the crest of the long waves before reaching the critical breaking limit.
Theoretical estimates of the reduction of the wind wave energy showed very good agreements not only with their measurement but also our measurement. Therefore, it seemed that the problem of the attenuation of the short waves by long waves was solved, though some disagreement between the theory and experiment was reported on the wind-speed dependence of the attenuation of wind waves by long waves (Wright, 1976).9) Our later study, Mitsuyasu and Maeda (2002)10) on the interaction between winds and waves which was done in a big wind-wave tank generally supported the Phillips and Banner’s mechanism too.
Further problems.Mitsuyasu and Yoshida (2005)11) studied the attenuation of swell (mechanically-generated long waves) by opposing wind. In the experiment an unexpected phenomenon was found: wind waves did not attenuate but rather were augmented by the opposing swell. This is contradictory to the mechanism of Phillips and Banner (1974),8) because their mechanism predicts the attenuation of wind waves even for the opposing swell. The result showed the need for the study of another mechanism.
Chen and Belcher (2000)12) proposed another mechanism for the attenuation of wind waves by a long wave. Their idea is that when a long wave travelling in the wind direction comes into a wind area, the long wave absorbs a part of the wind momentum for its development, leading to the reduction of the momentum transferred into wind waves. The theoretical prediction obtained from their model agreed fairly well with the observations by Mitsuyasu (1966)7) and Phillips and Banner (1974).8) However, their model fails to explain the effect of the opposing swell, because the attenuation of the opposing swell requires a transfer of wind momentum, the momentum available for transfer into the wind waves is reduced, leading to the attenuation of wind waves, which contradicts the observation by Mitsuyasu and Yoshida (2005).11)
Another mechanism for the suppression of wind waves by swell, the nonlinear interaction between wind waves and swell can be considered. Masson (1992)13) studied the effects of this mechanism more than twenty years ago. However, her results showed that the interaction is generally negligible unless the two spectral peaks are so close that the spectrum can be hardly qualified as bimodal. The condition for the effective interactions is very different from that in the laboratory experiments. Furthermore, her study was concerned with the phenomenon without wind action. Therefore her results cannot explain the phenomena observed by Mitsuyasu (1966)7) and Phillips and Banner (1974).8)
Regarding the interaction between swell and wind waves, recently Tamura et al. (2009)14) published an interesting paper showing that nonlinear coupling between wind waves and swell under the wind action lead to a freakish sea state. The result suggests that if the nonlinear coupling between wind waves and swell under the wind action is studied by changing their frequency difference, their directional difference and the wind speed, we will be able to get some interesting results.
In connection with the interaction between wind waves and swell, Cheng and Mitsuyasu (1992)15) found a very curious phenomenon that the surface drift current generated by the wind was augmented by opposing swell of large steepness.
Although the studies on the interaction among the wind, short wind waves and long waves (swell) have a long history, the problem seems to be unsolved theoretically and further studies are needed: interactions between winds and waves, nonlinear interaction between short waves and long waves, and interactions between waves and currents (organized or turbulent) will provide some breakthrough.16),17)
To a first approximation wind waves which have random properties can be considered as a linear superposition of infinitely many wave components with different amplitudes, frequencies, phases and propagation directions as shown schematically in Fig. 2. Such a linear model describes fairly well statistical and dynamical properties of wind waves and it makes the spectral description of wind waves effective.
Spectral model of wind waves. Wind waves (left) are composed of infinitely many elementary waves with different amplitudes, frequencies, phases and propagation directions respectively (right).
However, there are several important phenomena which require consideration of nonlinear effects. The nonlinear energy transfer in the spectrum of wind waves is one of those that play an important role in the evolution of wind waves.
Emergence and progress in the theory.In 1961 a conference Ocean Wave Spectra,19) was held at Easton, Maryland in the US. In the conference the theories on the nonlinear energy transfer in random water waves were presented by Phillips (1963),20) and Hasselmann (1963).21) Phillips (1981)22) described the atmosphere of the meeting as follows: the reaction from some well-known and senior people in the field was, to my astonishment, vigorous and hostile, the flat disbelief that different components could exchange energy at all. My mathematics was certainly very primitive but Michael Longuet-Higgins, to whom I had sent a copy of the manuscript, offered his cautious and most welcome support and Klaus Hasselmann, whom I met here for the first time, was working quite independently along the same lines but with greater generality, so that sharp, encounter ended with a stand-off.
Apart from the disbelief of his theory by quite senior people, it is interesting that Phillips did not know the work of Hasselmann at all. Such a thing is difficult to appreciate in the present day when information can be transferred quickly and widely. This is probably due to the fact that Hasselmann did not publish his now-famous papers on nonlinear energy transfer (Hasselmann, 1962, 1963)23),24) until a year or so after the meeting.
After the meeting and the publications of the papers, Phillips (1963)20) and Hasselmann (1962, 1963),23),24) the topics of the nonlinear interaction in random water waves attracted the interests of many fluid-dynamicists and physical oceanographers and many papers were published (cf. Phillips, 1977 and Komen et al., 1994).25),26)
Experimental studies.In contrast to the progress in theoretical studies on nonlinear wave interactions, quite a few experimental studies were done to see whether the evolution of the spectrum of water waves follows to the theoretical prediction or not. Simultaneously Longuet-Higgins and Smith (1966)27) and McGoldrick et al. (1966)28) clarified theoretically and experimentally the existence and consequence of the nonlinear resonant interaction in two crossing waves. However, there were no studies which compared the Hasselmann’s theory23),24) with the evolution of the wave spectrum, though the inference was made later by Hasselmann et al. (1973).29)
One day when I was reading a paper of Uberoi (1963)30) on the experimental study of homogeneous turbulence, his technique in the study of turbulence gave me an idea that the same technique can be applied to the study of the nonlinear energy transfer in random water waves. His technique is that of measuring turbulence spectra at two cross sections in a wind tunnel. From the difference of the spectra at the two sections he determined the nonlinear energy transfer in the turbulence spectrum after correction for viscous energy dissipation.
Almost the same techniques can be used for the determination of the nonlinear energy transfer in the spectrum of random water waves. I selected the data of the wave spectra measured in a decay area where the wind was not blowing (Mitsuyasu and Kimura, 1964).31) In this case, since the energy input from the wind to the waves and the energy dissipation due to wave breaking were absent, and the effects of the wind-induced drift current were very small, we can determine the nonlinear energy transfer in the spectrum of wind waves in the decay area by taking the difference of the spectra in two sections and by compensating for viscous energy dissipation.
Figure 3(a) shows an example of the result of our study (Mitsuyasu, 1968),18) where the nonlinear energy transfer in the spectrum of wind waves in decay area, which is determined through the above procedures, is compared with the theoretical computation. In the figure the wave spectrum at the upward station is also shown. As shown in Fig. 3(a) the agreement between the theory and the experiment is fairly good and we can see the important role of the nonlinear energy transfer in the evolution of the spectrum of random water waves.
(a) Comparison of the computed nonlinear energy transfer with observational data.18) ϕ(f): frequency spectrum at the upward station, $(\frac{\partial \phi }{\partial \text{t}})_{\text{measured}}$: measured nonlinear energy transfer, $(\frac{\partial \phi }{\partial \text{t}})_{\text{calculated}}$: theoretical one computed by using the spectrum at the upward station (Note that ϕ(f) is used in this figure instead of Ψ(f) for frequency spectrum). (b) Comparison of the computed nonlinear energy transfer with observational data.33) Ψ(f): frequency spectrum at the upward station, $(\frac{\partial \Psi }{\partial \text{t}})$: nonlinear energy transfer. The solid curve is measured transfer, broken curve and dotted curve are theoretical ones computed by using respectively the spectrum at the upward station and the mean spectrum of those at the upward station and the downward station.
However, one problem remained. At that time since I hadn’t a program for the accurate computation of the nonlinear energy transfer in the Hasselmann’s model, I used a parameterized equation by Barnett (1966)32) for the computation of the nonlinear energy transfer. However, the parameterized equation was not necessarily specialized for our spectrum, and Barnett himself pointed out this problem in his private communication. I was very anxious about this problem for the long time after the study.
The problem was solved fourteen years later by the study of Masuda (1980).33) He developed a very accurate computation method of the nonlinear energy transfer for the Hasselmann’s model and computed accurately the nonlinear energy transfer in our measured spectra. In Fig. 3(b) wave spectrum Ψ(f) at the upward station, the observed nonlinear energy transfer determined with the same techniques used in the previous study, and the nonlinear energy transfers computed by using two different spectra: the spectrum at the upward station (dotted line) and the mean spectrum of those at the upward and the downward stations (broken line). As shown in Fig. 3(b) the agreement between the theory and the measurement is much better than that in Fig. 3(a). The result shows clearly that the nonlinear energy transfer plays an important role in the evolution of the spectrum of random water waves.
Further problem.After the study of Mitsuyasu (1968)18) there were quite a few similar studies. As far as I know, Zhao et al. (2006)34) is the only study similar to Mitsuyasu (1968).18) They measured the spectra of wind waves in the decay area of their laboratory tank and determined the nonlinear energy transfers using the techniques similar to those of Mitsuyasu (1968).18)
Comparisons of their measured results with the theoretical computations showed that some data agreed well with the theoretical computation but many other data were different from the theoretical computations, though the reason was not clear. Furthermore, even for the data showing good agreement between the theory and experiment, the peak frequency for the computed positive transfer is always a little higher than that for the measured one. The similar trend can be seen in the result obtained by Masuda (1980)33) shown in Fig. 3(b).
These results show the need for further theoretical and experimental studies to clarify the phenomena in the nonlinear energy transfer in the wind wave spectrum. In the laboratory experiment, in order to study the purely nonlinear energy transfer in the wave spectrum, we need to take care of other phenomena affecting the evolution of the wave spectrum, such as the drift current, the viscous energy dissipation of the composite waves and the contamination of the water surface.
In the theoretical study, we will need to develop a new theory in which the wind effects such as the wind stress and the wind-induced surface current are considered. Such wind effects are inevitable for the wind-induced waves in generation area.
Since Neumann (1952)37) and Pierson (1953)38) introduced the use of a spectral model to describe oceanic wind waves in the 1950s, many studies had been done on the spectrum of wind waves. In addition to Phillips’ (1958)39) famous equilibrium spectrum and the Pierson and Moskowitz’s (1964)40) spectrum for the fully developed wind waves in the ocean, there were quite a few studies on the detailed properties of the wind wave spectrum and its generating conditions.
We planned to clarify the growth of the spectrum of wind waves under accurately controlled conditions (Mitsuyasu, 1968).35) The wave tank used for the study is the same wave tank used in the study of Mitsuyasu (1966).7) In this study, however, by setting a side wall and a ceiling to a part of the wave tank, it was partially converted into a conventional wind-wave flume to simplify the generating condition. Furthermore, in order to obtain the wind and wave data in the longer fetches, observations of the wind and waves were done in Hakata bay, Fig. 4(a), by using a observation platform, Fig. 4(b), where we could get the wind and wave data over the fetch of about 4.5 km in the direction of the north wind. We obtained great many data of the wind and wind waves both in the tank and in Hakata Bay.
(a) Site of the wind and wave observations: Hakata bay.35) D.C.M.O.: Fukuoka district central meteorological observatory. (b) The platform for the wind and wave observations in Hakata bay.
In order to show the growth of the wind wave spectrum by the wind quantitatively we selected two scale parameters, spectral peak frequency fp and total spectral energy E of the wind wave spectrum, where E does not include ρg (ρ: density of water, g: gravitational acceleration) and corresponds to $\overline{\eta ^{2}}$ (η: elevation of the wind wave surface). These two parameters of the wind wave spectrum were related, in dimensionless forms, to the fetch X and the friction velocity of the wind U* which was determined from the wind profile over the water surface. We obtained the following simple relations;
| \begin{equation} \frac{\mathrm{g}\sqrt{\mathrm{E}}}{\mathrm{U}_{*}^{2}} = 1.31 \times 10^{-2}\left(\frac{\mathrm{gX}}{\mathrm{U}_{*}^{2}} \right)^{0.504}, \end{equation} | [1] |
| \begin{equation} \frac{\mathrm{U}_{*}\mathrm{f}_{\text{p}}}{\mathrm{g}} = 1.00 \times \left(\frac{\mathrm{gX}}{\mathrm{U}_{*}^{2}} \right)^{-0.330}, \end{equation} | [2] |
These relations are usually called the fetch relations, and the relations are satisfied almost universally for the wind waves in the generation area. Equation [1] means that the total energy E of the wind waves is proportional to the fetch X and $\text{U}_{*}^{2}$ (= τ wind stress/ρa density of the air), and these properties are physically reasonable. Equation [2] means that the wave age Cp/U* (the ratio of the wave velocity and the wind friction velocity) increases slowly with the increase of the dimensionless fetch using the equation g/2πfp = Cp where Cp is the phase velocity of the spectral component at the spectral peak frequency fp.
The fetch relations for the significant wave height and period were obtained firstly by Sverdrup and Munk (1946).1) Later Wilson (1965)41) improved the relations by using many reliable data of the wind and waves. In order to compare the newly obtained fetch relations for the wave spectrum, [1] and [2], with the fetch relations for the significant waves by Wilson (1965),41) Mitsuyasu (1973)42) converted the fetch relations [1] and [2] to those for the significant waves by using statistical theory and an empirical relation. It was found that the fetch relations converted from [1] and [2] were almost the same to those obtained by Wilson (1965).41)
JONSWAP (the Joint North Sea Wave Project).29)While we were studying the growth of the spectrum of wind waves in the laboratory tank and in Hakata Bay, a big project observing wind waves in the ocean was going on in the North Sea. The project JONSWAP was organized by Kraus Hasselmann in Germany and many well-known wave researchers in the Europe and the US attended the project.
Five years after the publication of our papers35),36) on the growth of the spectrum of wind waves, the results of the JONSWAP were published by Hasselmann et al. (1973).29) In the report they presented the fetch relations obtained by using the wind and wave data in the North Sea. In the relations they used the 10 m height wind speed U10 as a representative wind speed. We convert the 10 m height wind speed U10 into the friction velocity of the wind U* by using the empirical relation $\text{U}_{*}^{2} = 10^{ - 3}$ U102 (Hasselmann et al., 1973)29) then their relations become
| \begin{equation} \frac{\mathrm{g}\sqrt{\mathrm{E}}}{\mathrm{U}_{*}^{2}} = 1.26 \times 10^{-2}\left(\frac{\mathrm{gX}}{\mathrm{U}_{*}^{2}} \right)^{0.50}, \end{equation} | [3] |
| \begin{equation} \frac{\mathrm{U}_{*}\mathrm{f}_{\text{p}}}{\mathrm{g}} = 1.08 \times \left(\frac{\mathrm{gX}}{\mathrm{U}_{*}^{2}} \right)^{-0.33}. \end{equation} | [4] |
The report (Hasselmann et al., 1973)29) contains many important results. They proposed a standard spectrum, JONSWAP spectrum, for the fetch-limited wind waves, which simulate well the sharp spectral form in the limited fetches as compared to the fully developed spectrum obtained by Pierson and Moskowitz (1964).40) Furthermore they showed the important effects of the nonlinear energy transfer in the evolution of the wind wave spectrum by examining the downwind evolution of the measured spectra and by applying the theory of Hasselmann (1962, 1963).23),24) The presentation and discussion of the energy balance equation for the evolution of the wind wave spectrum also greatly contributed to the progress in the study of wind waves.
3/2 power law.A little before the publication of the JONSWAP report, Toba (1972)43) clarified an important property of the wind waves in the generation area: the 3/2 power law. The law can be written, using the original expression, as
| \begin{equation} \mathrm{H}^{*} = 0.059\mathrm{T}^{*3/2}, \end{equation} | [5] |
The 3/2 power law can be obtained from the fetch relations [1] and [2] by eliminating the dimensionless fetch $\text{gX}/\text{U}_{*}^{2}$ after the approximations 0.504→1/2 and 0.330→1/3, and using the relations $\text{H} = 4.01\sqrt{\text{E}} $ (Longuet-Higgins, 1952)44) and T = 1/1.05fp (Mitsuyasu, 1968).35) The 3/2 power law obtained from [1] and [2] through the above procedures is
| \begin{equation} \mathrm{H}^{*} = 0.057\mathrm{T}^{*3/2}, \end{equation} | [6] |
Although the 3/2 power law is very important law, we need to know its limitation. Kusaba and Masuda (1988)45) studied the relation between $\text{E}\omega _{\text{p}}^{4}/\text{g}^{2}$ and ωpU*/g, where the former corresponds to the square of the wave steepness and the latter corresponds to the inverse of the wave age (or dimensionless wind speed). They found that the 3/2 power law is satisfied, that is, $\text{E}\omega _{\text{p}}^{4}/\text{g}^{2}$ is linearly proportional to ωpU*/g, in the region 0.4 ≦ ωpU*/g ≦ 1.0, but in the region 1.0 < ωpU*/g, the parameter $\text{E}\omega _{\text{p}}^{4}/\text{g}^{2}$ tends to saturate gradually approaching the value corresponding to the breaking limit of the Stokes wave.
Recently Takagaki et al. (2012)46) studied the air-sea fluxes under the action of the very high wind speed up to 68 m/s (wind speed at 10 m height). They also found that the 3/2 power law was not satisfied in the wind waves under the very strong wind action. The reason is the same as that in Kusaba and Masuda (1988):45) the wind waves saturated due to wave breaking under the very strong wind action.
Toba’s spectrum.47)In 1973, when the report of the JONSWAP was published, Toba (1973)47) presented an important study on the spectrum of wind waves. He proposed a spectral form which has a high frequency tail proportional to gU* f−4, which is very different from the equilibrium spectrum of Phillips (1958)39) which has the high frequency tail proportional to g2 f−5. Toba’s spectrum took a little time to be approved. This was probably because the equilibrium spectrum of Phillips (1958)39) was too popular and because a difference between slopes f−4 and f−5 in the high frequency tails of the measured spectrum was not very clear due, in many cases, to the statistical scatters of the data. However, many spectral data supported the high frequency tail of gU* f−4 (e.g. Kawai et al., 1977, Mitsuyasu et al., 1979: Forristall, 1981: Donelan et al., 1985)48)–51) and Phillips (1985)52) himself revised his theoretical equilibrium spectrum to show the high frequency tail proportional to gU* f−4. Responding to such a result the JONSWAP spectrum was revised to have the high frequency tail proportional to gU* f−4 (Mitsuyasu, 1984: Donelan et al., 1985).53),51)
Directional spectrum F(f, θ) shows the energy distribution of the spectral components (Fig. 2) as a function of wave frequency and the directional angles for the wave propagation. It is usually expressed as
| \begin{equation} \mathrm{F}(\mathrm{f}, \theta) = \Psi(\mathrm{f}) \mathrm{G} (\mathrm{f}, \theta), \end{equation} | [7] |
Until the beginning of the 1970s, many spectral forms for the frequency spectrum were presented: the Phillips’ equilibrium spectrum,39) the Pierson-Moskowitz spectrum,40) the JONSWAP spectrum29) and Toba’s spectrum.47) However, there were no reliable data for the angular distribution function. This is largely due to the difficulties in its measurement compared to the measurement of the frequency spectrum.
Wave observation project in the RIAM.In 1970 the ocean research group in the RIAM started a project to measure the directional spectrum of wind waves in open seas (Mitsuyasu et al., 1975).54) As a portable wave recorder to measure the directional wave spectrum in the ocean, we selected and developed the cloverleaf buoy which was originally developed in the National Institute of Oceanography in the UK (Cartwright and Smith, 1964).55)
The cloverleaf buoy has three floats arranged in a triangular form resembling a usual clover leaf and accurately follows to the motion of the wave surface (Fig. 5). By detecting and analyzing the motion of the buoy (heaving, pitching, rolling, etc.) by using the sensors installed in the buoy, we can determine the directional wave spectrum (Cartwright and Smith, 1964).55)
Observation of ocean waves by using a cloverleaf buoy in the North Pacific Ocean.
The reason for the selection of this buoy was its performance of high directional resolution of the directional wave spectrum. Although the technical level in electronics in Japan was not very high in 1970, fortunately the cloverleaf buy worked more than ten years without troubles. We continued wave observation by using the buoy for about ten years in various locations including the North Pacific Ocean (Fig. 5) and the East China Sea.
A standard form of the angular distribution function.After great many trial and error we selected, as a general form for the angular distribution function, the following form proposed by Longuet-Higgins et al. (1963)56) and fitted it to the measured directional spectra,
| \begin{equation} \mathrm{G} (\mathrm{f}, \theta) = \mathrm{G}'(\mathrm{s}) [\cos \theta/2]^{\text{2s}}. \end{equation} | [8] |
A model function for the angular distribution of wind waves.54)
In order to determine the properties of s and to obtain the standard form of the angular distribution function we selected, from great many wave data, the data obtained in simple conditions in which the wind continued to blow for a long time with constant speed and direction.
From the analysis of the selected data we finally obtained the following interesting properties of the parameter s (Mitsuyasu et al., 1975),54)
| \begin{equation} \mathrm{s} = \mathrm{s}_{\text{p}}(\mathrm{f}/\mathrm{f}_{\text{p}})^{5}\quad \text{for $\mathrm{f} \leq \mathrm{f}_{\text{p}}$}, \end{equation} | [9] |
| \begin{equation} \mathrm{s} = \mathrm{s}_{\text{p}}(\mathrm{f}/\mathrm{f}_{\text{p}})^{-2.5}\quad \text{for $\mathrm{f}_{\text{p}} \leq \mathrm{f}$}, \end{equation} | [10] |
| \begin{equation} \mathrm{s}_{\text{p}} = 11.5(\mathrm{U}_{10}/\mathrm{C}_{\text{p}})^{-2.5}, \end{equation} | [11] |
The form of the angular distribution function determined in the present study is given by the Eqs. [8]–[11], and it shows the following properties: the angular distribution is narrowest for the spectral component at the peak frequency fp, the angular spreading becomes wider both for the low frequency side and for the high frequency side, and the angular spreading at the spectral peak frequency becomes narrower with the increase of the wave age Cp/U10, i.e., with the development of the wind waves.
Five years later Hasselmann et al. (1980)57) presented another form of the angular distribution function which was determined from the wave data obtained by using their pitch and roll buoy at the JONSWAP site. Their angular distribution function has a form similar to ours. Another five years later Donelan et al. (1985)51) presented another form of the angular distribution function which was determined by using a different technique: they used the linear array of the wave recorders to determine the directional spectrum. The angular distribution function obtained by Donelan et al. (1985)51) is different from either Mitsuyasu et al. (1975)54) or Hasselmann et al. (1980),57) but has qualitatively similar properties.
In 1974 Ramamonjiarisoa in France sent me a very thick paper on the experimental study of wind waves. He was working at the Institut de Méchanique Statistique de la Turbulence de l’Université de Provance, and the paper was his doctoral thesis.60) The paper contained many results on the growth of the spectrum of wind waves obtained by using a big wind-wave tank of his institute. Although the results on the growth of the wind wave spectrum were similar to ours (Mitsuyasu, 1967, 1968),35),36) the paper contained a very interesting result on the phase velocities of the spectral components of wind waves: the spectral components near 2 fp did not follow the linear dispersion relation.
This is a very interesting but serious problem for the spectral model of wind waves, because the linear model as shown graphically in Fig. 2 is based on the assumption that spectral components propagate with their own phase velocities depending on their frequencies. If such a curious phenomenon happens frequently applications of the linear spectral model will be limited.
The dispersion relation of random waves.In order to clarify the curious phenomenon discovered by Ramamonjiarisoa, our group started theoretical and experimental studies on this problem. Masuda et al. (1978)58) studied the nonlinear theory of two dimensional random waves and Mitsuyasu et al. (1978)59) made a laboratory experiment to clarify the phenomena.
In the experiment we measured the wind waves in a decay area where the wind was not blowing (Fig. 7). This was done to simplify the phenomenon by measuring the random waves without wind action. The phase velocities of the spectral components and the directional spectrum of the wind waves in the decay area were measured by using a linear array of wave staffs and the directional spectrum was determined by rotating the linear array.59)
Schematic diagram of the wind-wave flume (units in cm).59)
The measured results were compared with the theoretical predictions of Masuda et al. (1978).58) A typical example of the result is shown in Fig. 8 where the measured phase velocity of the spectral component C is compared with the theoretical computations of different approximations: [1] is the result of the one-dimensional linear theory, [2] is that of the second order approximation of the two-dimensional nonlinear theory, [3] is that of the third order approximation of the two-dimensional nonlinear theory.
Comparison of the phase velocity computed by nonlinear theory with observational data.59) Ψ(f): frequency spectrum of random waves, C: the measured phase velocity, [1]: the phase velocity of random waves computed by a one-dimensional linear theory, [2] and [3]: the phase velocities computed by a second-order approximation and a third-order approximation respectively to the two-dimensional nonlinear theory.
From Fig. 8 we can see that our measured phase velocity C has properties similar to those found by Ramamonjiarisoa: the phase velocities of the spectral components at around 2 fp ($ \doteqdot $ 4 Hz) are nearly equal to those at the spectral peak frequency fp ($ \doteqdot $ 2 Hz). Furthermore the curious phenomenon can be explained by the nonlinear theory of the two dimensional random waves.
Our theory58) and experiment59) also clearly showed that the spectral components near 2 fp are not the free waves propagating with their own speed but mostly the nonlinear bounded waves propagating with the speed of the spectral components near the spectral peak frequency fp. Even near the spectral peak frequency fp the measured phase velocity is different from the theoretical one given by the one-dimensional linear theory. This is due to the effect of the angular spreading and the phase velocity computed based on the theory of the two-dimensional random waves agrees with the measured one.
Later we measured the phase velocity of ocean waves at the oceanographic tower in Tuyazaki (Kuo et al., 1979).61) Although the number of the data were limited (wave steepness H/L: 0.02, 0.03), the measured phase velocity followed to the two-dimensional linear theory. This is due to fact that the steepness of the measured ocean waves was relatively small and the nonlinear effect for the phase velocity was negligible. Although the number of the data is limited, we are able to say that the linear dispersion relation can be used ordinarily for the ocean waves except for the special cases where very steep waves are generated by extremely strong wind.
One of our previous studies, “Interactions between water waves and wind (1)” 7) was originally designed to clarify the wind-induced growth of water waves. Although we found the very interesting phenomenon that the wind waves attenuate by the co-existing steep swell (Mitsuyasu, 1966),7) the original purpose was not attained due to the unsuitable performance of the facility. Later we built a wind-wave flume with high performance and planned again to study the same problem, the wind-induced growth of water waves, by using the new wind–wave flume.
In the experiment to clarify the fundamental properties of the wind-induced growth of water waves we planned to study the phenomena by using the following three different waves:
The third experiment was intended to clarify the following problem. In the present numerical wave model for the computation of the evolution of wind waves, we generally assume that each spectral component receives the wind energy independently without the effects of the other spectral components. That is, we assume the linear process in the energy transfer from the wind to wind waves, but this linear process is not yet proved. Regarding the third experiment, we made only a preliminary study and did not reach definite conclusions (Kusaba and Mitsuyasu, 1984).63) But for the first and second experiments we obtained many important results as described below.
Wind-induced growth of water waves co-existing with wind waves (1st experiment).The energy E of the mechanically-generated waves co-existing with wind waves was obtained from the wave records by using spectral filtering techniques.62) The energy E of the mechanically-generated wave in a wind area developed exponentially with the fetch X. The exponential growth rate β was determined after the correction of the viscous energy dissipation.
The exponential growth rate β shows a simple relation with the friction velocity of the wind U*. The relation in a dimensionless form is shown in Fig. 9 where the following best fit relation is also shown by a solid curve (Mitsuyasu and Honda, 1982);62)
| \begin{equation} \beta/\mathrm{f} = 0.34(\mathrm{U}_{*}/\mathrm{C})^{2},\quad 0.1\leqq \mathrm{U}_{*}/\mathrm{C} \leqq 1.0, \end{equation} | [12] |
The mechanically-generated waves with smooth surface even under the wind action were obtained by suppressing the generation of wind waves with a surfactant. Short discussions on the effect of the surfactant for the generation of wind waves will be given later in the section 9.
By the action of the wind the energy E of the mechanically-generated waves with smooth surface increased also exponentially with the fetch X, and the exponential growth rate β was determined after the correction of the energy dissipations due to the effects of the water viscosity and the surfactant.
The relation between the dimensionless growth rate β/f and the dimensionless friction velocity of the wind U*/C for the waves with smooth surface was compared in Fig. 10 with that for the waves co-existing with wind waves. It is very interesting that the relation between β/f and U*/C is almost the same for the both cases. This is due to the following reason. The measured growth rate for the waves with smooth surface was actuary smaller than that for the waves co-existing with wind waves, but the friction velocity of the wind was also smaller in this case than that for the case of the waves co-existing with wind waves, and almost the same relation was satisfied.
From the result mentioned above we can say that the wind-induced growth of water waves is uniquely determined by the friction velocity of the wind. Here we call it as the U*-similarity for the wind-induced growth of water waves. In this connection Kato et al. (2002)65) presented an interesting result that the U*-similarity for the wind-induced growth of the water waves was satisfied even when the atmospheric stability changed. That is, the growth rate increases in such a way, (β in stable stratification) < (β in neutral stratification) < (β in unstable stratification), but the friction velocity U* also increases in a similar way, and the growth rate of wind waves by the wind is uniquely determined by the friction velocity.
The formulae of Plant (1982).66)When we submitted the paper on the wind-induced growth of the waves to Journal of Fluid Mechanics, Plant, who was working in the Naval Research Laboratory in the US, published a paper on similar topics (Plant, 1982)66) in Journal of Geophysical Research. He assembled the previous typical data on the growth rate of water waves by the wind, which included not only laboratory data but also oceanic data. By using the assembled data he derived the formula for the growth rate of water waves by the wind. His formula, with the same expression with ours, is given by
| \begin{equation} \beta/\mathrm{f} = (0.25 \pm 0.13)(\mathrm{U}_{*}/\mathrm{C})^{2}\cos \theta, \end{equation} | [13] |
Phillips (1985)52) questioned the formula [12] for the growth rate which gives slightly larger values than Plant’s formula [13]. We consider that the difference in the two formulas [12] and [13] can be attributed to the difference in the range of U*/C: Plant (1982)66) used the laboratory and the field data in the range, 0.05 ≦ U*/C ≦ 3.0, while Mitsuyasu and Honda (1982)62) used only their laboratory data in the range, 0.1 ≦ U*/C ≦ 1.0. When we look at the data used by Plant (1982)66) carefully we can see that there are some data in the range U*/C ≦ 0.1, which are much smaller than his regression line. Those data would affect the formula [13] and made the numerical coefficient in the formula smaller than that of the formula [12].
In fact the data themselves are not much different from those assembled by Plant (1982)66) in the same rage of U*/C (cf. Fig. 1 in Phillips, 1985).52) Furthermore, Kato et al. (2002)65) also derived a formula similar to [12] in their laboratory study.
Therefore, we can say that the formula derived from the laboratory data gives a little larger growth rate than the formula derived from the data including both laboratory and field data, and the difference can be attributed to the difference in a range of U*/C: in the range, U*/C ≦ 0.1, the relation between the dimensionless growth rate β/f and (U*/C)2 is not linear but curves to the lower side (to smaller β/f). Miles’ (1985)67) revised theory gives a curve which fits very well to the data assembled by Plant (1982).66)
Laboratory study by Peirson and Garcia.Peirson and Garcia (2008)68) made a laboratory experiment on the wind-induced growth of water waves by using a quite similar technique to that of Mitsuyasu and Honda (1982).62) One of the most interesting results of their study was that they found a dependence of the growth rate on the wave steepness H/L: with the decrease of the wave steepness H/L the growth rate increases rapidly. In our study we tried to find the effect of the wave steepness on the growth rate of water waves, but we couldn’t find any significant effect as shown in Fig. 9 (Mitsuyasu and Honda, 1982).62) The reason may be that according to their study the effect of the wave steepness on the growth rate is very small for the waves with relatively large steepness, and we used the waves with relatively large steepness (0.02 ≦ H/L, 0.063 ≦ ak), where a is the wave amplitude and k is the wave number defined as 2π/L.
Some problem on the growth rate of water waves.The growth rate of water wave by the wind is determined with various techniques. In the laboratory study the growth rate is usually determined by directly measuring the wind-induced growth of water waves along the fetch and correcting the viscous energy dissipation. In the ocean the same technique is practically impossible and we determine the momentum flux from the winds to water waves by measuring pressure fluctuation P(t) above the wave surface with a wave-following sensor simultaneously with the measurement of the water surface elevation η(t).64)
Both techniques are usually thought to give the same value but, as far as I know, there have been no studies that have confirmed this idea. In laboratory studies, since we can apply both techniques simultaneously for the measurement of the growth rate, we can establish whether the results agree with each other or not. Furthermore, the accurate comparison may give deeper understanding of the phenomenon of the interaction between wind and water waves.
Another problem is whether or not we can apply the growth rate given by the empirical formula to each spectral component of wind waves independently. As far as I know, there are no studies which determine the effects of interferences that the other spectral component have on the growth rate of a particular spectral components. One of the purposes of our preliminary study was to clarify this problem, but the study was terminated without reaching a definite conclusion (Kusaba and Mitsuyasu, 1984).63)
In April 1983 a symposium honoring Professor Robert O. Reid of Texas A&M University was held, and his former students and colleagues attended the symposium. The author spent one year, from 1963 to 1964, in his laboratory and joined his research project on the study of air-sea interactions.
Although I could not attend the symposium, I had an opportunity to submit a paper to the Journal of Geophysical Research (JGR), where it was planned to publish the symposium papers. Mitsuyasu (1985)69) is the paper published in that occasion.
The momentum transfer from winds to waves.Mitsuyasu (1985)69) studied the momentum balance in the water waves under the steady wind action where the waves are locally stationary in time (cf. Fig. 11). In such a condition the momentum balance of the water waves can be shown as
| \begin{equation} \frac{\Delta\mathrm{S}_{11}}{\Delta\mathrm{x}}\left[= \frac{\mathrm{C}_{\text{g}}}{\mathrm{C}}\frac{\Delta \mathrm{E}}{\Delta \mathrm{x}} \right] = \tau_{\text{w}} + \tau', \end{equation} | [14] |
Momentum balance in the water waves under wind action.69) S11: downwind flux of the wave momentum, Us: surface drift current, τs: the total downward flux of the wind momentum, τw: the downward flux of the wind momentum goes into the water waves, τt: the turbulent stress in the air, τ′: the turbulent stress in the water corresponding to the momentum loss from the water waves due to viscous dissipation and wave breaking.
In a previous study Mitsuyasu and Honda (1982)62) determined τw by measuring directly the left-hand side in Eq. [14] and τ′ due to viscous dissipation; stable waves were used in the experiment and there was no dissipation due to wave breaking. Although the result was expressed in the form for the exponential growth rate β as shown in Eq. [12], by using the relation τw = βE/C, we could obtain the momentum flux τw entering into the water waves from the wind as
| \begin{equation} \tau_{\text{w}} = \left[2.2 \times 10^{2}\left(\frac{\mathrm{H}}{\mathrm{L}} \right)^{2}\right]\tau_{\text{s}}, \end{equation} | [15] |
The Eq. [15] is the relation between the total downward flux of the wind momentum τs ($ = \rho _{\text{a}}\text{U}_{*}^{2}$) and the momentum flux entering into the water waves τw, and the Eq. [15] shows that the momentum flux entering into the water waves τw increases with the increase of the wave steepness H/L. If we assume the wave steepness say H/L = 0.05 (the large side but reasonable value), it becomes
| \begin{equation} \tau_{\text{w}} = 0.55\tau_{\text{s}}, \end{equation} | [16] |
Mitsuyasu (1985)69) also computed the left-hand side of Eq. [14], the divergence of the horizontal momentum flux, for the fetch-limited wind waves, by assuming the cos2 θ type directional wave spectrum and using the fetch relation [1]. The result was given by
| \begin{equation} \frac{\Delta\mathrm{S}_{11}}{\Delta\mathrm{x}} = 0.054\tau_{\text{s}}. \end{equation} | [17] |
From the large difference between the absorbed momentum given by Eq. [16] and the advected momentum given by Eq. [17] in the wind waves, we can get the following image on the growth of the wind waves by the wind: the large part of the momentum transferred from wind to the sea surface are absorbed by wind waves but they lose most of it through wave breaking and only a few percent of the absorbed momentum is advected away by the wind waves.
The image discussed above is shown schematically in Fig. 11. This surprising result is supported indirectly by the study of Kusaba and Mitsuyasu (1986),71) where the nonlinear instability and the evolution of steep water waves under wind action were studied. Their study showed that steep water waves in some transition stage did not develop even under the wind action: they kept nearly constant values of their energies or decreased their energies slowly due probably to the wave breaking (Figs. 9b and 9c in Kusaba and Mitsuyasu, 1986).71) Furthermore, beyond the curious phenomenon observed by Kusaba and Mitsuyasu (1986),71) yet more complicated processes such as studied by Tulin and Waseda (1999)72) and Waseda and Tulin (1999)73) may exist, but their detailed discussions will be left for future study.
A study of Melville and Rapp (1985).74)Immediately after the publication of our paper Mitsuyasu (1985),69) Melville and Rapp (1985)74) published a very interesting paper which support, to a certain extent, the above image on the momentum balance in the wind waves. Through an ingenious laboratory experiment they showed that nearly 30 percent of a wave momentum flux could be lost in a single large-scale wave breaking event.
The motivation of this study was to clarify the roll of the wind waves in the air-sea boundary processes. We used a technique to suppress the generation of wind waves by using a surfactant. Then we studied the phenomena at the air-sea boundary by using two different waters: usual tap water and the water containing a surfactant (sodium lauryl sulfate NaC15H25SO4). In the former water wind waves were generated as usual by the wind but in the latter water wind waves were suppressed by increasing the concentration of the surfactant, and we could study various phenomena at the air-sea boundary, such as the wind shear stress, water surface roughness and the wind-induced growth of mechanically-generated water waves, under the presence or the absence of the wind waves.
The experiment was done in the same wind-wave flume used in the study of the dispersion relation of random water waves (Fig. 7), though in this study the air flow was not allowed to exit the middle part of the flume. Before the studies of the phenomena mentioned above, we studied in a preliminary experiment, the effects of the concentration of the surfactant on the generation of wind waves.
The results of the study were presented in the conference Wave Dynamics and Radio Probing of the Ocean Surface held in Miami in 1981. However, the publication of the proceedings of the conference was much delayed until 1986 (Mitsuyasu and Honda, 1986).75)
Attenuation of wind waves by the surfactant.In a preliminary study we tested the effects of the surfactant on the generation of wind waves. The wind waves in the tank were gradually attenuated by increasing the concentration of the surfactant in the water as shown in Fig. 12, which is the result obtained at the fetch X = 6 m and the reference wind speed U = 7.5 m/s. Here the reference wind speed is the wind speed at the inlet of the wind-wave flume and nearly equal to the cross-sectional mean wind speed. From Fig. 12 we can see that the wind waves were almost completely suppressed at the concentration of 2.6 × 10−2%. Therefore we can measure various phenomena at the air-sea boundary under the condition of the smooth water surface.
The effect of the surfactant concentration on the growth of the wave spectrum.75) Concentrations (%): 0, 2.3 × 10−3, 5.0 × 10−3, 7.3 × 10−3, 2.6 × 10−2 from the top to the bottom.
It is interesting in Fig. 12 that in their attenuation stage, the spectra retain a similar form. This is contrary to our expectation that the high frequency components attenuate more strongly than the low frequency components and the spectra should therefore not keep a similar form. If we recall the U*-similarity in the growth of water waves by the wind, the phenomenon seems to be due to a gradual decrease of the friction velocity of the wind. Unfortunately however, the measurement of the friction velocity of the wind in each stage was not done, and detailed analysis of the phenomenon will be left for the future study.
Wind shear stress.The wind profiles over the water surface were measured both for the smooth water surface and for the water surface covered with wind waves. By applying logarithmic profiles to the measured winds, the friction velocity of the wind U* and the 10 m height wind speed U10 were determined for the both water surfaces: the smooth water surface and the water surface covered with wind waves, where the 10 m height wind speed U10 was obtained by extrapolating the logarithmic wind profile.
The relations between the friction velocity of the wind U* and the 10 m height wind speed U10 for the both water surfaces are shown in Fig. 13. The friction velocity for the water surface covered with wind waves is larger than that for the smooth water surface in a region of the relatively high speed wind, 8 m/s ≦ U10, although it is interesting that the friction velocities of the wind for the both water surfaces show similar values in a region of the low speed wind, U10 ≦ 8 m/s.
The result means that the water surface shows properties of a hydrodynamically smooth surface for the low speed wind, U10 ≦ 8 m/s, even when the small wind waves are generated on the water surface. Such a result supports the very old but interesting idea presented by Munk (1947):76) the water surface is hydrodynamically smooth below the critical wind speed of about U10 = 7 m/s, but it becomes rough above the critical wind speed.
The best fit curves for the relations between the friction velocity of the wind U* and the 10 m height wind speed U10 were given respectively by the following equations:
for the smooth water surface
| \begin{equation} \mathrm{U}_{*} = 4.20 \times 10^{-2}\ \mathrm{U}_{10}^{0.878},\ \mathrm{U}_{10} \leqq 20\,\text{m/s}, \end{equation} | [18] |
| \begin{equation} \mathrm{U}_{*} = 1.61 \times 10^{-2}\ \mathrm{U}_{10}^{1.327},\ 8\,\text{m/s} \leqq \mathrm{U}_{10} \leqq 35\,\text{m/s}. \end{equation} | [19] |
A neutral drag coefficient (referred 10 m) is defined as CD = (U*/U10)2. Therefore, the relations between CD, and U10 for the both water surfaces can be obtained directly from the Eq. [18] and Eq. [19]:
for the smooth water surface
| \begin{equation} \mathrm{C}_{\text{D}} = 1.77 \times 10^{-3}\ \mathrm{U}_{10}^{-0.244},\ \mathrm{U}_{10} \leqq 20\,\text{m/s}, \end{equation} | [20] |
| \begin{equation} \mathrm{C}_{\text{D}} = 2.60 \times 10^{-4}\ \mathrm{U}_{10}^{0.654},\ 8\,\text{m/s} \leqq \mathrm{U}_{10} \leqq 35\,\text{m/s}. \end{equation} | [21] |
These relations are shown, with the broken curves, in Fig. 14, where the individual data of CD are also shown as a function of U10. The straight curves in Fig. 14 are the best fit relations between CD, and U10 obtained directly from the data assuming respectively linear relation.
Neutral drag coefficient CD versus U10.75) ○: for tap water, ●: for water containing the surfactant. The symbols ○ and ● with error bars are oceanic data from Garratt (1977).79) The dashed curves are Eq. [20] and Eq. [21]. The solid curves are linear relations between CD and U10 (cf. original paper). The short dashed curve is a formula proposed by Yelland and Taylor (1996),82) which is newly added in the present paper.
The relation between CD and U10 for the usual water surface covered with wind waves is not much different from many typical previous relations shown in Mitsuyasu and Kusaba (1984),80) which contains concentrated discussions on the drag coefficient over water surface. Within a range of our measured wind speed, U10 ≦ 35 m/s, the saturation level for the drag coefficient reported by Donelan et al. (2004)81) is not clear.
However, the smooth water surface, where the wind waves were suppressed almost completely by using the surfactant, clearly shows a property of hydrodynamically smooth surface, and the drag coefficient CD decreases with the increase of the wind speed.
The CD at the low speed wind and the study of Yelland and Taylor (1996).82)As can be seen from Fig. 13 and Fig. 14, in a region of the low speed wind, U10 ≦ 8 m/s, the water surface has the drag on the wind as a smooth surface even when the wind waves are generated, and the drag coefficient in this region increases rapidly with the decrease of the wind speed as shown in Fig. 14. A similar result on the drag coefficient over the ocean was reported by Yelland and Taylor (1996)82) based on their observations of the wind stress in the open seas. Through the careful measurements and analysis of the sea surface wind they derived a formula for the drag coefficient over the sea surface.
Their formula for the drag coefficient shows that there is a critical wind speed at U10 = 6 m/s. In the region 6 m/s ≦ U10, the drag coefficient increases with the increase of the wind speed as usual. However, in a region, U10 ≦ 6 m/s, the drag coefficient increases rapidly with the decrease of the wind speed. In Fig. 14 their formula is shown with a short dashed curve. Their formula agrees very well with our observations particularly in a region of the low wind speed, even though their formula was obtained by using the field data and our data were obtained in the laboratory. Recent laboratory data of the drag coefficient obtained by Donelan et al. (2004)81) shows the property qualitatively similar to those of Yelland and Taylor (1996)82) and Mitsuyasu and Honda (1986).75)
In this review eight topics were selected from our studies which were done mostly at the RIAM, Kyushu University in the period 1965 to 1992. In each stage of the study the selection of the topic was done depending on my interest and it was not necessarily systematic. Concluding the discussions I would like to talk about the significance or perspective of the selected topics in the study of wind waves.
The evolution of the wind waves in space and time is governed by the energy balance equation,
| \begin{equation} \frac{\partial \mathrm{F}}{\partial \mathrm{t}} + \mathbf{C}_{\text{g}} \cdot \nabla \mathrm{F} = \mathrm{S}(= \mathrm{S}_{\text{in}} + \mathrm{S}_{\text{nl}} + \mathrm{S}_{\text{ds}}), \end{equation} | [22] |
Our studies on the wind waves were designed to identify and quantify the source functions represented in Eq. [22] by S. Since the source functions are based respectively on the complicated dynamical processes we need to clarify each elementary process and express it in a computable form. Many studies on wind waves completed or still continuing follow this approach.
Taking this viewpoint, we conclude our studies discussed in this review as follows:
In recent years many important new advances in the study of wind waves have been made. The current numerical wave models, supported by many fundamental studies, enable us to compute ocean surface waves on a global scale with sufficient accuracy for practical purposes. We can obtain wind and wave data in wide areas of the ocean by using a technique of satellite remote sensing. This situation gives an impression that there are few important problems in the study on wind waves. However, the physical process controlling the energy balance of wind waves is still not completely understood and we still depend on empirical knowledge in practical problems such as the numerical wave model. Many of the studies discussed in the present review are, as it were, classic studies but I have tried to show and to draw attention to the unsolved problems relating to the phenomena discussed in each study.
Hisashi Mitsuyasu was born in Hiroshima, Japan, in 1929. He graduated from the Department of Physics, Hiroshima University in 1952, and pursued his research on ocean waves at the Port and Harbour Research Institute; Ministry of Transportation from 1952 to 1965. During this period, he received Dr. of Science degree in 1961 from Hiroshima University and spent one year in Department of Oceanography, Texas A&M University from 1963 to 1964 as a visiting scientist. In 1965 he moved to the Research Institute for Applied mechanics, Kyushu University and concentrated on the fundamental study of air-sea boundary process focused on wind waves until his retirement from Kyushu University in 1992. He was awarded the Sverdrup Gold Medal of the American Meteorological Society in 1988 for pioneering experimental work on ocean wave dynamics and its applications to wave modeling and forecasting. He was also awarded the Prize of the Oceanographic Society of Japan in 1998 for his study on the air-sea boundary process focused on wind waves. He is an honorary member of the Oceanographic Society of Japan. He is also an honorary member of the Japan Society of Civil Engineers. He is now an Emeritus Professor of Kyushu University (Research Institute for Applied Mechanics).
I am particularly grateful to Prof. K. Horikawa, M.J.A., who invited and encouraged me to write this review article in the Proceedings of the Japan Academy Series B. I am also thankful to the many colleagues and friends for their help in the preparation of the review. Prof. S. A. Thorpe read the manuscript very carefully and provided penetrating comments and important suggestions. Prof. Y. Toba and Prof. T. Waseda provided invaluable comments on an early draft of the review. Prof. H. Honji helped me in every stage in writing the manuscript providing encouragement and valuable comments. Prof. S. Mizuno provided technical support for the preparation of the manuscript and helpful comments. Prof. A. Masuda and Prof. M. Tanaka provided valuable comments on the review from the theoretical view point. Dr. Y. Suzuki provided recent information on the numerical wave model. During my studies on wind waves at the RIAM I was supported by many members of our group. In particular the late Dr. T. Kusaba (early name: T. Honda) was an outstanding co-worker in almost all my studies. Penetrating discussions with Prof. S. Mizuno and Prof. A. Masuda were very helpful to advance my study on wind waves. Many experiments in the ocean and in our laboratory tanks were technically supported by K. Marubayasi and M. Ishibashi. I would like to thank all members of our group. I am grateful as well for the valuable comments and discussions of the reviewer.
