£S*~. ap; XT/?£ -**? - GuTsr* /? c .^ T 0f .% \ I" (J /" ^r E s o* * NOAA Technical Report ERL 408-GLERL 14 Spectral Growth and Nonlin Characteristics of Wind Waves in Lake Ontario Paul Chi Liu November 1979 U. S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration Environmental Research Laboratories Digitized by the Internet Archive in 2013 http://archive.org/details/spectralgrowthnoOOIiup NOAA Technical Report ERL 408-GLERL 1 4 ^•Jjgg^ Spectral Growth and Nonlinear Characteristics of Wind Waves in Lake Ontario Paul Chi Liu Great Lakes Experimental Research Laboratory Ann Arbor, Michigan November 1979 2*5 * o a, o U. S. DEPARTMENT OF COMMERCE Philip M. Klutznik National Oceanic and Atmospheric Administration Richard A. Frank, Administrator Environmental Research Laboratories Boulder, Colorado Wilmot Hess, Director NOTICE Mention of a commercial company or product does not constitute an endorsement by NOAA Environmental Research Laboratories. Use for publicity or advertising purposes of information from this publica- tion concerning proprietary products or the tests of such products is not authorized. This report is based on a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Oceanic Science) at the University of Michigan, 1977. CONTENTS Page Symbols Used iv Abstract 1 1. Introduction 1 2. Theoretical Considerations 2 3. Observational Considerations 3 4. Data Acquisition and Processing 5 5. Spectral Computations 6 6. Results and Discussion 7 6.1 The Episodes 7 6.2 The Unispectra 8 6.3 Temporal Growth of Unispectral Components 9 6.4 The Source Function — Empirical and Theoretical 11 6.5 The Bispectra 16 6.6 Temporal Growth of Bispectral Components 16 6.7 The Trispectra 17 6.8 Temporal Growth of Trispectral Components 18 6.9 Further Remarks on Bispectra and Trispectra 19 7. Summary and Concluding Remarks 23 8. References 36 Appendix A. Relationships Between Higher-Order Covariances and Higher-Order Spectra 39 Appendix B. Testing for Stationarity 43 Appendix C. Application of Barnett's Parameterization of Nonlinear Source Function 45 Appendix D. Trispectra During the Episodes on 30 September 1972 and 7 October 1972 47 in SYMBOLS USED A(k,X;a,,T) Fourier coefficient of f(x.X;f, T) used in appendix A A(t,u)) a deterministic modulating function, which is unity for a stationary process D a parameter used in Barnett's parameteri- zation of nonlinear energy transfer that represents the part receiving energy from other components E total energy / linear frequency = io/2ir /o mean frequency g acceleration due to gravity G source function; also used in Barnett's parameterization to represent the part that transfers energy to other com- ponents h,, a counting symbol used in stationarity tests i (-1)" 2 in complex representation, also used as an index counter / index counter k wave number vector with magnitude k = |k|; k also is used as an index counter K counter for data points index counter L counter for data points M counter for data points N counter for data points Ni N(k,) action density at wave number k, used in the Boltzmann integral in ap- pendix C p index counter P pressure on the free surface resulting from air motion; also used as a counter for data points r position vector R„ nth order covariance function S„ nth order spectrum; S without subscript n represents a second-order spectrum T total time; also representing a coupling function in the Boltzmann integral in appendix C x space vector in Cartesian coordinates X a space parameter used in appendix A; also used as a Fourier transform, X(w), for data f r (f) y Cartesian coordinate Y complex random function for surface f used in the Fourier-Stieltjes represen- tation z Cartesian coordinate, directed verti- cally upward f free surface displacement 8 wave direction 7T 3.14159 .... q density of water t time velocity potential $ complex random function for 4> used in the Fourier-Stieltjes representation co radian wave frequency IV SPECTRAL GROWTH AND NONLINEAR CHARACTERISTICS OF WIND WAVES IN LAKE ONTARIO Paul Chi Liu ABSTRACT. Recent studies have shown that the growth processes of wind waves are primarily associated with the nonlinear energy flux due to wave-wave interactions. A detailed empirical examina- tion of these interactions uses calculated unispectra, bispectra, and trispectra of continuously recorded wave data during three episodes of growing waves. While the unispectrum provides information on the energy content of the frequency components, the bispectrum and trispectrum generally provide informa- tion on the interactive relations between two- and three-frequency components respectively. These higher-order interactive relations can be considered characterizations of nonlinear interactions. The results indicate that the peak-energy frequency transfers more energy to the lower frequency components than to the higher ones, which is confirmation that unispectral peaks shift progressively toward lower fre- quencies during wave growth. 1. INTRODUCTION The study of wind-generated waves has experi- enced significant and extensive development during the last 20 years or so. Ursell's (1956) review, in which he found the state of our knowledge of wind- wave generation profoundly unsatisfactory with re- spect to both theory and observation, has often been credited with providing the major stimulation for modern studies. After two decades and vast theoreti- cal and experimental efforts, a recent review by Bar- nett and Kenyon (1975) observed that At the time Ursell reviewed the field, the body of theoretical work exceeded that of the experimental work, but both were in an unsatisfactory state. Today the same ratio holds in that the theoretical ideas are still ahead of the experimental testing. In par- ticular, field observations relevant to wave generation and dissipation in the oceans in 1955 were nearly nonexistent. Today they are simply very scarce. Theory often leads experiment in science. In the study of wind waves, however, the difference is so large that experiment cannot interact with theory very efficiently. Theoretical analyses result in mathe- matical complexity, which prevents exact and practi- cable solutions; at the same time a lack of basic knowledge of the actual wave processes limits the ability to develop new models. This report presents a detailed empirical examination of the temporal evo- lution processes of the energy spectrum of wind- generated surface waves. Our measurements and analyses, presented here, help satisfy the need for basic information about wave processes. Because several recent studies (Hasselmann et al., 1973; Longuet-Higgins, 1976; Fox, 1976) stress the importance of nonlinear energy transfer in the wave-growth processes, we have examined these im- plications empirically. Since the nonlinearity comes from the higher-order terms in the equations of motion, the first step in evaluating the nonlinearity of a set of wave data recorded from a single station is to analyze the higher-order moments of the process. In practice, this is equivalent to performing bispec- tral and trispectral analyses of the data. A bispectrum is the two-dimensional Fourier transform of the third-order covariance function of the data. A trispectrum is the three-dimensional Fourier transform of the fourth-order covariance function of the data. As two- and three-dimensional Fourier transforms of the corresponding covariance functions are very cumbersome, we have found a way to calculate bispectra and trispectra by using a fast Fourier transform algorithm directly on the wave data. (The details will be discussed in section 5.) Physically, just as the unispectra provide informa- tion on the energy content of the frequency compo- nents, the bispectra and trispectra provide informa- tion on the interactive relations between two-fre- quency components and between three-frequency components, respectively. We consider these higher- order relations to be estimates or characterizations of nonlinear interactions. Hasselmann, Munk, and McDonald (1962) and Garrett (1970) have demonstrated that calculations of observed bispectra of ocean waves correlate rea- sonably well with theoretically derived bispectra. These studies, however, are not extensive enough to provide much insight into the detailed behavior of the physical processes. Using continuous wave data recorded in Lake Ontario, this report examines, identifies, and re- solves the temporal characteristics of linear and non- linear interactions during wave growth. Our demon- stration that the mean energy of the waves is closely related to the unispectra, bispectra, and trispectra of the data provides some analytical background to our basically intuitive approach. 2. THEORETICAL CONSIDERATIONS All theoretical studies of surface waves start from basic equations of fluid dynamics with varied idealizations or assumptions for obtaining the solu- tions. We assume irrotational motion of a hori- zontally unbounded incompressible fluid with infi- nite depth and a free surface at z = f (x, f), where x = (x, y) and z are Cartesian coordinates and the z- axis is directed vertically upward; then a velocity potential (x, z, f) exists and the motion is governed by the Laplace equation V 2 = 0, forz < f . (1) Further neglect ng the surface tension, the kinetic and dynamic boundary conditions at the free surface z = f are given, respectively, by _^+vr-v0 = o dt and £+*+>>" (2) (3) where p represents the pressure on the free surface re- sulting from the air motion, q the water density, and g the acceleration due to gravity. Now if we assume that the random surface displacement is statistically stationary with respect to both space and time, we can use the Fourier-Stieltjes representation (Phillips, 1966) as f(x,f) dY(k,u)e i(k-x-o)() (4) Under the same assumptions and the further assump- tion that —■ as z — -oo, the solution to (1) can be represented by 0(x,z,f) = J k i (o d$(k / co)e kz -'' (k - x -" t) . (5) Here cM>(k,co) and dY(k,u>) are complex random func- tions of the horizontal wave number vector k and fre- quency co with k = |k|. Analogously, the pressure on the surface resulting from air motion can be repre- sented by p(x,t) = J J dP(k / u)e- ;(k,x -" rf) . (6) The representations (4), (5), and (6) are quite general and appropriate for studying stationary random wave processes. Most of the studies in the literature consider processes that are stationary only with re- spect to space and thus reduce the equations of motion into differential equations with time deriva- tives only. Examples are Phillips (1960), Hasselmann (1962), and Benney (1962). Assuming the process is stationary with respect to both space and time, how- ever, further reduces the equations of motion into algebraic operations. Substituting (4), (5), and (6) into (2) and (3); expanding the factor e k ^ in (5) as a power series; and using (4) again in the series, we obtain -iWY(k,u) - /«M>(k,co) - f k , \ u ,k' 2 d$(k' ,«' )dY(k - k',« - co' ) -\ v \J v A^d^k',<,')dY{k"^") x dY(k-k' -k",o>-aj' -co") + J k ,J u , -k' • (k-k')eM>(k',co') x dY(k-k',co-a>') + \ k ,\J y ,\ ia „-k>.k"k>d*{k',o>') xdY(k",w")dY(k-k' -k", go -co' -co") = 0, and - iW*(k,w) + gdY(k,co) + $ k , J w ,{-xV k'd$(k',w' )dY(k - k\ co - co' ) +j[k'\k-k'\-k' .(k-k')]d$(k,o}) x d$(k-k' , co- co')} + k>L'k"L" [-*« / ^y^(k',« / )dY(k", w ") X dY(k - k' - k", co - co' - co" ) + *(*:' + k")(k'k" -k'.k")rf$(k',co') X c/$(k",co")^Y(k-k' -k",co-co' -co")"| = --dP(k,co) (7) (8) Equations (7) and (8), retaining terms to the third order, can be solved for dY(k,w) and d$(k,co) if the representation dP{k,u>) is given. From the point of view of the empirical study, the higher order terms in (7) and (8) clearly point to the need for study of higher order spectra. Most of the nonlinear studies in the literature assume the processes are undisturbed by air motion; hence dP(k,co) = in (8). Linearized analyses by Phillips (1957) and Miles (1957) con- sidered dP(k,o)) to be primarily associated with the turbulent wind field and to consist of two parts: the part produced by the turbulent eddies in the wind and in random phase with the wave field, and the part directly induced by and phase-locked with the wave field. A combination of the two mechanisms (Phillips, 1966) shows that the growth of wave energy is initially linear with time under the turbulent pressure alone and subsequently becomes expo- nential because of the induced pressure acting on the growing waves. A more realistic representation for dP(k,(k,cd), solvable from (7) and (8), are closely related to the distribution of energy per unit among the com- ponents of the wave field. The mean energy of the waves per unit projected surface area is F = F ■ , + F, "- '-potential ' kinetic = ^egf 2 +flLe(VcA) 2 rfz, (9) where the bars denote ensemble averages. It can be shown (e.g., Phillips, 1961) that hence, substituting (4) and (5) into (10), we obtain E =ycl k IJ k .J <1) {g^(k',«')dY(k-k', w - w ') + i'w'dY(k',u')d*(k-k',u-w') + LA „zo>'fc"rfY(k',a>')rf*(k",w") J k J a; X dY(k - k' - k",w -co' - cu" ) ...], (11) The relation between d(k,co) and dY(k,co), correct to the first order, is given by d(k,co) ~7^Y(k,co). k (12) Furthermore, since we assume stationarity with re- spect to both time and space, we expect non-zero contributions to the integral in (11) to occur only when both k = and co = 0; thus (11) becomes = y e J k j u [(2g)dY(k, U )dY*(k, u ) + J„ J uw 1 rfY(k / «)dY(k 1 ,«i)dY*(k + k a ,co + Wl ) ki, J o)i WCOi ^K)M^g dY ^ )dY{v ^ ] x dY(k 2 ,co 2 )dY*(k + k 2 + k 2 ,co + Ul + co 3 ) + ...], (13) where dY*(k,co) is the complex conjugate of dY(k,co). From appendix A, we see th at the averages dY(k,co)dY(ki,coi) . . . dY*(k n ,w n ) relate to the nth order spectrum. Thus, to study the complete energy distribution, it is necessary to examine the higher order spectra. 3. OBSERVATIONAL CONSIDERATIONS The results presented in section 2 and appendix A are derived for the characterization of the whole lake surface. To correlate the results with actual wave observations, usually made from a wave gage installed at a single selected location, the equations are integrated and normalized over all directions and wave numbers; this explicitly depends on time and wave frequency only. We can rewrite equations A. 8 and A. 9 as dY(w,)rfY(a> 2 ) . . . rfY(co„_ 1 )rfY*(a) fI ) du)idu) 2 ■ ■>n-l . . \R„(t 1m . . . ,7-„_!)e n-\ II 1=1 (2tt)"-i X dri . . . dr„_i = S n (w!,co 2 , . . . ,«„_].), if co„ = coi + co 2 + . . . + w n _i; and dY{^)dY{u> 2 ) . . . dYju^dY*^) du)ido) 2 ■ ■ ■ ^M-2 \ l \) 1 2 K I I I I I L *3 2 ) and their physical implica- tions. (3) A different approach to analyzing a non-sta- tionary process, perhaps more rigorous statistically, was developed by Priestly (1965). He introduced a representation of f(f) = \ u A(t,a)dY(a)e- (21) where A(t,oi) is a deterministic modulating function that approaches unity when the process approaches stationarity. This representation could have been used in our analyses. However, since A{t,o>) is not a known function and its application inevitably re- quires further assumptions and complications, we did not use it. It is of interest to note that Priestly's model leads to the conclusion that "the evolutionary spectrum at each instant of time may be estimated from a single realization of a process." This is exactly what the local stationarity assumption implies. (4) We define local stationarity here in its literal sense. Silverman (1957) introduced the concept of a locally stationary random process with a locally sta- tionary covariance that can be written as the product of a stationary covariance and a nonnegative func- tion. Since we allow our local covariance to vary from segment to segment and the variation is gener- ally smooth and gradual, it seems analogously pos- sible that a nonnegative function could depend on time segments and that a stationary covariance can be deduced from our consecutive local covariances. As we expect the property to apply to third- and fourth-order covariances also, our assumption car- ries a different sense than Silverman's rigorous process. Figure 2. — Location of wind and wave gages in Lake Ontario. were deployed for the IFYGL programs. The wave data in this study were recorded from the two Wave- riders designated as OS-1 and OS-2 (figure 2); the corresponding wind data used here were recorded from PDCS buoy 11. The Waveriders were deployed in 150 m of water and freely moored to a chain sinker with a mooring line approximately twice the depth of the water. The Waverider Buoy, manufactured by Data- well, Holland, is of spherical shape 1 m in diameter and weighs about 100 kg. It contains two main com- ponents: an accelerometer and a transmitter. The ac- celerometer, mounted on a pendulous system, meas- ures the vertical component of acceleration as the buoy moves with the waves. Two electronic inte- grators in cascade then transform the output into a voltage that represents the vertical displacement of the buoy. This voltage controls the frequency of an audio oscillator, which in turn modulates a crystal- controlled transmitter that transmits the signal by telemetry to a shore receiver. The telemetered wave data were recorded continuously on analog magnetic tapes. The analog wave data tapes were subsequently processed through a computer digitization and edit- ing system (Liu and Robbins, 1974) to obtain final data tapes, which were digitized at a sampling rate of approximately three per second. This sampling rate is more than sufficient to avoid an aliasing problem, since the buoy response is such as to adequately damp waves having a frequency of > 1.0 Hz. On the other hand the Waverider's frequency range is given as between 0.065 Hz and 0.50 Hz; hence, in the ac- tual computations we use a sampling period of Af = 2/3 s to yield a Nyquist frequency of 0.75 Hz for the computed spectra. Referring to the scheme discussed in the last section, we use the following numbers in the analysis: 4. DATA ACQUISITION AND PROCESSING The data used in this study were recorded in Lake Ontario from 1 April 1972 to 31 March 1973, the International Field Year for the Great Lakes (IFYGL). Seven Waverider Buoys and a large number of Physical Data Collection System (PDCS) buoys N = 15975 M = 64, L = 1800, K = 225. Consequently each selected episode is 177.5 minutes long. The episodes are subdivided into 64 equal seg- ments of 20 minutes each, with an overlap of 17.5 minutes from one segment to the next. 5. SPECTRAL COMPUTATIONS In the previous discussion we have been using general nth order spectra. For the actual applications, however, we shall concentrate only on n = 2, 3, and 4 for the unispectrum, bispectrum, and trispectrum, respectively, defined as S 2 (c dY(u)dY*(co) = \ R 2 (T)e- ,ulT di (22) R 4 (-T 2 ,T3 ~ T 2 ,Ti - T 2 ) R*(ti - T 2 ,-T 2 ,T 3 - T 2 ) R*(ti ~ T 2r T 3 - T 2 ,-7 2 ) RiW 3 - T 2 ,~T 2 ,Ti - T 2 ) R*{t 3 ~ 7 2 ,Ti - T 2 ,-T 2 ) Ri{-Ti,Ti - T 3 ,T 2 - T 3 ) R^{-T 3 ,T 2 - T 3 ,Ti - T 3 ) Ro(t 2 ~ T 3 ,-T 3 ,Ti - Ti) Ri(T 2 ~ T 3 ,Ti - T 3 ,-T 3 ) Ra(Ti ~ Ti,-Ti,T 2 - T 3 ) R 4 (Ti - Ti,T 2 - Ti,-T 3 ). (30) The corresponding transforms of the symmetry rela- tions are S 3 (Wi, and U 2 ) dYMdYiwiWiw* + oj 2 ) S 2 (o)) = S 2 (-oo), (31) du)idoj 2 = i i Ri(TuT 2 )e-' { ^ + ^^d Tl dT 2 , (23) S 4 (a>i,co 2 ,u) 3 ) — rfY(a) 1 )^Y(w 2 )c/Y(uj 3 )rfY*(co 1 + Ui 2 + co 3 ) do)idu) 2 du)i = J J j R,(T ll T 2 ,Ti)e~' { ^ T ' + ^ T ' + UJ ' T ' ) a)i OJ2 to xdTidr 2 dT 3 where Rz(t) - f(f)f(f + r), (24) (25) R3(ti,t 2 ) = r(t)r(* + Tx)f(f + t 2 ), (26) and R*(T lf T if T 3 ) - f(f)f(f + Ti)f(r + r 2 )f(f + T,).(27) The above definitions lead to the following sym- metry relations: R 2 (t) «K 2 (-t), (28) Ri(Ti,T 2 ) = R 3 (t 2 ,ti) = R 3 (-T 2 ,Ti - T 2 ) = R 3 (Ti - T 2 , - T 2 ) = R 3 (-tut 2 - n) = R 3 (t 2 - n, - n), (29) and RATi,T 2l T 3 ) = Ri{Ti,T 3 ,T 2 ) = Ri{T 2 ,Tx,T 3 ) = R i {T 2l T 3 ,T l ) = R^T 3 ,T- i ,T 2 ) = R i {T 3 ,T 2 ,T l ) = R i {-T l ,T 2 - T U T 3 - 7i) = Ri(-T lr T 3 - T U T 2 - Ti) = Ro(t 2 - T U - T x ,Ti - Ti) = R 4 (t 2 - Ti,T 3 - Ti,-Ti) = Ra(t 3 - T lf -T lf T 2 - Ti) = R 4 (t 3 - ti,t 2 - n,-Ti) = Ri(-T 2 ,T-i - T 2 ,Ti - T 2 ) ■>i{U) u O) 2 ) = S 3 (co 2 ,a;i) = S 3 (co 2 ,-Wi -co 2 ) = S 3 (-uh - CU 2 ,0> 2 ) = S 3 (a)i,-C0i -co 2 ) = S 3 (-OJi o) 2 ,03i), (32) and S 4 (coi,w 2 ,co 3 ) = S 4 (aJi,co 3 ,a; 2 ) = S 4 (co 2 ,ah,a> 3 ) = S 4 (cj 2 ,co 3 ,oji) = S 4 (co 3/ u>i,co 2 ) = S 4 (co 3 ,a; 2 ,Wi) = S 4 (-u>i - a> 2 - co 3 ,a) 2 ,a;3) = S 4 (-wi - a> 2 - cj 3 ,w 3 ,w 2 ) = S 4 (cJ 2/ -G0i - C0 2 - C0 3 ,O) 3 ) = S 4 (w 2 ,O)3,-C0i - oo 2 - co 3 ) = S 4 (w 3 ,-a)i - a> 2 - oj 3 ,o) 2 ) = S 4 (c0 3 ,C0 2 ,-GJi - U) 2 - (jj 3 ) = S 4 (-Wi - w 2 - co 3 ,coi,a; 3 ) = S 4 (-C0i - lo 2 - co 3 ,co 3 ,a)i) = S 4 (co 1 ,-a) 1 -cu 2 - C0 3 ,U> 3 ) = S 4 (aJi,O) 3 ,-C0i - co 2 - a> 3 ) = S 4 (aj 3 ,-a)i - co 2 - co 3 ,uh) = S 4 (co 3 ,o)i,-a)i - co 2 - co 3 ) = S 4 (-a)i - co 2 - co 3 ,oji,oj 2 ) = S 4 (-C0i - C0 2 - C0 3 ,C0 2 ,O)i) = S 4 (a)i,-o)i - a> 2 - OJ 3 ,C0 2 ) = S 4 (c«Ji,a; 2 ,-coi - co 2 - co 3 ) = S 4 (c0 2 ,-Wi - C0 2 - C0 3 ,C0i) = S 4 (co 2 ,a)i,-cOi - a) 2 - oj 3 ). (33) Because of these symmetries, we need only to esti- mate the S n 's, n = 2, 3, and 4, within a fundamental region. The fundamental region for unispectrum S 2 (co) is the line segment < w < u) N ; for bispectrum S 3 (wi,co 2 ) is the triangle defined by < co 2 < ah, and < ooi < w N ; and for trispectrum S 4 (co 1 ,a) 2 ,a; 3 ) is the tetrahedron defined by < co 3 < w 2 , < co 2 < coi, and < o»i < w w , with oo N = 2ir/(2At) representing the Nyquist frequency. Comparing the definitions (22), (23), and (24) with (A. 19) in appendix A, we see that for a finite segment of a time series the coefficients of its Fourier transform X(u>) can be used as approximations for the theoretical values c/Y(oj) and hence for the appro- priate products of Fourier coefficients for estimating corresponding spectral densities. We used the follow- ing approach generally similar to those discussed by Haubrich (1965) and Hinch and Clay (1968) . Starting with f(f At), 2 = 1, 2, . . ., L, we sub- divide the series into P non-overlapping groups each of length K, such that for p = 0, 1, . . ., P - 1 and k = 1, 2, . . ., K we have f p (fcAf) = f[(pX + /c)Af], and the K complex Fourier coefficients for each group are given by X>) = ^) EfpO'Ary 2 ^, fc = 1, 2 K. (34) We can then sum and obtain the average estimates by wind conditions during these periods, recorded at 10- minute intervals at 4 m above the lake surface on PDCS buoy 11, are shown in figures 3, 4, and 5. The group of short and straight lines plotted on the fig- ures under wind direction and wind speed indicate the locations in time of the 64 overlapping segments of wave data analyzed for each episode. Each seg- ment is 20 minutes long and has a 17.5-minute over- lap with the next segment. Although each of the epi- sodes represents a growing, nonstationary wave field, the 20-minute segments are assumed to be locally stationary, and thus we can calculate unispec- trum, bispectrum, and trispectrum for each segment using equations (35), (36), and (37), respectively. The three episodes are representative of growing wave conditions in the Great Lakes. Because we are interested in the early stage of wave growth, none of the episodes is under severe storm conditions. The 9 August episode started at a wind speed of 8 m s" 1 and increased to more than 11 m s" 1 in 3 hours. The l L S 2 (oo) =— L, X (w)X u (u), (35) -l S 3 (coi,w 2 ) =-p S X p (co 1 )Xp(co 2 )X p (coi + to 2 ), (36) 1 p = l and 1 P S 4 (coi,w 2 ,a) 3 ) —~jS ^ X p (co 1 )Xp(a) 2 )Xp(aj 3 ) 1 p=\ XX p (coi + w 2 + a> 3 ). (37) Equations (34)-(37) represent the basic approach in spectral computations used in this report. The ap- proach is quite efficient and feasible, especially with the fast Fourier transform algorithm (Cooley and Tukey, 1965) available as a computing subroutine. In the actual computations, we use P = 30 for each data segment of L = 1800 to obtain smoothed spec- tral estimates with 60 degrees of freedom and a 95 percent confidence interval between 1.48 and 0.72. 6. RESULTS AND DISCUSSION 6.1 The Episodes From the ample supply of wave data recorded in Lake Ontario during IFYGL, we selected three epi- sodes for our study: 9 August 1972 EST 1215-1515, 30 September 1972 EST 0540-0840, 7 October 1972 EST 0000-0300. The August and October episodes were recorded from Waverider OS-2 shown in figure 2; the Septem- ber episode was recorded from Waverider OS-1. The 9 August 1972 Figure 3. — Wind conditions for the episode of 9 August 1972. The series of short lines in the lower part indicates the locations in time of the 64 overlapping segments of wave data analyzed. 30 September 1972 7 October 1972 Figure 4. — Wind conditions for the episode of 30 September 1972. The series of short lines in the lower part indicates the locations in time of the 64 overlapping segments of wave data analyzed. Figure 5. — Wind conditions for the episode of 7 October 1972. The series of short lines in the lower part indicates the locations in time of the 64 overlapping segments of wave data analyzed. westerly wind direction provided long and approxi- mately constant fetches during the episode. The 30 September episode included the passage of a steep low-pressure center directly over eastern Lake On- tario, and wind speeds increased from 4 m s" 1 to more than 14 m s -1 within an hour, while wind direc- tions were changing rapidly from south through west to north during the early part of the episode. The 7 October episode was milder with wind speed in- creased from 2.5 m s "' to 8.5 m s "' under south- westerly direction. The total wave energy under these three wind fields was growing steadily in each of the three cases. 6.2 The Unispectra We first plot the computed unispectral density versus frequency versus time for three episodes as shown in figures 6, 7, and 8. These three-dimensional perspective figures present a clear overview of the spectral growth of the episodes. Several basic charac- teristics can be observed from these figures: (1) The growth activity is mainly concentrated over the low-frequency side; the high-frequency side does not change much during the episode. (2) During the growth period, the peaks of the spectra tend to shift toward lower frequencies. (3) The growth rate varies for different frequency components. (4) Once a frequency component grows to be the spectral peak, it reaches a relative equilibrium and its growth rate tends to diminish. (5) A comparison of these local spectral growth episodes with their respective wind conditions shown in figures 3, 4, and 5 indicates that dominant growth of the spectra happens during increasing wind speeds. The above features are generally known; similar results were obtained in laboratory studies, erg., Jacobson and Colonell (1972) and more recently by Wuetal. (1979). Figure 6. — Perspective view of the unispectrum during the episode of 9 August 1972. Figure 7. — Perspective view of the unispectrum during the episode of 30 September 1972. With respect to the well-known shifting of spec- tral peaks during wave growth, Hasselmann et al. (1973) suggested that nonlinear energy transfer plays a dominant role in the process. They concluded that the spectral form results from the self-stabilizing property of the nonlinear interactions that con- tinually readjust the energy distribution within the spectrum. We shall further explore these im- plications. 6.3 Temporal Growth of Unispectral Components To examine the time-dependent behavior of the frequency components individually, we first smooth them in time by hanning and then plot them semi- logarithmically. The components have different time dependences, but they can be grouped into three fre- quency ranges. The results are shown in figure 9 for the 9 August episode, in figure 10 for the 30 Septem- ber episode, and in figure 11 for the 7 October episode. (1) Figures 9(a), 10(a), and 11(a) show the low-fre- quency group that contains components with fre- quencies less than all the peak-energy frequencies during the episode. The components in this group can be characterized by their sensitivity to the wind field. Their growth seems to follow directly with the increases in wind speed. The approximate linearity during the growth as shown in the logarithmic plot indicates the exponential growth that Miles' theory predicts. (2) Figures 9(c), 10(c), and 11(c) show the high-fre- quency group that contains components with fre- quencies beyond the peak-energy frequencies during the episode. The components in this group represent the portion of the spectrum that is usually considered Figure 8. — Perspective view of the unispectrum during the episode of 7 October 1972. to be the equilibrium range. They tend to parallel each other in time and are insensitive to either in- creasing wind speed or time duration. Some anoma- lous behavior shown in figure 10(c) during the early stage may be due to the fact that wind direction dur- ing this short time interval was changing con- tinuously and thus further complicated the whole picture. (3) Figures 9(b), 10(b), and 11(b) show the compo- nents, lying between groups 1 and 2, that have been peak frequency during the episode. The behavior of components in this group is complicated since they have mixed properties of both of the first two groups. This group contains a large part of the total spectral energy. N 3 -* 2 3 CO o o o o o o o o ■■'••••• !■■■■■______■■■• »•••■■•■■■■ •• •••••••• >■■■ ■■■■■"•••• ■ __ - ••••••••• •• ■ ■ • a>/2ir Hz 0.100 • 0.125 ■ 0.150 o -i l I I I I I I I I I I I I I I I H I I I I I I I I I II I I I I 30 60 90 120 150 Minutes from beginning of episode Figure 9a. — Growth of low-frequency unispectral components during the episode of 9 August 1972. N 3 -+ 2 3 CO o x S 8 x x x g 5 fr- V * * * * * * * * * + + + + + + £ + + + ■ * + * + ** 100000000 i < W/27T Hz 0.175 2 0.200 * 0.225 + 0.250 * -i l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 [ 1 1 1 1 II 1 1 1 1 1 1 1 1 I 30 60 90 120 150 Minutes from beginning of episode Figure 9b. — Growth of middle-frequency unispectral components during the episode of 9 August 1972. 10 N 3 -^ 2 3 o 4444444. „„,, 44444 4444444444444*****444.,, 4444444 .444444 44444 44444 «.«.«« ^ _ _ *** t,, 44444 ••••••••*•••••••••»• •••-. ..-••••••* • --■■■"---- ■.«..*,»;• ■■■■■"l»Il2«""--.888?8S88885.."SS5oo"o"ooo c ooo°° „ '" nnnnoooooo ooaa OOO Jb«C (iBf3»j«B;ej 8 0C , — * * * * SSS4 *SS*SS «S4- «SS ^ * + &• * •••••• ************* * * ?....&*&***** ••••••ts±± & & S m ttf/27T Hz 0.175 0.200 * 0.225 + 0.250 * 0.275 4 0.300 • & & & & • ••• -T i I I I I I I I I I I I I I I I M I I I I I I I I I I I I I I I I I I I I I I II 30 60 90 120 150 Minutes from beginning of episode Figure 10b.— Growth of middle-frequency unispectral components during the episode of 30 September 1972. 12 N 3 -* 2 3 o l". 'oooooooo '00 oo„""!, BjOOOOO "oooooooo ooo ■ 530000000§fcg000 ea „ xxxxxxx xx$ $£$$$+¥¥¥ + _ 5 xxxxxxxxx xxxxxx +***** +XX+++ ****** RvIIIxxx»v2 X+ + + J . J .±tittt*tt ++ iti**********AAAA* J2+ &&&&•• •• ■■■oooooooooo 0000XX„ -«X X X + + + + + + H ******** && &&*! * * O < °v + ± + - ± &0 ' & & & S55 XXX + * (D/2TT Hz 0.325 ■ 0.350 o 0.375 a 0.400 X 0.425 + 0.450 * 0.475 & 0.500 • -i ll 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 [ I 30 60 90 120 150 Minutes from beginning of episode Figure 10c. — Growth of high-frequency unispectral components during the episode of 30 September 1972. N 3 — 2 3 i— i o 1 - W/21T Hz 0.100 • 0.125 ■ 0.150 0.175 e 0.200 x X X X X X 00 OOOO ...••mill 88 " 88 -••••••••••••■•■B5 ..iiiii.loooooo ■■■■■■■■00 00° ..III" __■•-- ■■"••••••••• f ■■■ 000000 -1 l 1 1 1 1 1 1 1 I Ill 1 1 1 1 1 1 1 1 1 1 1 30 60 90 120 150 Minutes from beginning of episode Figure 11a. — Growth of low-frequency unispectral components during the episode of 7 October 1972. 13 N X ~2 -i o ooo oooooooo oooooo „.„00 0OOOOOO„„ „„„„„„„oooooooooogooooo 0030 00 0000000300 www^^^ v^wwwv „ OOOOOOOxx „ „ 000;;? za z 32000x00000 «* 01 0.2 03 04 0.5 06 7 Frequency w/2nHz Figure 13. — Unispectra ( — ) and empirical ( — ) and Barnett's (- — -) source functions during the episode of 30 September 1972. The numbers 2.5, 32.5, . . ., and 152.5 are minutes from beginning of episode. our subsequent comparisons. (More recently, Wu et al. (1979) also used Barnett's parametric equation in calculating nonlinear energy transfer and found good comparisons in the low and intermediate frequency region of a wave spectrum.) We ignore the high fre- quency part of the parameterized results because it was based on Neumann's (1953) assumption that energy density is proportional to aT 6 at the high fre- quency side rather than to co" 5 , which most recent studies have confirmed as correct. The results for the three episodes are shown in figures 12, 13, and 14. Because of gradual changes in the process, we present only six spectra and source functions for each episode at 30 minutes apart to show the essential features. In the figures we plot the logarithm of unispectra log S 2 (cj), and the empirical and calculated d[log S 2 (u>)]/dt all with respect to the frequency of cc/2-k. The distinction between the empirical and calculated "theoretical" source functions is quite evident. The theoretical source function consistently has large positive lobes at frequencies below the peak-energy frequency. The empirical source function develops similar positive lobes only toward latter stages of the episode when the growth activity is intense. At the beginning of the episode when the growth is gener- ally slow or when the waves are well developed and further growth is slowing down, the empirical source function tends to be much less pronounced in its lobes, positive and negative. This behavior of the empirical and theoretical source functions is not particularly surprising since the theoretical results are dependent on the shape of the spectrum whereas the empirical results are de- pendent on temporal changes in the spectrum. This also provides a possible explanation for the domi- nance of nonlinear interactions during growth. If we assume that the empirical results represent the com- plete source function and the theoretical results only the nonlinear interactions, their differences then rep- resent the other processes, such as wave breaking and 15 2.5" 4 - | - N X II II •^ II *X 3 F u J ? 1 '/ 1 1 \ i J 1 - o , _l I / | 1 / I 1 / \ - ■ / 1 1 1 1 i i i X 3- E 32.5" 1 H J I ^r^" ■ lYrU* i 01020304050607 Frequency w/2n Hz 01020304050607 Frequency u/2nHz Figure 14.— Unispectra ( — ) and empirical ( — ) and Barnett's ( ) source functions during the episode of 7 October 1972. The numbers 2.5, 32.5, . . ., and 152.5 are minutes from beginning of episode. dissipation. At the beginning of the episode, when growth is generally slow or when the waves are well developed and growth is also slowing down, the non- linear interactions must be balanced by some equally important dissipation process in order to yield the smaller total source function. During rapid growth, on the other hand, both the nonlinear interactions and the total source functions have large positive peaks, which implies that dissipation is less signifi- cant and that most of the growth is through non- linear interaction. 6.5 The Bispectra There are three published bispectral studies on wind-generated waves: Hasselmann, Munk, and McDonald (1962), and Garrett (1970) compared four bispectra computed from actual wave data with those theoretically derived and found satisfactory agreement, and Houmb (1974) presented three examples of computed wave bispectra. In the present study we are interested in the higher order interac- tions during wave growth. As a first step we have computed 64 consecutive bispectra for each of the three growth episodes. Because the temporal varia- tions are gradual, we present only six bispectra from each episode at 30 minutes apart as representative of the whole process. Figures 15, 16, and 17 show the sample bispectra for episodes 9 August, 30 Septem- ber, and 7 October. Since the bispectrum S 3 (cdi,to 2 ), computed from equation (36), is complex, it is convenient to express it as a bispectral amplitude rather than as co-bi- spectra and quadrature-bispectra separately. Figures 15, 16, and 17 show the contours of the logarithms of bispectral amplitude plotted in the fundamental re- gion of < u> 2 ^ w 3 and < «i < co w . Several main features can be seen in these figures: (1) The bispectrum has a hill at (co p ,go p ), where cj p is the frequency of the spectral peak in the correspond- ing unispectrum. (2) Each bispectrum also shows two ridges ap- proximately parallel to the two frequency axes and sloping down from the bispectral hill toward both higher and lower frequencies. (3) The magnitude of the bispectral amplitude in- creases and the hills and ridges migrate toward lower frequencies during wave growth. (4) These observed characteristics are qualitatively similar to those depicted by Hasselmann, Munk, and McDonald's (1962) deep-water wave co-bispectrum. It is not readily discernible that these features re- sult from a nonlinear process. However, from the qualitative resemblance to Hasselmann, Munk, and McDonald's results and the fact that their results agree with those derived from a theoretical nonlinear process, we may conjecture that these features are characteristic of a weakly nonlinear wave growth process and hence proceed to examine their temporal behavior. 6.6 Temporal Growth of Bispectral Components Since the predominant bispectral interactions occur when the frequency of the spectral peak in the corresponding unispectrum interacts with itself and these interactions migrate toward lower frequencies during wave growth, it is sufficient that we examine the time-dependent behavior of those bispectral com- ponents on the 45° line whose frequencies interact with themselves. We proceed in a manner similar to that for unispectral components by temporally smoothing them with hanning and then plotting them semi-logarithmically as a function of time. The re- sults are quite similar to those unispectral compo- nents. We again separate them into three groups for the three episodes (figures 18, 19, and 20). 16 0.3 0.2 0.1 i i i i i ! i i i i i i i i — i — r~ r 62.5 'fit- •!• fit > ' ■<■»■- 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Frequency w/2ir (Hz) 0.3 i i i i i i : i i 1 ! 1 1 1 1 i I 1 1 i 1 I 1 1 1 1 1 ! ! g 92.5: 0? 0.1 Jfi^ ^ '^r ■ n 0.3 0.2 0.1 I I ! I I I I I ,'.,■,.)., .S^;^. 03 i i i i i i i i i III; i l l l I I I I 152.5: ;■■ ■■*■■ t 0.2 0.1 . .' ,_ — v ' n >'-' : 'T f '■: ' i' f ■ ; ■*'- 0.1 0.2 3 0.4 0.5 0.6 7 Frequency w/2ir (Hz) Figure 15. — Bispectra during the episode of 9 August 1972. The numbers 2.5, 32.5, . . ., and 152.5 are minutes from beginning of episode. Several points should be noted here: (1) The figures are plotted with log[S 3 (co,co)] versus linear time as before but with a smaller range. Conse- quently the exponential growth rates shown in fig- ures 18, 19, and 20 that appear comparable with those shown in figures 9, 10, and 11 are actually about four times smaller. (2) We have seen that in unispectral growth the spectral peak moves to a lower frequency during wave growth. The bispectra provide some indication of phenomena that can be attributed to this transfer, namely, that the interactions of the peak frequency with the next lower frequency grow consistently stronger relative to the interactions with the next higher frequency during the latter part of the growth. Examples are shown in figures 21, 22, and 23. As growth continues, the component of the next lower frequency eventually becomes the spectral peak. (3) If the bispectral amplitudes represent the sec- ond-order nonlinear interactions in the wave growth process, as we have conjectured, then our results show that the nonlinear interactions should also have functional relationships with respect to time. This is not unexpected. Since the theoretical nonlinear source term is a function of the unispectrum, it must vary with time if the unispectrum does. 6.7 The Trispectra Brillinger and Rosenblatt (1967) presented one set of calculated trispectra for daily sunspot num- bers. Their work represents the only trispectral calcu- lations available in the literature. They did not, how- ever, provide an interpretation of their results. Tri- spectral analysis has remained relatively unexplored. By extending the calculation scheme Haubrich (1965) used in bispectral calculations one step further and using equation (37), we have been able to calcu- late a trispectrum for each of the 64 data segments for each episode. It is not immediately clear how to visualize and present the trispectra effectively. It is even more cumbersome to try to present them in time. Since the fundamental region for the trispec- trum is the tetrahedron defined by < oo 3 ^ w 2 , < , because of coupling through the nonlinear boundary conditions at the 19 N X 14 3. 10 n (/) o ooo OBIH 00000000 • ••••" !•• o| 8 |8f •••••• 00000. Jj , ooo u |»t»«iiiii u>i/2ir\ + + ; * <» i e + + + + ******* ++++++++ ++ + + + ++ + + + + + * + + + ++ + + + ++** * * * x X X X X X X Z. i X X ** + **XXXXXX ** *******00 XXXXXXXXX X X X X + + + + + - I 000000000000 00000000 CO &> o o ttillinUKMltlSltlliUlSttiiiitU, "• • i O ( IIS!*ff 8888888888888888 88 DODOOOOOOOOOOOOOOa2H> 0000 A-*-- OOOOOQQ •••••••• '000000000000 SIM" 0;" • ••• __■■■■*• ? ? c 2 ; : - 10 2 /27r Hz 0.100 • 0.125 ■ 0.150 I I 1 1 1 [ I 1 1 1 1 1 1 1 1 1 1 30 60 90 120 Minutes from beginning of episode 150 Figure 19a. — Growth of low-frequency bispectral components during the episode of 30 September }972. 18 N I ^ f 14 1— 1 *"■* 3_ 4 444 • ••• iio + + + + co CO o X X X X x X x X X X + + + + 4 * & i \ t • *•• ox VVi X X X X X X X X + + + + + + + + + 444441 4 1 J isi 4144«4S14it*«4., 'si* •••••••••••••••• ••••••••&£ 4 4 — 4 4 444 * * +++++++ x x : W 1 /27T|<0 2 /27r Hz 0.175 0.200 x 0.225 + 0.250 * 0.275 4 0.300 • II [ 1 1 1 1 1 1 1 1 I I II 1 1 1 1 1 1 1 1 1 1 1 30 60 90 120 Minutes from beginning of episode 150 Figure 19b. — Growth of middle-frequency bispectral components during the episode of 30 September 1972. N X 14 - 3_ -2- 10 \f°<. co CO o -I o a °£ SS ooooo0 0^.« ss .ooo 5 §§ o ggggg <80Ba ° 8 °°°i°° 88?88 3???§ • 0000 00000000 00000 .15' W 1 /27T|w 2 /2ir Hz 0.325 ■ 0.350 o 0.375 « I II II I I I 1 1 1 1 1 1 1 1 1 30 60 90 120 Minutes from beginning of episode 150 Figure 19c. — Growth of high-frequency bispectral components during the episode of 30 September 1972. 21 18 N X f 14 a 10 V) o <0 1 /27r|<0 2 /27T Hz 0.100 • 0.125 ■ 0.150 0.175 a 0.200 " aeaeeaaa a a xxxxxxxo2l oooooo o_i oo ?Sg" a „.oooo»»o .••s:s ■ x** aBBzaiigrS* :. o :w ****** 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 mi 1 1 1 1 1 1 30 60 90 120 Minutes from beginning of episode 150 Figure 20a. — Growth of low-frequency bispectral components during the episode of 7 October 1972. N 14 3,10 CO D> O ****++ * * .+ + + +++ + + + H ******** nstustjis. :::::■■■■.:•*■* ssitttuu ., «***++**** isiiaait((( -- ..... «&***ii***±± + ***********. *r sa ***** & &£ t * * i|| iiii •■•••■• mill,, + + + + -. + ■ €0, / 27T | <0 2 / 27T Hz 0.225 + 0.250 * 0.275 & 0.300 • 0.325 ■ I 1 1 1 1 1 1 1 1 1 1 Ill 1 1 1 1 1 1 1 1 1 1 30 60 90 120 Minutes from beginning of episode 150 Figure 20b. — Growth of middle-frequency bispectral components during the episode of 7 October 1972. N 14 a a a a „ a a a a a ooe a ' 9, °°°°°°°°OQOooogQoo OO! OOO Soooogooo „„ooooooooooo„„„„„oqqqi „ „ „ ae o ooo o a a a a a a a a a _ oooog 88J a a a a a a a a a a a e4.aa„__aa 3. 10 CO (0 OJ o W 1 /27T| Hz 0.350 0.375 a 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 I 30 60 90 120 Minutes from beginning of episode 150 Figure 20c. — Growth of high-frequency bispectral components during the episode of 7 October 1972. 22 N X 14 - 3 it m CO en o Q OO OOO 0°° B "" •* oooooooooo ogg"" "■ ■■■■•••* ...•..»■ ■■ *•• I I I I ! I I I II M II o) 1 /27r|a» 2 /2fl - " Hz 0.150 | 0.175 • 0.175 0.175 B 0.175 | 0.200 M Ml I I I 30 60 90 120 Minutes from beginning of episode 150 N I E 14 o £10 ■s i * , en UJ o ;86 2> OOO tssSc'SsSaaaa >aa0^xxxx >< *oo 00° oooooooooooo ooooooooooo 0) 1 /27r|w 2 /27r Hz 0.175 0.200 0.200 0.200 0.200 0.225 x 1 1 1 1 I 1 1 1 I II I 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 I II 30 60 90 120 Minutes from beginning of episode 150 Figure 21. — Growth of bispectral components during the episode of 9 August 1972. Above: 0.175 Hz vs. 0.150, 0.175, and 0.200 Hz; Below: 0.200 Hz vs. 0.175, 0.200, and 0.225 Hz. free surface, energy is transferred to the side-band frequencies to (1 ± f). Our results appear to re- semble, to some extent, this side-band energy trans- fer theory. We may conjecture that, analogously, the stronger interaction with the lower side-band fre- quency component than with the higher side-band frequency component is also due to the complicated couplings of the nonlinear boundary conditions at the free surface. Furthermore, Longuet-Higgins's (1976) recent theory, indicating that the transfer of energy tends to reduce any symmetry in the spec- trum, can also be qualitatively attributed to these un- equal side-band interactions. 7. SUMMARY AND CONCLUDING REMARKS In this report we set out to empirically examine the temporal growth processes of wind-generated waves using data recorded in Lake Ontario. We have examined unispectra, bispectra, and trispectra com- puted consecutively for the three selected episodes. The scheme used for our study is based on the as- sumed property of local stationarity. This assump- tion allows us to apply successfully the analysis method developed for stationary processes to study nonstationary wave growth processes. 23 N X 14 - ^ 10 - V) o __«■■■■■■_■■■■ ., H 8«?!! . . .3B51«««8 a 8 ' • O Q ■ O O 0° o_ ■ .;«,".. sS iS 2 /2ir Hz 0.150 0.175 • 0.175 0.175 ■ 0.175 0.200 I I I I I ! I l I I I I ! I I I I I I I I I I I I I I II I I II I I I I I II 30 60 90 120 Minutes from beginning of episode 150 N I 14 m (/) o ■ ' 0° 00 x x §8° a o x x x x 3 o 30000^00 O X „0 O ^g^WSgOOOOOO ! /2 / 7r|o) 2 /2'7r Hz 0.225 0.250 * 0.250 0.250 4 0.250 0.275 • 6 1 I I I I I''' I I I I I I I I I 30 60 90 120 150 Minutes from beginning of episode Figure 23. — Growth of bispectral components during the episode of 7 October 1972. Above: 0.225 Hz vs. 0.200, 0.225, and 0.250 Hz; Below: 0.250 Hz vs. 0.225, 0.250, and 0.275 Hz. becomes less significant. (5) During the latter part of wave growth, the in- teractions of the peak-energy frequency component with the next lower frequency component grow con- sistently stronger than the interactions with the next higher frequency component, and the next lower fre- quency subsequently becomes the peak-energy fre- quency. This result, as demonstrated clearly from an examination of the temporal growth of bispectral and trispectral components, provides an explanation for the well-known fact of unispectral peaks shifting toward lower frequencies during wave growth. These results are consistent with our primary in- terest of exploring the empirical aspects. Since the de- tailed process of wave growth is still far from being completely understood, we hope our results will pro- vide some insight that can be useful for further understanding of it. 25 0.3 0.2 0.1 i : i i l ! : — i i ! i i i i i i ! i i i 0.200 J i l I i i I L I I I I i I I "l — i — i — l — i — I — I — n — f— | — i 0.175 03 02 ■0.1 II ii 1_| L_l_ ! ! I I I i I I I i I ! 0.125 "- , . . 0.3 -02 ■0.1 N I CM 1 i I I I I i i i i i i i I I I \ ■ I I I ! I I ! ! — FT 0.075 u -.3 I a (V " / '■■ <-^ ■0.2 -01 j : i i i i i i i i i i l J I I I L_ I I I I I I n — I I I : I I ' i — r 0.025 03 ■0.2 0.1 "^ . V x . .^cT^ 1 0.2 3 0.4 0.5 0.6 0.7 Frequency > O § 0.1 J #- / . ( J I L_l I i I I I I i I l_ n ! i ! r i i i i r 0.050 • I ! : I I i I I I 0.175 0.3 0.2 0.1 \ fmSw- _J I i i i L I i i ! I I 0.1 02 0.3 0.4 0.5 0.6 0.7 Frequency u)/2tt (Hz) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Frequency w/2?r (Hz) Figure 24b.— Trispectra during the episode of 9 August 1972, 32.5 minutes from beginning of episode. 27 CM O c 3 N £ o c 0> 3 0) Frequency 3 a> 0.2 0.3 0.4 0.5 0.6 0.7 Frequency O oooooo± S 0° e on 00 ° O 00 ■ ■■ ■ ■ ■ m" -■"".—. ■■••••• • • • • _• * ••!•■ :• •••••• • 1 / 21T | U) 2 1 27T | 0> 3 / 27T - Hz 0.100 • 0.125 ■ 0.150 o 0.175 0.200 x 0.225 + I I Ill 1 1 I I I I I I I 1 1 30 60 90 120 Minutes from beginning of episode 150 Figure 25. — Growth of trispectral components during the episode of 9 August 1972. 26 N I 22 + + + + + + * 18 3_ £ t— J 2> 14 o 10 .**!o| 00 ag8|*« ■«"• + °£lt*iii "■■■■■ xx + 00 I* xxx x + 9m XX + 0O«* O** ■ + + . -..000.. oooo ft** -« ■■ .••.....••'• «j 1 /2ir|to 2 /27r| 14 o 10 o> 1 /27r| 14 10 o2gg 8 52288 88o °oooooooo 00 ooooo fig ,■■■■■„.■"••• ooooo "__■■ _• ...: •••••••••••• ••••••••••• l /27r|<02/2ir| 14 10 3 0?g x ?? 0000000000°. xxxxxxxxx ,» » O 1 O .0 X X X X X O 1 n (J « O ooooooo 000 o ft), / 27T|a> 2 / 27r|<0 3 / 27T Hz 0.200 0.225 0.225 0.225 0.225 0.225 0.225 0.250 0.250 x « l I I I I I I I I I 1 1 I 1 1 1 i 1 1 1 1 I I 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 I 1 1 1 1 I I I 1 1 I I 1 1 I 1 1 I 1 1 30 60 90 120 150 Minutes from beginning of episode Figure 29b. — Growth of trispectral components during the episode of 30 September 1972: Third-order interactions among components 0.200, 0.225, and 0.250 Hz. 22 Eif 3 „ u 3^ £ to o oo 00 fiS 8S 88"8 88 !..* # * # ooo° ■■ •• 0° " •* _■ • o !•• o° .»«' _■:• O - # < CO, / 27T | fa» 2 / 21T | ft) 3 / 2ir Hz 0.175 0.200 0.200 • 0.200 0.200 0.200 ■ 0,200 0.225 0.225 « l ! I I I I I I I I I I I I I I I I I I I I I I I I 30 60 90 120 150 Minutes from beginning of episode Figure 30a.— Growth of trispectral components during the episode of 7 October 1972: Third-order inter- actions among components 0.175, 0.200, and 0.225 Hz. 35 22 N X 3^ 3_ i (/> 1—1 Dl O 14 10 xxxxxxxxxxx xxxqOOox HO . oe e x o o e x a x 00000000 X and If I > T which satisfies the condition for a Fourier integral to exist. The corresponding Fourier coefficient is A(k,X;co,T) = tt^tJ \t(x,X;t,T)e +i ^™»dxdt. (A.2) From Weiner's generalized harmonic analysis, it can be shown that as the parameters X and T extend to oo, the limit exists as [y(k,c)f ; :: = lim £,$«. J*. r'AiKX^Vdk.dkzdhdu X-oo T-oo (A.3) l t f(x 3 ,t)(n (2tt) 4 Jx X {- -£ Jdxdt. e + ikjXj_ e + ikjXj + ix; We can write dY(k,cc) = [Y(k,o))] k + dV, co + do) k,co x (n^- 1 )(^) « ^ y + idk:X: i\ / „-idut which is the inverse relation of f(x,f) = j . ( dY(k,u)e-'^*-« f >. (A.4) (A.5) Now we can obtain the ensemble average of an nth order product of c/Y(k,co) as dYiKaJdYik^) . . . dY(k„. 1 ,w„_ 1 )dY*(k„,w„ 1 (2x) s;j .-. . j r(x 1 ,f 1 )r(x 2/ f 2 ) . . . f(x„) (A.6) +; x g E (k ; • X; - U:t:) - (k n • % n - (i>„t„) 1=1 / n e +idk i™ x i>n - 1\ / e - /rfa) i f i - j \ vm=l 3 / ^ e + ,dfc n-l,m jc n-l,m _ l\ife- ,d6, n-l f n-l _ 1 \ \m = l ~^~i- x n-l,m /\ _z '^i-l / / JL e - idk nm x nm - 1 \ / e + "H f n - l\ X dx 1 dx 2 ■ ■ .dx n (dt) n . Now for stationarity with respect to both space and time, the right side can be written as (2tt) t -j . . . j f(x + r x ,f + Tl ) f(x + r„_ a ,f + T„_ a )f(x,f) +i x e ] E [ky • (X + ry) " £dy(f + 7y)] - (k„ • X + U>„t) 7 = 1 , + irffc lm U m + r lm ) _ 1 \ / g-'^l (f+^i) _ ^ lm>-l\ / e -«'^(' + '-l)_l\ rim) ) \ -»(f + r„_ a ) / ' { n ( n + f(x m + r r n-l,m> - 1\ / e - Ida, M-l (t + T «-l' - 1\ «-l,m) / \ -1'U + T «-l) / 3 e -idk nm x„ + ido),, t -IX _1\ / g^'^" -1 \ X dxdii . . . dr n _ 1 dtdT- li . . . dr,,^ 1 (27r) 4 <"- ■R n ( r l'- • •' r n-l' T l'- ■ ■ T n-V + i x e ~n-\ E (ky-ry 7 = 1 - COyTy) — (A.7) x -![(k n -k : - . . . -k„_ 1 )-x-(u n -a> 1 - . . . -co„_ a )f] (^1 = 1 +X>lm j ^ ~ iT l ) , + idk, y m = i +ir„- hm J \ -n n -i J X dr x . . . dr^idr-y . . . dr n _ x . 39 Since the stationary process is independent of both x and r, the above relation is identically zero unless and k„ - ki - k 2 - ... - k„_j — o„ - ojj - co 2 - . . . - ovi = 0. Furthermore we let dk's and du's approach zero to obtain -I E CtfyTy X e 7=1 dTi and R«(ti M-l ■/ ?Vi) i E Vi X e =i • do)! ■ dr„_ 1# . JS„(w : , . . ., w„_x) . rfw„_i. (A.ll) (A.12) dk 1 dk 2 . . . dk^du^do^ • • ■ du n _i (2tt) 4(w-l) [i?„(r 1/ r 2/ . . . r^T^,. . .t^) +i x e E {kj't-OljTj) ;-l x dr x . . . dr„_ x dr^ . . . dr n .j = S n (Ki, . . . k„_i, ojj, . . . fc^-i), ll K n — K| i K2 1 . . . K- n _] and o>„ = &1 + u>i . . . + ii) n -i and (A.8) dYjk^dYjk^) . . . rfY(fc n . 1 ,cj,,- 1 )rfY*(fc n> a),,) rfkjdkz . . . dk n _id(x)jdo^2 ■ ■ ■ du> n _i = (A. 9) if k„# kj + . . . + k„_i and w„ ^ a?! + . . . + co M _i. The last expression in (A.8) gives the general defini- tion of the nth order energy spectrum, where i?„(r lx . . ., r„_ i; n, . . ., r n _ : ) J • • • $S„(k a k„_ i; a?!, . <«Vi) x e E (k.- • r ( - «y7p X dkj . . . dk n _ido}j . . . d n _i. (A.10) For observations at single points, we may integrate and normalize the above equations over all directions and wave numbers to obtain For n = 2, we have the following familiar pair of classic spectral analyses: and S 2 (co) = f- \R 2 {r)e-™dT, 1 2tt l R 2 (t)= \S 2 (o))e™dw. (A. 13) (A.14) Detailed discussions corresponding to the formal definitions (A.ll) and (A.12), as well as conditions for existence and convergence of estimates, are beyond the scope of our present study and can be found in Brillinger and Rosenblatt (1967). In practical applications, we may define f T (r) = f(f), < t < 7; = 0, t> T. (A. 15) Let the Fourier transform of f _(f) be X(co); then we have X(w) = \^ T (t)e-^dt, (A.16) and f 7 (f) = $ u X(a>)e iut du. (A. 17) Now for the nth order covariance we have R„(T! • ■ -, r,,.,) = r(f)f(f + t x . . . ttt + 7„_ T ) lim y j f f T (f)f T (f + 7,) . . . f T (f + 7„_ a )^ 7-00 7-00 T i f f T Wf ul ...L„.X(a, 1 )...X(c„_ 1 ) x e' |lw >- X rfcO] . . . do)„_- [ dt = lim — ( ... T w : 7— 00 "n-1 X(oj 1 ) . . . X(o)„_ 1 ) x e l(u>i 7. +. . . +0)„.iT 1 ,. 1 ) S„(C0 1/ . . ., W n _!) 27T' (n-1) \K(^ T n _ x ) x j ( f r (f)e' (uJ i + ' • ■+«*'-i>yfdw] . . .dco„_ 40 1 lim j j ui . . . j M X(o)!) . . . X(ova) X X*(co 1 + . . . + w n _J X e ;( "i T i + ' ■ ■+«»-iV-i>d Ul . . . rfaj„_ x . (A. 18) Comparing (A. 18) with integrated (A. 12), we have formally S„(o>i w„-i) = lim y [X(u> a ) . . . Xiu^X*^ + . . . + a)„_a)]. 7 ^°° (A.19) Thus, in practical applications we can use the product of Fourier transforms of fr(f), with the aid of the fast Fourier transform algorithm, to facilitate the calculations of higher order spectra. 41 Appendix B: Testing for Stationarity In this report we have assumed local stationarity to study the temporal growth of spectral components that are in general nonstationary. To test the assump- tion of local stationarity, we adopted a testing method similar to that used by Bjerkaas (1976). The scheme divides each data set into P equal-length sub- segments; calculates the f , f 2 , f 3 , and f 4 for each sub-segment; and then applies two nonparametric tests, the run test, and the reverse arrangement test (Bendat and Piersol, 1966) to the data set to examine its stationarity. In the run test, the test parameter for each inter- val is compared to the median value of the test parameter for all intervals. If the test parameter is greater than or equal to the median value, +1 is as- signed to the interval; otherwise, -1 is assigned. A se- quence of consecutive +l's or -l's is called a run, and the number of runs N r in the P intervals is deter- mined. N r gives an indication as to whether or not re- sults are independent random observations of the same random variable. The reverse arrangement test is most useful in detecting monotonic trends in the time series. Consider the test parameter A\, with i = 1, 2, . . ., P. If Ai > Aj for i < j, the pair of param- eters is called a reverse arrangement. The total number of reverse arrangements, N a , is defined as follows: Let Then and N a N = p / = / + ! P-l L N a Based on P and the level of significance, the ac- ceptance ranges for N r and N a can be calculated (Bendat and Piersol, 1966) and used as a basis to ac- cept or reject the stationarity assumption. Accordingly, in our analysis eight separate tests were performed for each data set. The results, pre- sented in terms of the percentage of data failing the tests for the three episodes studied, are shown in table B.l. At a level of significance a = 0.10, the overall failure rate was 29 percent; the rate reduced to 19 percent for a = 0.02. Every data set passed at least two of the eight tests performed. A total of 27 percent passed all eight tests. The parameter f 3 posed the best results; £* 4 , on the other hand, posed the worst results. Although, ideally, absolute sta- tionarity requires that the data pass all the tests, we find that for a basically nonstationary process these results generally tend to enhance the acceptability of our assumption of local stationarity. Thus, without pursuing more complicated physical and statistical nonstationary analysis, we can employ available sta- tionary analysis methods to study an otherwise non- stationary process. Table B.l. Stationarity test results Percentage of Data Failed Au gust 9 September 30 October 7 All Data Level of a = 0.10 a = 0.02 a = 0.10 a = 0.02 a = 0.10 a = 0.02 a = 0.10 a = 0.02 Significance ? 45 17 45 14 16 6 35 13 r 2 16 8 36 25 25 16 26 16 r 3 3 6 3 4 r 4 98 92 73 63 52 41 74 65 T 2 2 1 r 28 17 53 30 50 33 44 27 f 3 8 2 5 4 r 34 20 53 38 48 33 45 30 29 19 34 21 24 16 29 19 Run Test Reverse Arrangement Test All Parameters 43 1200 1000 O 800 o g 600 u. JJ 400 ,55 200 c o o 'w £ "200 E TJ -400 41 O "600 -800 -1000 Sell & Hasselman Barnett Appendix C: Application of Barnett's Parameterization of a Nonlinear Source Function Theoretical results have shown that the non- linear energy transfer of gravity waves can be ex- pressed by the Boltzmann integral as |^- 4 = j . . . jr(k 1 ,k 2/ k 3 ,k 4 )[(N 3 + NJNyN z - (N : + N 2 )N 3 N ii ] 5(a)! + a> 2 - w 3 - co 4 ) x 5(ki + k 2 - k 3 - k 4 )dkjdk 2 dk 3 (C.l) where N, = N(k;) = S(k,)/co, with co, = (g/c,) 1/2 and S- (k,) is the two-dimensional wave spectrum with re- spect to wave numbers |k,| = k\. The delta functions express conditions for resonance between waves. The coupling coefficient T(k 1/ k 2 ,k 3 ,k 4 ) is a very compli- cated function and its precise form does not provide any physical interpretation. While the Boltzmann integral has been applied to other fields of study, Hasselmann (1962, 1963a, b) first derived equation (C.l) for gravity waves. Other derivations have been given by Benney and Saffman (1966) and Watson and West (1975). Recently Longuet-Higgins (1976) presented a simplified model and showed that T(k 1 ,k 2 ,k 3 ,k 4 ) is equal to 47r when the four wave numbers k 1# k 2 , k 3 , and k 4 are nearly equal. This re- sult implies that the exchange of energy within the peak of the spectrum is of dominant importance. No practical application of this simplified model has been developed, however. Attempts to evaluate the integral (C.l) have been made by Hasselmann (1963b) and more recently by Sell and Hasselmann (1972). The latter computa- tions, typically 30 minutes per spectrum on a CDC6600 computer, lead to interpretations of the JONSWAP measurements that strongly suggest that the nonlinear transfers of energy play an essential role in the development of the wave spectrum, par- ticularly in the growth of the wave energy at low frequencies. Because of the complexity of the direct calcula- tions of (C.l), Barnett (1966), using a Neumann spec- trum, obtained a parameterization of (C.l) as part of a wave prediction scheme based on the earlier Hassel- mann (1963a, b) results. Mitsuyasu (1968) applied this parameterization to his studies on nonlinear energy transfer in wave spectrum and obtained fair agreement between calculated and observed non- linear source functions. Resio and Vincent (1976) satisfactorily applied Barnett's prediction scheme for hindcasting waves in the Great Lakes. Barnett's parameterization was based on Hassel- mann's (1962) theoretical results, which demon- strated that the wave-wave interaction of a spectral component consists of an active part that transfers its energy to other components and a passive part that receives energy from other components. Thus, 0.0 .5 10 1.5 2.0 2.5 3.0 3 5 4.0 4 5 5.0 5.5 Frequency in Hz Figure C.l. — A white spectrum and its computed source functions. with n 4.4 X 10 8 EV 4 ,_ ../, rt .,/o\ 3 G= \ — cos (0- O )(1 -0.42^ 1 (C.3) for/> 0.42 / , and |0-0 O | < G = 0, otherwise; and 7.5 x 10 7 £ 2 /q 7 2 ' D V &M =G-DS(f,9) (C.2) for/> 0.53 f ; D = 0, otherwise, 1 + 16|cos(0-0 o )|](/-O.53/ o ) 3 , (C.4) 45 3 O 52 o.ooc CO c o m « -.0000005 E ■5 i> c O - 0000010 l\ ^— — Barnett I I i i i i i 00 .6 8 10 Frequency in Hz Figure C.2. — Pierson-Moskowitz spectrum and its computed source functions. ■ 000002 ■ 000004 000006 Sell & Hasselman Barnett 10 20 40 50 60 70 30 Frequency in Hz 90 1 00 1 10 Figure C.3. — JONSWAP spectrum and its computed source functions. where E, f , and 9 are total energy, mean frequency, and mean direction, respectively, defined by and E = \\S(f,d)dfdd, f =h\fS{f.e)dfde, es(f,d)dfdd. (C.5) (C.6) (C.7) In the above equations, is the wave direction and f = w/2ir is the linear wave frequency. We applied the above parameterization by further assuming a directional spreading factor of (8/37r) cos 4 to compare some of the results given in Sell and Hasselmann (1972) that led to the major con- clusions based on JONSWAP measurements. The re- sults for a fairly white spectrum, a Pierson-Mosko- witz spectrum, and a mean JONSWAP spectrum are presented in figures C.l, C.2, and C.3, respectively. The JONSWAP study (Hasselmann et al., 1973) con- cluded that, as the sharpness increases from figure C.l to figure C.3, the evolution of a sharp peak is pri- marily controlled by the nonlinear energy transfer evidenced by the shifting of the positive lobe in Sell and Hasselmann's calculations toward lower fre- quencies. While not necessarily profound, the same conclusion can also be drawn from the shifting of the positive lobe as a result of Barnett's parameteriza- tion. Therefore, with an overwhelming savings in effort and computer time, Barnett's parameterization can be used approximately for the examination of theoretical nonlinear energy transfers, particularly with respect to the location of the positive lobes of the parameterized nonlinear source function. 46 Appendix D: Trispectra During the Episodes 30 September 1972 and 7 October 1972 We present here the calculated trispectra for the tral sheets located 30 minutes apart, corresponding in episodes of 30 September and 7 October as figures time to the bispectra shown in figures 16 and 17. D.l and D.2. Each figure contains six sets of trispec- 03 0.2 0.200 I I ! i ! I I l ! I 1 I ! I ££S N I CM O c 3 0.1 N X CM 3 >< o c 4> 3 CT 0) t, U. l-t^XCY,) 0.2 0.3 0.4 0.5 0.6 0.7 Frequency u c 4) 3 0.1 0.2 0.3 0.4 05 0.6 07 Frequency 1 3 4 0.5 0.6 0.7 Frequency to/2ir (Hz) J 1 0.2 0.3 4 5 6 Frequency ml lit (Hz) Figure D.lc. — Trispectra during the episode of 30 September 1972, 62.5 minutes from beginning of episode. 49 0.1 0.2 0.3 0.4 5 0.6 0. Frequency Frequency <»/2ir (Hz) 1 2 0.3 0.4 5 6 0.7 Frequency ml lit (Hz) Figure D.lf. — Trispectra during the episode of 30 September 1972, 152.5 minutes from beginning of episode. 52 N I CM 0.3 0.2 0.1 0.3 0.2 0.1 03 ! I I I I I I I i ! { I I I ! I I 0.200 I I ! ! I ;* J I I I I I I I I I I I L I i ; i i i 0.150 ■Is 1—1—1— I— 1—1— I— I o c 9) 3 0.1 CT 0) Im u. 0.3 I i ! I I i i ! I i I I ' 0.100 ^ /-\. J I i I I I I i ! I I l— J L 0.2 0.1 03 0.2 0.1 0.050 i i : i i i i j : i L_l_ ~i i : i i r T (#& \ L I ' I I ' I I I I i ' I i ! i I I ! I i i ! I I : \ I I ! ! I I : I 0.000 th — i — r~r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Frequency u/lir (Hz) J 1 I ! L_i_J i i L_ I ! ! I ! lilt 0.175 : ! I . I 0.125 J i ! ! L. 0.075 0.3 0.2 0.1 0.3 0.1 ^ CN *>» 3 > O c 10.3 ® 0) 0.2 £ 0.1 j i : i i i ^j i_ i ; i ; i l l i i i 0.025 J L_! I I I i I L -1 i i I I l_ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Frequency u)/2tt (Hz) -10.3 0.2 0.1 Figure D. 2a. — Trispectra during the episode of 7 October 1972, 2.5 minutes from beginning of episode. 53 03 0.2 0.1 0.3 2 1 N £ 0.3 CM 3 0.2 > U a> 01 3 cr 0) £ 03 2 01 "I — I I I I ! I I I i I I I I i I I I I ! I I i ! I I ! ! I 0.200 A J I I i I l I I i I I i I I I I I I I I I I I I i I I L I I i ! I I I I I ! ! 0.150 I i ! I I I m 0.100 o J L_l I ! I I i I I I I I l_ 0.050 J I I I l i I I I I I i I I I I ! I I I I I I I I L I ■ _l I CM > o c 0) 3 O" 0.1 0.2 0.3 0.4 0.5 06 07 Frequency /2ir (Hz) Figure D.2b. — Trispectra during the episode of 7 October 1972, 32.5 minutes from beginning of episode. 54 0.3 0.2 - i i i i i ; i i i i i i i i i i : i i i i i 0.200 0.1 0.2 0.3 0.4 0.5 06 0.7 Frequency /2ir (Hz) J 0.1 0.2 0.3 0.4 0.5 06 0.7 Frequency talltr (Hz) Figure D.2e. — Trispectra during the episode of 7 October 1972, 122.5 minutes from beginning of episode. 57 CM 0.3 02 0.1 0.3 0.2 0.1 03 I I I I I I I I ! I I ! ! I I I I ! I I I I I I 0.200 J LJ I ; I I I i_L I ! ' I I ! | — rT" i i ; i i i i i i i i — i i i ; i i i : i i 0.150 J I i I I I ! i I I l I I I I i I L ' l . ■ ' ' ; - i • 02 >. O I 0.1 o- k. "■ I I ! ■ I I [ r 0.100 J I I i I I I I I I I ; I I l i i i i I i I I i i i i j i ii i .iii 0.3 2 1 0.050 iii — r~ i — i— r I i ! I I ! 0.175 0.3 02 n i 0.125 _i_j i i : ' i i i i ' i 0.075 i l i l l . : i r I I I i I I I ■ i I 0.3 0.2 01 *— CN — , 3 >> O 0.3 § 0) 0.2 £ 0.1 J I I i I i I I I I I i I I I I i I I i I I I I I I i L I I i I I I I l I 0.1 0.2 0.3 4 5 6 0.7 Frequency