Hasselmann Etal DHZ1973

7/22/2019 Hasselmann Etal DHZ1973

1/96

Measurements of Wind-WaveGrowth and Swell Decay duringthe Joint North Sea Wave Project

(JONSWAP)UDC 551.466.31; A N E German Bight

B yK . Hasselmann, T. P. Barnett, E.Bouws, H.Carlson,D . E. Cartwright, K . Enke, J. A . Ewing, H. Gienapp,

D . E. Hasselmann, P, Kruseman, A . Meerburg, P. Mi i l l e r ,D . J. Olbers, K. Richter, W. Sell, H . Walden

Erganzungsheft zur Deutschen Hydrographischen ZeitschriftReihe A (8), Nr. 12

1973

D E U T S C H E S H Y D R O G R A P H I S C H E S I N S T I T U T H A M B U R G


2/96

Erganzungsheftzur

Deutschen HydrographischenZeitschriftHerausgegeben vom Deutschen Hydrographischen Institut

Reihe A (8), Nr. 12

Measurements of Wind-Wave Growth and Swell Decayduring the Joint North Sea Wave Project (JONSWAP)

UDC 551.466.31; A N E German Bight

1973



3/96

Measurements of Wind-WaveGrowth and Swell Decay during

the Joint North Sea Wave Project(JONSWAP)

UDO 551.466.31; A N E German Bight

B yK . Hasselmann, T. P. Barnett, E. Bouws, H . Carlson,D . E. Cartwi-ight, K. Enke, J. A . Ewing, H. Gienapp,

D.E.Hasselmann, P. Kruseman, A.Meerburg, P.Mller,D . J. Olbers, K. Richter, W. Sell, H . Walden

1973



4/96

Ansohriften der VertasserK . Hasselmann, Insti tut fr Geophysik, Hamburg, Germany; 1970-72 also Doherty Professor atWoods Hole Oceanographio Institution, Woods Hole, Mass., U.S.A.T . P. Barnett, Westinghouse Ocean Research Laboratory, San Diego, Calif., U S A - now at

Scripps Institution of Oceanography, La Jolla, Calif., U.S.A.B . Bouws, K o n i n k l i j k Nederlands Meteorologisch Instituut, De B i l t , The NetherlandsH . Carlson, Deutsches Hydrographisches Institut, Hamburg, GermanyD . E . Cart-iwight, National Institute of Oceanography, Wormley, Great B r i t a i nK . Enke, Institut fr Geophysik, Hamburg, GermanyJ. A. Ewing, National Institute of Oceanography, Wormley, Great B r i t a i nH . Gienapp, Deutsches Hycbographisches Inatitut, Hamburg, GermanyD . E. Hasselmann, Insti tut fr Geophysik, Hamburg, GermanyP. Kruseman, K o n i n k l i j k Nederlands Meteorologisch Insti tuut, De B i l t , The NetherlandsA . Meerburg K on i n l d i j k Nederlands Meteorologisch Insti tuut, De B i l t , The Netherlands; now atDutch Minis t ry of Foreign Affa i rs , The NetherlandsP. Mller, Institut fr Geophysik, Hamburg, GermanyD . J. Olbers, Institut fr Geophysik, Hamburg, GermanyK . Richter, Deutsches Hydrographisches Institut, Hamburg, GermanyW . Sell, Institut fi ' Geophysik, Hamburg, GermanyH . Walden, Deutsches Hydrogi'aphisches Institut, Hamburg, Germany

Deutsches Hydrographisches Institut, Hamburg 1973Schriftleitung: Dr. Helmut Madler, 205 Hamburg 80, Sichter 4

und Walter Horn, 2 Hamburg 6, Pehx-Dahn-StraBe 2Druck: J. J. Augustin, Glckstadt/Blbe


5/96

C O N T E N T SSummary - Zusammenfassung - Rsum 7Part 1. The experhnent 101.1. Introduction 101.2. The wave profile 151.3. Other field measui-ements 161.4. Intercomparison of frequency wave spectra 191.5. Dh'eotional measurements 191.6. Cases studied 221.7. Logistics 25Aclinowledgments 26Part 2. Wave growth 272 .1 . Generation cases 272.2. Kitaigorodskii's similarity law 282.3. W i n d stress and turbulence measiu-ements 282.4. Fetch dependence of one-dimensional spectra 322.5. The dhectional distribution 382.6. The mean som-oe function 392.7. Analysis of individual generation cases 422.8. O r i g i n of t l i e spectral peali 482.9. The energy and momentum balance of the spectrum 492.10. V a r i a b i l i t y of the spectrum 522.11. Conclusions 57Part 3. Swell Attenuation 713 .1 . Conservation of action and energy flux 713.2. Attenuation due to bottom f r i c t i o n 733.3. Energy flux analysis of swell data 743.4. Lateral variations of bottom topography 813.5. Dispersion characteristics 873.6. Conclusions 89

Appendix: Statistical analysis of the attenuation parameter F 90Notation 02References 94


6/96


7/96

S u m m a r y : Wave spectra were measm-ed along a profile extending 160 kilometers into the NorthSea westward f r om Sylt for a period of ten weeks in 1968 and 1969. During the main experiment inJuly 1969, thirteen wave stations were in operation, of which six stations continued measiu'ementsinto the first two weeks of August. A smaller pi lo t experiment was carried out in September 1968.CWrents, tides, air-sea temperatme differences and turbulence in the atmospheric bomidarylayer were also measured. The goal of the experiment (described in Part 1) was to determine thestructm-e of the source f imc t ion governing the energy balance of the wave spectrum, w i t h particularemphasis on wave gi-owth imder stationary offshore w i nd conditions (Part 2) and the attenuationof swell in water of finite depth (Part 3).The som'ce fmictions of wave spectra generated by offshore winds exhibit a characteristicplus-minus signature associated w i t h the shift of the sharp spectral peak towards lower frequencies.The two-lobed distribution of the source function can be explained quantitativeljr b j ' the nonlineartransfer due to resonant wave-wave interactions (second order Brag g scattering).'The evolutionof a pronomiced peak and its shift towards lower frequencies can also be understood as a self-stabilizing feature of this process. Por small fetches, the principal energy balance is between theinput by w i n d in the central region of the spectrum and the nonlinear transfer of eiiergj' awayf r om this region to short waves, where it is dissipated, and to longer waves. Most of the wavegrowth on the forward face of the spectrum can be attributed to the nonlinear transfer to longerwaves. For short fetches, approximately (80+ 20) % of the momentum transferred across the air-sea interface enters the wave field, in agreement w i t h Dob son's direct measurements of the workdone on the waves by surface pressirres. About 80-90% of the wave-induced momentum fluxpasses into currents via the nonlinear transfer to short waves and subsequent dissipation; the restremains in the wave field and is advected away. At larger fetches the interpretation of the energybalance becomes more ambiguous on account of the milmown dissipation in the low-frequencypart of the spectrum. Zero dissipation in this frequency range yields a minimal atmosphericmomentum flux into the wave field of the order of 20 % of the total momentmn transferacross the air-sea interface - but ratios up to 100 % are conceivable i f dissipation is important. Ingeneral, the ratios (as inferred f r om the noiflinear energy transfer) lie w i t h i n these l im i t s over awide (five-decade) range of fetches encompassing both wave-tank and the present field data,suggesting that the scales of the spectrum continually adjust such that the wave-wave interactionsjust balance the energy input f r om the w i nd . This may explain, among other featiu'es, the observeddecrease of Phillips' "constant" w i t h fetch.The decay rates determined for incoming swell varied considerably, but energy attenuationfactors of two along the length of the profile were t j rpica l . This is in order of magnitude agi'eementw i t h expected damping rates due to bottom f r i c t i o n . However, the strong tidal modulation predictedby theory for the case of a quadratic bottom f r i c t i o n law was not observed. Adverse winds didnot affect the decay rate. Computations also rule out wave-wave interactions or dissipation dueto turbulence outside the bottom boundary layer as effective mechanisms of swell attenuation.We conclude that either the generally accepted f r i c t i o n law needs to be significantly modified orthat sorne other mechanism, such as scattering by bottom u-regularities, is the cause of theattenuation.The dispersion characteristics of the swells indicated rather nearby origins, for which thoclassical 5-event model was generally inapplicable. A strong Doppler modulation by tidal cm'rentswas also observed.Messuiigen des Wind-Wellen-Waehstums uiid des Dnungszeralls whreiid des Joint North SeaW a v e P ro je c t ( J O N S W A P ) (Z u s amme n las s u n g ) . Walxrend einer Zeitspanne von insgesamt zelmWoohen in den Jaliren 1968 und 1969 wurden langs eines Profils, das sich 160 km weit seewartswestlich Sylt in die Nordsee hinaus erstreckte, Seegangsspektren gemessen. Walirend des Haupt-experiments im Juli 1969 arbeiteten dreizelm WellemneBstationen, an sechs Stationen wurdendie L'lessmigen audi in den ersten zwei Augustwoohen weitergefiUu't. Ein kleineres Vorexperimentwtu'de im September 1968 ausgeflirt. S t r m u nge n , Gezeiten, Wasser-Luft-Temperaturdifferenzenund Tm'bulenz in der atmospharischen Grenzschicht wurden zusatzlich gemessen. Das Ziel desExperiments (besclirieben in Te i l 1) war es, die Quellfunktion zu ermitteln, die die Energiebilanzdes Seegangsspektrums bestimmt, wobei besonders das Wellenwaohstum bei stationaren ablandigenWinden (Te i l 2) m id die Dampfung der Dnung bei encUicher Wassertiefe imtersucht -ivm-den(TeU 3).Die Quel l f iml i t ion des von ablandigen Winden erzeugten Seegangsspektrums zeigt ernecharakteristische Plus-Minus-Signatiu, verbunden mit einer Verschiebung des scharfen spektralenPeaks zu niedrigeren Frequenzen. Diese zweiteilige Strulitur der Quellfunlrtion kaim quantitativ


8/96

8 Erganzungsheft zur Deutschen Hydrographischen Zeitschrift. Reihe A (8), Nr. 12, 1973durch nichthneare Energieiibergange infolgeresonanter Wellen-Wellen-Wechselwirkungen (Bragg-Streuung zweiter Ordnung) erklart werden. Die Entwicldung eines betonten Peaks uird seine Verschiebung zu niedrigeren Frequenzen kann audi als Selbststabilisierung dieses Prozesses verstanden werden. Fr kurzen Fetch w i r d die Energiebilanz im wesentlichen ausgeglichen zwiscliender Zufu lu - durch den W i n d im Hauptteil des Spektrmns und dem nichtlinearen Transport vonEnergie aus diesem Bereich zu krzeren Wellen, wo sie dissipiert whd, und zu langoren Wellen.Der Hauptanteil des Wellenwaclistums an der Vorderflanlie des Spektrums kami auf nichthnearetibergange zu langoren Wellen zurckgeflirt werden. Fr km"zen Fetch flieBen (80 + 20) % desImpulses, der an der Grenzflache Luft-Wasser bertra-gen w i r d , in das Wellenfeld; das steht inbereinstimmung mit Dobson's d'ekten Messungen der Arbeit, die vom Oberflachendruok anden Wellen geleistet w i r d . Ungefalu' 80-90% des welleninduzierten Impvilsflusses w i r d ber nichtlinearen Transport zu kurzen Wellen und folgende Dissipation an die Strmung abgegeben, derRest bleibt im Wellenfeld m id w i r d von diesem fortgefhrt. Bei langerem Fetch wh-d die Interpretation der Energiebilanz wegen der unbekannten Dissipation im niederfrequenten T e i l desSpektrums ungewisser. Keine Dissipation in diesem Frequenzbereicli liefert eine imtere Grenze/ 4- 20des atmospharischen Impulsflusses in das Wellenfeld von etwa 20 _ % des gesamten Impuls-flusses durch die Grenzflache Luft-Wasser - aber Werte bis 100% sind vorstellbar, wenn dieDissipationwesentlich ist. Im allgemeinenliegen die Werte (bestnmt aus den nichtlinearen Energie-bergangen) innerlialb dieser Grenzen, und zwar fi ' einen weiten Bereich (fnf Dekaden) des Fetch,wobei sowohl Daten aus Wellentanks als auch von den hier dargestellten Feldmessungen einge-schlossen sind. Dadurch wu-d nahegelegt, daB sich die Skalen des Spektrums kontinuierhch derarteinstellen, daB die Wellen-Wellen-Wechselwirkungen gerade die Energiezufulir dm'ch den W i n dausgleichen. Dies kann imter anderem die beobaclitete Abnalnne der Phillipssohen , , K on stante" erklaren.Die Zerfallsraten, die fr einlaufende Dnung bestimmt wurden, variierten betraehtlich,aber Energiedampfungsfaktoren von 2 langs der gesamten Lange des Profils waren tj^pisoh. Diesstimmt grBenordnungsmaBig berein mit Dampfungsraten, wie sie infolge von Bodem'eibungerwartet werden. Jedoch ^vurde die starke Gezeitemnodulation, die von der Theorie f r einquadratisches Bodenreibungsgesetz vorhergesagt w i r d , nicht beobachtet. Gegenwinde battenkeinen EinuB auf die Zerfallsrate. Durch Berechnungen lassen sich auch Wellen-Wellen-Wechsel-wirkimgen oder turbulente Dissipation auBerhalb der Bodengrenzscliicht als wirksame Mechanismen der Dnungsdampfmig ausschlieBen. Wir folgern daraus, daB entweder das allgemeinakzeptierte Reibungsgesetz wesentlich geandert werden muB oder daB ein anderer Mechanismus,wie z.B. Streuimg durch BodenunregelmaBigkeiten, die Ursache der Dampfung ist.Die Dispersionscharakteristiken der Dnmigen deuteten auf recht nahe gelegene Ursprmigs-gebiete bin, f r die das klassische (5-Ereignis-Modell im allgemeinen nicht anwendbar war. Einestarke Doppler-Modulation durch Gezeitenstrme wurde beobachtet.Jlesures de la croissance des vagues dues au vent et du comblement de la houle dans Ie Projet commuurelatit aux vagues do la mer du Nord (Joint North Sea Wave Project - J O N S W A P ) ( R s u m ) .On a mesiu' les spectres des vagues, Ie long d'ime coupe s'tendaut sm- 160 ki lomt r e s , dans lamer du Nord , l'Ouest de Sylt, pendant une priode de dix semaines en 1968 et 1969. Au corn's del'exprience principale, en j u i l l e t 1969, treize stations d'observation des vagues furent mises enservice, dont six continurent les mesures pendant les deux premires semaines d'aot. On a f a i tune experience pilote plus rduite en septembre 1968. On a galement mesiu' les comants, lesmares , les differences de tempratm-e entre Fair et la mer ainsi que la tm'bulence dans la coucheatmosphrique voisine de la surface. Le hut de l'exprience (indic[u dans la premire partie)tait de dterminer la structure de la fonction-origine determinant l'quilibre de l'nergie du spectrede la vague, en insistant particulirement sur la croissance de la vague quand le vent souffle deterre de fajon constante (deuxime partie) et sur l'attnuation de la houle par fonds de profondeurlimite (troisime partie).Les fonctions-origines du spectre de la vague souleve par des vents de terre ont une allureplus ou moins ca rac t r i s t ique , en rapport avec le dcalage du sommet escarp du spectre vers lesfrquences plus basses. La distribution de la fonction-origine selon deux lobes peut s'expliquerquantativement par le transfert non linahe du aux actions rciproques vague-vague, en rsonance(Dispersion de Brag g du second ordre). L'volution d'un sommet accentu et son dcalage versles frciuences plus basses peuvent aussi tre interprts comme mie caractristique de stabilisationde ce processus. Pom' les petits dp lacements , l'quilibre de l'nergie se realise principalement entrecelle qu'apporte le vent dans la region centrale du spectre et le transfert non lineaire d'nergieloin de cette region: - Vers les vagues courtes, o elle se dissipe, ^ et vers les vagues plus tongues.La plus grande partie de la croissance de la vague, sur la face avant du spectre, peut tre attribueau transfert non lineaire vers les vagues plus longues. Pom' les dplacements courts, environ(80 + 20) % de la force vive transfre k travers l'interface air-mer entre dans le champ de la vague,d'aprs les mesures d-ectes de Dobson, relatives au travail que les pressions extrieures effectuentsm' les vagues. Environ 80 a 90 % du f l u x de force cre par la vague passe dans les coiu'ants parla voie d'un transfert non lineaire dans les vagues ooiu'tes et d'une dissipation u l t r i e u r e , le restedemeui'e dans le champ de la vague et se trouve emport au lo in . Pour les dplacements plus


9/96

Hasselmann et al., JONSWAP 9amples, l'interprtation de Tquilibre de l ' nergie devient plus incertaine car on ne sait r ien de ladissipation dans la rgion des basses f r q u e nc e s du spectre. Une dissipation nuUe dans oette gammede f r q u e nc e s donne un f l ux min ima l de force v ive a tmosphr ique dans le champ de la vague, del'ordre de 20 ^ 10)'^ '" ^ transfert total de force vive , travers l'interface air-mer, mais des rapportsallant jusqu'a 1 0 0 % sont concevables si la dissipation est importante. En gnra l , les rapports(dcoulant du transfert non lineaire d 'nergie) restent en dedans de ces limites dans mie largegamme (cinq dcades) de dp lacements cernant Fenvironnement de la vague et, aussi, les donnesprsen te s du champ. Cela laisse penser que les chelles du spectre s'ajustent d'une fagon continue,de sorte que les actions r c ip roques des vagues qu i l ib r en t exactement l ' ne rg ie appor te par levent. On peut expliquer ainsi, entre autre pa r t i cu la r i t s , la dcroissance observe de la constantede Phill ips avec le d p l a c e m e n t .Les taux de comblement d te rmins pom la houle arrivant sm des fonds l imi ts variaient no r m m e n t mais les facteurs d'attenuation des deux nergies siu- la longueur de la coupe t a i e n tca r ac t r i s t iques . Leur ordre de grandeur correspond aux taux d'amortissement escompts pourle frottement sm' le fond . Toutefois, on n'a pas observ la forte modulation due k la m a r e ,t h o r i q u e m e n t p r v u e dans le cas d'une lo i quadratique du frottement sur le f ond . Les ventscontrahes n'ont pas af fec t le taux de comblement. Les calculs l im inen t aussi, comme moanismesrels de l ' a t t n u a t i o n de la houle: - Les actions reciproques vagues-vagues, - ou la dpe rd i t iondue . la turbulence hors de la couche voisine du fond . Nous oonclm'oiis, ou bien c[u'il faut modifierp r o f ond m e n t la loi gnra l ement admise pour le frottement, ou que quelqu'autre mcani sme ,telle la dispersion cause par les i r rgula r i ts du fond , est la cause de l ' a t t nu a t i on .Les ca rac t r i s t iques de dispersion des boules indiquent p l u t t des origines voisines, pourlesquelles le modle classique du cas ta i t gnra l ement inapplicable. On a ga lement oliservime forte modulation Doppler due aux courants de mare .


10/96

1. The Experiment1.1. Intioductiou

The last fifteen j ears have seen considerable progress in the theory of \\dnd-generatedocean waves. These advances have been paralleled hj the development of improvedmathematical techniques and computer facilities for numerical wave forecasting. However,critical tests of theoretical concepts and numerical forecasting methods have been severelyl i m i t e d in the past by the lack of detailed f i e ld studies of wave growth and decay. Thusconsiderable effort has been devoted to the theoretical anatysis of many of the processeswhich could affect the spectral energy balance of the wave field - the generation of waves byw i n d (bj^ a variety of mechanisms), the nonlinear energj' transfer due to resonant wave-waveinteractions, the energy loss due to white capping, interactions between very short wavesand longer waves, and the dissipation of waves in shallow water by bottom f r i c t i o n , to l i s tsome of the more important contributions - but only very few Lave been subjected toquantitative field tests. Even where field experiments have been specifically designed tostudy these questions, limitations in available instrumentation have often led to incompletecoverage of the wave field, thus handicapping the interpretation of the data. Consequently,present wave prediction methods have necessarily been based on rather subjective conjecttu'esas to the f o r m of the som'ce function governing the fundamental energy balance ec(uation.

The Joint North Sea Wave Project (JONSWAP) was conceived as a cooperative ventm-eby a number of scientists in England, Holland, the United States and Germany to obtainwave spectral data of sufficient extent and densit}^ to determhie the strnctiue of the soiu-cefunction empu'ically. I t was decided that an arraj' of sensors operating c[uasi-continuouslj'for a period of several weeks was the most straightforward method of obtaining adecjuatesampling clensitj' of the wave f ie ld wi th respect to frec[uenc3', propagation direction, spaceand time, under the deshed wide variety of external geophysical conditions. The array thatwas eventually used consisted of thirteen wave stations spaced along a 160 km profile extendingwestward f r o m the Island of Sylt (North Germany) into the North Sea. A l l instrumentsprovided the usual frec[uency power spectra, while six of the stations (5 pitch-roll buoj^sand 1 six-element array) yielded directional information in addition. The complete profilewas in operation for four weeks during July 1969; another six weeks of data were obtainedf i ' om a smaller number of stations during the two weeks immediately f o l l o w i n g the mainexperiment and in a p i l o t experiment in September 1968.

The enthe set of time series was spectral anatyzed, but the interpretation of the datais restricted in the present papers to selected cases f a l l i n g into either of two partioularlj'simple categories: the evolution of the spectrum under stationary, offshore w i n d conditions(Part 2), and the attenuation of incommg swell (Part 3). Over 2000 wave spectra weremeasm-ed; about 300 of these represented reasonably well-defined c|uasi-stationary generationcases, of which 121 con-esponded to "ideal" stationary and homogeneous 'wind conditions;654 spectra contained well-developed peaks suitable for the swell attenuation study.Wave Growth

The present wave generation study may be regarded as a natural extension of thepioneering work of R. L. Snyd er and C. S. Cox [1966] and T. P. B arnet t and J. C. Wilker-son [1967]. Both of these experiments utilized a single mobile wave station to determinethe evolution of the wave spectrum in space and time.


11/96

Hasselmann et al. , JONSWAP H_ Snjrder and Cox measured the growth rate of a single frequeney component (0.3 Hz) bytowmg an array of wave recorders downwind at the group velocity of the component. Theyfound exponential growth in accordance mt h the linear instability mechanisms proposedby J. Je f f r eys [1924] and J. W. Miles [1957], but at a rate almost an order of magnitudegreater th an predicted by either theory - or by the later instability theories which took moredetailed account of the turbulent response characteristics of the atmospheric boundary layer

ggC M . P h i l l i p s [1966], K. Hass el mann [1967, 1968], R. E. Davi s [1969, 1970],P. Long [1971], D. E. Hassel mann [1971]). Moreover, assuming that the observed growthrates were due enthely to a linear input f r o m the atmosphere, Snyd er and Cox inferred aniomentum tran.sfer f r o m the atmosphere to the wave field which was several times greaterthan the knomi total momentum loss f r o m the atmosphere to the ocean. As shown below,i t appears that this paradox can be largely resolved by taking account of the nonlinear waveinteractions, which we found to be responsible for the major part of the observed wave growthon the forward face of the spectrum.The evolution of the complete (one-dunensional) wavenumber spectrum under fetoh-l i m i t e d conditions was fii'st measured by T. P. B arnet t and J. C. Wi l kerson [1967] usingan au-borne radar ahuneter*. Only two wave profiles were AoMm, but again exponentialgrowth rates were found in general agreement Awth Sny der and Cox's results. I n addition,the measurement of the complete spectrum revealed a significant overshoot effect; the energyat the peak of the spectrum was found to be consistently higher by factors between 1.2 and2 than the asymptotic equilibrium level approached by the same frequency at large fetchesA similar effect was observed by A. J. Sut h erlan d [1968] and by H. Mi tsuyasu [1968a,1969] m a wind-wave tank (see also T. P. B arnet t and A . J . Sut h er lan d [1968]) Al lauthors speculated that the effect may be due to nonlmear processes. This interpretationfound fm-ther support through H. Mit suyasu 's [1968b] wave t a i f i i mea.surements of thedecay of a random wave field, which could be explained quite w e l l by T. P. Barnett's [1968]parametrical estmiates of the nonlinear energy transfer due to resonant wave interactions( l i ^ Hassel ma nn [1962, 1963a, b]). (In then Pacific swell study, P. E. SnodgrassG. W. Grov es, K. E. Hassel mann et al. [1966] had sunilarly concluded that the decay

o l new swell close to a generation area was in accordance w i t h this process.)Useful i i f f o r m a t i o n on wave growth can also be obtained indirectly f r o m measurementsat a single fixed location under different w i n d conditions. Using dimensional argumentsS A. Ki ta igorodski i [1962] has suggested that for a stationary, homogeneous w i n d fieldblowing orthogonally off a straight shore, the appropriately nondimensionalized wavespectrum shoid be a universal function of the nondimensional fetch x = gxjul only where xIS the (dimensional) fetch, g is the acceleration of gravity and is the f r i c t i o n velocity Thuswave measurements obtained at a fixed location for different w i n d speeds can be translatedinto an eqmvalent dependence on fetch at a fixed -\vind speed.

H . Mitsuyasu [1968a, 1969], H. Mitsuyasu, R. Nakayama and T. Komor i [1971],P. C. L i u [1971] and others have applied Kitaigorodski i ' s scaling law successfully to relatewave tank and field measm-ements and characterize the basic properties of fetch-limitedspectra. The JONSWAP data is in general consistent w i t h Kitaigorodskii's scalinghypothesis and confirms many of the spectral featui-es summarized by these authors:pronounced overshoot factors, narrow peaks (considerably sharper than the fully-developedPi er so n- Mos ko wi tz [1964] spectrum), very steep forward faces ( w i t h associated rapidgrowth rates of wave components on the face), and an /-^ high frequency range w i t h a fetch-dependent 0. M. Phi l l ips ' [1958] "constant".A principal conclusion of om- wave growth study is that most of these properties canbe explained quantitatively or qualitatively by the nonlinear energy transfer due to resonantwave-wave interactions (second order Bragg scattering). Thus the evolution of a sharp spectral

\T T * ^ " ^ i l ' " " ! ' experiments using a laser altimeter have been repeated recently by D. B . Ross,De L e o r i b u s l i m ] Conaway [1970] and J. R. Scbule, L. S. Simpson and P. sl


12/96

12 Erganzungsheft zui- Deutschen Hydi-ographischen Zeitscln-ift. Reihe A (8), Nr. 12, 1973peak is found to be a self-stabilizing feature of this process. The continual shift of the peaktowards lower frequencies is also caused by the nonlinear energy transfer, and explains thelarge growth rates observed for waves on the forward face of the spectrum.Por small fetches, the energy balance in the main part of the spectrum is governed bythe energy input f r o m the atmosphere, the nonlinear transfer to lower and higher frec[uenoies,and advection, dissipative processes apparently playing only a minor role. This balancedetermines the energy level of the speotnim, i.e. the value of Phi l l ips ' constant, and therate at which the peak shifts towards lower frequencies. Assuming a f r i c t i o n coefficient c oof the order of 1.0 x 10-^ this pictm-e implies that nearly all of the momentum lost from theatmosphere must be entering the wave field - in agreement ^^dth P. W. Dobson's [1971]correlation measm-ements of wave height and surface pressure. About 10 % of the momentumfiux f r o m the atmosphere to the wave field is transfen-ed to lower-fr-equency waves, wherei t is advected away. The rest is transferred to short waves, where it is converted to cm-rentmomentum by dissipative processes.W i t h increasing fetch, the wave-wave momentum transfer to high wavenumbersdecreases, representing only 13 % of the to tal momentum fiux across the ah-sea interfacei n the I h n i t of a fully-developed Pierson-Moskowdtz spectru,m. The net momentum fiuxT f r o m the atmosphere to the wave field at large fetches depends on the unknown dissipationi n the main part of the spectrum. Zero dissipation in this range implies a lower l i m i t for T , ^of about 20 % of the to ta l ah-sea momentum flux - but this value could also lie closer to100 %, as i n the small-fetch case, i f dissipation is an important factor in the energy balanceof the principal wind-sea components.Although the present study has helped to c l a r i f y the general struotm-e of the energybalance of fetch-limited wave spectra, a number of questions remain. The mechanism bywhich energy and momentum is transferred f r o m the w i n d to the waves could not beinvestigated. Measm-ements w i t h hot-wu-e instruments, w i n d vanes and cup anemometersyielded atmospheric tm-btence spectra and the net momentum flux fr-om the atmosphereto the ocean, but the instruments were too far above the surface to resolve the interactionsbetween the atmospheric boundary layer and the wave field. A second open problem concernsthe dissipative processes balancing the nonlinear energy flux to short waves. This questionis of considerable interest not only w i t h regard to the wind-wave energy balance, b t i t alsofor the application of microwave techniques to the measurement of sea state fr'om sateUites(cf. K. Hasselmann [1972]). Finally, the transition from fetch-limited spectra to a f u l l ydeveloped eqiulibrium spectrum poses a number of questions which could not be investigatedw i t h i n the finite range of fetches available in this study.Swell At te n u a t io n

For the generation studies, the water depth along the entire profile was sufficientlydeep to be regarded effectively as i n f i n i t e . However, much longer incoming swell componentsnormally " f e l t bottom" along most of the profile, thus affording a good opportunity toinvestigate the effects of bottom topography on swell propagation.I t is w e l l known that bottom effects play a significant role in the energy balance of wavesi n relatively shoal areas such as the North Sea or other parts of the continental margins.Most investigators have attributed the observed modifications in the f o r m of the f u l l ydeveloped spectrum (e.g. J. D arbysh i re [1963]) and the enhanced damping of the swellto bottom f r i c t i o n . Assuming a c^uadratic friction law, C. L. Bret sohnei der and R. 0. Reid[1954] estimate a f r i c t i o n coefficient of the order 10"^. Similar values were found byK . Hasselmann and J. I . Collins [1968], who applied a more detailed spectral formulationof the bottom fr-iction theory to hm-ricane swell data i n the G u l f of Mexico. The considerablylarger decay rates found by H. Walden and H . J. Rub ac h [1967] in the North Sea are alsoconsistent w i t h the Hassel mann and Co l li ns theory i f the tidal cm-rents are taken intoaccount.However, the extensive swell data available through this experiment, while i n order ofmagnitude agreement w i t h previous observations of swell attenuation rates, contradict a


13/96

Hasselmann et al., J ONSW A P


14/96

Erganzungsheft zur Deutscfien Hydrograptiischen Zeitschrift. Reihe A (8), Nr. 12,1973


15/96

Hasselmann et al., JONSWAP 15significant prediction of the theory. A quadratic (or any nonlinear) f i ' i c t i on law should leadto a strong modulation of the swell decay rates by tidal currents. Experimentally, no suchvariation is f o i u i d .We suggest two possible explanations - neither of which we have been able to test i nthe present experiment. I t is conceivable that the quachatic f r i c t i o n law, although shown tobe an acceptable fii'st-order approximation for sinusoidal waves in a wave tank (J. A. Putnamand J. W. Johnson [1949], P.P. Savage [1953], K. K aj i u r a [1964], Y. Iwagaki ,Y . Tsuchiya and M . Sakai [1965], I . G. Jonsson [1965]), is not v a l i d for the time-dependentboundary layer generated by the superposition of a random wave field on a tidal currenti n the real ocean. An objection to this explanation is th at a complete independence of theswell attenuation on the tidal cuiTents implies a linear f r i c t i o n law, which seems improbablefor a tm'bulent boundary layer.

Alternatively, the observed swell decay may be due to processes other than bottomf r i c t i o n . Estimates of the damping due to interactions ^vith turbulence distributed throughoutthe water column (K. Hasselmann [1968]) indicate that this process is too small by severalorders of magnitude. A more promising possibility is the baokscattering caused by u-regularitiesof the bottom topography at scales comparable w i t h the wavelength of the swell. Recentcomputations by R. Long [1972] show that this interaction could account for the observedattenuation rates assuming r.m.s. bottom hregularities in the appropriate wavelength rangeof the order of 20 om. Experimentally, this process should be identifiable by the presence ofoffshore-propagating swell components whose energy increases w i t h distance f r o m shore (incontrast to the decreasing energy of swell which may be refieoted fi-om the shore). Unfortunately, the dh'eotional resolution in the present experiment was insufficient to resolvesecondary swell components propagating in a dhection opposite to the main swell,1 . 2 . The Wave Profile

The site of the wave study and the 160 km long measming profile are shown in Figures 1.1and 1.2. The area was chosen on account of its relatively smooth bottom topography, moderatetidal cm-rents and convenient logistics.The main experiment ran f r o m July 1 u n t i l July 31, 1969, but measurements continuedf r o m August 1 to August 15, 1969 along a reduced profile (Stations 1-5 and 8). Diu-ing thesesix weeks, 30-minute wave recordings were obtained contmuously at 2 or 4 hom-ly intervals.Fm-ther measm-ements were also made along a 10-station profile dm-ing a pot experhnentf r o m September 1 to September 30, 1968. I n this earlier experiment measm-ements werel i m i t e d to "hiteresting" cases of wave generation or incoming swell, and the times of recordmgat different stations were staggered to allow for the propagation time of the expected prmeipalwave components. This mode of operation proved to be too demanding on the reliabUity ofweather forecasts and commmiications, so that in the main experiment the f o l l ow i ng yeara more r i g i d , predetermined schedule of recordings was introduced, in which the only on-spotdecisions involved changes from standard four-hourly recording to two-hourly measm-ementsdm-ing east-wind generation conditions.The depth distribution along the profile lay in a convement range enabling the studyof both the generation of waves dm-ing offshore winds and the attenuation of long swellsrmunng inshore. For short, fetch-limited waves generated by local east muds, the depthcould be regarded as essentiaUy i n f i n i t e along most of the profile, cf. Figm-e 1.3. The om-ves

indicate the depth H = at which the principal wind-sea components would begin to beinfiuenced appreciably by the bottom; the wavelength = ^ for a given w i n d speedand fetch was evaluated f r o m the appropriate empu-ical fetch dependence of the frecjuencyof the spectral peak. Fig. 2.6. The bottom influence is seen to be negligible for all w i n dspeeds relevant for this study (the maximum east winds occiu-ring dm-ing the experimentwere of the order of 15 meter per second). On the other hand, a typical swell component of


16/96

16 ErganzLingsheft ziu' Deutschen Hj ch-ographischen Zeitschrift. Reihe A (8), Nr. 12, 197310 seconds period, corresponding to a deep water wavelength of about 150 n i , is appreciablyinfluenced by the bottom along the enthe profile, so that bottom f r i c t i o n or bottom scatteringeffects shoidd be easily observable.A variety of wave instruments were used in the experiment ( F i g . 1.2). A t Stations 1, 2, 3, 5and 8 (we refer here and in the f o l l ow i ng to the July 1969 profile) the wave motion wasmeasm-ed by a small fioat moving w i t h i n a perforated pipe mounted on a post. A t Station 4 alinear dhectional array of six wave recorders was installed at variable separations along aline orthogonal to the profile (of. Section 1.5). The wave sensors consisted of sub.sm'facepressm-e transducers mounted on posts, w i t h the exception of a closely spaced pah of resistancewave staffs used to resolve the short wave components. Tethered "waverider" buoys, whichconverted measm-ements of vertical acceleration internally to wave height, w ere used atStations 7, 9 and 10. Al l of these nearshore instruments were unattended and/or remotelycontrolled and generated FM output signals which were telemetered to a shore station atthe base of the profile. This technique was not readily applicable to the more distant Stations11-13 and Station 6, which lay w i t h i n a shippmg lane. These stations were occupied bypitch-roll buoys deployed f r o m ships. The pitch-roll buoy measm-es the vertical accelerationand both components of wave slope, thus giving a dn-ectional resolution of the wave fieldroughly comparable to the array at Station 4 (cf. M. S. Longuet -Higgins, D. E. Cartwrigh t and N . D. Smi th [1963]).A l l instruments had high-frec[uency cut-offs between 0.7 and 1 Hz, corresponding towavelengths between 3 and 1.5 m. Digitizmg rates v.'ere normally 2 per second, ecpuvalentto 1 Hz Nyquist frequency.For iiitercalibration pm-poses Station 10 was occupied dm-ing the f irst half of the 1969experiment by two instruments, a waverider and a pitch-roll huoy. This also provided aredundancy safeguard, and was useful in occasionally freeing the ship at station 10 to assisti n servicing the unattended stations. I n the second h a l f of the experiment the pitch-rollbuoy of Station 10 was moved to Station 7 and the waverider of that station was switchedto Station 5 for comparison w i t h the wave fioat instrument. I n addition, a number of intercomparison nms between various combinations of instraments were made during the 1968experiment. A continual check on calibration consistency was fm'thermore provided by theintermeshed distribution of different types of instruments along the profile (cf. Section1.4).1.3. Other Field Measm-ements

To define the envhonmental parameters relevant for the energy balance of the wavefield, -winds, atmospheric momentum fiuxes, cmrents, tidal heights and afr-sea temperatm-eswere monitored at various stations along the j i r o f i l e .W i n d speeds and dfrections and tidal heights were recorded contmuously at Stations 2and 8. Winds were also measmed concm-rentty w i t h ah-sea temperatm-es every two hourson the ships.A t Station 8, atmospheric turbulence was measm-ed at two levels, 5 and 8 m above meansea level. The sj'stem at each level consisted of a combmation of standard hot-whe instruments,together vdth cup anemometers and vanes intended for calibration purposes (K. Enke[1973]). A l l tm-bience data were telemetered to shore together w i t h the concurrently takenwave data (of. Section 2.3).Contmuous cm-rent recorduigs were made throughout the 1968 and 1969 experimentsat a number of depths and stations (Fig. 1.3). Emphasis on current measurements was basedon the theoretical anticipation (K. Hasse lmann and J. I . Collins [1968]) that eim-entsnear the bottom would have a strong influence on the attenuation of long swell by bottomf r i c t i o n . I t was also conceivable that vertical cmTent shear would influence wave generation.However, no detectable current influence was found in either the generation or the swellattenuation study. Since in most cases the observed ciUTents did not deviate appreciablyfrom the tabulated tidal cm-rents (particularly the near-bottom cm-rents) it proved adequate


17/96

Hasselmann et al, JO NS WA P 17

ST . 5JULY 2 2 1 9 6 9 2 0 0 0X WAVE RI DE R+ WAVE F L OA T

0 . 4 0 rST. 5JULY 2 2 1 9 6 9 1 6 0 0X WAVE RIDER+ WAVE F L OAT

ST. 4JU LY 2 2 1 9 6 9 1 8 3 0X PITCH - R O L L+ RESI STANCE WIRE

1.00

080

0 6 0

0 . 4 0

O 2 0

0.00

ST. 4J UL Y 2 2 1 9 6 9 1 6 3 0X PITCH - R O L L-t- RESISTANCE WIRE

0. 6 0.0FREQUENCY (Hz)

Fig. 1.4 a-d. Intercomparison measurements of frequency wave spectra


18/96

PES

(H

PWES

MmHoo

1

1

PITCH

1

l )

- ROLL /y

'/"A

\m

-/'

S t. 5Sep t 6 1 9 6 8 1 2 3 0


19/96

Hasselmann et al., JONSWAP 19for the negative ontcome of the eorrelation analysis described in Parts 2 and 3 to w o r l i w i t ht l i e tabulated rather than the observed currents.Independent of the wave project, however, the measui'ement of oun-ents was alsomotivated by interest in the baroclmio response of the sea to a variable w i n d field. Por thispm-pose a continuous hydro-station program was carried through f r o m the ships at theoutermost stations 11, 12, and 13. This work v d l l be reported elsewhere (G. Becker ,K . P. Ko l ter mann , G. Pr ah m et al. [ i n prep.].1.4. Intercomparison of Frequency Wave Spectra

The choice of a variety of wave measmdng technic[ues was necessitated by the limitationsof VHF and UHF telemetry ranges, the depths at which wave posts could be economicallyinstalled, and in part by the prior experience of participating groups. A prime concern ofthe experiment was therefore to establish the compatibility of the different technic[ues used.Figs. 1.4a-h show a series of one-dimensional frequency spectra measm-ed at the same locationand time by pahs of different instruments. I n general, the agreement is excellent. Not al lcombinations of different instrument pahs were compared, but an independent conthmalcheck on the iiitercalibration consistency of all instruments was provided by the dovetaileddistribution of different instrument types along the profile: at least one instrument of eachgroup was fianked at some point in the profile by instruments of another type. In all casesstudied, the spectral distributions showed a smooth variation with station number, withoutdiscermble discontinuities at the instrument transition points ( w i t h the exception of onegeneration case on September 11, 1968, i n which thi'ee of the runs indicated an - mrresolved -systematic error in the pitch-roll buoy measm'ements at Station 6, which were then discarded).

Although c o i i t r a r j ' to standard experimental practice, the simultaneous use of a numberof independent measmement teohiiic[ues proved to be an asset i n establishing a broadcalibration base and providing continual calibration monitorhig thi-oughout the experiment.The restriction to a smgie type of instrument has advantages in linear problems involvingthe comparison of energy levels at the same frequency. I n this case the common instrumentaltransfer function can be eliminated. However, in the present experhnent the evolution ofthe wa.ve spectrum was f o m i d to be strongly dependent on noifiinear processes for which thedetermination of absolute energy levels in different frecjfuency bands was essential.I t appears that Avith very few excejitions the one-dimensional wave spectra measuredby al l instruments were acom-ate to ^v i th in the attauiable statistical resolution. I n a fewcases the records were contaminated by noise, most frequently due to radio interference inthe telemetry links. The noise spectrum was generally white, and i f sufficiently low wassimply .subtracted. For noise levels greater than 20 or 30 % of the total energy, the measm'ementwas discarded. About 90 % of the data yielded useable spectra in this sense. I n the case ofthe swell studj' (Part 3), the figm-e was somewhat reduced by the inherent low-frequencynoise of the pitch-roll buoys, which sometimes masked low-energy swell peaks of long period.1.5. Dhectional Measm'ements

The dh-ectional distribution of the wave field was determined f r o m the pitch-roll buoys(Stations 6, 10, 11, 12 and 13) and the linear array at Station 4. Both instruments yieldedoifij^ a l i m i t e d number of moments of the energy distribution (, 0) w i t h respect to propagationdhection 9 for a fixed frec(uenoy . The moments obtained from the two types of instrumentdiffer, so that an intercomparison is, in general, possible only in terms of a given parametricalmodel of the angidar distribution function.The dh-ectional characteristics of the pitch-roll buoy (h i its earliest instrumental form)are described by M . S. Lo ng ue t- Hi ggi ns et al. [1963] and by D. E. C ar tw r i gh t andN . D. Smith [1964]. Essentially, the instrument records three channels, two components (1)and (2) of the wave slope, and the vertical acceleration (3). A fourth (compass) channelenables (1) and (2) to be re-oriented to geographical axes. The auto-spectrum of (3), Ogg, is


20/96

20 Erganzungsheft zur Deutschen Hych'ographisehen Zeitschrift. Reihe A (8), Nr. 12, 1973easily translated into the total energy spectrum of vertical displacement, E{f), and theautospeotra of (1) and (2) combine to give a dhect measm-e of total wavenumber k, throughthe relationThree of the cross-spectral elements C,j., Qij, have zero expectancy, while the remainingthree, together mth Oj^ C a, provide four explicit and independent angular moments ofthe dhectional spectrum, of the f o r m

]" F ( f , e ) ( k e y d e , n=l,2. 71

Eor reasons given below we in fact found use only for moments of order n = i, and thesemay be expressed as moments of the normalised angular distribution s{f, ), where ! ' ( / , 0) =s (/, 0) E ( / ) , Avith Y s (/, 0) d0 = 1 ; thus

It

where (/cj, k^) = (k cos 0, k sin 0), and k; = J kjSdO. nI n the case of the linear array, the wave heights d and l^j measured at any two locationsspaced at a distance Vjj normal to the profile y i e l d dhectional moments

where the angle of wave propagation 0 is measm-ed relative to the profile dhection.The array consisted of a superposition of two 4-element arrays differing by a factorof 4 i n scale (Fig. 1.5). The coordinates of each 4-element array were chosen as Xj = 0, D,

L I N E A R A R R A Y , S T A T I O N 4z1-

A 1J (1J 1' "1

L I N E A R A R R A Y , S T A T I O N 4z1-

A 1 2D 1 3D I D I , ' * ^1 2D' 3D' . 1 . .1to0WAVE STAFF Oe PRESSURE TRANSDUCER ^D=7m t)D'=4D = 28m

F i g . 1.5. Tl i e linear array at Station 44D, GD {D = 7 m and 28 m) yielding the 6 lags r^j = D, 2D, .. . 6D. Each subarray provideddhectional information (depending somewhat on the distribution assumed) in wavelengthbands between about 10 m and 150 m, and 40 m and 600 m, respectively.On account of theh mhTor sj^mmetry linear arrays are unable to distinguish betweenwaves approaching the array at the same angle but f r o m opposite sides of the array, i.e. at


21/96

Hasselmann et al., JONS^VAP 21anoies 9 and 7 0 - 0 . This ambigui ty was not regarded as a serious limhation in the presentexperiment. The smaller-scale array was designed to resolve the dhectional distribtition ofshort locally generated weaves imder east-wind condhions, whereas the larger-scale arraywas iised to determine the propagation dhection of incoming swell. I n both cases only oneof the two waveimmber half-planes contained signhicant energy.

A number of techniques have been developed for the dhectional analysis of muhisensorarrays I n our case, the details of the dhectional distributions tm-ned out to be unerhicalfor the interpretation of the data. Thus in the wave-growth study, values of the meanpropagation dhection and mean square dhectional spread proved adequate to compute andinterpret the somce function to wdthin the error limits imposed by other factors of the ex-periraent The same parameters were also sufficient to characterize an incoming swell beam.(Spectra containmg two swell components w i t h the same frequeney but different propagationdhections were very seldom.) n . .1 . ^ iConsecpiently, the dhectional data were analyzed and compared prunariiy w i t h respectto the two simplest dhectional parameters, defined (in terms of the phch-roll data) as

0 = arotan^ ^]''s{9)9d9,

1^ J s{9){9-9j'd9I n the Ihnh of a narrow beam, these are identical w i t h the mean dhection and r.m.s. dhectionalspread, as defined by the integrals on the right hand side*.Although this made o i f iy partial use of the avafiable dhectional resolution, no attemptwas made at a systematic, high-resolution analysis, except for occasional spot checks of theconsistency of the remahihig dhectional moments w i t h respect to the two-parameter dhectional

Apart fromthe insensitivity of om- conclusions to the details o f the dhectional distribution,this approach was also motivated by experimental difficuhies. Although extensively testedi n individual measurements, the compass units of the pitch-roll buoys were apparentlyunable to withstand the continual mode of operation requhed for this experhnent, and onlyone of the five instruments yielded rehable dhectional data throughout the enthe period.Also an early failm-e of the dighal data logging system for the linear array meant that muchof this data had to be recovered from analog traces by pamstaking semiautomatic digitaliza-tion. The restriction to two dhectional parameters had the advantage in this case that theycould be derived from only two sensors of the array.Analogous variables to 0 and 0 , for two wave sensorsseparated by a distance r orthogonal

to the profile may be defined as/arctan QIG9' = arcsin' kr

i - R ' 1 .1 1 J, f(^'+Q-, where the coherence M ={kr cos 0 ; ) ^and we have dropped the subscripts ij on C, Q and r. For a narrow beam 9'^, 9', - > 0 ^ , 0 ^

* I n contrast to the definitions in terms of kj, the latter integrals represent meaningfulquantities only m the narrow beam I rn i i t . On acooimt of the non-periodicity of the "^tegrand,they are not invariant w i t h respect to redefinition of the 0 interval, say from (-Tt, n) to (, Zn),for arbitrary distributions s ( 0 ) .


22/96

22 Erganzungsheft zur Deutschen Hydrographischen Zeitschrift. Reihe A (8"), Nr. 12, 1973To compare the dhectional estimates f r o m the pitch-roll and linear array measm-ements,a standard spreading function model was assumed,

where / j is a normalismg parameter, making+ 11 s d = 1. 71

Pig. 1.6 a shows the relation between 9^ and 9'^ for hr = 1 and different beam widths 9^,as characterized by the parameter p. The corresponding relation between 9^ and 9' as afunction of the propagation dhection 9^ is shown in Eig. 1.6b.Shmlar curves were computed for other models of the spreading function, some includinga skewness factor. The transformation was f o m i d to be rather insensitive to the choice ofmodel.W i t h the aid of Figs. 1.6 a and b, the linear-array dh-ectional data could be convertedto equivalent pitch-roll dhectional parameters. A typical intercomparison run at Station 4,shoAving satisfactory agreement of the dh-ectional parameters after conversion, is presentedi n F i g . 1.7.1.6. Cases Studied

F i g . 1.8 shows the general w i n d history for the three periods of the experiment. W i n ddhections are defined relative to the proffie, arrows pointing vertically upwards representingoffshore winds parallel to the j i r o f i l e . The arrows represent averages of the winds measm-edat given time along the profile.Wave groivth was investigated for the periods marked G; the symbols I G refer to "idealgeneration" conditions of high stationarity and homogeneity, which were selected for moredetailed study. A more complete time history of the vnnd fields during these periods andweather maps are presented i n Part 2.Periods marked S were selected for the swell attenuation study. The term swell is normallyapplied to waves no longer being actively generated by the w i n d , i.e. in the present case,waves travelling inshore at phase speeds exceeding the onshore component of the local w i n dvelocity. Usually, these conditions apply only after the waves have traveUed a sufficientdistance fi-om the wave som-ce for dispersion to have converted the swell spectrum into theclassical f o r m of a narrow low frequency peak. However, in the present experiment severalcases also occurred w^here the local -wind field died down rather rapidly, leaving a locallygenerated, broad-band swell spectrum. We have investigated here only "classical" swellcases i n which the w i d t h of the swell peak was narrow compared w i t h the swell frecjuenoy.This was because om- main interest concerned the bottom interactions, which we wished toseparate as far as possible f r o m other dynamical processes. Computations (K. Hasselmann[1963b]) and experiments (F. E. Snodgrass et al. [1966], H. Mitsuyasu [1968b]) indicatethat noifiinear wave-wave interactions are important for the energy balance of a broadband, newly created swell specti-um, whereas they can be neglected for a w e l l dispersed swelltrain* (pro-sdded the frequency lies below the frequencies of the local background w i n d sea).The generation and swell cases studied in this paper have been chosen f r o m a muchlarger ensemble of measured spectra. They represent the simplest situations for investigatingthe basic processes that presumably also control the evolution of the wave spectrum in thegeneral case of an arbitrary vnnd field and boundary condition. The interpretation of the* Tlie essential condition is that the swell energy is small relative to a newly created swell.I n sorne of the quasi-stationary swell cases considered in Part 2, this was caused by angulardispersion alone, rather than the usual combination of angular and frequency dispersion.


23/96

Hasselmann et a l, J O N S W A P 2 3

es (PITCH - R O LL BU O Y )Fig. 1.6. a,b. Relation between parameters ctiaracterizing ttie mean direction (a) and directional spread (b) for ttie pitcti-roll buoys

and the linear array. Acos^e spreading factor is assumed


24/96

Fig18WnhsodnSp1aJyA11Wnawrp

a

aohpe

Ohwnpeohpepnuw


25/96

Hasselmann et al., JONSWAP 25remaining data would require more sophisticated analysis technic[ues than used here,includmg the numerical integration of the energy balance ec[uation for time and spacedependent v / i nd fields.Edited data tapes of the complete set of wave spectra are available on recpiest f r o m theNational Oceanographio Data Center, U.S.A., the I nst itut fr Geophysik, University ofHamburg, or the Deutsches Hydrographisches Institut, Hamburg. W i n d and other envhonmental data are available f r o m the latter two institutions.1.7. Logistics

Overall logistic coordination of the experiment was carried out by the Deutsches Hydro-graphisohe Institut ( H W ) * , Hamburg, w i t h assistance f r o m the Institut f i h Geophysik,Umversity of Hambirrg ( I ^ H ) . Site preparation and technical coordination of field operationswere handled by A. Hechich through the Deutsches Hydrographisches Inst itut . Scientificcommunication dmdng the experhnent was maintained f r o m the shore-based telemetry andradio station (1968: K H ; 1969: HW, KH) for which accomodation was k i n d l y provided byU . Jessel, Institut fih Bioklimatologie und Meeresheilkmade. W i t h h i this general logisticframework, each participatmg group canded individual responsibility for its contributionto the experhnent, as follows (see also T. P. B arnet t [1970]):

Deutsches Hydrographisches Institut. 1968: Wave posts at Stations 2 and 8 (Carstens).3 phch-roll buoys operated f r o m "Walter K r t e " (KR), "0T 2" (HC, H . Erentz), and"Gauss" (H . J. Rubach, P. Lmdner). Tide gauges at Stations 2 and 8. Cm-rent meters.1969: Wave posts at Stations 1, 2, 3, 5 and 8 (U . Carstens, Rossfeld, HG) ; 4 phch -rollbuoys operated from "Waher K r t e " (KR), "0T2" (HC, H. Erentz), "Gauss"(H.J . Rubach, P. Lindner), and "Holnis" (R. Pirhrmann); (one of the phch-rollbuoys w as loaned f r o m the National Institute of Oceanography). The telemetry anddata acquisition system for the wave posts and turbience instmmentation were bu tby U . Carstens. Wave posts for the D H I wave instraments were b u i l t and installed(by jetting into the sandy bottom) by Harms & Co. (S. Knabe). Tide gauges atStations 2 and 8.Institut fih- Geopliysik, Hambm-g. 1969: Tm-bience measurements, Station 8 (O. Kertel-hem, KE, D O ) . The turbulence instramentation was but by 0. KerteUiem.K o n m k l i j k Nederlands Jleteorologisch Instituut. 1968: Thi-ee waverider buoys (AM).1969: Waverider buoys at Stations 7, 9 and 10 (EB). The waverider system, complete-with telemetry and data acqirisition system, was b u i l t by Datawell NV.National Insthute of Oceanography. 1968: Two phch-roll buoys, operated f r o m HMS"Enterprise" (JE, D. Bishop) and HMS "BuUdog" (N. Smith, P. Cofiins). 1969:Pitch-roU buoy at Station 13, operated f r o m "Moray Eh-th" (DC, JE, P. Collins,C. Clayson). A second pitch-roll buoy was loaned to the Deutsches HydrographischesInstitut.Westhighouse Research Laboratories. 1968: Six-instrament array at Station 4, pressm-etransducers at Stations 1 and 3 (R. Bower, H . M a r t i n , T B ) . 1969: Sis-instrument arrayat Station 4 (R. Bower, H . M a r t i n , G. Bowman, TB). The wave instrumentation andtelemetry were b u i l t by H. Martm, Ocean Applied Research Corp., San Diego.During the experhnent the telemetering Stations 1-5, 7-10 were serviced by the tugs"Ehr" (1968) and "Eriechioh Voge" (1969), -with support f r o m ah-sea rescue launches basedi n the Bundesmarine Station in L i s t and occasionally the l i f e vessel "Hindenbm-g".The prelhninary editing and spectral analysis of the data was carried out by each groupindividually. A general system for assembling and processing the complete set of spectra

* L i i t i a l s refer to authors.


26/96

26 Erganzungsheft zur Deutschen Hj^ckographisohen Zeitschrift. Reihe A (8), Nr. 12, 1973was developed by WS. The analysis and interpretation of the data was then carried outchn-ing two workshops at the Woods Hole Oceanographio Institution f r o m February toA p r , 1971 (WS, K H , JE, DH, PM, TB , EB , HC) and May-August, 1971 (WS, KH , EB ,HC, DO, P M ) . Computations of the nonlinear energy transfer rates were made on the CDC 6600and 7600 computers at the National Center of Atmospheric Research in Boulder (WS, K H) .Woods Hole Ooeanographie Institution provided assistance dmdng the preparation of a firstdi-aft of this paper ( K H ) * .Acknowledgments

The work was supported by the Mhfisterimn fr Bdung und Wissenschaft, the Ver-teidigmigsmhiisterium, and the Verkehi'sministerimn (ERG); the Office of Naval Research,the National Science Poundation [ N C A R ] , and the Woods Hole Oceanographio Institution[Doherty Eund] ( U S A ) ; the Natmal Envh-onment Research Counc and the B r i t i s h M i n i s t r yof Defence (UK); and the N A T O Science Commhtee. Ships were made available throughthe Verteidigung.sministerium, Bundesmarine ("0T2", "Holms", " P h i - " , "Eriechich Voge",ah sea rescue boats); the U K Hydrographie Dept., M . O . D . (HMS "Enterprise" and "Bulldog");the U K Natural Environment Research Council (charter for "Moray Pirth") ; the Wasser- undSchiffahrtsdhektion Hambm-g ("Waher K r t e " ) ; the Deutsches Hydi-ographisohes Instfiut("Gauss"); and the Deutsche Gesehschaft zm- Rettung Schfbrchiger ("Hindenbm-g").

We are grateful to these agencies and the individuals mentioned in the previous section,and numerous others not listed e x p l i c i t l y whose generous assistance and support made thisproject jDossible.* Tl i i s paper also appears in the Collected Reprints of the Woods Hole Oceanogr;Institution as contribution no. 2911.


27/96

2. Wave Growth2 . 1 . Generation Cases

The primary pm-pose of the wave generation measm-ements was to determine the som-cehmotion S' in the spectral energy balance (radiative transfer) equation*dF dF , + Vi =S' (2.1.1)dt 3. ^fo r the two-dimensional wave spectrum F(f,9; x, t), where i i . = = ^ is the groupvelocity. "^^i ^^f ^'^

Much of the discussion mil, in fact, center on the contracted f o r m of this equationobtamed by integrating over the propagation dhection 6,dE dE + ^ =S (2.1.2)at dxiwhere E = E{f; x, t) and S are the one-dimensional frec[uenoy spectram and soiu-ce function,respectively, and = ^VjF AdfE.I n either eciuation the som-ce function can be determmed f r o m wave data taken alongthe profile only i f the gradient of the wave spectrum transverse to the profile is small compared-with the derivative parallel to the profile. Tor an ideal geometry consisting of a profileperpendicular to an i n f i n i t e straight shore, this w i l l be the case i f the field is homogeneousw i t h respect to the parallel-to-shore coordmate x^ = y. The dhection of the -wind and the-ivhid-field variations -with respect to the coordinate x-^ = x parallel to the profile or time tmay be arbi trary. However, in practice, the length of straight shoreline perpendicular tothe profile was finite (cf. Figs. 1.1 and 1.2) and we f e l t reasonably confident that transversegradients could be neglected in Eqs. (2.1.1) or (2.1.2) only for offshore muds blowing w i t h i nabout + 30" of the profile dhection. Although no restrictions were necessary w i t h respectt o the fetch or time variabfiity of the w i n d field, the conditions on the w i n d dhections werethen normally satisfied only for w i n d fields which were also fahty homogeneous w i t h respectto X and t.

S. A. K i ta i go rod sk i i [1962] has suggested that under these conditions the wave datafo r different wdnd speeds and fetches should be expressible in terms of a shigie nondimensionalfetch parameter x = gxju^, where u.^, = (XIQ^Y'- is the f r i c t i o n velocity and T is the momentumtransfer across the ah-sea interface {Q^ = density of ah). We found Ki ta i go ro d sk i i ' srelation to be fahly w e l l satisfied, and the som-ce functions computed for indi-^ddual casesagreed reasonably w e l l w i t h the mean som-ce f m i c t i o n inferred f r o m the '-dependence ofthe wave spectra after averaging over all "ideal" cases of stationary, homogeneous w i n dfields. Many of our general conclusions have accordingly been derived fi-om the mean spectraldistributions after scaling w i t h respect to nondimensional fetch (Sections 2.3-2.6).However, a number of indi-vidual generation cases were also analyzed (Section 2.7) toindicate the variability of the data and the representativeness of the mean results, incluclingpossible limitations of Kitaigorodski i ' s sealing law. I t appeared th at most of the observedvariability of our data was associated -with the small scale gustiness of the w i n d field ratherthan systematic deviations f r o m Kitaigorodski i ' s law. Nevertheless, there is some indicationthat further parameters not included in Ki tai gorod sk i i ' s analysis, such as boimdary-layer scales, must be taken into account in comparing laboratory and field data.

* We ignore refractive effects in t l i is section, but w i l l consider the f u l l equation in Part 3.


28/96

28 Erganzmigsheft ziu- Deutschen Hydrographischen Zeitsclu'ift. Eeilie A (8"), Nr. 12, 19732 . 2 . Kitaigorodskii's Similarity Law

Ki ta igorodski i has pointed out that i f the geometry of the w i n d field and the lawsof wave generation are sufiieiently simple that the wave spectrum can be imicpiely specifiedby a single fetch parameter x, the local f r i c t i o n velocity u.. and the gravitational acceleration g,then by dimensional arguments the spectrum must be of the general f o r m

F(f,0) = g^r'F(f,0;x) (2.2.1)where the parameters = fu^jg, x = (7. /w| and the function are non-dimensional cpiantities.Clearly the general three-dimensional space-time dependence of the wave field w i l lreduce to a single fetch parameter x i f the w i n d field and boundarj^ conditions are stationaryand homogeneous v d t l i respect to the dhection y perj)endicular to x, e.g. for a constantoffshore -ndnd blowing perpendicular to an i i r f i n i t e straight shore.However, i t is less obvious that the spectrum shoidcl depend in addition only on thelocal fi'iction velocity (besides g). A l t h o t i g h i t appears reasonable to assume that thef r i c t i o n velocity can adec(uately characterise the local interaction between the atmosphereand the wave field - a.t least to first order - this is relevant only for the local rate of changeof the spectrum. The siiectrum i t s e l f must be determined by integrating the radiative transferecjuation and thus represents a net response to the enthe upwind mnd field. To avoid thisd i f f i c i f i t j r , one could assume that M,,, is constant, i.e. independent of fetch. Although thisfinds some observational support, i t is d i f f i c u l t to find a convuicmg physical reason whythis should be the case a p r i o r i , uifiess i t is assumed that the wave field plays a negligiblerole i n the momentum transfer across the ah-sea interface - which is not supported byexperiment. I t appears more consistent to regard instead as an internal variable of theproblem which has to be determined as a function of fetch in the same way as the spectrum.Adopting this viewpoint, Ki tai gorod sk i i ' s spectrum (2.2.1) can nevertheless bederived hj the foUoAvhig, slightly revised argiiment. As relevant external parameter wespeoifj' the constant "wind speed at some height above the sea surface which is no longersignificantly affected by the wave field - ideally, outside the planetary boundary layer. Thewave spectrum and u.^, are then governed bj ' ' the couiiled radiative transfer and atmosphericboundary-layer ecjuations under appropriate upmnd i n i t i a l conditions. Assummg that thewave spectrum is zero at the upwmd shore line x = 0 and that at a sufficient distance f r o mthe coast the i n i t i a l properties of the boundary laj^er at x = 0 are no longer important, i tfollows that the spectrum and f r i c t i o n velocity can be fimctions of x, U and g only. Henceby the same dimensional argument as employed by Ki ta igorodski i , the spectrum mustbe of the general f o r m (2.2.1) w i t h / and x replaced by /' = fUJg and x' = gxjU^. Shnilaiiy,i'JUo} must be a universal fitnetion i.i{x') of the nondimensional fetch x'. However, bothscaling laws are equivalent, as is seen by mdting J'(/, 0, x) =F[f'n, 0, x'j.r'^) = f (/', 0, x'), say.Clearly, the same argument would have applied i f had been replaced by some othercharacteristic bouudary-laj^er velocity, e.g. the velocity at 10 m height above the smface.Since U-^^ is easier to measirre than and was routinelj' determmed at most stations alongthe profile, we have sealed al l om- data in the f o l l o w i n g w i t h respect to U^^g rather than tt,.,,denotmg the appropriate variables by .r = g.r/Z7JQ, f = jU-^Jg, etc. Por comparison w i t h dataf r o m other authors, however, we have also inchxded axes for the corresponding variablesscaled w i t h respect to u... using the relation M | = c g [7jg Avith a constant drag coefiieient= 1.0 X 10-3.2.3. Wind Sti'ess and Tm-bidcuce Measm-ements

Dm-ing the main 1969 experiment, chag coefiicients were determmed at Station 8 f r o mturbulent momentixm fiux measurements using both hot-whe and otxp-and-vane anemometers(cf. 1. 3. A more detailed description of the apparatus is given in K. Enk e [1973]). Eor neutralor stable conditions the average value of C K , was found to lie fahly close to the value 1.0 X 10^^


29/96

Hasselmann et al., J O N S W A P 2 9

10"

6.H

2.H

10

10'_ l

K R A U S(1968)

to

10- 10' 10 10'

O UPPER ANEMOMETERl STABLE LOWER ANEMOMETER! STRATIFICATIOND UPPER ANEMOMETE Rl UNSTABLE LOWER ANEMOMETERl STRATIFIC ATION

LI U (1971)W E I L E R e BURLING(1967)

LARG E FET CH( T A B L E D

10 10 10Fig. 2.1. Drag coetficients c, versus non-dimensional fetchx= gxl U'. Large-fetch measurements are listed in Table 1

Fig. 2.2. Reynolds stress sprectrum [k = 2ii U L/,, = anemometer height, U U,)


30/96

Erganzungsheft zur Deutschen Hydrographischen Zeitschrift. Reihe A (8), Nr. 12,1973


31/96

Hasselmann et al., JONSWAP 31applied i n the previous section. Under these conditions no systematic dependence of thedrag coefficient on x was found, cf. M g . 2.1. Slightly larger di-ag coefficients were found forunstable ah-sea temperature differences, "with indication of a decreasmg C J Q trend vdthincreasing x. Since most of our wave data were obtained under stable conditions, we haveignored the temperature dependence in convertmg f r o m l /^ p to u.^. This procedme is consistent"with Ki ta i go ro d sk i i 's scaling law, which also disregards temperatm'e elfeots.Figure 2.1 and Table 1 indicate considerable v a r i a b i t j ' in the drag coefiicients measuredby difi'erent groups. This causes difficulties in applying Kitaigorodski i ' s scaling law,which can be expected to be v a l i d only i f C ^Q is a miic[ue function of the non-dimensionalfetch X. I n the literature, wave parameters are tisually presented in nondimensional f o r mw i t h respect to u,^. To compare wave data f r o m other sources w i t h orrr data in Figs. 2.6, 2.7(Section 2.4) we have simply plotted all parameters w i t h respect to our axes - although thechag coefficient C j g = 10"^ used to convert f r o m U to may differ f r o m that used in theoriginal source. Slightly different plots woid have resulted i f al l data had been nondimensionalized vdth respect to Z / J Q . (In practice, however, variations of C J Q by a factor of2 or 3 are not very noticeable in the log-log plots of Figs. 2.6, 2.7.)

Table 1Large Fetch Values o c

Author Cio 10 Stratification MethodSmith [1967] 0.9 0.24

m/s3-13 correlation

Smith [1970] 1.35 0.341.41 0.261.20 0.48 6-15 stable correlationHasse [1968] 1.21 0.24 3-11 correlationBrooks et al. [1970] 1.300.18 3-13 neutral profileDeLeonibus [1971] 0.8 0.41.2 0.41.3 0.6 3-15

stableneutralunstable correlationPond et al. [1971] 1.520.26 4-7 correlationJONSWAP 1.0 0.41.2 0.4 3-11 stableunstable correlation

The mean spectrum of the shear stress and its r.m.s. variation over the set of measm-ements are presented in Fig. 2.2. Figures 2.3 and 2.4 show the corresponding spectra for thehorizontal and vertical velocities. The distributions agree Cjuite w e l l mt h measurements byother workers.Cross spectra between atmospheric tm'bulence fiuetuations and the sm'face displacements,which were measmed at the same position, were also analyzed, but no significant correlationswere detected. The ratio anemometer height to the wavelength of the principal wave componentwas typically of the order 0.2. J. E l l i o t t [1972] also found no correlation for comparableratios*. I t appears that turbulence measurements considerably closer to the sm'face areneeded to study wave-induced velocity fiuetuations.

* However, both E l l i o t t and we observed correlations for swell com ionents whose phasevelocities exceeded the w i nd speed - as have a niunber of other authors. The phases were inaccordance w i t h potential theory.


32/96

32 Erganzungsheft zm- Deutschen Hydrogi-aphischen Zeitschi-ift. Eeihe A (8), Nr. 12, 19732.4. Petcli Dependence of One-Dimensional Spectra

The fetch dependence of the one-drmensional frequency spectra were investigated byparametrizing them w i t h a least-square fitted analytic function. Various functional formswere tried. A uniform good fit to nearly al l of the spectra observed during "ideal" generationconditions was finally attained by the function(/ - i JE (/) = ag' {2n)-* exp ( - ; { ^ Y ) y ' " ^ 2cr /; (2.4.1)

(Ta for

contahnng five free parameters a, and a^,.Here represents the frequency at the maximum of the spectrum and the parameter acorresponds to the usual Ph i l l i ps constant (the spectrum approaches the usual f-'^ powerlaw fo r large / /,). The remaining three parameters define the shape of the spectrum: y is theratio of the maximal spectral energy to the maximum of the corresponding Pierson-Moskowitz [1964] spectrmn*5 / (24.2)

-with the same values of a and f^; and define the l e f t and right sided widths,respectively, of the spectral peak. The functional f o r m (2.4.1) was obtained by midtiplyinga Pierson-Moskowitz spectrum w i t h the "peak enhancement" factor

The pm-pose of the parametrization was not to contribtite a fm-ther emphical spectrum,but rather to reduce the 100 frequeney points of the measured spectra to a manageable, non-redmidant subset of parameters whose dependence on fetch, windspeed and other factorscould be systematically investigated. (The parametrization also had advantages in d i f ferentiating the spectra to determine the source functions, cf. Sections 2.6, 2.7). Pormulaefor fetch-limited spectra proposed by other authors, while perhaps adequate to describecertain properties of mean spectra, were f o m i d to contain too few parameters to characterizethe variations encountered in the individual measm-ed spectra.Pig. 2.5 shows a typical series of spectra measm-ed under "ideal" generation conditions,together -ndth theh- parametrical approximations. Plots of the five spectral parameters forthe 121 spectra measured under "ideal" generation conditions agahist nondimensional fetch

x = gx\'U\g are shown in Pigures 2.6, 2.7 and 2.8. Pigures 2.6 and 2.7 also include scaleparameters / , and a f r o m other experiments.The distribution exhibits the smahest scatter. I n part this may be due to the factthat , is the most accm-ately defined spectral parameter. However, the smaller variabilityof the distribution as compared mt h the a data or the s t i l l more scattered shape parameterscan also be understood f r o m the o r i g i n of the scatter, which we attribute in Section 2.9 tothe gustiness of the w i n d .

* The exponent of thePierson-Mosliowitz spectrum is normally -written as 0.74 (//)"''w i t h reference to the frequency = gl-KTJ.^^ for winch the phase speed of the waves is equal tothe -wind speed. The two expressions are identical, but i n our case - or equivalently - is treatedas an adjustable parameter.


33/96

Hasselmann et al., J O N S W A P 33

0 . 7 - ,

HzFig. 2.5. Evolution of wave spectrum with fetch for offshore winds (11 -12^, Sept. 15,1968). Numbers refer to stations (of. Rg. 1.2.).Th e best-fit analytical spectra (2.4.1) are also shown. The Inset illustrates the definition of the five free parameters in form (2.4.1)


34/96

34 Erganzungsheft zur Deutschen Hydrographischen Zeitschrift . Reihe A (8), Nr. 12,1973

5H

2 -

10l _ 10" 10= 10''I

x=xg/U,o

, 2

10'I

1 Burling(1959)- Pierson(1960)" KitaigorodskiitStrekolov(1962 PiersoncMoskowitz(l9e4) Hidye Plate (1966)" Mitsuyasu (1968) Volkov(1968) Kononkova(1970) Li u (1970)Pl^Pierson Moskowitz (1964)+ This Experiment PM

05

02

01

Fig. 2.6. Peak frequency v. fetch scaled according to Kitaigorodskii. Data at small fetches are obtained from wind-wave tanks.(Capillary-wave data was excluded where possible.)

10' 10'

2 h.1 ha

05

02 h

.005 I -

ABurling(1959)i. Pierson{1960)V Hicks (1960)0 Kinsman (1960) Longuet Higginsct a((l963)X Sutherland (1967)o(|itsuyasu(1968) L iu (1970)o Toba (1971)PM Pierson Moskowitz (196A)^S cl i ul ee t al (1971) Pierson & Stacy (1971)+ Tt is Experiment1tf' 1 10 10" 10 10" 10=

Fig, 2.7. Phillips constant v. fetch scaled according to Kitaigorodskii. Small-fetch data are obtained from wind-wave tanks.(Capillary-wave data was excluded where possible.) Measurements by Sutherland (1967) and Toba (1971) weretaken from W.J. Pierson and R A Stacy [19731


35/96

P E A K S H A PE P A R A M E T E R S10 '

I

/ !X 10=

10^ 1 0 -X

100 .3

0 .2

0 1

0 .0

0 .2 J

0 .1 H

0 .0IQ

10 =

+ +

10'

+ + +++ +


36/96


37/96

Hasselmann et al., JONSWAP .37Oiu- a parameters in Fig. 2.7 are seen to lie a l i t t l e higher than the data h-om some ofthe other groups. This may be due partly to the function-fitting technic^ues used to evaluate a.H . Mi tsuyasu [1969] for example, mentions three methods, of which the one used for the avalues shown in Fig. 2.7 yielded systematically smaller estimates by 10 to 20% than theother two. Om' technic[ue of simultaneously optimizing five spectral parameters tended toy i e l d a values about 20 % higher than, say, a straight-line fit to the function E (/) f for allfrec[uencies / greater than f^, which corresponds more closely to the standard method.A significant featiu'e of F i g . 2.7 is that a is clearly n o t a universal constant, as requhedby Phillips' original dimensional argument. Om' analj^sis of the source functions presentedlater in Sections 2.6-2.9 indicates th at the energy balance in the region of the sj)ectrumbetween and about 3 , is indeed not governed by white-capping, as envisaged byPhi l l ips , but primarily by the energy input f r o m the atmosphere and the nonlinear energytransfer by wave-wave interactions towards lower and higher frequencies*.The straight line

/ , = 3.5 x-"-'' (2.4.3)i n Pig, 2.6 represents the best fit to the data ( H . Mitsuyasu [1969] finds the same exponent);the line

a = 0.076 x--^^ (2.4.4)i n Fig. 2.7 is based on dynamical scaling considerations (Section 2.8). Although it jdelds areasonable overall fit for all field and wave-tank data, an exjionent of 0.4 is a betterapproximation to the average slope of om' a distribution.The peak shape parameters y, and (Fig. 2.8) are more highly scattered than thescale parameters. Since they characterize the properties of a rather narrow spectral peak,some variability is to be expected f r o m the statistical indeterminacy associated "with thespectral measm'ement. However, this accounts for an r.m.s. relative variability of only about20%, which is considerably smaller than observed. Attempts were made to reduce thevariability by subdividing the data w i t h respect to additional parameters such as the ah-seatemperatm'e difference, currents, and the dhection of the w i n d relative to the profile. Noreduction of the scatter was achieved. However, local non-stationaries (defined by the timederivative estimated f r o m three neighboring spectra) above a certain thr'cshold were foundto account for some of the variabfiity, and Figs. 2.6, 2.7 and 2.8 accordingly contain onlygeneration cases for which the non-stationaritj' was below a critical value, determinedemph'ieally by the inecpiality ^ < where represents the group velocity of the wavesat the spectral peak. ot dxI t ap23ears that the residual scatter of om' data in Figs. 2.6, 2.7 and 2.8 is geophysicallyreal and cannot be accounted for in a simple manner by additional parameters characterizingthe mean generation conditions. The analysis of indi-vidual generation cases discussed i nSection 2.7 shows that a variabfiity of the same order occurs vdthin each generation ease,suggesting that it may be associated w i t h small scale inhomogeneities of the TOnd field(Section 2.9).A clear trend of the shape parameters w i t h fetch is not discernible w i t h i n the scatterof the data, although there is some indication of y peaking to a maximum around x = 10 .Accordingly, we shall assume as first approximation that the spectral shape is self-preservingand consider in much of the foUowing a mean spectrum defined by the parameters**

y = 3.3, cr = 0.07, = 0.09. (2.4.5)* I t could he sm'mised that the fetch dependence of om- a data is induced by the fetchdependence of o\a averaging band, defined w i t h respect to e.g. due to an exponent in thespectral power law which diifers slightly f r om 5. This is not the case. Essentially the same resultswere obtained using fixed freciuency bands.** A more detailed dynamical analysis suggests that this is not strictly tenable, cf. Seotion2.9.


38/96

38 Erganzungsheft zur Deutschen Hych'ographisehen Zeitschrift. Reihe A (8"), Nr. 12, 1973F i g . 2.9 shows a comparison of the normalized shape functions

(p(p\={2nta-'g-'f'E(f) (2.4.6)f o r this spectrum, the Pierson-Mosliowitz spectrum and a schematic empirical spectrumproposed by S. A. Kitaigorodskii [1962]. The f o r m corresponding to (2.4.5) is cjualitativelyvery similar to the shape functions observed by T. P. B ar ne t t and A. J . Su th er land[1968] and H. Mitsuyasu [1968a, 1969].To conclude om' summary of emphical fetch relationships. Fig. 2.10 sliows a j j l o t of thenondimensional energy i = S X c/ 17^^,^, where S = 1 E (/) d/. The data is w e l l representedby the linear relationship

# = 1 . 6 x l O - ' - T . (2.4.7)For a seh-simikr speetmm, S ~ so that the exponents Hj-, and w ^ i n the power lawsf o r a and S should satisfj^ the relation

4 ? i j . - H ^ + ? j ^ =0. (2.4.8)The values given in equations (2.4.3), (2.4.4) and (2.4.7) y i e l d

4 ( - 0 . 3 3 ) - ( - 0 . 2 2 ) + ! = - 0 . 1 (2.4.9)which lies w i t h i n the error of about + 0.2 estimated f r o m the error bands of the individualexponents.2.5. The Dhectional Disti'ibution

The majority of investigated generation cases ocom-red dm-ing the latter part of July1969, when the dhectional data of all but one of the p i t c h - r o l l buoys was questionable, ordming the period August 1-August 15, 1969, when only the innermost section of the profile(without p i t c h - r o l l buoj^s) was in operation. Unfortunately, the directional data f r o m thesingle p i t c h - r o l l buoy and the linear array at Station 4 did not encompass a sufficiently widerange of dimensionless fetch values x to permit a systematic investigation of the fetch dependence of dhectional parameters. Thus we have assumed - in analogy w i t h the similar resultf o m i d for the hequency spectra - that the scaled dh-ectional distributions s{6, fifj,,) areindependent of fetch.

Averaging over all ideal generation cases, the linear array yields a mean beam -widthf o r frec[uencies near the spectral peak of 0 ^ 19 + 5". This corresponds to an ec^uivalentp i t c h - r o l l beam w i d t h of 6^ x 27" + 8" (/ x 0.4 Hz) which is reasonably close to the value31" for a cos 9 distribution. At frequencies appreciably higher than the peak frec[uency,the linear an-ay was normally too strongly aliased to be interpreted reliably. The p i t c h - r o l lbuoy at Station 10 also jdelded beam widths 9^ x 30" for f = f^.Examples of the dependence of beam w i d t h on frecptencies greater than is shown in

Pig. 2.11. Data f r o m both the fimctioning and questionable p i t c h - r o l l instruments are includedi n the fig-ure. The malfunctioning instruments were characterized by random variations inthe mean dhection f r o m one recording to another. Rather sm'prisingly, however, the meansc uare angular spread for all instruments were found to be reasonably consistent. A possibleexplanation is that the compass was sticking and the orientation of the buoj^ remainedrelative^ constant dm'ing any given recording. This interpretation finds some support throughthe fact that for the same wave conditions the frequency dependence of the mean angle 0 ,was generally similar for all p i t ch - r o l l instruments except for random shifts of the zeroreference angle. We infer f r o m Fig. 2.11 that the dhectional distribution maj' be modeled bycos 9 ( 0 s K, 30") near the peak of the spectrum - in accordance w i t h the linear arraj' data -changhig gradually to an approximately uniform half-plane radiation ( 0 ^ 4 3 " ) at highfrequencies, > 2 to 3 .


39/96

Hasselmann et al., JONSWAP 30

60a's

30i S T A . 6 J U L Y 26 22 '2 S T A . 6 J U L Y 27 00'3 S T A , 7 J U L Y 27 00'

30,30,30

Fig. 2.11. Typical frec[uency dependence of t ieroot-mean-sc[uai'e directional sju'ead 0 s measiu'edfo r fetch l i m i t e d spectraSince most of the wave energy is concentrated near the peak we have assumed forshnp l i c i t j f a frequency independent eos 0 spreading function in computing net som-cefunctions, nonlinear transfer rates, etc. I t w as found that the prhieipal conclusions regardingthe energy balance were in fact rather insensitive to the precise f o r m assumed fo r the sjireadingf i m c t i o n .

2.6. The Mean Source FunctionThe emphical average dependences (2.4.3), (2.4.4) of the spectral parameters and on fetch x = gxlUlg, together -with the (constant) average shape parameters (2.4.5), define amean one-dimensional spectrum at each x, f r o m which a mean one-dimensional som-cef m i c t i o n can be computed through ecpiation (2.1.2),

Fig. 2.12 shows the mean JONSWAP spectrum and the associated mean som-ce function S(a cos 0 spreading factor was assumed in evaluating v). The plus-minus distribution of thesom-ce function is associated w i t h the transition of the spectral peak towards lowerfrec[uencies, the positive lobe corresponding to the rapidly growing waves on the forwardface of the spectrum and the negative lobe to the attenuation of the waves on the rear slopeof the peak as they approach equilibrium (the overshoot effect - cf. also Fig . 2.5).Several authors have pointed out that the f o r m of the som-ce function, in particular theovershoot effect, is d i f f i c u l t to explain in terms of a linear (or any frequency decoupled)theory, and have suggested that cross-spectral energy transfer due to resonant wave-waveinteractions, white-capping or some other f o r m of nonlinear couplhig may be significant(T . P. Barnett and A.J. Sutherland [1968], B . V , Korvin-Kroukovsky [1967], H .Mitsuyasu [1969], A. J. Suther land [1968]). Our computations of the nonlinear energytransfer due to resonant wave-wave interactions, also shown in Pig. 2.12, indicate that thisprocess can indeed account for the prmeipal features of the observed som-ce function,including the plus-minus signature, the major part of the wave growth on the forward faceof the peak, and the overshoot phenomenon. We attribute the difference between the observed som-ce function S and the nonlinear energy transfer *S, to the positive energy inputf r o m the atmosphere, which is discussed fm-ther in the context of the overall energybalance of the spectrum in Section 2.9.

(2.6.1)


40/96

40 Erganzimgsheft zur Deutschen Hydrographischen Zeitschrift. Reihe A (8), Nr. 12, 19730.4 !-

0.0 0.1 0,2 0.3 0.4 0,5 0,6 0.7 0,8 0,9 1.0

Eig. 2.12. Mean JONSWAP spectrum, source function S computed from power laws (2.4.3), (2.4.4)using (2.7.1), and computed nonlinear energy transfer Snj

The spectra and source functions shown in Fig. 2.12 correspond to particular values ofthe fetch x and windspeed U^^. I n general the ratio of the soiu'ce functions S and may beexpected to vary w i t h the nondimensional fetch x. However, under certain scaling conditions -which we return to in Section 2.9 - the ratio is approximately independent of x. From theBoltzmann-integral expression for the nonlinear energy transfer (K. Hasse lmann [1962,1963 a]) - namely

StX 8 {coi + (O2 - (O3 - 0 ) 4 ) dfej dfeg

where = hj^ + k^, Nj = N{kj) = {kj)lcOj, C0j=]/gkj, F{k) is the two dimensionalwave spectrum w i t h respect to wavenumber, and the transfer coefficient T is a ciuadratichomogeneous funct ion of the four wavenumbers fej, . . . k^ - it follows that for a family ofself-similar spectra of the form E'{f, 9) = ag^iZiz)-^ f-^ ^' dUm' ^) nonlinear transferscales w i t h respect to a and according toS, = Ulg-'a'fZ'ilyUIL) (2.6.2)

where !/'(//) is a dimens

Documents

Hasselmann Etal DHZ1973