The Winds, Currents and Waves at the Site of the Floating City Off Waikki

T V 44

THEWINDS, CURRENTS AND WAVES AT THESITE QF THE FLOATING CITY OFF WAIKIKI

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CIRCUI.ATING COPY Sea Grant Depository

HAWAII'S FI.OATING CITY DEVELOPMENT PNX;HAM

Technical Report No. 1

UNIHI-SEAG HANT-C R-75-01

THE WINDS, CURRENTS AND WAVES AT THE SITE OF THE FLOATING CITY OFF WAIKKI

Manley St. Dcnis, D. Eng. H AI I 0 HA LSEA Ci R "i H TP t'P "', ~ p~ p y PELt L!9 -','iity ! I! December 1974 0

for

National Sea Grant Program National Oceanic and Atmospheric Adxninist ration U.S. Department of Commerce Hockville, Maryland 20852

Thispublication was funded by Grant No. 04-8-158-17 -8- 158- 17 of NOAANO Officeof SeaGrant, Department ofCommerce. The U. S. Cmvcrnm~nt purposesnotwithstanding anycopyright notatton thatmay app~ar hereon. PROGRAM MANAGER'S NOTE

Dr, St. Denisperformed the w'ork reported uponin this document as an adjunctto our FloatingCity program. However,hc actedsolely as a member of the facultyof the Ocean EngineeringDepartment of the University of Hawaii, and his efforts therefore must be consideredas matching the Sea Grant work rather than being directly supported by Sea Grant funds. Sea Grant did, however, support the preparation and printing of this report by Oceanic Institute as weII as some of the technical assistance Dr. St. Dcnis received.

As wc have all come to expect of Manley St. Denis, the work described here is a unique and systematic application of a sophisticated technique which perhaps yields new insights into Iong-term environmental prediction. At any rate, as Dr. St. Denishimself says, "se non8 vero, e ben trovato. "

Joe A. Hanson The Oceanic Foundation TABI.E OF CONTENTS

l. INT RODUC T ION

2. WINDS

2. 1 WIND TYPES 2. 2 TRADE AND KONA WINDS 2. 3 KONA STORMS AND TROPICAL CYCLONES

3. CU RRENTS

3. I OVERVIEW ll 3. 2 THE GENERAL CIRCU LATION 12 3. 3 TIDAL CURRENTS 16 3. 4 EKMAN OR WIND DRIFT 17 3. 5 LARGE SCALE OCEANIC FLOW 18 3. 6 VKRIFICATION OF THE RESULTANT CURRENT 18 3. 7 LONG-TERM PREDICTION OF THK GENERAL SURFACE CIRCU LATION 18

4. WAVES 21

4. 1 INTRODUCTION 21 4. 2 PRIMARY WIND WAVES AND SWELL 23 4. 3 CYCLONIC WAVES 31 4.4 ALL WAVES 31

APPENDIX A DATA ON WINDS IN THE HAWAIIAN AREA 33 APPENDIX B THE PREDICTION OF EXTREME VALUES BASED UPON GUMBE L'S THIRD ASYMPTOTIC DISTRIBUTION APPENDIX C PREDICTION OF THE EXTREME VALUES OF WIND INTENSITIES IN THE HAWAIIAN AREA APPENDIX D ON THE SEVERITY OF TROPICAL CYCLONES IN THE HAWAIIAN AREA 51 APPENDIX K THE EKMAN DRIFT 67 APPENDIX F VERIFICATION OF THE RESULTANT CURRENT 69 APPENDIX G THE ENERGY SPECTRUM OF THK SEAWAY 71 APPENDIX H DATA ON HAWAIIAN WIND WAVES AND SWELL 73 APPENDIX I CYC LONE-GENERATED WAVES 89 APPENDIX J ON COMBINING LONG-TERN TRENDS 93

PRE FACE

5e ~~a e i.'era, e ben trouato

The attexnpt is made in this report to present long-tenn forecasts of winds, currents and waves at a station some three miles off Waikiki selected as a possiblesite for a floating platforxnto housea communityof oceanic denizens. The for ecasts are to be used to design the platform.

The reader wiQ, undoubtedly, find disappointxnent to be his reward. The observational data upon which to base the forecasts are always far from abundant; usually their quantity falls somewhere between the scanty and the scarce; occasionallythe data are altogether rare. But the reader can take comfort in the thoughtthat his disappointxnentis exceededby the author's f rustration.

The insufficiency of data has led the author to improvise and to use prediction techniquesthat are far from rigorous and at best only plausible. But then, this state of affairs ia usual with the engineer, for if he were to wait for all the inputs to be in before starting s.project, only a precious few would ever get underway.

Italian saying: If it is not true, it is well thoughtup. I. INT RODUC T ION

"... that bide the peKting of thT'.e pi.tigress eton'."

Shakespeare"

Determtnation of the environmentis the first step in the process of obtainingthe responseof the floating communityto the natural elementsto which it is subject. The three most important environmentalaspects to be predicted are winds, currents, and waves. For the design of the platform it is necessary to have in handdata on both short-tenn snd long-term mean, peak and extreme values of wind intensity and duration, current strength and wave heights and periods.

One may well believe that there is in hand a body of data on the Hawaiian environment sufficient to reliably predict those characteristics which are of concernin specific projects; but the situation is quite otherwise, as will become apparent upon further reading. Particularly serious is the insufficiency of observations in support af the long-term forecasts which are basic to the design of important marine structures. One is forced in such cases to design for conditions which are little morc than the outcome of informed guesswork. Of course, the designs may still prove to be quite satisfactory: the factors of safety, hopefully selected with judgment, will insure that. But then, no city has yet been designedto float permanently on the surface of the oceanand withstand with poise the combined efforts of the elements.

2. WINDS

"... the mirrors that ~~it.f i.-e hauling at al L hears,..."

W' f. 1 i an K-,r dsu<. r .. h '

2. 1 WIND TYPES

The winds blowing over the Hawaiian Archipelago are classable as trade winds, Kona winds, tropical storms, and tropical cyclones. Inasmuch as this classification is not accepted by all interested members of the meteorological coxnmunity, it may be weil to define and justify the terms. The problem is not with the trade and Kona winds, which blow steadily with gentle force unless temporarily overwhelmed by their stronger counterparts, but with the latter.

An initial distinction is made between winds, on the one hand, and tropical storms or cyclones, on the other. Winds are atmosphericmotions characterized by the apparent absenceof a core while storms and tropical cyclonesare characterized by its apparentpresence. The further distinction between trade and Kona winds is one of direction: between ttepicai storms and cyclones, it fs one oi core temperature, this being cold in the tropical storm and hot in thc tropical cyclone. Suchcore temperature differences manifest themselvesin differences of wind intensities: the hot, rapidly rising air core in a tropical cyclone xnakcs for winds of an intensity comparatively greater than occurs with a tropical storm. To be sure, a cold core tropical storm occasionally transforxns itself into a warm core tropical cyclone sometimes referred to, with etyxnological inaccuracy, as a hurricane" !. Miscellaneous Sonnets, p. II, xxxiii The terxn hurricane is derived from the Taino and Mayan hurakan; the terzn typhooncoxnes to us frozn the Cantonesetai fang which has the meaningof great wind: theyboth describe the saxnegeneric pheonmenon, and the distinction between them is made on a purely geographical basis. The former is u.sed when the area is that of the Gulf of Mexico, the Caribbean Sca and the Western Atlantic; the latter is usedwhen the area is that of the China Sea and adjacentwaters. The geographicalfact thatthe HawaiianIslands are almostas distant froxn one of these areas as from the other poses a dilemma as to which terxn to use. Of course, one could suggestthe Hawaiian term make'~ii rru'i or maka'ni pukiki, meaningrespectively grea.t or strongwind, as fitting andproper. It is strange that sucha suggestiondoes not appear to have been znadc. But one fs faced with a serious problem: that of separating the observatfons of winds froxn those of tropical storxns and cyclones so as to permitthe separatestatistical treatment of eachgroup. The supporting informationby whichto accoxnplishthis has been obtainedonly for wind conditionsoccurring since 1939. The axnountof datathat canbe grouped is hardly abundant.

A completedescription of the windBeld at a stationis givenby the tfmehfstory of thewind velocity vector. To designstructures exposed to windaction, the further knowledge is requf red as to howthe velocity vector varies spatially over the extent of the structure. 3ut such a descriptionhasthe intolerable inconvenience oi'numerical imxnensity. Also,the prediction ofwind fields in suchdetail, particularly any long- termprediction, is of impractical realization. One escapes the suffocation ofnumbers byfnvoking statistical averages. If these are meaningfully selected,the overwhelxning ponderosity ofthe complete description reducestoefficient statements interxns of sixnple manageable parameters. Andalthough theprediction problem is still a difficultone, it is attemptable, andin certaincases solvable fn a probahilisticsense. Theaverages ofconcern are both temporal and spatial and relate to thexnagxdtude andvariability ofthe wind strength andto the angle and variabilityofthe wind direction over both short and long runs. Unfortunately, thewind field fs not routinely measured ina xnanner designed toyield such averagesdirectly snd the reported data are at best sufficient only to provide a pointof departurefor speculations.

2.2 TRADEAND KONA WINDS

2.2.1 Short-TermMean Wind Intensi Thisinformation fsrequfred topredict the strength ofwind-induced currentsand the characteristics ofwind-waves. Thepertinent dataavailable arepresented inTable A.I ofAppendix A. theyare statements off requency ofwind direction andspeed measured at HonoluluInternational airport,where islocated theweather-recording stationclosest tothe contemplated sitefor the floatin coxnxzaxnity. tablespresent a 29-year sumxnary reducedfrom almost a quarter ofa millionobservations. Similarsummaries areavailable foreach xnonth. The wind intensities given are onc-minute averages. A pertinent piece of information that does not appear to have been recorded is the persistence or steadiness of the wind, Without an idea of steadiness, it is fairly difficult to predict with an adequate measure of confidence the strength of the wind-generated sea. Another omitted itezn is the short-term variabiUty in wind direction.

It is a justifiable question whether the wind strength measured at thc airport can be assumed to represent validly that obtaining at the prospective site. Since relevant measurements are lacking, the question cannot be answered objectively, But the conditions that the airport is just off the sea and that the site planned for the floating city is but ten miles distant from the airport tower should support the belief that any difference in wind data should not be of serious import.

2. 2. 2 Short- Term

The behavior of craft approaching or entering the floating city is governed in part by the peak intensities of the wind to which they are subject, the influenceof peak wind intensities being the more serious the lighter the craft. It is obviousthat sail craft suffer most in this regard. The designof mooring lines and the capacity of dynamicalznooring systezns are also based on the peak wind intensity.

Peak wind intensities are the greatest one-minute average values observed over an interval of reference. Table A. 2 presents peak wind intensities observedover a month monthly maxima!. The derived yearly znaxizna are also listed.

2.2. 3 Short- Term Gust Intensi

These are important for the same reasonas the short-term peak wind intensities and, in addition, for the designof wind-exposedstructure. By gustintensity is meantthe maximuminstantaneous values of wind intensity. The available data are the maximum values of gust intensities observedlisted by month,direction and year of occurrence. Theseare presentedin Table A. 3. There is an obviouslack of parallelisznbetween this and the first two tables. The observations do not form a continuous sequ.ence,and this tendsto blur soznewhatthe statistics. The values shown are the instantaneous znaxiznum values registered by the wind recorderduring the oneminute every hour duration of samplingfor mean wind. The peak available value of gust intensity 8 knots! stands to the peak value of mean wind intensity SS knots! in the ratio of 1. 10.

2.2.4 Ron -Term Trend of Mean %1nd Intensit

The data of Table A. 1 can be extended to yield statements of expected maximum value of mean wind intensity for any desired return period. The method employed for the prediction is known as that of Gumbel's Third Asymptotic Distribution: it is presented in Appendix B. Application of the method to the specific data available is discussed in Appendix C. The results are given in Figure 2. 1. The expected maximum value of mean wind intensity corresponding to a return period of 100years is 40 knots. There }s a 50 percent probaMUty that this expected value will be exceeded.

law

ji i 'R$ p 'N e 810 >02 10~104 Return period years!

Fig. 2. 1 Long-termtrend in the expectedmaximum valueof the meanwind intensity. 2.2. 5 Lon -Terxn Trend of peak blind intensity

The five-year maxima of Table A.2 arc hardii sufAcicnt to establish a long-terxntrend in whichone can have xnuchconfidence. The extrapolation is xnadein Figure 2.2 whichyields expected maxixnum values of peakwind intensityas a functionof returnperiod. For a 100-yearreturn period the expectedvalue is 82 knots. The extrapolationmethod employed is the saxne as for the mean wind intensity.

0 O g H~ 8 m

P 102 10 zo' Return period years! ~ng-terxn trendof theexpected xnaximum value of peak wind intensity.

2. 2. 6 Lon Terxnof GustIntensit No data are availableto estabhshthis trend directly. py reference to TableA. 3, onecan simply estixnate that the gustintensities will be aboutl. 10times the peak ones. Thisis, adxnittedly,a gross w y to estimate. 2.3 KONA STORMS AND TROPICAL CYCLONES

2.3. I The Observational Material

Kma storms and tropical cyclones occur infrequently in the Hawaiian area; the expectancyof the former is oncein aboutfive years; that of the latter, once in abou.tfifty years if the simple empirical .~le is followed that if the pressure drop at the center of the cyclonic wind field is one inch of mercury or greater, the a.txnosphericdisturbance is a tropical cyclone; if less, a Kona storm. The expectancy is a mean value but the variability aboutthe expectedvalue is great. The infrequencyof occurrence coupled with the almost complete absence of xneasurements of central pressure drop and kinematics makes for the unfortunate necessity of having to improvisecertain steps of the process for forecastingthe long-terxn trend of the descriptive parameters of these powerful disturbances.

Up to the present, measurement of the essential parameters of central pressure drop, xnaximuxnwind intensity and radius of maximuxn wind have been carried. out for only two tropical cyclones or storms: NINA in 1957 and DOT in l959, No core temperatures were obtained, and since both were at the threshold between tropical storm and cyclone when judged by the criterion of central pressure drop, there is a resultant vagueness of categorlxation.

The tracks of these cyclones are shown in Figure 2, 3. Also shown in the same figure are the tracks of somehistorical cyclones. Relevant dataon the cyclonicparameters are containedin AppendixD.

The scarcity of observationaldata on Kona storms and tropical cyclonesis to some extent compensatedfor by the availability of data onAtlantic and Gulf of Mexico hurricanes. These data are, if not abundant,at leastadequate for establishingthe trendswith return period of theexpected values of the essentialparameters of centralpressure drop, maximum wind intensity and radius of maximuxn.wind. Conversion of such datato the Hawaiianregion could be madeupon knowledge of the applicable scalinglaws. Unfortunately,the laws are not known. To overcoxnethis disadvxmtage,hypotheses are introduced.These are difficultto justify andone is left to hopethat theirplausibility is evidentto the reader. Fig. 2. 3 Tropical cyclonetracks in the HaesHanarea. I Tracks of the tropicaL cycloneof 1921and earlier years after 'S.S. Visher, courtesy of SishopNuseo~- l 2.3.2 Lo -Term Trend in C clonic Severi

The statistical techniquefor forecasting the long-terzet trend in tbs severity of cyclones is the same as that employed for winds and described in AppendixB. In Appendix 0 is presented an empirical techniquefor forecasting the wind field given the cyclonic parameters. Also discussed ia that appendix are the application of the method to the Hawaiian scene and some consequent. results.

The final result of the prediction process is contained in Figure 2. 4. It is seen that the maximum wind intensities developed in a Kona storm or Hawaiian cyclone are higher than the peak wind intensities of the quasi- steady field of trade snd Kona winds.

150

g 130

o 110

I' 70

10 10 10 10

Return period years! ~ng-«rm trend of the expected maximum valuesof the mean, peakand gust intensities of cyclonic winds.

10 3. C U RRZNTS

"SLY;Lp

S. T. O'-Lewis'pe~

3. 1 OVERVIEW

The current enters into the design of the floating coznznunitythrough its dynamical action on the underwater body, an action which must be overcome by the mooring or positioning systezn.

What is required for the design of the floating community is the long- term trend of the current strength vector as a function of depth at the contemplated site. But no measurements have been made at the site: the three closest stations for which current data are in hand are:

a! off DiamondHead at a point where the water depthis 80 m,

b! off Waikiki, at a point where the water is 9 m deep,

c! off SandIsland, at a point where the water is 125 zn deep.

Thc observations of current at these sites last 15, 21 and 80 days, respectively.

Since a long-term trend can be establishedonly upona duration of observationssufQciently long to yield a. set of data that is statistically significant,one can hardly view a spanof tizneof at most8G days as being adequate.

There is, thus, justifiableconcern as to the applicabilityof such datato the designproblem of interest,. But concerndoes not providean answer: it only makesone suspiciousof anyestimate that maybe advanced.

Currentis a complexphenomenon which results from a plurality« causes.Inasmuch as somecauses operate in a mannermore predictable

The Rhyme of the Ancient Mariner than others, it is advantageousto forecast separately the components of current to which each cause gives rise and then obtain their resultant. However, there is inherent in such a procedure a grave difficulty: observations can be made only of resultant current, and the task of their resolution into componentsis, at least for the site of interest, far from being simple and straightforward.

The current in the area of the Hawaiian Archipelago can be resolved into three main coxnponentstermed respectively large scale cix'culation, wind or Ekman drift, and tidal flow.

It is usefulto precedea discussionof currentcomponents with some x'emarkson the resultant current or general circulation.

3.2 THE GENERAL CIRCULATION Thegeneral circulation is characterized bygreat complexity and by remarkablystrong variations with season, location and depth. Anidea ofthe variations is conveyed byFigures 3. 1 through3.4, whichpresent expectedvectors of surfacecurrent close to the islandof Qahufor the fourcoxnbinations ofwinter and summer seasons with Qooding and ebbin g tides. Thevectors are for thehorizontal coxnponent of current velocityinthe surface layer extending down to90 xn or 300 ft!. The verticalcomponents ax'enegligible by comparison. The figures make evidentthe strong roles played by the tide and by the season. MeasurementsbyWyrtki et al.* ofcurrent velocity at a stationoff DiamondBead hence, inthe vicinity ofthe site contemplated forthe floatingcitybut closer inshore, where the depth isshallower! are reproducedin Figure 3,5. Note the frequency distribution ofthe observedcurrent direction Figure 3.5. a!: its symmetry attests toa largetidal component.

~ KlausWyrtki, Volker Graefe e andand Wxn. patzert, CurrentObservations in thee HawaiianArchipelago, HIGRep. 69-15.

12 Fig. 3. 1 Most probable current pattern occurring between Oct! Nov- Feb Mar! in the 0 to 500 ft layer during a Qooding tide.

Fig. 3. 2 Most probablecurrent pattern during ebbing tide Oct! Nov- Feb Mar!.

13 Fig. 3. 3 Most probable current pattern during flooding tide, Mar! Apr - Sept Oct!.

Fig. 3. 4 Moatprobable current pattern during ebbingtide, Mar! Apr - S~t Oct!. ai 1'requency distributin ix rcent per ck gree of angh !.

0 20 40 60 80 100 1/0 crri lii F'requencvdistribution of observedcurrent streni~h,

Data: Latitude 20o 14.O' 'A Longitude157" 4K. 4' 9;. Period of obsercation from 1115 hiiurs un 10 Mal . 10fi7, io 0400 huurson 25 Mai.. 10ii7. Tiitai time «f ebs< rest jun . 14 days. lti hours, 45 min.

From K, WX-rtki. V. Graefe a W, patrert. Current E»scrsations in the HawaiianArchi la o, HawaiiInstitute of Geophysi

~~3.4 EKMAN OR %'IND DRIFT~~

~~The methodfor estimating the Ekman, or wind, drift is containedin AppendixE. For the latitudeof Oahu,its surfacevalue is givenby~~

Ue! = 0. 021 Ua and its direction is 45 deg clockwiseto the wind. Its strengthdiminishes expon.entiallywith depthand is virtuaHyzero at the depthof 12.6 U m, wherethe windspeed is in m sec l. Notethat the meanwind intensity is to be used.

The extreme values of Ekmandrift are in principle derivable from the extreme values of mean wind intensity Figure 2. 1! provided these are complemented by correlated values of wind duration, for the Ekman drift builds up but slowly under persistent stiznulation. Unfortunately, correlated data on wind duration are not in hand. If, to make progress, one assumes boldly that the wind is always of sufficient durationto generatethe virtually full strength of the Ekman drift, the extreme values that obtain are those of Figure 3.7. 1. 0

~~c5 Q 88 g e 0~~

~~1 10 10 10 10 Return period years! Fig. 3. 7 Long-term trend in the expectedmaximum value of the Ekman drift.~~

17 3. 5 LARGE SCALE OCEANIC FLOW Accordingtothe U.S. Navy Hydrographic Office 947!*, the large scaleoceanic flow through the Hawaiian Islands has a strengthof about 25cm aec-I and ia westwardly directed. Subsequent observations by Seckel962! « leadto theconclusion that the direction of theoceanic flow issomewhat uncertain while the strength ia,if notnegligibly small, perhaps 10to 20 cm aec-1. Later work by Seckel etal, 967!+ and Wyrtki ct al. 969!+supports these conclusions. Itmay be observed thatthe large scaleoceanic flow is difficultto measure because it is dominatedbythe complementarytidalcurrent and sometimes bythe wind drift.

3.6 VERIFICATIONOFTHE RESULTANT CURRENT If thelogic is correct, it should bepossible torecombine thecurrent componentsintoa resultant current which would correlate favorably with observations.A firstorder verification baaed onexpected values iacarried outin Appendix F. The error in correlation iaon the order of 6 percent by deficit.

$.7LONG-TERM PREDICTION OFTHE GENERAL SURFACE CIRCULATION A iong-termprediction ofthe general surface circulation canbe made onthe aasulnption thatthe large scale current remaiIla constant andthat ' U.S.Navy Hydrographic Office947!, Cur rent Atlas ofthe Northeast Pacific,Hydrographic OfficePublication no.570, 12 charta. ««G.R,Seckel {1962!, Atlasof the Oceanographic Climateofthe Hawaiian Regton,U.S.Fish and Wildlife Service Bulletin 193,p. 371-427. + G,R.Seckel, B.L.Charnell andD.W.K. Au967!, The Trade Wind Zone,Oceanography PilotStudy, parts I toVI: TOWNSEND CROMWELL Cruisesl-6,8-l7 and 21, February 1964toJanuary 1966,U,S. Fish 22and Wildlife pp!~ Service, SpecialScientific Report:Fish. No. 552 to557, ~op. cit. Section3.2

18 the amplitudesof thetidal components r emain invariant. This leaves the Ektnandrift as the singlecotnponent to be Predicted and then joined to the others. The Predictiontechnique is that of Appendix11; the input cia~ are thoseof Figure2. 1. The resultinggeneral surface circulation is plotted in Figure 3. 8.

~~2 CJ O~~

~~50 2 2~~

~~R CC~~

~~2 U 47 O g 0' o~~

~~10 10 10'~~

~~Return period year s!~~

~~Fig. 3, 8 Long-term trend in I,hc expected maximum strength of the general oceanic c ireulation~~

~~4. WAVES~~

~~"The ~eau" arm' the holt an~! tlirt hurl cp8 ~>n.'.cpcls~ i ~ d>lc '-'c'"i&a'1'~>l,';c.-~i 7u~t is~~~~

~~Cl&t A~~ t~~

~~4. 1 INTRODUCTION~~

Thewave pattern in theregion of the Hawaiian Islands is thevery complexresult of local anddistant generativeactions which are seasonal in nature;these are the windwaves and swell. Occasionally,but quite infrequently,the seasonal trend is upsetby a transientcyclonic disturbance of brief duration.

~~To designthe structure of theOoating community, two important aspects of the seaway need be estabBshcd:~~

~~o The expectedlong-term trend of theparameters by which a seaws.y is described.~~

~~o The resultingshort-term or quasi-instantaneous!description given such parameters.~~

The foregoing is what is required; what is available falls far short of this andit becomesnecessary to spanthe gapof ignorancewith a bridge constructedeither of empirical relations derived from wave observationsin other regionsor, failingthese, of plausibleassumptions. These are made explicit in the next section.

The wind and currents have been describedby certain short-term statistical averages and long-term trends of their velocities. When seeking to design the platform to supportthe floating community, further descriptive refinement is not necessary or even utilizable should it be available which it is not!. The sufficiencyof simple statistical averages and trends of velocities to describe the complex motions of winds and currents obtains because: a! There is virtually no mutual interaction betweenthe actions that winds and currents exercise on the system and its structural or kinetic responses thereto: the latter do not affect the former; and b! The system responsesare simply related to the statistical averages mean and peak value! of the wind and current velocities.

~~21 The absenceof mutualinteraction betweenwind and current excitation andresponses thereto arises from the virtual imznobilityof the platform with respect to the movementof wfnds and currents.~~

Thesimple relationships between excitation and response obtain because. a! Theelastfc response of the structureis, for practically every term of whichis is composed,so quick that the strains follow faithfuUy,quasi-instantaneously, andwith directproportionality the fluctuationsin the excitations, Thus, the statistics of elastic responseare identical to thoseof the excitation and it results that whatis of import in the design of the structure is the long-termtrend in the peak andznaximum values of the velocities of the media. b} Thekfnetic response ofthe platform heel, drift! is so lethargicthat only the trend of the short-termznean values of theexcftations of vrfnd and current is of significance. Butwaves requfre a znorecomplex description, for there is strong interactfonbetween their hydrodynaznical excitation ofa floatinghull andits kineticresponse. This mutual interaction depends onthe period ofthc exciting wave should this be regular and, indeed varies strongly therewith.A seawayis vfewed asthe superposition in random phase of a vastpluralfty in principle, of an infinity! of regularwave trains of smaU ia principle, infinitesimal! amplitude andof different period. Thfsconc~t has led to the short-term description ofa seawaybyits energyspectzum. Such a descriptionis a statementofhow the energy ofthe seaway fspartitioned among the component wave systems. Knowledgeofthe partitfonfng is essential toa determinationof hull-wave fntezactfon,hence toa pzedictionof the oscillations and drift whicha Qoatfnghull will experience fna seaway. Theshozt-term description ofa seawaybyfts energy spectrum fs outlinedinAppendix G. The appendix also contains a statement ofthe derivativesignificant waveheight and period and of the resulting distd~~mofthe expectancies ofmaximum wave height and period. Thecharacteristics ofthe waves ofthe sea depend onthe nature of thewin< towhich they owe their generation andcan, consequently, be categorfEedby wind type. Therefore, following thepattern introduced

22 in thesection on winds, waves will be grouped according to whetherthey are causedby' the primary wind circulation trade winds goxla storm d distants o ms! or by a cyclonicwind. Theformer are described " as p"ry windwaves and swell, thesecond as cyclonic waves.

~~4.2 PRIMARY WIND WAVES AND SWELL~~

~~4. 2. 1 The ObservationalMaterial~~

~~The only reportsavailable at presenton Hawaiianwind wa "es ~ swell in the open sea are:~~

~~o A compilation by P. S, Homer 964!»~~

~~o A study by Ho and Sherretz 969!»»~~

~~A brief outlineof thesereferences follows. A fuller exposition, which includes numerical data, is to be foundin AppendixH.~~

~~The Homer Co ilation~~

Most reports of recent date bearing on the characteristics of waves in the Hawaiian area quotea studyby Homer 964!, the apparent reason being that it is the only compilation of its kind a railable. T'be central part of the repo rt is a tabulation of probabilities of occur rortce of significant wave hei@ts and periods groupedaccording to the sector of theirorigin. The original data on which the study is based vvere either wave recordings off Kahuku and off Barber's Point or the results of hindcasts. The cornyilationis a laboriouswork oi synthesisin whichjudgment has entered strongly and the outcomeis a statementof values r~resentatfvc of a typicalyear. Yet theoutcome falls somewhatshort of providing a reliable basis for making long-rangepredictions. Suchpredictions can be made only upon estabBshmentof the long-term trend in wave

~~P. S. Homer 964!, Characteristics of DeepWater Waver in pahu~~

~~Area for a TypicalYear, preparedfor theHoard of Harkx>r Commissionerst State of Hawaii, Contract No. 57'72. Marine Advisers, ~ Jolia~~

~~I'. P, Ho andL.A. Sherretz 969!, A PreliminaryStudy of ~e~ Wavesin theHawaiian Area, HawaiiInstitute of GeophysicsHeart No. HIG-69-16, Universityof Hawaii.~~

23 c~teristics, while the report gives only a distribution of expected values Also theseare raw, so to speak, i. e., they have not beenfaired to yield probabi]ity distributions ha,ving well-behaved trends and the consequeoceis that oneis forced to refer to a plethora of numbers rather thanto a fewparameters definitive of distribution~. But the conversionof figures into parameters is not of easy accomplishment--if it can be accomplished at aH.

~~The Ho snd Sherretz Stud~~

Thisstudy is a condensationof wave characteristics height, period, anddirection! visually observedfrtom two stations. The visual estimates weremade by U.S. CoastGuard personnel at the MakapuuI.ight House onOahu 40 feetabove sea level! and at the K ilaueaLight Houseon Kauai 80 feetabove sea level!: theyrepresent sea conditions prevailing at a, distance of 300 to 400 yards offshore.

It is the contentionof the authors that the plotted wave characteristics are representativeof conditionsin the opensea. If this statement canbe accepted,the observationsfrom Makapuu Point would be statistically closeto those that obtain at the site of the floating community. Thereare, however,two points of concernwith these studies: the accuracyofthe visual estimates and the defmitions employed to express the results. Oneis boundtowonder whether wave heights can be reported in values differingbyan increment ofone to two feet when they are observed from a highelevation anda considerabledistance, orwhether thereported values are,indeed, anoptical hence, scientific! illusion. Unfortunately, resolutio n ofthis point must await the realization ofa sufficiencyofinstrumented measurements. Thesecond point of concern lies in the suspected variance ofthe terminologyemployed inthe Ho and Sherretz report from that customarily used in discussssingwaves. The authors give values of meanwave heights. but it is inferredredfrom. the substance ofthe report that they are referring to the mean valueses of f the observed, hence tothe s~ttftcant, wave heights.

~~The tableof p robahili babilitiesof wave height, period and di 148 entries.~~

~~24 This is not a pedanticremark: the significant observed!wave height standsto the xneanwave height in the ratio of l. 6 tounity,~~

It mightbe well to observethat these publications do not represent muchof an inputfor initiating a designwhose configuration -- perhaps even a judgmentof its feasibility will dependin an essentialmanner on the predicted extreme state of the sea. That they are, nonetheless, extrexnely valuableis evident: oneneed but supposetheir non-existence to appreciate the seriousness of the difficulty in which one would be in seeking to speak objectively and in quantitative measures about the waves that flow past the Hawaiian Archipelago,

~~4.2.2 Short-Term Descri tion~~

If a coxnparison is xnadeof the yearly mean values of significant wave height containedin the Homer snd in the Ho and Sherretz reports, one is led to the ixnxnediate conclusion that there is considerable vs.riance between the two: the data contained in the forxner report yield 6. 25 as the measure while those contained in the latter report reduce to 4. 69 ft off Makapuu Point and 4. 64 ft off Kilauea Point.

An attempt has been made to seek the reason for the apparent discrepancy and, indeed, it has been found that the figures can be reconciled if it is assumed that the observers at the Makapuu and Kilauea Points did not detect swell when it underlay the steeper wind- driven waves. The assumption is not ixnplausible inasmuch as the average period of the swell derived from the Homer compilation! is 13. 55 sec, a period that corresponds to a wave length of almost a thousand feet. lt is. I suppose, fairly obvious that the rise and fall of a fat swell but a few feet in height and of a considerable length on the order of a thousand feet can hardly be perceived by eye froxn a distant station especially if the surface of the sea is roughened by shorter and steeper wind sea. This assumption is reinforced by the results that the yearly mean significant height of the wind waves of the Homer compilation is very close to that of all the Ho and Sherretz waves, naxnely 4. 3l ft against 4. 69 ft off Makapuu Point and 4. 64 ft off Kilauea Point.

~~The analysis to which reference is made requires the initial separation of the Homer data into four sets -- Trade wind waves, North Pacific swell, Kona storm waves, and Southern swell in accordance~~

~~25 with a pattern suggestedby Moberly and Chamberlain~, see Figure 4.1.~~

~~g. 4.1 Hawaiianwave pattern showing range of directionand mean directionfor eacbwave grojp. I,Derived from Homer compilation!~~

~~L Moberly,Jr.and T. Chamberlain 964!, Hawaiian Beach Systems, HawINfnstitnte ofGeophysics ReportFHG-64-2, University ofHawaii.~~

2$ A first separation into northern wind waves and swell and southern wind waves and swell the range cf the forxner extendingfrom the coxnpass angle of 281 deg to that of 324 deg; while that of the latter, from 124 deg to 259 deg! is facilitated by the gapin wavedirectionality to the west and by the very dixninished values of probability of occurrence of waves approaching from the southeast. Within each of these two sets a second separation into wind waves snd swell is basedon period. Some results are contained in Table 4. l.

~~Table 4. l Some results derived from the Homer and the Ho and $herretz studies on Hawaiian waves.~~

~~Homer Ho and Sherretz EXpected frequency Sfsnfffcant Sfgnffzcant Stgntf to ant Wave type of occurrence %! hetght t! pertod sec! hetght ft!~~

~~Trade wind waves 75. 3 4. 79 8. 63 Kona storm waves IG. 3 3. 52 6. IS~~

~~Wfnd waves 85. 6 4. SI 11. 45~~

~~North PactBc sweU 74. 0 4,79 13. 89 Southern swe!l 53. 0 2. 60 13. 07~~

~~swell 127. 0 4. 53 13. 55~~

~~All waves 212. 6 6,25 4. 69 Makapuu Potnt~~

~~4. 64 Ktlauea Point~~

The analysis reveals that sweQ is prevalent almost all the time and that its mean height is coznxnensuratewith that of the wind sea. Since the averageperiod of the swell is approximately8/5 that of the wind waves, the result is a soxnewhat complex seaway.

Conversion of a. statement of significant wave height and period into an energy spectrum can be readily znadeif the shape of the spectrum Is known. WMe a large number of instruxnentedobservations znostly ln the

The directiOnal range quotedherein deviateasomewhat from tlxat givezI by Moberly and Chamberlain and iS the authOr's own iltterpretation of the Hoxner data. Atlantic Gcean! have resulted in estabUshing the general shape of the energy spectrum of wind seas of modest severity and of swell, there is insufficient resmm to believe that the shape will rexnain unaltered as the seaway becoxnes heavy. Nevertheless, it is usual to make the convenient assuxnption that the spectral shapedoes, indeed, remain independentof the vigor of the seaway, There is reason to believe that such an assuxnption leads to conservative overestimates of extrexne maximum wave heights. But reason and belief constitute no proof. The problexn is parallel to that of predicting statistical parameters which describe the shape of adults froxn measurexnents of the same parameters made on infants.

In Hawaiian waters, the seaway being normally a xnixture of wind sea and swell, its descriptive energy spectrum is the combination by superposition of the separate spectra of these coxnponentseas. lf the yearly mean values of heightand period of wind sea and swell are introduced, the energy spectrum that results50 is the double-peaked one of Figure 4.2. 0

8 12 l6 20 24 Wave period, T sec! Fig, 4.2 Energyspectruxn of the averageseaway in Hawaiianwaters. Givena set of waveparazneters, such as significantwave height and period or equivalent, for any combinationof w}ndwaves and swell, a similar energyspectrum can be constructedwhich wiII be descriptiveof the short-term characteristics of the sea surface and which will be valid for the duration of the selected set of parameters,

~~4.2.3 Lon -Term E ectation of Si ficant Wave Hei ht and Period~~

No data are available on the long-tenn trends in wave height and wave period and one is, consequently,faced with the necessity to construct such trends. Since the waves are wind-generatedand since the long-term trend in the zneanwind intensity has been establishedin section2.2. 4, the long-terzn trends in wave height and period can be derived therefrom by application of the empirical relations of Sverdrup and Munk 947!~.

For a fetch length and persistence of steady conditions both sufficiently great to result in a fully developedseaway, the mean wind intensity Va andsignificant wave height Hs are relatedby the simpleformulation

gHs/Ua = 0.30 where g is the gravitational acceleration.. If the long-term trend in expected maximuzn mean wind of Figure 2, I is used, the trend of Figure 4. 3 results, However, it should be recognized to be an over- estimate, for winds of great intensity do not persist steadily over an interval of time sufficient to develop a wind sea to its full intensity.

An alternative method consists in extrapolating the trend exhibited by the Hozner data. This extrapolation, which is carried out by the technique of Appendix 8, is discussed in Appendix H. The resulting long-term trends in the expected maximum wave heights and periods are shown in Figu.res 4. 4 and 4. 5 for each of the four primary wave types.

* H. U. Sverdrup and W. 1Vl,Munk I947!, "Wind,Sea and Swell: Theory of Relations for Forecasting."Hydrographic Office Publication No. 60I, U.S, Navy Department, 1947. 10 lO 10 104 Return period yenrs! Fig, 4.3 Long-term trend in expectedmaximum wave height in Hawaiian waters based on the empirical relations of Sverdrup and Munk for fuQy developed sea states.

~~~ 30~~

~~~ 20 6~~

~~8 I 10~~

~~7 0 10 10~ 10 104 Return period years! Fig.4.4 Long-termtrend inthe expected maximum value of slgIgficant waveheights based an the Homer compilation.~~

~~30 c 40 030~~

~~20 Q~~

~~,~ lO~~

~~Fig. 4. 5 Long-terxn trend in the expected maximum value of significant period based on the Homex compilation.~~

~~4, 3 CYC LONIC WAVES~~

~~4. 3.1 Lon -Term E ectation of Si ficant Wave Hei ht and Period~~

The infrequent tropical cyclonescan generatewaves of great steepnessand considerableheight. The methodfor determiningthe significant wave height and period of suchwaves is presentedin Appendix The outcomeof the analysis basedon the cyclonicwind intensities of Appendix D is shown in Figure 4, 6.

~~4.4 ALL WAVES~~

The long-termtrend in significantwave height and period of the four types of primary windwaves and swelland of the cyclone-generated wavesare combinedby the xnethodof AppendixJ to yield the long-term. trends of aH waves. The result is presented in Figure 4.7.

~~3l 40 8 'g~~

Fig. 4.6 Lang-terjn trend in the expected maximum nQue of significant wave height and period A bD at the radius of a 2P maximum wind of a Hawaiian cyclone translating at average speed of ~ 10 10 knots.

~~10 102 103 104 Return period yea.rs!~~

~~Ftg. 4.7 Mng-term trend in 40 the expected maximum value of significant wave height andperiod. L~ 30 g 8 8 20 10 10 10 104 Returnperiod years! APPENDIX A~~

~~DATA ON WINDS IN THE HAWAIIAN AREA~~

This appendixcontains the wind data obtainedfor predictingthe environmentat the site selected for the floating community. The data were obtained through the cou.rtesy of the Honolulu Bureau of the U.S. Weather Service.

~~Table A. 1 contains a listing of mean wind intensities. These are averages of one-minute observations taken every hour.~~

~~Table A. 2 is a listing of the peak values of mean wind intensity. Again, the values are based on one-minute observations made every hour. Unfortunately, the records are fairly incomplete.~~

~~Table A. 3 is a lisWg of maximum values of gust intensity, i. e., of the instantaneous peaks of wind intensity observed during a month. ~ I 8~~

~~I q -.~~

~~9 59 gXg o~~

~~0 0 w~~

~~S,g Ig3~~

~~~>8 6~~

~~I I g M e o 2 E X o M~~

~~o o o ai o 0 4 lg 2 foal4l I~~

~~c N 41 X fal44 B 43 ERR 04o 04 VJ 41lal lal XZK~~

~~cC m~~

~~Q eO Ionn O z M C4W NX K W lsl~~

~~o ~ o 5~~

~~I iK I n ~ c n v V W W K Z f4 L1~~

~~ct o R S A 414l CDR R 2 z o g n e lO~~

~~R mls~~

~~Ifl ~I@I o4 cD TableA.3 -- Maximumvalues of gust intensity observed at the Honolulu airport.~~

~~Year Gust intensity ktonth of msstmnm ~knots Dtroctton~~

January 1,959 58 SW February 1957 55 March 1951~ 51 SK April 1961 35 NE May 1953 31 June 1954 34 E July 1.959 30 August 1959~ DOT! 45 SE September 1963 31 SE October 1,961 35 SE November 57 NANNA! NE December 1957+ 51 NE

~~~ tropical cyclones Source:U. S. Air Force, DataProcessing Division, Asheville, N. C. 28801 APPENDIX B~~

~~THE PREDICTION OF EXTREME VALUES BASED UPON GUMBEL' S THIRD ASYMPTOTIC DISTRIBUTION~~

~~I, Introduction~~

Several distributions have been employed to predict the maximum values of a variate; of these, two have found favor in environmental predictions, namely, Guxnbel's first and third asymptotic distributions the second asymptotic distribution is for xninimuxnvalues!. Both asymptotic distributions are of the exponential type: but the first approaches infinity as the variate along which the entrapolation is ordered, namely, the logarithm of the return period, increases indefinitely, whereas the second levels off and converges to an asyxnptotic value as the return period approaches infinity.

To be sure, the trend to infinity of the first asymptotic distribution is lethargic, and in most applications to forecast ng this tendency of growth does lead to predictions of reasonable accuracy. This obtains when the observed values, when ranked by magnitude and related to a probability of occurrence, crowd the regression line of expected values. There exist, however, cases for which this crowding does not obtain and for which there is, indeed, a marked tendency of the observed values to deviate increasingly from the regression line as the retu.rn period increases. Such cases occur when physical causes operate to impose constraints of ever greater severity as the variate tends to grow in magnitude. An example occurs with the central pressure drop in tropical cyclones. The reliable treatment of such problems requires extremal probability distributions somewhat more flexible than the first asymptotic distribution.

Of the available alternatives, a convenient one is Gumbel's third asymptote distribution. As against the first, whichis a tw~parameter distribution mode and slope!, the third is a three-parameter distribution asymptote, characteristic value and exponent!. However, the enhanced flexibility introducedby the third parameter is paid for in increased complexity, and a relatively simple and direct deterxninationof the paraxneters, as in the case of the first distribution, is not quite in hand. The way out is to have recourse to a computer and derive the parameters by iteration. A method for accomplishingthis is ouQinedherein. Whichdistribution is the one to employ in a specificapplication is a matterofjudgment. Such a judgmentcanbe exercised byconsidering how wellthe inputs fit thedistribution and by taking into account any factor whichmay tend to influence the trend toward the extreme of theznaxiznum values.While goodness of5t isa statisticalproblezn, the criterion by which it isjudged depends onthe intended application and no a priori rulecan beformulated inits regard. The deviation lntrend to extremes is usually a physicalproblem and its prediction cannot always be made on a purely statisticalbasis but may require insight into the nature of the phenomenon to be predicted.

2. ~ts Theprediction ofextreme maxima is made by carrying out certain statisticalopezations ona setof maximum values of thevariate to be predicted.Such a set isobtained either by saznpling subsets ofa totality e.g.,maximum deviations from specifications of5 ball bearings from eachbatch of100 comprising a total lot of l0, 000!, or by the sequential measureznentofa process over specific intervals of time suitably spaced e.g.,one-minute average wind intensities measured every hour and reportedasmonthly maxima!. The method isthe sazne ineither case, butsince the environmental phenomena ofintez'est areessentially 5zne- dependent,theexposition thatfollows is expressed inthe parlance pertinent to mme functions.

Thus,the inputsrequired are: a! a coznpilationof set zzuudma, x b! thetotal number of set maxizna, N thederivationOrdinarily, thatthefoQowsset maxima thisuniformityare for a is uniformassumed saznplingto obtain. interval However, and in thereexist cases forwhich thesaznpling interval isnot uniform. These casesrelate toshort transient phenomena ofvery infrequent occurrence. Theprediction ofthe intensity ofa tzopical cyclone ina specificareais an example-Thepaucity ofthe observational material inhand enhances the valueofeach rare observation sothat it isiznportant tomake use of every single event. Theprocedure thatfoDows isvalid only when thenumber ofmaximuzn valuesinhand isnot small, i.e., when N isgreater than about 32. The calculationbecoznes znoreand more complex asN decreases. Ordering index, m

WhenN is large, the adjusted!set maximaare orderedby magnitude, either ascendingor descending,+ ~hat folio~a ~ascendfnorder is used. Thegeneric value of the rankingindex of the set maxima is denotedby x; thus,

~~To eachset maximum x corresponds the ordinal frequency~~

~~m N+l~~

~~2 c! N' N' N~~

~~xN= l g x N m=l~~

~~The corresponding variance is~~

~~N 2 l 2 xN N g [m N m=1~~

~~The cor responding dispersion,~~

~~x,N~~

~~39 d! ReducedVariate, y~~

~~Theset mKdmuxn x is plottedat theposition given by the ~ed variate~~

y = -8n - Sn y !! Notethat the subscript m is employedto define either an input x ! ora valuedirectly related to an input y~, om,etc. !. e! ParametricS ence of the Characteristic Value v Thecharacteristic value is obtainedby iteration. A new chLrtuWristicvaluevp+I l isrelated toits previous value vpthrough the recurrence relation

v p+1! = v p! + f-0.5] c p! ~ av wherep isan iteration indexf c p!is a signchange count to bedered belowand 4v is anarbitrary incremental value. The initial valueof v shouldbetaken close to the value of x correspondingtoy = 0; that of av canbe taken at aboutone-fourth tRs value. f! Parametxic8 enceof theForm Factor k Theform factox k is alsoobtained by iteration. A newvalue k q+>!is rebated toits previous value k q!through the recurrence x'elation

k«+» =k q!+ -0.5]'«' ak q isan iteration index, c q!is a signchange count to bedeGned belowand Ak is an arbitrary incremental value. The initial values ofk and ak can be set equal tounity. g! paramnetricence of the As totew Theasymptotic valueof the vaxiate w is also obtained byiteration Anew vahte ofthe asymptote w r+1!is related toits previous value w r! throughthe recurrence relation

~~w x+1!= w r! + f-0.5] ! ' aw~~

40 where r is an interaUon index, c r! is a sign change count to be dered below and aw is an arbitrary incremental value. The initial value of w can be estimated from the plot of x vs y; that of aw sbouldbe taken at approximately one-fourth the difference w-x h! Function x ym, v p!, w r!!

~~The function x corresponding to the input value x and located at the sax' value of the reduced variate as the latter, nmnep y, but determined from values of the parameters~~

~~v p!, k q! and w r! is~~

~~x m; p,q,r! =- x y, v p!, k q!, w r!! = w r! - I w r! - v p!] ' exp -ym/k q!! i! Difference between Function and t~~

~~d m' p q r! = x m p q r! - xm j! Variance of Function from t N a psqsr!=- I Zc d ml p q r! N m=1 k! Criterion~~

~~2 2 a p, q, r+1! ~ a p, q, r!- the sign change count~~

~~c r+1! = c r!~~

~~a p q, r+l!! a p, q r!- the sign change count~~

~~c r+I! = c r! +1~~

~~In these expressions, c is an arbitrary tolerance. l! Asignptote w p, q!~~

~~Return to atop g!, introduce the next sequential value of w, f. e,, w r+1! and repeat steps h! through k! with w r+1! replacing w r!. Continue to convergence, i.e., untG~~

~~2 2 a p, q, r+I! =< p, q!~~

~~and w p, q, r+1! =-w p, q! m! Criterion If ~2 2p, 0+1!-:0 p, q! - ck~~

~~the sign change count~~

~~c 9+1! = c q!~~

~~the sign change count~~

~~c q+1! = c q!+ 1 In theseexpressions, ck ia anarbitrary tolerance. n! E nen.t k and A tote w~~

~~Returnto step f!, introducethe next sequentialvalue of k, i. e., k q+1!and repeat steps g! through m! with k q+1!replacing k q!. Continue to convergencewhen !g 2p, 9+1! ~2 !P'9!I~~

~~' 2p = ~ e1!=~ 2 p! k p, q+1! k p! w p, q+1! nw p!~~

~~4.1 M@djma of IndkvikuQ Excursions~~

~~Theadjusted vahe is the original value multlpUed by the Rayleigh factor f~= fN!= ~gn 2 N!~~

~~whereN, the number of excursionsabove the mean during the interval of interest, is givenwith sufficient accuracyby~~

whereT is the interval of interest and T is the meanperiod of the 0 m excursions. The interval of interest is the duration of the steady state of phenomenonbeing analyzed. This is largely a matter of judgro.ent; but notethat the factors grow very slowly with N so that large errors in estimating this number are tolerable.

The correction is applicableprovided N ~ 8; it is to be neglected otherwise Suchan ~ustment wouldbe applicableto the prediction of the extremevalues of gustsand in the prediction of maximum wave heights.

~~4.2 Maxima of Mean Values~~

~~Themax maof meanvalues varies even more slowlythan doesthat of themaximum excursions. The factor by which to multiplythe values of the original set is~~

~~fm = f f !/f l! wherefr is nowthe Bayleigh factor with the ratio of samplinginterval to samPlingthration,s/d,rePlacing N, i.e., fr = fr s/d!,and where~~

j f != t r! o!l fr' exp -fr2 /2! I exp -f 2 /2! in whichexpression o{ ! is thecumulative probability of non-exceedance andfr = f fr=1!= 0.638.The hnction has the asymptotic value of fm = 1.963corresponding tof = ~ . It is bythis function that a variate whichis intermittentlysampled is to be multiplied to derivedits expected value. APPENDIX C

~~PREDICTION OF THE EXTREME VALUES OF WIND INTENSITIES IN THE HAWAIIAN AREA~~

~~1. Introduction~~

~~In this appendix the data of Appendix A are extrapolated by the technique of AppendixB to yield the trend in the expectedmaxixnum values of the mean and peak wind intensities.~~

~~2. Mean Wind~~

Table A. 1 of Appendix A lists the frequency of occurrence of parametric intervals of wind intensity both as a. function of wind direction and independently thereof, the second set of values being obtained by sixnple summation over direction of the forxner. It is this set that is of interest in the context of tMs study and for convenience of reference, the pertinent data are reproduced in Table C.1 herein.

~~The wind intensities are reported in integer values of speed knots! and the groupings by intervals reflect this. If a. reported wInd intensity Ua is interpreted to xnean a speed ranging from~~

U - 0.5 ~U ~ U + 0.5 knots! the intervals become continuous. Such a plausible intexp retation is xnathematicallyconvenient. The resultingintervals of wind intensity are listed parenthetically in column 1 of Table C.1.

~~The observed mean wind intensity whenplotted against the reduced variates e Ua have the trend of Figurc C. 1 Heres U ! is the cuxnulativefrequency of the observed intensity in Table C. j.~~

The ted of Figure C.1 is for the raw observationaldata uncorrected for the consequenceof interxnittent samplingduration. Since the reported dataare one-minuteaverages taken every hour, theywill be xnodifiedand TableC.1 Percentagefrequency of meanwind intensityat Honoluluairport.

~~Fr uenc of occurrence of wind intensi Speed interval probabilitydensity cumulativeprobability~~

0 tcalm] 0 - 0. 5! 4. 7 7 1-3 . 5 3. 5! 8. 3 13,0 4-6 .5 6. 5! 16. 4 29. 4 7-10 . 5 10, 5! 30. 1 59. 5 11-16 0. 5 - 16.5! 29. 3 88. 8 17-21 6. 5 - 21.5! 9.4 98. 2 22-27 {21. 5 - 27, 5! 1. 5 99. 7 27-33 {2'7. 5 - 33. 5! 0. 1 99. 8 upgradedtothe probable value they would have if thesampling had been continuous.It is, however,preferable to workwith the raw data and applythe modificationto the regressionline to be derived. 40

~~jo 30~~

-2 0 2 4 6 8 Reducedvariate y' Fig.C.1 Input data snd regression linefor mean wind intensity The table lists ninevalues of windintensity, and although these are sufficient to establish the long-term trend in extreznevalues, the znetbod is no longer so simple as when an abundanceof pointsis availaMe. appearspreferable, therefore, to manufacturethis abundance by a scheme which is, indeed, heuristic but which is accuratein somemean sense. ln essence,it consistsin assumingthat a valueof meanwind intensity xs in handfor eachper mille pointof the cumulativeprobability; there are, therefore, N = 998 meanwind intensities. Thesefall ona straight line connecting the values corresponding to the nine known wind intensities locatedalong the axis of the reducedvariate y at theposition set by their cuxnulativeprobability. This line is shownin dashesin Figure C. l. Note that inasznuch as in the last speed interval 27. 5 to 33. 5 knots there is only a single point, this is arbitrarily placed at the mean ordinateUa = 30.5 knots!.The effectof this arbitrarinessis negHgiblysznall.

~~The solution by the technique of Appendix B yields the foHowing long-term trend in the maximum values of mean wind intensity, namely, * U = U ~U - Ua~ exp -y/k!~~

~~where: Ua asymptoticvalue of the meanwind Ua characteristicvalue of the zaeanwind~~

~~k farm factor~~

~~The data of Table C. l yield the following regression line for the raw data, h Ua = 38- 30.65' exp -y/6,25!~~

~~Ua = 7.35!. This line is plottedas solidin FigureC. 1, as canbe seen, the fit is, with the single exc~tion of the extreme point, quite good.~~

~~Two steps zemain: that of adjustingthe plot to read in returnperiods measured in years and that of aggradationof the windintensities to take into account the shortness in sampling.~~

~~The adjustmentin return period is made as follows. The return period is linked to the reducedvariate and to the cumulative probability through the relations~~

~~T = ~ y!- =cm -~~ -k a!!!- r 2 2~~

~~47 For theextreme point, xn = 998,one has 4 =998/999 0. 9990, y = 6.906, Tr = 998fntervals of observation.~~

~~Sincethe observations were made over a span of 29years, this intervalamounts to 29/99S= 0.0290 years. To expressthe returnperiod in years modifythe reducedvariate by the amount~~

~~ay = Q -@ 9/30!! + Q -4 998/999!!~~

~~Thus, the new reduced variate becomes~~

z ~ y+ 4y =y-3.522 TheRayleigh factor based on sampling interval with s =60 and d =1 xnln. is fr s/d! = l. 43and the aggradation factor for themean wind results as f = 1.34. m Theresulting trend in thexnaximum value of meanwind intensity is given by the relation h U = 50.9 - 41.1 ' exp -z/6.25! Uponsubstituting return period for z T = exp z!+ 1/2!,one has A 0 = 50.9 - 23.56fT + I] l Thistrend is plottedin Figure2. 1 ofthe text.

3. Peak Wind InTable A.2 arelisted the maximuxn values inpeak wind intensity obtainedduring the Bve years 1938, 1942, 1945, 1947, and 1953. More speci5cally,theyare yearly xxuudmum valuesof one-minute saxnplings taken every houx. Themethod ofreduction is the same as for the mean wind. The data areplotted inFigure C,2 agILinst thereduced variate. The regression Hne is expressedas

~~7 U = U [U - U j exp -y/k! Where: asyxnpto5cvalueofpeak wind characteristic value of peak wlad~~

~~The best fit to the pointsis givenby Uz = 71 - 28. 5 exp -y/4. 24! and is plottedsolid in the figure. In termsof returnperiod, U a=- 71 28.5G [T+ 2 1~~

~~Reduced variate y Ftg. C.2 Inputdata and regression line for peak ~~nd intensity~~

~~APPENDIX D~~

~~ON THE SEVERITY OF TROPICAL CYCLONES IN THE HAWAIIAN AREA~~

~~l. Introduction~~

A cycloneis chars.cterizedby two features: an advancingvortical wind andcloud! pattern when viewed from aboveand a dropin atxnospheric pressurewithin the core of the vortex. The characteristicradius of the vortex is relatively small when cyclonesare generatedin tropical latitudes; also, solong as a,tropical cycloneremains over the ocean, its speedof advance never exceeds a critical value. Tropical cyclones are further classable as warm core and cold core, the latter beingalso termed tropicalstorxns to differentiatethem from theformer which then retain the naxneof tropical cyclonesbut with a morerestrictive meaning. In the Hawaiian area, tropical storms are usually referred to as Konastorxns and this nomenclatureis followed herein. Althoughboth tropical cyclones andKona storms are generatedover the ocean,the first comeinto being at the peripheryof the zoneof doldrums,which normally extends some 6 degreesnorth and south of the equator,the seconddevelop somewhat farther north or south!but neverin excessof some35 degreesof latitude.

The dropin atmosphericpressure is that obtainingat the centerof the vortex, while the characteristicradius is that of the maxixnumwind. To be sure, the vortexis not necessarilycircular: whenthe cyclone entersan otherwisesteady windfield, there is a compoundingof velocities with consequentdistortion of the circularwind pattern. This is, indeed, what obtainsin the area of the HawaiianIslands where there is a steady occurrence of trade winds: the vortical radius xneasuredorthogonally to the course of the trade wind on a generally northern direction can be about three timesthe correspondingone measured in a generallysouthern direction.

The critical velocityof advanceor progressionof the tropicalcyclone is that of the groupvelocity of the generatedwaves. Whenthe velocityof advanceexceeds the critical, the generatedwaves result comparatively smaller thanthose generated by thecyclone moving at or belowthe critical velocity. Theprediction of theintensity of the cyclonicwinds follows in part a procedureby Graham and Nunn~ which is basedon two parameters being in hand:the drop in pressureat the centerof the cyclone, ap in. of mercury!,and the radiuscorresponding to the maximumwind intensity, R naut. mi. !. Theseparameters must be separatelypredicted. Givena sufficiencyofmeasurements ofcentral pressure drop, predictionofthe long-term trend can be made by the technique of Gumbel discussedin Appendix 3, suitablytnodified totake into account the condition thatthe observations ofcentral pressure drop are not made at uniformly spacedtime intervals, 3ut the data in handare far frombeing statistically sufficient. Indeed,one has reliable data for only two tropical cyclones occurring in theHawaiian area: theseare NINA and DOT. Theformer lasted from 1 to3 Decemberl957 while the latter lasted from 5 to6 August1959. Pertinentdata on these tropical cyclones are reported inTable D. 1; theircourses are plottedin Figures2. 3 andD. I.

~~TableD. 1 Cyclonicparameters~~

~~Centralpressure drop Radius to maximum En.ergy index Cyclone name ap inchesof mercury! wind R naut. rni. !~~

~~NINA 1.0 20 20 DOT 1.0 1.8 0.9 20 18 Comparisons:~~

~~Labor Day, 1935 hurricane S. 55~~

~~4 ~ 01~~

Farthe purpose ofdesign, it is the long-term prediction ofthe extreme valuesofthese parameters that is required. H.K, Graham andD. E. Nunn 959!, "Meteorological Considerations pertinenttoStandard project Hurricane: Atlantic and Gulf Coasts ofthe UnitedStates." National Hurricane Research Project Report No.33, U. S. %cather Bureau.

~~52 22 8~~

~~21'N~~

~~20 !4~~

~~1IPN~~

~~FigureD, I -- Tracksof the tropicalcyclones NINA and DOT.~~

The necessarycondition for the reliableprediction of anysct of environmentalparameters is that therebe availablean adequate number of pertinentmeasurements. But in thecontext of the intended application of the method,there is such a paucityof observationaldata that at best only a.Qrst orderprediction method can be justified, Theprocedure that follows is to be viewed in this light.

~~2. Lon - Term Prediction of C clonic Wind Parameters~~

The expectedintensity of cyclonesis predictableupon knowledge of the kong-termtrend in the radiu.sof themaximum wind R ! andin the central pressuredrop hp!. Theseparameters are notmutually independent bui are, in fact, statisticallycorrelated. Their product R ' ~p!, ~own a-

~~53 2.1 Central Pressure Dxo 4~~

It is obviousthat only two xneasurementsof central pressure drop are not statistically significant and eoxnearguments must be xnadeto interpret the data and allow for their extension. The data on GuUof Meaico hurricanes presented by Riedel indicate that the central px essure drop follows the trends of Figure D. 2.

The centra.l pressure drops of Hawaiian tropical cyclones are considerably sxnaller than those obtaining in Floridian hurricanes, and since only two values C are available, some 3 plausible arguments must be adduced to establish a relation between central pressure drop and return period.

The argument made is as follows: The trend with return period of Hawaiian cyclones is one of constant proportionality when compared with that of Floridian hurricanes. 10 10 103 10 The factor of proportionality is fixed by estixnating Iteturnperiod years! the return periods of Fig. D.2 Long-termtrend in centralpressure tropical cyclones NINA drop of Floridian and Hawaiian snd/or DOT. cyclones. ~ J,T. Riedel968!, "InterimReport - MeteorologicalCharacteristics of the ProbableMaxixnuxn Hurricane, Atlantic and Gulf Coastsof the United States",Weather Bureau, Environmental Sciences Services Adxninistration U,S.Dept. of ConunerceMemoranduxn HUR7-97 to Corpsof Engineers, ReferenceW211 7 May 1968. Bretschneider*has estimated on the basisof publishedinformation on past Hawaiiancyclones that NINA is to be regardedas a fifty-year tropical cyclone. However,the logic hasnot been made explicit in a publicationso that oneis left to regardthe statementas authoritativebut subjective opinion. If for want of greater objectivity,Bretschneider's value is acceptedthe resulting trend in central pressure drop vs returnperiod is that of Figure D. 2. The pressure drop in Floridian hurricanesused for reference in determining the pressure drop in Hawaiiancyclones is the geometric mean of that in Atlantic Coast and Gulf of Mexico hurricanes at the same return period.

~~2. 2 Radius of Maximum Wind R~~

Graham and Nunn have plotted the variation of the radius of maximum wind with central pressure drop for Atlantic and Gulf of Mexico hurricanes. In the former case, the variation is a weak function of latitude while in the latter case, it appears to be independentthereof; therefor, the values closest to the latitude of Hawaii are used. ! The trend is shownin Figure I!. 3. The empirical formulae of Table D. 3 obtain. They applyto expectedvalues: the dispersion about such values is quite large.

The following commentson the symbolismof the table are pertinent: In a!, the centralpressure drop is writtenas a functionof the mean radius of maximum wind; hence, the latter is the independentvariable. In b! the opposite obtains.

~~Hence,the relationslinking expected central prcssure drop and radius of maximum wind in the Hawaiian area are:~~

~~hp R ! = 1.47 l. 52 log R /10!~~

~~R hp! = 10 exp.23- 1.52 hp!~~

~~See Figu.re D.4.~~

~~C. L. Bretschneider, verbal communication. Op. cit., Section l.~~

~~55 0 0 10 20 30 40 50 Radius of xnaximuxnwind, R naut. xni. ! Fig. D. 3 Relation between central pressure drop snd and radius of maxixnum wind~~

~~Table D. 3 -- Mean values and dispersion of radius of xnaxixnuxnwind.~~

~~a. Central pressure drop~~

~~hp R ! = A - B log R /10!~~

~~b, Mean radigxs of maximum wind~~

~~R hp! = 10' exp~~

~~The constants depend on the geograpjxical location and are:~~

~~Atlantic coast 23'N Gulf of Mestco Averageof two Hawaii A B 2. 82 2. 82 2. 82 l. 47 2. 76 3. 15 2. 94 1. 52 C 2. 36 2.24 2.20 2. 23 0. 83 0. 79 0. 78 l. 52~~

56 80 In Figure D.2, the expected central pressure drop in Hawaiian tropical I 60 cyclones is plotted as a 6 function of return period, i e, hP ~ hp Tr!. When this relation is '9 40 introduced in the last given formula, one obtains the dependence upon return period of 8 120 the mean radius to maximum wind, i. e., 8 ~R hp T!! 00 = Rm Tr!. The resulting plot is shown 0 in Figure D. 5. Central pressure drop, hp in. Hg! Fig. D.4 Radiusof maximuxnwind vs. central pressure drop for the Hawaiian area

~~~a 20 O 4 0 10 102 Return period years! Fig. D.5 Expectedradius of maximumwind v return pe riod~~

~~57 2. 3 Maximum Sustained Wind Intensi~~

~~The average value of the maximum sustainedwind intensity is from Graham and Nunn»~~

~~Ua R! = Ky4p - 0.5f R+0'5Ut knots! where;~~

~~U R! = maximum sustained wind intensity knots! at the a distance R naut. mi.! from the center of the cyclone. Note that the foregoing formula is valid for R ~ R~~

~~K: a coefficientwhich is a very weak function of latitude q!. For +~ 25 deg, K = 67.~~

~~f = Coriolis parameter, f = 0.525 sin e! rad hr -1 for Hawaii 4 = 21 deg, hencef = 0.19~~

~~Ut = translationalspeed of cyclone knots!~~

Thespeed of translationof the cycloneis a functionof the latitude. NumerousFloridian observationsxnade at latitudes close to that of Hawaii give 4 < Ut < 16 knots! withthe mean value Ut! = 10knots, Thatsuch figures are transferable tothe Hawaiian area may be brought out by observing that cyclone NINA hadan average speed of tx anslation of 11 knots while that of cycloneDOT was B.

~~UponintrodLxcing mean values corresponding to Hawaii, one has~~

U R! = 67~ap G.095 R+ 5 knots If therelation between central pressure drop and radius to xnaximum wind ofFigure D. 4 isentered into this relation, the trend that results is that of Figures D.6 sndD.7.

~~Grahamand Nunn, op. cit., Section1~~

~~58 100~~

~~80 C 'g m ~ ~so R8& 6 40 10 20 30 40 p Mean radius of maximum wind, 8 naut. mi. ! Fig. D.6 Maximum sustainedwind intensity vs. mean radius of maximum wind for the Hawaiian area~~

~~Inasmuch as the two~~

~~100 variables upon which the maximum sustained wind intensity, namely central pressure drop and radius~~

~~SO of maximum wind, are known in terms of return period, this wind intensity can be similarly expressed. The re suiting p 60 plot is that of Figure D.8.~~

The maximum sustained wind speed is a mean wind intensity averagedover an 0.4 0. 8 1. 2 1.6 2. 0 interval of some ten Central pressure drop, hp in. Iig! Ddnutes duration. This implies that there will Fig. D. 7 Maximum sustained wind intensity be winds of greater vs. central pressure drop for the Hawaiian area,.

59 100 intensity for intervals of briefer I5 duration one-minute C d! peaks and 80 instantaneous gus ta!. U g 0 The factors by w'hichpeak wind and gust intensities are derived froxn the maximum sustained cyclonic wind intensitydo not 10 102 108 10 appear to have been set down and once Return period years! again one is left to Fig. D. 8 Maximum sustained wind intensity vs the exercise of return period for the Hawaiian area judgxnent. If for want of superior argument these factors are taken to be identical to those used for the quasi-steady winds, the results are those of Figure 2.4 of the text. Of course, the trend shown therein is for guidance.

~~2.4~~

~~The energyindex is dered as the productR '~~

Bretschneider«presentsdata on the expectationof occurrenceof Atlantichurricanes according to energyindex. In FigureD. 9 the trend givenby Bretschneideris reploffedagainst return period. A plausible trendfor Hawaiiancyclones obtained by direct proportionand satisfying thecondition R ~p = 20for Tr = 50years!, is suggestedin the same Qgure.

~~«C. L. Bretsehneider959!, "Hurricane Design WavePractices", Transactions, American Society of Civil Zngineers, 124:39-62,~~

~~60 120~~

~~40 g~~

~~0 10 1 10 10 10 Return period years! Fig. D.9 Cyclonicenergy index vs. returnperiod~~

2.5 E ectedExtreme!ntensi of C clonicWinds at a Secific Site Theforegoing isa predictionofmaximum expected intensities ofcyclonic windsinthe Hawaiian area. It nowremains topredict the long-term trend in cyclonicwind intensity atthe specific site of the floating city. This step is madenecessary because ofthe relatively highlocal variabQity ofthe wind field and,consequently, of the wave field. Theproblem suffers from vagueness. Onecan relate expected rnaxiznum windintensity toreturn period for cyclones appearing inthe Hawaiian area, butthe bounds ofthe area are themselves undefined. Oneis forced atthis pointinto making anarbitrary assumption: byHawaiian areais intended an expanseofthe globe centered atthe site and having a width of400 nautical miles,the measurement beingmade normally tothe trajectory ofthe cyclone. A cyclonesweeping through such an area is a Hawaiiancyclone. A cyclonehasa profile ofwind intens.ity which rises rapidly from calmat the center to a peak at the radius of maximum wind and then tapers offslowly toinfinity. See Figure D.10, Bretschneider's expression for the wind field is*

~~Op. cit., Section2.4~~

~~61 62 U~ R! f R R U~ R ! 2 U~ ~!~~

~~R R R U R ! 3~~

~~Sucha profile is plotted in Figure D. 11 for the casewhen the Coripljs parameterf = 0. 19, the valuepertaining to theHawaiian area, andthe Ua R ! relationshipof FigureD.6 obtains.~~

To eachpoint on the curve of Figure D. 6 thereis associateda return period,since both Ua Rm!and R are functionsthereof. The return periodis relatedto an expectationof occurrence.The curves of FigureD. 11

~~100~~

~~60 C~~

~~40 '020~~

~~100 150 200 Radius, R naut. mi.! Fig. D.11 profileof cyc]onic wind intensity in theHawaiia»r fromcenter for parametricvalues of the returr!pe~~~

63 2G are replotted in Figure D. 12 for parametric values of the return period. cC C The curves extend to infinity but they are cut off at the point where the cyclonic wind intensity has 10 fallen off and attained the value of the steady ambient wind, assumed in this case to be ten knots. Rather than employ the foregoing Return period years! curves in the prediction process, it is Fig. D. 12 Equivalent radius of a Hawaiian tropical convenient to replace cyclone vs. return period them with their definite range equi valents. These are attained from 8 U g~~JR @Ua=10!+ J I U R!~ dR 0 R Ua=10! + 1 j U R! dR Ua Rm! J

Separationof the integral into twoterms reflecting range of validity is madefor the following reason; The wind intensity profile of Figure D. 11 is quasi-rotationallysymmetric abouta vertical axis through its center: it resemblesa volcano, whosecrater is containedin the region 0 ~ R < R whoserim is locatedat R and whoseoutward slopes fall off for R ~ R As the cyclonesweeps through an area, that portion of the area contained within the awe~ width 2 Rm, i. e., the area sweptby the crater, will experiencethe maximumwind intensity Ua R ! whilebeyond this region, the maximumwind intensity is that associatedwith the position radius R. Consequently, the first integral is simply 8 T>e results of the integration carried out by xnechanicalquadrature} for the relations linking U H ! to R and Tr of Figures D,6and D.S and for those connectingU R! to R of Figure D. 11 are presentedin Figure D.12.

Inasmuchas the sweepwidth 2Re of a Hawaiiancyclone is less thLn the transverseextent of whathas been defined as the Hawaiianarea, namely, four hundrednautical miles, the returnperiods heretoforequoted and based on the expectationof a tropical cycloneoccurring in this area xnust be modified by the factox'

Transverseextent of area 400naut. mi. 200/R Cyclonic sweepwidth R ! see Figure 2.4 of the text. The results of this operationyield the expected maximum wind intensity at any site within the Hawaiian area.

The soxnewbat arbitrary choice of a 400 nautical mile width is defensibleonly on the groundof plausibility: any other more acceptable value can be introduced, but note that the evenstrong departures from the 400 nautical miles value wQl have but a minor impact on xnodifyingthe return period.

~~APPENDIX E~~

~~THE EKMAN DKFT~~

~~The fully developed Kkman drift or wind-genera.ted oceanic current at the depth z below the surface has the intensity~~

~~Ue z!e =U ! exp - bz!- exp -i[bz+ /4]! where: U != Ekman drift at the surface == mixing parameter~~

~~The surface drift is proportional to the windintensity Us ci= 0.0120~csc 0 1' U~~

~~where 0. 0126 is Thorade's empirical value and e is the latitude. For the intendedCity Site, 42= 21.25 deg, henCe~~

~~U ! = 0. 021 UR~~

~~The mixingparameter is obtainedfrom~~

~~~sepsis0 1 0 248 = I Vs61 Ua Ua~~

~~whereU is in m/sec . The depthof the windlayer amountsto~~

~~3X zU zd 4~~

~~A parametricrepresentation ofthe Ekman drift vs depth is givenin Figure E. 1.~~

~~67 ct>on 0 80~~

~~Figure E. 1 Ekxnan d.rift vs d epth,th parametric represeutsti~. APPENDIX F~~

~~yEIQFICATION OF THE RESULTANT CUItRKNT~~

To gain confidencein the prediction of current strength,it is desirable to verify whether the three componentsof current can be separately deterxninedand assembledto yield a measuredstrength distribution. This is carried out for the site off Diamond Head. Of the three components, the tidal current is predictable deterministically and the expected strength and direction of the wind drift is sixnilarly derivable from the xneanwind intcnsitv and direction. However, the large scale oceanic flow can only be crudely approximated.

In principle, sucha verificationshould be madeby hindcastingthe continuoushistory of the current, or at least of the tidal andwind-drift components,but a first approximationcan be based on mean values,

The periodof observation0 to 25 May, 196'|!barely misses the occurrence of a new moon;therefore, a springtide. From Figure 3, 5 the xnsxixnumtidal current experiencedaxnounts to the sumof the semi- diurnal and diu.mal components,with insignificanterror, henceto 59 + 9 = 68 cxn sec 1. The direction is 290deg from the north.

The maximum value of the mean windintensity attained during the observationalperiod is 27 knots,i. e., 14m sec1, whenthe wind direction is from the NE, ZNK, andE sectors. Thefrequencies of occurrence correspondingto such wind vectors are respectively0.4, 0.5, and0. 2 percent. The resultingxnaximum strength of the surfacewind drIft is 400!.021!= 29cm sec 1, At therecording depth of 20 m, thisstrength dixninishesto 29 exp [-20/!4!! = 15cm sec 1; the rotation f rom the winddirection s.t this depthis approximately + s = + 20 =1 15rad=66deg 4 4 UR 4 44! andis clockwise.This gives wind drift, directions of291, 314, and 336 deg fromthe north when the wind bearings a.re from the NE, KNE, and E respect.vely.The first of these directions coincides almost exactly with that of the flood b.de. 9 ~ay at 14 hrs 56 m 20 hr, 23 min ephemeral tixne.

69 Such a strength yields a resultant of 108 cm scc directed 286 deg from the north. The magnitudeof the resultant current is to be compared withthe Inaximummeasured value of 115 cm scc 1 Figure 3.G!. The error is 6 percent,

~~70 APPENDIX G~~

~~THE ENERGY SPECTRUM OF THE SEAWAY~~

The description of the seawayover the short term is providedby its energy spectrurrr, if tMs is known, so a.re certain statistical averagesof the unduiatipnspf the sea surface, suchas the mean andsignificant wave heightsand periods and the maximumvalues that these parameters attain over a given interval of time,

The sea spectrum S T,e ! definesthe partition of energyas s function of wave Period T. It is usuallyexpressed in terms of a scalar spectrum, and a wave spreading functionf T,e!, i.e. S T,s! =. $ T! f T s!

~~In recent years there has beena convergencetoward a scalar spectrum of the form~~

~~S T! =- AT ' exp -BT !~~

sepa.ratelyproposed by Bretschneider~and by Piersonand Moskowitz~~ andusually identified by their names, In theBretschneider spectrum the constantsare A = 4H /T andB = Ts whereHs andTs are thesignificant s wave height and period respectively+.

The foregoing expression appliesto a single-peakedspectrum descriptiveof a pure wind-seaor swell, butnot of a concurrentwind-sea andswel1. The spectrumof the latter is obtainableby simplesuperposition of the two single peakedspectra of wind-seaand swell.

C. L. Bretschneider 963!, "A One-DimensionalWave Spectrum" in oceanWave H~ectra prentice-Hallpebl., EngletvccccliHa, H.J. ~*%.J. Piersonand L. Moskowitz ]964! ~ "Aproposed Spectral Form for Fully-DevelopedWind SeasBased on the Similarity Concept of KitaigorodskiÃ',Journal of GeophysicalResearch, N. 29. +Theterm "significant'l rnearts in this context "the average of theone-third highest" It is the mathematicalinterpretation of "observed".

~~71 Severalspreading functions have been proposed, the mostrecent by St. Denis* which is~~

f T,e! = A cos t ! where: o s! ~ 0 ~ + < s! 9 = angle relative to dominantwave direction a s! = 2/z As As - I.O.795 s +0. 021] 1/2 s = O. 878T or, preferably, s = 0.878T 2 sin xy! wherey = [ 7 +~3. 82T 7 ] /2 T~l. 53 sec

Givenan observedwave height H and periodT, the scalar and directionalspectra ofa purewind sea or sweU are Ierived as follows: a! equatethe observed and wave height and period to thesignificant waveheight and period Hs andTs b! scalarspectrum S T!= 4Hs/T ~ exp -T/T 4 !

~~c! wavespreading function f T,s!~~

~~d! directionalspectrum S T, e! = S T! ' f T, e!~~

~~e! mean wave height H = 0. 625 Hs~~

~~f! meanperiod T = 0.707 Ts~~

g! introduce interval of interest T.1 h! Rayleigh factor f T ! = i! expectedmaximum wave height during period of interest H Ti! = fr T ! H *M. St. Denis973!, "SomeCautions on theEmployment ofthe Spectral TechniquetoDescribe the Waves of the Sca and the Response Thereto of OceanicSystems". OfishoreTechnology Conference, Paper 1819. APPENDIX H

~~DATA ON HAWAIIAN WIND WAVES AND SWEI.I.~~

~~1. The Homer Com ilation~~

~~The Homer compilationis presentedas a tabulationof expectedfrequency of occurrence of waves grouped in cells by significant height, period and sector of provenance.~~

~~In this report, the Homer data are presented follo~Mg a pattern suggested by Moberly and Chamberlain~ according to which, wind wa.vesand swell in the Hawaiian area can be grouped into;~~

~~Northeast trade waves North Pacific swell Kona storm waves Southern swell~~

~~See in this connection Figure 4. 1 of the text which is based on an interpretation of the Homer data and is similar to but not quite congruent with the corresponding figure by these authors.~~

The Homer wave data are not separated according to this pattern; such a separationis madein this reportfor possiblegreater usefulness and case of interpretation. This step hasbeen accomplished by introducingsome arbitrary but plausible assumptions,namely;

a! The trade-windwaves and Kana storm waves are mutuallyexclusive; whenthe sumof the expectationof their occurrenceis subtracted from unity, the remainderis theexpectation of occurrenceof seas that are either calm or in which only swcD is present. However, the North Pacific swell and the Southernswell are not mutually exclusive;they may occur either separately or jointlyand they are independentof any windwaves that may be present. b! For thosesectors in whichboth wind waves and swell may appear, a separationbetween the two types is madeon the basis of period: wind waves trade-wind andKona storm! are less than 11 sec in periodwhild swell North Pacific and Southern! 1sof period greater than this value,

R. IVIoberly,Jr. andT. Chamberlain964!, "Hawaiian Beach Systems". ReportHIG-64-2, Hawaii Institute of Geophysics,University of Hawaii. Thefrequencies ofoccurrence quoted by Homer add up to a totality t ~ greaterthan unity, e.g.,

Trade-wind waves occur 75.3 Percent of the time Kona storm waves occur 10. 3 percent of the time North Pacific swell occurs 74.0 percent of the time Southern sweG occurs 53.0 percent of the time The~lanation given for this apparent anomaly is that wind waves and swell can run concurrently. TheHomer compilation ofprobability distributions ofsignificant height andsignificant period bydirection oforigin has been resolved soas to reflect thedistributions ofthe four wave groups: see Tables H. 1 throughH.4, A listingofthe averaged statistical characteristics foreach gmup and for thetotality ofgroups isgiven inTable H. 5. Soxnccomments maybe per@vent: a!The weighted meandirections oforigin, significant heights and periods,and frequency ofoccurrence result as follows: DirectionAverage Average Frequency of oforigin significantsignificant occurrence %ave group degtrue! height ft! period sec! percent!

8. 63 75. 3 NE trade- 78 4. 79 generatedwaves 74. 0 13. 89 Northpacific swell 320 4. 79 10. 3 6. 18 Konastorm waves 187 3. 52 53. 0 2. 60 13. 07 Southern swell 194 b!The wind waves occur85. 6 percentofthe time; only swell orcabn obtainsduring the remaining 14.4 percentofthe time, c!North pacific andSouthern swellare simultaneously presenta large fractionof the time.

~~74 O~~

~~4" Zl8~~

~~'N g~~

~~~ ~~~

~~ZlI~~

~~at ]~~

~~1 oO t 8 8~~

~~g j 76 0 e 4 8~~

~~4 A gg 9 lII 5 ~~~

~~0 g ~~~

~~P' CQ 0 O O~~

~~II 4" 4 8 9 8 4 «! CO 9 O O~~

~~g I I g 5 9~~

~~a e 78 Table H. 5 Average characteristics of waves in Hawaiian waters.~~

~~Group index 1 Sec~~

~~Ave Ave Fre~~

~~Sec Ave Ave Fre~~

~~Sect Ave Ave Fre~~

~~Sec Ave Ave Fre~~

Grandaverages; k* S.17 = 71.28' Hs 6.2Sft Tv 11.4S' sec d! Theweighted mean direction of all wavesis 4 4 !=Z p;ki/Zp, i=1 i=1 wherepi is thefrequency ofoccurrence ofwaves of group index i. The calculation gives Tc= 71. 28 deg. e! Thewind waves generated bythe Northeast trades snd the North Pacific swellhave the same average height but are of widelydiffering period!; theKona storm waves are of lesser height and have the shortest period, whilethe Southern swell has the shortest height and a periodalmost as longas that of the North Pacific swell. f! Theaverage significant wave height is obtainedfrom 4 2 1/2 H= ' i i=1

~~onthe assumption that it is thewave energy proportional to H ! that is additive. The result is~~

~~8 = 6.25 ft g! Inthe same way the average significant height of the wind waves i = 1,3! is H = 4.31 ft~~

~~whilethe averagesignificant height of the swell i = 2,4! is~~

~~H=4.53 ft h! Theaverage significant period of all thewaves is givenby 4 ~=g pi i=1~~

~~End amounts to~~

~~T = ll. 45 sec i! Theaverage significant period of thewind waves i = I, 3!is~~

~~and amounts to 8. 33 sec. j! Theaverage significant period of swell i = 2,4!is similarly calculated and results as~~

T = 13. 55 sec k! Suchan average swell period corresponds toa wave940 ft in length. Thereis noquestion as to whetherswell of sucha lengthcan be detectedvisually; indeed, the opinionis expressedthat it cannot. Thispoint should be borne in mindwhen comparing observational data on waves with hindcasts. lt is possibleto havean idea of the Iong-terxn trend in significant, waveheight by plotting the Homer data against the reduced variate y =- k -& o!!, seeAppendix B. Theresulting trends of the data are plottedin FigureH. 1 foreach of the four wave groups. Except for the trade-wind waves 2p at the lower values of y, the trend is linear, which implies an asymptotic value of infinity. ~ Since the Homer ~ data are for a ca lp typical year, the trend can be readily v extended to any L s return period. This trend is shown in Figure 4.2 of the text. 0 2 Reduced variate, y Fig- H. 1 Trend in sigai6cantwave height according to the Homer compilation.

81 Thegame procedure can be employed for the wave period; the trends of thedata are shown in FigureH. 2 andthe corresponding trends with return periodofthe xnaximuxn waveperiods are given in Figure 4. 3 ofthe text.

~~0 ~20~~

~~Reduced variate y Fig.H. 2 Trendin significantwave period according to the Homer coxnpilation.~~

~~2. Ho Sherretz Data on Hawaiian Waves~~

The observationson whichthe Ho andSherretz report is basedwere carriedout twice daily over a spanof sevenyears by U.S. CoastGuard personnel.The wave heights were estimated in feet, convertedinto a WeatherBureau code and then reconverted into feet andrecorded as taking onvalues, in thediscrete and somewhat irregular! sequence of 1. 5, 3, 5> 6.5, 8, 9, 11,13, 14, 16, 17.5,19, 21, 22.5, 24 feet, Theobserved periods were estimatedin secondswhile the true directionsof waveapproach were recordedby octants. Ho and Sherretz grouped the databy xnonthand deterxninedthe monthly!mean observed heights, the standarddeviations andthe means of the maxixnum observed heights. The results are presented graphicaiiyin Figures H. 3 andH. 4 forthe Makapuu paint and Kilauea Point stations.Figure H. 5 givesthe xnean value of thewave direction for both station.

Notethat thereis a substantialdifference in the wavestatistics at thetwo stations not in themesa values of wave height, and dispersion= theseare aU quite close but in the values of xnaximum wave heights and in the direction of arrivaL 120F J

~~M A M J J A S 0~~

~~Fig. H.3 "All-time" mmdmum,mean maximum, and mean oceanwave heights in feet crest to trough! by month for Makapuu Point, oahu, 20~~

~~J F N A M J J A S 0 Ã~~

~~Fig. H.4 "All-time" maximuro.,mean maximum, and znean oceanwave heights in feet crest to trough! for Kilauea Point, Kauai.~~

~~84 000~~

~~020~~

~~040~~

~~060~~

~~080~~

~~100~~

~~l20~~

~~140~~

~~l60 J F M A M J J A Fig.H. 5 Meanmonthly ocean vrave direction in degrees true! for KilaueaPoint, Kauai and Nakapuu Point, Oahu.~~

~~85 The analysisthat followsis basedon the observationsmade from MakspuuPoint, inasmuchs.s these are of greater relevancy to the project.~~

~~The mean value of the mean of all the observedwave heights H is M 5110 H M!= g H = g H =4.69ft M 1 5110 =1 where N = 5110 is the total number of observations M 2 x 365 x 7!.~~

Giventhe meanvalue, the expectedvalue of the meanof the monthly maxima is 12 h «H» = 12Q f f r iN !' H. i Ni j]r = H' fr N! i=1 whereNi representsthe number of observations made during a generic monthi, N is the averagevalue of N., whilef N ! and f N! are the factors i Ni! ~f'm Ãi!~2i N! =~X H!/2 sinceN =2 x 365x 7/12= 426, fr H! = 1,74 andthe expectedmean of the maxima is

~~A «H» = .69!..74! '= 8.16 ft~~

This valueis to be comparedwith that derivedfrom I'igure H.3 whichis computedto be 7. 78 or 5 percent lower. Thesame procedure is followedin obtainingthe expectedvalue of the monthlymaxima «H» . All oneneeds to dois replaceN withthe total numberof observationsM. TheRayleigh factor fr M! = 2. 07and the expectedmaximum wave height results as

~~«H» = . 69! . 07! = 9. 71 ft~~

Th}.svalue is to be comparedwith derived from Figure H. 3 which is calculated to be 9. 90 or 2 percent higher, Such limited coxnparisons suggest that prediction of expected values through application of the Hayleighfactor can be xnadewith a fair mcasur, of confidence.

One further prediction rexnains: that of the maximum value of wave height over the interval of observation. For a narrow energy spectrum of the seaway thc maximum wave height H is related to the mean wave height H throughthe Hayleighdistribution h 4 H! = 1 - exp -H/H!

~~where e ! is the cumulative probability. By order statistics,~~

~~IL q H! = 1 = = 0.9998 M+1 5111~~

~~This valueyields Hh =13.71 ff, a valuethat is to becompared with the maxixnum observed value of H = 14 ft. Note, however, that the obscrvcd values are reported in integer values. !~~

3. Co arison of Observed Wave Data A predictionof thewave conditions at theprospective site of the Qoatingcommunity is facilita.tedby a comparisonof the Homer with the Ho and Sherretz data. The pertinent valuesare containedin Table H. 6.

~~Table H.6 Comparisonof Homerand of Hoand Shex retz waveheights~~

~~Homer Ho 5 Sherretz~~

I'rade-wind waves 4.79 ft Kong storm waves 3 ~52 wind waves 4. 31 North Pacific swell 4. 79 Southern swell 2. 60 swell 4. 53 4 69 ft >Iakapuu All waves 4. 64 Kilauca Theaverage Ho and Sherretz wave height is closeto the average heightof the Homer wind waves. Such a quasi-coincidenceis explainab]e onlyif it is assumedthat: a!the observers were unable to identify swell whenit underlaythe locally generated wind waves, and b!the Homer swe]1characteristics werc derivedin part from hindcasts. The Homer reportis notsufficiently explicit on this point!.

Ho andSherretz suggest that swell from distantstorms is not fully felt at Makapuubecause of its moresheltered location in comparisonwith thatof Kilauea. Suchan argumentreinforces the assumptionthat swellis not observablewhen it underlieswind sea, for the averagewave height at Makapuuis practicallyidentical to thatat Kilaueawhere the relatively highNorth Pacific swell occurs 74 percent of thetime.

The followingcourse which is reasonablyjustifiable onlyso longas oneworks with overaHaverages, is to combineby superpositionthe scalarspectrum of the windwaves and that of the swell. Thc Homer characteristics will be used. Thus, S~ T! 44~ 31 T exp -T/!8.33! ! 8. 33!

~~44.53~ . T exp -T 4 /34 55! ! 3. 55~~

~~= 0.01543T3 exp -T /4815!+ 0. 002435T ~ cxp -T /33,700!~~

~~Theplot is shownin Figure4.2 of thetext. As previouslystated, the dominantdirec5on of the seawayis about 71 deg true.~~

~~88 APPENDIX I~~

~~CYCLONE-GENERATED WAVES~~

~~1. Introduction~~

If the cyclonic windfield is known, it is possibleto derive the characteristics of the waves to which it gives rise. A method by Bretscbneider*is availablefor thispurpose. In thisappendix such a method is applied to the determination of the long-term characteristics of Hawaiian cyclonic waves,

~~2. Procedure~~

~~The inputs to a determination of significant wave height and period are those resulting from the processoutlined in AppendixD, namely R, U R !, Rm' ap.~~

~~These variables are all functions of the return period T and this functionality is carried throughin the solution.~~

~~2. 1 Maximum Value of Si cant Wave Hei~~

The maximum value of the significantwave height obtains at the radius of maximum wind and is given by H R !=K f, ~,p!'~R' lip In thefactor K f, R, 9 !, f is theCoriolis factor and 0 the wind incurvatureangle, which varies with the normalized radius R/R . Based onvalues recommended byGraham and Nunn, thisvariation can be expressed by 0 = 25- 15 exp -1.10 R/R ! deg! TheBretschneider analysis based on a =25 deg and the values of K f, R, 0 !that result are given by the empirical relation 0. 544 K f,R, 25'!= 7.5 exp -1.4lifR /Ua R !] } C.L. Bretschneider972!, "A Non-DimensionalStationary Hurricane cavo Model,"Offshore Technology Conference, paper 1517 **Op. cit., AppendixD, Sectionl.

89 Thesignificant wave height at the radiusof maximumwind obtaining in the Hawaiian area is plotted in Figure I. 1. Thesignificant wave height at the genericradius R is relatedto thatat the radiusof maximumwind Rm throughthe empirical formula

~~H R ! la where R ~ R , /so The relation is valid for a stationary cyclone. 2~~

~~~ 6~~

~~10 20 30 40 ~ H hg Mean radius of maximum wind, R K naut. mi. !~~

~~Fig. i. 1 Significant wave height at the radius of maximuxn wind for the Hawaiian area~~

~~2. 2 Si icant Wave Period~~

~~The significantwave period of a stationarycyclone is relatedto the significantwave height through the empiricallink of Bretschneider ~~~= O.s~i.ov ~re~ "~~I!! Ua Rm! K u~~~

~~90 2. 3 Modificationto the Si icantWave Hei ht and Periodfor a~~

~~The speedof translationof tropicalcyclones does not exceed the critical speed. The latter is givenby Ucr = 1. 515Tcr R ! knots!~~

~~-1 1.515 a T ~ sec! U Itm!~~

~~where: T R ! = significantwave period at theradius of maximum wind whenthe cyclone movesat the critical speed.~~

~~empiricalfactor whose suggested value is 0. 5~~

~~If the cyclonehas a speedof translationUt ~ Ucr, thesignificant waveheight and period are respectivelyincreased to~~

~~R R,V,! = R R! ! + Ut 2 'Ua~ M~~

~~T R, Ut!= T R! + t ~'v ~a~~

~~Qf course H R ! = H R, 0! andsimilarly for T R ! ~ If thespeed of translation can be taken as 10 knots, the trends in significantwave height and period become those of Figure I.2.~~

~~91 30~~

~~@ O~~

~~10 2 10~ 10 Return period years! Fig. L2 Long-termtrend in significantwave height andperiod of wavesgenereated hy tropical cyclonesin thc klawaiianarea. APPENDIX J~~

~~ON COMBINING LONG-TERN TRENDS~~

~~l. Introduction~~

Thc long-term trendsin the expectedvalues of maximumsignificant waveheight and period are determinedin AppendixH for eachof the four typesof Hawaiianwaves; the correspondinglong-term trends of cyclonic waves are determinedin AppendixI; it now remains to combineth~ m into long-termtrends of the totality. The simpletechnique is described in thi s appendix.

~~2. Method~~

~~Consider a set of P long-termtrends versus returnperiod in the maximumvalue of a variate x suchas the significantwave height or period.!. Denotethis set by~~

x T! p r whereT is the returnperiod andp is the indexof the set e.g., p-l denotestrade-wind waves; p=2, NorthPacific swell; p=3, Konastorm waves;p=4, Southern swell; p=5, cyclonic waves; hence P=G !. The problemis to derivethe return period of the variate corresponding toth whole set. Thesolution is as follows;Denote a parametricvalue of the set xp Tr!byx . Eachvariatc attains such avalue at omcreturn period denotedbyTr p,i!. Thereturn period ofthe value xi corresponding to the whole set is simply T i!= /QP /Tz P, !! p=l Applicationofsuch an expression leads to the trends of maximum significant height andperiod of Figure4. 6 of the text.