NPS ARCHIVE 1969 CHACE, A. OCEAN SURFACE WAVES: ATTENUATION AND A FIELD TEST OF DSA II by Alden Buffington Chace, Jr. O DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY, CA 93943-5101 United States Naval Postgraduate School THESIS OCEAN SURFACE WAVES ATTENUATION AND A FIELD TEST OF DSA II " by Alden Buf f ington Chace , Jr. 7 13 803 October 1969 Tku document ka& bzen appsiovcd ^on pubLLc kz.- Izcat and &atz; itt> dUVUbutlon Is unJUmitzd. US; ":l dnfSnia 93940 Monterey, iaui Ocean Surface Waves : Attenuation and a Field Test of DSA II by Alden Buffington Chace , Jr. Lieutenant, United States Navy B.S., United States Naval Academy, 1962 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL October 1969 ih^cx.^ A, ABSTRACT Several models of ocean surface wave attenuation were employed in making hindcasts for comparison with spectra obtained from a wave recorder located at Point Sur , California. Quantitative decay models were reduced to common variables. The French Meteorological Service DSA II generation model was used. The attenuation model which best fits the data obtained is n - exp (-20 T~2 t), where n is a ratio of energy at the start of decay to that remaining afterwards, T is wave period in seconds, and t is decay duration in hours. The period range considered was 8.5 to 23.5 seconds. U s £f i x, DUDLEY KNOX LIBRARY Monterey rfi°f^radUat6 SchdQl NAVAL P°STGRADUATE SCHOOL Monterey, California 93940 MONTEREY, C A 93943-5101 TABLE OF CONTENTS I. INTRODUCTION 11 II. ATTENUATION THEORIES AND MODELS 13 A. PROPOSED MECHANISMS OF DECAY 13 1. Molecular Viscosity 13 2. Wave Breaking 13 3. External Turbulence 13 4. Air Resistance 14 5. Parasitic Capillaries 14 6. Wave-Wave Interaction 15 7. Scattering Due to Turbulence 15 B. EXPERIMENTS 16 1. San Clemente Study 16 2. New Zealand to Alaska Attenuation Study 17 3. Small Amplitude Deep Water Wave Study 17 4. Spectro-Angular Attenuation 18 5. Eddy Viscosity Application 18 6. Laboratory Underwater Turbulence Study 18 III. DATA AND ANALYSIS 20 A. HINDCAST PREPARATION 20 1. Wind Field Plot 21 2. Propagation Diagrams 21 3. Energy Computation 23 B. ANALYSIS OF WAVE RECORDS 28 IV. GENERATION 29 V. SUMMARY OF RESULTS 30 VI. CONCLUSIONS .......... 36 APPENDIX A: EQUATION OF ENERGY REPRESENTATIONS 37 APPENDIX B: TABUIAR DISPIAY OF SPECTRAL COMPONENTS 38 APPENDIX C: ATTENUATION MODEL TABLES 44 REFERENCES CITED 49 INITIAL DISTRIBUTION LIST ...... 51 FORM DD 1473 53 LIST OF ILLUSTRATIONS Figure Title Page 1. Sample Wind-Field Plot 22 2. Sample Propagation Diagram 24 3. Propagation Line Overlay 25 4. Observed Energy vs Hindcast Energy Using Unmodified DSA II Equation Decay 31 5. Observed Energy vs Hindcast Energy Using DSA II Decay Equation with l/10th Actual Duration 32 6. Observed Energy vs Hindcast Energy Using DSA II Decay Equation with l/100th Actual Duration 33 7. Observed Energy vs Hindcast Energy Using Carswell Decay 35 TABLE OF SYMBOLS AND ABBREVIATIONS Symbol Definition Unit DSA Spectro-angular density. X Incremental distance of generation or decay naut . mile L Deep water wave length. ft L Wave length at sensor position. ft s T Wave period. sec t Time required for wave energy of a given hr period to travel a distance X. The duration interval for generation or decay. g Acceleration of gravity. r sec W Wind speed (nm/hr) . knots 0 Angle between wind direction and direction degrees of energy propagation under consideration, 0° < 0 < 180°; 0 = 180° is an adverse wind. f Wave frequency. hertz p Density of sea water (2.0). s lugs 3 ft' 2 E Total wave energy for band indicated. ft 2 E(f) Wave energy as a function of frequency. ft 2 E(0) Initial wave energy present in a band of ft periods at the start of an increment of generation or decay. 2 E(X) Wave energy in the same band of periods ft as E(0) after propagating a distance X in time t. tw~n T . . -. , /t, . \ centijoules R(0) Initial energy density (French) pro- "^ portional to (1/2) p g E(0). See cm Appendix A. R(X) Final energy density (French) at end of centijoules distance X. Corresponds to E(X). 2 cm E(X) R(X) n Attenuation function = „)'. = p/rtN • 1/6 td Groen [1954] and [1950] attenuation meter coe f f icient . -5/3 K Paquin constant (1.26). sec d Depth of pressure sensor (65). ft dl Refraction coefficient. The ratio of orthogonal separation in deep water to that at the sensor location. x Incremental distance of generation or meters decay. ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to advisors Associate Professor J. B. Wickham, who originally suggested the topic, and Dr. T. Green III, both of whom provided immeasurable assistance of all types. I am indebted to Miss Diane Budde for color coding the wave propagation diagrams. My particular thanks are due to Dr. R. Gelci of the French Meteorological Service for several letters and publications on the subject of DSA II. I. INTRODUCTION Neither the magnitudes nor the mechanisms of ocean wave attenu- ation have been well determined for ocean surface waves with periods of 8.5 to 23.5 seconds. The standard wave forecasting method in the United States, for nearly 15 years, has been the Pierson, Neumann, and James [1955] method. The 'PNJ1 technique allows for angular spreading but fails to consider mechanisms of attenuation. Wave attenuation studies have been conducted using various methods of attack. Theoretical and experimental approaches lead to a variety of proposed mechanisms for the damping of wave energy. Wave forecasters have devised empirical methods of estimating loss of wave energy during propagation from a specified fetch area to an observation point. The French DSA series of forecasting methods attempted to improve on the limited fetch methods of 'PNJ' by considering wave energy approaching the forecast point from several directions. The DSA method number two (DSA II) [Gelci, Cazale, and Vassal 1957] employs an attenuation model which allows for decay as a function of wave period, decay distance, and cross-wind condition. This method was developed in the North Atlantic Ocean. Field use in the Pacific Ocean provided an opportunity to evaluate DSA II under different geographical conditions. The objectives of this thesis were two-fold. First, to test DSA II in a region in the Northeast Pacific extending roughly 2000 miles from Point Sur, California and, second, to attempt to evaluate some 11 of the proposed attenuation models. The method employed was compari- son of hindcasts (made using DSA II) with analyzed wave records from Point Sur, California for January 1948. 12 II. ATTENUATION THEORIES AND MODELS When ocean waves propagate out of a region of active generation a period of attenuation commences. It is generally believed that longer waves decrease in height more slowly than their shorter counterparts. The following mechanisms and experiments provide insight into the processes which may be involved. A. PROPOSED MECHANISMS OF DECAY 1 . Molecular Viscosity Viscous dissipation has been suggested. However, it has been noted theoretically that viscous amplitude decay time for ocean waves varies as the square of wave length [Kinsman 1965]. Except for very minute ocean waves, such as parasitic capillaries, viscous decay is several orders of magnitude too small to explain observed decay. 2 . Wave Breaking Breaking of waves is a method of converting wave energy into turbulence. It is the most important mechanism in the generating area [Barnett 1966] and on the final beach but is rare elsewhere [Cartwright 1967]. Wave-wave interaction, as discussed below (II. A. 6.), may lead to breaking. 3 . External Turbulence Groen [1954] and Bowden [1950] ascribed wave damping primarily to externally caused turbulence. This turbulence is produced inde- pendently of the wave. An artificial "eddy vicsosity" may be used to deal with this mechanism as long as the size of turbulent eddies is small compared to the length of the waves [Barber and Ursell 1948]. 13 Bowden assumed monochromatic significant waves. Groen dealt with wave spectra to arrive at an energy decay expression of the following -7/6 form: n = exp - (axL ). Here n is the ratio of energy at the start of a region of decay to that present at the end of the region of length x (meters); L is the deep water wave length in meters; a is Groen's 1/6 attenuation coefficient in units of meter . The energies were in a band centered at wave length L. Groen [1954, p. 7] stated that turbulence is frequency selective and attenuates the shorter period waves more rapidly. Thus it is better to employ wave spectra rather than assume, as a representative system, only the significant waves present. 4. Air Resistance Darbyshire [1957] examined air resistance and concluded that it did not provide a dominant mechanism of decay since he found actual attenuation to be independent of wind speed for a following wind . Phillips [1966, p. 148] concluded that the condition of 9 greater than 90 results in a very weak coupling between wind and swell which leads to an exponential form of decay. Here, 9 is the angle between the wind direction and the direction of energy propa- gation under consideration. He suggested that the attenuation is proportional to COS 9. 5. Parasitic Capillaries Phillips [1966, p. 134] discussed capillary waves which tend to form as standing waves in advance of the sharp crests of short gravity waves due to local surface tension effects. The capillary waves are small enough to cause viscous attenuation to 14 be significant. This mechanism is important in the case of steep gravity waves of length 5 to 30 cm. Therefore it is significant in small lakes but probably not in the ocean where most energy is con- centrated in lower frequencies. 6 . Wave-Wave Interaction Barnett [1966] suggested that, once the wind has ceased its active generation, the spectrum looses energy mainly through wave- wave interactions. Gelci and Cazale [1962, p. 30] have developed a model (DSA V) to consider losses due to the interaction between wave trains. DSA II does not consider this effect. Yet the decay observed in DSA V is quite similar to that resulting in DSA II. Gelci (personal correspondence) reports that wave-wave interaction does not significantly affect waves with a period of 14 seconds or greater. The details of the wave-wave interaction mechanism are complex, but the effect is the transfer of energy from three active wave components to a fourth, passive component. The magnitude of the effect appears to be comparable to wave generation processes [Barnett 1966], For a typical fully developed spectrum, energy is transferred from a broad mid-frequency band to both higher and lower frequencies. The energy transferred to higher frequencies could be lost via breaking and viscosity. 7 . Scattering Due to Turbulence Phillips [1961] proposed that some ocean turbulence is generated by gravity waves. This is a type of turbulence not included in the externally caused turbulence (see II. A. 3.). Phillips [1959] developed a theory to explain attenuation and directional scattering 15 s of surface gravity waves which appears to result from interaction between wave turbulence and wave motion. His conclusion was that scattering would predominate over viscous attenuation for wave lengths greater than about 3 meters (periods of 1.4 sec). It is not certain that this would be a significant effect since viscous effect themselves are small for the 8.5 to 23.5 second periods covered by this thesis. Phillips' theory results in the following model for -5/3 which values are tabulated in Appendix C: n = exp -(0,0087 T t). B. EXPERIMENTS The first serious application of modern analysis techniques to ocean waves occurred in the early 1940's [Cartwright 1967]. It was discovered that random fluctuations in sea level from wave records can be resolved with spectral analysis into the sum of a number of quasi -{ • riodic oscillations of various frequencies or periods. Most of the wave measurements made during the next ten years were used to develop wave generation models and parameters for energy-density spectra . 1. San Clemente Study During the 1950's, Munk , Miller, Snodgrass , and Barber [1963] examined wave energy received at San Clemente, California from storm waves generated in distant regions of the North Pacific and in the Southern Ocean. They detected significant amounts of energy only -2 at frequencies below 5.0 x 10 hz (periods larger than 20 sec). Wave energy at higher frequencies was heavily attenuated somewhere along the great circle path [Cartwright 1967]. This result was used in justification of hindcasting waves on the basis of surface winds within 2000 nm of Point Sur for this thesis. 16 2 . New Zealand to Alaska Attenuation Study Munk and his colleagues set up one of the most ambitious series of measurements in the history of oceanography [Snodgrass et al. 1966]. During the summer of 1963 they tracked swell along a great circle path from New Zealand to Alaska using six observation _2 stations. At frequencies above 7.0 x 10 hz (period 14.3 sec) there was a tendency for spectral energy to decrease at a rate of 1/10 db per degree along the great circle path. This is equivalent to the equation : n = exp [-(0.011 sec"1 hr"1) T t]. -2 Comparative values are presented in Appendix C. Below 7.2 x 10 hz the energy either remained constant or increased. Cartwright [1967] indicated that the major result of this experiment was that attenu- ation above 5.0 x 10 hz (period 20 sec) in the first 20 or so from the generation ares was much greater than that observed over the rest of the path to Alaska. 3 . Small Amplitude Deep Water Wave Study Carswell [1968], of the Naval Postgraduate School, looked at very small waves (frequencies 2 to 2.4 hz) in a lake. Following a suggestion of Professor J. B. Wickham, Carswell used his data to N fit an f exponential decay function. An N of three gave the best r 3,n fit and resulted in the following: n = exp - (5 ) f X . Here, r L nm J f is frequency in hertz; X is decay distance in nautical miles. This expression was modified to the following equivalent form for evalu- ation in this thesis: r 2 ? i n = exp [-(7.575 -^-) 1~l tj. 17 [-<100 + 0.38416 (^) sin2 Q 9 -45°)|>sec2 hr" ' t T~' 4. Spectro-Angular Attenuation DSA II has its own attenuation function, which was developed for use in the North Atlantic and is applicable where the angle between the wind direction in the decay region and the direction of energy propagation (9) exceeds 60 or wind velocity (W) is not greater than ten knots. This function is: -<100 + 0.38416 (H When 9 does not exceed 60 and W is greater than ten knots DSA II assumes no decay. Gelci (personal correspondence) points out that this is nearly true for an isolated component and DSA II neglected "interferential decay" due to multiple wave trains. The case of 9 no greater than 60 and W in excess of 10 knots is taken care of by either maintaining steady-state or generation as applicable for the wave period under consideration. 5. Eddy Viscosity Application Groen and Dorrestein [1950] evaluated the attenuation coefficient (a), in the equation previously presented (II. A. 3.), to get an empirical value (8.5 x 10 meter ). Their equation has been modified to a more convenient form for this thesis: n = exp -2 4/3 -1 -4/3" (1.92 x 10 sec hr ) t T ' 6 . Laboratory Underwater Turbulence Study Paquin [1968] studied the effect of grid-generated underwater turbulence on small waves (1.2 to 12.3 hz) in a wave tank. His results showed that turbulent attenuation would have the following form: n = exp 1/3] — }X I , where K is a constant (1.26 sec ) X is a decay distance (in units compatible with g) ; g is gravitational acceleration; and f is frequency in hz . This was converted to vari- ables convenient to DSA II as shown below: n = exp (4.55 x 103 sec"2/3 hr"1) t T2/3] A table of values is not provided as this model is not applicable to the ocean. The values of n approach zero very rapidly. 19 III. DATA AND ANALYSIS Air Weather Service, Northern Hemisphere Historical Weather Maps were used to select a time interval with desirable conditions for this study during the period for which wave records were avail- able. The interval 5 January 1948 through 12 January 1948 was used because of a cyclonic disturbance located approximately 1500 nm northwest of Point Sur. This disturbance, which varied in strength, produced time varying wave energy. Microfilm reproductions of United States Weather Bureau, North American Surface Weather Charts were used as a source of wind-field data. Only actual ship wind reports were used and given a minor amount of smoothing. The result was a realistic wind field com- parable to that employed for DSA II by the French Meteorological Service . Wave records were obtained from a Mark III pressure wave sensor installed near the bottom at Point Sur, California by the Department of Engineering, University of California as part of an investigation of surface waves conducted for the Bureau of Ships , United States Navy. The sensor was located in sixty-five feet of water [Isaacs and Saville 1949] on a bearing of 240 true from Point Sur. The recorder operated at a tape speed of three inches per minute for 20 minutes out of each eight hour interval [Wiegel 1949], A. HINDCAST PREPARATION Unlike the ' PN J ' method DSA II requires no fetch definitions but instead considers the action of the space- and time-varying winds 20 on each wave component. Therefore results are much less subject to the interpretation of the hindcaster. The procedure for hindcast preparation is summarized by the following expression: *s = {l(-[H[^]}[S where E is the hindcasted energy in feet square for a given period band at the sensor position off Point Sur ; .E is a summation over the various azimuths of energy approach to Point Sur; G is the generation and decay function which, when integrated over duration, yields the 2 energy in centi j oules/cm arriving from a specified azimuth; 2 com- pensates for a 20 separation between azimuths instead of the 10 specified in DSA II; C is a unit conversion (see Appendix A) from 2 dl0 centijoules/cm to the feet squared energy representation; -77— is L0 the correction for refraction; and - — corrects for shoaling, s 1. Wind-Field Plot Ship wind reports were transferred from microfilm to a plot as shown in Figure (1), which illustrates typical data density for the six hourly wind-field. Due to the limited detail of the wind-field, azimuths from Point Sur were positioned every 20 true instead of every ten degrees as called for in DSA II. Azimuths were limited to the north by Point Reyes which bears 325 true from Point Sur. Bearing 320 true from Point Sur is a 300 ft shoal off Point Reyes which refracts waves whose period is greater than 11 seconds. 2 . Propagation Diagrams The propagation diagram is an extremely valuable means of establishing the wind history of waves arriving at an observation 21 Figure 1. Sample Wind Field Plot 1 145" 140 West Longitude Date : Time : 11 January If 48 0030Z Point Sur California 60l 55v 50l 45v 40* - 35 - 30l 25^ 20l 15 22 point along a particular azimuth. Figure (2) depicts a portion of one diagram. Actual diagrams were plotted continuously for the interval of 5 through 12 January 1948 to a range of 2000 nm from Point Sur. Each propagation diagram gives the wind conditions as a function of time and distance from the observation point along one azimuth in the wind field plot. For each increment of range (100 nm) and time (6 hrs) both 9 and W were plotted. Here, 0 is the angle between the wind direction and the direction of energy propagation along an azimuth and W is the wind velocity in knots. Isotachs were contoured every five knots of W. Wind directions were color coded into the following DSA II categories : Description Q or W Limits Generating 9 < 20° Favorable 20° < 9 < 60° Favorable-Fair 60° < 9 < 90° Contrary-Fair 90° < 9 < 120° Contrary 120° < 9 < 180° Calm W < 10 kts 3 . Energy Computation Wave record availability within each eight hour interval dictated hindcast times. DSA II is designed to consider waves whose central periods are 20, 14, 10, and 7 seconds respectively. The 7 second waves were not used as they were obscured by hydrodynamic filtering and noise on the wave records. A propagation line overlay was prepared on the basis of the group velocity of the three wave bands considered (see Figure 3). 23 Figure 2. Sample Propagation Diagram 500 400 Range (nm) 300 200 100 n 1 r V//////////////£//S 10 r KEY © Generating CD Favorable © Favorable/Fair © Contrary/Fair © Contrary O Calm 1230 00 o> 0630 £ 3 C CO 0030 1830 - 1230 CO > u % c CO - 0030 24 25 The advantages of the propagation diagram in combination with the overlay are evident. The corresponding ranges and times of each wave-energy packet are shown, and a hindcast can be computed for any given time on the diagram. Generation was presumed to occur when 9 < 60 and one of the following minimum W conditions was met or exceeded: T (sec) Generating 8 Favorable 9 20 40/45 kts 45 kts 14 30/35 kts 35 kts 10 25 kts 25/30 kts When minimum generating or favorable winds were present the DSA II generation diagram for the appropriate period was entered with 9, W, and t. Subsequent intervals with 9 < 60 either increased the energy density or left it unchanged depending upon W. When 9 was greater than 60 or W < 10 knots decay was pre- sumed to occur. One of the four attenuation models below was then employed for the appropriate conditions : Unmodified DSA II equation (see II. B. 4.). DSA II equation with 1/I0th of actual duration. DSA II equation with l/100th of actual duration. Carswell decay function (see II. B. 3.). This stepwise process was continued until the energy arrived at the observation point. Due to the large nearly stationary high pressure system to the southwest all hindcasted wave energy came from azimuths 270 , 290 , and 310 . The reported winds were not strong enough at any of the hindcast times to generate waves with a period of 20 seconds according to the DSA II generation model. 26 The refraction coefficients (dl /dl) below were applied to the energy arriving from the various azimuths [Wiegel 1964, chapter 7]. The coefficient (dl /dl) is the ratio of orthogonal separation in deep water to that at the sensor location. Azimuth Wave Period (sec) 10 sec 14 sec 310° 1.00 0.95 290° 1.05 0.83 270° 0.99 0.66 Within each period band the energy from the various azimuths was summed. The DSA II method considered azimuths of wave approach for every ten degrees. However in this problem the azimuths were 20 apart and therefore the energy in each band was doubled to obtain the correct total. The resulting energy density in each band was converted to the feet squared representation (see Appendix A). The arriving energy was corrected for shoaling through multiplication by the appropriate shoaling factor (L_/L ) below, where L is deep water wave length and L is wave length in 65 feet of water (at the sensor position). The result is E for comparison to the observed value of E at the sensor position. Period (sec) 10 14 20 Shoaling Factor 1.38 1.68 2.34 27 B. ANALYSIS OF WAVE RECORDS A grid overlay was used to sample the analogue wave records at two-second intervals. This provided a time series of 512 data points per 20 minute wave record. The data points were read in arbitrary units and converted to feet of head. The mean reading was computed and subtracted from each value, the result being a series of zero mean. Fourier amplitudes (coefficients) were obtained from the raw data using subroutine RHARM on the Naval Postgraduate School IBM 360 com- puter. RHARM is based upon the fast Fourier transform developed by Coo ley and Tukey [1965]. The amplitudes were squared and summed into the following period bands : DSA II Wave Peric id Band Limits Bandwidth Descri| ation (sec ) (sec) (sec) Long 20 16.5-23.5 7 Me d i urn 14 11.5-16.5 5 Short 10 8.5-11.5 3 2 The energy in each band is multiplied by COSH (2nd/L ) [Wiegel 1964, Appendix 1] to obtain the energy at the surface, where L is the wave length at the sensor position and d is the S or depth of the pressure sensor (65 ft). The resulting energy is the observed energy for comparison to the hindcasted energy. 28 IV. GENERATION Two assumptions were made that affect the amount of energy generated. First, the reported wind field was presumed to represent the actual winds present at a constant reporting height above the water suitable for use with DSA II. Second, the DSA II generation model was presumed to be correct and was used throughout so that decay parameters could be varied. The first assumption was examined to define the effect of errors in the wind field. The following table indicates the approximate percentage error in wave energy which would result from a reported "generating" wind being lOjo higher than the actual wind for a duration of 10 hours. Actual Wind (kts) Reported Wind (kts) Period Long Medium Short 25.0 27.5 0 0 211 30.0 33.0 0 00 25 35.0 38.5 0 130 23 40.0 44.0 CO 75 10 45.0 49.5 105 30 3 If the reported wind is considered as a mean for the six hour period, then speed variations during the interval would tend to make the hindcasted energy too small. The effect of this phenomena seems to be minor. It is apparent that a prerequisite for this type of study is an accurately defined wind field. 29 V. SUMMARY OF RESULTS Figure (4) presents the results of the analyses done using the unmodified DSA II equation. The line marked "equal values" represents the position that should be occupied by all points if generation, decay and wind-field were correct and no scatter existed. This line is shown on each figure as a reference since various scales are used for the hindcast energy depending on its magnitude. The DSA II equation led to hindcasts that, with the exception of two points, were zero. It was concluded that this model provided too rapid an energy decay if the generation function is appropriate. Figure (5) shows the effect of using the DSA II decay equation with duration reduced to 1/lOth of the actual duration. This was the best set of hindcasts made. Four of the five points farthest from the equal value line came from two hindcast times only six hours apart. This makes it appear that the reported wind-field for this area may have been stronger than the actual wind-field. Figure (6) is the final hindcast made with a form of the DSA II equation. One hundredth of the duration was used in the DSA II equation and the hindcasts are generally too large. A decay model slightly more rapid than the DSA II equation with l/10th duration would give the best results as far as general magni- tude is concerned. The DSA II decay showed a good balance as far as period is concerned. No one band of periods forms a predominant grouping away from the line of equal values. 30 Figure 4. Observed Energy vs Hindcast Energy Using DSA II Decay Equation (U > OJ cu CO > > 3 CO CO » 5 E 3 u $H oo •H u u c •o o U o CD -C h5 £ co X a <3 CO ] J CO > -H-»- 3.0 2.4 1.8 1.2 00 CNI CN O CM >£> CN co aO U Q) C u CO CO U T) C ao O o .6 Observed Energy (ftsq) 31 Figure 5. Observed Energy vs Hindcast Energy Using DSA II Decay Equation with l/10th Actual Duration tsi i> OJ to > > a « tO 3 s E fH 3 4J u 00 •M u u c •o o o 01 si ►J X Cfl X D < 00 00 u CD 01 co > > 3 CO co at » E >* 3 ■u u 60 •r-i U s c -o o o 01 ja fj X w X o <3 u-i cr 41 C << < < n CO •0 u 1-1 CM 3.0 2.4 1.8 Observed Energy (ftsq) — 33 Figure (7) presents hindcasts made with Carswell decay. Like the " -2 DSA II equation, decay is modeled as a function of exp [-(T )]. The difference is the removal of the wind dependent term, which did not significantly degrade the quality of the hindcast. Snodgrass et al. [1966, p. 493] also observed in the Pacific Ocean that swell attenu- ation seems to be independent of the winds encountered during propagation outside the generating area. The Groen decay function and the Phillips' scattering model were not used since visual inspection showed that both would make the hindcasts too large due to their very slow rate of decay. The model based on Snodgrass et al. [1966] (see II. B. 2.) was not used for hindcasting. In this model, n varies as exp [-(T)] and thus attenuates small period waves more rapidly than is observed. No energy was hindcast in the 20 second period band but a small amount was observed. It is probable that this energy arrived from distant regions in a manner similar to that noted by Snodgrass et al. [1966]. 34 Figure 7. Observed Energy v» Hindcast Energy Using Carsmell Decay a 4 • m d pi CM cr a 4J Ed B r* Long Wave D Medium Wave > CO u 1-4 o J3 < Energy 00 •u m m u *o c o □ CN i-4 D D <3 < an cr « w > a vO — r- ^_ 'O 3.0 2 .4 1.8 1.2 .6 ( ) Observed Energy (ftsq) 35 VI. CONCLUSIONS The attenuation model which best fits the data obtained is -2 n = exp (-20 T t), where n is a ratio of energy at the start of decay to that remaining afterwards, T is wave period in seconds, and t is decay duration in hours. The advantage of a wind dependent term, as used in DSA II, in the decay model is unclear from the data obtained. It was not possible to relate attenuation models to theories of decay in the ocean in this thesis. Future work should employ a more detailed and more precise wind field. Also it is possible to reduce the influence of the generation model through analysis. Recent innovations in computer prepared wind field products may be a partial answer to the wind-field problem. Density of reporting ships has also increased since 1948 which is an important gain. Knowledge of the rate of ocean wave attenuation may provide the key to understanding the mechanisms of decay. 36 APPENDIX A EQUATION OF ENERGY REPRESENTATIONS The French Meteorological Service representation of energy as R(X) 2 in centi joules/cm can be converted to the amplitude squared notation 2 1 of E(X) in ft using the following: R(X) = k t pg E(X) where pg 3 (sea water) is 64 lb/ft and k is a constant unit conversion. Intro- ducing appropriate unit conversions provides the following: [E(X)ft ] = 64 lb Ttj/va centijoules~| |" joule 3J |_K.W 2 J L 2 ft cm 10 centi i 10 centijoules 104 cm' "meter 0.0929 meter ft2 2-, 0.7376 ft-lb joule ] E(X)ft = 0.214 R(X) centijoules/cm2 37 APPENDIX B TABULAR DISPLAY OF SPECTRAL COMPONENTS All dates refer to January 1948. Values given are energy in units of feet squared within the period band indicated. 1. Observation. 2. Hindcast using unmodified DSA II equation. 3. Hindcast using DSA II equation with l/10th of actual duration. 4. Hindcast using DSA II equation with l/100th of actual duration. 5. Hindcast using Carswell decay function. 38 1. Observation, Date/Time Period Long Medium Short 07/1610 07/2215 08/0415 08/1015 08/1615 09/1000 10/1645 11/1615 12/1045 0.05 0.59 0.05 0.06 0.09 0.03 0.02 0.03 0.00 0.83 1.31 2.96 1.24 2.15 1.69 1.07 0.63 0.27 1.70 0.65 0.93 0.79 0.91 1.00 0.79 0.74 0.70 39 2. Hindcast using unmodified DSA II equation, Date/Time Period Long Medium Short 07/1610 0.0 0.0 0.0 07/2215 0.0 0.0 0.0 08/0415 0.0 0.24 0.0 08/1015 0.0 0.03 0.0 08/1615 0.0 0.0 0.0 09/1000 0.0 0.0 0.0 10/1645 0.0 0.0 0.0 11/1615 0.0 0.0 0.0 12/1045 0.0 0.0 0.0 40 3. Hindcast using DSA II equation with l/10th of actual duration. Date/Time Period Long Medium Short 07/1610 0.0 0.0 0.0 07/2215 0.0 1.80 0.62 08/0415 0.0 5.49 2.52 08/1015 0.0 13.70 5.60 08/1615 0.0 7.13 3.46 09/1000 0.0 2.22 4.36 10/1645 0.0 16.17 1.03 11/1615 0.0 2.49 0.61 12/1045 0.0 2.35 0.44 41 4. Hindcast using DSA II equation with l/100th of actual duration. Date/Time Period Long Medium Short 07/1610 0.0 0.0 6.49 07/2215 0.0 6.54 14.63 08/0415 0.0 10.08 14.08 08/1015 0.0 21.55 17.93 08/1615 0.0 29.05 27.90 . 09/1000 0.0 14.42 28.15 10/1645 0.0 77.00 23.45 11/1615 0.0 19.50 26.91 12/1045 0.0 9.08 17.68 42 5. Hindcast using Carswell decay function. Date/Time Period Long Medium Short 07/1610 0.0 0.0 0.95 07/2215 0.0 7.26 7.39 08/0415 0.0 7.30 4.68 08/1015 0.0 13.29 5.87 08/1615 0.0 14.51 7.44 09/1000 0.0 4.51 4.44 10/1645 0.0 41.92 2.21 11/1615 0.0 5.32 1.53 12/1045 0.0 3.28 1.21 43 APPENDIX C ATTENUATION MODEL TABLES 1. Carswell Attenuation Function. 2. Groen Attenuation Function. 3. New Zealand - Alaska Attenuation Observations 4. Phillips1 Scattering Decay. 44 1. Carswell attenuation function. Values of n are presented as functions of period (T) and decay duration (t). Carswell attenuation was applied when neither favorable nor generating winds of greater than 10 knots were reported. t (hours) Pe r i od Long Medium Short 1/2 0.99 0.98 0.96 1 0.98 0.96 0.93 2 0.96 0.93 0.86 5 0.91 0.82 0.68 10 0.82 0.67 0.47 20 0.68 0.46 0.22 30 0.57 0.31 0.10 50 0.39 0.14 0.02 45 2. Groen attenuation function. Values of n are presented as functions of period (T) and decay duration (t). This function was not used since it was apparent by inspection that the resulting hindcasts would be much too large. t (hours) Period Long Medium Short 1/2 1.00 1.00 1.00 1 1.00 1.00 1,00 2 1.00 1.00 1.00 5 1.00 1.00 1.00 10 1.00 0.99 0.99 20 0.99 0.99 0.98 30 0.99 0.98 0.97 50 0.98 0.97 0.96 46 3. New Zealand - Alaska attenuation observations. Values of n are presented as functions of period (T) and decay duration (t). This function is slightly more rapid in its attenuation of energy than the Carswell model. Compared to the results of this study decay is of the right order of magnitude. However, the frequency distribution does not appear correct. This study shows that short waves attenuate more rapidly than medium waves rather than as shown below. t (hours) Period Medium Short 1/2 0.9259 0.9465 1 0.8573 0.8958 2 0.7349 0.8025 5 0.4630 0.5770 10 0.2144 0.3329 20 0.0460 0.1108 30 0.0099 0.0369 50 0.0004 0.0041 47 4. Phillips Scattering Decay Values of n are presented as functions of period (T) and decay duration (t). This function was not used for hindcasts since, by inspection, the rate of attenuation it models is much slower than that actually observed. Period t (hours) Long Medium Short 1/2 1.0000 1.0000 0.9999 1 0.9999 0.9999 0.9998 2 0.9999 0.9998 0.9996 5 0.9997 0.9995 0.9991 10 0.9994 0.9989 0.9981 20 0.9988 0.9979 0.9963 30 0.9982 0.9968 0.9944 50 0.9970 0.9947 0.9907 48 REFERENCES CITED Barber, N. F. and Ursell, F., "The Generation and Propagation of Ocean Waves and Swell," Phil. Trans. Roy. Soc . (London) A, v. 240, p. 527- 560, 24 February 1948. Barnett, T. P., On the Generation, Dissipation and Prediction of Ocean Wind Waves , Ph.D. Thesis, University of California, San Diego, Univ. Microfilms, Ann Arbor, Michigan (No. 67-6155), 1966. Bowden, K. F., "The Effect of Eddy Viscosity on Ocean Waves," The Philosophical Magazine, series 7, v. 41, p. 907-917, September 1950. Carswell, M. S., Attenuation of Surface Waves in Deep Water, MS Thesis, Naval Postgraduate School, Monterey, California, December 1968. Cartwright, D. E., "Modern Studies of Wind-Generated Ocean Waves," Contemp. Phys . , v. 8, no. 2, p. 171-183, March 1967. Cooley, J. W. and Tukey, J. W. , "An Algorithm for the Machine Calcu- lation of Complex Fourier Series," Mathematics of Computations, v. 19, p. 297, April 1965. Darbyshire, J., "Attenuation of Swell in the North Atlantic Ocean," Quar. J. Roy. Met. Soc. , v. 83, no. 357, p. 351-359, 1957. Gelci, R. and Cazale, H., "Une Equation Synthetique de L'Evolution de L'Etat de La Mer," Journal de Mechanique et de Physique de L'Atmosphere , series II, no. 16, p. 15-41, October-December 1962. Gelci, R. , Cazale, H. , and Vassal, J., "Prevision de la Houle , La Methode des Densities Spectro-Angula ires , " Bulletin d ' Information du Comite Central d ' Oceanographie et d 'Etude des Cotes, series IX, no. 8, p. 416-435, September-October 1957. Groen, P., "On the Behavior of Gravity Waves in a Turbulent Medium, with Application to the Decay and Apparent Period Increase of Swell," Netherlands. Meterologisch Instituut, Mededee lingen en Verhande lingen , series A, no. 63, p. 5-23, 1954. Groen, P. and Dorrestein, R. , "Ocean Swell: Its Decay and Period Increase," Nature, v. 165, no. 4194, p. 445-447, 18 March 1950. Isaacs, J. D. and Saville, T. , Jr., "A Comparison Between Recorded and Forecast Waves on the Pacific Coast," Ocean Surface Waves, Annals of the New York Academy of Sciences, v. 51, art. 3, p. 502-510, 13 May 1949. Kinsman, B., Wind Waves; Their Generation and Propagation on the Ocean Surface , p. 501, Prentice-Hall, Inc., 1965. 49 Munk, W. H., and others, "Directional Recording of Swell from Distant Storms," Phil. Trans. Roy. Soc. (London) A, v. 255, no. 1062, p. 505-584, 1963. Paquin, J. E., A Laboratory Experiment on Surface Wave Attenuation Due to Underwater Turbulence, MS Thesis, Naval Postgraduate School, Monterey, California, December 1968. Pierson, W. J., Jr., Neumann, G. , and James, R. W. , Practical Methods for Observing and Forecasting Ocean Waves by Means of Wave Spectra and Statistics . , p. 80-137, U. S. Navy Hydrographic Office Pub. No. 603, 1955. Phillips, 0. M. , The Dynamics of the Upper Ocean, p. 134-154, Cambridge University Press, 1966. Phillips, 0. M., "A Note of the Turbulence Generated by Gravity Waves," J. of Geophysical Res. , v. 66, no. 9, p. 2889-2893, September 1961. Phillips, 0. M. , "The Scattering of Gravity Waves by Turbulence," J. of Fluid Mech. , v. 5, no. 2, art. 12, p. 177-192, 1959. Snodgrass, F. E., and others, "Propagation of Ocean Swell Across the Pacific," Phil. Trans. Roy. Soc. (London) A, v. 259, p. 431-497, 1966, Wiegel, R. L. , "An Analysis of Data from Wave Recorders on the Pacific Coast of the United States," Amer. Geoph. Union Trans., v. 30, no. 5, p. 701, October 1949. Wiegel, R. L. , Oceanographica 1 Engineering, p. 516-524, Prentice-Hall, Inc., 1964. 50 REFERENCES CITED Barber, N. F. and Ursell, F., "The Generation and Propagation of Ocean Waves and Swell," Phil. Trans. Roy. Soc . (London) A, v. 240, p. 527- 560, 24 February 1948. Barnett, T. P., On the Generation, Dissipation and Prediction of Ocean Wind Waves , Ph.D. Thesis, University of California, San Diego, Univ. Microfilms, Ann Arbor, Michigan (No. 67-6155), 1966. Bowden, K. F., "The Effect of Eddy Viscosity on Ocean Waves," The Philosophical Magazine, series 7, v. 41, p. 907-917, September 1950. Carswell, M. S., Attenuation of Surface Waves in Deep Water, MS Thesis, Naval Postgraduate School, Monterey, California, December 1968. Cartwright, D. E., "Modern Studies of Wind-Generated Ocean Waves," Contemp. Phys . , v. 8, no. 2, p. 171-183, March 1967. Cooley, J. W. and Tukey, J. W. , "An Algorithm for the Machine Calcu- lation of Complex Fourier Series," Mathematics of Computations, v. 19, p. 297, April 1965. Darbyshire, J., "Attenuation of Swell in the North Atlantic Ocean," Quar. J. Roy. Met. Soc. , v. 83, no. 357, p. 351-359, 1957. Gelci, R. and Cazale, H., "Une Equation Synthetique de L'Evolution de L'Etat de La Mer," Journal de Mechanique et de Physique de L'Atmosphere , series II, no. 16, p. 15-41, October-December 1962. Gelci, R. , Cazale, H. , and Vassal, J., "Prevision de la Houle , La Methode des Densities Spectro-Angula ires ," Bulletin d ' Information du Comite Central d ' Oceanographie et d 'Etude des Cotes, series IX, no. 8, p. 416-435, September-October 1957. Groen, P., "On the Behavior of Gravity Waves in a Turbulent Medium, with Application to the Decay and Apparent Period Increase of Swell," Netherlands. Meterologisch Instituut, Mededee lingen en Verhande lingen, series A, no. 63, p. 5-23, 1954. Groen, P. and Dorrestein, R. , "Ocean Swell: Its Decay and Period Increase," Nature, v. 165, no. 4194, p. 445-447, 18 March 1950. Isaacs, J. D. and Saville, T. , Jr., "A Comparison Between Recorded and Forecast Waves on the Pacific Coast," Ocean Surface Waves, Annals of the New York Academy of Sciences, v. 51, art. 3, p. 502-510, 13 May 1949. Kinsman, B., Wind Waves; Their Generation and Propagation on the Ocean Surface, p. 501, Prentice-Hall, Inc., 1965. 49 Munk, W. H., and others, "Directional Recording of Swell from Distant Storms," Phil. Trans. Roy. Soc. (London) A, v. 255, no. 1062, p. 505-584, 1963. Paquin, J. E., A Laboratory Experiment on Surface Wave Attenuation Due to Underwater Turbulence, MS Thesis, Naval Postgraduate School, Monterey, California, December 1968. Pierson, W. J., Jr., Neumann, G. , and James, R. W. , Practical Methods for Observing and Forecasting Ocean Waves by Means of Wave Spectra and Statistics . , p. 80-137, U. S. Navy Hydrographic Office Pub. No. 603, 1955. Phillips, 0. M. , The Dynamics of the Upper Ocean, p. 134-154, Cambridge University Press, 1966. Phillips, 0. M., "A Note of the Turbulence Generated by Gravity Waves," J. of Geophysical Res. , v. 66, no. 9, p. 2889-2893, September 1961. Phillips, 0. M. , "The Scattering of Gravity Waves by Turbulence," J. of Fluid Mech. , v. 5, no. 2, art. 12, p. 177-192, 1959. Snodgrass , F. E., and others, "Propagation of Ocean Swell Across the Pacific," Phil. Trans. Roy. Soc. (London) A, v. 259, p. 431-497, 1966 Wiege 1 , R. L. , "An Analysis of Data from Wave Recorders on the Pacific Coast of the United States," Amer. Geoph. Union Trans., v. 30, no. 5, p. 701, October 1949. Wiegel, R. L. , Oceanographica 1 Engineering, p. 516-524, Prentice-Hall, Inc., 1964. 50 INITIAL DISTRIBUTION LIST No. Copies 1. Defense Documentation Center 20 Cameron Station Alexandria, Virginia 22314 2. Library, Code 0212 2 Naval Postgraduate School Monterey, California 93940 3. Oceanographer of the Navy 1 The Madison Building 732 N. Washington Street Alexandria, Virginia 22314 4. Dr. T. Green III 3 Department of Meteorology University of Wisconsin Madison, Wisconsin 57306 5. Associate Professor J. B. Wickham 3 Department of Oceanography Naval Postgraduate School Monterey, California 93940 6. Dr. R. Gelci 1 E. E. R. M./M. M. 196 rue de 1' Universite Paris (Vile) France 7. Department of Oceanography 3 Code 58 Naval Postgraduate School Monterey, California 93940 8. Dr. T. Laevastu 1 Fleet Numerical Weather Facility Naval Postgraduate School Monterey, California 93940 9. Lieutenant A. B. Chace , Jr., USN 1 USS HARDHEAD (SS 365) FPO New York, N. Y. 09501 10. Associate Professor J. J. von Schwind 1 Department of Oceanography Naval Postgraduate School Monterey, California 93940 51 11. Lieutenant Commander R. E. Ettle, USCG Coast Guard Oceanographic Unit Building 159-E Navy Yard Annex Washington, D. C. 20390 52 Security Classification DOCUMENT CONTROL DATA -R&D (Security classification of title, body of abstrmcl and indexing annotation must be entered when the overall report Is classified i originating activity (Corporate author) Naval Postgraduate School Monterey, California 93940 2a. REPORT SECURITY CLASSIFICATION Unclassified 2b. GROUP 3 REPORT TITLE Ocean Surface Waves: Attenuation and a Field Test of DSA II 4. DESCRIPTIVE N o T E S (Type of repot t and. inclus i ve dates ) Master's Thesis; October 1969 S. authoR(S) (First name, middle initial, last name) Alden Buffington Chace , Jr. 6- REPORT DATE October 1969 8a. CONTRACT OR GRANT NO. 6. PROJEC T NO. 7a. TOTAL NO. OF PAGES 52 76. NO. OF REFS 22 9a. ORIGINATOR'S REPORT NUMBER(S) 9b. OTHER REPORT NOISI (Any other numbers that may be assigned this report) ** 10. DISTRIBUTION STATEMENT This document has been approved for public release and sale its distribution is unlimited. II. SUPPLEMENTARY NOTES 13. ABSTRACT 12. SPONSORING MILI TARY ACTIVITY Naval Postgraduate School Monterey, California Several models of ocean surface wave attenuation were employed in making hindcasts for comparison with spectra obtained from a wave recorder located at Point Sur, California. Quantitative decay models were reduced to common variables. The French Meteorological Service DSA II generation model was used . The attenuation model which best fits the data obtained is n = exp (-20 T"2 t), where n is a ratio of energy at the start of decay to that remaining afterwards, T is wave period in seconds, and t is decay duration in hours. The period range considered was 8.5 to 23.5 seconds. DD I NOV 65 I *T / *J S/N 0101 -807-681 1 (PAGE 1) 53 Security Classification A-31408 Security Classification KEY WORDS Wave forecasting Wave decay DSA II DD ,F.T..1473 back) S/N 0101 -807-6821 54 Security Classification thesC33857 Ocean surface waves 3 2768 002 09675 2 DUDLEY KNOX LIBRARY