CHAPTER 8 APPLICATIONS OF EMPIRICAL FETCH-LIMITED SPECTRAL. FORMULAS to GREAT LAKES WAVES 1. Paul C. Liu, Mi ASCE

Size: px

Start display at page:

Download "CHAPTER 8 APPLICATIONS OF EMPIRICAL FETCH-LIMITED SPECTRAL. FORMULAS to GREAT LAKES WAVES 1. Paul C. Liu, Mi ASCE"

Shavonne Carpenter
5 years ago
Views:

1 CHAPTER 8 APPLICATIONS OF EMPIRICAL FETCH-LIMITED SPECTRAL FORMULAS to GREAT LAKES WAVES 1 Paul C. Liu, Mi ASCE ABSTRACT Tw episdes f Great Lakes waves fr which bth wind and wave data are simultaneusly available are used t examine the applicability f the empirical fetch-limited spectral wave frmulas develped by JONSWAP, Mitsuyasu, Liu, and Sverdrup-Munk-Bretschneider. Cmparing the results hindcast frm the frmulas with thse recrded shws that, fr hindcasting significant wave heights, Liu's frmula gives better results fr less than fully develped Waves, while frmulas by JONSWAP, Mitsuyasu, and Sverdrup-Munk-Bretsehnelder give better results fr fully develped waves. In hindcasting average wave perids and peakenergy frequencies, all the frmulas result in a deviatin f up t 2 s and 0.5 rad s~*, respectively. These results can he used as a reference in evaluating and interpreting wave predictins made by these frmulas as applied t the Greak Lakes. 1. INTRODUCTION Varius surface wave studies In recent years have results in the develpment f several sets f empirical fetch-limited spectral frmulas f practical interest in wave predictins. These frmulas ate all characterized by the dimensinless fetch parameter X 0 = gx/u* 2, where g is the acceleratin f gravity, X the fetch distance, and U* the frictihal wind Velcity. In terms f X 0, the significant wave height H1/3, peak-energy radian frequency up, and spectral energy density S(uj) as a functin f frequency m can be expressed by U* 2 H 1/3 = A X 0 a (1) "P " B f A X 0 - b (2) 13 t S(u) = C g2a)-5 X -C exp^dx -d ( _ _)-4]jj (3) in which A, B, C, and D = the empirical cefficients; a, b, c, and d = the empirical expnentials; and F = an empirical spectral shape functin. Several authrs have develped varius sets f numbers fr these empirical cnstants. Table 1 represents a summary f the results btained frm these studies. Data GLERL Cntributin N Physical Scientist, Great Lakes Envirnmental Research Labratry, Natinal Oceanic and Atmspheric Administratin, Ann Arbr, Michigan

2 114 COASTAL ENGINEERING-1976 a) u V 3 B M I H a,, 15! a - CO rh 3 > r~ CO c^ a) rp S Ai ^ rh ^ 03 CO c 3 CO ^ O r-. CO CO O C c^ 3 4-i t >-> O.-1 H H Hi ^ «^ _1 s 3 B r,c H 4-t X en T) m H <? s iw B rh O 3 CO >-i S-l 4J 4J H u O 4~> ai 0) a) (i a U CO CO II II,- e>

3 FETCH-LIMITED FORMULAS 115 cllected frm many ceans and lakes, as well as labratry wave tanks, were used. Table 1 reveals that sme f the cnstants derived by these authrs are quite clse; sme f the cnstants, hwever, differ significantly. It is f interest t nte that, althugh the authrs used their wn wave measurements in develping their empirical frmulas, they all made an effrt t incrprate the results f well-knwn wave studies published during the past 20 years. Obviusly, sme f the differences in the cefficients can be attributed t the authrs' subjectivity in adjusting the regressin line t fit their wn data. Because f these differences and because f the simple nature f the frmulas, case studies have been made t examine the applicability f the frmulas by using available wave data frm the Great Lakes. This paper presents tw episdes f wave cnditins fr which recrded data are cmpared with thse hindcast by the frmulas f JONSWAP, Mitsuyasu, and Liu. In additin, cnventinal hindcasts frm the Sverdrup-Munk-Bretschneider frmulas (Shre Prtectin Manual, Vl I, 1973) are als included fr further cmparisns. 2. CHARACTERTISTICS OF THE FORMULAS Befre prceeding t the case studies, it is f interest t examine sme f the basic characteristics f the frmulas that use hypthetical wind cnditins. Because frictinal wind velcity was nt available fr this study, lgarithmic wind prfile and a drag cefficient, C^Q f 10" 3 have been used in this paper fr cnverting wind speed at different levels t 11*. Fig. 1 presents a set f examples f wave spectra calculated by.empirical frmulas (3) fr 10-m level wind speeds f 5 and 40 m s -1 and fetch distances f 10 and 100 km. As the fur cases shwn in the figure cmbine lng and shrt fetches with high and lw wind speeds, respectively, the magnitude f spectral energy and peakenergy frequency varies significantly frm case t case. Hwever, the relative differences amng the three empirical spectra are similar fr all cases. In general, JONSWAP and Mitsuyasu spectra are fairly clse, especially at high wind speeds, while Liu spectra yield much lwer energy cntent. The energy at the spectral peaks fr JONSWAP and Mitsuyasu are apprximately an rder f magnitude larger than thse f Liu. An explanatin fr this great difference lies in the fact that the wave data used in develping the JONSWAP and Mitsuyasu frmulas are frm fully develped seas, whereas the wave data frm Lake Michigan used in Liu's frmula are nt. With energy spectra calculated frm the empirical frmulas, the spectral mments can be readily btained. The basic parameters, such as significant wave heights and average wave perids, can then be derived frm the mments, based n the theretical results f Lnguet-Higgins (1952) and Rice (1945), as: and H_ - 4m %, " 4m 1/2 (4) T 2n(m 0 /m 2 ) 1/2 (5) m 0,2

4 I I 116 COASTAL ENGINEERING-1976 r^ 1 X-10Q km M,0=40 ms-' 10^ I zz: Liu Jnswap Mitsuyasu 10 1! r>" - 10 ll! f!l \ r 1 - OTf FREQUENCY rad s< 1(V 1 II I L_ FREQUENCY ra,d s n X'10km U'5 ms" 1 n // 1 ' ' / u - ^ i \\ II \ n i[ [! \ \\ - ii i V FREQUENCY rad s FREQUENCY rad s Fig. 1 Examples f wave spectra calculated by empirical frmulas.

5 FETCH-LIMITED FORMULAS 117 in which the nth mment f a wave spectrum is given by m n = / 0 u n S(i) di. (6) Here the average wave perid Tj^ is presumably equivalent t the wave perid given by the Sverdrup-Munk-Bretschneider methd. The significant wave height H m(. calculated by Eq. (4) shuld apprximate Hi/3, calculated by (Eq. 1). A crrelatin f calculated H m _ and H1/3 is given in Fig. 2. In the case f the JONSWAP methd, H^ tends t be larger than ^i/j- A slightly greater trend is als shwn in Mitsuyasu's case. Since Liu's empirical equiatin fr HJ/J was a direct derivatin frm the spectral mments, the clse crrelatin shwn is nt surprising. In the fllwing discussins, H1/3, <i) p, and T m, will be calculated frm the varius empirical frmulas and cmpared with actual data. 3. APPLICATIONS OF THE FORMULAS Tw episdes f Great Lakes waves, with bth wind and wave data available, are used t examine the applicability f the varius empirical frmulas. The first episde ccurred in Lake Michigan. Wave data were recrded by a staff gage installed n a research twer lcated 2 km ffshre frm Muskegn, Michigan, in 16 m f water (Fig. 3). The wind anemmeter was lcated at 10 m abve the lake surface n the same twer. The episde cvered apprximately 4 days during Octber 25th thrugh 28th f The wind cnditins, which are typical f the Great Lakes, are shwn in Fig. 4. The wind speeds during this episde were quite unsteady with the predminant directin frm the suth and suthsutheast prviding mderate fetch distances f the rder f 50 km. «. Fig. 5 represents the significant wave heights H^/3 recrded during this episde as cmpared with thse hindcast by empirical frmulas. Here and in the fllwing figures in this paper, the heavy slid lines indicate recrded data; the dtted lines, the lng-shrt dashed lines, the dashed lines, and the light slid lines indicate the hindcast data by Sverdrup-Munk-Bretschneider, Mitsuyasu, JONSWAP, and Liu, respectively. The results f Sverdrup-Munk-Bretschneider, Mitsuyasu, and JOJUJSWAP, shwn in Fig. 5, are strikingly clse. In this episde, they all verestimated the recrded data by a factr f 2. Liu's results are relatively lwer than the thers since r che energy cntents given by Liu's frmula are generally less than the thers. In this case, Liu's results are clser t the recrded results than the thers. Fig. 6 shws the cmparisns fr the average wave perid T m0 2- Tne relative differences between the results f different authrs seem quite similar and distinctive. Sverdrup-Munk-Bretschneider, fr the mst part, furnished the highest estimate, fllwed in rder by Mitsuyasu and JONSWAP. Liu hindcast relatively lwer average wave perids fr the same wind cnditin. The results fr the peak-energy frequency Wp are shwn in Fig. 7. The Sverdrup-Munk-Bretschneider frmula des nt give estimatins fr a ; hence,

6 118 COASTAL ENGINEERING-1976 Fig. 2 Crrelatins f H and H,.. calculcated by empirical frmulas. 5 m Q 1/3

7 FETCH-LIMITED FORMULAS 119 I I : m O i u O 1 1 % CO > z CD - > CM

8 120 COASTAL ENGINEERING-1976 Recrded Liu Jnswap Mitsuyasu SMB OCTOBER, 1971 Fig. 5 Cmparisn f hindcast and recrded H 1.. during Lake Michigan episde. S<

9 FETCH-LIMITED FORMULAS OCTOBER, 1971 Fig. 6 Cmparisn f hindcast and recrded T during Lake Michigan episde. 0, OCTOBER, 1971 Fig. 7 Cmparisn f hindcast and recrded OJ during Lake Michigan episde.

10 122 COASTAL ENGINEERING-1976 nly JONSWAP, Mitsuyasu, and Liu are cmpared. The relative differences are again quite similar and distinctive. Liu hindcast a higher m, fllwed by JONSWAP and then Mitsuyasu. p The secnd episde is frm Lake Ontari. The wave gage, a waverider buy manufactured by Datawell f Hlland, was lcated 30 km nrtheast f Osweg, New Yrk, In 150 m f water. The slid circle labeled 0sweg-2 n the map, Fig. 8, indicates the waverider lcatin. The pen circle n the map shws the lcatin f an instrumented buy where the wind data used in this analysis were recrded. The wind anemmeter n the buy was 4 m abve the lake surface. Fig. 9 gives the wind cnditins during this 4-day episde f Octber 7-11, The wind speeds were typically unsteady, with directins starting frm the nrth, with a mderate fetch f 40 km, changing t west with a lng fetch f 200 km r mre when the strm intensified, and gradually switching back t nrth after the strm subsided. The cmparisns fr the hindcast and recrded significant wave heights fr this episde are shwn in Fig. 10. The relative differences between the empirical results remain the same. The results f this episde are different frm the first episde, hwever, in the sense that Liu's hindcasts, which were clser t the recrded results than the thers in Fig. 5, are clearly underestimatins in Fig. 10. The hindcasts f Sverdrup-Munk-Bretschneider, Mitsuyasu, and JONSWAP are again clse t each ther in this episde and als clse t the recrded data during the grwth part f the strm. All the empirical frmulas significantly underestimated H1/3 when the wind field decayed and directins switched frm lng fetches t shrt fetches. Figs. 11 and 12 shw the cmparisns fr T m Q and (Up, respectively, fr this episde. The results are generally similar t thse discussed in Figs. 6 and 7. It seems that, at any given time, ne f the frmulas tends give a better hindcast than the thers, but nne f the frmulas is able t maintain the clse hindcast thrughut the episde. The incnsistency, therefre, casts dubt ver the applicability f all the frmulas. Tw examples f cmparing cmputed spectra with the recrded spectra are shwn in Fig. 13. The accuracy f the spectrum hindcast by the frmulas depends mainly n the accuracy f crrespnding frmulas in hindcasting significant wave height and peak-energy frequency. If the hindcast significant wave height and peak-energy frequency values are clse t thse recrded, then the hindcast spectrum will certainly be clse t that recrded and vice versa. The cmparisns f spectra shwn in the figure represent the typical results that can be expected frm these empirical frmulas. 4. SUMMARY AND CONCLUDING REMARKS It has been the intent f this paper t present an assessment f the usefulness and applicability f the varius empirical frmulas in cnnectin with surface waves in the Great Lakes. The assessment has been perfrmed by cmparing waves that were hindcast by the frmulas with thse actually recrded during tw strm episdes in Lakes Michigan and Ontari. Based n the detailed

11 FETCH-LIMITED FORMULAS 123 Fig. 8 Lcatin f wave gage in Lake Ontari Fig. 9 OCTOBER, 1972 Wind cnditins during Lake Ontari episde.

12 124 COASTAL ENGINEERING-1976 OCTOBER, 1972 Fig. 10 Cmparisn f hindcast and recrded H.. during Lake Ontari episde.

13 FETCH-LIMITED FORMULAS OCTOBER, 1972 Fig. 11 Cmparisn f hindcast and recrded T during Lake Ontari episde. ra 0,2 8 9 OCTOBER, 1972 Fig. 12 Cmparisn f hindcast and recrded a) during Lake Ontari episde.

14 126 COASTAL ENGINEERING-1976 s z w AOH3N3

15 FETCH-LIMITED FORMULAS 127 cmparisns fr the three parameters Hi,,, T m(. «, and (»)_, results can be summarized as fllws: ' a. Significant Wave Height, 1^/3. The hindcasts by the JONSWAP, Mitsuyasu, and Sverdrup-Munk-Bretschneider methds are generally clse t each ther and greater than Liu's hindcast by abut 50 percent. During the Lake Michigan episde, Liu's results were clser t the recrded results than the thers; thus, the ther frmulas yielded verestimatins. During the Lake Ontari episde, JONSWAP, Mitsuyasu, and Sverdrup-Munk-Bretschneider's results were clser t the recrded results than Liu's during the grwth part f the strm; hence Liu's results appear t be underestimatins. At first glance, the results rendered by the tw episdes seem t be incnsistent. The incnsistency, hwever, can be clarified by further examinatin f the tw respective wind fields. During the Lake Michigan episde, the 10-m wind speeds started at 2 m s _ l and gradually increased t 10 m s~l and higher; thus, the wave spectra were mstly under develpment. During the Lake Ontari episde, n the ther hand, the wind speeds at the 4-m level had reached ver 10 m s _ i at a very early stage f the strm. The wave spectra under this wind field tended t be fully develped. Frm these facts, it appears that Liu's frmula is applicable fr the less than fully develped waves, whereas JONSWAP, Mitsuyasu, and Sverdrup-Munk-Bretschneider frmulas are applicable fr the fully develped waves. This is cnsistent with the characteristics f the data frm which the crrespnding frmulas were derived. The abve discussins apply nly t the grwth part f the strms. All the frmulas underestimated the significant wave height when the wind field decayed, especially when the decay invlved changes in wind directin as well as reductins in fetch distances. As the frmulas were neither derived frm, nr intended fr, decaying prcesses, the results are by n means unexpected. b. Average Wave Perid, T mq 2 - The difference between high and lw estimates as hindcast by the frmulas'fr a given time is abut 2 s. The recrded data generally lie within this range. Therefre, the errr in hindcasting average wave perids by any frmula is abut 2 s r less. Nne f the empirical results can be cnsidered substantially clse t the recrded results thrughut either f the tw episdes. c. Peak-Energy Frequency, up. The difference between high and lw (Dp hindcast fr a given time is abut 0.5 rad s~l. With the exceptin f the lw wind prtin f the Lake Michigan episde and the decaying prtin f the Lake Ontari episde, the recrded u p lies within the range f the empirical hindcasts. An accurate hindcast f Up will lead t accurate hindcasting f wave spectra if the hindcast f significant wave height is als accurate. Nne f the frmulas seems t be able t prvide an accurate i. During bth episdes, Liu's hindcasts were relatively clse t the recrded results thrugh a majr prtin f the strm. JONSWAP and Mitsuyasu's hindcasts came clser t the recrded results at the peak f the strm. These results, while generally fulfilling mst f the bjectives f this paper, are by and large bth disappinting and encuraging. These results are disappinting because they demnstrate that simple fetch-limited spectral frmulas have nt prvided substantial applicability t the predictin f Great

16 128 COASTAL ENGINEERING-1976 Lakes waves. Undubtedly, parameters neglected by the frmulas, such as duratin and atmspheric stability, play imprtant rles nt knwn at the present time and need t be further explred. On the ther hand, these results are encuraging because they prvide infrmatin n the limitatins and errr ranges that can be expected in the applicatin f these frmulas t predicting Great Lakes waves. Until further develpments and additinal studies are made, the results presented in this paper will cntinue t be useful. 5. REFERENCES Garratt, J.R., "Studies f Turbulence in the Surface Layer Over Water (Lugh Neagh). III. Wave and Drag Prperties f the Sea-Surface in Cnditins f Limited Fetch," Quart. J. R. Met. Sc., Vl. 99, 1973, pp Hasselmann, K., T. P. Barnett, E. Buws, H. Carlsn, D. E. Cartwright, K. Enke, J. A, Ewing, H. Gienapp, D. E. Hasselmann, P. Kruseman, A. Meerburg, P. Miiller, D. J. Olbers, K. Richter, W. Sell, and H. Walden, "Measurements f Wind-Wave Grwth and Swell Decay During the Jint Nrth Sea Wave Prject (JONSWAP)," Deutsche Hydrg. Z. Suppl. A(8), N. 12, 1973, 95 p. Knnkva, G. E., E. A. Nikitina, L. V. Pebrchaya, and A. A. Speranskaya, 1970, "On the Spectra f Wind Driven Waves at Small Fetches." izvestiya, Atmspheric and Oceanic Physics, Vl. 6, 1970, pp Liu, P. C, "Nrmalized and Equilibrium Spectra f Wind Waves in Lake Michigan," J. Phys. Oceangraphy, Vl. 1, 1971, pp Lnguet-Higgins, M. S., "On the Statistical Distributin f the Heights f Sea Waves," J. Marine Res., Vl. II, 1952, pp Mitsuyasu, H., "On the Frm f Fetch-Limited Wave Spectrum," Castal Eng. in Japan, Vl. 14, 1971, pp Rice, S. 0., "The Mathematical Analysis f Randm Nise," Bell System Tech. J., Vl. 24, 1945, pp U.S. Army Castal Engineering Research Center, "Shre Prtectin Manual," Vl. I, U.S. Gvernment Printing Office, Washingtn, D.C., 1973.

COASTAL ENGINEERING Chapter 2

COASTAL ENGINEERING Chapter 2 CASTAL ENGINEERING Chapter 2 GENERALIZED WAVE DIFFRACTIN DIAGRAMS J. W. Jhnsn Assciate Prfessr f Mechanical Engineering University f Califrnia Berkeley, Califrnia INTRDUCTIN Wave diffractin is the phenmenn