Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UF00076133/00001
## Material Information- Title:
- Wind-wave hindcasting and estimation of bottom shear stress in Lake Okeechobee
- Series Title:
- UFLCOEL
- Creator:
- Ahn, Kyungmo, 1957- (
*Dissertant*) Sheng, Y. Peter (*Thesis advisor*) University of Florida -- Coastal and Oceanographic Engineering Laboratory - Place of Publication:
- Gainesville, Fla.
- Publisher:
- Coastal & Oceanographic Engineering Dept., University of Florida
- Publication Date:
- 1989
- Copyright Date:
- 1989
- Language:
- English
- Physical Description:
- x, 108 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Amplitude ( jstor )
Boundary layers ( jstor ) Lakes ( jstor ) Modeling ( jstor ) Orbitals ( jstor ) Sediments ( jstor ) Shallow water ( jstor ) Shear stress ( jstor ) Water depth ( jstor ) Waves ( jstor ) Coastal and Oceanographic Engineering -- Dissertations, Academic -- UF ( local ) Coastal and Oceanographic Engineering thesis M.S ( local ) Wind waves -- Mathematical models ( lcsh ) Wind waves -- Okeechobee, Lake -- Florida ( lcsh ) Lake Okeechobee ( local ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt ) theses ( local )
## Notes- Abstract:
- Three wind-wave models are compared with measured field data from Lake Okeechobee, Florida. The wind-wave models tested in this study are the shallow-water SMB model, the finite-depth wind-wave spectral model, and the GLERL-Donelan deep-water wind-wave prediction model. The wave models were tested using idealized steady and uniform wind fields and also using measured wind fields over Lake Okeechobee with realistic bottom topography. The comparisons of model results with measured field data include significant wave height, wave period, and one dimensional spectral shape. By assuming the significant wave height and the peak wave period as a representative monocromatic wave, the bottom shear stresses are computed through linear wave theory and Kajiura's oscillatory boundary layer formulation in Lake Okeechobee. A discussion about correlations between the computed bottom shear stress and the distribution of the surficial sediment is presented.
- Thesis:
- Thesis (M. Eng.)--University of Florida, 1989.
- Bibliography:
- Includes bibliographical references (leaves 103-107).
- General Note:
- Typescript.
- General Note:
- Vita.
- Funding:
- This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
- Statement of Responsibility:
- by Kyungmo Ahn.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- All rights reserved, Board of Trustees of the University of Florida
- Resource Identifier:
- 22140389 ( OCLC )
## UFDC Membership |

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Full Text |

UFL/COEL-89/027
WIND-WAVE HINDCASTING AND ESTIMATION OF BOTTOM SHEAR STRESS IN LAKE OKEECHOBEE by Kyungmo Ahn Thesis 1989 WIND-WAVE HINDCASTING AND ESTIMATION OF BOTTOM SHEAR STRESS IN LAKE OKEECHOBEE By KYUNGMO AHN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING UNIVERSITY OF FLORIDA 1989 ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my advisor Dr. Y. Peter Sheng for his continuous guidance and encouragement throughout this study. I would also like to extend my thanks and appreciation to my thesis committee members, Dr. Robert Dean and Dr. Hsiang Wang, for their patience in reviewing this thesis. I would like to thank Dr. Hans Graber of Woods Hole Oceanographic Institution for his kind permission to use his finite-depth wind-wave model. Also I would like to thank Dr. David Schwab of Great Lakes Environmental Research Laboratory, National Oceanic and Atmospheric Administration, for providing me with the deep-water GLERL/Donelan wind-wave model. Appreciation is extended to Jei Choi, who contributed so much toward the sucessful application of the GM model to Lake Okeechobee. Special thanks go to all Lake Okeechobee crew members, Joaquim, Steve, Victor, P.F., Dave, and members of the Coastal Engineering Laboratory who did an excellent job in the synoptic surveys and the deployment of in-situ instruments. Financial support provided by the South Florida Water Management District, West Palm Beach, Florida through the Lake Okeechobee Phosphorous Dynamics is appreciated. I would like to dedicate this thesis to my parents. I hope they would like what I have achived so far. Finally I would like to thank my wife Eunhye for her support, encouragement, and patieni.e and my sons, Taewook and Hyunwook, for sharing joyful and happy smiles. TABLE OF CONTENTS ACKNOWLEDGEMENTS..................................... 1 LIST OF FIGURES.......................................... v LIST OF TABLES.......................................... ix ABSTRACT............................................... x CHAPTERS 1 INTRODUCTION......................................... 1 2 WIND-WAVE HINDCASTING MODELS......................... 4 2.1 Introduction.......................................... 4 2.2 The Shallow-Water SMB model ............................. 5 2.3 A Simple Numerical Wave Prediction Model ...................... 6 2.4 A Finite Depth Wind Wave Model by Graber and Madsen............ 9 2.4.1 Theoretical Description of GM model.................... 10 2.4.2 Atmospheric Forcing by the Wind....................... 13 2.4.3 Nonlinear Energy Transfer by Resonant Wave-Wave Interactions 14 2.4.4 Dissipation by Bottom Friction......................... 15 3 WAVE BOUNDARY LAYER AND BOTTOM SHEAR STRESS........... 16 3.1 Introduction......................................... 16 3.2 Bottom Boundary Layer................................. 17 3.3 Jonsson's Integrated Momentum Approach..................... 22 3.4 Kajiura's Model of the Bottom Boundary Layer in Water Waves.........2 4 MODEL PERFORMANCE IN IDEALIZED WIND FIELD. .. .. .. ... ...28 4.1 Introduction. .. .. .. .. .. .. ... .. ... .. ... ... .. ... ....28 4.2 Comparison of Wave Models in Idealized Wind Fields .. .. .. ... .....28 5 MODEL PERFORMANCE IN REAL WIND FIELD....................52 5.1 Introduction........................................... 52 5.2 Field Data Analysis.......................................52 5.3 Comparison of Model Results vs. Measured Data................... 55 5.3.1 Comparison of the GM Model results with Measured Data .. .. ....56 5.3.2 Comparison of the SMB Model Results with Measured Data 65 6 SUMMARY AND CONCLUSION. .. .. .. ... ... .... ... ... ......3 APPENDICES A RADIATIVE-TRANSFER EQUATION .. .. .. .. .... ... ... ... ....76 B DEPTH TRANSFORMATION FACTOR 'I(Wh). .. .. ... ... ... ......80 C JONSSON'S FRICTION COEFFICIENT FORMULA. .. ... ... ... ....83 D TABLES................................................. 85 E ONE-DIMENSIONAL FREQUENCY SPECTRA......................90 BIBLIOGRAPHY............................................. 103 BIOGRAPHICAL SKETCH...................................... 108 LIST OF FIGURES 3.1 Comparison of different formulas for the friction coefficient f varies with a/k (Sleath, 1984) ....... .............................. 24 4.1 Geometry and bottom topography of Lake Okeechobee ............ 31 4.2 Location of research towers A, B, C, D, E, and F, with wind station L, and a 2 km computational grid used in Lake Okeechobee .......... 32 4.3 Bottom sediment characteristic of Lake Okeechobee (from Reddy et al. 1988) ......... .................................... 33 4.4 Comparison of wave models for 5 m/sec easterly wind. Variations of significant wave height, H,, peak wave period, Tp, and bottom shear stress, Tb, with fetch along the cross section AA' ................. 34 4.5 Comparison of wave models for 10 m/sec easterly wind. Variations of significant wave height, H,, peak wave period, Tp, and bottom shear stress, rb, with fetch along the cross section AA' .................. 35 4.6 Comparison of wave models for 15 m/sec easterly wind. Variations of significant wave height, H,, peak wave period, Tp, and bottom shear stress, rb, with fetch along the cross section AA' .................. 36 4.7 Comparison of wave models for 20 m/sec easterly wind. Variations of significant wave height, H,, peak wave period, Tp, and bottom shear stress, Tb, with fetch along the cross section AA' .................. 37 4.8 Contours of the significant wave height, H3, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the GM model for 5 m/sec easterly wind ....................... 40 4.9 Contours of the significant wave height, H, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, b, obtained from the GM model for 10 m/sec easterly wind ....................... 41 4.10 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the GM model for 15 m/sec easterly wind ...................... 42 4.11 Contours of the significant wave height, H3, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, b, obtained from the GM model for 20 m/sec easterly wind ...................... 43 4.12 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, 7b, obtained from the SMB-LD model for 5 m/sec easterly wind .................. 44 4.13 Contours of the significant wave height, H8, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, Tb, obtained from the SMB-LD model for 10 m/sec easterly wind ................... 45 4.14 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-LD model for 15 m/sec easterly wind ................... 46 4.15 Contours of the significant wave height, H8, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, Tb, obtained from the SMB-LD model for 20 m/sec easterly wind ................... 47 4.16 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, n, obtained from the SMB-AD model for 5 m/sec easterly wind .................... 48 4.17 Contours of the significant wave height, H, peak wave period, Tp, bottom orbital amplitude, A6, and bottom shear stress, Tb, obtained from the SMB-LD model for 10 m/sec easterly wind ................... 49 4.18 Contours of the significant wave height, H., peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, Tb, obtained from the SMB-LD model for 15 m/sec easterly wind ................... 50 4.19 Contours of the significant wave height, H, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, Tb, obtained from the SMB-LD model for 20 m/sec easterly wind ................... 51 5.1 Typical wave spectrum from Lake Okeechobee ................... 55 5.2 Comparison of the calculated significant wave heights and peak wave periods with measured data for friction coefficient of 0.01, 0.02,0.04,0.06, and 0.08 at stations B, C, and E on October 7, 1988 ............... 57 5.3 Comparison of computed wave spectral shape with measured data for friction coefficients of 0.02, 0.04, 0.06, and 0.08 at station E on October 7, 1988 ....... ...................... ............ ..58 5.4 Measured wind bar graph at station L005 and the comparison of the GM model results (-) and measured (+++) H, and Tp at station E ..... .59 5.5 Measured wind bar graph at station L005 and the comparison of the GM model results (-) and measured (+++) H. and Tp at station C ..... .60 5.6 Measured wind bar graph at station L005 and the comparison of the GM model results (-) and measured (+++) H, and Tp at station B ..... .61 5.7 Comparison of hourly hindcast by the GM model and measured significant wave height H8, and peak wave period Tp, and zero up-crossing wave period T, at station E ............................... 5.8 Comparison of hourly hindcast by the GM model and measured significant wave height H8, peak wave period Tp, and zero up-crossing wave period T, at station C ................................... 5.9 Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) H, and Tp at station E ....... .................................. 5.10 Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) H, and Tp at station C ....... .................................. Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) H, and Tp at station B ....... .................................. 5.12 Comparison of hourly hindcast by the SMB-AD model and measured significant wave height H, and peak wave period Tp at station E..... 5.13 Comparison of hourly hindcast by the SMB-LD model and measured significant wave height H, and peak wave period Tp at station E ..... 5.14 Comparison of hourly hindcast by SMB-AD model and measured significant wave height H8, and peak wave period Tp, and zero up-crossing wave period T_. at station C .............................. 5.15 Comparison of hourly hindcast by SMB-LD model and measured significant wave height H8, and peak wave period Tp, and zero up-crossing wave period T, at station C .............................. Hindcasted and measured one-dimensional frequency C in Lake Okeechobee. (continue) ............ Hindcasted and measured one-dimensional frequency C in Lake Okeechobee. (continue) ............ Hindcasted and measured one-dimensional frequency C in Lake Okeechobee. (continue) ............ Hindcasted and measured one-dimensional frequency C in Lake Okeechobee. (continue) ............ ttindcasted and measured one-dimensional frequency C in Lake Okeechobee. (continue) ............ Hindcasted and measured one-dimensional frequency C in Lake Okeechobee .................... spectra at station s p t a s t a t i spectra at station spetr a s a t o n spectra at station spectra at station spectra at station spectra at station 5.11 E.7 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) ..... ..................... 97 E.8 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) ..... ..................... 98 E.9 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) ..... ..................... 99 E.10 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) ..... ..................... 100 E.11 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) ..... ..................... 101 E.12 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee ...... ............................ 102 LIST OF TABLES 3.1 Nikuradse eqivalent sand-grain roughness height for flat bed. DP is the diameter of the grain corresponding to p % finer in weight. (van Rijn, 1982) ......... ..................................... 19 5.1 Locations of the pressure transducers deployed in Lake Okeechobee 53 5.2 Comparison of hindcast by the GM model and measured values of significant wave height, peak wave period, and zero up-crossing wave period at station E ........ .................................. 65 5.3 Comparison of hindcast by the GM model and measured values of significant wave height, peak wave period, and zero up-crossing wave period at station C ....... .................. ............. ..65 5.4 Comparison of hindcast by the SMB-AD and SMB-LD model and measured values of significant wave height and peak wave period at station E .................................................. 66 5.5 Comparison of hindcast by the SMB-AD and SMB-LD models and measured values of significant wave height and peak wave period at station C .......... ........................................ 72 D.1 Model comparison for 5 m/sec wind. Variations of H8, Tp, and 7b with fetch along the cross section AA'... .......................... 86 D.2 Model comparison for 10 m/sec wind. Variations of H5, Tp, and Tb with fetch along the cross section AA' ............................. 87 D.3 Model comparison for 15 m/sec wind. Variations of Hs, Tp, and b with fetch along the cross section AA' ............................. 88 D.4 Model comparison for 20 m/sec wind. Variations of H8, Tp, and rb with fetch along the cross section AA' ............................. 89 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Engineering WIND-WAVE HINDCASTING AND ESTIMATION OF BOTTOM SHEAR STRESS IN LAKE OKEECHOBEE By KYUNGMO AHN December 1989 Chairman: Dr. Y. Peter Sheng Major Department: Coastal and Oceanographic Engineering Three wind-wave models are compared with measured field data from Lake Okeechobee, Florida. The wind-wave models tested in this study are the shallow-water SMB model, the finite-depth wind-wave spectral model, and the GLERL-Donelan deep-water wind-wave prediction model. The wave models were tested using idealized steady and uniform wind fields and also using measured wind fields over Lake Okeechobee with realistic bottom topography. The comparisons of model results with measured field data include significant wave height, wave period, and one dimensional spectral shape. By assuming the significant wave height and the peak wave period as a representative monocromatic wave, the bottom shear stresses are computed through linear wave theory and Kajiura's oscillatory boundary layer formulation in Lake Okeechobee. A discussion about correlations between the computed bottom shear stress and the distribution of the surficial sediment is presented. CHAPTER 1 INTRODUCTION Nutrients and hazardous chemicals are transported into lakes through surface runoff, river inflow, and precipitation, thus causing many environmental problems including enhanced eutrophication of lakes and deterioration of water quality in lakes. Fine sediments, with their small settling velocity and affinity to adsorb contaminants, act as an agent for recycling of contaminents. Thus it is important to understand the transport mechanisms of fine sediment particles ( i.e., silt and clay size range) in various water bodies (e.g., Sheng and Lick, 1979, Somly6dy, 1983, Pettersson and Bostr6m, 1984, and Luettich et al. 1989). Understanding the dynamics of fine sediment transport in a shallow-water body, such as Lake Okeechobee, is of particular importance because the bottom sediment and associated contaminants can be readily resuspended and transported by waves and currents. A mathematical model capable of simulating the transport of suspended sediment in a shallow water body is a useful tool for the study of practical problems. For example, Sheng and Lick (1979) developed a three-dimensional fine sediment transport model, which includes the effects of wind-driven currents, wind waves, turbulent mixing, and erosion and deposition at the bottom. One-dimensional sediment transport models, which couple the bottom boundary-layer dynamics with the sediment-transport problem, have been developed for non-cohesive sediments (Grant and Madsen, 1986) and cohesive sediments (Sheng and Villaret, 1989). Bottom boundary layers are regions where intensive turbulent mixing of mass and momentum and frictional dissipation of energy take place. Close to the shallowwater bed, shear stresses and turbulent kinetic energy are generated by both waves and currents. The combined action of waves and currents increases the bottom shear stress and 2 enhances resuspension of sediments (Sheng, 1984, Trowbridge and Madsen, 1984). Because the rate of sediment resuspension is a function of both bottom shear stress and sediment characteristics, bottom shear stress under waves and currents must be accurately calculated. As a first step, the wind-driven currents and wind waves should be accurately determined. As a part of the "Lake Okeechobee Phosphorous Dynamics" project funded by the South Florida Water Management District, the Coastal and Oceanographic Engineering Department of the University of Florida is studying the circulation patterns, wave climate, and sediment transport processes in Lake Okeechobee under the general supervision of Dr. Y. Peter Sheng. This thesis deals with the modeling of wind-generated wave in Lake Okeechobee. Three wind-wave hindcasting models were tested with wind and wave data from Lake Okeechobee. Excluding the Great Lakes, Lake Okeechobee has the largest surface area (1700km2) of any lake in the United States. Its depth ranges from 0.5 to 6 meters. Due to its long fetch and shallowness, Lake Okeechobee is a good location to calibrate shallow-water wind-wave models. The wind-wave models tested in this study are as follows: 1. The shallow-water SMB model presented in the Shore Protection Manual (U.S. Army Coastal Engineering Research Center, 1984) which is based on a semi-empirical approach. 2. The GLERL-Donelan deep-water wind-wave prediction model developed by Donelan (1977) and used successfully to predict wave height and direction in Lake Erie (Schwab et. al., 1984), Lake Michigan (Liu et. al., 1984), and Lake St. Clair (Schwab and Liu, 1987). 3. The finite-depth wind-wave model developed by Graber and Madsen (Graber, 1984, Graber and Madsen, 1988) which is a parametric windsea model for arbitrary water depth. The calculation of bottom shear stress on a natural lake bed under unsteady oscillatory 3 flow is a complex problem due to the irregular bottom roughness and random nature of the waves. However, the estimation of bottom shear stress may give some useful insight for understanding the transport and fate of sediment particles in the lake. As part of this study, the bottom shear stresses in Lake Okeechobee were estimated using the boundary layer methods of Kajiura (1964, 1968) and Jonsson (1966, 1978). In Chapter 2, the three wind-wave models are described. Theoretical descriptions of the models as well as the limitations on their application are presented. In Chapter 3, a literature review and a theoretical description of the oscillatory boundary layer is presented. The bottom shear stress formulae of Kajiura and Jonsson are also presented. In Chapter 4 the three wind-wave models are compared using steady, uniform wind conditions and Lake Okeechobee bottom topography. In Chapter 5, wave data-obtained from subsurface pressure gages at three stations in Lake Okeechobee are analyzed and the finite depth wind-wave model by Graber and Madsen and the shallow-water SMB model are compared with the measured wave data in detail. Summary and conclusions are presented in Chapter 6. CHAPTER 2 WIND-WAVE HINDCASTING MODELS 2.1 Introduction The commonly utilized models for wind-wave prediction include the empirical model based on dimensional analysis, the discrete spectral model, and the parametric model. There are major theoretical differences between the various models. However, a rational framework has been established in the radiative-transfer equation, and the details of windwave generation, evolution, and dissipation mechanism have been clarified. Each model has numerical advantages and disadvantages. The Sea Wave Modelling Project (SWAMP) Group (1984) compared nine operational deep water wave models and recommended future model improvements. More recently, the SWIM (1985) study compared three operational shallow-water wave models for idealized wave generational cases and for a severe North Sea storm. In the SWIM project, the three wave models tested were the model of the Meteorological Office; UK, (BMO) (Golding, 1983), the GONO-model of the Royal Netherlands Meteorological Institute (Janssen et al. 1984), and the HYPAS model of the University of Hamburg, FRG (Giinther et al. 1979). According to the SWIM study, there exist no significant differences between the model performances. Because of the shallow depth of Lake Okeechobee, the comparison of model performances in this study is focused on two shallow-water wind-wave models: the SMB model and the Graber-Madsen model. The deep-water version of GLERL-Donelan model is also tested in this study, however, because of the successful application of the model to shallow lakes such as Lake St. Clair (Schwab and Liu, 1987). Many different wind-wave hindcasting models are currently being used and developed. In order to select a particular model for a specific study, the questions that need to be 5 addressed include (1) which precise wave characteristics need to be known, (2) what are the detailed practical questions to be answered, and (3) what are the basic physical principles and the inherent limitations of the models. In this regard, this chapter presents a detailed description of the wind-wave hindcasting models being tested in this study. 2.2 The Shallow-Water SMB model The shallow-water SMB (Svendrup, Munk, and Bretshneider) model which will be denoted hereafter as SMB, is an empirical model based on dimensional analysis. The SMB model presented in the US Army Corps of Engineers Shore Protection Manual (1984) is a simplified wave hindcast model. Waves are generated by a uniform wind blowing over a known fetch for a given duration. Wave energy is added by wind stress and pressure and lost to bottom friction (Putnam and Johnson, 1949) and percolation (Putnam, 1949). The basic assumptions are spatially uniform wind velocity and uniform depth over the fetch for a given duration. Consequently, in shallow water, the evolution of waves such as refraction, shoaling, and dissipation are poorly described by a single dimensionless depth parameter. No directional wave information is available from the method, so it is not useful when the combined effect of waves and currents is needed. Despite these severe limitations, the SMB method is still used extensively for giving quick, order-of-magnitude estimates. Assuming a bottom friction coefficient of 0.01, the significant wave height H in meters, and the wave period T in seconds, are given by U 0.283 tanh anh'3410006 (A) (2.1) [ ( ) ]0"39(g)/3tanh JJ 2 7.54 tanh [0.833 (d)] tanh I 1 (2.2) 0tanh [0.833 ( 3/8 and gt 7) gT 7/3 (2.3) UA .37x 102 A 6 where g is the gravitational acceleration (9.81 m/sec2), d is the uniform water depth (in meters) which is often taken as the average water depth over the fetch or the local water depth of interest, F is the fetch (in meters), t is the duration of the wind over the fetch (in seconds), UA is an adjusted wind velocity or wind-stress factor given by UA = 0.71U1.23 where U = RTU(10), U(10) is the mean hourly wind velocity (in m/sec) at 10 m above the mean sea level, and RT is an air-sea temperature difference adjustment factor. In the absence of temperature information, RT = 1.1 is generally assumed. For practical wave predictions, it is usually satisfactory to assume uniform wind if the variation in wind direction is less than 150 and the variation in wind velocity is less than 2.5 m/sec. Hurdle and Stive (1989) reported discontinuities in the SMB model at the transition region between deep-water developing seas and deep-water fully-developed seas, and that between shallow and deep waters. Further, Hurdle and Stive questioned the relationship between critical duration and fetch in very shallow water. They proposed a modified version of equations which removed discontinuities in the transition regions, and hence should be valid for both deep and shallow waters. The revised equations are gH 0.25[tanh0.6 -,3/4] tanhl/2 f A (2.4) 0.2 tah= r f\3/41 [ Utanh [0.6 U ) gT ~ [ (gd' ).37,5]1 4.1 x 10-5 ( F) 1 p 8.3 anh [076 2j~ tanh' / 3 [0.7 ~) W375]J (.5 where Tp is the period of peak spectral energy in seconds. Despite the recommended revised equations, we used the original shallow water SMB model because of its widespread use. 2.3 A Simple Numerical Wave Prediction Model A simple numerical wave prediction model which will be denoted hereafter as the deepwater GD (GLERL-Donelan) model, was originally developed by Donelan (1977) and used successfully by the Great Lakes Environmental Research Laboratory (GLERL) of NOAA to predict wave height and direction in Lake Erie (Schwab et al. 1984), Lake Michigan (Liu et al. 1984), and Lake St. Clair (Schwab and Liu, 1987). This model is a parametric model based on a momentum conservation equation for the wave field. The model predicts the two components of the wave momentum vector and the phase speed of peak energy waves. From these variables, one can then derive significant wave height, wave period, and wave direction. In the model formulation, the waves are assumed to obey the deep-water dispersion relation. Refraction and bottom dissipation are ignored. Schwab and Liu (1987) reported that the shallow-water GD model was developed by incorporating the Kitaigorodskii et al. (1985) shallow-water spectrum along with a depthdependent group velocity and a simple form of bottom friction. The results of the shallow water version and the deep water version of the model were compared to observations at six towers in Lake St. Clair with depths ranging from 3.7 to 7.0 meters. The shallow water version of the model was found to underestimate the highest waves at all stations. Although the shallow water model could be adjusted to better match the observed wave heights by decreasing the bottom friction parameter, even the best results were no better than those of deep water model. Schwab and Liu speculated that the possible reason why the deepwater model works in Lake St. Clair better than shallow-water model is that "The wind momentum input function in the model is oversimplified and if it were formulated more realistically, the deep-water model would tend to overestimate the highest waves." Because of this, the deep-water GD model was applied to Lake Okeechobee in this study. In the deep-water GD model, the time rate of change of the wave momentum supplied from the wind is modeled as follows: O M OTx= 8T~~ r: ' + (2.6) 5T + x a y p. oM +-Ty + OTy -y (2.7) O x 8~Y P" where p.. is the density of water and M and My are the x and y momentum components, respectively, which defined as = FY ) cosOdOdf (2.8) M 1.=g~ Jo iC(f-" o 8 My = I F(f, ) sin OdOdf (2.9) M 1g0 0o C(f) where F(f, 0) is the wave energy spectrum as a function of frequency f and direction 0, and C(f) is the phase speed. If we assume that the deep-water linear wave theory applies, then the group velocity is C(f) C9(f) = 2 (2.10) From equations 2.8, 2.9 and 2.10, the components of the momentum flux tensor are g oo 21r F( f, 0) T uz = cos2 OdOdf (2.11) 2 o o C(f) T = Ty0 F(f, o) 2sin 0 cos OdOdf (2.12) Tx ~ =O JO C(f) T,, Joo 2 sin2 Od~df (2.13) Furthermore, if we assume that the wave energy E(f) is distributed about the mean wave direction 00 with cos2 and there is no energy for 10 0ol > 7r/2, then 2 F(f, 0) = -2 E(f)cos2(0 00) (2.14) If 00 is independent of frequency, then the momentum fluxes can be expressed in terms of 00 and variance oo a2 = E(f)df (2.15) The integration of Equations 2.11, 2.12 and 2.13 yields g O2 C 2 0 2 T = cos 0 + (2.16) Ty = TY g cos 0o sin Oo (2.17) TYY = g 42 sin2 00 + -2 (2.18) The relationship between the variance a2 and the momentum components can be derived by assuming the average JONSWAP spectrum (Hasselmann et al. 1973) E(f)= ag'(2r)-4f5exp +n3.3exp (f -1) (2.19) 9 o=0.07 f fm a=0.09 f > fm (2.20) where a is the Phillips equilibrium range parameter and fm is the peak frequency. Donelan (1977) eliminated the fetch from the problem to obtain a = 0.0097 (U- )2/3(2.21) where C, = g/27rfm is the group velocity of peak energy frequency and U is the wind velocity at 10 m height. Substituting equation 2.21 into equation 2.19 and integrating the JONSWAP spectrum approximately yields = L (2.22) IMI g and a2 = 0.30ag2(27r)-4f,4 (2.23) where IMI is the magnitude of the momentum vector (M,My). The right-hand sides of equations 2.8 and 2.9 repesent the source of momentum from the wind. Donelan (1977) used the following empirical formula - = 0.028DlU 0.83Cp[(U 0.83Cp) (2.24) PW where Df is the form drag coefficient defined here as Dy = [0.4/ln(50/a)]2 with a in meters. The empirical factor of 0.028 is the fraction of stress that is retained by the waves. 2.4 A Finite Depth Wind Wave Model by Graber and Madsen The finite-depth wind-sea model, developed by Graber (1984) and verified with data obtained during a complex North Atlantic frontal system by Graber and Madsen (1988), will be denoted hereafter as the GM model. The GM model is an outgrowth of the deepwater Hybrid Parametric (HYPA) wave model developed by Hasselmann et al. (1976) and Gunther et al. (1979). The GM model is a decoupled parametric wave model based on the conservation of energy flux for arbitrary water depth. The windsea is described by 10 the JONSWAP parameter set and by a directional parameter representing the mean direction of the windsea spectrum. The depth dependent frequency spectrum is assumed to have similarity shape which is related to the deep water energy spectrum by multiplying a depth dependent transformation factor. This factor affects the spectral shape of the highfrequency part which is proportional to f-s in deep water and to f-3 in shallow water. The directional dependence of the spectrum is assumed to be cos2(0 00), where 00 is the mean wave direction. The windsea model explicitly accounts for finite depth effects such as shoaling,refraction, dissipation by bottom friction, finite depth modification of atmospheric input, and nonlinear wave-wave interaction source terms. Based on an energy flux, transport equations of the full set of prognostic parameters is derived including finite depth effects. The equations of the prognostic parametric variables are solved on a finite difference grid by means of the Lax-Wendroff method. Swell is treated in a decoupled spectral fashion and swell characteristics are considered straight and effects of refraction are disregarded for simplicity. However, shoaling and dissipation of energy by bottom friction are included. 2.4.1 Theoretical Description of GM model The surface wave field is generally described in terms of the variance spectral density F(k) of the surface gravity waves in directional wave-number space. F is assumed to be a slowly varying function of position F and time t. The radiative-transfer equation for the surface wave field with variable bottom topography was originally given by Hasselmann (1960) OF d:F dk -t- + Tt-VF+ T-V F = T~;i )(2.25) where = Cg is the group velocity,k= (ks, k,) and F = (x, y) are the wave number and position vector, respectively, Vj and VE are the horizontal gradient vector operator of the position vector E and wave number vector k, respectively. The source function T(k; 5, t) represents the net transfer of energies to, from, and within the spectrum at the wave number due to all interaction processes which affect the component k. The terms of the left-hand 11 side of equation 2.25 represent, respectively, local accumulation (term 1), propagation (term 2), and combined refraction and shoaling (term 3). In addition, the kinematics of wave propagations are described by ray theory -- + VW = 0 (2.26) and the dispersion relationship for linear surface gravity waves w2 = gk tanh kh (2.27) where h = h(E) is local water depth. In practice, it is easier to collect data on the directional frequency spectrum E(f, 0; E, t) rather than the directional wave number spectrum F(k;i,t). These alternative spectral representations are related by F(k;i,t)dk = F(k,0;5,t)kdkdO = E(f, 0; i, t)dfdO (2.28) where 0 = tan-'(km/ky) is the direction of the wave number vector k (k = k cos 0, ky = k sin 0) and k = I$. From equation 2.28, the following relation can be deduced F(k;5,t) = J E(f,0;5,t) (2.29) where J is the Jacobian of the transformation S= O(fO) CC g = 10, (2.30) g(kk,) 2rw' C k Substitution of equation 2.29 into 2.25 gives a transport equation for the directional frequency spectrum 10 8 8 (CCgE) + cos 08-(CCgE) + sinO (CC9E) 11 DC 9C\ 8 22rw Cg (Ct E + x FoYO + sin 0C cosO ) a(CC9E) = 2rw S (2.31) S Ox -y 8 C9 12 where S(f,0;5,t) becomes the forcing term. The detailed derivation of equation 2.31 is shown in Appendix A. The energy flux density C is given by the product of depth-dependent group velocity and wave energy spectrum as follows: (f,0,h) = Cg(f,h)E(f,0,h) (2.32) Formulation of the problem is completed by specifying the source term S = S(f, 0, F, t). Consistent with linear wave theory, the source term is considered as the linear superposition of a number of mechanisms including atmospheric input by the wind, nonlinear transfer by resonant wave-wave interactions, and energy dissipations. The GM model adopted the depth-dependent frequency spectrum which was originally developed by Kitaigorodskii et al. (1975) and verified in the experiment by Bouws et al. (1985) as follows = ~ 22 E(f, 0, h) = T(wh)Ej(f) cos (0 0o) (2.33) where the depth dependent transformation factor is defined as 1 T(Wh) =(2.34) X(h) = x2[1 + W(x22- 1)] (2.34) where x is the solution to the transcendental equation X tanh(whX) = 1 and wh = 2irf(h/g)1/2. The derivation of equation 2.34 is shown in Appendix B. The one-dimensional windsea spectrum, Ej(f) has the JONSWAP spectral shape which is determined from the set of five free parameters ai = [fmO, a,7,a ,aI] Ej(f; ai) = EpM(f; ai)7exp[-((f/fm-1)/22] (2.35) where EpM is the Pierson and Moskowitz (1964) spectrum: Epm(f; ai) = ag2(27r)-f-5 exp [-4 (2.36) As in equation 2.33 the cosine-squared angular spreading function about the mean windsea direction (-( < 0 < Z) is adopted. The mean windsea direction, 00 which is a sixth spectral parameter, is defined as follows. 00 = tan-1 (0) (2.37) where . = 1 ,"1.jI1 sin Odfd9 = J. j 1 cos Odfd6 (2.38) 2.4.2 Atmospheric Forcing by the Wind The GM model adopts a parameterization of Miles' mechanism based on the results of Snyder et al. (1981) for atmospheric forcing by wind: Ti.(f,0,h) = (f,0,h)E(f,0,h) (2.39) where the growth function 08(f, 0, h) is given by 0, <1C (2.40) where p,, and pw are the densities of air and water, respectively, w = 27rf is the wave angular frequency, C is the depth-dependent phase velocity, 0 and 0 are the wave and mean wind directions, respectively, and U11 = U10 cos(0 9) is the wind speed at 10 m parallel to the wave direction 0. The coefficient B was determined by Snyder et al. (1981) to vary from 0.2 to 0.3. This was determined from direct measurements of the work done by wave-induced air pressure fluctuations over the sea surface. The heuristic lower limit of the windsea peak frequency is generally accepted as the frequency of a fully-developed sea (Pierson and Moskowitz, 1964). At the Pierson-Moskowitz frequency, fPM, the parameter U11/C = 0.82 in deep water. Adopting this concept also for fully developed sea states in finite depth leads to fPM = 0.13gtanh kpMh (2.41) U10 cos(Oo 0) In the deep-water limit this expression reduces to the original Pierson-Moskowitz relation. In the shallow-water limit, for which Cma-(fPM) = (gh)'l2 (2.42) it would not be possible to reach fully developed conditions if U11 > 0.82(gh)'/2. 2.4.3 Nonlinear Energy Transfer by Resonant Wave-Wave Interactions Atmospheric forcing provides energy to gravity waves and resonant wave-wave interactions among wave components redistribute energy within the spectrum. The general form of the "exact" nonlinear transfer terms Ta; is given by the Boltzmann integral which expresses the rate of change of energy of the wave spectrum (k) at wave number k4 due to nonlinear wave-wave interaction (Hasselmann and Hasselmann, 1985) as: T.(k4) '= 46(k + k2 3 k4) x 6(W +W2 w3 4) [N1N2(N3 + N4) N3N4(N1 + N2)]dkldk2dk3 (2.43) where Ni =- N(ki) = #(k)/wj stands for wave action densities, wi = (gki tanh kih)'/2 are the wave frquencies corresponding to the ith wavenumber ki, and a represents a complex scattering coefficient describing the coupling strength of four-way resonant interacting wave modes. Hasselmann et al. (1973) deduced the general form of the nonlinear transfer scales as Tnt(f,0) = 3 2 f-40(f/fn,O) (2.44) where P is a dimensionless function describing the spectral shape. Based on similarity arguments, Herterich and Hasselmann (1980) calculated the finite-depth interaction for a narrow-band wave spectrum. The finite-depth source function of the one-dimensional nonlinear transfer can be scaled by a depth-dependent factor, R, i.e., S.I(f,h) = R(whm)Snl(f, oo) for Whm > 0.7 (2.45) Confirmation of Equation 2.45 and the validity of shape similarity for directional distributions was made by Hasselmann (1981) who computed the interaction rates for a representative set of spectra F(f, 0, h) from the "exact" nonlinear transfer integral 2.43. Examination 15 of the results of Hasselmann and Hasselmann (1985) leads to the following expression for the proportionality factor X4 R = R(whm) X (2.46) - [1 + .(X2 1)]2 (2.46) 2.4.4 Dissipation by Bottom Friction The average rate of energy dissipation in an oscillatory wave boundary layer (Kajiura, 1968), or for random wave field (Hasselmann and Collins, 1968) can be expressed as BE --= -Edi, = -F- u (2.47) where the overbar denotes averaging over the wave field. Equation 2.47 can be expressed in terms of the frequency-directional spectrum as: OE 1 fw2 2= 1 2E(f,O). < ub > (2.48) at 2 g sinh 2kh where the root mean square bottom velocity < Ub > = Ubr = 2 2 sinh kh E(f, 0, h)dfdO) (2.49) is representative of the near-bottom velocity field. CHAPTER 3 WAVE BOUNDARY LAYER AND BOTTOM SHEAR STRESS 3.1 Introduction The flow field in a shallow lake such as Lake Okeechobee is determined by a number of mechanisms, including wind-driven currents, wind waves, density differences due to changes in water temperature and suspended sediment concentration, and the water-surface slope, etc.. Sediment particles are suspended from the bottom due to turbulent shear stress at the sediment-water interface and transported within the water column by advection and diffusion of the flow field. The bottom shear stress due to slowly varying currents and oscillatory waves is affected by the highly nonlinear processes within the bottom boundary layer. For sediment transport studies in shallow waters, the bottom boundary layer dynamics is extremely important. The presence of waves is known to greatly enhance the bottom shear stress and resuspension of sediments, moreover, the surface waves in shallow waters are affected by the boundary-layer dynamics through the dissipation of wave energy. For non-cohesive sediments, the volume flux (q,) of sediment transported in the bottom boundary layer in terms of bed load or suspended load is related to the n-th power of the friction velocity, u., i.e. q, = Const(u.)n, where u. = b; n is bottom shear stress, and p is the fluid density. Various measurements have shown that the exponent n normally lies in the range 3 < n < 7 (Dyer and Soulsby, 1988). For cohesive sediments, the rate of erosion of sediments is proportional to the n-th power of the excess bottom shear atress with n ranging from 1.2 to 4 (Sheng, 1986, Lavelle and Mofjeld, 1987). Thus, accurate determination of sediment transport rate relies on the accurate estimation of bottom shear stress. Basic difficulties in estimating the bottom shear stress are associated with the following facts: " The wind sea is essentially a random wave field. Therefore, the bottom shear stress calculated from such deterministic wave parameters as significant wave height (H11/3), and root mean square wave height (Hrms), and wave period (e.g., peak wave period, significant wave period, zero up-crossing wave period, etc. ) could be inadequate for the quantitative estimate of sediment transport. It should be noted that the resuspension of sediment is by the instantaneous bottom shear stress, and not by the maximum bottom shear stress or mean bottom shear stress of representative waves. " One of the most important parameters in the bottom boundary layer dynamics is the roughness height zo, at which the time mean velocity vanishes. For a natural bottom, there is great uncertainty in quantifying the roughness height because of its dependence on sediment properties, shape of bed form, and even flow parameters such as depth and velocity (Grant and Madsen, 1986). This chapter presents the basic concept and literature review of the oscillatory wave boundary layer, a description of the turbulent wave boundary layer according to the models of Jonsson (1963, 1966, and 1978) and Kajiura (1964, 1968). 3.2 Bottom Boundary Layer Boundary layers are typically described in terms of characteristic length and velocity scales that divide the boundary layer into at least two regions: one dependent on the absolute velocity and is directly influenced by the wall boundary, and the other dependent only on the relative external driving velocity and the overall scale of the boundary layer. Dimensional analysis leads to the following result for the velocity: u~z -7-z (3.1) where 0 is a universal function to be determined, 6 is boundary layer thickness scaled by u./w, r. is von Kirmgn's constant, and w is the wave angular frequency. 18 The outer limit (z/zo -+ o, z/6 finite) corresponds to a velocity-defect law: U U00 u = 01(z/6) (3.2) The inner limit ( z/6 --+ 0, z/zo finite) corresponds asymptotically to a constant-stress layer - = 02(Z/Zo) (3.3) The actual form of the velocity profile can be found by assuming that there exists an overlap layer, zo < z < 6, where both equations 3.2 and 3.3 are valid simultaneously (Yaglom, 1979). It follows from equating the derivatives of equations 3.2 and 3.3 within this overlap layer that both functions are logarithmic: u Aln(z/6) + B (3.4) U.,. and = Aln(z/zo) (3.5) where 1/A is the von Kirmin constant r (r ranges from 0.38 to 0.41 for fully rough turbulent flows). The value of the constant B is less well known. Yaglom (1979) suggested that B is slightly above 2 (with a mean of about 2.35 for all suggested values) in boundary layer flows. For oceanic boundary layers, no direct determinations of the values of B have been reported. In a time-dependent flow due to waves and/or currents, the linearized equation for the boundary layer flows is: Ou 1 Op +O -/p) (3.6) Ot p Ox Oz where u is the ensemble mean velocity, p is the ensemble mean pressure, r is the shear stress, and x is the flow direction and z is the vertical axis upward from the bottom. Invoking the usual boundary layer assumption that the pressure gradient is imposed by the free stream flow, u00, equation 3.6 becomes O U U) 9(7/p) at (9p (3.7) Ot O Table 3.1: Nikuradse eqivalent sand-grain roughness height for flat bed. Dp is the diameter of the grain corresponding to p % finer in weight. (van Rijn, 1982) Ackers-White (1973) ks = 1.25D35 Einstein (1950) k3 = D65 Engelund-Hansen (1967) k. = 2D65 Hey (1977) k, = 3.5D84 Kamphuis (1975) k. = 2.5D90 Mahmood (1971) ks = 5.1D84 van Rijn (1982) ks = 3D90 Close to the wall boundary (z/zo -- 0), unsteadiness vanishes (o2 -+ 0) and the solution to equation 3.7 is the quasi-steady law-of-the-wall as shown in Equation 3.5. The logarithmic velocity region depends on the roughness length zo and requires the presence of an overlap layer in z, where z0 < z < 6, to exist. As mentioned earlier, one of the most important parameters in the bottom boundary layer is the bottom roughness height zo, at which the mean velocity is zero. The roughness height is not measurable but is related to the geometrical scales of various roughness elements on the bottom. The mean velocity near the wall can be written in general form z- -, d., U =Uf(Zdk v (3.8) where d. is the displacement height or zero displacement (by analogy to the concept of the displacement thickness in boundary layer theory), and k is the mean height of the roughness elements (Monin and Yaglom, 1971). The roughness height, k, is usually related to Nikuradse equivalent sand-grain roughness height, k,. In general, k' is determined experimentally in terms of a measurable roughness dimension, and k is then determined from k8/k which is invariant for geometrically-similar roughness arrays (Wooding et al. 1973). For flat beds of sand, it is expected that the roughness height would be related to the diameter of the largest grains (D65, Ds4, and Dgo). Table 3.2 shows the different values of k, suggested by many authors (van Rijn, 1982). 20 The flow velocity at some distance above the bed is related to the bottom shear stress through the quadratic friction law: rb = pf lublub (3.9) where f.. is the wave friction factor. Alternatively, one can write: -- = f, /2 (3.10) ttb where u. = %/f-/p, ro, is the maximum bottom shear stress, and ub is the amplitude of the wave orbital velocity at the bottom. Based on laboratory measurements, Jonsson (1963, 1966, and 1978) and Jonsson and Carlsen (1976) developed a simple, semi-empirical formula for the friction factor f". Kajiura (1964, 1968) developed an analytic model by using a time-invariant eddy viscosity for steady turbulent boundary layers. Analytical models similar to that of Kajiura, based on a time-invariant eddy viscosity, have since been proposed by Grant (1977), Smith (1977), Grant and Madsen (1979), Brevik (1981), and Long (1981). Kamphuis (1975) reported an extensive, purely empirical study in which the maximum boundary shear stress in a turbulent oscillatory flow was measured directly. Subsequent discussion by Grant (1975) and Jonsson (1976) indicate that the above semiempirical, theoretical, and purely empirical studies give essentially the same maximum bed shear stress over a wide range of bed and flow conditions. (Refer to Trowbridge and Madsen, 1984) Horikawa and Watanabe (1968) found that the eddy viscosity does vary significantly during the course of a wave cycle. Trowbridge and Madsen (1984) developed a time-varying eddy viscocity model which treated the turbulent oscillatory boundary layer at a level of approximation consistent with linear wave theory. To this degree of accuracy, prediction of the time-averaged energy dissipation rate are 20 30 % smaller than those obtained from time invariant eddy viscosity models. Trowbridge and Madsen (1984) also advanced the solution to second order in wave steepness. The second-order, wave-induced mass transport is shown to depend critically on temporal variation of the eddy viscosity. The reversal of 21 the mass transport produced by relatively long waves is qualitatively simulated. This result is believed to explain the experiments of Inman and Bowen (1962) which measured the rate of sediment transport resulting from progressive waves combined with a very weak net flow rate in the direction of wave propagation. They found cases in which an increase in the net flow rate caused a decrease in the sediment transport rate. This result cannot be reproduced by a time-invariant eddy viscosity model. The Trowbridge and Madsen (1984) results demonstrate that the first-order solution depends only slightly on time variation in the eddy viscosity and is more sensitive to the more proper treatment of vertical variability. Most oscillatory turbulent boundary-layer models are based on the eddy-viscosity assumption. If sufficient data are available to establish the validity of the required parameters in the models, then the predictions of models in that particular application give reasonably acceptable results. However, when sufficient data axe not available and the parameters for a specific application must be extrapolated from much different situations, the resulting predictions are highly speculative (Sheng, 1982). Sheng (1984) modified a turbulent transport model (often called "second-order closure model"), which was originally developed by Donaldson (1973), for application to coastal environment. In addition to predicting the mean flow variables in bottom boundary layers, the second-order closure model can predict turbulent quantities such as the Reynolds stresses, u~,the turbulent kinetic energy, q'/2, and the thickness of log layer. Due to the robust physics contained in the model, the second-order closure model is particularly valuable in dealing with sediment transport in shallow waters where highly oscillatory flow with appreciable density stratification due to variations in temperature, salinity and suspended sediment concentration. Although the second-order closure model gives very accurate results, the model is rather complicated and requires more computational effort than simpler eddy viscosity models. Thus, Sheng and Villaret (1989) developed a simplified (T.K.E. closure) version of the comprehensive second-order closure model to examine the effect of flow-sediment interaction on the erosional behavior of fine sediments. Since the present study is focused on the applicaton and comparison of different wave models and the estimation of resulting bottom shear stresses, we decided to use the simplest turbulent wave boundary layer models of Kajiura and Jonsson for the calculation of bottom shear stresses. 3.3 Jonsson's Integrated Momentum Approach Based on the velocity measurements in a large oscillatory water tunnel, Jonsson (1966) and Jonsson and Carlson (1976) developed a simple theory for the calculation of the boundary layer thickness, the friction factor, and the dissipation of wave energy in an oscillatory rough turbulent boundary layer. This theory is based on the assumptions that the velocity profile is logarithmic throughout the entire wave boundary layer and that the phase shift between maximum shear stress and maximum amplitude of orbital velocity is negligible. Jonsson's integrated momentum approach differs from others in that the momentum equation need not be satisfied directly. A brief description of Jonsson's approach is given in the following. In the presence of a viscous sublayer, the shear stress near the bottom is described as: 7- = PV-Oz puIwI (3.11) where v is the kinematic viscosity, and u' and w' are the fluctuating velocity components in the x and z directions, respectively. The boundary conditions for equation 3.7 are u=O for z=O (3.12) U = UOO= Uoom coswt for z o (3.13) where uo,,m is the maximum free stream velocity and w is the angular frequency. From equation 3.7 and boundary conditions 3.12 and 3.13, we have: az=O"=) du (3.14) 9z zo= di 23 It is interesting to note that the shear stress gradient at the wall is determined exclusively by the flow outside the boundary layer (free stream) in equation 3.14. Integration of equation 3.7 with boundary condition 3.13 yields: 7 = (noo U)d - = O(u dz (3.15) If the velocity field u(z,t) is known, the shear stress can be calculated everywhere from equation 3.15. Adopting the quadratic friction law for the bottom shear stress, Jonsson developed a formula which related the friction coefficient f,, and the Nikuradse sand roughness height k, for a given wave condition. For hydraulically rough walls, the following expressions of boundary layer thickness 6 and the friction coefficient f, were obtained: 306 306 12 (3.16) klogxo k, = 1.20 (316 1 1 (aoom 4V + log10 4V- -0.08 + loglo k (3.17) 4Vf 45 k, The detailed derivation of equations 3.16 and 3.17 are shown in Appendix C. Figure 3.1 shows the variations of f,, with aoom/ks as given by equation 3.17. Based on the test results of Bagnold (1946), Jonsson suggested a constant value of f = 0.30 for aoom/ks < 1.57 as shown in figure 3.1. For hydraulically smooth beds, Jonsson (1967) suggested the following expressions 6 0.0465 (318) = (3.18) aoom /-10 f,,, = 0.09Re-0.2 (3.19) where Re = uoomaeoom/v is the 'amplitude Reynolds number'. For laminar flow, the last term on the right-hand side of equation 3.11 is zero, and a simple analytical solution to equation 3.7 was obtained (Lamb, 1932, pp 622-623) U = Uom costt e- z cos(wt fiz)] (3.20) Figure 3.1: Comparison of different formulas for the friction coefficient f varies with a/ks (Sleath, 1984). with 8 = /= V7{) and T is the period of wave. The shear stress is obtained by combining equations 3.11 and 3.20 S= Vvooume-IPz cos(wt Oz + 7r/4) (3.21) P From equation 3.20, the boundary layer thickness is b = = V (3.22) The maximum bottom shear stress is defined by: 1 ,m6 = fw PUlomIUoo m (3.23) For laminar flow, the friction coefficient is deduced from equations 3.21 and 3.23 2 f,- (3.24) 25 In order to apply the preceding equations to a variety of flow conditions, the transition regimes among the laminar flow, smooth turbulent, and rough turbulent flow need to be clarified. Jonsson (1978) concluded that the laminar-smooth turbulent transition regime goes from Re = 10 to Re = 3 x 10', give or take a factor of two at both ends of intervals. For design purposes one can probably use Re = 105 as a transition value. For the transition between the hydraulically smooth bed and rough bed, Jonsson suggested aoom/ks = 45 which corresponds approximately to Re = 1.9 x 10'. 3.4 Kajiura's Model of the Bottom Boundary Layer in Water Waves One of the most consistent and detailed theoretical boundary layer theories available is by Kajiura (1968). Kajiura developed a model that calculates various characteristics of the wave boundary layer, such as the friction coefficient and universal profiles of stress and velocity in the defect layer. Introducing time-mean properties of the turbulence, he assumed a time-invariant eddy viscosity compatible with the two-layer model (wall and defect layer) of the turbulent boundary layer. By further assuming the shear stress to vary sinusoidally, he introduced a modified friction velocity S(r/p) (3.25) where fii is the amplitude of the bottom friction velocity, i.e. fu. = V in which -fb is the real part of the amplitude of the bottom shear stress. Differentiation of equations 3.7 and 3.25 with respect to t and z, respectively, yields: &92Urn u =0 (3.26) 19Z2 Ct Kajiura solved equation 3.26 together with: au r = p9tz (3.27) and variable eddy viscosity, Et, in the inner, overlap, and outer layer: v, Kfi.z, and Kfi.A, respectively, for hydraulically smooth beds and 0.185nfi~k,, Kit.z, and K6*uoo, respectively, 26 for hydraulically rough beds, where A = 0.05,t./w is the thickness of the wall layer, is the von karmin constant (= 0.04), K is the Clauser constant (= 0.02), and V* is the displacement thickness defined as * Amp[ (u, u)dz (3.28) The constants of integration are evaluated by making the velocity and velocity gradient to be continuous at the boundaries between the layers. The solution is straightforward but algebraically very laborious. Therefore, only the procedure to determine the bottom shear stress is given briefly as follows: 1. Determine wave properties, H. and Tp, water depth, d, and Nikuradse sand-roughness height, k,. 2. Compute wave number k from the linear dispersion relation W = gk tanh(kd) (3.29) 3. Compute the maximum orbital velocity, um, and orbital amplitude of the water particle orbital motion, am just outside the boundary layer. i'H8 1 ucom Tp sinh kd (3.30) aoom = -- (3.31)" 4. Compute the amplitude Reynolds number, Re U ,maOm (3.32) 5. Assume the initial value of maximum bottom shear stress during the wave cycle, Tbm 6. Compute the thickness of the viscous sublayer _12v -D = -12 = 12v (3.33) fi. AM 27 7. If k,/D < 0.4, use the hydraulically smooth bed expression for fu 2 f = for Re <2x 105 1 + log- -0.135+1ogv e for Re > 2 x 105 (3.34) 8.1VIj V7 If k,/D > 0.4, use the hydraulically rough bed expression for f, , = 0.25 for o < 1.67 k9 aom -2/3 acmf. = 0.35 aom / for 1.67 < a om -< 30 k, k8 0.98 +log 1 -0.25 + log m for am > 30 (3.35) 4v" + log 0.25 + log (aco o cr 4 4 4V k, k,8. Compute the bottom shear stress, 1 1m = Pfluom (3.36) 9. Iterate until the solution converges to within a certain limit, e.g., (9m)new (Tbm)old If (-m w 5 0.0001, then stop, otherwise go back to step 5 (nm)new I CHAPTER 4 MODEL PERFORMANCE IN IDEALIZED WIND FIELD 4.1 Introduction The three wind-wave models (SMB, GD, and GM) described in previous chapters will be compared for the case of steady and uniform easterly wind field with wind velocities of 5, 10, 15, and 20 rn/sec over Lake Okeechobee. The results showed significant spatial variation in wave parameters in Lake Okeechobee. Using the significant wave height and the peak wave period to represent a monocromatic wave, the bottom shear stress everywhere were computed through linear wave theory and Kajiura's oscillatory turbulent boundary layer model. 4.2 Comparison of Wave Models in Idealized Wind Fields The geometry and the bottom topography of Lake Okeechobee are shown in Figure 5.1. As can be seen in Figure 5.1, Lake Okeechobee is an extremely shallow lake with a maximum depth of about 5 meters. The depth variation in a cross section AA' shows that the eastern and central portions of the lake are relatively deep While the western part of th e lake is shallow with a very mild slope of approximately 0.00016. Since the -seasonal variation in water surface elevation can be as much as 2 meters in Lake Okeechobee and the shallow water wave models are sensitive to the change in water depth, it is necessary to use measured water level to adjust the water depth for wave model. The water depth shown in Figure 5.1 correspond to the water depth in fall of 1988. As shown in Figure 4.2, Lake Okeechobee sediments are composed of mostly fine-grained (silt and clay size) materials in the relatively deeper region and sand and shell in the shallower region of the lake. As was mentioned in Chap 3, the determination of roughness 29 heights in a natural lake is a difficult task especially over the muddy bottom. For the estimation of bottom shear stress, however, the Nikuradse roughness height, k,, was assumed to be 1 mm all over the lake. In order to compare the three wave models, the same computational grid with Ax = Ay = 2km was used as shown in Figure 4.3. The wind data of Lake Okeechobee during the deployment period, fall of 1988, showed that the strong winds with velocity about 10 in/sec were mostly blowing from east, north-east, and north. Based on this observation, the easterly winds with velocities of 5, 10, 15, and 20 in/sec were chosen to test the performance of the wave models in the Lake Okeechobee bottom topography, especially to compare the shallow water effects in the shallow western region. The SMB model assumes that the spatially uniform wind is blowing over uniform water depth over a sufficiently long distance for the waves to reach a steady state at the point of interest (i.e. fetch limited condition). Bretschneider (1958) suggested the procedure for computing wind waves generated onto the continental shelf from deep water shoreward with varying depth. According to the procedure, the bottom profile along the fetch is known and the traverse cross-section is divided into equal segments of about -5 to 10 miles or less in length, depending on the bottom slope. In order to get the wave height and period at the end of the each segment, the average depth, the refraction coefficient, shoaling coefficient, and the energy dissipation due to bottom friction are calculated over each segment. This procedure, however, can become quite involved for a large lake like Lake Okeechobee. Since Lake Okeechobee has a maximum fetch of about 50 km and has very mild bottom slope, a simpler method may be developed by using the average depth over the fetch to compute the wave height and period at the point of interest. This method may overpredict the wave height at the shallow water when the wind is blowing from deep to shallow water. Another simple method is to use the local water depth at the point of interest instead of the average water depth over fetch. The SMB model using average water depth will be denoted as SM B-AD model and the SMB model using local water depth will be denoted as SMB-LD model. These two methods will be compared with GM model and GD model. 30 The significant wave height, H,, peak wave period, Tp, and the maximum bottom shear stress, rb, during a wave cycle are computed at the center of the grid cell. Figures 4.4, 4.5, 4.6 and 4.7 show the variations of H8,Tp, and rb with the fetch distance along the cross section AA' shown in Figure 5.1. The numerical values of results are shown in Appendix D. For the 5 m/sec wind, the wave doesn't feel the bottom except in the very shallow western part of the lake where the water depth is less than 2 meters. As can be seen in Figure 4.4, there are no significant differences in H8, Tp, and 7b among the results of the 3 models. Compared to the other models, the GD model gives slightly lower values of significant wave heights and peak wave periods. It is interesting to note that the GM model and the SMB-LD model give almost the same significant wave height along the entire fetch. On the other hand, the GM model and the SMB-AD model yield almost the same peak wave period along the fetch. These trends remain quite the same for the stronger wind cases. For the case of 10 m/sec wind as shown in Figure 4.5, significant wave heights given by the various models show significant difference in the shallow region, but the peak wave periods given by the various models remain almost the same. As expected, the deep water GD model shows an increase in significant wave height as the fetch increases. The SMB-AD model shows no variation in H, when the fetch exceeds 30 km and the water depth is less than 3 meters. This is due to the fact that the increase in H, due to the longer fetch is balanced by the decrease in H, due to the shallower average depth. As was shown in Figure 4.5, the bottom shear stresses produced by the GD model and the GM model are different by almost an order of magnitude. Since wind velocities on the order of 10 m/sec can be commonly observed on moderately windy days over Lake Okeechobee, using the GD model will lead to excessive bottom shear stress and erosion of sediment. The results for wind velocities 15 m/sec and 20 m/sec are shown in Figures 4.6 and 4.7, respectively. It is interesting to note the similar behavior of H, given by the GM model and the SMB-LD model, and the nearly indentical behavior of Tp produced by the GM model and the SMB-AD model. It should be reminded that it is a very crude approximation to 4 A 0 M -2 -3 4 5 Figure 4.1: Geometry and bottom topography of Lake Okeechobee 2 KM 4. /D -2 KM / B 0 LC 05 E/ Figure 4.2: Location of research towers A, B, C, D, E, and F, with wind station L, and a 2 km computational grid used in Lake Okeechobee PEAT MARSH SAND/SHELL/MARL MUD ROCK Figure 4.3: Bottom sediment characteristic of Lake Okeechobee (from Reddy et al. 1988) D 0. 1. 2. 3. 4. 5. 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 7. 6. 5. 2. 1. 0. 1000.0 100.0 10.0 1.0 0.1 0.0 0. 5. 10. 15. 20. 25. 30. 35. 0. 5. 10. 15. 20. 25. 55. 40. 45. 50. 55. 30. 35. 40. 45. 50. 55. i (km) Figure 4.4: Comparison of wave models for 5 m/sec easterly wind. Variations of significant wave height, H3, peak wave period, Tp, and bottom shear stress, no, with fetch along the cross section AA' I I I I I I I I I I I I I I I I I I I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 0. E 2. 3. . 4. ILl 5MB-RD :....... 5. SMB-L: GO GM -.... 1.4 i i 1.2 1.0 0.8 .... -. (0 0.6 0.2 00 I I I I I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 7. 6. (D) 4 m - a- 2. 0. 0. I I I I I i I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 1000.0 o100.0 100.0~ ~ ....:...:..-g -11 10.0 0.1 I 0,0 I I I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. FETCH (km) Figure 4.5: Comparison of wave models for 10 m/sec easterly wind. Variations of significant wave height, II,, peak wave period, Tp, and bottom shear stress, n, with fetch along the cross section AA' 36 0. E" 2. :: 3. I - 4. UiJ" SMB-PD ............ C SMB-LD G/D G& 2.0 1.6 E 1.2 0.4 0.0 ; ' 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 7. 6. o 5. 0-. 2. I 1. 0. 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 1000.0 L 100.0 ,r ...... ..... ......... " 10.0 C ~' 1.0 0.1 I0 0 .0 I I I I I I I I I I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. FETCH (km) Figure 4.6: Comparison of wave models for 15 m/sec easterly wind. Variations of significant wave height, H,, peak wave period, Tp, and bottom shear stress, rb, with fetch along the cross section AA' I I I I I I I I I I I I I I I I I I I 37 0. 2. = 3. 0. 4. LU 5MB-RD 5. SMB-LD G/D G&M 2.0 7 1.6 -- E 1.2 ...... o 8_' ""S... N \ 0.4.. 0.0 I I I I I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 7. 6. - 5. (U 2. i 1. 0. 0 1000.0 o 100.0 o 10.0 C r 1.0 S 0.1 Cr I- i I I i i i i I i i I I I I I I I I I I 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55 ...........--I I I I I I I I I I I. 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. FETCH (km) Figure 4.7: Comparison of wave models for 20 m/sec easterly wind. Variations of significant wave height, H,, peak wave period, Tp, and bottom shear stress, rb, with fetch along the cross section AA' 38 use the average water depth over fetch or the local water depth at the point of interest to obtain the H, and Tp in the SMB model. The similarity between H, given by the GM model and the SMB-LD model is due to the fact that the depth-dependent transformation factor 1P(Wh) of Equation 2.34 employed by the GM model is only a function of local water depth and frequency. This means that the transformation of finite depth spectrum is independent of the path of the waves and consequently the significant wave height is strongly dependent on the local water depth in the GM model. It is generally accepted that in deep water the spectral peak of wind waves shifts toward lower frequencies as a consequence of the nonlinear energy transfer. Input from the wind occurs over the central region of the spectrum and wave-wave interaction processese rapidly redistribute the enhanced energy level towards lower and higher freqencies. In the high-frequency range, the input and nonlinear transfer terms are balanced by whitecapping dissipation processes. As surface wave propagates into waters of finite depth, the low-frequency spectral components interact with the bottom, thus leading to frictional dissipation of lower frequency wave components. This frictional dissipation tends to slow down the rate of wave energy migration towards lower frequencies or even totally reversing the trend of the shift (Graber and Madsen, 1988). The similar behavior of peak wave period along the fetch as given by the GM model and the SMB-AD model might be a coincidence. It is likely that the migration of peak wave period produced by a balance between the nonlinear energy transfer and the frictional bottom dissipation in the GM model is similar to the change of peak wave period due to a balance between the increase of fetch and the decrease of the average depth in the SMB-AD model. The relative performance of the various wave models might be evaluated in terms of the bottom shear stress produced from the wave parameters. As can be seen in Tables D-1, D.2, D.3, and DA4 (Appendix D), the different models give significantly different bottom shear stresses in the shallow water region (water depth less than 3 meters) especially for 39 high wind velocities. For example, the GM model gives bottom shear stresses of 16.98 and 18.36 dyne/cm2 for 15 and 20 m/sec wind velocities, respectively at 30 km fetch and 2.6 meters water depth. At the same location, the SMB-LD model gives bottom shear stresses of 26.94 and 38.45 dyne/cm2 for 15 and 20 m/sec wind velocities, respectively. The SMB-AD model gives 45.43 and 66.43 dyne/cm2 for 15 and 20 m/sec wind, respectively, and the GM model gives 100.70 and 241.21 dyne/cm2 for 15 and 20 m/sec wind velocities, respectively. Figures 4.8, 4.9, 4.10, and 4.11 show the contour plots of H3, Tp, the bottom orbital amplitude, Ab, and Tb obtained from the GM model for steady and uniform easterly wind velocities 5, 10, 15, and 20 m/sec, respectively. As the wind velocity increases, the regions with peak H, shift eastward to the deeper central part of the lake indicating that the wave heights become more constrained by the water depth in the shallow region. As shown in the bottom shear stress contours in Figure 4.8 and 4.9, the low bottom shear stress region is very closely correlated with the mud zone shown in Figure 4.2. Particularly strong bottom shear stresses occur in the marsh and sand/shell regions. This may imply that the fine sediments are resuspended due to the moderately strong wave action and carried by wind-driven currents to the deep eastern part of lake where most of fine sediments are deposited. Sheng et al. (1989) showed that under the easterly wind, opposing currents driven by the surface slope are found in the bottom layer. For strong winds or episodic storm events as shown in Figure 4.9 and 4.10, bottom shear stresses in the deeper mud zone appeared to be strong enough to resuspend the sediment particles. Figures 4.12, 4.13, 4.14, and 4.15 show the contours of H,, T, Ab, and Tb obtained from the SMB-LD model and Figures 4.16, 4.17, 4.18, and 4.19 are the model results of the SMB-AD model for wind velocities 5, 10, 15, and 20 m/sec, respectively. The spatial distributions of significant wave height -and bottom shear stresses are similar between the GM model and SMB-LD model. The spatial distributions of peak wave period are similar between the GM model and the SMB-AD model. SIGNIFICANT WAVE HEIGHT (PetersV ORBITAL AMPLITUOE ( eters) BOTTOM SHEAR STRESS (dyne/sq. cWJ Figure 4.8: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, n, obtained from the GM model for 5 m/sec easterly wind. PERK WAVE PERIO0 (sec) SIGNIFICANT HAVE HEIGHT metersi ORBITAL AMPLITUOE (meters) BOTTOM SHEAR STRESS (dyne/sq. cm) Figure 4.9: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, b, obtained from the GM model for 10 m/sec easterly wind. PEAK HAVE PERIOD (sec) SIGNIFICANT H VE HEIGHT (meters) ORBITAL AMPLITUDE ( meters) BOTTOM SHEiAR STRESS (djne/sq. cm) Figure 4.10: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, ir, obtained from the GM model for 15 m/sec easterly wind. PEAK HAVE PERIOD (sec) SIGNIFICANT WAVE HEIGHT (Meters) ORBITAL AMPLITUDE (Meters) BOTTOM SHEAR STRESS (dyne/sq. cm) Figure 4.11: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, ,, obtained from the GM model for 20 m/sec easterly wind. PEAK HAVE PERIOD (sec) SIGNIFICANT WAVE HEIGHT (metersW ORBITAL AMPLITUDE (meters) BOTTOM SHEAR STRESS (dyne/sq. cm) Figure 4.12: Contours of the significant wave height, H8, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-LD model for 5 m/sec easterly wind. WAVE PERIOD (3ec) SIGNIFICANT WAVE HEIGHT (eters3) ORBITAL AMPLITUDE meterss) BOTTOM SHEAR STRESS (dyne/sq. cm) Figure 4.13: Contours of the significwnt wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-LD model for 10 m/sec easterly wind. HAVE PERIOD (sec) SIGNIFICANT HAVE HEIGHT (eters) ORBITAL AMPLITUDE (meters BOTTOM SHEAR STRESS (dyne/sq. cm) Figure 4.14: Contours of the significant wave height, H, peak wave period, T,, bottom orbital amplitude, Ab, and bottom shear stress, ri, obtained from the SMB-LD model for 15 m/sec easterly wind. WAVE PERIOD (sec) SIGNIFICANT HAVE HEIGHT (xMeters) ORBITAL AMPLITUOE (meters BOTTOM SHEAR STRESS (dyne/sq. cm) Figure 4.15: Contours of the significant wave height, H., peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, 7b, obtained from the SMB-LD model for 20 m/sec easterly wind. WAVE PERIOD (sec) SIGNIFICANT HAVE HEIGHT (Peters) ORBITAL AMIPLITUODE (eters) BOTTOM SHEAR STRESS (dyne/sq. cm) Figure 4.16: Contours of the significant wave height, IIH,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, 'r, obtained from the SMB-AD model for 5 m/sec easterly wind. HAVE PERI0 (sect SIGNIFICANT NAVE HEIGHT (metLers) ORBITAL AMPLITUDE (meters) BOTTOM SHEAR STRESS (dyne/sq. cm) Figure 4.17: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, 6, obtained from the SMB-LD model for 10 m/sec easterly wind. I WAVE PERIOD (sec) SIGNIFICANT HAVE HEIGHT (meters) ORBITAL AMPLITUDE meters BOTTOM SHEAR STRESS (dyne/sq. cal Figure 4.18: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-LD model for 15 m/sec easterly wind. HAVE PERIOD (sec) SIGNIFICANT WAVE HEIGHT (meters) ORBITAL AMPLITUDE (Peters BOTTOM SHEAR STRESS (dyne/sq. cal Figure 4.19: Contours of the significant wave height, H, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, m, obtained from the SMB-LD model for 20 m/sec easterly wind. WAVE PERI0 (sec) CHAPTER 5 MODEL PERFORMANCE IN REAL WIND FIELD 5.1 Introduction The GM model and the SMB model are tested with the measured wave data obtained from subsurface pressure transducers at three stations in Lake Okeechobee. The wind data collected from Anemometer at the wind station are used to force the finite-depth GM model and the 5MB model. Comparisons between the hindcast and measured waves include the significant wave height, peak wave period, zero up-crossing wave period, and one-dimensional wave energy spectral shape. 5.2 Field Data Analysis For Lake Okeechobee Phosporous Dynamics Study, wave, current, turbidity, and temperature data were collected at six platforms in Lake Okeechobee during 20 September to 5 November, 1988, by the Coastal and Oceanographic Engineering Department of the University of Florida (Sheng et al. 1989). The subsurface pressure data were collected at six locations, (site A, B, C, D, E, and F) in Figure 4.2. Detailed information of the wave gages is shown in Table 5.1. The wind data were measured by anemometer at 8 meters above the water surface at the wind station L005 in Figure 4.2, which has been maintained by the South Florida Water Management District. The subsurface pressure data obtained from Lake Okeechobee were analyzed using program PREANL.FOR, which was developed at the Coastal and Oceanographic Engineering Department of the University of Florida. The surface waves are recovered from subsurface time series- pressure data through linear wave theory. Although the measurement of waves with pressure transducers has been used widely, controversy still exists over the adequacy of Table 5.1: Locations of the pressure transducers deployed in Lake Okeechobee Gage Water Location of Station pressure depth depth ho/h station transducer ho h Latitude Longitude A COE 55695 1.33 3.9 0.34 27 06.31 80 46.21 B COE 55696 0.71 2.7 0.26 27 02.78 80 54.31 C COE 48228 0.79 4.6 0.17 26 54.10 80 47.36 D COE 55694 1.04 4.3 0.24 26 58.47 80 40.34 E COE 55699 0.61 2.7 0.23 26 52.81 80 55.96 F COE 55697 0.61 1.8 0.34 26 51.90 80 57.09 the transfer function from subsurface pressure to surface wave height using linear wave theory. However, according to Bishop and Donelan (1987), linear theory is generally adequate to compensate pressure records to give surface wave heights to within five percent. The pressure data measured from the subsurface gage are considered to be a linear summation of various contributing components: P(t) = P. + pg(h ho) + pgKpi7(t) (5.1) where P(t) is the measured pressure at the gage and P. is the atmospheric pressure at the surface, h and ho denote the mean water depth and gage height above bottom, respectively, i7(t) is the surface wave, Kp is the pressure response function, p is the water density and g is the gravitational acceleration. The pressure induced by surface gravity waves is P = pgKp7 = P(t) (5.2) where T denotes the time-averaged mean pressure at the gage and the pressure response function, Kp is usually defined through linear wave theory Kp(wn) = coshk,,(h + z)/coshk.(h) (5.3) where z is the depth of gage location and w,, and 54 kn are wave frequency and wave number, respectively. Although equation 5.2 relates 7 and P in the time domain, the surface wave information is commonly recovered in the frequency domain (in terms of wave energy spectrum) because Kp can be more readily evaluated in this domain. The spectral technique assumes that P(t) is expressible as a summation of harmonics: P(t) = pg An cos(wnt En) (5.4) n where en is the phase angle. The surface wave is then expressible as: A. cos(Wt Cn) (5.5) through linear wave theory. At present, the values of An and Kp(wn) for every value of w" are most efficiently obtained from the Fast Fourier Transform (FFT) of the pressure data. The major difficulty of applying FFT_ method for real field data is that 1/Kp(w) increases almost exponentially with frequency in the high-frequency range. Therefore, without a prior knowledge of noise level, it is difficult to predetermine a cutoff frequency (Lee and Wang, 1984). However, the present program imposed the cutoff frequency of 0.7 Hz. The subsurface pressure data were sampled every half hour for 17 minutes at a sampling rate of 2 Hz. Thus, each burst of data contains 2048 digitized data points. Bad data such as flat points and anomalous peak points have been removed by the bad-point correction algorithm of PREANL.FOR. The averaged energy spectrum are obtained such that the total of 2048 data points axe divided into 32 segments of data using 50 % overlapping factor, therefore each segment contains 128 data points. In order to reduce the undesirable effects related to spectral leakage, the time series of each segment with length of 64 seconds was analyzed by using the cosine bell window. Also low frequency components of less than 0.1 Hz due to currents were eliminated by using low pass filter. The resultant spectral estimates have 48 degrees of freedom, with the expected spectral value with in a factor of 0.7 to 1.23 of the sample value at 80 % confidence limits. The typical energy spectrum obtained from analysis is shown in Figure 5.1. From the spectrum, the significant, wave 55 0.4 C.CU 0. L LUQ E 0 I 0.2 z I. 0.0 0.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure 5.1: Typical wave spectrum from Lake Okeechobee. height, peak wave period, and zero up-crossing wave period were determined as follows: H, 4/-m (5.6) T= Fm2 (5.7) where T, is the zero up-crossing wave period and mr and m2 are'the first and second order moment of the spectrum. 5.3 Comparison of Model Results vs. Measured Data For the bench mark test of the GM model and the SMB model vs. measured wave data, the period from October 7 to October 12 was selected because of the sustained wind condition. During this period, only three pressure gages at stations B, C, and E were working properly. As shown in Figure 4.2, the station B is located in 2.7 m depth of water 56 close to the northwest vegetation area. It should be mentioned that the shaded area in Figure 4.2 is actually a samll island, but for the sake of computational convenience, it is treated as very shallow area (0.5 m depth). Stations C and E are located in 4.6 m and 2.7 m depth of water, respectively. 15 minutes averaged wind data were available at station L005. Wind velocities were vector averaged each hour to force the wave models and the wind fields were assumed to be spatially uniform over the lake for every hour. 5.3.1 Comparison of the GM Model results with Measured Data In the GM model, the bottom friction coefficient needs to be specified as input. Figure 5.2 shows the sensitivity of model results to variation in friction coefficient at stations B, C, and E on Octorber 7, 1988. During this period, the GM model overestimated the smaller waves but underestimated the larger waves. The variation in significant wave height due to the varying friction coefficients is most notable during high winds from hour 19 to 24 on Octorber 7. During the high wind period, the wave spectral shapes at station E corresponding to various friction coefficients are shown in Figure 5.3. Since the friction coefficient is a function of the bottom roughness and the flow characteristics at the bottom, it would have been more proper to provide the roughness height as the model input. However, due to the uncertainty in determining the roughness height and also for the sake of simplicity, the GM model uses the friction coefficient instead of roughness height as an input parameter. As shown in Figure 5.2 and 5.3, lower friction coefficient gave better estimation of significant wave height yet higher friction coefficient gave better result of wave spectral shape. The friction coefficient of 0.02 was chosen as an input to GM model for Lake Okeechobee application. For the six day period from October 7 to October 12, the hindcast results of the GM model are compared with measured data at stations B, C, and E in Figures 5.4, 5.5, and 5.6. The hindcast time series of significant wave height and peak wave period show good overall agreement with measured data at stations C and E. At station B, however, the hindcast H and Tp are overestimated compared with measured data. The poor agreement at station 57 . 0 4 8 12 HOURS 5. 4. C3. 2. 1. 0. 16 20 24 0 4 8 12 HOURS 16 20 24 0.8 0.6 7 0.4 0.2 0.0 0 4 8 12 HOURS 5. 4. 9 3. 2. 1. 0. 16 20 24 5. 4. t3. N.2. 1. 0. STATION I I I I I I I I I I I 0 4 8 12 16 20 24 HOURS I I I I I I I. I I I I 0 4 a 12 HOURS 16 20 24 0 4 8 12 HOURS 16 20 24 Figure 5.2: Comparison of the calculated significant wave heights and peak wave periods with measured data for friction coefficient of 0.01, 0.02, 0.04, 0.06, and 0.08 at stations B, C, and E on October 7, 1988. 0.8 0.6 7 -0.4 0.2 0.0 STATION : 5 4. I I I I I I I I I : 0.02 : f z 0.04 E88100701 4 E88100706 0.2 I" 0.1 0.0 -j 0.3 E88100713 0.2 I 0.0 I4T.mi - f a 0.06 - -zF. I f a 0.08 E88100703 E88100708 E88100715 - : msaued E88100705 E88100711 EB8100716 a' 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure 5.3: Comparison of computed wave spectral shape with measured data for friction coefficients of 0.02, 0.04, 0.06, and 0.08 at station E on October 7, 1988. 59 STATION : L005 10 mn/sec Ivnd speed) 281 282 283 284 285 286 287 1.0 I I 1.0 STATION : E 0.8 E 0.6 + + C 0.4 0.2 + ++ +II ++ ++ 0.0 281. 282. 283. 284. 285. 286. 287. 6.STATION : E 5. ., 4. 281. 282. 283. 284. 285. 286. 287. * N ._., STAT ION : E C- 2 E 1. 0 . -z, N I I. I I 281 282 283 284 285 286 287 JULIAN DAY Figure 5.4: Measured wind bar graph at station L005 and the comparison of the GM model results (-) and measured (+++) H8 and T, at station E. STATION : L005 10 a/sec (vind speedl 281 282 283 284 285 286 21 1.0 .IIIII 1.0 STATION: C 0.8 0.6 0.4 0.2 + + 0.0 I I I 281. 282. 283. 284. 285. 286. 28 6. STATION C 4. 2. 1. A I I I I I 281. 282. 286. 281 282 283 284 285 286 JULIAN DAY Figure 5.5: Measured wind bar graph at station L005 and the comparison of the GM model results (-) and measured (+++) H, and Tp at station C. 61 STATION : LOS 10 o/sec (wind sp@ 281 282 283 284 285 286 287 1.0 STATION : 0.8 -E 0.6 o 0.4 0.2 + +++ + ++ E+ 0.0 281. 282. 283. 284. 285. 286. 287. 6. 1 1 1 i I STATION : B 5. - 3. C.. 2. 0. I 281. 282. 283. 284. 285. 86. 287. N STATION : 8 N > S 0 x C E 0 N I I I I I 281 282 283 284 285 286 287 JULIRN DRY Figure 5.6: Measured wind bar graph at station L005 and the comparison of the GM model results (-) and measured (+++) H, and Tp at station B. 0.2 0.4 0.6 MEASURED Hs 0.8 (m) 5. -4. U +3.+ U ++ M ++ 3 4. Z 2. + + + 0. I 0. 1. 2. 3. 4. MEASURED Tp (sec) 5. - 4. o' N 3. I 2z. CO) 0. I 0. 1. 2. 3. 4. 5. MEASURED Tz (sec) Figure 5.7: Comparison of hourly hindcast by the GM model and measured significant wave height H,, and peak wave period Tp, and zero up-crossing wave period Tz at station E. 0.0 V0.0 0.2 0.4 0.6 0.8 MEASURED Hs (m) 5. -4.+ .2 ~ 4 1. * CL 3. L 2 . S++ T -rI. O 1. 2. 3. 4. MEASURED Tp {sec) s. N - 4.4 CU)2. + + C. + CE + 0. I 0. 1. 2. 3. 4. MERSURED Tz (sec) Figure 5.8: Comparison of hourly hindcast by the GM model and measured significant wave height H,, peak wave period Tp, and zero up-crossing wave period T at station C. E 0.6 U0) 50.4 z = "A 0.0 V 0.0 64 B is primarily due to the elimination of an island in the model which is located at 7 km northeast of station B. Since no measured mean wave direction is available, the hindcast mean wave direction and the measured wind direction are shown in Figures 5.5 and 5.6. The deviations of the hindcast mean wave direction from the measured wind direction are mostly less than 15 degrees. Figures 5.6 and 5.7 show the comparison of hindcast and measured significant wave height, peak wave period, and zero up-crossing period at stations E and C, respectively. The correlation coefficient and a root-mear-square error normalized by the root-mear-square value of measured data are used for the statistical analysis between the hindcast and measured data. The correlation coefficient and the normalized root-mean-square error between hindcast xi and measured yi are defined as follows: PZ= -1(x 7) i -1(yj (5.8) and = -Y )/l (5.9) "=1 i=1 where the overbar denotes the mean value. As can be seen in Figure 5.6, the agreement between the hindcast and measured significant wave height, at station E is quite good with a normalized root-mear-square error of 0.22 and a correlation coefficiant of 0.90. From Figure 5.6, we can see that the hindcast significant wave height at station E systematically overestimates the lower waves and underestimates the higher waves. Table 5.2 shows the correlations and the normalized root mean square error between hindcast and measured significant wave height, peak wave period and zero up-crossing wave period at station E. Figure 5.7 and Table 5.3 show the comparison between the hindcast and measured significant wave height, peak wave period, and zero up-crossing wave period at station C. Table 5.2: Comparison of hindcast by the GM model and measured values of significant wave height, peak wave period, and zero up-crossing wave period at station E. Correlation Normalized root coefficient mean square error H. 0.90 0.22 Tp 0.84 0.14 T, 0.85 0.14 Table 5.3: Comparison of hindcast by the GM model and measured values of significant wave height, peak wave period, and zero up-crossing wave period at station C. Correlation Normalized root coefficient mean square error H. 0.88 0.21 0.74 0.17 Tz 0.83 0.16 Since station C is located in the deeper central part of the lake, the waves are less influenced by the shallow water effect. The comparison of significant wave height between station C and E indicate that the systematic underestimation of higher waves and overestimation of lower waves is due to the improper treatment of shallow water mechanism in the GM model. The comparisons between hindcast and measured spectra st stations C and E are shown in Appendix E. The hindcast spectra have sharper peak than the measured ones. The overall agreement between hindcast and measured spectral shape seems good. 5.3.2 Comparison of the SMB Model Results with Measured Data The significant wave height and peak wave period hindcast by the SMB-AD model and the SMB-LD model were compared with measured data at stations B,C, and E. Figures 5.9, 5.10 and 5.11 show the comparisons of the time series of significant wave height and peak wave period hindcast from the SMB-AD and SMB-LD model with measured Table 5.4: Comparison of hindcast by the SMB-AD and SMB-LD model and measured values of significant wave height and peak wave period at station E. Correlation Normalized root coefficient mean square error SMB-AD H, 0.81 0.27 Tp 0.73 0.22 SMB-LD H 0.83 0.25 ITp 1 0.68 0.22 data at stations E,C, and B, respectively. The overall agreements are considered good at stations C and E. Again, model overestimates the significant wave height at station B due to the ignorance of the small island located at 7 km to the northeast of station B. As can be seen in Figure 5.9, when the waves are propagationg from deep to shallow water, results of the SMB-LD model agree well with measured data (note the strong northeast wind event during the first day of the period, Julian day 281). This is consistent with the idealized uniform easterly wind case where the SMB-AD model gave higher wave heights than the SMB-LD model and the GM model. On Julian day 281, the GM model slightly underestimates the significant wave height (Figure 5.4) and the SMB-LD model slightly overestimates significant wave height (Figure 5.9) at station E. The hindcast zero wave height and zero peak wave period shown in Figure 5.9 are due to the zero wind velocities. However, the data showed that wave height and wave period can be non-zero even for zero wind velocities due to the remnant effect of wind at previous hours. Figures 5.12 and 5.13 show results obtained from the SMB-AD model and the SMB-LD model, respectively, where hindcast and measured significant wave height and peak wave period at atation E are compared. Table 5.4 summarizes the correlation coefficient and the normalized root mean square error between the hindcast and measured significant Wave height and peak wave period at station E. The results are consistent with the idealized wind 67 STRTSM ~10 /sec (wira speed) 281 282 283 284 285 286 287 1.0 IIII SI-:O 1.0 STATION : E SMB-AD 0.8 SMB-LD : .......... E 0.6 (n 0.4 . S.2B- r. + 0.0 281. 282. 283. 284. 285. 286. 287. 6.I STATION : E S5. SHB- . 4. + o I I I I 1. 0. 281. 282. 283. 284. 285. 286. 287. *2 N C -' S c. E 0 0:I NI! 1 I I 281 282 283 284 285 286 287 JULIRN DAY Figure 5.9: Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) HI and Tp at station E. 68 STRTION : LOOS 10 R/sec (vind peed) 281 282 283 284 285 286 287 T1.0 NATION :C S0.8 .. SMB-LD : --------- E 0.6 , ++ 0.2 0.0 281. 282. 283. 284. 285. 286. 287. 6.II H-P.I STATION : C SMB-A S. SB-LD : --- O 4. E 3. ++-+ M. 2. + 1. 0. o. I I I .I 281. 282. 283. 284. 285. 286. 287. * N L '$TATION : C *.me 1 C .0 C3 " 2 I I I I 281 282 283 284 285 286 287 JULIAN DAY Figure 5.10: Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) H, and Tp at station C. 69 STATION : LOS 10 n/sec (wind speed) 281 282 283 284 285 286 287 1.0 I I I IH-0 I____1.0 STATION :B SMB-AO 0.8 sB-Lo : .......... 2 0.6 5. q -," J -s, :.:,* i ' S0.2 + + + + 0.0 281. 282. 283. 284. 285. 286. 287. 6. STATION : 8 SH-AD . 5. SiB-LD : ..........-(O) 4. S+ 3.. + 2) .........,, 1. 0. 281. 282. 283. 284. 285. 286. 287. . N - STATION : 8 S S x x: N U) 281 282 283 284 285 286 287 JULIAN DRY Figure 5.11: Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) H, and Tp at station B. 0.2 0.4 0.6 MEASURED Hs 0.8 1.0 (m) 5. V.) 3. + 2. + 0. 1. 2. 3. MEASURED Tp (sec) CC* + z + + = 1+ 0. I 0. 1. 2. 3. aMEASURED Tp (sec) Figure 5.12: wave height Comparison of hourly hindcast by the SMB-AD model and measured significant H, and peak wave period Tp at station E. 0.2 0.4 MEASURED 0.6 0.8 Hs (in) 5. - 4. 0.) 3. 2. + (::4. 0.--. MES4.4 4. O.4. 4.I 0. 1. 2. MEASURED Figure 5.13: Comparison of hourly hindcast by the SMB-LD model and measured significant wave height H, and peak wave period Tp at station E. E 0.6 I- U z 0.4 z = 0.0 V 0.0 1.0 0.8 E ,n 0.6 I- U (.04 z =0.42 0.0 0.0 3. 4. Tp (sec) 1.0 5. 0.8 -- 4. + + + 0.6 + + + 3.4 + I- *44F- +I- + (J) O..- ." j,' +:: +) + M0 ++.*+ U ++ 2. 4. 44 + Z C:41 + Z 0.2 1. +. 4 0.0 + I I 0.0 0.2 0.4 0.6 0.8 1.0 0. 1. 2. 3. 4. 5. MEASURED Hs (m) MEASURED Tp (sec) Figure 5.14: Comparison of hourly hindcast by SMB-AD model and measured significant wave height H,, and peak wave period Tp, and zero up-crossing wave period T at station C. case in that the SMB-LD model does a good job in hindcasting significant wave height and the SMB-AD model does a good job in hindcasting peak wave period. Figures 5.14 and 5.15 show comparisons between the hindcast from the SMB-AD and SMB-LD models and measured significant wave height and peak wave period at station C. Table 5.5 shows the correlation coefficiant and normalizeds root-mear-square error of the SMB-AD and SMB-LD model at station C. Since station C is located at the deep water, there is little difference between the results obtained from the SMB-AD and the SMB-LD models. Luettich and Harleman (1989) applied the 1973 version of the shallow water SMB model (1973, Shore Protection Manual) and the 1984 version of the shallow water SMB model using local water depth to Lake Balaton, Hungary, and compared the hindcast results with measured data. According to their study, the 1973 version of the SMB model estimates significant wave height very well and the 1984 version of the SMB model overestimates significant wave height by up to 20 %. Both versions of the SMB model consistently underestimate the peak wave period. This finding is consistent with our experience with the SMB-LD model on Lake Okeechobee. 1.0 0.8 E I (20.4 0.2 0.0 0.0 5. CI+ ++ + - 3. + O44 + + LO2. z+ =1. -+ + + + + + 0. I I I 0. 1. 2. 3. 4. 5. MEASURED Tp (sec) Figure 5.15: Comparison of hourly hindcast by SMB-LD model and measured significant wave height H, and peak wave period Tp, and zero up-crossing wave period Tz at station C. Table 5.5: Comparison of hindcast by the SMB-AD and SMB-LD models values of significant wave height and peak wave period at station C. Correlation Normalized root coefficient mean square error SMB-AD H8 0.81 0.30 Tp 0.70 0.25 SMB-LD H8 0.81 0.31 Tp 0.69 0.24 and measured 0.2 0.4 0.6 0.8 1.0 MEASURED Hs (m) CHAPTER 6 SUMMARY AND CONCLUSION The main objective of this study is to compare various wind-wave models with measured field data obtained from Lake Okeechobee, Florida. The wind-wave models tested in this study are: 1. The shallow-water SMB model presented in the Shore Protection Manual (CERC,1984). For the computation of significant wave height and wave period, two different methods were tested. One uses fetch and the local water depth at the point of interest (SMB-LD model) and the other uses fetch and the average water depth along the fetch (SMB-AD model). 2. The finite-depth wind-wave model developed by Graber and Madsen (GM model). 3. The GLERL Donelan deep-water wind-wave model originally developed by Donelan and extensively used by GLERL (GD model). The wave models were tested using idealized steady and uniform wind fields and measured wind fields over Lake Okeechobee with realistic bottom topography. In the idealized wind case, the four wave models are compared in terms of significant wave height, peak wave period, and bottom shear stress with easterly wind and velocities 5, 10, 15, and 20 m/sec, respectively. The GD model overpredicts the significant wave height and peak wave period for wind velocities greater than 10 m/sec and results in excessive bottom shear stresses. The GM model and the SMB-AD model show nearly identical wave period along the entire fetch. The SMB-AD model overestimates the significant wave height and hence excessive bottom shear stress at the Western end of the lake where the fetch is long and the depth is shallow. 74 The SMB-LD model and the GM model gave similar significant wave height along the entire fetch. The SMB-LD model shows consistently 10, 20, and 30 cm higher wave height than the GM model for 10, 15, and 20 m/sec wind velocities, respectively. The SMB-LD model underestimates the wave period in the shallow water. In the real wind case, the GM, the SMB-LD, and the SMB-AD models are tested with the measured data obtained from subsurface pressure gage at three stations B, C, and E in Lake Okeechobee. At station B which is located close to the small island and vegetation area, the hindcasted results show poor agreement with measured data due to the ignorance of the vegetation area and the island in model simulation. At station C located in 4.6 meters deep center part of the lake, the GM model gave the best results compared with measured data. The SMB-LD and SMB-AD models gave good results in agreement with measured data. At station E, located in 2.7 meter depth of water in the south-west shallow part of the lake, the GM model gave best results in agreement with measured significant wave height and peak wave period. The comparisons of results of the SMB-AD model and SMB-LD model with measaured data confirm the finding of the idealized wind case: the SMB-LD model does a better job in estimating wave height than the SMB-AD model, and the SMB-AD model does a better job in eastimating wave period than the SMB-LD model. The GM model performed the best in estimating both the wave height and the period. It also gave additional such useful information as mean wave direction and one-dimensional frequency spectrum. The GM model shows that deviation of the mean wave direction from the mean wind direction are generally less than 15 degrees. The SMB model performed fairly well in shallow lake where the swell doesn't contribute significantly. If the local water depth is used for computing wave height and the average water depth is used for computing wave period, then the SBM model gives results comparable with the GM model in Lake Okeechobee. The major advantage of the SMB model is its simplicity. For the 6-day simulation in Lake Okeechobee using real wind field, the SMB model needs only about 10 minutes CPU time on Vax 8350 computer compared with the 75 GM model's 40 hours CPU time for the same simulation. By using the significant wave height and the peak wave period to represent a monocromatic wave, the bottom shear stresses were computed through linear wave theory and Kajiura's oscillatory boundary layer formulation. The computed bottom shear stress over the entire Lake Okeechobee correlates well with the spatial distribution of the surficial sediment. For the estimation of the bottom shear stresses, Kajiura's and Jonsson's formulas give essentially the same maximum bottom shear stresses. The computed bottom shear stress contains additional uncertainty due to uncertainties in defining roughness height in natural bottom. Most wave boundary layer models are based on the sinusoidal monocromatic wave assumption. Proper inclusion of random waves on the bottom shear stress calculation needs to be explored. For using these formulas over a wide range of bed and flow conditions, it is necessary to improve the clear transition between the hydraulically smooth bed and rough bed and transition between laminar flow and turbulent flow rather than improving the friction factor formula itself. APPENDIX A RADIATIVE-TRANSFER EQUATION Suppose that the total energy density of the waves per unit phase volume [dkodi*] is initialy F(ko, Y, to) at t = to. At time t = to + bt, the energy balance can be expressed in the integral form F(k, 9, t) = F(ko, io, to) + Q(,', A,t')dt' (A.1) where E is the wave number vector and Q(',, t') represents the net rate of change of energy per unit phase volume resulting from any sources and sinks of wave energy. k', xI, and t' vary along the path of a wave group from an initial value ko, i, and to to k,i, and t. Expanding the energy spectrum F(k, t) in a Tayor series gives dk d5 F(k, t) = F(ko + dt, o + --dt, to + dt) OF di dk = F(o, to) + dt + VFdt + .VFdt (A.2) where Vg and VE are the horizontal gradient vector operator of the position vector Y and wave number vector E, respectively. Substituting of equation A.2 into Equation A.1 and approximating the integral by T(E, 5, t)dt, we have the radiative-transfer equation OF di dk Wt + -t VzF + VEF = T(k, 9, t) (A.3) where dt (k,') (A.4) For equations A.3 and A.4 to be valid, it is assumed that the both the amplitude and the wave number vectors of individual spectral components are slowly varying functions of k and t, such that 1 OF 1 8F 1 OF 1, 1 1 (A.5) kF wF Ot 76 where k = Iki and w is the angular wave frequency. Futhermore, it is assumed that the water depth h(i) is also slowly varying, i.e., 1 OF - < 1 (A.6) kh Bz so that the geometrical-optics approximation is valid. In addition, the wave number vector k and the angular wave frequency w = w(k, h) are related by the conservation-of-crests equation -+ Vzw = 0 (A.7) and by the dispersion relationship for linear surface gravity waves w = w(k,i) = vgktanhkh (A.8) An equation for the variation of i along the ray path can be deduced by introducing w above in equation A.7 and writing the result in the tensor notation aki Ow Oki Ow S+ -_ + = 0 (A.9) at Oki Ox Oxi Note that k is irrotational, i.e. Ok = Ok (A.10) Oxi Ox Then from equation A.9 and A.10, we have + *9 V =(A.11) or = Vh (A.12) dt x Also, by taking the scalar product of equation A.7 with Cq, we have aw Ot + C- VW = 0 (A.13) or d - = 0 (A.14) dt 78 Equations A.12 and A.14 represent Eulerian time rate-of-change equations for both k and w and have the same characteristic curve d-i C (A.15) dt 8k 9 The directional wave number spectrum E(k; 9, t) can be transformed into the directional frequency spectrum E(f, 0, i, t) as follows: F(k;Y,t) = J. E(f,0;F,t) (A.16) where J is the Jacobian of the transformation o(f, 0) cc CC W J O(kk) = 2r-w' C9 = jCg, C = k(A.17) a(ks,, ky) 2nuw k Substitution of equation A.16 into A.3 gives a transport equation for the directional frequency spectrum D 8(CCEE) OW t (CC9E) (CCE) + Vz(CC9E) V(CC9E) = 2rwS (A.18) where S(f, O; t) becomes the forcing term. The refraction term in equation A.18 can be expressed in terms of the wave direction 0 in cartesian coordinate as follows: 8w 8w 8098 - V(CC9E) = (CCgE) (A.19) O O _.-' =-: (C Ok OtC E(.9 Since 0 is defined as 0 = cos-'(k./k) = sin-'(k./k). We have 89 sin0 cose) (A.20) ..-: (A .20) 8k k 'k Also along the wave ray, a moving particles experiences no change of phase. dw Oiw Ow Ok d + O 0 (A.21) di O 8O 85kO or OW C- Ok (A.22) Oz 51z From the dispersion relation, we know Ok 0 W Dz 82 C o9k 0 w ay y C W OC C2 z w OC C28y k OC C Oz k OC COy Substitution of equations A.20, A.22 and A.23 into equation A.19 yields e, 800 -w aa(CCgE) DEI ak0 a Ow 9Ok 0 8 = (CCE) C9 OC 8 C) = ( sin 0- cos 0 C Ox Dy ( a0(CCE) (A.24) Introducing equation A.24 in equation A.18 and dividing by Cg we have 1 89 a o9 C -(CCgE) + cos0 (CC.,E) + sin0 C ZC9-t Ox oy( gE 1+ OC OC +sin 0 Cos 0 8 C 8z53i 0 27rw S(CCgE) = 2r S 0 C, (A.23) (A.25) APPENDIX B DEPTH TRANSFORMATION FACTOR T(wh) A consistent approach on the self-similarity of depth dependent frequency spectra was originally demonstrated by Kitaigorodskii, Krasitskii and Zaslavskii (1975). Adopting space and time scale ranges suggested by Phillips (1958) F(k) = Pk-4 p(0) (B.1) E(w) = ag2w-s (B.2) where F(k) and E(w) are, respectively, the wave number and frequency spectra, k is the modulus of the wave-number vector k = (k cos 0, k sin 0), w the angular frequency, 0 the angle characterizing the direction of wave propagation, a and P are universal non-dimensional constants, and V(0) is a certain universal function describing the angular distribution of wave component energy within the equilibrium range and satisfying the standard normalization condition o(O)dO = 1 (B.3) It follows from equation B.1 and B.3 that F(k) = F(k)dk = F(k, O)0k-4p(0)kd0 = k-3 (.4) where F(k) is the spectrum of wavenumber moduli. In the absence of any mean currents, the dispersion relation of the linear wave is isotropic w(k) = [gk tanh(kh)]'1/2 (B.5) where h is local water depth. The use of dispersion relation B.5 leads to a formulation of the relation between F(E) and E(w). 100 OO F(k)dk = F(k,O)kdkdO 8OO 0 80 81 = F(k)dk 0 = E(w)d =< 72 > (B.6) where < 72 > is the variance of the sea surface. From relation B.5, the frequency and wave number spectrum are related as dk E(w) = F(k) (B.7) The wave group velocity is C dw 1 w (+ 2kh (B.8) Cg -k 2k1 + si~f2h](B.8) k 2 k sinh 2kh Introducing equations B.4 and B.8 in equation B.7 gives a general expression for the frequency spectrum, E(w) = k-3 1 2kh (B.9) 1 + sn2kh) Because of dimensional reasoning, a more general expression for the finite-depth frequency.spectrum might be of the form E(w) = ag2W 5-t(Wh) (B.10) where Wh = wV/I7g and T(wh) is some universal nondimensional function. From equation B.5, the solution for k is W32 k(w) = -X(Wh) (B.11) 9 where x(wh) is the solution of the trascedental algebraic equation xtanh(w X) = 1 (B.12) Introducing equation B.11 in equation B.9 and comparing with equation B.10 gives an expression for P(wh) a(h) = X-2 sinh[2w X] = -X21+We(X2 1)-1 (B.13) 82 It is easily ascertained that in deep water lim Q(Wh) = 1 (B.14) Wh0 This asymptotic behavior gives the following relation between constants a and/3 - = (B.15) a 2 and in shallow water, lim Q(wh) = .- (B.16) W-h"O 2 Introducing equation B.16 in equation B.10 leads to the similarity form in shallow water E(w) = Oghw-3 (B.17) APPENDIX C JONSSON'S FRICTION COEFFICIENT FORMULA The bottom shear stress ro can also be found by assuming that the steady-state expression for a turbulent velocity profile over a rough bottom is valid near the bottom (where the accelerations are small) u 30z 30z - 2.5n = 5.751og (C.1) where u. = v'%Tp is the friction velocity, k, is the Nikuradse roughness parameter, and z is the height over theoretical bed. The eddy viscosity -t is defined as - = E (C.2) p O Since u=0 at z = k,/30 for fully developed turbulent flow, equation ?? yields the bottom shear stress as follows: To 6 8(u- )dz p Jk/30 at (0.3) where b = b(t) is the thickness of wave boundary layer. u,. and 6 are interrelated by u" = 2.5 In 30b (C.4) U* ka Subtraction of equation C.1 from equation C.4 yields uoo u = 2.5 u. In(6/z) (C.5) Substitution of equation C.5 into C.3 and integration by assuming that k < 306 and 6 = 0 for t = 0 gives. u,6 = 0.40u! dt (C.6) 84 Now, squaring equation C.4 and substituting the result into equation C.6 with uo = u.oom sin wt, we have it sinkit u.6 = 0.403 Um sin2 Wt dt (C.7) 00 n0 2 30--"6d C7 oIn k, km Since ln2(306/k,) is slowly varying function with time t, it may be neglected in the integral. Equation C.7 becomes 0.403 s u.6 = om (t 2wt (C.8) 2 In2 2w ko where 6 is the wave boundary layer thickness corresponding to uo = uim ( i. e. wt = 7r/2 ). From equations C.4 and C.8, we have 3061 3061 aim In = m (C.9) k, k. k. where aim = uoom/ww and m6 (1.2r/ In 10 = 1.64) must be determined from experimental data. The wave friction factor f, is defined from the maximum bed shear stress as follows: 1 o w1PUcom (C.10) From equation C.4 and C.10 ( u.m )2 0.06050 306i f, um In2 k (C.11) Now maximum bed shear stress and maximum free stream velocity are assumed to occur simultaneously. Elimination of 3061/k, from equations C.9 and C.11 gives 1 1 _aim 1 + In 1 m + Inam (C.12) m4 is a constant determined to be -0.08 from experimental data. m1 is a constant determined to be -0.08 from experimental data. as APPENDIX D TABLES Fetch 3 7 11 15 19 23 27 31 35 39 43 47 (kin) Depth 4.3 4.5 4.8 4.9 4.9 4.4 4.1 3.0 2.6 1.9 1.5 0.6 (m) H, SMB-AD 0.15 0.23 0.28 0.31 0.34 0.37 0.38 0.40.. 0.41 0.41 0.41 0.41 in SMB-LD 0.16 0.23 0.28 0.32 0.35 0.37 0.38 0.36 0.35 0.30 0.26 0.14 m GM 0.14 0.22 0.28 0.32 0.36 0.39 0.41 0.40 0.40 0.32 0.27 0.15 GD 0.12 0.18 0.24 0.27 0.27 0.30 0.33 0.33 0.33 0.33 0.36 0.43 T, SMB-AD 1.5 1.8 2.1 2.2 2.3 2.4 2.5 2.5 2.6 2.6 2.6 2.6 in SMB-LD 1.5 1.9 2.1 2.3 2.4 2.4 2.5 2.5 2.4 2.3 2.2 1.8 sec GM 1.4 1.9 2.1 2.4 2.5 2.7 2.8 2.9 2.9 2.9 2.9 3.0 GD 1.2 1.7 1.9 2.0 2.2 2.2 2.3 2.3 2.4 2.4 2.5 2.7 it, SMB-AD 0.0 0.02 0.08 0.14 0.24 0.56 0.87 3.35 5.15 9.67 14.2 42.6 dyne SMB-LD 0.0 0.03 0.08 0.17 0.28 0.57 0.86 2.49 3.40 4.89 5.72 6.97 per GM 0.0 0.02 0.08 0.24 0.44 1.18 1.98 4.72 6.33 7.54 7.70 8.80 sq.cm GD 0.01 0.01 0.04 0.04 0.11 0.22 0.47 1.73 3.05 6.13 11.2 45.4 Table D.A: Model comparison for 5 m/sec wind. Variations of H, Tp, and rb with fetch along the cross section AA'. Fetch 3 7 11 15 19 23 27 31 35 39 43 47 (kin) Depth 4.3 4.5 4.8 4.9 4.9 4.4 4.1 3.0 2.6 1.9 1.5 0.6 (m) H. SMB-AD 0.53 0.74 0.87 0.96 1.02 1.06 1.08 1.09 1.08 1.05 1.02 0.98 in SMB-LD 0.57 0.80 0.94 1.03 1.09 1.08 1.07 0.89 0.81 0.65 0.54 0.27 In GM 0.43 0.64 0.77 0.86 0.9 0.92 0.88 0.73 0.59 0.42 0.36 0.26 GD 0.49 0.82 1.06 1.22 1.34 1.46 1.55 1.61 1.67 1.73 1.79 2.07 T, SMB-AD 2.3 2.9 3.3 3.6 3.8 3.9 4.1 4.2 4.2 4.3 4.3 4.3 in SMB-LD 2.4 3.0 3.4 3.7 3.9 4.0 4.1 3.9 3.9 3.7 3.5 2.6 sec GM 2.2 2.9 3.3 3.5 3.8 3.9 4.1 4.2 4.3 4.3 4.2 5.0 GD 2.4 3.2 3.8 4.1 4.4 4.6 4.7 4.9 5.0 5.1 5.2 5.6 m SMB-AD 0.71 4.53 7.62 10.6 13.4 18.9 23.1 38.0 45.4 62.9 77.4 177. dyne SMB-LD 0.94 5.63 9.28 12.6 15.5 19.7 22.6 26.1 26.9 27.1 25.9 21.6 per GM 0.42 3.34 5.99 8.65 10.8 14.9 16.5 19.6 17.0 13.8 13.8 18.1 sq.cm GD 0.82 7.38 15.1 20.8 27.1 39.1 48.8 79.7 101. 151. 203. 612. Table D.2: Model comparison for 10 m/sec wind. Variations of H., T2, and m with fetch along the cross section AA'. Fetch 3 7 11 15 19 23 27 31 35 39 43 47 (km) Depth 4.3 4.5 4.8 4.9 4.9 4.4 4.1 3.0 2.6 1.9 1.5 0.6 () H. SMB-AD 0.34 0.49 0.59 0.66 0.71 0.75 0.77 0.79 0.79 0.78 0.76 0.74 in SMB-LD 0.36 0.52 .0.62 0.69 0.75 0.76 0.76 0.66 0.61 0.50 0.42 0.21 m GM 0.29 0.44 0.55 0.62 0.67 0.69 0.69 0.60 0.54 0.38 0.31 0.19 GD 0.27 0.46 0.58 0.67 0.73 0.79 0.82 0.85 0.88 0.91 0.94 1.09 T. SMB-AD 2.0 2.5 2.8 3.0 3.2 3.3 3.5 3.5 3.6 3.6 3.6 3.6 in SMB-LD 2.0 2.5 2.9 3.1 3.3 3.4 3.4 3.3 3.3 3.2 3.0 2.3 sec GM 1.9 2.5 2.8 3.1 3.3 3.4 3.5 3.7 3.7 3.7 3.8 4.0 GD 1.8 2.5 2.9 3.1 3.3 3.5 3.6 3.6 3.7 3.8 3.9 4.2 SMB-AD 0.10 0.86 2.13 3.40 4.71 7.73 10.1 19.3 24.3 36.3 46.1 112. dyne SMB-LD 0.14 1.29 2.55 3.95 5.48 7.96 9.84 13.5 14.8 16.2 16.2 14.3 per GM 0.06 0.70 1.90 3.17 4.53 7.11 8.96 12.9 13.6 11.5 10.7 11.6 sq.cm GD 0.03 0.80 2.41 3.76 5.41 9.37 12.2 22.3 29.9 48.0 66.9 215. Table D.3: Model comparison for 15 m/sec wind. Variations of H8, Tp, and rb with fetch along the cross section AA'. Fetch 3 7 11 15 19 23 27 31 35 39 43 47 (kin) Depth 4.3 4.5 4.8 4.9 4.9 4.4 4.1 3.0 2.6 1.9 1.5 0.6 (m) H, SMB-AD 0.71 0.97 1.12 1.22 1.29 1.33 1.35 1.34 1.32 1.28 1.24 1.19 in SMB-LD 0.79 1.07 1.24 1.34 1.40 1.36 1.32 1.08 0.97 0.77 0.65 0.33 m GM 0.57 0.82 0.97 1.04 1.10 1.08 1.01 0.78 0.61 0.46 0.39 0.26 GD 0.61 1.22 1.52 1.82 2.13 2.13 2.43 2.43 2.74 2.74 2.74 3.34 T. SMB-AD 2.6 3.3 3.7 4.0 4.2 4.4 4.6 4.7 4.7 4.8 4.8 4.8 in SMB-LD 2.7 3.4 3.8 4.1 4.3 4.4 4.5 4.4 4.3 4.1 3.8 2.9 Sec GM 2.5 3.2 3.6 3.9 4.2 4.4 4.5 4.7 4.7 4.7 4.7 5.0 GD 2.8 3.9 4.6 5.0 5.3 5.6 5.8 6.0 6.1 6.2 6.4 7.0 m SMB-AD 2.67 10.4 15.6 20.1 24.2 32.1 37.7 57.2 66.4 89.5 108.4 243. dyne SMB-LD 3.71 13.2 19.4 24.4 28.5 33.4 36.1 38.7 38.5 37.1 36.0 30.0 per GM 1.54 7.33 11.7 15.1 18.2 22.2 23.1 23.1 18.4 16.5 15.9 18.2 sq.cm GD 3.18 22.4 36.6 52.4 71.2 84.8 118. 169. 241. 331. 416. 1362 Table D.4: Model comparison for 20 m/sec wind. Variations of H, Tp, and rb with fetch along the cross section AA'. |

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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EJCOYOT41_T4OMW2 INGEST_TIME 2017-12-01T19:31:37Z PACKAGE UF00076133_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES PAGE 1 UFL/COEL-89/027 WIND-WAVE HINDCASTING AND ESTIMATION OF BOTTOM SHEAR STRESS IN LAKE OKEECHOBEE by Kyungmo Ahn Thesis 1989 PAGE 2 WIND-WAVE HINDCASTING AND ESTIMATION OF BOTTOM SHEAR STRESS IN LAKE OKEECHOBEE By KYUNGMO AHN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING UNIVERSITY OF FLORIDA 1989 PAGE 3 ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my advisor Dr. Y. Peter Sheng for his continuous guidance and encouragement throughout this study. I would also like to extend my thanks and appreciation to my thesis committee members, Dr. Robert Dean and Dr. Hsiang Wang, for their patience in reviewing this thesis. I would like to thank Dr. Hans Graber of Woods Hole Oceanographic Institution for his kind permission to use his finite-depth wind-wave model. Also I would like to thank Dr. David Schwab of Great Lakes Environmental Research Laboratory, National Oceanic and Atmospheric Administration, for providing me with the deep-water GLERL/Donelan wind-wave model. Appreciation is extended to Jei Choi, who contributed so much toward the sucessful application of the GM model to Lake Okeechobee. Special thanks go to all Lake Okeechobee crew members, Joaquim, Steve, Victor, P.F., Dave, and members of the Coastal Engineering Laboratory who did an excellent job in the synoptic surveys and the deployment of in-situ instruments. Financial support provided by the South Florida Water Management District, West Palm Beach, Florida through the Lake Okeechobee Phosphorous Dynamics is appreciated. I would like to dedicate this thesis to my parents. I hope they would like what I have achived so far. Finally I would like to thank my wife Eunhye for her support, encouragement, and patience and my sons, Taewook and Hyunwook, for sharing joyful and happy smiles. ii PAGE 4 TABLE OF CONTENTS ACKNOWLEDGEMENTS ................................ ii LIST OF FIGURES .................................... v LIST OF TABLES .................... ................ ix ABSTRACT ........................................ x CHAPTERS 1 INTRODUCTION .................... ............... 1 2 WIND-WAVE HINDCASTING MODELS ........................ 4 2.1 Introduction .................... ................ 4 2.2 The Shallow-Water SMB model .......................... 5 2.3 A Simple Numerical Wave Prediction Model ................. .6 2.4 A Finite Depth Wind Wave Model by Graber and Madsen .......... 9 2.4.1 Theoretical Description of GM model ................. 10 2.4.2 Atmospheric Forcing by the Wind ................ ... 13 2.4.3 Nonlinear Energy Transfer by Resonant Wave-Wave Interactions ..14 2.4.4 Dissipation by Bottom Friction ................. .. .. .15 3 WAVE BOUNDARY LAYER AND BOTTOM SHEAR STRESS ......... 16 3.1 Introduction .................... ............... .16 3.2 Bottom Boundary Layer ................... .......... 17 3.3 Jonsson's Integrated Momentum Approach .................. 22 3.4 Kajiura's Model of the Bottom Boundary Layer in Water Waves ...... 25 4 MODEL PERFORMANCE IN IDEALIZED WIND FIELD ........... 28 4.1 Introduction ................... .................. .28 iii PAGE 5 4.2 Comparison of Wave Models in Idealized Wind Fields ............ 28 5 MODEL PERFORMANCE IN REAL WIND FIELD ................ 52 5.1 Introduction ..................... .............. .52 5.2 Field Data Analysis .................... ........... 52 5.3 Comparison of Model Results vs. Measured Data ............... 55 5.3.1 Comparison of the GM Model results with Measured Data ..... ...56 5.3.2 Comparison of the SMB Model Results with Measured Data .... 65 6 SUMMARY AND CONCLUSION ........................ .. 73 APPENDICES A RADIATIVE-TRANSFER EQUATION ................... ..... 76 B DEPTH TRANSFORMATION FACTOR (wh) ................. 80 C JONSSON'S FRICTION COEFFICIENT FORMULA .............. .. 83 D TABLES ........................................ 85 E ONE-DIMENSIONAL FREQUENCY SPECTRA ................. .. 90 BIBLIOGRAPHY ...................................... .103 BIOGRAPHICAL SKETCH ............................... 108 iv PAGE 6 LIST OF FIGURES 3.1 Comparison of different formulas for the friction coefficient f varies with a/k, (Sleath, 1984). ............................ .. 24 4.1 Geometry and bottom topography of Lake Okeechobee ......... 31 4.2 Location of research towers A, B, C, D, E, and F, with wind station L, and a 2 km computational grid used in Lake Okeechobee ........ 32 4.3 Bottom sediment characteristic of Lake Okeechobee (from Reddy et al. 1988) ............ ......... ...... .. ...... .. .. 33 4.4 Comparison of wave models for 5 m/sec easterly wind. Variations of significant wave height, Hs, peak wave period, Tp, and bottom shear stress, Tb, with fetch along the cross section AA' ............. 34 4.5 Comparison of wave models for 10 m/sec easterly wind. Variations of significant wave height, Hs, peak wave period, Tp, and bottom shear stress, rb, with fetch along the cross section AA' ............. 35 4.6 Comparison of wave models for 15 m/sec easterly wind. Variations of significant wave height, H,, peak wave period, Tp, and bottom shear stress, rb, with fetch along the cross section AA' ............. 36 4.7 Comparison of wave models for 20 m/sec easterly wind. Variations of significant wave height, H,, peak wave period, Tp, and bottom shear stress, Tb, with fetch along the cross section AA' ............. .37 4.8 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the GM model for 5 m/sec easterly wind. ................ .40 4.9 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, Tb, obtained from the GM model for 10 m/sec easterly wind.. ............... .41 4.10 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the GM model for 15 m/sec easterly wind.. ............... .42 4.11 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, Tr, obtained from the GM model for 20 m/sec easterly wind.. ............... .43 v PAGE 7 4.12 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, Tb, obtained from the SMB-LD model for 5 m/sec easterly wind. .............. 44 4.13 Contours of the significant wave height, Hs, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-LD model for 10 m/sec easterly wind. .............. 45 4.14 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-LD model for 15 m/sec easterly wind ............... 46 4.15 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-LD model for 20 m/sec easterly wind ............... 47 4.16 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, n, obtained from the SMB-AD model for 5 m/sec easterly wind. ....... ....... .. .48 4.17 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, A6, and bottom shear stress, Tr, obtained from the SMB-LD model for 10 m/sec easterly wind ............... 49 4.18 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, 7b, obtained from the SMB-LD model for 15 m/sec easterly wind ............... 50 4.19 Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, Tb, obtained from the SMB-LD model for 20 m/sec easterly wind ............... 51 5.1 Typical wave spectrum from Lake Okeechobee. ............... 55 5.2 Comparison of the calculated significant wave heights and peak wave periods with measured data for friction coefficient of 0.01, 0.02,0.04,0.06, and 0.08 at stations B, C, and E on October 7, 1988 ............ 57 5.3 Comparison of computed wave spectral shape with measured data for friction coefficients of 0.02, 0.04, 0.06, and 0.08 at station E on October 7, 1988 ................... .. .............. .58 5.4 Measured wind bar graph at station L005 and the comparison of the GM model results (-) and measured (+++) H, and Tp at station E. ....59 5.5 Measured wind bar graph at station L005 and the comparison of the GM model results (-) and measured (+++) Hs and Tp at station C. ....60 5.6 Measured wind bar graph at station L005 and the comparison of the GM model results (-) and measured (+++) H, and Tp at station B. ....61 vi PAGE 8 5.7 Comparison of hourly hindcast by the GM model and measured significant wave height H,, and peak wave period Tp, and zero up-crossing wave period Tz at station E .......................... .. 62 5.8 Comparison of hourly hindcast by the GM model and measured significant wave height H,, peak wave period Tp, and zero up-crossing wave period Tz at station C ................... ......... 63 5.9 Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) H, and Tp at station E. .................. ............. ..67 5.10 Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) H, and Tp at station C. .... .. ... .. .... ... ..... .. .... ... .68 5.11 Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) H, and Tp at station B ................. .... ............. .69 5.12 Comparison of hourly hindcast by the SMB-AD model and measured significant wave height H, and peak wave period Tp at station E. ....70 5.13 Comparison of hourly hindcast by the SMB-LD model and measured significant wave height H, and peak wave period Tp at station E. ....70 5.14 Comparison of hourly hindcast by SMB-AD model and measured significant wave height H,, and peak wave period Tp, and zero up-crossing wave period Tz at station C. ........................ 71 5.15 Comparison of hourly hindcast by SMB-LD model and measured significant wave height H,, and peak wave period Tp, and zero up-crossing wave period Tz at station C. ........................ 72 E.1 Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. (continue) ................... ..91 E.2 Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. (continue) ................... ..92 E.3 Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. (continue) ...................... 93 E.4 Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. (continue) ................... .... 94 E.5 Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. (continue) ................... ..95 E.6 Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. ........................... 96 vii PAGE 9 E.7 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) ................... ..97 E.8 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) ................... ..98 E.9 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) ................... ..99 E.10 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) ................... ..100 E.11 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) ................... ..101 E.12 Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. ........................... 102 viii PAGE 10 LIST OF TABLES 3.1 Nikuradse eqivalent sand-grain roughness height for flat bed. Dp is the diameter of the grain corresponding to p % finer in weight. (van Rijn, 1982) .....................................19 5.1 Locations of the pressure transducers deployed in Lake Okeechobee ..53 5.2 Comparison of hindcast by the GM model and measured values of significant wave height, peak wave period, and zero up-crossing wave period at station E. .... ... .... ....... .... ... ... .... .65 5.3 Comparison of hindcast by the GM model and measured values of significant wave height, peak wave period, and zero up-crossing wave period at station C. .................. ................ 65 5.4 Comparison of hindcast by the SMB-AD and SMB-LD model and measured values of significant wave height and peak wave period at station E. ........................................... 66 5.5 Comparison of hindcast by the SMB-AD and SMB-LD models and measured values of significant wave height and peak wave period at station C ........................................72 D.1 Model comparison for 5 m/sec wind. Variations of H,, Tp, and Tb with fetch along the cross section AA'... .................... 86 D.2 Model comparison for 10 m/sec wind. Variations of H,, Tp, and Tb with fetch along the cross section AA'....................... 87 D.3 Model comparison for 15 m/sec wind. Variations of H,, Tp, and Tb with fetch along the cross section AA'. ................. ..... 88 D.4 Model comparison for 20 m/sec wind. Variations of H,, Tp, and rb with fetch along the cross section AA'. ................. ..... 89 ix PAGE 11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Engineering WIND-WAVE HINDCASTING AND ESTIMATION OF BOTTOM SHEAR STRESS IN LAKE OKEECHOBEE By KYUNGMO AHN December 1989 Chairman: Dr. Y. Peter Sheng Major Department: Coastal and Oceanographic Engineering Three wind-wave models are compared with measured field data from Lake Okeechobee, Florida. The wind-wave models tested in this study are the shallow-water SMB model, the finite-depth wind-wave spectral model, and the GLERL-Donelan deep-water wind-wave prediction model. The wave models were tested using idealized steady and uniform wind fields and also using measured wind fields over Lake Okeechobee with realistic bottom topography. The comparisons of model results with measured field data include significant wave height, wave period, and one dimensional spectral shape. By assuming the significant wave height and the peak wave period as a representative monocromatic wave, the bottom shear stresses are computed through linear wave theory and Kajiura's oscillatory boundary layer formulation in Lake Okeechobee. A discussion about correlations between the computed bottom shear stress and the distribution of the surficial sediment is presented. x PAGE 12 CHAPTER 1 INTRODUCTION Nutrients and hazardous chemicals are transported into lakes through surface runoff, river inflow, and precipitation, thus causing many environmental problems including enhanced eutrophication of lakes and deterioration of water quality in lakes. Fine sediments, with their small settling velocity and affinity to adsorb contaminants, act as an agent for recycling of contaminents. Thus it is important to understand the transport mechanisms of fine sediment particles ( i.e., silt and clay size range) in various water bodies (e.g., Sheng and Lick, 1979, Somlybdy, 1983, Pettersson and Bostr6m, 1984, and Luettich et al. 1989). Understanding the dynamics of fine sediment transport in a shallow-water body, such as Lake Okeechobee, is of particular importance because the bottom sediment and associated contaminants can be readily resuspended and transported by waves and currents. A mathematical model capable of simulating the transport of suspended sediment in a shallow water body is a useful tool for the study of practical problems. For example, Sheng and Lick (1979) developed a three-dimensional fine sediment transport model, which includes the effects of wind-driven currents, wind waves, turbulent mixing, and erosion and deposition at the bottom. One-dimensional sediment transport models, which couple the bottom boundary-layer dynamics with the sediment-transport problem, have been developed for non-cohesive sediments (Grant and Madsen, 1986) and cohesive sediments (Sheng and Villaret, 1989). Bottom boundary layers are regions where intensive turbulent mixing of mass and momentum and frictional dissipation of energy take place. Close to the shallowwater bed, shear stresses and turbulent kinetic energy are generated by both waves and currents. The combined action of waves and currents increases the bottom shear stress and 1 PAGE 13 2 enhances resuspension of sediments (Sheng, 1984, Trowbridge and Madsen, 1984). Because the rate of sediment resuspension is a function of both bottom shear stress and sediment characteristics, bottom shear stress under waves and currents must be accurately calculated. As a first step, the wind-driven currents and wind waves should be accurately determined. As a part of the "Lake Okeechobee Phosphorous Dynamics" project funded by the South Florida Water Management District, the Coastal and Oceanographic Engineering Department of the University of Florida is studying the circulation patterns, wave climate, and sediment transport processes in Lake Okeechobee under the general supervision of Dr. Y. Peter Sheng. This thesis deals with the modeling of wind-generated wave in Lake Okeechobee. Three wind-wave hindcasting models were tested with wind and wave data from Lake Okeechobee. Excluding the Great Lakes, Lake Okeechobee has the largest surface area (1700km2) of any lake in the United States. Its depth ranges from 0.5 to 6 meters. Due to its long fetch and shallowness, Lake Okeechobee is a good location to calibrate shallow-water wind-wave models. The wind-wave models tested in this study are as follows: 1. The shallow-water SMB model presented in the Shore Protection Manual (U.S. Army Coastal Engineering Research Center, 1984) which is based on a semi-empirical approach. 2. The GLERL-Donelan deep-water wind-wave prediction model developed by Donelan (1977) and used successfully to predict wave height and direction in Lake Erie (Schwab et. al., 1984), Lake Michigan (Liu et. al., 1984), and Lake St. Clair (Schwab and Liu, 1987). 3. The finite-depth wind-wave model developed by Graber and Madsen (Graber, 1984, Graber and Madsen, 1988) which is a parametric windsea model for arbitrary water depth. The calculation of bottom shear stress on a natural lake bed under unsteady oscillatory PAGE 14 3 flow is a complex problem due to the irregular bottom roughness and random nature of the waves. However, the estimation of bottom shear stress may give some useful insight for understanding the transport and fate of sediment particles in the lake. As part of this study, the bottom shear stresses in Lake Okeechobee were estimated using the boundary layer methods of Kajiura (1964, 1968) and Jonsson (1966, 1978). In Chapter 2, the three wind-wave models are described. Theoretical descriptions of the models as well as the limitations on their application are presented. In Chapter 3, a literature review and a theoretical description of the oscillatory boundary layer is presented. The bottom shear stress formulae of Kajiura and Jonsson are also presented. In Chapter 4 the three wind-wave models are compared using steady, uniform wind conditions and Lake Okeechobee bottom topography. In Chapter 5, wave data-obtained from subsurface pressure gages at three stations in Lake Okeechobee are analyzed and the finite depth wind-wave model by Graber and Madsen and the shallow-water SMB model are compared with the measured wave data in detail. Summary and conclusions are presented in Chapter 6. PAGE 15 CHAPTER 2 WIND-WAVE HINDCASTING MODELS 2.1 Introduction The commonly utilized models for wind-wave prediction include the empirical model based on dimensional analysis, the discrete spectral model, and the parametric model. There are major theoretical differences between the various models. However, a rational framework has been established in the radiative-transfer equation, and the details of windwave generation, evolution, and dissipation mechanism have been clarified. Each model has numerical advantages and disadvantages. The Sea Wave Modelling Project (SWAMP) Group (1984) compared nine operational deep water wave models and recommended future model improvements. More recently, the SWIM (1985) study compared three operational shallow-water wave models for idealized wave generational cases and for a severe North Sea storm. In the SWIM project, the three wave models tested were the model of the Meteorological Office; UK, (BMO) (Golding, 1983), the GONO-model of the Royal Netherlands Meteorological Institute (Janssen et al. 1984), and the HYPAS model of the University of Hamburg, FRG (Giinther et al. 1979). According to the SWIM study, there exist no significant differences between the model performances. Because of the shallow depth of Lake Okeechobee, the comparison of model performances in this study is focused on two shallow-water wind-wave models: the SMB model and the Graber-Madsen model. The deep-water version of GLERL-Donelan model is also tested in this study, however, because of the successful application of the model to shallow lakes such as Lake St. Clair (Schwab and Liu, 1987). Many different wind-wave hindcasting models are currently being used and developed. In order to select a particular model for a specific study, the questions that need to be 4 PAGE 16 5 addressed include (1) which precise wave characteristics need to be known, (2) what are the detailed practical questions to be answered, and (3) what are the basic physical principles and the inherent limitations of the models. In this regard, this chapter presents a detailed description of the wind-wave hindcasting models being tested in this study. 2.2 The Shallow-Water SMB model The shallow-water SMB (Svendrup, Munk, and Bretshneider) model which will be denoted hereafter as SMB, is an empirical model based on dimensional analysis. The SMB model presented in the US Army Corps of Engineers Shore Protection Manual (1984) is a simplified wave hindcast model. Waves are generated by a uniform wind blowing over a known fetch for a given duration. Wave energy is added by wind stress and pressure and lost to bottom friction (Putnam and Johnson, 1949) and percolation (Putnam, 1949). The basic assumptions are spatially uniform wind velocity and uniform depth over the fetch for a given duration. Consequently, in shallow water, the evolution of waves such as refraction, shoaling, and dissipation are poorly described by a single dimensionless depth parameter. No directional wave information is available from the method, so it is not useful when the combined effect of waves and currents is needed. Despite these severe limitations, the SMB method is still used extensively for giving quick, order-of-magnitude estimates. Assuming a bottom friction coefficient of 0.01, the significant wave height H, in meters, and the wave period T in seconds, are given by /i ) 3/4] 0.00565 (F) gH-= 0.283tanh 0530 tanh 05 -A)3/ (2.1) UA U tanh 0.530 ( 3J I A \ T gd 3/8-0.0379 (F) -= 7.54 tanh 0833 r tanh 1 (2.2) UA tanh 0.833 ()A2/ and gt 53 ( gT 7/3 (2.3) UA= 5.37x 102 PAGE 17 6 where g is the gravitational acceleration (9.81 m/sec2), d is the uniform water depth (in meters) which is often taken as the average water depth over the fetch or the local water depth of interest, F is the fetch (in meters), t is the duration of the wind over the fetch (in seconds), UA is an adjusted wind velocity or wind-stress factor given by UA = 0.71U1'23 where U = RTU(10), U(10) is the mean hourly wind velocity (in m/sec) at 10 m above the mean sea level, and RT is an air-sea temperature difference adjustment factor. In the absence of temperature information, RT = 1.1 is generally assumed. For practical wave predictions, it is usually satisfactory to assume uniform wind if the variation in wind direction is less than 150 and the variation in wind velocity is less than 2.5 m/sec. Hurdle and Stive (1989) reported discontinuities in the SMB model at the transition region between deep-water developing seas and deep-water fully-developed seas, and that between shallow and deep waters. Further, Hurdle and Stive questioned the relationship between critical duration and fetch in very shallow water. They proposed a modified version of equations which removed discontinuities in the transition regions, and hence should be valid for both deep and shallow waters. The revised equations are : 0. 252d3/4 4.3 x 10-s f ) gH 0.25 tanh 0.6 tanh/2 3 (2.4) UA U J tanh 2 0.6 3/4 l d\.3751 4.1 x 10-5 ) -= 8.3 tanh 0.76 2 tanh3l/3 3) (2.5) UA = 8tan [ tanh3 [0.76 (g) 2 J where Tp is the period of peak spectral energy in seconds. Despite the recommended revised equations, we used the original shallow water SMB model because of its widespread use. 2.3 A Simple Numerical Wave Prediction Model A simple numerical wave prediction model which will be denoted hereafter as the deepwater GD (GLERL-Donelan) model, was originally developed by Donelan (1977) and used successfully by the Great Lakes Environmental Research Laboratory (GLERL) of NOAA to predict wave height and direction in Lake Erie (Schwab et al. 1984), Lake Michigan PAGE 18 7 (Liu et al. 1984), and Lake St. Clair (Schwab and Liu, 1987). This model is a parametric model based on a momentum conservation equation for the wave field. The model predicts the two components of the wave momentum vector and the phase speed of peak energy waves. From these variables, one can then derive significant wave height, wave period, and wave direction. In the model formulation, the waves are assumed to obey the deep-water dispersion relation. Refraction and bottom dissipation are ignored. Schwab and Liu (1987) reported that the shallow-water GD model was developed by incorporating the Kitaigorodskii et al. (1985) shallow-water spectrum along with a depthdependent group velocity and a simple form of bottom friction. The results of the shallow water version and the deep water version of the model were compared to observations at six towers in Lake St. Clair with depths ranging from 3.7 to 7.0 meters. The shallow water version of the model was found to underestimate the highest waves at all stations. Although the shallow water model could be adjusted to better match the observed wave heights by decreasing the bottom friction parameter, even the best results were no better than those of deep water model. Schwab and Liu speculated that the possible reason why the deepwater model works in Lake St. Clair better than shallow-water model is that "The wind momentum input function in the model is oversimplified and if it were formulated more realistically, the deep-water model would tend to overestimate the highest waves." Because of this, the deep-water GD model was applied to Lake Okeechobee in this study. In the deep-water GD model, the time rate of change of the wave momentum supplied from the wind is modeled as follows: O M 9 OT, OTy r7 S+ + -(2.6) 5t + ay p" + Ty= Tyy -(2.7) t tzx y p, where p, is the density of water and M. and My are the x and y momentum components, respectively, which defined as oo g 2j F( f os)2 M = g F Icos Oddf (2.8) Jo Jo C(f PAGE 19 8 My = g F(f,) sin OdOdf (2.9) where F(f, 0) is the wave energy spectrum as a function of frequency f and direction 0, and C(f) is the phase speed. If we assume that the deep-water linear wave theory applies, then the group velocity is C(f) C(f) = (2.10) From equations 2.8, 2.9 and 2.10, the components of the momentum flux tensor are T = c F( os2 OdOdf (2.11) 2 o Jo C(f) T = Ty = F(f,) sin 0 cos dOdf (2.12) T Y Jo J= C(f) T o sin2 Oddf (2.13) Furthermore, if we assume that the wave energy E(f) is distributed about the mean wave direction 00 with cos2 and there is no energy for 10 -0ol > 7r/2, then F(f, 0) = 2E(f)cos2(0 -00) (2.14) If 00 is independent of frequency, then the momentum fluxes can be expressed in terms of 00 and variance a2 = E(f)df (2.15) The integration of Equations 2.11, 2.12 and 2.13 yields T. = ( cos o0 + (2.16) Ty = Ty = cos 0o sin 02 (2.17) TyY = g sin o + -(2.18) The relationship between the variance a2 and the momentum components can be derived by assuming the average JONSWAP spectrum (Hasselmann et al. 1973) E(f)= ag2(2r)-4fexp ( -+ +n3.3exp -(ff1)2 (2.19) 4 f,. 2a2 PAGE 20 9 S=0.07 f PAGE 21 10 the JONSWAP parameter set and by a directional parameter representing the mean direction of the windsea spectrum. The depth dependent frequency spectrum is assumed to have similarity shape which is related to the deep water energy spectrum by multiplying a depth dependent transformation factor. This factor affects the spectral shape of the highfrequency part which is proportional to f-s in deep water and to f-3 in shallow water. The directional dependence of the spectrum is assumed to be cos2(0 -00), where 0o is the mean wave direction. The windsea model explicitly accounts for finite depth effects such as shoaling,refraction, dissipation by bottom friction, finite depth modification of atmospheric input, and nonlinear wave-wave interaction source terms. Based on an energy flux, transport equations of the full set of prognostic parameters is derived including finite depth effects. The equations of the prognostic parametric variables are solved on a finite difference grid by means of the Lax-Wendroff method. Swell is treated in a decoupled spectral fashion and swell characteristics are considered straight and effects of refraction are disregarded for simplicity. However, shoaling and dissipation of energy by bottom friction are included. 2.4.1 Theoretical Description of GM model The surface wave field is generally described in terms of the variance spectral density F(k) of the surface gravity waves in directional wave-number space. F is assumed to be a slowly varying function of position F and time t. The radiative-transfer equation for the surface wave field with variable bottom topography was originally given by Hasselmann (1960) OF d5 dk -+ dtV iF + -VgF = T(; t) (2.25) where : = Cg is the group velocity, k = (ks, k,) and Â£ = (x, y) are the wave number and position vector, respectively, Vj and VE are the horizontal gradient vector operator of the position vector E and wave number vector k, respectively. The source function T(k; ,t) represents the net transfer of energies to, from, and within the spectrum at the wave number k due to all interaction processes which affect the component k. The terms of the left-hand PAGE 22 11 side of equation 2.25 represent, respectively, local accumulation (term 1), propagation (term 2), and combined refraction and shoaling (term 3). In addition, the kinematics of wave propagations are described by ray theory -+ Vw = 0 (2.26) and the dispersion relationship for linear surface gravity waves w2 = gk tanh kh (2.27) where h = h(x) is local water depth. In practice, it is easier to collect data on the directional frequency spectrum E(f, 0; E, t) rather than the directional wave number spectrum F(k;i,t). These alternative spectral representations are related by F(k;i,t)dk = F(k,0;5,t)kdkdO = E(f,0;, t)dfdO (2.28) where 0 = tan-'(k,/ky) is the direction of the wave number vector k (k = k cos, ky = k sin 0) and k = Ikl. From equation 2.28, the following relation can be deduced F(k;i,t) = J E(f,0;5~,t) (2.29) where J is the Jacobian of the transformation O(f,O) CC (2.30) J= O ) = Cg = 10, C= -(2.30) (k) 2w' k Substitution of equation 2.29 into 2.25 gives a transport equation for the directional frequency spectrum (CCE) + coso 0 (CCE) + sinO (CCE) 1 DC 9C\ 0 2ruw + sin 0 -cos a(CC9E) = S (2.31) _CO PAGE 23 12 where S(f,0; ,t) becomes the forcing term. The detailed derivation of equation 2.31 is shown in Appendix A. The energy flux density C is given by the product of depth-dependent group velocity and wave energy spectrum as follows: Â£(f,0,h) = Cg(f,h)E(f,0,h) (2.32) Formulation of the problem is completed by specifying the source term S = S(f, 0, t). Consistent with linear wave theory, the source term is considered as the linear superposition of a number of mechanisms including atmospheric input by the wind, nonlinear transfer by resonant wave-wave interactions, and energy dissipations. The GM model adopted the depth-dependent frequency spectrum which was originally developed by Kitaigorodskii et al. (1975) and verified in the experiment by Bouws et al. (1985) as follows 2 22 E(f, O, h) = T(wh)Ej(f) cos (0 -0o) (2.33) where the depth dependent transformation factor is defined as T(Wh) = (2.34) X2[1 + W2(X2 -1)] where X is the solution to the transcendental equation X tanh(whX) = 1 and wh = 2irf(h/g)1/2. The derivation of equation 2.34 is shown in Appendix B. The one-dimensional windsea spectrum, Ej(f) has the JONSWAP spectral shape which is determined from the set of five free parameters ai = [fmi, a,7,a,,ia E(f; a) = Ep; a= EP( ai)"exp-(ffm-1)/22] (2.35) where EpM is the Pierson and Moskowitz (1964) spectrum: EpM(f; a) = g2(27r)-4f-5 exp [( ] (2.36) As in equation 2.33 the cosine-squared angular spreading function about the mean windsea direction (< < 0< Â ) is adopted. The mean windsea direction, 00 which is a PAGE 24 13 sixth spectral parameter, is defined as follows. 0o = tan~1 (I (2.37) where I E= ,I sin Odfd0 Jo 0 Iy= ICl cos dfdO (2.38) 2.4.2 Atmospheric Forcing by the Wind The GM model adopts a parameterization of Miles' mechanism based on the results of Snyder et al. (1981) for atmospheric forcing by wind: Tin(f, 0, h) = 3(f, 0, h)E(f, h) (2.39) where the growth function 0(f, 0, h) is given by 0, <1 (2.40) where pa and pw are the densities of air and water, respectively, w = 2irf is the wave angular frequency, C is the depth-dependent phase velocity, 0 and 0 are the wave and mean wind directions, respectively, and UII -Uo1 cos(O -0) is the wind speed at 10 m parallel to the wave direction 0. The coefficient B was determined by Snyder et al. (1981) to vary from 0.2 to 0.3. This was determined from direct measurements of the work done by wave-induced air pressure fluctuations over the sea surface. The heuristic lower limit of the windsea peak frequency is generally accepted as the frequency of a fully-developed sea (Pierson and Moskowitz, 1964). At the Pierson-Moskowitz frequency, fPM, the parameter U11/C = 0.82 in deep water. Adopting this concept also for fully developed sea states in finite depth leads to S= 0.13gtanh kpMh (2.41) U10 cos(00 -0) In the deep-water limit this expression reduces to the original Pierson-Moskowitz relation. In the shallow-water limit, for which PAGE 25 14 Cmaz(fPM) = (gh)1/2 (2.42) it would not be possible to reach fully developed conditions if U11 > 0.82(gh)/2. 2.4.3 Nonlinear Energy Transfer by Resonant Wave-Wave Interactions Atmospheric forcing provides energy to gravity waves and resonant wave-wave interactions among wave components redistribute energy within the spectrum. The general form of the "exact" nonlinear transfer terms TnI is given by the Boltzmann integral which expresses the rate of change of energy of the wave spectrum #i(k) at wave number k4 due to nonlinear wave-wave interaction (Hasselmann and Hasselmann, 1985) as: Tl(k4) = aW46(ki + k2 -k -k4) x 6(wl + 2 -3 -4) [N1N2(N3 + N4) -N3N4(N1 + N2)]dkldk2dk3 (2.43) where Ni = N(kfi) = f(k,)/w; stands for wave action densities, wi = (gki tanh kih)'/2 are the wave frquencies corresponding to the ith wavenumber ki, and a represents a complex scattering coefficient describing the coupling strength of four-way resonant interacting wave modes. Hasselmann et al. (1973) deduced the general form of the nonlinear transfer scales as Tnt(f, ) = a32f2~4V(f/fmn,) (2.44) where I is a dimensionless function describing the spectral shape. Based on similarity arguments, Herterich and Hasselmann (1980) calculated the finite-depth interaction for a narrow-band wave spectrum. The finite-depth source function of the one-dimensional nonlinear transfer can be scaled by a depth-dependent factor, R, i.e., Sn.(f,h) = R(whm)Snl(f, oo) for Whm > 0.7 (2.45) Confirmation of Equation 2.45 and the validity of shape similarity for directional distributions was made by Hasselmann (1981) who computed the interaction rates for a representative set of spectra F(f, 8, h) from the "exact" nonlinear transfer integral 2.43. Examination PAGE 26 15 of the results of Hasselmann and Hasselmann (1985) leads to the following expression for the proportionality factor R = R(wh,,) [ (X1) (2.46) 2.4.4 Dissipation by Bottom Friction The average rate of energy dissipation in an oscillatory wave boundary layer (Kajiura, 1968), or for random wave field (Hasselmann and Collins, 1968) can be expressed as aE -= -Edi,, = -" uT (2.47) where the overbar denotes averaging over the wave field. Equation 2.47 can be expressed in terms of the frequency-directional spectrum as: OE 1 fow2 = -2gsinh E(f,O). < ub > (2.48) at 2 g sinh kh where the root mean square bottom velocity < Ub > = Ubr 121 ,oo 2 1/2 = 2 sin kh E(f,0,h)dfdO) (2.49) is representative of the near-bottom velocity field. PAGE 27 CHAPTER 3 WAVE BOUNDARY LAYER AND BOTTOM SHEAR STRESS 3.1 Introduction The flow field in a shallow lake such as Lake Okeechobee is determined by a number of mechanisms, including wind-driven currents, wind waves, density differences due to changes in water temperature and suspended sediment concentration, and the water-surface slope, etc.. Sediment particles are suspended from the bottom due to turbulent shear stress at the sediment-water interface and transported within the water column by advection and diffusion of the flow field. The bottom shear stress due to slowly varying currents and oscillatory waves is affected by the highly nonlinear processes within the bottom boundary layer. For sediment transport studies in shallow waters, the bottom boundary layer dynamics is extremely important. The presence of waves is known to greatly enhance the bottom shear stress and resuspension of sediments, moreover, the surface waves in shallow waters are affected by the boundary-layer dynamics through the dissipation of wave energy. For non-cohesive sediments, the volume flux (q,) of sediment transported in the bottom boundary layer in terms of bed load or suspended load is related to the n-th power of the friction velocity, u., i.e. q, = Const(u.)n, where u. = Tb; 7b is bottom shear stress, and p is the fluid density. Various measurements have shown that the exponent n normally lies in the range 3 < n < 7 (Dyer and Soulsby, 1988). For cohesive sediments, the rate of erosion of sediments is proportional to the n-th power of the excess bottom shear atress with n ranging from 1.2 to 4 (Sheng, 1986, Lavelle and Mofjeld, 1987). Thus, accurate determination of sediment transport rate relies on the accurate estimation of bottom shear stress. Basic difficulties in estimating the bottom shear stress are associated with the following 16 PAGE 28 17 facts: The wind sea is essentially a random wave field. Therefore, the bottom shear stress calculated from such deterministic wave parameters as significant wave height (H1/3), and root mean square wave height (Hrm,,,), and wave period (e.g., peak wave period, significant wave period, zero up-crossing wave period, etc. ) could be inadequate for the quantitative estimate of sediment transport. It should be noted that the resuspension of sediment is by the instantaneous bottom shear stress, and not by the maximum bottom shear stress or mean bottom shear stress of representative waves. One of the most important parameters in the bottom boundary layer dynamics is the roughness height zo, at which the time mean velocity vanishes. For a natural bottom, there is great uncertainty in quantifying the roughness height because of its dependence on sediment properties, shape of bed form, and even flow parameters such as depth and velocity (Grant and Madsen, 1986). This chapter presents the basic concept and literature review of the oscillatory wave boundary layer, a description of the turbulent wave boundary layer according to the models of Jonsson (1963, 1966, and 1978) and Kajiura (1964, 1968). 3.2 Bottom Boundary Layer Boundary layers are typically described in terms of characteristic length and velocity scales that divide the boundary layer into at least two regions: one dependent on the absolute velocity and is directly influenced by the wall boundary, and the other dependent only on the relative external driving velocity and the overall scale of the boundary layer. Dimensional analysis leads to the following result for the velocity: u(z) (Z = -(-) (3.1) u. \6 Zo where 4 is a universal function to be determined, 6 is boundary layer thickness scaled by Ku./w, .is von Kirmgn's constant, and w is the wave angular frequency. PAGE 29 18 The outer limit (z/zo -oo, z/6 finite) corresponds to a velocity-defect law: U -U00 S = 1 (z/6) (3.2) The inner limit ( z/6 -+ 0, z/zo finite) corresponds asymptotically to a constant-stress layer -= 02(Z/Zo) (3.3) The actual form of the velocity profile can be found by assuming that there exists an overlap layer, zo < z < 6, where both equations 3.2 and 3.3 are valid simultaneously (Yaglom, 1979). It follows from equating the derivatives of equations 3.2 and 3.3 within this overlap layer that both functions are logarithmic: u -= Aln(z/6) + B (3.4) and -= Aln(z/zo) (3.5) Us, where 1/A is the von Kirmin constant K (K ranges from 0.38 to 0.41 for fully rough turbulent flows). The value of the constant B is less well known. Yaglom (1979) suggested that B is slightly above 2 (with a mean of about 2.35 for all suggested values) in boundary layer flows. For oceanic boundary layers, no direct determinations of the values of B have been reported. In a time-dependent flow due to waves and/or currents, the linearized equation for the boundary layer flows is: Ou 1 Op o(r/p) = -F+ (3.6) Ot p Oz Oz where u is the ensemble mean velocity, p is the ensemble mean pressure, r is the shear stress, and z is the flow direction and z is the vertical axis upward from the bottom. Invoking the usual boundary layer assumption that the pressure gradient is imposed by the free stream flow, uo, equation 3.6 becomes 8(u -u) 9(r/p) -(3.7) dt dz PAGE 30 19 Table 3.1: Nikuradse eqivalent sand-grain roughness height for flat bed. Dp is the diameter of the grain corresponding to p % finer in weight. (van Rijn, 1982) Ackers-White (1973) k, = 1.25D35 Einstein (1950) k, = D65 Engelund-Hansen (1967) k, = 2D6s Hey (1977) k, = 3.5D84 Kamphuis (1975) k, = 2.5D90 Mahmood (1971) ks = 5.1D84 van Rijn (1982) k, = 3D90 Close to the wall boundary (z/zo -0), unsteadiness vanishes (o -* 0) and the solution to equation 3.7 is the quasi-steady law-of-the-wall as shown in Equation 3.5. The logarithmic velocity region depends on the roughness length zo and requires the presence of an overlap layer in z, where zo < z < 6, to exist. As mentioned earlier, one of the most important parameters in the bottom boundary layer is the bottom roughness height zo, at which the mean velocity is zero. The roughness height is not measurable but is related to the geometrical scales of various roughness elements on the bottom. The mean velocity near the wall can be written in general form U= f( -d, u,'v (3.8) where d. is the displacement height or zero displacement (by analogy to the concept of the displacement thickness in boundary layer theory), and k is the mean height of the roughness elements (Monin and Yaglom, 1971). The roughness height, k, is usually related to Nikuradse equivalent sand-grain roughness height, k,. In general, k, is determined experimentally in terms of a measurable roughness dimension, and k is then determined from k,/k which is invariant for geometrically-similar roughness arrays (Wooding et al. 1973). For flat beds of sand, it is expected that the roughness height would be related to the diameter of the largest grains (D65, Ds4, and Dgo). Table 3.2 shows the different values of k, suggested by many authors (van Rijn, 1982). PAGE 31 20 The flow velocity at some distance above the bed is related to the bottom shear stress through the quadratic friction law: rb = pfwllublub (3.9) where f, is the wave friction factor. Alternatively, one can write: u = f/2 (3.10) where u. = \/~7m/, rom is the maximum bottom shear stress, and ub is the amplitude of the wave orbital velocity at the bottom. Based on laboratory measurements, Jonsson (1963, 1966, and 1978) and Jonsson and Carlsen (1976) developed a simple, semi-empirical formula for the friction factor f,. Kajiura (1964, 1968) developed an analytic model by using a time-invariant eddy viscosity for steady turbulent boundary layers. Analytical models similar to that of Kajiura, based on a time-invariant eddy viscosity, have since been proposed by Grant (1977), Smith (1977), Grant and Madsen (1979), Brevik (1981), and Long (1981). Kamphuis (1975) reported an extensive, purely empirical study in which the maximum boundary shear stress in a turbulent oscillatory flow was measured directly. Subsequent discussion by Grant (1975) and Jonsson (1976) indicate that the above semiempirical, theoretical, and purely empirical studies give essentially the same maximum bed shear stress over a wide range of bed and flow conditions. (Refer to Trowbridge and Madsen, 1984) Horikawa and Watanabe (1968) found that the eddy viscosity does vary significantly during the course of a wave cycle. Trowbridge and Madsen (1984) developed a time-varying eddy viscocity model which treated the turbulent oscillatory boundary layer at a level of approximation consistent with linear wave theory. To this degree of accuracy, prediction of the time-averaged energy dissipation rate are 20 -30 % smaller than those obtained from time invariant eddy viscosity models. Trowbridge and Madsen (1984) also advanced the solution to second order in wave steepness. The second-order, wave-induced mass transport is shown to depend critically on temporal variation of the eddy viscosity. The reversal of PAGE 32 21 the mass transport produced by relatively long waves is qualitatively simulated. This result is believed to explain the experiments of Inman and Bowen (1962) which measured the rate of sediment transport resulting from progressive waves combined with a very weak net flow rate in the direction of wave propagation. They found cases in which an increase in the net flow rate caused a decrease in the sediment transport rate. This result cannot be reproduced by a time-invariant eddy viscosity model. The Trowbridge and Madsen (1984) results demonstrate that the first-order solution depends only slightly on time variation in the eddy viscosity and is more sensitive to the more proper treatment of vertical variability. Most oscillatory turbulent boundary-layer models are based on the eddy-viscosity assumption. If sufficient data are available to establish the validity of the required parameters in the models, then the predictions of models in that particular application give reasonably acceptable results. However, when sufficient data are not available and the parameters for a specific application must be extrapolated from much different situations, the resulting predictions are highly speculative (Sheng, 1982). Sheng (1984) modified a turbulent transport model (often called "second-order closure model"), which was originally developed by Donaldson (1973), for application to coastal environment. In addition to predicting the mean flow variables in bottom boundary layers, the second-order closure model can predict turbulent quantities such as the Reynolds stresses, uu' the turbulent kinetic energy, q2/2, and the thickness of log layer. Due to the robust physics contained in the model, the second-order closure model is particularly valuable in dealing with sediment transport in shallow waters where highly oscillatory flow with appreciable density stratification due to variations in temperature, salinity and suspended sediment concentration. Although the second-order closure model gives very accurate results, the model is rather complicated and requires more computational effort than simpler eddy viscosity models. Thus, Sheng and Villaret (1989) developed a simplified (T.K.E. closure) version of the comprehensive second-order closure model to examine the effect of flow-sediment interaction PAGE 33 22 on the erosional behavior of fine sediments. Since the present study is focused on the applicaton and comparison of different wave models and the estimation of resulting bottom shear stresses, we decided to use the simplest turbulent wave boundary layer models of Kajiura and Jonsson for the calculation of bottom shear stresses. 3.3 Jonsson's Integrated Momentum Approach Based on the velocity measurements in a large oscillatory water tunnel, Jonsson (1966) and Jonsson and Carlson (1976) developed a simple theory for the calculation of the boundary layer thickness, the friction factor, and the dissipation of wave energy in an oscillatory rough turbulent boundary layer. This theory is based on the assumptions that the velocity profile is logarithmic throughout the entire wave boundary layer and that the phase shift between maximum shear stress and maximum amplitude of orbital velocity is negligible. Jonsson's integrated momentum approach differs from others in that the momentum equation need not be satisfied directly. A brief description of Jonsson's approach is given in the following. In the presence of a viscous sublayer, the shear stress near the bottom is described as: 7 = pv-pu'w' (3.11) where v is the kinematic viscosity, and u' and w' are the fluctuating velocity components in the z and z directions, respectively. The boundary conditions for equation 3.7 are u = for z=0 (3.12) U = uoo = Uom coswt for z -00 (3.13) where u,, is the maximum free stream velocity and w is the angular frequency. From equation 3.7 and boundary conditions 3.12 and 3.13, we have: 9(r/p) du, z=O dt (3.14) 9z di PAGE 34 23 It is interesting to note that the shear stress gradient at the wall is determined exclusively by the flow outside the boundary layer (free stream) in equation 3.14. Integration of equation 3.7 with boundary condition 3.13 yields: T = [ (noo -U) -= ( dz (3.15) If the velocity field u(z,t) is known, the shear stress can be calculated everywhere from equation 3.15. Adopting the quadratic friction law for the bottom shear stress, Jonsson developed a formula which related the friction coefficient f, and the Nikuradse sand roughness height k, for a given wave condition. For hydraulically rough walls, the following expressions of boundary layer thickness 6 and the friction coefficient f, were obtained: 306 306 aoo (. -logo = 1.20 (3.16) 1 1 aoom 14V + log10o --0.08 + loglo k (3.17) 4Vf@ 4] k, The detailed derivation of equations 3.16 and 3.17 are shown in Appendix C. Figure 3.1 shows the variations of fw with a,,m/k, as given by equation 3.17. Based on the test results of Bagnold (1946), Jonsson suggested a constant value of f, = 0.30 for aoom/ks < 1.57 as shown in figure 3.1. For hydraulically smooth beds, Jonsson (1967) suggested the following expressions 6 0.046538) =(3.18) aoom 10 f, = 0.09Re-0.2 (3.19) where Re = uoomaoom/v is the 'amplitude Reynolds number'. For laminar flow, the last term on the right-hand side of equation 3.11 is zero, and a simple analytical solution to equation 3.7 was obtained (Lamb, 1932, pp 622-623) U = UOm [coswt -e-zcos(wt -f3z)] (3.20) PAGE 35 24 10 \ Key f Kamphuis --Kajiura \--.--. Kajiura(exact) 10 --Jonsson -modified curves suggested by Jonsson and Kajiura -2 10 -110-1 100 10 102 a 103 10 ks Figure 3.1: Comparison of different formulas for the friction coefficient f varies with a/k, (Sleath, 1984). with 3 = vl(2v) = r/-(ir) and T is the period of wave. The shear stress is obtained by combining equations 3.11 and 3.20 r = Vvuoom,,e-p cos(wt -/z + r/4) (3.21) P From equation 3.20, the boundary layer thickness is 6 = = V/T (3.22) The maximum bottom shear stress is defined by: n6m = fwPILUoomloom, (3.23) For laminar flow, the friction coefficient is deduced from equations 3.21 and 3.23 2 f =(3.24) PAGE 36 25 In order to apply the preceding equations to a variety of flow conditions, the transition regimes among the laminar flow, smooth turbulent, and rough turbulent flow need to be clarified. Jonsson (1978) concluded that the laminar-smooth turbulent transition regime goes from Re = 103 to Re = 3 x 105, give or take a factor of two at both ends of intervals. For design purposes one can probably use Re = 105 as a transition value. For the transition between the hydraulically smooth bed and rough bed, Jonsson suggested aoom/ks = 45 which corresponds approximately to Re = 1.9 x 105. 3.4 Kajiura's Model of the Bottom Boundary Layer in Water Waves One of the most consistent and detailed theoretical boundary layer theories available is by Kajiura (1968). Kajiura developed a model that calculates various characteristics of the wave boundary layer, such as the friction coefficient and universal profiles of stress and velocity in the defect layer. Introducing time-mean properties of the turbulence, he assumed a time-invariant eddy viscosity compatible with the two-layer model (wall and defect layer) of the turbulent boundary layer. By further assuming the shear stress to vary sinusoidally, he introduced a modified friction velocity S(r/) (3.25) u* where f., is the amplitude of the bottom friction velocity, i.e. f. = f/p in which ib is the real part of the amplitude of the bottom shear stress. Differentiation of equations 3.7 and 3.25 with respect to t and z, respectively, yields: -= 0 (3.26) 9z2 Ct Kajiura solved equation 3.26 together with: au 7 = Etz (3.27) and variable eddy viscosity, Et, in the inner, overlap, and outer layer: v, Ki.z, and Kfi.A, respectively, for hydraulically smooth beds and 0.185nf.k,, Kit.z, and K6*uo, respectively, PAGE 37 26 for hydraulically rough beds, where A = 0.05iL./w is the thickness of the wall layer, K is the von Karmin constant (= 0.04), K is the Clauser constant (= 0.02), and 6' is the displacement thickness defined as 1 6* Amp (ul -u)dz (3.28) The constants of integration are evaluated by making the velocity and velocity gradient to be continuous at the boundaries between the layers. The solution is straightforward but algebraically very laborious. Therefore, only the procedure to determine the bottom shear stress is given briefly as follows: 1. Determine wave properties, H, and Tp, water depth, d, and Nikuradse sand-roughness height, k,. 2. Compute wave number k from the linear dispersion relation w2 = gk tanh(kd) (3.29) 3. Compute the maximum orbital velocity, uoom, and orbital amplitude of the water particle orbital motion, aom just outside the boundary layer. r'H, 1 toom (3.30) u Tp sinh (330) Uoom aoom = (3.31)" 4. Compute the amplitude Reynolds number, Re = Uoomaoom (3.32) 5. Assume the initial value of maximum bottom shear stress during the wave cycle, Tbm 6. Compute the thickness of the viscous sublayer 12V -P D = 12 = 12 v (3.33) ". VArbm PAGE 38 27 7. If k,/D < 0.4, use the hydraulically smooth bed expression for fu 2 fw = for Re < 2 x 105 81f+ log= -0.135 + log VRe for Re > 2 x 105 (3.34) 8.1iVJ V7w If k,/D > 0.4, use the hydraulically rough bed expression for f, f, = 0.25 for ao< 1.67 k, f. = 0.35 (ao -3 for 1.67 < a < 30 \ k, k, 0.98 1 -0.25 + log for a > 30 (3.35) 4" + log -05+og (aaorn 4Vf 4Vf7w k, k, 8. Compute the bottom shear stress, rbm = ~fwU2om (3.36) 9. Iterate until the solution converges to within a certain limit, e.g., (7m)new -(Tbm)old If (m)new -0.0001, then stop, otherwise go back to step 5 (Anm)new PAGE 39 CHAPTER 4 MODEL PERFORMANCE IN IDEALIZED WIND FIELD 4.1 Introduction The three wind-wave models (SMB, GD, and GM) described in previous chapters will be compared for the case of steady and uniform easterly wind field with wind velocities of 5, 10, 15, and 20 m/sec over Lake Okeechobee. The results showed significant spatial variation in wave parameters in Lake Okeechobee. Using the significant wave height and the peak wave period to represent a monocromatic wave, the bottom shear stress everywhere were computed through linear wave theory and Kajiura's oscillatory turbulent boundary layer model. 4.2 Comparison of Wave Models in Idealized Wind Fields The geometry and the bottom topography of Lake Okeechobee are shown in Figure 5.1. As can be seen in Figure 5.1, Lake Okeechobee is an extremely shallow lake with a maximum depth of about 5 meters. The depth variation in a cross section AA' shows that the eastern and central portions of the lake are relatively deep while the western part of the lake is shallow with a very mild slope of approximately 0.00016. Since the seasonal variation in water surface elevation can be as much as 2 meters in Lake Okeechobee and the shallow water wave models are sensitive to the change in water depth, it is necessary to use measured water level to adjust the water depth for wave model. The water depth shown in Figure 5.1 correspond to the water depth in fall of 1988. As shown in Figure 4.2, Lake Okeechobee sediments are composed of mostly fine-grained (silt and clay size) materials in the relatively deeper region and sand and shell in the shallower region of the lake. As was mentioned in Chap 3, the determination of roughness 28 PAGE 40 29 heights in a natural lake is a difficult task especially over the muddy bottom. For the estimation of bottom shear stress, however, the Nikuradse roughness height, k, was assumed to be 1 mm all over the lake. In order to compare the three wave models, the same computational grid with Ax = Ay = 2km was used as shown in Figure 4.3. The wind data of Lake Okeechobee during the deployment period, fall of 1988, showed that the strong winds with velocity about 10 m/sec were mostly blowing from east, north-east, and north. Based on this observation, the easterly winds with velocities of 5, 10, 15, and 20 m/sec were chosen to test the performance of the wave models in the Lake Okeechobee bottom topography, especially to compare the shallow water effects in the shallow western region. The SMB model assumes that the spatially uniform wind is blowing over uniform water depth over a sufficiently long distance for the waves to reach a steady state at the point of interest (i.e. fetch limited condition). Bretschneider (1958) suggested the procedure for computing wind waves generated onto the continental shelf from deep water shoreward with varying depth. According to the procedure, the bottom profile along the fetch is known and the traverse cross-section is divided into equal segments of about -5 to 10 miles or less in length, depending on the bottom slope. In order to get the wave height and period at the end of the each segment, the average depth, the refraction coefficient, shoaling coefficient, and the energy dissipation due to bottom friction are calculated over each segment. This procedure, however, can become quite involved for a large lake like Lake Okeechobee. Since Lake Okeechobee has a maximum fetch of about 50 km and has very mild bottom slope, a simpler method may be developed by using the average depth over the fetch to compute the wave height and period at the point of interest. This method may overpredict the wave height at the shallow water when the wind is blowing from deep to shallow water. Another simple method is to use the local water depth at the point of interest instead of the average water depth over fetch. The SMB model using average water depth will be denoted as SMB-AD model and the SMB model using local water depth will be denoted as SMB-LD model. These two methods will be compared with GM model and GD model. PAGE 41 30 The significant wave height, H,, peak wave period, Tp, and the maximum bottom shear stress, rb, during a wave cycle are computed at the center of the grid cell. Figures 4.4, 4.5, 4.6 and 4.7 show the variations of H,,Tp, and rb with the fetch distance along the cross section AA' shown in Figure 5.1. The numerical values of results are shown in Appendix D. For the 5 m/sec wind, the wave doesn't feel the bottom except in the very shallow western part of the lake where the water depth is less than 2 meters. As can be seen in Figure 4.4, there are no significant differences in H,, Tp, and Tb among the results of the 3 models. Compared to the other models, the GD model gives slightly lower values of significant wave heights and peak wave periods. It is interesting to note that the GM model and the SMB-LD model give almost the same significant wave height along the entire fetch. On the other hand, the GM model and the SMB-AD model yield almost the same peak wave period along the fetch. These trends remain quite the same for the stronger wind cases. For the case of 10 m/sec wind as shown in Figure 4.5, significant wave heights given by the various models show significant difference in the shallow region, but the peak wave periods given by the various models remain almost the same. As expected, the deep water GD model shows an increase in significant wave height as the fetch increases. The SMB-AD model shows no variation in H, when the fetch exceeds 30 km and the water depth is less than 3 meters. This is due to the fact that the increase in H, due to the longer fetch is balanced by the decrease in H, due to the shallower average depth. As was shown in Figure 4.5, the bottom shear stresses produced by the GD model and the GM model are different by almost an order of magnitude. Since wind velocities on the order of 10 m/sec can be commonly observed on moderately windy days over Lake Okeechobee, using the GD model will lead to excessive bottom shear stress and erosion of sediment. The results for wind velocities 15 m/sec and 20 m/sec are shown in Figures 4.6 and 4.7, respectively. It is interesting to note the similar behavior of H, given by the GM model and the SMB-LD model, and the nearly indentical behavior of Tp produced by the GM model and the SMB-AD model. It should be reminded that it is a very crude approximation to PAGE 42 31 0.60 A A' Fig 1 -2 3 F5 Figure 4.1: Geometry and bottom topography of Lake Okeechobee PAGE 43 32 2 KM -i 2KM SB D ------___--------------t E km computational grid used in Lake Okeechobee ----/c 5 --------. _ _ _ _ _ ----------------------^ ^ ^ ^ ^ Y I:---^ -----/----B, km copttoa/rduedi aeOeco PAGE 44 33 K L1 J A -. N d/ ,. :.3 .: G :AJ^ i -__Li--^ lF ~ .....'.. .l t... H -*.* .. .... .-. -, !:-~ -4 .e .. i MARSH .. I : SANDISHELL/MARL -/..'"'" ;*/ ,: -:'J "" | } --" Dl -..MUD ROCK Figure 4.3: Bottom sediment characteristic of Lake Okeechobee (from Reddy et al. 1988) .;.... ..b .,.'. PAGE 45 34 0. -1. 2. S3. 4. U .SMB-AD ..... : ---D 5. -SMB-L : --GD Go GM : -1.4 i -I -\ 1.2 1.0 5 0.8 0.6 .0. 2. 0 .0 ii I I I I I I I 0.2 0.0 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 7. .---------6. .5. I0. 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 1000.0 100.0 M 0.1 i.o 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. FETCH (km) Figure 4.4: Comparison of wave models for 5 m/sec easterly wind. Variations of significant wave height, H,, peak wave period, Tp, and bottom shear stress, 7,, with fetch along the cross section AA' PAGE 46 35 0. 2. S3. 0 4. L SMB-R : .... M 5. SMB-LO : GO GM : --1.4 --------1.2 1.0 0.8 -...................... (0 0.6 S 0.4 0.2 0 0 I I I I I I I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 7. 6. o 5. () 4. l_2. 1. 0 .I I I I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 1000.0 I I E -100.0 -1 10.0 cr (I 0.0 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. FETCH (km) Figure 4.5: Comparison of wave models for 10 m/sec easterly wind. Variations of significant wave height, II,, peak wave period, Tp, and bottom shear stress, n,, with fetch along the cross section AA' PAGE 47 36 0. --0. r---------^____ -.2. = 3. I-Q 4. U 5 SMB-PD : ............ *. SMN-LD : G/D G& : 2.0 1.6 E 1.2 S0. 8 -0.4 0.0 I I I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 7. 6. 4. a. 2. -1. 0. I I I I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 1000.0 100.0 ............. 10.0 C 1.0 r 0.1 I0.0 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. FETCH (km) Figure 4.6: Comparison of wave models for 15 m/sec easterly wind. Variations of significant wave height, H,, peak wave period, T,, and bottom shear stress, Tb, with fetch along the cross section AA' PAGE 48 37 0. -1. 2. r 3. o0 4. LU SHB-D : ----M 5. -SMB-LD : G/D G&M : 2.0 1.0 --------------i------| --|--|----0.4 0.0 S. I-------------------0.0 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 7. 6. S 5.5) -z -N-o. 2. 1. o-I,-----------0 .I I I I I I I I I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. 1000 .0 5 -| -1 -| -| -\ -I --I -I -1 -1 -j 100.0 .r ..........----. S10.0 .0 1.0 0.1 r CC 0 .0 I I I I I I I I 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55. FETCH (km) Figure 4.7: Comparison of wave models for 20 m/sec easterly wind. Variations of significant wave height, H,, peak wave period, Tp, and bottom shear stress, Tb, with fetch along the cross section AA' PAGE 49 38 use the average water depth over fetch or the local water depth at the point of interest to obtain the H, and Tp in the SMB model. The similarity between H, given by the GM model and the SMB-LD model is due to the fact that the depth-dependent transformation factor ((wh) of Equation 2.34 employed by the GM model is only a function of local water depth and frequency. This means that the transformation of finite depth spectrum is independent of the path of the waves and consequently the significant wave height is strongly dependent on the local water depth in the GM model. It is generally accepted that in deep water the spectral peak of wind waves shifts toward lower frequencies as a consequence of the nonlinear energy transfer. Input from the wind occurs over the central region of the spectrum and wave-wave interaction processese rapidly redistribute the enhanced energy level towards lower and higher freqencies. In the high-frequency range, the input and nonlinear transfer terms are balanced by whitecapping dissipation processes. As surface wave propagates into waters of finite depth, the low-frequency spectral components interact with the bottom, thus leading to frictional dissipation of lower frequency wave components. This frictional dissipation tends to slow down the rate of wave energy migration towards lower frequencies or even totally reversing the trend of the shift (Graber and Madsen, 1988). The similar behavior of peak wave period along the fetch as given by the GM model and the SMB-AD model might be a coincidence. It is likely that the migration of peak wave period produced by a balance between the nonlinear energy transfer and the frictional bottom dissipation in the GM model is similar to the change of peak wave period due to a balance between the increase of fetch and the decrease of the average depth in the SMB-AD model. The relative performance of the various wave models might be evaluated in terms of the bottom shear stress produced from the wave parameters. As can be seen in Tables D.1, D.2, D.3, and D.4 (Appendix D), the different models give significantly different bottom shear stresses in the shallow water region (water depth less than 3 meters) especially for PAGE 50 39 high wind velocities. For example, the GM model gives bottom shear stresses of 16.98 and 18.36 dyne/cm2 for 15 and 20 m/sec wind velocities, respectively at 30 km fetch and 2.6 meters water depth. At the same location, the SMB-LD model gives bottom shear stresses of 26.94 and 38.45 dyne/cm2 for 15 and 20 m/sec wind velocities, respectively. The SMB-AD model gives 45.43 and 66.43 dyne/cm2 for 15 and 20 m/sec wind, respectively, and the GM model gives 100.70 and 241.21 dyne/cm2 for 15 and 20 m/sec wind velocities, respectively. Figures 4.8, 4.9, 4.10, and 4.11 show the contour plots of H,, Tp, the bottom orbital amplitude, Ab, and Tb obtained from the GM model for steady and uniform easterly wind velocities 5, 10, 15, and 20 m/sec, respectively. As the wind velocity increases, the regions with peak H, shift eastward to the deeper central part of the lake indicating that the wave heights become more constrained by the water depth in the shallow region. As shown in the bottom shear stress contours in Figure 4.8 and 4.9, the low bottom shear stress region is very closely correlated with the mud zone shown in Figure 4.2. Particularly strong bottom shear stresses occur in the marsh and sand/shell regions. This may imply that the fine sediments are resuspended due to the moderately strong wave action and carried by wind-driven currents to the deep eastern part of lake where most of fine sediments are deposited. Sheng et al. (1989) showed that under the easterly wind, opposing currents driven by the surface slope are found in the bottom layer. For strong winds or episodic storm events as shown in Figure 4.9 and 4.10, bottom shear stresses in the deeper mud zone appeared to be strong enough to resuspend the sediment particles. Figures 4.12, 4.13, 4.14, and 4.15 show the contours of H,, T,, Ab, and rb obtained from the SMB-LD model and Figures 4.16, 4.17, 4.18, and 4.19 are the model results of the SMB-AD model for wind velocities 5, 10, 15, and 20 m/sec, respectively. The spatial distributions of significant wave height 'and bottom shear stresses are similar between the GM model and SMB-LD model. The spatial distributions of peak wave period are similar between the GM model and the SMB-AD model. PAGE 51 40 SIGNIFICANT WAVE HEIGHT (mteers) PEAK WAVE PERIOD (sec) HIND : HIND : S M/SEC S M/SEC 90 DEGREES 90 DEGREES ORBITAL AMPLITUOE (Meters) BOTTOM SHEAR STRESS (dyne/sq. cml HIND : HIND : 5 W/SEC 5 H/SEC 90 DEGREES 90 DEGREES Figure 4.8: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rn, obtained from the GM model for 5 m/sec easterly wind. PAGE 52 41 SIGNIFICANT HAVE HEIGHT (meters) PEAK HAVE PERIOD (sec) WIND : WINO : 10 SEC 10 M/SEC 90 DEGREES 90 DEGREES ORBITAL AMPLITUOE Metersl) BOTTOM SHEAR STRESS (dyne/sq. cml HIND : MIND : 10 M/SEC 10 MISEC 90 DEGREES 90 DEGREES Figure 4.9: Contours of the significant wave height, Hs, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, Tb, obtained from the GM model for 10 m/sec easterly wind. PAGE 53 42 SIGNIFICANT HAVE HEIGHT (meters) PEAK HAVE PERIOD (sec) NINO : HIND : 15 M/SEC 15 H/SEC 90 DEGREES 90 DEGREES ORBITAL AMPLITUDE (meters) BOTTOM SHEAR STRESS (dyne/sq. ca) HIHND : HIND : 15 M/SEC IS K/SEC 90 DEGREES 90 DEGREES Figure 4.10: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, ir, obtained from the GM model for 15 m/sec easterly wind. PAGE 54 43 SIGNIFICANT WAVE HEIGHT (meters) PEAK HAVE PERIOD (sec) HIND : HIND : 20 M/SEC 20 M/SEC 90 DEGREES 90 DEGREES ORBITAL AMPLITUDE (Melers) BOTTOM SHEAR STRESS (dyne/sq. cm) IND : HIND : 20 H/SEC 20 M/SEC 90 DEGREES 90 DEGREES i Figure 4.11: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, b, obtained from the GM model for 20 m/sec easterly wind. PAGE 55 44 SIGNIFICANT WAVE HEIGHT (meters) WAVE PERIOD (sec) WIND : WIND : 5 M/SEC 5 M/SEC 90 DEGREES 90 DEGREES o3 ORBITAL AMPLITUDE (meterL) BOTTOM SHEAR STRESS (dyne/sq. ca) HIND : WIND : 5 M/SEC S H/SEC 90 DEGREES 90 DEGREES Figure 4.12: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-LD model for 5 m/sec easterly wind. PAGE 56 45 SIGNIFICANT WAVE HEIGHT (neters) HAVE PERIOO (sec) HIND : HINO : 10 K/SEC 10 HISEC 90 DEGREES 90 DEGREES ORBITAL AMPLITUDE (Meters) BOTTOM SHERR STRESS (dyne/sq. cn) HIND : IND : 10 M/SEC 10 M/SEC 90 DEGREES 90 DEGREES Figure 4.13: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-LD model for 10 m/sec easterly wind. PAGE 57 46 SIGNIFICANT HAVE HEIGHT (eters) HAVE PERIOD (sec) HIND : HIND : 15 M/SEC [ 15 H/SEC 90 DEGREES 90 DEGREES ORBITAL AMPLITUOE (neters BOTTOM SHEAR STRESS (dyne/sq. cr) HIND : WIND : IS H/SEC IS M/SEC 90 DEGREES 90 DEGREES Figure 4.14: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-LD model for 15 m/sec easterly wind. PAGE 58 47 SIGNIFICANT HAVE HEIGHT (Meters) HAVE PERIOD (sec) HINO : WIND : 20 M/SEC 20 M/SEC 90 DEGREES 90 DEGREES ORBITAL AMPLITUDE (meters BOTTOM SHEAR STRESS (dyne/sq. ca) HIND : WHIND : 20 H/SEC 20 M/SEC 90 DEGREES 90 DEGREES Figure 4.15: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, 7b, obtained from the SMB-LD model for 20 m/sec easterly wind. PAGE 59 48 SIGNIFICANT HAVE HEIGHT (meters) HAVE PERIOO (sec) NINO : WIND : 5 M/SEC 5 M/SEC 90 DEGREES 90 DEGREES ORBITAL AIPLITUOE (meters) BOTTOM SHEAR STRESS (dyne/sq. cm) WIND : W IND : S N/SEC 5 M/SEC 90 DEGREES 90 DEGREES Figure 4.16: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-AD model for 5 m/sec easterly wind. PAGE 60 49 SIGNIFICANT AHVE HEIGHT (meters) WAVE PERIOD (sec) HIND : HIND : 10 M/SEC 10 M/SEC 90 DEGREES 90 DEGREES ORBITAL AMPLITUOE (meters) BOTTOM SHEAfi STRESS (dyne/sq. cn) WIND : W: HIND : 10 M/SEC .ae \ \ 10 M/SEC 90 DEGREES \ 90 DEGREES Figure 4.17: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, 7, obtained from the SMB-LD model for 10 m/sec easterly wind. PAGE 61 50 SIGNIFICANT HAVE HEIGHT (meters) HAVE PERIOD (sec) WIND : WIND 15 M/SEC 15 M/SEC 90 DEGREES 90 DEGREES I00 ORBITAL AMPLITUDE Ieters BOTTOM SHEAR STRESS (dyne/sq. c)l HIND : WIND : IS M/SEC IS M/SEC 90 DEGREES 90 DEGREES Figure 4.18: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, rb, obtained from the SMB-LD model for 15 m/sec easterly wind. PAGE 62 51 SIGNIFICANT HAVE HEIGHT (meters) WAVE PERIOO (sec) HIND : WIND : 90 DEGREES 90 DEGREES ORBITAL AMPLITUDE (meters BOTTOM SHERR STRESS (dyne/sq. cal HIND : W IND : 20 M/SEC 20 M/SEC 90 DEGREES 90 DEGREES Figure 4.19: Contours of the significant wave height, H,, peak wave period, Tp, bottom orbital amplitude, Ab, and bottom shear stress, mr, obtained from the SMB-LD model for 20 m/sec easterly wind. PAGE 63 CHAPTER 5 MODEL PERFORMANCE IN REAL WIND FIELD 5.1 Introduction The GM model and the SMB model are tested with the measured wave data obtained from subsurface pressure transducers at three stations in Lake Okeechobee. The wind data collected from anemometer at the wind station are used to force the finite-depth GM model and the SMB model. Comparisons between the hindcast and measured waves include the significant wave height, peak wave period, zero up-crossing wave period, and one-dimensional wave energy spectral shape. 5.2 Field Data Analysis For Lake Okeechobee Phosporous Dynamics Study, wave, current, turbidity, and temperature data were collected at six platforms in Lake Okeechobee during 20 September to 5 November, 1988, by the Coastal and Oceanographic Engineering Department of the University of Florida (Sheng et al. 1989). The subsurface pressure data were collected at six locations, (site A, B, C, D, E, and F) in Figure 4.2. Detailed information of the wave gages is shown in Table 5.1. The wind data were measured by anemometer at 8 meters above the water surface at the wind station L005 in Figure 4.2, which has been maintained by the South Florida Water Management District. The subsurface pressure data obtained from Lake Okeechobee were analyzed using program PREANL.FOR, which was developed at the Coastal and Oceanographic Engineering Department of the University of Florida. The surface waves are recovered from subsurface time series pressure data through linear wave theory. Although the measurement of waves with pressure transducers has been used widely, controversy still exists over the adequacy of 52 PAGE 64 53 Table 5.1: Locations of the pressure transducers deployed in Lake Okeechobee Gage Water Location of Station pressure depth depth ho/h station transducer ho h Latitude Longitude A COE 55695 1.33 3.9 0.34 27 06.31 80 46.21 B COE 55696 0.71 2.7 0.26 27 02.78 80 54.31 C COE 48228 0.79 4.6 0.17 26 54.10 80 47.36 D COE 55694 1.04 4.3 0.24 26 58.47 80 40.34 E COE 55699 0.61 2.7 0.23 26 52.81 80 55.96 F COE 55697 0.61 1.8 0.34 26 51.90 80 57.09 the transfer function from subsurface pressure to surface wave height using linear wave theory. However, according to Bishop and Donelan (1987), linear theory is generally adequate to compensate pressure records to give surface wave heights to within five percent. The pressure data measured from the subsurface gage are considered to be a linear summation of various contributing components: P(t) = Pa + pg(h -ho) + pgKp7(t) (5.1) where P(t) is the measured pressure at the gage and Pa is the atmospheric pressure at the surface, h and ho denote the mean water depth and gage height above bottom, respectively, r7(t) is the surface wave, Kp is the pressure response function, p is the water density and g is the gravitational acceleration. The pressure induced by surface gravity waves is P = pgKp7 = P(t)-(5.2) where P denotes the time-averaged mean pressure at the gage and the pressure response function, Kp is usually defined through linear wave theory Kp(wn) = coshkn(h + z)/coshkn(h) (5.3) where z is the depth of gage location and w, and PAGE 65 54 kn are wave frequency and wave number, respectively. Although equation 5.2 relates 7 and P in the time domain, the surface wave information is commonly recovered in the frequency domain (in terms of wave energy spectrum) because K, can be more readily evaluated in this domain. The spectral technique assumes that P(t) is expressible as a summation of harmonics: P(t) = pg Ancos(wnt -en) (5.4) n where en is the phase angle. The surface wave is then expressible as: () = cos(wnt -Cn) (5.5) through linear wave theory. At present, the values of An and Kp(wn) for every value of w, are most efficiently obtained from the Fast Fourier Transform (FFT) of the pressure data. The major difficulty of applying FFT method for real field data is that 1/K,(w) increases almost exponentially with frequency in the high-frequency range. Therefore, without a prior knowledge of noise level, it is difficult to predetermine a cutoff frequency (Lee and Wang, 1984). However, the present program imposed the cutoff frequency of 0.7 Hz. The subsurface pressure data were sampled every half hour for 17 minutes at a sampling rate of 2 Hz. Thus, each burst of data contains 2048 digitized data points. Bad data such as flat points and anomalous peak points have been removed by the bad-point correction algorithm of PREANL.FOR. The averaged energy spectrum are obtained such that the total of 2048 data points are divided into 32 segments of data using 50 % overlapping factor, therefore each segment contains 128 data points. In order to reduce the undesirable effects related to spectral leakage, the time series of each segment with length of 64 seconds was analyzed by using the cosine bell window. Also low frequency components of less than 0.1 Hz due to currents were eliminated by using low pass filter. The resultant spectral estimates have 48 degrees of freedom, with the expected spectral value with in a factor of 0.7 to 1.23 of the sample value at 80 % confidence limits. The typical energy spectrum obtained from analysis is shown in Figure 5.1. From the spectrum, the significant wave PAGE 66 55 0.4 CU c 0.3 0.2 0 3 >I02 LU LU 0.1 _1) a-0.0 0.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZl Figure 5.1: Typical wave spectrum from Lake Okeechobee. height, peak wave period, and zero up-crossing wave period were determined as follows: H, 4V/-m (5.6) Â• Tz, = (5.7) where T, is the zero up-crossing wave period and mo and m2 are the first and second order moment of the spectrum. 5.3 Comparison of Model Results vs. Measured Data For the bench mark test of the GM model and the SMB model vs. measured wave data, the period from October 7 to October 12 was selected because of the sustained wind condition. During this period, only three pressure gages at stations B, C, and E were working properly. As shown in Figure 4.2, the station B is located in 2.7 m depth of water PAGE 67 56 close to the northwest vegetation area. It should be mentioned that the shaded area in Figure 4.2 is actually a samll island, but for the sake of computational convenience, it is treated as very shallow area (0.5 m depth). Stations C and E are located in 4.6 m and 2.7 m depth of water, respectively. 15 minutes averaged wind data were available at station L005. Wind velocities were vector averaged each hour to force the wave models and the wind fields were assumed to be spatially uniform over the lake for every hour. 5.3.1 Comparison of the GM Model results with Measured Data In the GM model, the bottom friction coefficient needs to be specified as input. Figure 5.2 shows the sensitivity of model results to variation in friction coefficient at stations B, C, and E on Octorber 7, 1988. During this period, the GM model overestimated the smaller waves but underestimated the larger waves. The variation in significant wave height due to the varying friction coefficients is most notable during high winds from hour 19 to 24 on Octorber 7. During the high wind period, the wave spectral shapes at station E corresponding to various friction coefficients are shown in Figure 5.3. Since the friction coefficient is a function of the bottom roughness and the flow characteristics at the bottom, it would have been more proper to provide the roughness height as the model input. However, due to the uncertainty in determining the roughness height and also for the sake of simplicity, the GM model uses the friction coefficient instead of roughness height as an input parameter. As shown in Figure 5.2 and 5.3, lower friction coefficient gave better estimation of significant wave height yet higher friction coefficient gave better result of wave spectral shape. The friction coefficient of 0.02 was chosen as an input to GM model for Lake Okeechobee application. For the six day period from October 7 to October 12, the hindcast results of the GM model are compared with measured data at stations B, C, and E in Figures 5.4, 5.5, and 5.6. The hindcast time series of significant wave height and peak wave period show good overall agreement with measured data at stations C and E. At station B, however, the hindcast H, and Tp are overestimated compared with measured data. The poor agreement at station PAGE 68 57. ......... : f x 0.01 : f = 0.02 : = 0.04 : f = 0.08 : f = 0.10 0.8 r 5. STATION : E STATION : E 0.6 -4. C; 3. 2. 0.2 0.0 0. ---' -I-I0 4 8 12 16 20 24 0 4 8 12 16 20 24 HOURS HOURS 0.8 5. STATION : C STATION : C 0.6 4. lot 2. S'V 21. 0.2 1 0.0 --tt I I I -0. t-, -I 0 4 8 12 16 20 24 0 4 8 12 16 20 24 HOURS HOURS 0.8 5. STATION : B STATION : B 0.6 4 ...... 3. -CN 2. 0.2 1. + 4 1. -, 0.0 "0. O 4 8 12 16 20 24 0 4 8 12 16 20 24 HOURS HOURS Figure 5.2: Comparison of the calculated significant wave heights and peak wave periods with measured data for friction coefficient of 0.01, 0.02, 0.04, 0.06, and 0.08 at stations B, C, and E on October 7, 1988. PAGE 69 58 ......... t 0.02 f ta 0.06 -: : ued -: r 0.04 ---~ f a 0.08 0.3 E88100701 E88100703 E88100705 0.2 .o L I I 0.1 0.0 0. 0.3 E88100706 E88100708 E88100711 <' 0.2 0.0 I Z LJ 0.3 E88100713 E88100715 E88100716 >-. J 0.2 z LlJ -j I* -^---I -H----I I-------I cr 0.0 I0.3 SE88100718 0 E88100720 E88100723 0.2 Â• 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure 5.3: Comparison of computed wave spectral shape with measured data for friction coefficients of 0.02, 0.04, 0.06, and 0.08 at station E on October 7, 1988. PAGE 70 59 STATION : L005 10 n/sec Ivnd speed) 281 282 283 284 285 286 287 1.0 I STATION : E 0.8 E 0.6 -+ + C O 0.4 0.2 + ++ + + 0.0 281. 282. 283. 284. 285. 286. 287. STATION : E 5. ., 4. Cn 2 + 0. I I--281. 282. 283. 284. 285. 286. 287. C N .. STATION : E C S .'3 E N .I I. I 281 282 283 284 285 286 287 JULIAN DRY Figure 5.4: Measured wind bar graph at station L005 and the comparison of the GM model results (-) and measured (+++) H. and Tp at station E. PAGE 71 60 STATION : LOS5 10 a/sec (wind sped) 281 282 283 284 285 286 287 1.0 STATION : C 0.8 S 0.6 + S++ 0.2 0.0 281. 282. 283. 284. 285. 286. 287. STATION : C 5. CL 2. ++ + + I0. 281. 282. 283. 284. 285. 286. 287. .N L STATION : C C E results (-) and measured (+++) H, and Tp at station C. PAGE 72 61 STATION : L005 0 M/sc (wind sp0. 281 282 283 284 285 286 287 STATION : B 0.8 E 0.6 co 0.4 + ++ + 0.2 + ++ +04+H 4 + + + 0.0 281. 282. 283. 284. 285. 286. 287. 6. 1-I STATION : B 5. U 4. -s3. 0. ^2. 0. !---l-------------281. 282. 283. 284. 285. 86. 287. N STATION : 8 > S 0 C E 0 S N I I I I I 281 282 283 284 285 286 287 JULIRN DRY Figure 5.6: Measured wind bar graph at station L005 and the comparison of the GM model results (-) and measured (+++) H, and Tp at station B. PAGE 73 62 1.0 5. 0.8 -4. IS0.6 3. -' ) 0.4 t 2. +* t^>+ Ct +) Â§ + + 0 + Z 0.2 E + -. 0.0 0 .' I 0.0 0.2 0.4 0.6 0.8 1.0 0. 1. 2. 3. 4. 5. MEASURED Hs (m) MEASURED Tp (sec) 5. 4. N 3. t 2. cc CO) 0o. 0. 1. 2. 3. 4. 5. MEASURED Tz (sec) Figure 5.7: Comparison of hourly hindcast by the GM model and measured significant wave height Hs, and peak wave period Tp, and zero up-crossing wave period T, at station E. PAGE 74 63 1.0 5. 0.8 -4. E (u .46, f 0.6 -+ 3. + 3+ .. + + "1 1 ++ + + +.-+ S0.4 + S 4. + m 2 ++4+ + Z 0.2 + 0.0 ------0 MESURED Hs UR MESURED Tp (sec)sec Figure 5.8: Comparison of hourly hindcast by the GM model and measured significant wave height H, peak wave period T., and zero up-crossing wave period T, at station C. N 3. U 2. + 4: 0. p--i --0. 1. 2. 3. 4. 5. MERSURED Tz (sec) Figure 5.8: Comparison of hourly hindcast by the GM model and measured significant wave height H5, peak wave period Tp, and zero up-crossing wave period T, at station C. PAGE 75 64 B is primarily due to the elimination of an island in the model which is located at 7 km northeast of station B. Since no measured mean wave direction is available, the hindcast mean wave direction and the measured wind direction are shown in Figures 5.5 and 5.6. The deviations of the hindcast mean wave direction from the measured wind direction are mostly less than 15 degrees. Figures 5.6 and 5.7 show the comparison of hindcast and measured significant wave height, peak wave period, and zero up-crossing period at stations E and C, respectively. The correlation coefficient and a root-mear-square error normalized by the root-mear-square value of measured data are used for the statistical analysis between the hindcast and measured data. The correlation coefficient and the normalized root-mean-square error between hindcast zx and measured yi are defined as follows: l-1[(z -)E(y -(5.8) = Ei"_-1(Kx -E7-=I(yl and i=l i= E=< E(xY01 .)2 (5.9) where the overbar denotes the mean value. As can be seen in Figure 5.6, the agreement between the hindcast and measured significant wave height, at station E is quite good with a normalized root-mear-square error of 0.22 and a correlation coefficiant of 0.90. From Figure 5.6, we can see that the hindcast significant wave height at station E systematically overestimates the lower waves and underestimates the higher waves. Table 5.2 shows the correlations and the normalized root mean square error between hindcast and measured significant wave height, peak wave period and zero up-crossing wave period at station E. Figure 5.7 and Table 5.3 show the comparison between the hindcast and measured significant wave height, peak wave period, and zero up-crossing wave period at station C. PAGE 76 65 Table 5.2: Comparison of hindcast by the GM model and measured values of significant wave height, peak wave period, and zero up-crossing wave period at station E. Correlation Normalized root coefficient mean square error IHI 0.90 0.22 Tp 0.84 0.14 Tz 0.85 0.14 Table 5.3: Comparison of hindcast by the GM model and measured values of significant wave height, peak wave period, and zero up-crossing wave period at station C. Correlation Normalized root coefficient mean square error H. 0.88 0.21 Tp 0.74 0.17 Tz 0.83 0.16 Since station C is located in the deeper central part of the lake, the waves are less influenced by the shallow water effect. The comparison of significant wave height between station C and E indicate that the systematic underestimation of higher waves and overestimation of lower waves is due to the improper treatment of shallow water mechanism in the GM model. The comparisons between hindcast and measured spectra st stations C and E are shown in Appendix E. The hindcast spectra have sharper peak than the measured ones. The overall agreement between hindcast and measured spectral shape seems good. 5.3.2 Comparison of the SMB Model Results with Measured Data The significant wave height and peak wave period hindcast by the SMB-AD model and the SMB-LD model were compared with measured data at stations B,C, and E. Figures 5.9, 5.10 and 5.11 show the comparisons of the time series of significant wave height and peak wave period hindcast from the SMB-AD and SMB-LD model with measured PAGE 77 66 Table 5.4: Comparison of hindcast by the SMB-AD and SMB-LD model and measured values of significant wave height and peak wave period at station E. Correlation Normalized root coefficient mean square error SMB-AD H, 0.81 0.27 Tp 0.73 0.22 SMB-LD H, 0.83 0.25 T _0.68 0.22 data at stations E,C, and B, respectively. The overall agreements are considered good at stations C and E. Again, model overestimates the significant wave height at station B due to the ignorance of the small island located at 7 km to the northeast of station B. As can be seen in Figure 5.9, when the waves are propagationg from deep to shallow water, results of the SMB-LD model agree well with measured data (note the strong northeast wind event during the first day of the period, Julian day 281). This is consistent with the idealized uniform easterly wind case where the SMB-AD model gave higher wave heights than the SMB-LD model and the GM model. On Julian day 281, the GM model slightly underestimates the significant wave height (Figure 5.4) and the SMB-LD model slightly overestimates significant wave height (Figure 5.9) at station E. The hindcast zero wave height and zero peak wave period shown in Figure 5.9 are due to the zero wind velocities. However, the data showed that wave height and wave period can be non-zero even for zero wind velocities due to the remnant effect of wind at previous hours. Figures 5.12 and 5.13 show results obtained from the SMB-AD model and the SMB-LD model, respectively, where hindcast and measured significant wave height and peak wave period at atation E are compared. Table 5.4 summarizes the correlation coefficient and the normalized root mean square error between the hindcast and measured significant wave height and peak wave period at station E. The results are consistent with the idealized wind PAGE 78 67 mnaSo ~~~~~"------"--------------'S SThRTti S ---10 r/sec (wind sped) 281 282 283 284 285 286 287 1.0 1. 1. STATION : E SMB-A : 0.8 -SMB-LD : --...... E 0.6 (n 0.4 -I-1 .-0.2+" .* -* + 0.0 281. 282. 283. 284. 285. 286. 287. 6. STATION : E 8-: : 5. SMB3. 4. o 2. I I 0. 281. 282. 283. 284. 285. 286. 287. .STATION : E C c. E O, I N I -1 I281 282 283 284 285 286 287 JULIAN DAY Figure 5.9: Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) Ha and Tp at station E. PAGE 79 68 STATION : L005 10 R/Lc (vind qped) 281 282 283 284 285 286 287 1.0 1I I -1 STATION : C SMB-RO : 0.8 -.. SM-LD : ---------. E 0.6 -,, 0.2 0.0 281. 282. 283. 284. 285. 286. 287. 6. IIII----. STATION : C SMB-O : 5. SIB-L : -------U 4. .. 2. I-2. +I I 1 .I 281. 282. 283. 284. 285. 286. 287. 0 N S'$TATION : C C 0) S .1 I I I I 281 282 283 284 285 286 287 JULIAN DAY Figure 5.10: Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) H. and Tp at station C. PAGE 80 69 STATION : L005 10 n/sec (wind speed) 281 282 283 284 285 286 287 1.0 I I -\--____I STATION : B SMB-AO : 0.8 -SMB-LO : ....... S0.6 CA 0.4 -, ,, ,, ... 0.0 281. 282. 283. 284. 285. 286. 287. 5. -SMB-LD : .......*-S30. o. I----I-----I-----I281. 282. 283. 284. 285. 286. 287. .N -STATION : B im N |I I I I I 281 282 283 284 285 286 287 JULIAN DRY Figure 5.11: Measured wind bar graph at station L005 and the comparison of the SMB-AD and SMB-LD model results and measured (+++) H8and Tp at station B. PAGE 81 70 1.0 5. 0.8 4 U.) 0.6 3. I+ 3.:0.4 ++ 2. 0 + + + L 2. C) .+ + + + CC .41+ + +/ S++ M +4+4 + 0.2 -,.* + + + 0.0 I I -I 0. 0.0 0.2 0.4 0.6 0.8 1.0 0. 1. 2. 3. t. 5.. MERSURED Hs (m) MEASURED Tp (sec) Figure 5.12: Comparison of hourly hindcast by the SMB-AD model and measured significant wave height H, and peak wave period Tp at station E. 1.0 5. 0.8 -4. E + n 0.6 -3. + + ++ z + 0 V +. 0.0 0.2 0.4 0.6 0.8 1.0 0. 1. 2. 3. 4. 5. MEASURED Hs (m) MEASURED Tp (sec) Figure 5.13: Comparison of hourly hindcast by the SMB-LD model and measured significant wave height H. and peak wave period Tp at station E. wave height Hs and peak wave period Tp at station E. PAGE 82 71 1.0 5. 0.8 --4. + + S0.6+ + 3. + f4 S0.+4+ S) + Z 4 C: I..+ u 4 + Z O 0.0 0. 0.0 0.2 0.4 0.6 0.8 1.0 0. 1. 2. 3. 4. 5. MERSURED Hs (m) MERSURED Tp (sec) Figure 5.14: Comparison of hourly hindcast by SMB-AD model and measured significant wave height H,, and peak wave period Tp, and zero up-crossing wave period T, at station C. case in that the SMB-LD model does a good job in hindcasting significant wave height and the SMB-AD model does a good job in hindcasting peak wave period. Figures 5.14 and 5.15 show comparisons between the hindcast from the SMB-AD and SMB-LD models and measured significant wave height and peak wave period at station C. Table 5.5 shows the correlation coefficiant and normalizeds root-mear-square error of the SMB-AD and SMB-LD model at station C. Since station C is located at the deep water, there is little difference between the results obtained from the SMB-AD and the SMB-LD models. Luettich and Harleman (1989) applied the 1973 version of the shallow water SMB model (1973, Shore Protection Manual) and the 1984 version of the shallow water SMB model using local water depth to Lake Balaton, Hungary, and compared the hindcast results with measured data. According to their study, the 1973 version of the SMB model estimates significant wave height very well and the 1984 version of the SMB model overestimates significant wave height by up to 20 %. Both versions of the SMB model consistently underestimate the peak wave period. This finding is consistent with our experience with the SMB-LD model on Lake Okeechobee. PAGE 83 72 1.0 5. 0.8 -4. + o S0.6 -+ + + +:+ ++ ." I--. 4 + (n + + 0.4 ++. ++ L 2. + S+ ++ 4 + S+ + i Tz 0.2 --+ + + + + 0.0 .Il 0. 0.0 0.2 0.4 0.6 0.8 1.0 0. 1. 2. 3. 4. 5. MERSURED Hs (m) MEASURED Tp (sec) Figure 5.15: Comparison of hourly hindcast by SMB-LD model and measured significant wave height H,, and peak wave period Tp, and zero up-crossing wave period Tz at station C. Table 5.5: Comparison of hindcast by the SMB-AD and SMB-LD models and measured values of significant wave height and peak wave period at station C. Correlation Normalized root coefficient mean square error SMB-AD H, 0.81 0.30 Tp 0.70 0.25 SMB-LD H, 0.81 0.31 Tp 0.69 0.24 PAGE 84 CHAPTER 6 SUMMARY AND CONCLUSION The main objective of this study is to compare various wind-wave models with measured field data obtained from Lake Okeechobee, Florida. The wind-wave models tested in this study are: 1. The shallow-water SMB model presented in the Shore Protection Manual (CERC,1984). For the computation of significant wave height and wave period, two different methods were tested. One uses fetch and the local water depth at the point of interest (SMB-LD model) and the other uses fetch and the average water depth along the fetch (SMB-AD model). 2. The finite-depth wind-wave model developed by Graber and Madsen (GM model). 3. The GLERL Donelan deep-water wind-wave model originally developed by Donelan and extensively used by GLERL (GD model). The wave models were tested using idealized steady and uniform wind fields and measured wind fields over Lake Okeechobee with realistic bottom topography. In the idealized wind case, the four wave models are compared in terms of significant wave height, peak wave period, and bottom shear stress with easterly wind and velocities 5, 10, 15, and 20 m/sec, respectively. The GD model overpredicts the significant wave height and peak wave period for wind velocities greater than 10 m/sec and results in excessive bottom shear stresses. The GM model and the SMB-AD model show nearly identical wave period along the entire fetch. The SMB-AD model overestimates the significant wave height and hence excessive bottom shear stress at the Western end of the lake where the fetch is long and the depth is shallow. 73 PAGE 85 74 The SMB-LD model and the GM model gave similar significant wave height along the entire fetch. The SMB-LD model shows consistently 10, 20, and 30 cm higher wave height than the GM model for 10, 15, and 20 m/sec wind velocities, respectively. The SMB-LD model underestimates the wave period in the shallow water. In the real wind case, the GM, the SMB-LD, and the SMB-AD models are tested with the measured data obtained from subsurface pressure gage at three stations B, C, and E in Lake Okeechobee. At station B which is located close to the small island and vegetation area, the hindcasted results show poor agreement with measured data due to the ignorance of the vegetation area and the island in model simulation. At station C located in 4.6 meters deep center part of the lake, the GM model gave the best results compared with measured data. The SMB-LD and SMB-AD models gave good results in agreement with measured data. At station E, located in 2.7 meter depth of water in the south-west shallow part of the lake, the GM model gave best results in agreement with measured significant wave height and peak wave period. The comparisons of results of the SMB-AD model and SMB-LD model with measaured data confirm the finding of the idealized wind case: the SMB-LD model does a better job in estimating wave height than the SMB-AD model, and the SMB-AD model does a better job in eastimating wave period than the SMB-LD model. The GM model performed the best in estimating both the wave height and the period. It also gave additional such useful information as mean wave direction and one-dimensional frequency spectrum. The GM model shows that deviation of the mean wave direction from the mean wind direction are generally less than 15 degrees. The SMB model performed fairly well in shallow lake where the swell doesn't contribute significantly. If the local water depth is used for computing wave height and the average water depth is used for computing wave period, then the SBM model gives results comparable with the GM model in Lake Okeechobee. The major advantage of the SMB model is its simplicity. For the 6-day simulation in Lake Okeechobee using real wind field, the SMB model needs only about 10 minutes CPU time on Vax 8350 computer compared with the PAGE 86 75 GM model's 40 hours CPU time for the same simulation. By using the significant wave height and the peak wave period to represent a monocromatic wave, the bottom shear stresses were computed through linear wave theory and Kajiura's oscillatory boundary layer formulation. The computed bottom shear stress over the entire Lake Okeechobee correlates well with the spatial distribution of the surficial sediment. For the estimation of the bottom shear stresses, Kajiura's and Jonsson's formulas give essentially the same maximum bottom shear stresses. The computed bottom shear stress contains additional uncertainty due to uncertainties in defining roughness height in natural bottom. Most wave boundary layer models are based on the sinusoidal monocromatic wave assumption. Proper inclusion of random waves on the bottom shear stress calculation needs to be explored. For using these formulas over a wide range of bed and flow conditions, it is necessary to improve the clear transition between the hydraulically smooth bed and rough bed and transition between laminar flow and turbulent flow rather than improving the friction factor formula itself. PAGE 87 APPENDIX A RADIATIVE-TRANSFER EQUATION Suppose that the total energy density of the waves per unit phase volume [dkodio] is initialy F(ko,,oto) at t = to. At time t = to + 6t, the energy balance can be expressed in the integral form F(k, t) = F(ko, i, to) + t Q(,',At')dt' (A.1) where k is the wave number vector and Q(k',x,t') represents the net rate of change of energy per unit phase volume resulting from any sources and sinks of wave energy. k',x', and t' vary along the path of a wave group from an initial value ko, X:, and to to c, 9, and t. Expanding the energy spectrum F(k, t) in a Tayor series gives dik d5 F(k, t) = F(ko + dt, o + -dt,to + dt) OF di dk at dt dt = F(ko, t, PAGE 88 77 where k = \kl and w is the angular wave frequency. Futhermore, it is assumed that the water depth h(i) is also slowly varying, i.e., 1 OF k1 (A.6) kh 'z so that the geometrical-optics approximation is valid. In addition, the wave number vector k and the angular wave frequency w = w(k, h) are related by the conservation-of-crests equation S+ Vw = 0 (A.7) and by the dispersion relationship for linear surface gravity waves w = w(k, 9) = gk tanh kh (A.8) An equation for the variation of k along the ray path can be deduced by introducing w above in equation A.7 and writing the result in the tensor notation aki Ow Okj Ow + + = 0 (A.9) it ki axi O xzi Note that k is irrotational, i.e. Ok = Ok (A.10) Oxi OXj Then from equation A.9 and A.10, we have + 9,V V = (A.11) or = -h (A.12) dt Ox Also, by taking the scalar product of equation A.7 with Cq, we have aw or d 0 (A.14) -= 0 (A.14) dt PAGE 89 78 Equations A.12 and A.14 represent Eulerian time rate-of-change equations for both k and w and have the same characteristic curve di aw d-C (A.15) dt 9k -k The directional wave number spectrum E(k; 9, t) can be transformed into the directional frequency spectrum E(f, 0, 5, t) as follows: F(k;Y,t)= J. E(f,0;9,t) (A.16) where J is the Jacobian of the transformation o(f,0) cc CC J O(k, ) -2 C' =jI, C = (A.17) Substitution of equation A.16 into A.3 gives a transport equation for the directional frequency spectrum D (CCE) OW t (CCgE) -9 t + V(CCE) -O V;(CC9E) = 2rS (A.18) where S(f, 0; t) becomes the forcing term. The refraction term in equation A.18 can be expressed in terms of the wave direction 0 in cartesian coordinate as follows: Ow Owo 0 -.V(CC9E) = -.(CCE) (A.19) Since 0 is defined as 0 = cos-'(k/k) = sin-'(ky/k). We have 00 sin 0 cos ) (A.20) 9k k 'k Also along the wave ray, a moving particles experiences no change of phase. duw Qw Ow Ok d + Ok -0 (A.21) or --C9 Ok (A.22) 0B. 8 PAGE 90 79 From the dispersion relation, we know Ok 0 w" w OC k OC Ox OZx \C C2 zx C Oz Ok 0 lw w OC k OC (A.23) Oy Oy \C) C2y C y Substitution of equations A.20, A.22 and A.23 into equation A.19 yields (w 000 O w 9k 00 8 --O(CCE) = (CC (CCE) Cg 0--cos C' = sin a -cs ) '(CCgE) (A.24) C Ox 7y T0 Introducing equation A.24 in equation A.18 and dividing by Cg we have 1 9 i a C (CC,E) + cos O (CC,E) + sin (CC9E) Ssin 0 -os 0 (CCE) = S (A.25) C 0y To C, PAGE 91 APPENDIX B DEPTH TRANSFORMATION FACTOR ((wh) A consistent approach on the self-similarity of depth dependent frequency spectra was originally demonstrated by Kitaigorodskii, Krasitskii and Zaslavskii (1975). Adopting space and time scale ranges suggested by Phillips (1958) F(k) = fk-4p(8) (B.1) E(w) = ag2w-5 (B.2) where F(k) and E(w) are, respectively, the wave number and frequency spectra, k is the modulus of the wave-number vector k = (k cos 0, k sin 0), w the angular frequency, 0 the angle characterizing the direction of wave propagation, a and / are universal non-dimensional constants, and V(0) is a certain universal function describing the angular distribution of wave component energy within the equilibrium range and satisfying the standard normalization condition p(O)dO =1 (B.3) It follows from equation B.1 and B.3 that F(k) = F(k)dk = F(k, )0k-4p(0)kdO = 3k-3 (B.4) where F(k) is the spectrum of wavenumber moduli. In the absence of any mean currents, the dispersion relation of the linear wave is isotropic w(k) = [gk tanh(kh)]1/2 (B.5) where h is local water depth. The use of dispersion relation B.5 leads to a formulation of the relation between F(k) and E(w). J F(k)dk = F F(k,O)kdkdO 80 PAGE 92 81 = F(k)dk = E(w)dw =< 72 > (B.6) where < 72 > is the variance of the sea surface. From relation B.5, the frequency and wave number spectrum are related as dk E(w) = F(k) (B.7) The wave group velocity is d 1 w ( 2kh (B.8) C -dk -2 k 1 + sinh 2kh(B8) Introducing equations B.4 and B.8 in equation B.7 gives a general expression for the frequency spectrum, E(w) = pk-3 (B.9) w + sinh2kh Because of dimensional reasoning, a more general expression for the finite-depth frequency.spectrum might be of the form E(w) = ag2W-5t(Wh) (B.10) where Wh = wh/g and TI(wh) is some universal nondimensional function. From equation B.5, the solution for k is 02 k(w) = -X(wh) (B.11) 9 where X(wh) is the solution of the trascedental algebraic equation xtanh(wX) = 1 (B.12) Introducing equation B.11 in equation B.9 and comparing with equation B.10 gives an expression for T(wh) ) = sinh[2w X] = X-21+ (X21)(B.13) 1 +W PAGE 93 82 It is easily ascertained that in deep water lim '(wh) = 1 (B.14) Wh-OO This asymptotic behavior gives the following relation between constants a and / = 1 (B.15) a 2 and in shallow water, lim Q(wh) = -(B.16) W"h-O 2 Introducing equation B.16 in equation B.10 leads to the similarity form in shallow water E(w) = ogh-3(B.17) PAGE 94 APPENDIX C JONSSON'S FRICTION COEFFICIENT FORMULA The bottom shear stress To can also be found by assuming that the steady-state expression for a turbulent velocity profile over a rough bottom is valid near the bottom (where the accelerations are small) u 30z 30z S= 2.51n = 5.751og (C.1) U, k, k, where u. = vfro7pis the friction velocity, k, is the Nikuradse roughness parameter, and z is the height over theoretical bed. The eddy viscosity et is defined as -1= (C.2) p Oz Since u=0 at z = k,/30 for fully developed turbulent flow, equation ?? yields the bottom shear stress as follows: ro 6 8(uoo -u) p = /30 dz (C.3) p= Jk/30 at where 6 = 6(t) is the thickness of wave boundary layer. u. and 6 are interrelated by S= 2.5 In (C.4) U* ka, Subtraction of equation C.1 from equation C.4 yields uoo -u = 2.5 u. In(6/z) (C.5) Substitution of equation C.5 into C.3 and integration by assuming that k, < 306 and 6 = 0 for t = 0 gives. u.6 = 0.40 u2 dt (C.6) 83 PAGE 95 84 Now, squaring equation C.4 and substituting the result into equation C.6 with uo = Uoom sinwt, we have u.6 = 0.403 U 'm sin2 wt d (C.7) kIn k Since ln2(306/k,) is slowly varying function with time t, it may be neglected in the integral. Equation C.7 becomes 0.403 sin 2 2 In2 3062w u.6 1--^ \t 2w ] (c.8) ko where 6 is the wave boundary layer thickness corresponding to uo = Uim ( i. e. wt = 7/2 ). From equations C.4 and C.8, we have 3061 3061 aim In -= m (C.9) k, k, k. where aim = uoom/w and m6 (1.2r/ In 10 = 1.64) must be determined from experimental data. The wave friction factor f, is defined from the maximum bed shear stress as follows: rOm wfu m (C.10) From equation C.4 and C.10 f ( u.*m 2 0.06050 3061 (C. S \uoom/ In2 k, Now maximum bed shear stress and maximum free stream velocity are assumed to occur simultaneously. Elimination of 3061/ki from equations C.9 and C.11 gives 1 1 am (C.12) -+ In -= m + In (C.12) 4vm 4 f mf is a constant determined to be -0.08 from experimental data. ae PAGE 96 APPENDIX D TABLES 85 PAGE 97 86 Fetch 3 7 11 15 19 23 27 31 35 39 43 47 (km) Depth 4.3 4.5 4.8 4.9 4.9 4.4 4.1 3.0 2.6 1.9 1.5 0.6 (m) H, SMB-AD 0.15 0.23 0.28 0.31 0.34 0.37 0.38 0.40.. 0.41 0.41 0.41 0.41 in SMB-LD 0.16 0.23 0.28 0.32 0.35 0.37 0.38 0.36 0.35 0.30 0.26 0.14 m GM 0.14 0.22 0.28 0.32 0.36 0.39 0.41 0.40 0.40 0.32 0.27 0.15 GD 0.12 0.18 0.24 0.27 0.27 0.30 0.33 0.33 0.33 0.33 0.36 0.43 T, SMB-AD 1.5 1.8 2.1 2.2 2.3 2.4 2.5 2.5 2.6 2.6 2.6 2.6 in SMB-LD 1.5 1.9 2.1 2.3 2.4 2.4 2.5 2.5 2.4 2.3 2.2 1.8 sec GM 1.4 1.9 2.1 2.4 2.5 2.7 2.8 2.9 2.9 2.9 2.9 3.0 GD 1.2 1.7 1.9 2.0 2.2 2.2 2.3 2.3 2.4 2.4 2.5 2.7 i, SMB-AD 0.0 0.02 0.08 0.14 0.24 0.56 0.87 3.35 5.15 9.67 14.2 42.6 dyne SMB-LD 0.0 0.03 0.08 0.17 0.28 0.57 0.86 2.49 3.40 4.89 5.72 6.97 per GM 0.0 0.02 0.08 0.24 0.44 1.18 1.98 4.72 6.33 7.54 7.70 8.80 sq.cm GD 0.01 0.01 0.04 0.04 0.11 0.22 0.47 1.73 3.05 6.13 11.2 45.4 Table D.1: Model comparison for 5 m/sec wind. Variations of H,, Tp, and m with fetch along the cross section AA'. PAGE 98 87 Fetch 3 7 11 15 19 23 27 31 35 39 43 47 (kn) Depth 4.3 4.5 4.8 4.9 4.9 4.4 4.1 3.0 2.6 1.9 1.5 0.6 (m) H, SMB-AD 0.53 0.74 0.87 0.96 1.02 1.06 1.08 1.09 1.08 1.05 1.02 0.98 in SMB-LD 0.57 0.80 0.94 1.03 1.09 1.08 1.07 0.89 0.81 0.65 0.54 0.27 m GM 0.43 0.64 0.77 0.86 0.9 0.92 0.88 0.73 0.59 0.42 0.36 0.26 GD 0.49 0.82 1.06 1.22 1.34 1.46 1.55 1.61 1.67 1.73 1.79 2.07 T, SMB-AD 2.3 2.9 3.3 3.6 3.8 3.9 4.1 4.2 4.2 4.3 4.3 4.3 in SMB-LD 2.4 3.0 3.4 3.7 3.9 4.0 4.1 3.9 3.9 3.7 3.5 2.6 sec GM 2.2 2.9 3.3 3.5 3.8 3.9 4.1 4.2 4.3 4.3 4.2 5.0 GD 2.4 3.2 3.8 4.1 4.4 4.6 4.7 4.9 5.0 5.1 5.2 5.6 b SMB-AD 0.71 4.53 7.62 10.6 13.4 18.9 23.1 38.0 45.4 62.9 77.4 177. dyne SMB-LD 0.94 5.63 9.28 12.6 15.5 19.7 22.6 26.1 26.9 27.1 25.9 21.6 per GM 0.42 3.34 5.99 8.65 10.8 14.9 16.5 19.6 17.0 13.8 13.8 18.1 sq.cm GD 0.82 7.38 15.1 20.8 27.1 39.1 48.8 79.7 101. 151. 203. 612. Table D.2: Model comparison for 10 m/sec wind. Variations of H,, Tp, and mb with fetch along the cross section AA'. PAGE 99 88 Fetch 3 7 11 15 19 23 27 31 35 39 43 47 (km) Depth 4.3 4.5 4.8 4.9 4.9 4.4 4.1 3.0 2.6 1.9 1.5 0.6 (m) H. SMB-AD 0.34 0.49 0.59 0.66 0.71 0.75 0.77 0.79 0.79 0.78 0.76 0.74 in SMB-LD 0.36 0.52 0.62 0.69 0.75 0.76 0.76 0.66 0.61 0.50 0.42 0.21 m GM 0.29 0.44 0.55 0.62 0.67 0.69 0.69 0.60 0.54 0.38 0.31 0.19 GD 0.27 0.46 0.58 0.67 0.73 0.79 0.82 0.85 0.88 0.91 0.94 1.09 T, SMB-AD 2.0 2.5 2.8 3.0 3.2 3.3 3.5 3.5 3.6 3.6 3.6 3.6 in SMB-LD 2.0 2.5 2.9 3.1 3.3 3.4 3.4 3.3 3.3 3.2 3.0 2.3 sec GM 1.9 2.5 2.8 3.1 3.3 3.4 3.5 3.7 3.7 3.7 3.8 4.0 GD 1.8 2.5 2.9 3.1 3.3 3.5 3.6 3.6 3.7 3.8 3.9 4.2 m SMB-AD 0.10 0.86 2.13 3.40 4.71 7.73 10.1 19.3 24.3 36.3 46.1 112. dyne SMB-LD 0.14 1.29 2.55 3.95 5.48 7.96 9.84 13.5 14.8 16.2 16.2 14.3 per GM 0.06 0.70 1.90 3.17 4.53 7.11 8.96 12.9 13.6 11.5 10.7 11.6 sq.cm GD 0.03 0.80 2.41 3.76 5.41 9.37 12.2 22.3 29.9 48.0 66.9 215. Table D.3: Model comparison for 15 m/sec wind. Variations of H,, Tp, and 7b with fetch along the cross section AA'. PAGE 100 89 Fetch 3 7 11 15 19 23 27 31 35 39 43 47 __ (km) Depth 4.3 4.5 4.8 4.9 4.9 4.4 4.1 3.0 2.6 1.9 1.5 0.6 (m) H. SMB-AD 0.71 0.97 1.12 1.22 1.29 1.33 1.35 1.34 1.32 1.28 1.24 1.19 in SMB-LD 0.79 1.07 1.24 1.34 1.40 1.36 1.32 1.08 0.97 0.77 0.65 0.33 m GM 0.57 0.82 0.97 1.04 1.10 1.08 1.01 0.78 0.61 0.46 0.39 0.26 GD 0.61 1.22 1.52 1.82 2.13 2.13 2.43 2.43 2.74 2.74 2.74 3.34 T, SMB-AD 2.6 3.3 3.7 4.0 4.2 4.4 4.6 4.7 4.7 4.8 4.8 4.8 in SMB-LD 2.7 3.4 3.8 4.1 4.3 4.4 4.5 4.4 4.3 4.1 3.8 2.9 sec GM 2.5 3.2 3.6 3.9 4.2 4.4 4.5 4.7 4.7 4.7 4.7 5.0 GD 2.8 3.9 4.6 5.0 5.3 5.6 5.8 6.0 6.1 6.2 6.4 7.0 b SMB-AD 2.67 10.4 15.6 20.1 24.2 32.1 37.7 57.2 66.4 89.5 108.4 243. dyne SMB-LD 3.71 13.2 19.4 24.4 28.5 33.4 36.1 38.7 38.5 37.1 36.0 30.0 per GM 1.54 7.33 11.7 15.1 18.2 22.2 23.1 23.1 18.4 16.5 15.9 18.2 sq.cm GD 3.18 22.4 36.6 52.4 71.2 84.8 118. 169. 241. 331. 416. 1362 Table D.4: Model comparison for 20 m/sec wind. Variations of HI, Tp, and rb with fetch along the cross section AA'. PAGE 101 APPENDIX E ONE-DIMENSIONAL FREQUENCY SPECTRA The wave spectra that summarized here include the one-dimensional frequency spectra hindcasted by the finite depth G&M model and measured at the stations C and E in Lake Okeechobee. The eight-digit number shown in each figure indicates the time when spectra occured, and corresponds to the year, month, day, and hour, each represented by two digits, respectively. The further information shown in each figures are the significant wave height and zero up-crossing period corresponding to the hindcasted and measured spectra, respectively. 90 PAGE 102 91 ........................... COMPUTED MEASURED 0.4 C88100700 ......... C88100702 """" ....C88100705 ....-... HS= 0.53 0.38 HS= 0.54 0.42 HS= 0.52 0.45 0.3 TZ= 2.59 2.48 TZ= 2.65 2.53 TZ=2.56 2.58 0.2 0.1 0.0 0 0 0.4 0U C88100707 ....... -C88100708 .."". .-C88100710 ......... L HS= 0.48 0.51 HS= 0.53 0.45 HS= 0.52 0.48 W 0.3 TZ=2.582.64 TZ=2.61 2.59 TZ=2.63 2.71 E 0.2 U) -0.1 z uJ 0.4 C3 C88100712 -" """" C88100715 "......C88100717 >-S= 0.50 0.42 1H= 0.39 0.24 HS= 0.52 0.56 0.3 TZ= 2.63 2.50 TZ=2.32 2.18 TZ=2.45 2.66 z Li LU 0.2 cE 0.4 CB8100718 ..""""C88100720 I C88100722 HS= 0.63 0.62 HS= 0.63 0.68 HS= 0.63 0.66 0.3 TZ= 2.63 2.86 :. TZ=2.77 3.01 TZ= 2.74 2.82 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure E.1: Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. (continue) PAGE 103 92 ........................... COMPUTED MEASURED 0.4 C88100801 -.... C88100803 -.... -C88100804 ...... HS= 0.55 0.61 HS= 0.63 0.58 HS= 0.62 0.56 0.3 TZ= 2.69 2.84 TZ= 2.73 2.72 TZ= 2.78 2.77 0.2 0.1 0.0 C) 1) 0.4 C88100806 """ -I0C88100808 -C -088100811 L HS= 0.54 0.49 HS= 0.61 0.52 HS= 0.39 0.36 0.3 TZ=2.68 2.66 TZ=2.71 2.60 TZ= 2.37 2.41 0.2 (n z LW 0.4 0. C88100813 """ ...... C88100814 ".....C88100816 .... SHS= 0.45 0.31 HS=0.31 0.31 HS= 0.33 0.28 S 0.3 TZ=2.31 2.26 TZ= 2.23 2.29 TZ= 2.04 2.23 LL z Z w 0.2 -I CI 0.1 '-U 00.0 0.4 C88100818 -"".. -C88100821 ......... C88100823 HS= 0.46 0.40 HS= 0.51 0.59 HS= 0.55 0.46 0.3 TZ= 2.33 2.47 TZ= 2.59 2.79 TZ= 2.61 2.54 0.2 0.1 A 0.0 L .A-, 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure E.2: Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. (continue) PAGE 104 93 ........................... COMPUTED MEASURED 0.4 C88100900 ....... C88100902 ......... C88100904 ........ HS=0.54 0.44 MS=0.54 0.37 HS=0.48 0.34 0.3 TZ=2.63 2.42 TZ=2.65 2.48 TZ=2.57 2.37 0.2 0.1 0.0 S 0.4 0 C088100907 .......... C88100909 "" -...... C88100910 ...... ( HS=0.41 0.29 HS=0.31 0.31 HS=0.28 0.26 S 0.3 TZ=2.43 2.33 TZ= 1.56 2.18 TZ= 1.92 2.10 E 0.2 0S 0.1 0.4 z C0 0.3 TZ= 2.07 2.17 TZ= 1.84 1.94 TZ= 1.92 1.94 LLJ a Z WJ 0.2 _CE S 0.1 LLU 00 --. A-I --}-.-I S0.0 0.4 C88100918 ..."". -C88100920 -"*... -C88100922 HS=0.28 0.36 HS=0.43 0.34 HS=0.46 1.31 0.3 TZ= 1.98 2.42 TZ=2.22 2.20 TZ=2.43 2.69 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure E.3: Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. (continue) PAGE 105 94 *....... .COMPUTED MEASURED 0.4 C88101001 -......... C88101003 ...... -C88101004 .. HS= 0.38 0.78 HS= 0.42 0.61 HS=0.47 0.82 0.3 TZ=2.33 2.48 TZ=2.33 2.40 TZ=2.41 2.56 0.2 0.1 A 0.0 ) 0.4 (0 C88101006 """..... -C88101008 ...... C88101011 ... L HS= 0.51 0.75 HS=0.52 0.58 HS=0.49 0.49 : 0.3 TZ=2.56 2.44 TZ=2.60 2.59 TZ=1.55 2.38 E 0.2 -0.1 0.0 ,-) z LU 0.4 SC88101013 """ -.... C88101014 ......... C88101016 .. -HS= 0.24 0.39 HS= 0.23 0.23 HS= 0.25 0.24 S 0.3 TZ= 1.47 2.35 TZ= 1.87 2.04 TZ= 1.96 2.03 LU z L 0.2 -J CC ir 0.1 a_ 0.0 U-) 0.4 C8810108 881010121 C....... -C88101023 HS= 0.25 0.34 HS= 0.36 0.34 S= 0.32 0.32 0.3 TZ= 1.95 2.22 TZ=2.24 2.23 TZ= 2.14 2.22 0.2 0.1 0.0 4 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure E.4: Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. (continue) PAGE 106 95 ........................... COMPUTED MEASURED 0.4 C88101100 ........... C88101100 -C88101102 """"" -C88101104 HS= 0.33 0.25 HS= 0.17 0.23 HS= 0.28 0.19 0.3 TZ= 1.94 2.18 TZ= 1.58 1.98 TZ= 2.06 1.98 0.2 0.1 0.0 0 (D 0.4 (n C88101107 ......... C88101109 """"..... .C88101110 L HS= 0.25 0.21 HS= 0.24 0.19 HS= 0.19 0.17 ) 0.3 TZ= 1.98 1.95 TZ= 1.48 1.91 TZ= 1.47 1.95 -J U) 0.2 0A 0.1 0.0 z LUJ 0.4 C3 C88101112 .-C88101114 "" -.C88101117 >HS= 0.15 0.22 HS= 0.10 0.20 HS= 0.11 0.21 3 0.3 TZ= 1.57 1.96 TZ= 1.52 1.98 TZ= 1.47 1.97 LII z LiJ 0.2 _J 0.1 L.J C0.0 0.4 88101119 -C88101120 .......C88101122 ... HS= 0.16 0.19 HS= 0.28 0.25 HS= 0.18 0.25 0.3 TZ= 1.64 1.95 TZ= 1.70 2.09 TZ= 1.56 2.09 0.2 0.1 0.0 I I 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure E.5: Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. (continue) PAGE 107 96 .......................... COMPUTED MEASURED 0.4 C88101201 .......... C88101203 """........ C88101205 ... HS=0.12 0.18 HS=0.23 0.20 HS= 0.32 0.36 0.3 TZ= 1.62 1.96 TZ= 1.54 1.91 TZ=2.07 2.27 0.2 0.1 0.0 ---I-----I 1 I I l'i I 0.0 0.4 C88101206 ........ C88101208 ".......... C88101210 .... L HS=0.37 0.45 HS= 0.48 0.40 HS=0.21 0.29 S 0.3 TZ=2.23 2.56 TZ= 1.81 2.49 TZ= 1.47 2.28 U) E 0.2 0.1 eA 0.0 z LLJ 0.4 0 C88101213 """". -C88101215 C""". -088101216 .>HS=0.14 0.17 HS= 0.16 0.20 HS= 0.19 0.19 S 0.3 TZ= 1.47 1.97 TZ= 1.65 1.98 TZ= 1.74 1.95 LU J 0.2 -J CC D 0.1 (n ,i 0.0 4 I I ---------------I I -----I 0.4 C88101218 "". "" .. C88101220 C.......88101223 HS= 0.30 0.16 HS= 0.39 0.32 HS= 0.30 0.27 0.3 TZ=2.02 1.98 TZ=2.28 2.32 TZ=2.12 2.28 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure E.6: Hindcasted and measured one-dimensional frequency spectra at station C in Lake Okeechobee. PAGE 108 97 ........................... COMPUTED MEASURED 0.4 E88100701 "" ...... E88100703 ..E88100705 ........ HS= 0.44 0.49 HS= 0.39 0.47 HS= 0.42 0.44 0.3 TZ=2.58 2.45 TZ=2.36 2.39 TZ=2.44 2.31 0.2 0.1 .0.0 0.4 E88100706 ..."......E88100708 "........ -E88100711 ""........ L HS= 0.44 0.44 HS= 0.45 0.46 HS=0.45 0.36 W 0.3 -TZ=2.48 2.40 TZ=2.47 2.38 TZ= 2.56 2.25 E 0.2 A L.J 0.4 C' E88100713 ........... E88100715 ~ ""-E88100716 ........... >HS=0.51 0.28 H= 0.380.20 HS= 0.400.37 3 0.3 TZ=2.03 2.19 TZ=2.48 2.01 TZ=2.38 2.16 IU. Z LJ 0.2 -J Cc S 0.1 i-J I -, E88100718 """" ...E88100720 ~E88100723 HS= 0.55 0.60 15S= 0.53 0.62 HS= 0.47 0.51 0.3 TZ=2.75 2.63 TZ= 2.99 2.69 TZ=2.80 2.51 0.2 ik 0.1 0.0 0.0 0.2 0.9 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure E.7: Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) PAGE 109 98 ........................... COMPUTED MEASURED 0.4 E88100801 "..."" -... E88100802 ."""" ...E88100804 ... HS= 0.45 0.52 HS= 0.45 0.58 HS= 0.50 0.54 0.3 TZ= 2.69 2.46 TZ= 2.64 2.51 TZ=2.71 2.53 0.2 0.1 0.0 (. QO 0.4 W) E88100806 .""."" ..E88100809 -"" ..... E88100811 ... ( HS= 0.49 0.53 HS= 0.44 0.51 HS= 0.37 0.36 S 0.3 TZ= 2.53 2.49 TZ= 2.62 2.42 TZ=2.46 2.27 E 0.2 ;A -0.1 I0.0 z LI 0.4 0 E88100812 """" .... E88100814 -""'-E88100816 """ -H= 0.38 0.35 HS= 0.35 0.21 HS= 0.40 0.24 C 0.3 TZ= 2.44 2.23 TZ= 2.30 2.09 TZ= 2.36 2.07 LLI Z LLJ 0.2 _J cI 0.1 U S 0.0 0.4 E88100819 "" .....E88100821 ---E88100822 ..... HS= 0.50 0.62 HS= 0.44 0.52 HS= 0.47 0.46 0.3 TZ= 2.65 2.62 TZ= 2.76 2.59 TZ= 2.73 2.40 0.2 A A 0.0 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure E.8: Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) PAGE 110 99 ........................... COMPUTED MEASURED 0.4 E88100900 ........ -E88100902 ""..""". E88100905 ..... HS= 0.46 0.44 HS= 0.46 0.40 HS= 0.38 0.33 0.3 TZ= 2.66 2.35 TZ= 2.67 2.39 TZ=2.54 2.25 0.2 A A 0.1 0.0 Q1) 0.4 W1 E88100907 .......E88100908 .."""" .E88100910 ..... L HS=0.36 0.28 HS=0.39 0.34 HS=0.30 0.30 W 0.3 TZ=2.44 2.20 TZ= 1.99 2.28 TZ= 1.79 2.24 -.j () 0.2 c-0.1 0.0 z LLJ 00.4 ..... C E88100912 -""""'" -E88100915 """" .... E88100917 >HS=0.36 0.31 HS= 0.28 0.23 11S=0.31 0.19 .D 0.3 TZ=2.23 2.22 TZ= 1.98 2.08 TZ=2.17 1.97 LLa Z UJ 0.2 0.1 CC c: 0.1 S0.0 --0.4 E88100918 .......E88100920 -. -E88100922 ..... HS= 0.33 0.26 11S=0.440.39 HS=0.44 0.43 0.3 TZ=2.19 2.10 TZ= 2.39 2.42 TZ= 2.60 2.36 0.2 A A 0.1 0.0 \ ":-. 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure E.9: Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) PAGE 111 100 ........................... COMPUTED MEASURED 0.4 E88101001 ........ -E88101003 "....... -E88101004 ... HS= 0.35 0.32 HS=0.40 0.25 HS= 0.43 0.33 0.3 -TZ=2.42 2.27 TZ= 2.42 2.17 TZ=2.48 2.22 0.2 A 0.1 0.0 (U 0.4 0) E88101006 ""..... -E88101008 -""......" E881010II .11 L HS= 0.45 0.39 HS=0.45 0.38 HS= 0.38 0.32 a 0.3 TZ=2.60 2.35 TZ=2.62 2.31 TZ= 1.54 2.19 () E 0.2 ( 3 0.1 z .L 0.4 C E88101013 """ ....E88101014 -."" E88101016 .. "" -HS= 0.24 0.22 HS= 0.23 0.15 HS= 0.20 0.11 .0.3 TZ=1.472.10 TZ 1.862.00 TZ= 1.821.85 LU z UJ 0.2 ,_ 0.1 U LJ C .0.0 1 -I_ J.___ !__ I I_ I 0.4 E88101018 """"" .-E88101021 """" -E88101023 .. HS= 0.24 0.21 HS= 0.35 0.25 25H=0.31 0.29 0.3 TZ= 1.94 1.97 TZ=2.25 2.04 TZ=2.25 2.25 0.2 0.1 0.0 I I/ .-. 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ] Figure E.10: Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) PAGE 112 101 ........................... COMPUTED MEASURED 0.4 E88101100 ....... E88101102 """E88101104 ..... HS= 0.37 0.27 HS= 0.26 0.22 HS= 0.25 0.14 0.3 TZ=2.02 2.23 TZ= 1.61 2.17 TZ= 1.97 1.94 0.2 0.1 0.0 i --_ I_-,-I W 0.4 W0 E88101107 .......... E88101109 "...""". E88101110 .... L HS= 0.22 0.14 HS= 0.21 0.13 HS= 0.16 0.11 S 0.3 TZ= 1.90 1.88 TZ= 1.48 1.84 TZ= 1.47 1.82 E U) 0.2 03 cn S 0.1 I--0 .0 I I I I _.. --I I I 0.0 z L.J 0.4 ........... 0 E88101112 """"E88101114 "".""". E88101117 >HS= 0.15 0.08 HS=0.110.08 HS= 0.14 0.09 0. 0.3 TZ= 1.57 1.78 1.57 1.7 .55 1.83 TZ= 1.47 1.80 LLJ w 0.2 .-J 0.1 (.) .0 .0 I I I I I I I I L I 0.4 E88101119 ...E88101120 -E88101122 .. HS= 0.18 0.09 HS= 0.28 0.20 HS= 0.14 0.13 0.3 TZ= 1.71 1.82 TZ= 1.70 1.90 TZ= 1.56 1.88 0.2 0.1 0.0 I I I 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure E.11: Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. (continue) PAGE 113 102 ........................... COMPUTED MEASURED 0.4 E88101201 .""""" --. E88101203 .""...... E88101204 ......... HS= 0. 12 0.09 HS= 0.21 0.09 HS= 0.22 0.25 0.3 TZ= 1.60 1.79 TZ= 1.54 1.80 TZ= 1.83 1.97 0.2 0.1 0.0 II 1 1 U 0 0.4 0" E88101206 """""....E88101208 "".......E88101211 ...L HS= 0.32 0.37 HS= 0.48 0.34 HS= 0.20 0.19 Q 0.3 TZ=2.11 2.30 TZ= 1.81 2.25 TZ= 1.54 2.06 Q) E 0.2 00.1 I--0.0 z ILl 0.4 C3 E88101213 ""........ E88101214 """....." E88101216 .. S1H= 0.11 0.12 HS= 0.10 0.09 HS= 0.18 0.09 0.3 TZ= 1.47 1.93 TZ= 1.51 1.83 TZ= 1.68 1.79 L! z Z LJ 0.2 -J w 0.1 Ll -0.0 I I I I I I I I 0.4 E88101218 """ .....E88101221 """"" ...E88101223 ..... HS= 0.26 0.15 HS= 0.35 0.32 HS=0.29 0.30 0.3 TZ= 1.92 1.80 TZ=2.23 2.15 TZ=2.11 2.21 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 FREQUENCY (HZ) Figure E.12: Hindcasted and measured one-dimensional frequency spectra at station E in Lake Okeechobee. PAGE 114 BIBLIOGRAPHY Bagnold, K.A., 1946, "Motion of Waves in Shallow Water. Interaction of Waves and Sand Bottoms," Proc. Roy. Soc. Ser. A., 187, 1-15. 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