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## Material Information- Title:
- Wave Attenuation and Mud Entrainment in Shallow Waters
- Creator:
- JAIN, MAMTA
- Copyright Date:
- 2008
## Subjects- Subjects / Keywords:
- Lakes ( jstor )
Modeling ( jstor ) Mud ( jstor ) Sediments ( jstor ) Shear stress ( jstor ) Velocity ( jstor ) Water depth ( jstor ) Wave attenuation ( jstor ) Waves ( jstor ) Wind velocity ( jstor ) City of Gainesville ( local )
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- Copyright Mamta Jain. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 11/30/2007
- Resource Identifier:
- 659815254 ( OCLC )
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PAGE 1 1 WAVE ATTENUATION AND MUD ENTR AINMENT IN SHALLOW WATERS By MAMTA JAIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 PAGE 2 2 Copyright 2007 Mamta Jain PAGE 3 3 To mom, dad, my son Shivam and my husband Parag PAGE 4 4 ACKNOWLEDGMENT I would like to express my deep est thanks to my advisor and chairman of the supervisory committee, Dr. Ashish Mehta, for guidance th roughout this study. Special thanks go to Dr. Robert Dean for his help and advice and his time on weekends and holidays. Thanks are also extended to the other members of the committee including Dr. Tom T.-J. Hs u, Dr. Robert Thieke and Dr. John Jaeger for their assistance. Help provided by Dr. Earl Hayter (of USEP A in Athens, GA) in operating the EFDC model is sincerely acknowledged. Thanks go to Sidney Schofield and Vik Adams, for carrying out fieldwork in Newnans Lake. I wish to acknowledge the assistance of the Coastal and Oceanographic Engineering Program staff and students for their emotional s upport and friendliness. I would like to thank my husband, Parag Singhal, my son Shivam Si nghal and my parents for their love. I would like to acknowledge the financial support from three sources: the St. Johns River Water Management District (Palatka, FL), the U.S. Environmental Protection Agency (Athens, GA) and the Water Resources Research Center (Univers ity of Florida). PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENT................................................................................................................. .4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .........9 LIST OF SYMBOLS................................................................................................................ .....14 ABSTRACT....................................................................................................................... ............20 CHAPTER 1 INTRODUCTION AND PROBLEM STATEMENT............................................................22 2 FIELD INVESTIGATION.....................................................................................................25 2.1 Site Description........................................................................................................ .....25 2.2 Data Collection......................................................................................................... .....26 2.2.1 Extended Field Campaign...................................................................................26 2.2.2 Short Field Campaign..........................................................................................27 2.3 Data Analysis........................................................................................................... ......28 2.3.1 Data Analysis fro m Extended Field Campaign...................................................28 2.3.2 Data Analysis from Short Field Campaign.........................................................34 2.4 Conclusions............................................................................................................. .......35 3 CIRCULATION, WAVES AND SEDIMENT TRANSPORT..............................................55 3.1 Introduction.......................................................................................................... ........55 3.2 Circulation and Sediment Transport..............................................................................55 3.2.1 Flow Circulation..................................................................................................55 3.2.2 Suspended Sediment Movement.........................................................................57 3.3 Wave Field.............................................................................................................. .......59 3.4 Estimation of Wave-Current Shear Stress.....................................................................60 3.5 Concluding Observations...............................................................................................62 4 WAVES OVER COMPLIANT MUD BOTTOM..................................................................73 4.1 Introduction............................................................................................................ ........73 4.2 Some Previous Studies..................................................................................................73 4.3 Problem Formulation.....................................................................................................75 4.4 Solution Approach....................................................................................................... ..77 4.5 First-Order Analytical Solution.....................................................................................78 4.6 First Order Full Semi-Analytical Model Solution (FSM).............................................80 4.7 Comparison of Analytical of Ng and FSM Solutions....................................................82 PAGE 6 6 4.8 Second-Order Solution..................................................................................................83 4.9 Validity of Second-Order Solution................................................................................87 4.10 Comparison with Data..................................................................................................88 4.11 Mud Behavior from FSM.............................................................................................90 4.12 Mass Transport Velocity...............................................................................................92 4.12.1 Mass Transport Phenomenon............................................................................92 4.12.2 Comparison with Data.......................................................................................95 4.13 Application of FSM to Newnans Lake.........................................................................96 5 MUD AS A SINGLEPHASE MEDIUM............................................................................108 5.1 Introduction............................................................................................................ ......108 5.2 Rheological Models.....................................................................................................110 5.3 Single-Phase System : Viscoelastic Model...................................................................110 5.4 Viscoelastic Analogs...................................................................................................113 5.4.1 Voigt and Maxwell Models...............................................................................113 5.4.2 Jeffreyâ€™s and Burgers Models............................................................................114 5.5 Viscoelasticity Characterization Tests.........................................................................115 5.5.1 Creep Test..........................................................................................................115 5.5.2 Oscillatory Test.................................................................................................116 5.6 Mud Behavior as a Vi scoelastic Solid and Fluid.........................................................118 5.6.1 Solid Behavior...................................................................................................118 5.6.1 Viscosity Dependence on Shear Rate................................................................119 5.6.2 Creep-Compliance under Cyclic Loading.........................................................119 5.6.3 Shear Stresses in Mud.......................................................................................122 5.7 Mud as a Viscoelastic Fluid.........................................................................................123 5.8 Empirical Models........................................................................................................ .125 5.9 Comments on Resonance.............................................................................................127 6 MUD AS A TWO-PHASE MEDIUM.................................................................................142 6.1 Introduction............................................................................................................ ......142 6.2 Literature Review....................................................................................................... .142 6.3 Comparison of Poroelastic Models..............................................................................145 6.4 Coulomb Friction and Poroelastic Model....................................................................147 6.5 Domains of Applicability of Constitutive Models.......................................................150 7 WAVE-MUD ISSUES IN NEWNANS LAKE...................................................................163 7.1 Wave Damping...............................................................................................................163 7.2 Wave Damping in Newnans Lake..................................................................................167 8 SEDIMENT ENTRAINMENT IN NEWNANS LAKE......................................................177 8.1 Introduction............................................................................................................ ......177 8.2 Modes of Entrainment.................................................................................................177 8.3 Entrainment in Newnans Lake.....................................................................................178 8.4 Auto-Entrainment........................................................................................................180 PAGE 7 7 8.5 Dye Study............................................................................................................... .....182 8.6 Chemical and Biological Processes.............................................................................183 9 CONCLUDING OBSERVATIONS....................................................................................193 9.1 Study Summary...........................................................................................................193 9.2 Main Observations.......................................................................................................194 9.3 Recommendations for Future Studies..........................................................................196 APPENDIX FIRST-ORDER ANALYTICAL SOLUTION...................................................197 LIST OF REFERENCES.............................................................................................................201 BIOGRAPHICAL SKETCH.......................................................................................................210 PAGE 8 8 LIST OF TABLES Table page 2.1 Instruments deployed in the lake.......................................................................................36 2.2 Instrument elevations...................................................................................................... ...36 2.3 Summary of data in set-2...................................................................................................37 2.4 Sediment settling lags..................................................................................................... ...37 2.5 SSC data summary........................................................................................................... ..37 3.1 Comparison of model out put of SSC and data...................................................................63 4.1 Gadeâ€™s experimental parameters........................................................................................97 4.2 Sakakiyama and Bijkerâ€™s experimental parameters...........................................................97 4.3 Jiangâ€™s experimental parameters........................................................................................97 4.4 Characteristic parameters for Newnans Lake....................................................................97 6.1 Comparison of poroelastic models...................................................................................155 6.2 Characteristic parameters for model comparison............................................................155 6.3 Characteristic parameters for poroela stic beds with Coulomb friction...........................155 6.4 Dimensionless velocity for different soils.......................................................................156 6.5 Fluid mud density and concentration...............................................................................156 7.1 Characteristic parameters for Newnans Lake..................................................................171 7.2 Wave attenuation coefficient and percen tage reductions of amplitude and wave energy......................................................................................................................... ......171 7.3 Characteristic dimensionless numbers for Newnans Lake..............................................171 7.4 Dimensionless number comparison for viscoelastic model.............................................171 7.5 Dimensionless number comparison for poroelastic model..............................................172 PAGE 9 9 LIST OF FIGURES Figure page 2.1 Newnans Lake with main creeks and watershed...............................................................38 2.2 Schematic drawing of platform tower and instrumentation...............................................39 2.3 Platform-tower............................................................................................................. ......39 2.4 Wind-rose for January 2003...............................................................................................40 2.5 Wind direction time series and spectrum...........................................................................40 2.6 Air and water temperatures time-series.............................................................................41 2.7 Power spectra of air and water temperatures.....................................................................42 2.8 Time-series of wind speed and SSC-3...............................................................................42 2.9 Time-series of wind speed and SSC-3 phase -shifted by a day to examine correlation.....43 2.10 Variation of SSC-3 with wind speed.................................................................................43 2.11 Power spectrum of SSC-3 and SSC-1................................................................................44 2.12 Coherence spectra of SSC-1 and SSC-3 with wind speed.................................................44 2.13 Spectra of time-lag between SSC and wind speed.............................................................45 2.14 Suspended sediment load variation with wind speed........................................................45 2.15 Variation of SSC-3 with wave height................................................................................46 2.16 Wave period varia tion with wind speed.............................................................................46 2.17 Wave height variation with wind speed.............................................................................47 2.18 Phase lag spectra between wind speed and wave height...................................................47 2.19 Coherence spectrum for wind speed and wave height.......................................................48 2.20 Current velocity versus wind speed...................................................................................48 2.21 Power spectra of wind speed and current velocity.............................................................49 2.22 Pore pressure versus wind speed........................................................................................49 2.23 Power spectra of wind speed and pore pressure................................................................50 PAGE 10 10 2.24 Coherence spectrum for wind speed and pore pressure.....................................................50 2.25 Data for Newnans Lake.....................................................................................................51 2.26 DO, pH, temperature and SSC plots for site P-2...............................................................52 2.27 DO, pH, temperature and SSC plots for site P-4...............................................................53 2.28 DO, pH, temperature and SSC plots for site P-1...............................................................54 3.1 Lake flow and sediment transport simulation grid............................................................64 3.2 Model Simulations.......................................................................................................... ...64 3.3 Instantaneous velocity vectors...........................................................................................66 3.4 Density profile of the muck layer......................................................................................68 3.5 Settling velocity as a functi on of sediment concentration.................................................68 3.6 Erosion flux versus bed shear stress..................................................................................69 3.7 Comparison of suspended sediment concentration from model and data..........................69 3.8 SWAN grid and bottom contours of the Lake...................................................................70 3.9 Comparison of wave heights from data and SWAN..........................................................71 3.10 Comparison of wave periods from data and SWAN.........................................................71 3.11 Significant wave height for fetch-lim ited waves at different wind speeds........................72 3.12 Wave period for fetch-limited waves at different wind speeds.........................................72 4.1 Schematic diagram of water-mud system..........................................................................98 4.2 Dimensionless wave number fr om models and Gadeâ€™s data.............................................98 4.3 Dimensionless wave attenuation coeffi cient from models and Gadeâ€™s data......................99 4.4 Wave attenuation coefficient from FS M and data of Sakakiyama and Bijker..................99 4.5 Yield stress versus density of kaolinitic mud..................................................................100 4.6 Dimensionless amplitude in mud from FSM and data of Sakakiyama and Bijker..........100 4.7 Wave attenuation coefficient fr om FSM and data of Jiang)............................................101 4.8 Horizontal acceleration am plitude for Jiangâ€™s data.........................................................101 PAGE 11 11 4.9 Horizontal wave orbital velocity amplitude.....................................................................102 4.10 Vertical wave orbita l velocity amplitude.........................................................................102 4.11 Surface amplitude profiles...............................................................................................103 4.12 Dependence of the phase of wave hor izontal velocity on mud properties......................103 4.13 Horizontal velocity variati on with density and viscosity.................................................104 4.14 Normalized Lagrangian, Stoke sâ€™ drift and Eulerian streaming mass transport velocity.104 4.15 Normalized Lagrangian, Stokesâ€™ drif t and Eulerian streaming velocities.......................105 4.16 Mass transport velocity profile and Jiangâ€™s data..............................................................105 4.17 Horizontal wave velocity.................................................................................................106 4.18 Mass transport velocity................................................................................................... .107 5.1 Flow behavior of different materials................................................................................131 5.2 Mechanical analogs for Voigt model and Maxwell model..............................................131 5.3 Mechanical analogs for Jefferyâ€™s -a model and Jeffreyâ€™s-b model...................................132 5.4 Mechanical analogs for Burger s-a model and Burgers-b model.....................................132 5.5 Rheometric tests .......................................................................................................... ....133 5.6 Creep test plot for Voigt model.......................................................................................133 5.7 Creep test plot for Maxwell fluid.....................................................................................133 5.8 Two-Voigt-elements model.............................................................................................134 5.9 Jiangâ€™s model (Jiang, 1993).............................................................................................134 5.10 Creep test data for AK mud.............................................................................................135 5.11 Viscosity variation with shear rate...................................................................................135 5.12 Schematic diagram showing changing cr eep-compliance response with increasing applied stress................................................................................................................. ...136 5.13 Initial compliance versus applied stress for AK mud......................................................136 5.14 Yield stress versus solids vol ume fraction for AK sediment...........................................137 5.15 Shear stress profile for AK mud......................................................................................137 PAGE 12 12 5.16 Phase of shear stress..................................................................................................... ....138 5.17 Variation of shear stre ss with water content....................................................................138 5.18 Definition sketch for stress-strain constitutive behavior.................................................139 5.19 Wave attenuation coefficient ve rsus frequency plot for AK mud...................................139 5.20 Mass transport velocities and Jiangâ€™s data.......................................................................140 5.21 Wave height variation with distance................................................................................140 5.22 Mud mass transport velocity profiles...............................................................................141 5.23 One-dimensional spring-das hpot-mass harmonic oscillator............................................141 6.1 Dimensionless wave attenuation coefficien t as a function of dimensionless water depth.......................................................................................................................... .......157 6.2 Dimensionless wave attenuation coeffici ent as a function of dimensionless bed thickness...................................................................................................................... .....158 6.3 Percolation loss: wave at tenuation coefficient as a f unction of wave frequency............160 6.4 Coulomb friction loss: wave attenuation coe fficient as a function of elastic modulus...160 6.5 Wave attenuation coefficient variatio n with frequency for viscoelastic and poroelastic beds............................................................................................................... .161 6.6 Domains of applicability of constitutive models ............................................................162 6.7 Bed liquefaction mechanisms..........................................................................................162 7.1 Wave attenuation coefficient as a function of water depth..............................................173 7.2 Wave attenuation coefficient as a function of mud thickness..........................................173 7.3 Wave number as a f unction of water depth.....................................................................174 7.4 Wave number as a function of mud thickness.................................................................174 7.5 Change in wave height.....................................................................................................175 7.6 Change in wave energy....................................................................................................175 7.7 Effective stress as a function of wind speed....................................................................176 8.1 Lake sediment concentration zones and entrainment modes...........................................184 PAGE 13 13 8.2 Wave Reynolds number as a function of wind speed in Newnans Lake.........................184 8.3 Wave diffusion coefficient as a function of wind speed in Newnans Lake.....................185 8.4 Scales of diffusion........................................................................................................ ....185 8.5 Bed shear stress as a function of wind speed in Newnans Lake......................................186 8.6 Auto-entrainment due to exchange of suspended sediment.............................................186 8.7 Suspended sediment concentration time-series...............................................................187 8.8 Eddy diffusivity as a function of wind speed in Newnans Lake......................................187 8.9 Comparison of spectra of SSC -1 from data and model...................................................188 8.10 Comparison of spectra of SSC -3 from data and model...................................................188 8.11 Coherence spectra of data and model..............................................................................189 8.12 Concentration profile from model and data.....................................................................190 8.13 Dye concentration time-series .........................................................................................190 8.14 Dye concentration variation with wind speed..................................................................191 8.15 Dye concentration variation with the outflow discharge.................................................191 8.16 Dissolved oxygen profiles at five locations.....................................................................192 PAGE 14 14 LIST OF SYMBOLS a surface wave amplitude a0 un-damped (deep water) surface wave amplitude b interfacial wave amplitude cs measure of the consistency of material C suspended sediment concentration (SSC) CD drag coefficient Cf, concentration limit below which settling velocity is assumed constant Cg group velocity of waves d mud thickness d dimensionless mud thickness eD Deborah number E wave energy E0 un-damped wave energy fw wave friction factor Fr Froude number h water depth i=1, 2 subscript denoting firstand sec ond-order solutions, respectively j=1,2 subscript denoting water and mud, respectively Jo instantaneous compliance due to purely elastic response J(t) creep-compliance G shear modulus of elasticity G storage modulus PAGE 15 15 G loss modulus H0 un-damped wave height Hs significant wave height k wave number ik wave attenuation coefficient ik dimensionless wave attenuation coefficient iCk wave attenuation coefficien t due to Coulomb friction ipk wave attenuation coeffi cient due to percolation rk real wave number ks Nikuradse bed roughness pK coefficient of permeability L wavelength me equivalent mass per unit area of the mud m virtual mass of soil v M a Mach number for viscoelastic bed p M a Mach number for poroelastic bed ne Porosity OC organic content OBS optical backscatter sensor pw pore pressure 1 free stream pressure jP dynamic pressure PAGE 16 16 TP total pressure q discharge velocity Q mean discharge of mud per unit width Rew wave Reynolds number SL total suspended load SSC-1,2,3 suspended sediment concentration at level 1-3 t time T wave period Te characteristic time of the deformation process, an extrinsic quantity u horizontal wave orbital velocity, a function of z ub bottom orbital velocity amplitude u horizontal wave orbital velocity amplitude jiu horizontal wave orbital velocity E u mean Eulerian streaming velocity umean depth mean current velocity Su Stokesâ€™ drift zu soil velocities U wind speed 1 U free stream velocity LU Lagrangian mass transport velocity Ur Ursell number w vertical wave orbital velocity, a function of z PAGE 17 17 zw pore fluid velocity w vertical wave orbital velocity amplitude w0 constant settling velocity jiw vertical wave orbital velocity ws settling velocity W water content of mud. z0 virtual origin of the logarithmi c velocity profile for turbulent flow i and i viscoelastic material constants shear strain rate of shear strain w unit weight of fluid c Coulomb energy loss parameter 1 Stokesâ€™ boundary layer thickness in water 2 Stokesâ€™ boundary layer thickness in mud h characteristic bottom water layer height u excess pore pressure wave steepness r erosion flux D mean rate of energy dissipation per unit time N erosion flux constant 1 free surface displacement 2 interfacial displacement PAGE 18 18 viscosity j viscosity of water (j=1) and mud (j=2) m viscosity of mud w viscosity of water p Poissonâ€™s ratio G retardation time dynamic viscosity of material measure of the elastic response of the material u nder oscillatory forcing. constant viscosity at high shear rate 1 kinematic viscosity of water 2 kinematic viscosity of mud ej equivalent kinematic visc osity of non-Newtonian fluid c bed shear stress due to current cw combined bed shear stress due to current and waves s bed shear strength against erosion w bed shear stress due to waves rate of shear stress bed displacement fluid (bulk) density s soil particle density 1 ,w water density PAGE 19 19 2 ,m density of mud wave angular frequency v total normal stress N resonance frequency v effective normal stress e characteristic time intrinsic to the material shear stress y apparent yield stress vs solids volume fractions PAGE 20 20 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy WAVE ATTENUATION AND MUD ENTR AINMENT IN SHALLOW WATERS By Mamta Jain May 2007 Chair: Ashish J. Mehta Major: Coastal and Oceanographic Engineering In shallow lakes with organic-rich, soft mud in the nepheloid layer, turbidity can result from wind-generated waves and circulation. We examined wave attenuation and mud entrainment in such environments with reference to Newnans Lake in north-central Florida. It is shown that soft mud behavior as a solid at low cyclic stresses changes to a fluid at higher stresses common to most laboratory and fi eld conditions. Accordi ngly, soft mud is modeled analytically as a (single-phase) Newtonian fluid and also as a viscoelastic fluid. Thes e representations of the constitutive behavior of m ud are coupled with a second-or der solution of the governing equations for flow in a two-layer mud-water system. Mud thickness is taken to be comparable to the Stokesâ€™ boundary layer height. Model results ar e shown to reproduce la boratory data on wave attenuation and mud mass transport with an acceptable degree of accuracy. Soft mud is then modeled as a two-phase, so lid-with-pore fluid poroe lastic medium with Coulomb damping. Wave attenuati on predicted by this model is shown to be compatible with laboratory data. However, since th e particle matrix is treated as a solid, mass transport cannot be calculated by this approach. The non-dimensional ratio of particle settling velocity to permeability is proposed as a parameter that, alon g with the solids volume fraction (or density), characterizes bottom type (gravel to clay) and its state of compac tion/consolidation. For gravel PAGE 21 21 the main mechanism of dissipation internal to the medium is pe rcolation. In sandy and coarse silty beds, both of which are two-phase media, the poroelastic model accounts for dissipation mainly through Coulomb friction. In finer si lts and clays, the singl e-phase viscoelastic description appears to be more appropriate, even though the poroelastic assumption has been made by others. Mud can be taken as a Ne wtonian fluid below dens ities of about 1,150 kg/m3. In the range of about 1,150-1,300 kg/m3, mud is better modeled as a viscoe lastic fluid, while beyond about 1,300 kg/m3 mud is not highly compliant. The characte ristic resonance frequency defining peak wave attenuation is shown to be dependent on the square-root of the shear wave velocity in mud and on a scale based on the mud depth or wave length. Physical parameters affecting wave attenuation are examined using dimensional anal ysis. It is shown that , depending on bed type, characteristic dimensionless groups which influence wave a ttenuation can be identified. Based on the application of a nu merical sediment transport model to the lake, it is shown that the combined current-wave bed shear stress usually does not exceed th e critical shear stress for erosion. It follows that the classical erosi on/deposition mechanism of sediment exchange at the bed does not to apply to this lake, where the wind speed seldom exceeds 8 m/s. An attempt has been made to explain the observed oscillatio ns of suspended sediment concentration (SSC) in the lake through the notion of auto-entrain ment, according to which SSC oscillations are generated by vertical exchange of sediment between the neph eloid layer and the upper dilute suspension, without bed erosion. Sediment advection, which is unimportant in calm (normal) conditions, appears to play a significant role in water column mixing during storm events. PAGE 22 22 CHAPTER 1 INTRODUCTION AND PROBLEM STATEMENT Wind-driven processes as a rule play a key role in circulating a nd overturning water in lakes and reservoirs, many of which are laden wi th fine-grained, organic-rich sediment often called muck. In recent decades several muck-laden lakes in Florida and el sewhere have attained high trophic levels, and their wa ter quality has become a matter of concern to the managing agencies. Such lakes can be repositories as we ll as sources of phosphor us and nitrogen, which are collectively responsible fo r the high trophic levels. Sin ce nutrients are adsorbed on particulate matter, they are released when sedi ment is entrained and causes water to become turbid. This release mechanism significantly cont ributes to, among other e ffects, the growth of algae and consequent poor water quality. As a resu lt, in various degrees the water quality of such systems is related to turbidity-g eneration in the lake by waves. In general, turbidity-generation by waves mu st be dealt in two parts, one of which concerns the interaction of wa ves with bottom muck, because th is interaction determines the wave height (and to an extent the period) for given wind speed, fetch and water depth. The other part concerns the mechanism of muck entrai nment by waves of this height and period. An opportunity to examine these two issues arose when the University of Florida was contracted by Floridaâ€™s St. Johns River Water Management District (S JRWMD) to provide a model of nutrient release to th e water column due to wind-driv en sediment entrainment in Newnans Lake. This lake is a shallow, 2,700hectare hypereutrophic body of water within the Orange Creek Basin in north-centra l Florida. Water quality in this lake has declined over the last two decades, and it is listed as impaired by the Florida Department of Environmental Protection (FDEP) due to elevated nutrient and chlorophyll levels. SJRWMD has been developing Pollutant PAGE 23 23 Load Reduction Goals (PLRGs) for the lake, to be used by FDEP to refine the Total Maximum Daily Loads (TMDLs) for the lake. To examine the behavior of suspended sedime nt, an instrumented tower was installed in the lake and data on wind, waves and suspended sediment concentration were collected over an eight-month period during 2003-2004. An effort to in terpret the data (Jain et al., 2005) led to several scientific questions th at require exploration for unders tanding Newnans Lake suspended sediment dynamics, and more broadly the nature of wave-induced entrainment of fine-grained sediment in the shallow marine environment. Out of these questions, the following are of interest to the present study: 1. Among the various models available for the calculation of wave energy dissipation by bottom mud, which one is applicable to Newnans Lake and similar environments? 2. In typically shallow-water environments, mud th ickness tends to be small, in contrast to thick, or even semi-infinitely thick, layers assumed in many models. How is one to model a thin mud layer, and what should be its rheology? 3. Applications of available entrai nment models to small lakes (of the size of Newnans) suggest that entrainment of mud by waves does not always conform to the classical erosiondeposition mechanism incorporated in these mo dels. Are there alternative approaches that could explain the observed be havior of the suspended sediment concentration? To answer the above question the thesis is organized in following manner. A brief description of Newnans Lake and da ta collection scheme is given in Chapter 2. This chapter also includes analysis of relevant data, which are mo re fully presented elsewhere (Jain et al., 2005). In Chapter 3 a flow circulation model, a wave model and a sediment transport model are applied to Newnans Lake. Based on these applications, argum ents have been presented stressing the need to understand wave-mud interaction, and the m echanism by which sediment is maintained in suspension in the lake. To explore wave-mud inte raction in Newnans Lake in a more general way, in Chapter 4 a first-order semi-analytica l solution is presented for the wave-mud boundary layer, and this solution is compared with laboratory experi mental data. A second-order nonlinear PAGE 24 24 solution is then derived using the well-known Stokesâ€™ perturbation approach. As a test of the applicability of the second-order model, in Chapter 4 model-calculate d mud mass transport is compared with available data. Also posed in this chapter is the question of correct representation of mud rheology. Chapter 5 discusses the sign ificance of mud rheol ogy for a single-phase system. Mud is modeled as a viscoelastic fluid, and its behavior under stress is compared with available data from rheometric tests. A three-co mponent viscoelastic flui d model is then applied to mud, and the resulting wave attenuation co efficient and mud mass transport are compared with experimental data. In Chapter 6, mud is modeled as a poroelastic bed as an alternative rheological description. Available poroelasti c models are summarized, and the domains of applicability of the poroelastic ve rsus viscoelastic mode ls are discussed with reference to bottom sediment composition and density. In Chapter 7, the wave energy di ssipation problem in Newnans Lake is treated in a general way by carrying out a dimensional analysis for characterization of the wave attenuation coe fficient. Some new dimensional numbers are introduced and their importance is discussed with respect to Newnans Lake. In Chapter 8, entrainment mechanisms are described very briefly in relation to Newnans Lake. A new concept describing the behavior of susp ended sediment concentration known as Auto-Suspension Model is proposed and applied to Newnans Lake. Chap ter 9 concludes the study with key observations related to wave-mud interaction in general, and the behavior of suspended sediment concentration in Newnans Lake. Recommendations have been made for future studies to continue and enhance the presented work. PAGE 25 25 CHAPTER 2 FIELD INVESTIGATION 2.1 Site Description Newnans Lake is located occurs approximately 8 km east of the city of Gainesville in Alachua County, Florida, at latitude 29o 38.700 and longitude 82o 13.133 (Figure 2.1). The lake has up to 2 m of muck (a highly organic fine soil made up primarily of humus from drained swampland), and portions of the lakeâ€™s litto ral zone are densely covered with floating macrophytes that have contributed to the orga nic component of muck. The maximum depth of water is about 3.9 m in the center of the lake a nd the mean depth is abou t 1.9 m (lake bathymetry is shown in Figure 3.8). The lake is between 5,000 and 8,000 years old (Holly, 1976; Brenner and Whitmore, 1998), and is underlain with Plioce ne Bone Valley formation below the northeast portion of the lake, and the Mi ocene Hawthorne formation belo w the southwest portion. Both geological formations contain deposits of phos phoric limestone with inter-bedded phosphoric pebbles and granules. The drainage area covers approximately 295 km2 (Figure 2.1), and supplies the majority of inflow through two tributaries, Hatchet Creek (H C) and Little Hatchet Creek (LHC), both at the northern end of the lake. Approxima tely 88% of the drainage basi n is rural, being made up of 55% upland forest, 17% wetland, 8% water, 6% agriculture, and 2% rangeland. Of the remaining 12%, 10% is urban, primarily associ ated with the eastern portion of the city of Gainesville, and 2% supports transportation associated with the nearby Gainesville Regional Airport. Overflow drains out of Prairi e Creek (PC) at the southern end and empties into Paynes Prairie and the Styx River through Camps Canal. Bee Tree Creek (BTC) (F igure 2.1) is a small tributary of HC. The majority of lake shorelin e is surrounded by a canopy of local forest trees such as cypress trees. PAGE 26 26 2.2 Data Collection Two field data collection campaigns were car ried out. The first campaign was intensive and included the deployment and operation of a three-legged aluminum-fra me platform tower (at the site marked by a red dot in Figure 2.1) sc hematically drawn in Figure 2.2. The tower was installed in December 2003, and fully operated fr om the beginning of January to early September in 2004 (data set-1). 2.2.1 Extended Field Campaign Figure 2.3 is a photographic view of the towe r. The zero reference level for instrument elevations was marked on a rod on which severa l of the instruments were deployed. Devices (details in Table 2.1) were included for meas uring wind velocity, air and water temperatures, total water pressure and pore water pressure within muck, horizontal and vert ical current velocity components in water, wave-induced acceleration in muck, suspended sediment concentration (SSC) in water (at three levels) and water sample s (two levels). Table 2.2 provides information on the instruments. Data on current velocity, tota l water pressure, pore water pressure and orbital acceleration were collected and st ored digitally. Sampling was at the top of each hour for 5 min at a bursting frequency of 10 Hz. Wind speed, air temperature, water temperature, air pressure and SSC data collection was in the analog mode, w ith 2-min mean values recorded at the top of each hour. All data were sent on real-time basis vi a a cell-phone connection to the University of Florida for storage and analysis. At stations BTC, HC, LHC and PC, daily discharges were re corded by the St. Johns River Water Management District (SJRWMD). In the la ke proper, daily lake water level data were reported by SJRWMD, and at the Gainesville Regi onal Airport, daily va lues of atmospheric parameters were recorded by the ai rport authority (Jain et al., 2005). PAGE 27 27 Two dye injection tests were also conducted in 2003. One was during a calm period (average wind speed of 4 m/s and outflow of 0.23 m3/s from Prairie Creek) beginning 3rd August, starting time 2:00 PM, to 9th August 2003, ending time 9:00 AM. The second test was conducted under stronger conditions (average wind speed of 8.4 m3/s and average outflow of 7.5 m3/s), from September 3rd, starting time 2:00 PM to September 9th, ending time 9:00 AM. For each test, 250 cm3 of Rhodamine-WT dye was placed in a syri nge. The syringe was inserted in a 3.0 m long PVS pipe with an interim diameter of 1.27 cm. A 3 m long rod with a diameter of 0.64 cm was also inserted into the pipe and was used as a push-rod to activate th e syringe plunger. The syringe was then lowered into th e water column and about 20 cm into the mud layer (at the same level as the pore pressure gage and accelerometer ; Figure 3.4). After the syringe was in mud, the dye was injected by pushing the r od. At the time of injecti on the two auto-samplers began collecting water samples, which allowed the time-hi story of dye concentrati on to be recorded at two levels, 0.8 m below water surface (which wa s 0.1 m higher than OBS-2) and 1.5 m (at the level of OBS-3) below water surface. A set of 24 water samples was collected at each depth every 6 hours using the auto-sampler. Dye c oncentration was measur ed in the Coastal Engineering Laboratory of the University of Fl orida using a fluorometer (Turner model 10-005). Analysis of the data is presented in Section 8.5. 2.2.2 Short Field Campaign A second, single-day, data collection cam paign (set-2) was ca rried out on the 23rd of February, 2006. These data were collected using a boat (a McK ee with a 90 hp motor), at five sites in the lake the tower, the deepest water in the lake, and the mouths of Prairie, Little Hatchet and Hatchet Creeks. The coordinates of these locations are given in Table 2.3. Water samples for SSC measurement were collected at th ree depths using a Niskin bottle. Water quality parameters including pH, temperature and di ssolved oxygen were measured using hydrolab PAGE 28 28 (Datasonde 4). A secchi disk was used to measure water transparency and depth. The top of the soft bed was defined as the location where the secch i disk firmly rested. Grab samples were also collected from each of the five sites. Grab samp le descriptions along with secchi disk values and water depths are given in Table 2.3. 2.3 Data Analysis In this section, analyses of data from the first 15 days of data se t-1 (January 14 to 30, 2004) and data set-2 (February 23, 2006) are provided. Analysis for the remainder of data set-1 is given in Jain et al. (2005). 2.3.1 Data Analysis from Extended Field Campaign The wind-rose for January 2004 is plotted in Figure 2.4. The maximum (2-min avg.) wind speed was over 7 m/s, and the mean in the range of 3-5 m/s. The wind-rose shows wind speed in all sectors of the pie chart, whic h is an indication of high directional variability. This variability is manifested in Figure 2.5A, which plots wind direc tion variation with days. Between days 16 and 18, and 20 and 22, the direction changed over the full circle, which in turn would suggest similar variability in wind-driven flow circulation. This variability is also highlighted in Figure 2.5B, which is a power spectrum of wind direction. The occurrenc e of peaks over a wide range of frequencies (0.7day-1 to 5.1 day-1) implies high wind variabili ty. The resulting directional variability in flow has been identified by using a numerical hydrodynamic model in Chapter 3 (Figures 3.3A-D). The time-series of air and water temperatures are plotted in Figure 2.6. Air temperature varied between 2oC and 22oC, and there is evidence of a cold front around day 20. Water temperature follows a similar trend; however, air temperature variation was as large as 20oC, whereas water temperature variation was limited to 5oC. PAGE 29 29 Power spectra of air and water temperatures are plotted in Figure 2.7. Frequency (days-1) is plotted on the abscissa and energy on the ordi nate. Since sampling was every hour, the Nyquist frequency (one-half the sampli ng frequency and also the maxi mum frequency that can be resolved) was 12 days. Water temperature spectrum has been scaled by a factor of 85 for it to be visually comparable with the ai r temperature spectrum. There are evident peaks in both air and water temperatures at frequencies of 1 day-1 and 2 day-1, corresponding to solar diurnal and semidiurnal periods, respectively. Figure 2.8 shows a time-series of wind speed (left ordinate) and one of SSC-3 (at level OBS-3) (right ordinate). In the similarly plotted Figur e 2.9, SSC-3 has been phase-shifted by a day, the time-lag evident in Figur e 2.8. In Figure 2.9, SSC-3 is s een to have responded efficiently to increases in wind speed, but not so to decreas es in speed. Peaks in wind at days 15.5, 17 and 18.2 occur synchronously with the SSC peaks, wh ereas the lows on days 16.2 and 19.2 in wind do not correspond to the lows in SSC. So the inference would be that SSC responds to energy increase in the system, but not to decrease. Since the sediment is highly orga nic, its settling velocity is very low, on the order of 0.1 mm/s (Section 3.2). To estimate the settling timelag, due to low SSC each particle may be assumed to settle independently of others. One se t of calculations is presented in Table 2.4 for SSC at different levels a nd frequencies. At 1 day-1 the settling lag for SSC-1 is 3.6 hours or 54o (with respect to sola r day), whereas for SSC-3 it is 0.7 hours (10o). This implies that for SSC-1, sediment suspended at a given time was in suspension three hour s later. Thus, SSC-1 cannot be expected to correlate with the instantaneous wind speed. In gene ral, such a response of SSC to wind can result in the superposi tion of resuspension behaviors at notable high frequencies such as 0.28 hr-1, 0.33hr-1 and 1.4 hr-1. PAGE 30 30 Another high frequency is the seiching fre quency. The seiching pe riod can be obtained from Eq (2.1) proposed by Raichlen, F., (1966) for elliptic basin with major axis as 2a and minor axis as 2b and constant depth h. 0.5 2 252 2 186sb a a T gh b a (2.1) where a = one-half the major axis and b = one-half the minor axis and h is the average depth. . The seiching period for Newnans Lake with a = 3.5 km and b = 2 km and average depth of 1.5 m would be 51 minutes, which corres ponds to frequency of 1.17 hr-1. Table 2.5 provides a summary of the SSC data . The ratio of mean SSC to its standard deviation decreases from 14 to 4% with an incr ease in water depth from 0.3 m to 1.5 m below the water surface, respectively, which in turn indicates that SSC oscillations decreased with depth. Also, SSC-3 had very low oscillations compared to SSC-1 and SSC-2, which reflects decay of wind energy with depth. This topic is further discussed in Chapter 8. Figure 2.10 is a plot of SSC-3 against wind sp eed, in which a best-f it line along with the 95% confidence interval is also shown. The cutoff at 2 m/s avoids the need to analyze data with high noise-to-signal ratios. The plot exhibits high level of scatte r, which suggests that wind speed and SSC-3 are non-uniquely related. The ratio of standard deviation to mean for SSC-3 for wind speed between 2 and 3 m/s is 2.8x10-2, while the same ratio in th e range of 8 to10 m/s is 3.7x102. These values indicate that the scatter incr eases with increase in wind speed. The mean relationship between SSC and wind speed U is given by Eq (2.2). SSC0.845133.7U (2.2) PAGE 31 31 This dependence of SSC on U, even though weak (and non-unique), is also manifested in the Figure 2.11, which gives power spectra of wind speed and SSC-3. There are predominant peaks at 1.2 days-1, 2.0 days-1 and at 3.7 days-1. Figure 2.12 plots the cohere nce (a measure of crosscorrelation in the frequency domain) spectra between wind speed, SSC-1 and SSC-3. Coherence is observed at several frequencies su ch as 1.25, 2, 2.6, and 3.75 days-1. As the distance between the OBS and water surface increased, coherence peaks shifted towards lower frequencies. In other words, correlation shifted fr om short term to longer term. Th e strongest coherence is at 2 days-1 (solar semi-diurnal period) for SSC-1, and at 1 day-1 (diurnal period) for SSC-3, implying that SSC-3 took 12 hours longer than SSC-1 to respond to wind. Figure 2.13 shows spectra of time-lag betw een SSC and wind. The phase lag of SSC increases from 10o at 1 day-1 to 170o at 2.5 day-1 for SSC-3, and the settling lag also increases (Table 2.4), suggesting that obser ved phase lags and settling lags are related to each other. Since the settling velocity was of the order of 0.1 mm/s , and at higher frequenc ies the phase lags were higher, entrained sediment (which would not correlate with wind at the time of entrainment), can be expected to have remained in su spension from 1-3 hr for SSC-3-SSC-1. The total suspended sediment load (SL) in the water column over a unit planar area can be found as the product of a weighted-mean concen tration and water depth, as given by Eq (2.3). 0.310.6(23) 1.5 SSCSSCSSC SL (2.3) The sediment load against wind speed plot in Fi gure 2.14 shows high degree of scatter. The ratio of standard deviation to mean for SL between wind speeds of 2 and 3 m/s is 3.0x10-2, while the same ratio between 8 and 10 m/s is 4.0x10-2, indicating that scatter incr eases with the increase in wind speed, as in Figure 2.10. PAGE 32 32 SSC-3 is plotted against th e significant wave height ( Hs) in Figure 2.15. Wave heights were obtained by spectral analysis of the measur ed pressure sensor da ta collected at 10 Hz bursting. This plot shows high degr ee of scatter. The ratio of sta ndard deviation to mean SSC for wave heights between 3 and 4 cm is 2.4x10-2, while the same ratio for heights between 5 and 6 cm is 4.4x10-2, which indicates that the scatter increased with increasing wave height. The bestfit line is given by Eq (2.4). SSC242126sH (2.4) Mean wave period versus wind speed data are pl otted in Figure 2.16. For wind of 2 m/s the period is around 0.7 s, and as the speed increases to 7 m/s the period increases to 1.2 s. The mean trend is given by Eq (2.5). 29.66100.555TxU (2.5) Wave heights versus wind speeds are plotted in Figure 2.17 and the equation of the mean line is given by Eq (2.6). 323.24103.01810sHU (2.6) Phase lags between wind and waves are plo tted in Figure 2.18. The phase lag at the frequency of 1 day-1 is 5o, which implies that it would ta ke about 20 minutes for the (windgenerated) waves of that frequency to devel op. The coherence spectrum between wave and wind in Figure 2.19 indicates that coherence is high, around 0.92 at frequencies of 0.75 day-1 and 1.3 day-1, 0.80 at 2.3 days-1 and 0.75 at 4.5 days-1. These values indicate that coherence declined with increasing frequency. In other words, short-term responses of waves to wind were less correlated than longer term responses. A possible qualitative explanation is that wave response to wind can be described in terms of a simple harmonic osci llator. This would mean that lower frequencies are selectively generated in preference to higher frequencies which are cutoff. PAGE 33 33 Current velocities measured at the tower ar e plotted in Figure 2.20. At wind speeds lower than 6 m/s the currents are of the order of fe w millimeters per second. Power spectra of current and wind speed are plotted in Figure 2.21. Since th e average water inflow into the lake was of the order of 0.2 m3/s (during calm periods), discharge-i nduced currents were negligibly small. The dominant direction of flow in the lake is from north to south (Figure 2.1). During winddriven circulation, current direct ions change, as evident from Fi gures 3.3A-3.3D. Since the lake is deeper on the northern end than at the southe rn end (contours in Figure 3.8), adverse bottom gradient from north to south tends to further slow the current speed. Pore water pressure is the water pressure over and above the effec tive normal stress. As mentioned, the pore pressure gauge was located 20 cm below the mud-water interface (Figure 3.4). In Figure 2.22, pore pressures between wind speeds of 2 and 4 m/s have a ratio of standard deviation to mean of 0.45, whereas between 4 an d 10 m/s the ratio is 0.88 . This indicates that data scatter increased with the increase in th e wind speed. However, there is a degree of correlation between pore pressure pw and wind speed U given by Eq (2.7). 1.814.4wpU (2.7) Power spectra of pore pressure and wind speed are plotted in Figure 2.23. Peaks for the two variables coincide at the frequency of 1.5 day-1; however, there is considerable noise at higher frequencies. This is clear from the spectra in Figure 2.24, in which coherence is 0.6 at 1.7 day-1, and decreases to 0.1 with increase in frequency. In Figure 2.25A, daily average of wind speed ti me-series is plotted, and in Figure 2.25B the corresponding total sediment load (SL) given by Eq (2.3), is plotted. Figure 2.25C plots the outflow discharge from Prairie Creek, and rainfall is plotted in Figure 2.25D. There is a correlative trend between outflow and SL during days 16-22. However, after day 22 no PAGE 34 34 noteworthy correlation occurs. Thes e trends suggest that during cal m periods when discharge is low (average only of 0.6 m3/s), the effect of advection on SSC is not significant, but is apparently so when the discharge is higher (> 2 m3/s). 2.3.2 Data Analysis from Short Field Campaign The second set of data was collected betw een 10:00 am and 12:00 pm on February 23rd 2006 at five locations (indicated in Table 2.3 a nd identified in Figure 2. 1). From these data, DO (dissolved oxygen), pH, water temperature and SS C are plotted for location P-2 (Table 2.3) in Figure 2.26. Near the water surface, DO was 9 mg /L (fully saturated su rface water), pH was around 6.5 likely due to leaching of organics (in the form of tanni c and fulvic acids) from the forested wetlands associated with Hatche t Creek (SJRWMD, 2004), water temperature was 20.15oC and SSC was 1 mg/L. At the depth of 2.8 m, which was also the top of the soft bed as measured by the secchi disk, DO concentration dropped to 5 mg/L, pH decreased to 6.58 and SSC increased to 2500 mg/L. A fu rther increase in depth by 20 cm reduced DO concentration to zero. This thickness could be a measure of the presence of fluid mud at that site. Similar plots as in Figure 2.26 are shown in Figure 2.27 for the shallower depth site P-4 (Table 2.3). DO changed from 9 mg/L at the water surface to 5 mg/L at 1.6 m. Figure 2.28 includes plots for the tower location. The top of soft sediment was 2.3 m below the surface where DO, pH and temperature started to decr ease with depth, sugge sting the presence of sediment (SSC was 300 mg/L). DO becomes zero at 2.7 m, which could mean that this nearbottom region may have contained fluid mud. The onset of seasona l anoxia in water overlying the sediment has been linked to an increase in phosphorus release from sediment (Caraco et al., 1991). Low DO favors the generation of methane, su lfide and ammonium, and release of ferrous iron from sediment, all of which can lower pH. PAGE 35 35 In order to obtain insight to the effects of sediment entrainment in the lake, flow and sediment transport modeling along with wave ge neration modeling are discussed in Chapter 3. 2.4 Conclusions Newnans Lake is fetch-limited with low bottom shear stresses. Due to low settling velocities, settling lags are of the order of 34 hours, which results in a superposition of SSC response at high frequencies. The amplitudes of SSC oscillations in the la ke are not large. The standard deviation of SSC normalized by mean SSC at 0.3 m depth was 14% and at 1.5 m it was only 3.6%. These values indicate that the effect of wind decays rapidly with water depth. There is a good correlation between wind and wave he ight (correlation coefficient of 0.8). PAGE 36 36 Table 2.1 Instruments deployed in the lake Property Instrument make Model no. Frequency of bursting (Hz) Wind RM Young anemometer 05103 â€” Air temperature Analog Devices gauge AC 2626 â€” Water temperature Analog Devices gauge AC 2626 â€” Current Sontek three-axes acoustic current meter Field ADV 10 Air pressure Trans-metrics gauge B020 â€” Total pressure Trans-metrics gauge P215L 10 Pore pressure Druck gauge PDCR-81 10 SSC Sea Point optical backscatter sensors (OBS) â€” 10 Fluid acceleration Analog Devices three-axes accelerometer ADXL 202 10 Water sampling ISCO auto-samplers 3700 â€” Data logging Campbell Scientific logger CR23X â€” Light Carmanah light 501 â€” Data transmission Airlink Communications modem Redwing CDMA â€” Table 2.2 Instrument elevations Instrument Elevation relative to zero reference* Anemometer and Air temperature gauge On the top of tower Water temperature -1.143 m Current meter -0.889 m OBS-1 -0.3048 m OBS-2 -0.9144 m OBS-3 -1.524 m Total pressure -1.0414 m Pore pressure -2.286 m Accelerometer -2.286 m * Zero reference was an arbitrary datu m marked on the rod on which several instruments were deployed. PAGE 37 37 Table 2.3 Summary of data in set-2 Point P-1 P-2 P-3 P-4 P-5 Location Tower Deep part of lake Prairie Creek mouth Little Hatchet Creek mouth Hatchet Creek mouth Latitude 29o 37â€™ 54â€ 29o 38â€™ 58â€ 29o 37â€™ 0.6â€ 29o 40â€™ 27.3â€ 29o 40â€™ 20.5â€ Longitude 82o 14â€™ 6.5â€ 82o 12â€™ 54â€ 82o 14â€™ 48â€ 82o 13â€™ 16â€ 82o 13â€™ 52â€ Secchi disk depth (m) 0.457 0.53 0.457 0.457 0.457 Water Depth (m) 2.28 2.89 1.68 1.67 1.75 Grab sample description Highly organic and dark brown Highly organic dark and like fluid mud Organic and dark brown lot of debris Dark and very stiff Sandy and light in color Table 2.4 Sediment settling lags OBS Distance from bed (m) Settling time (hours) Lag (degree) for frequency 1 Day-1 Lag (degree) for frequency 0.5 Day-1 1 (-0.3 m) 1.5 3.62 54.3 108.6 2 (-0.9 m) 0.9 2.17 32.5 65.1 3 (-1.5 m) 0.3 0.72 10.8 21.6 * Settling velocity of 1.15x10-4m/s is used in calculation. Table 2.5 SSC data summary OBS Mean (mg/L) Standard Deviation (mg/L) Std/Mean (%) 1 (-0.3 m) 6.3 0.87 14% 2 (-0.9 m) 14.6 0.99 6.8% 3 (-1.5 m) 136.7 4.93 3.6% PAGE 38 38 Figure 2.1 Newnans Lake with main creeks and watershed (adapted from Gao and Gilbert, 2003). Coordinates of discharge and water quality stations are shown Newnans Lake Landuse Urban & Open Low Density Residential Medium Density Residential High Density Residential Agricultural Rangeland Forested Water Wetlands Barren Land Transportation, Communications & Utilities Little Hatchet Creek North Branch station Lat. 29.69078 Long. -82.25568 Prairie Creek station Lat. 29.61094 Long. -82.24815 N 0 5 10 km Florida Newnans Lake region Newnans Lake station Lat. 29.63584 Long. -82.24365 Hatchet Creek station Lat. 29.72292 Lon g . -82.21499 Bee Tree Creek station Lat. 29.71600 Lon g . -82.18560 Hatchet Creek Water Quality station Lat. 29.68722 Long. -82.20670 P-1 P-2 P-4 P-5 P-3 PAGE 39 39 Figure 2.2 Schematic drawing of platform towe r and instrumentation. Initial (as shown) elevations of the submerged instruments were changed as necessary during the course of the study due to water level variation Figure 2.3 Platform-tower Elev. 0 m (+19.91 m NAVD88) NAntenna +4.4 m Anemometer +4.9 m Air temperature gauge +4.1 m Pressure gauge -1.1 m Water temp. gauge -1.1 m OBS-1 -0.3 m OBS-2 -0.7 m OBS-3 -1.0 m Accelerometer -2.3 m Pore press. gauge -2.3 m Muck surface â€“ 2.1 m Current meter -0.9 m Elevation +1.5 m Elevation +3.1 m Wire-mesh housing 60W Solar p anel N avigation ligh t 1.7 m 91 mm aluminum pipe Auto-sampler hoses Auto-samplers Data logger 12V Nav. li g ht batter y -0.74 m -1.35 m Density profile Water level PAGE 40 40 Figure 2.4 Wind-rose for January 2003 Figure 2.5 Wind direction time series and spectrum A) Wind direction va riation with days, B) Wind direction power spectrum A PAGE 41 41 Figure 2.5 Continued Figure 2.6 Air and water temperatures time-series Frequency (days1 ) B PAGE 42 42 Figure 2.7 Power spectra of air and water temperat ures (water temperature spectrum is scaled by a factor of 85) Figure 2.8 Time-series of wind speed (in blue, left ordinate) and SSC-3 (in green, right ordinate) Frequency (days1 ) PAGE 43 43 Figure 2.9 Time-series of wind speed (in blue, left ordinate) and SSC-3 (i n red, right ordinate) phase-shifted by a day to examine correlation Figure 2.10 Variation of SSC-3 with wind speed Cut off PAGE 44 44 Figure 2.11 Power spectrum of SSC-3 and SSC-1 Figure 2.12 Coherence spectra of SSC-1 and SSC-3 with wind speed Frequency (days1 ) Frequency (days1 ) SSC-1 SSC-3 SSC-1 SSC-3 PAGE 45 45 Figure 2.13 Spectra of time-lag between SSC and wind speed Figure 2.14 Suspended sediment lo ad variation with wind speed Cut off Frequency (days1 ) SSC-1 SCC-3 PAGE 46 46 Figure 2.15 Variation of SSC-3 with wave height Figure 2.16 Wave period va riation with wind speed Cut off PAGE 47 47 Figure 2.17 Wave height va riation with wind speed Figure 2.18 Phase lag spectra betw een wind speed and wave height Cut off Frequency (days1 ) PAGE 48 48 Figure 2.19 Coherence spectrum fo r wind speed and wave height Figure 2.20 Current veloc ity versus wind speed Cut-off Frequency (days1 ) PAGE 49 49 Figure 2.21 Power spectra of wi nd speed and current velocity Figure 2.22 Pore pressure versus wind speed Cut off Frequency (days1 ) PAGE 50 50 Figure 2.23 Power spectra of wi nd speed and pore pressure Figure 2.24 Coherence spectrum fo r wind speed and pore pressure Frequency (days1 ) Frequency (days1 ) PAGE 51 51 Figure 2.25 Data for Newnans Lake. A) Daily aver age wind speed, B) total daily sediment load, C) daily outflow discharge in Prairie Creek, and D) daily average rainfall in the lake area A Days Days Days Days Days B C D PAGE 52 52 Figure 2.26 DO, pH, temperature and SSC plots for site P-2 (d eepest location, Table 2.3) Top of the soft bed PAGE 53 53 Figure 2.27 DO, pH, temperature a nd SSC plots for site P-4 (Little Hatchet Creek mouth, Table 2.3) Top of the soft bed PAGE 54 54 Figure 2.28 DO, pH, temperature and SSC pl ots for site P-1 (Tower, Table 2.3) Top of the soft bed PAGE 55 55 CHAPTER 3 CIRCULATION, WAVES AND SEDIMENT TRANSPORT 3.1 Introduction A three-dimensional numerical model was used to simulate flow circulation and suspended fine sediment transport in the lake on a real-t ime basis. A summary of the model is given in Section 3.2. A wave-generation mode l (Section 3.3) was used to si mulate the wave field induced by wind. Figure 3.1 is a schematic of the bathymet ric grid, and inflow and outflow creeks. In order to obtain insight into the effects of entr ainment and advection on SSC, flow and sediment transport modeling was carried out. The main obj ective of using the wave generation model was to identify the effect of the presen ce of fluid mud on wave properties. 3.2 Circulation and Sediment Transport 3.2.1 Flow Circulation The numerical model Environmental Fluid Dyna mics Code (EFDC) was used to simulate windand creek discharge-induced circulation in Newnans Lake. EFDC was also used to simulate suspended fine sediment transport. The coordinate system in this model can be Cartesian or curvilinearorthogonal in the horizontal plane, and is stretched vertically to follow bottom topography and free-surface displacement (H amrick 1992). Further details on the model can be found in Hamrick (1992, 1996, 2000) , and also Park et al. (1995). The model grid is shown in Figure 3.1. The re ference coordinate system is geographical (latitudes and longitudes) and the vertical datum is NAVD 88. Each grid cellâ€™s planar size was selected to be 100 m in the x -direction and 150 m in the y -direction. These dimensions were selected on the basis of available bathymetry. Th e water column was divided into 3 layers, based on the use of three OBS sensors for SSC da ta collection. The flow model inputs were: PAGE 56 56 Real-time wind recorded at the tower. Discharge in two inflowing and one out-flowing creeks. Metrological data including rain fall, solar radiation, humidity and cloud cover recorded at the Gainesville Regional Airport. These data were used to calculate the loss of water due to evaporation. There were two inflow bounda ries (Figure 3.1) and one outflow boundary. Lake stage recorded at the tower was used as a comparison parameter between the data and the model. Comparison between measured and simulated water levels in Figure. 3.2A during the (hot-start) calibration model-run shows reas onable agreement, although by day 95, a 5 cm difference developed between the two time-series, with simu lation showing a lower water level than that measured at the SJRWMD gauge located near th e mouth of Prairie Creek . However, during the validation model run this difference begins to decrease (Figure 3.2B) until it reaches zero on day 124. Over the next 6 days (ending at day 130), th e sense of the difference changes so that the simulated level is about 2 cm higher than measured. In the second modeling scenario, two inflow boundary conditions at the north end and a water level boundary condition at the mouth of Prai rie Creek were set, and measured discharge in Prairie Creek was used as the calibrating pa rameter. In Figure 3.2C, a comparison is made between simulated and measured discharges in this creek for the calibration period. Simulated total volume of water flowing out over the duration was 7,619,823 m3, while the value from measurement was 7,731,790 m3, indicating an error of only 1. 45%. The three layers in the vertical are not enough to resolv e the vertical profile. Another te st run was also made with six layers in the vertical. With regard to circulation, instantaneous de pth-mean velocity field from the model is plotted in Figure 3.3A-3.3D. Zone s of circulation in the lake are identified by circles. In Figure PAGE 57 57 3.3A there are two small circula tion cells in the deeper zone a nd a large cell near the mouth of the outflow. As time progresses (day 70.13 to 70. 42, from Figures 3.3A-3.3D), the two small cells coalesce into one, leading to an increase in the size of the circulation cell, which moves northwards. There is some flow channelization as identified by th e rectangle. In general, the circulation pattern is not organi zed. In fact, from flow simulati ons in Figures 3.3A-3.3D, it can be inferred that the pattern is chaotic. 3.2.2 Suspended Sediment Movement The lake bottom is covered with a homoge neous layer of highly flocculent organic sediment (Holly, 1976). A description of sedi ment composition has been provided by Gowland et al. (2002). The grain size is in the range 0.3 to 1.3 m, and the organic content of bottom sediment is in the range of 20 to 50%. A typical density profile of bottom muck is s hown in Figure 3.4, which indicates that the bulk density ranged from 1,050 to 1,150 kg/m3. The dry density of sedime nt was in the range of 100-200 kg/m3. The settling velocity was obtained fr om laboratory settling column tests (Gowland et al., 2005), and also using the R ouse equation for low-concentration (with noninterfering particles) suspende d sediment (Henderson, 1966). The SSC data at three elevations mentioned in Chapter 2 were used for modeling purposes. The following function describes the de pendence of the settling velocity ws on SSC (designated as C) according to Hwang (1989): 22;,n s 0fsf maC w = w ,C CwC C bC (3.1) in which Cf, which typically ranges between 0.1 and 0.3 kg/m3, is the concentration limit below which ws is practically free of the ef fect of inter-particle collisi ons and is assumed constant ( w0). PAGE 58 58 Figure 3.5 shows the settling velocity plot for di fferent organic content. This plot is based on laboratory tests (Gowland et al ., 2005) on samples of lake se diment in a 2-m tall settling column. It is observed that settling velocities estimated by Jain et al. (2005) (using the Rouse equation and measured concentrations in the lake at three elevations) on the average agree with the 20% organic content ( OC ) curve. For this curve, which is assumed to be applicable to the present effort, the constants are: 0.17a, 2.4b , 1.8m 1.8n and 0.1fC kg/m3. This value of Cf corresponds to 00.000115 w m/s. The bed erosion fluxr due to waves and current combined is given by Eq (3.2) ()hNcwsdC r dt (3.2) where h is a characteristic bottom water layer height, C is the SSC within this layer, t is time, N is the erosion flux constant, cw is the bed shear stress and s is the bed shear strength against erosion (also called the critical bed shear stress). In general, cw is taken to be the combined shear stress due to waves and current, as obtained from Eq. (3.9). Gowland et al. (2005) report th at, in order to obtain Eq.(3.2), erosion tests were conducted on a variety of fine-grained sediments in a C ounter Rotating Annular Fl ume (CRAF), and in a Particle Erosion Simulator (PES). In both devices beds of 3 to 5 cm thickness were allowed to consolidate between 24 and 96 hr before each te st. The CRAF has been described in detail elsewhere (Stuck, 1996; Parchure and Mehta, 1985). In the PES, described by Tsai and Lick (1986), the bed was prepared inside a 15 cm diameter plexiglas cy linder. A porous vertical grid submerged in water over the bed was then oscillated at a selected angular speed (rpm), which caused the sediment to erode. The grid-associated shear stress was obtained from the calibration relationship cw =5.91x10-4 x rpm, derived by measuring and comparing the erosion of muck PAGE 59 59 from the Everglades Agricultural Area (EAA) in s outhern Florida, concu rrently in the PES and the CRAF (Stuck, 1996; Mehta et al. 1994). Gowla nd et al. (2005) obtaine d the plot shown in Figure 3.6 for the lake sediment. The values of N and s in Eq. (3.2) corresponding to the 20% OC line are 0.83 kg/m2/s/Pa and 0.12 Pa, respectively. Suspended sediment concentrations simulated by model (using a time step of 0.1 s) at three elevations are plotted in Figur e 3.7. The rating curves (relat ionship with di scharge and SSC) were developed for each of the inflow and outflow ; details are mentioned in Jain et al. (2005). Time-mean SSC values from the model and data ar e given in Table 3.1. The model predicts the â€œaverageâ€ sediment concentration within an erro r of 26% for SSC-3 and 46% for SSC-1. At the same time, the ratio of standard deviation to mean from the model ranges from 0.82 to 0.20, while the same ratio from data ranges from 0. 20 to 0.06, indicating that modeled SSC shows considerably higher degree of oscillations about the mean (factor of as much as 4 times) in comparison with data. 3.3 Wave Field In Section 2.4, it was indicated that the lake is shallow and is fetch-limited. To simulate the development of fetch-limited waves, the wave-generation model SWAN (S imulating WAves Nearshore), Ver 40.41, was used. The model is based on the wave energy balance equation (in the absence of current) with requ isite energy sources and sinks. Extensive model description has been documented elsewhere (Booij et al., 1999; Ris et al., 1999).T he wave field was simulated for a steady wind from six directi ons (north, north-east, north-west , south, south-east and south west) and five speeds (2.5, 5.0, 7.5, 10.0 and 12.5 m/s). Wind directions were selected with the intention of covering the smallest fetch (due to NE wind) to the longest fetch (due to NW wind) and the lowest (2 m/s) to the highest (12 m/s) wind speed. Relevant processes in SWAN include PAGE 60 60 wave generation by wind and di ssipation by bottom friction. Lake bottom contours and the grid used for SWAN are shown in Figure 3.8. Grid dimensions were 100 m by 100 m. Figure 3.9 plots significant wave height versus wind speed from SWAN and data. Measured heights are of the order of 4 cm, whereas the simulated hei ghts are of the order of 10 cm. Figure 3.10 shows the plot of the mean absolu te wave period measured at the tower and the same simulated by SWAN. At the wind speed of 5 m/s, SWAN predicts periods from 0.9 to 1.2 s, which are in reasonable agreement with data, in the range 1.1-1.2 s. SWAN simulated waves over an assumed rigid bottom in the lake. Although the predicted wave period agrees reasonably well with data, wave heights ar e 80-100% higher than data. The part of the difference can be attributed to the fact the SWAN model did not account for the current in the lake, which may sometimes oppose the growth of the waves. Also the SWAN model was run for the steady and constant wind speeds, whereas the wind speed was variable. Some the difference between SWAN and the model can be due to the fact that SWAN does not account for wave damping believed to be due to the presence of soft mud. 3.4 Estimation of Wave-Current Shear Stress Due to its simplicity, the combined wave-current bed shear stress cw formulation of Soulsby et al., (1993) was incorporat ed in EFDC as follows from Eq (3.3). 2cDmeanCu (3.3) wherec is the current-induced bottom shear stress, is the fluid density, CD is the drag coefficient and umean is the depth-mean current velocity. The drag coefficient is obtained from Eq (3.4). 20.4 ln(/)1D oC hz (3.4) PAGE 61 61 where h is the boundary layer height and z0 is the virtual origin of the logarithmic velocity profile for turbulent flow. The wave shear stressw is given by Eq (3.5) 2 0.5 wwb f u (3.5) where fw is the wave friction factor and ub is the bottom orbital velocity amplitude. This amplitude is calculated using the linear wave Eq (3.6) 2sinhbH u kh (3.6) where H is the wave height, 2/T is the wave angular frequency,2/kL is the wave number, T is the wave period and L is the wavelength. Given/ bAu , the wave friction factor is given by (Soulsby et al., 1993), Eq (3.7) 0.190.00251exp5.21;1.57w ssAA f kk (3.7) 0.3;1.57w sA f k (3.8) where ks = 30 z0 is the Nikuradse bed roughness. The wave and current shear stresses, w and,c respectively, are supe rimposed non-linearly using Eqs (3.9) and (3.10) 1(1)mn cw cw c cwYaXX Y X (3.9) 1234coscoslogII wDaaaaafC (3.10) PAGE 62 62 with analogous expressions for m and n . The quantity is the angle between the current stress vector and the wave stress vector . This angle was taken to be zer o for the present purposes. The coefficients a1-a4, m1-m4 and n1-n4 are given in Souls by et al. (1993). 3.5 Concluding Observations As mentioned, the sediment transport model predicts significant oscillations in the suspended sediment concentration. The ratio of standard deviation to mean SSC from the model ranges from 0.82 to 0.20, while the same ratio for the data ranges from 0.20 to 0.06. The total bed shear stresscw obtained from Eq (3.9) is plotted in Figur e 8.5. The stress was less than the critical bed shear stress (0.05 Pa for 20% OC obtained from the Figure 3.6) for the entire range of wind speed (2-10 m/s) , which means that there was no measurable erosion in the lake. It can t hus be inferred that the mechan ism causing SSC oscillations in the lake differs from the modeled (classica l) erosion and deposition mechanism. This inference leads to the need for a new explana tion of the behavior of measured SSC. This development is discussed in Chapter 8. Wave heights simulated by (ri gid-bed) SWAN are 80-100% higher than the measured data. A possible reason for the smaller lake wave he ights (than would occur over a rigid bed) could be that due to rapid changes in th e wind direction, the waves were not fully developed. Using the U.S. Army Corps of Engineers (2002) manual Figure II-2-23 (reproduced as Figures 3.11 and 3.12), the wave height calculated for the longest fetch of 5.5 km at the tower for a wind speed of 5 m/ s, is 18 cm, which are comparable with the value obtained from SWAN (Figure 3.9). Accord ingly, the wave height may vary from 8 to18 cm for fetches of 1 and 5.5 km, respectively. Another reason for the lower than predicted wave heights in the lake could be that the lake is covered with almost a meter thick layer of fluid mud, which is beli eved to attenuate the waves, while SWAN assumes a rigid bed. This conclusion leads to the need to examine wave-mud interaction and its potential effect on SSC. In that regard, various dissipation mechanisms and their importance to Newn ans Lake are described in Chapter 7. PAGE 63 63 Table 3.1 Comparison of mode l output of SSC and data Time-mean Std/Mean Suspended sediment concentration (SSC) Model (mg/L) Data (mg/L) Error (%) Model (mg/L) Data (mg/L) OBS-1 (-0.3 m ) 4.0 7.4 -46% 0.82 0.20 OBS-2 (-0.9 m) 12.8 15.2 -16% 0.47 0.06 OBS-3 (-1.5 m) 103.0 141.7 -26% 0.50 0.06 PAGE 64 64 17.4121.34Bottom ElevTime: 71.00 Figure 3.1 Lake flow and sedime nt transport simulation grid. Figure 3.2 Model Simulations A) Simulated water level time-series durin g calibration (days 7590), B) Simulated water level time-seri es during validation (days 102-128), C) Measured and simulated discharge time-ser ies in Prairie Creek (days 80-160). Spikes in simulation reflect Hatchet Creek and Little Hatchet Creek inflow character Little Hatchet Creek ( m ) Hatchet Creek Prairie Creek â€¢UF platform 100 m 150 m Grid Rectangle A PAGE 65 65 Figure 3.2 Continued B C PAGE 66 66 Newnans Lake .5 (m/s)Depth AveragedVelocities70.13 1498 Meters Newnans Lake .5 (m/s)Depth AveragedVelocities70.21 1498 Meters Figure 3.3 Instantaneous velocity vectors A) time 70.13 days, B) at time 70.21 days C) at time 70.31 days, D) at time 70.40 days. Circulati on cells are identified by a circle and an ellipse. Rectangle identifies channelized flow A B PAGE 67 67 Newnans Lake .5 (m/s)Depth AveragedVelocities70.31 1498 Meters Newnans Lake .5 (m/s)Depth AveragedVelocities70.40 1498 Meters Figure 3.3 Continued C D PAGE 68 68 Figure 3.4 Density profile of the muck layer Figure 3.5 Settling velocity as a f unction of sediment concentration C . Laboratory data and values estimated by Jain et al. (2005) from measured concentration profiles 1000 1020 1040 1060 1080 1100 1120 1140 1160 1180 Wet bulk density (kg/m3) 0.0 0.5 -2.5 -2.0 -1.0 -1.5 -0.5 Elevation relative to bed surface (m) Muck surface Level of accelerometer and pore pressure gauge 0.20 m 20%; 2.4 Concentration, C (kg/m3) Settling velocity, ws (m/s) 10-1 10-2 10-3 10-4 10-5 10-6 10-2 10-1 100101102 10% OC 30% 30%< OC 60% OC = 10%; b = 1.6 30%; 3.7 40%; 5.8 50%; 7.9 60%; 10 (Laboratory tests) Based on field data PAGE 69 69 Figure 3.6 Erosion flux versus bed shear stress wi th lines of constant OC; data from Newnans Lake Figure 3.7 Comparison of suspended sediment con centration from model and data at the tower. Red line represents measurement and blue line is model output Shear stress, b(Pa) 0 0.1 0.2 0.3 0 0.05 0.10 0.15 0.20 0.25 0.30 OC (%) 60 50 40 30 20 10 Erosion flux, r (kg/m2/s) PAGE 70 70 Figure 3.8 SWAN grid and bottom contours of the Lake x (m) y ( m ) PAGE 71 71 Figure 3.9 Comparison of wave heights from data and SWAN Figure 3.10 Comparison of wave periods from data and SWAN PAGE 72 72 Figure 3.11 Significant wave hei ght for fetch-limited waves at different wind speeds (from US Army Corp of Engineer, CEM, 2002) Figure 3.12 Wave period for fetch-limited waves at different wind speeds (in increments of 2.5 m/s, from US Army Corp of Engineer, CEM, 2002 PAGE 73 73 CHAPTER 4 WAVES OVER COMPLIANT MUD BOTTOM 4.1 Introduction As noted in Chapter 3, due to the presence of compliant (soft) mud in Newnans Lake, wave generation must be treated differently th an in the rigid-bottom case. To predict wave heights accurately, it is essential to understa nd soft mud interaction with waves, which is examined here in two parts. The first part de als with wave interaction with soft mud bottom assuming no interfacial mixing. The second part deals with entrainment of soft mud by waves. In this chapter the first part is presented. The second part is dealt with in Chapter 8. 4.2 Some Previous Studies Bottom mud damps waves, and even acts as an emergency harbor in the open sea during storms. One such location, known as â€œmud holeâ€ in the Gulf of Mexico, has been well recognized by investigators (e.g., Gade, 1958; S ilvester, 1974; p. 146). High wave attenuation rates have also been observed in the Mississipp i River delta area. Hurricane-generated waves with heights of 20-25 m in deep water reduc e to 3-5 m in depths of 12-20 m (Bea, 1974). Tubman and Suhayda (1976) measured waves in shallow water over the Mississippi River delta. Wave attenuation rates were much greater than could be attributed to bottom friction. Ewing and Press (1949) reported that wave motion on soft botto m sediment affect the characteristics of the overlying surface waves. Jiang (1993) observed that the wave attenuation coefficient in Lake Okeechobee (laden with fluid muck) in Florida was of the order 10-3 m-1, whereas for the same lake treated theoretically as one with a rigid bed, the wave attenuation coefficient was of the order of 10-5 m-1. The classical wave theory, when applied to soft mud beds, has a drawback due to the assumption of a non-porous bottom overlain by an i nviscid fluid. Soft mud interacts with waves PAGE 74 74 resulting in attenuation of wave height due to bottom friction, percolation losses and viscous damping within the sediment. These interactive modes are also manifested as changes in the wave length, water particle motion, and the el evation of the interface between the fluid and bottom sediment. To investigate wave damping, two-fluid sy stems (Figure 4.1) have been studied by researchers over past decades. Such a system wa s first considered by Gade (1958), who solved the problem analytically by si mplifying the governing equations of flow continuity and momentum for shallow water (/10kh ) application, where k is the wave number and h is the water depth. Gade verified his results by conduc ting experiments with kerosene on top of sugar solution in water. Dalrymple a nd Liu (1978) extended Gadeâ€™s (1958) solution to the intermediate water depth case. They simplified their system by assuming that mud thickness d is much greater than the Stokesâ€™ boundary layer thickness ( 222/ d ) where 2 is the kinematic viscosity of the mud, 2/T is the wave frequency and T is the wave period. Mei and Liu (1987) correctly pointed out that mud in the marine environment is typically much thinner than the overlying water layer. Accordingly, Ng (2000) obtained an analytical solution for the two-fluid system ha ving mud thickness of the order of 2 . His approach is discussed in Section 4.5. The so lution begins to deviate from Full Semi-Analytical model (FSM, discussed in Section 4.6) for 2d . Depending of the water depth (shallow to deep), wave energy dissipation reaches its peak value wh en the mud layer is 30-50% thicker than 2 . This is the region where both Ngâ€™s and Dalrymple and Li u (D&L) solutions are inaccurate (Figures 4.2 and 4.3. So there is a need to capture the corr ect magnitude of dissipati on and obtain a solution with greater accuracy in the range of 2d . Also, additional effort is required in solving the two problems separately, one (Ng) an alytically and the other (D&L) numerically. This extra effort PAGE 75 75 points to the need for a full solution that is valid over the entire mud depth. Lowes (1993) carried out the full first-order solution in clusive of boundary layer effects in thin mud. Her solution is described in Section 4.6. This solution converges to that of Ng (2000) for 2d , and to D&L for 2d (Figures 4.2 and 4.3). To capture the nonlinear effects of waves over mud a complete second-order solution is deri ved here from the first-order solution of Lowes. 4.3 Problem Formulation Consider the mud-water system of Figure 4. 1, in which water wave propagates in the positive x -direction. Water layer thickness is h , mud layer is d , free-surface displacement is1, interfacial displacement is2, water density is1, water viscosity is1, density of mud is2 and viscosity of mud is2. The two fluids are assumed to be Newtonian, laminar and incompressible, with no interfacial mixing. For a small disturbanc e to this system, the continuity and momentum equations can be derived from the linearized equation of motion (MacPherson, 1980; Kolsky, 1963) as follows from Eqs (4.1)-(4.3) 0jijiuw xz (4.1) 22 22 1jijijiji ej juPuu txxz (4.2) 22 22 1jijijiji ej jwPww tzxz (4.3) where jiu and jiw (functions of x , z and t ), are the horizontal and ve rtical components of the wave orbital velocity, respectively. Subscripts j= 1and j= 2 are for water and mud layers, respectively, and i= 1, 2 denote firstand second-order solutions, respectively. The quantityej is PAGE 76 76 the equivalent kinematic viscosity for water ( j= 1) and mud ( j= 2). The kinematic viscosity of the Newtonian fluid is its mate rial property, given by/ejejj . The dynamic pressure jP is given by Eq (4.4) 0T jjjjPPgzP (4.4) where TP is the total pressure in Eq (4.5) 0 210@1 @2jj P ghj (4.5) Ten boundary conditions are required to solv e the above system of equations [Eqs (4.1)-(4.3)]. At the rigid bottom, zhd , the no-slip and no-pen etration kinematic boundary conditions are, respectively in Eqs (4.6), (4.7). 2 0iu (4.6) 2 0iw (4.7) At the mud-water interface, zh , kinematic and dynamic boundary conditions must be satisfied. The continuity of both vertical and horizontal veloc ities in mud and water requires them to be equal at the interface. Also, there must be equality of normal and shear stresses in mud and water at the interface: 12iiuu (4.8) 12iiww (4.9) 2 12i iiww t (4.10) 12 1122 22TT ii ieieww PP zz (4.11) PAGE 77 77 1122 12iiii eeuwuw zxzx (4.12) At the free water surface, 0 z , the kinematic and dynamic conditions is given by Eq (4.13). 1 1i iw t (4.13) 1 11 20T i iew P z (4.14) 11 1 0ii euw zx (4.15) 4.4 Solution Approach The first-order problem is formulated usi ng the governing equati ons described in the preceding section. The solution is obtained in tw o ways. One is a simplified analytical solution by Ng (2000) for 2d , and other is the full semi-ana lytical solution of Lowes (1993). Second-order Stokesâ€™ (1847) pertur bation approach is then app lied to the first-order semianalytical solution. In this method, it is a ssumed that all variables can be expanded as a convergent power series of a sm all parameter such as the water surface slope. The variables can be written as summations of higher harmonics. 1111213() O (4.16) 2212223() O (4.17) 123jjjjuuuOu (4.18) 123jjjjwwwOw (4.19) 123jjjjPPPOP (4.20) PAGE 78 78 where, 11 is the first-order solution for water surface elevation, and 12 is the second-order solution for the same. Similarly, 21 and 22 are the firstand second-order solutions for the interfacial displacement, respectively. 4.5 First-Order Analytical Solution An analytical solution for the water-mud sy stem was rigorously derived by Ng (2000), based on the asymptotic theory for the kinematics of thin mud (2d ) under waves. His analysis was aimed at completing the soluti on of Dalrymple and Liu (1978). The basic assumption is that the wave amplitude a is comparable to2 , and also that a is much smaller than wavelength, i.e., 2() aOL . The small wave steepness kakdk can be used as the scaling parameter to understand the order an alysis of the basic eq uations (continuity and momentum equations [Eq (4.1)-Eq (4.3)]and the boundary conditions [Eq (4.6)-Eq (4.15)] for obtaining the wave attenuation coefficient. Since mud thickness is considered to be very small, the entire layer is subject to viscous shear. Un der this assumption the boundary layer equations become the governing equations, which are furthe r simplified by scaling analysis. The solution is divided in two parts, one for viscous flow and the other for (inviscid) pot ential flow. The viscous solution is obtained for the mud boundary layer, the potential solution for the water layer, and the two are asymptotically matched to solve for the unknowns. 4.5.1 Scaling Analysis Continuity of flow is given by Eq (4.1). As for momentum conservation, based on the following scaling of variables 1(), xOk ()(), zOdOx 1(), tO (), uOa (), wOa 21 /, POakOga the xmomentum conservation equation [Eq (4.2)] is scaled as follows: PAGE 79 79 22 2 22 1jijijiji ej juPuu txxz (4.21) 2(1)(1)()(1) OOOO This scaling suggests that the third tem, which is the diffusion term in the x -direction, can be ignored. The zmomentum equation [Eq (4.3)] is simplified to Eq (4.22) 2 1 0ji jP O z (4.22) which indicates that the pressure is constant to leading order. Given the local coordinate (pointed upward) for the boundary layernhzd , 0 n specifies the mud bottom, and th e mud-water interface is given by 2. nd The surface displacement , bOka is an order of magnitude smaller than a. The scaled boundary conditions [Eqs (4.6) â€“(4.12)] in terms of this coordinate convention are expanded using Taylor series about nd and are written as in Eqs (4.23) and (4.24) 2 12 1222 uu uuO nn (4.23) 211 OOOOO 2 12 1222 ww wwO nn (4.24) The kinematic boundary condition at the interface requires that 2 12 wOwO t (4.25) Normal stresses are ignored since they are of O . Balance of shear stresses at the interface demands that PAGE 80 80 22 2 1122 1222 22eeuuuu O nnnn (4.26) 2 12 12122212 PP PgPgO nn (4.27) Outside the boundary layer this solution is asymptotically matched with the inviscid potential solution 1111 ,, uPU as nd as zh (4.28) The governing equation for the potential solution is Eq (4.29) 111 1 1 1 UU U txx (4.29) Details of the solution are given in Appendix. The above analytical solution is compared in Section 4.7 with FSM obtaine d in the next section. 4.6 First Order Full Semi-Analy tical Model Solution (FSM) The full semi-analytical model (FSM) solution is obtained under same basic assumptions as for the analytical solution of Ng (2000), i.e., the wave amplitude a , is so small that all terms of 2()rOak can be ignored when compare to the terms of().rOak So, the kinematic and dynamic conditions can be linearized to obtain the firstorder solution. First-order equations are obtained from the governing equations [Eqs (4.1)-(4.3)] in Section 4.2. For doing so it is necessary to replace subscript i with 1 in Eqs (4.1)-(4.15). The surface wave profile is known from 111exp aikxt . Assume solutions of the type PAGE 81 81 211 11 11 11exp exp exp expjj jj jjbikxt uuikxt wwikxt PPikxt (4.30) The amplitudes in the above so lutions are functions of the z -coordinate only. Substituting Eq (4.30) into Eq (4.1) we have Eq (4.31) 1 1 j jdw i u kdz (4.31) From Eq (4.2) we obtain Eq (4.32) 3 11 2 1 23 ejjj jjdwdw P kdzdz (4.32) 22 j ejki (4.33) From Eq (4.3) we obtain a fourth-order ordinary differential equation. 42 11 2222 1 420jj jjjdwdw kkw dzdz (4.34) which has the following solution 1111 1111sinhcosh expexpwAkhzBkhz CzDhz (4.35) 2111 1212sinhcosh expexpwEkhdzFkhdz GhzHhdz (4.36) where 111111111,,,,,,,,, ABCDEFGHkb are unknowns to be dete rmined from the 10 boundary conditions, Eqs (4.6)-(4.15). PAGE 82 82 From the free-surface boundary condition of Eq (4.13), the wave dispersion relationship is given by Eq (4.37) 11111 1sinhcoshexp i AkhBkhCDh a (4.37) First, the complex part, ki of the wave number k , which is the wave a ttenuation coefficient, is obtained iteratively by sett ing the determinant from Eqs (4.6)-(4.15) to zero, and then solving the equations simultaneously to obtain the ot her nine unknowns. After these coefficients are known, the horizontal velocity and pressu re are calculated from the following Eqs (4.38)-(4.41) 11 11 1 1111coshsinh expexp AkhzBkhz ui CzDhz k (4.38) 11 21 2 1212coshsinh expexp EkhdzFkhdz ui GhzHhdz k (4.39) 22 1 11111coshsinhePkAkhzBkhz k (4.40) 22 2 21211coshsinhePkEkhdzFkhdz k (4.41) 4.7 Comparison of Analytical of Ng and FSM Solutions Since the analytical solution of Ng is obtained from an appr oximate dispersion relationship for the water layer of Eq A.19, it is only valid for 2d . The wave number is modified by the presence of mud, as shown from Figure 4.2. Ngâ€™s solution yields a constant wave number, while wave numbers from D&L and FSM vary with mud thickness. In Figure 4.3 (shallow water case), in Ngâ€™s solution, not only does the peak occur 0.5 d later than Gadeâ€™s data, but also dissipation is 17% higher than in data (Table 4.1). In D&L, the peak occurs 1.2d earlier and peak dissipation PAGE 83 83 is 17% higher than in data. The D&L solution begins to deviate from FSM for dimensionless mud depths 22 dd less than 2.5, and the analytical solution is applicable up to 1d . FSM not only patches the two solutions, but foll ows the data trend. It approaches zero when d tends to zero unlike D&L, which approaches a constant value at 0d . In the analytical solution of Ng, peak dissipation always occurs at 21.55 , while the occurrence of peak in data varies from 1.32 -1.52 depending on water depth. Since FSM appears to yield satis factory results based on the fi rst-order solution, it provides motivation for extending the solution to second-orde r so as to examine non-linear wave effects. 4.8 Second-Order Solution To examine wave-mud interaction more clos ely, e.g., to calculate the mass transport velocity in mud, it is necessary to ta ke the solution to a higher order. Higher-order solutions are obtai ned by extending FSM by the pe rturbation approach with respect to wave steepness. In the second-order m odel considered here, wave heights need not be infinitesimally small because a second-order2()rOak contribution to height is present. Second-order governing equations are obtained from Eqs (4.1)-(4.3) by replacing the subscript i by 2. Boundary conditions include forcing from first-order and are as follows. At the rigid bottom, zhd, the no-slip and no-pene tration kinematic boundary conditions are, respectively, 22 0 u (4.42) 22 0 w (4.43) PAGE 84 84 At the mud-water interface, zh , both kinematic and dynamic boundary conditions must be satisfied. These require that both, the vertical and the hor izontal, velocities in mud and water be equal, and also the equality of shear and normal stresses in mud and water: 12221 2111 121 uu uu zz (4.44) 12222 2111 221 ww ww zz (4.45) 22 123 1121 32111 w t w u zx 1121 1121& @uu wwzh (4.46) The condition that normal and shear stresses be equal at the mud-wa ter interface leads to Eqs (4.47) and (4.48). 1222 1212224 2111 421211121 22 22TT ee TT eeww PP zz ww PP zzz (4.47) 12122222 125 21211111 52121 ee eeuwuw zxzx uwuw zzxzx (4.48) At the free water surface, 0z, the kinematic and dynamic conditions is given by Eq (4.49). 12 126 1111 61111 w t w u zx (4.49) PAGE 85 85 12 1217 11 711111 2 2T e T ew P z w P zz (4.50) 1212 18 1111 8111 e euw zx uw zzx (4.51) where 12345678,,,,,,, are forcing terms from the first-order solution. In analogy with the first-order case, Eq (4.30), second-order variab les are taken to be second-order harmonics, Eq (4.52) 122 222 22 22 22exp2 exp2 exp2 exp2 exp2jj jj jjaikxt bikxt uuikxt wwikxt PPikxt (4.52) The amplitudes in the above e quations are functions of the z -coordinate only. Substituting Eq (4.52) into Eq (4.1) we have Eq (4.53). 2 22j jdw i u kdz (4.53) From Eq (4.2) we get Eq (4.54). 3 22 2 2 234ejjj jjdwdw P kdzdz (4.54) 2242j ejki (4.55) From Eq (4.3) we obtain a fourth-order or dinary differential equation, Eq (4.56). PAGE 86 86 42 22 2222 1 42440jj jjjdwdw kkw dzdz (4.56) which has the solution 1222 2121sinh2cosh2 expexp wAkhzBkhz CzDhz (4.57) 2222 2222sinhcosh expexp wEkhdzFkhdz GhzHhdz (4.58) where 2222222222,,,,,,,,, ABCDEFGHab are 10 unknowns, to be determined from the 10 boundary conditions [Eqs (4.42)-(4.51)]. Once the above coefficients are known, sec ond-order components of horizontal velocity and pressure are obtained from Eqs (4.59)-(4.62). 22 12 1 2121cosh2sinh2 expexp 2 AkhzBkhz ui CzDhz k (4.59) 22 22 2 2222cosh2sinh2 expexp 2 EkhdzFkhdz ui GhzHhdz k (4.60) 22 1 121222cosh2sinh2 2ePkAkhzBkhz k (4.61) 22 2 222222coshsinh 2ePkEkhdzFkhdz k (4.62) Finally, the total second-order velocity, pre ssure, surface elevation, etc., are obtained by adding the firstand seco nd-order solutions [Eqs (4.16)-(4.20)]. PAGE 87 87 4.9 Validity of Second-Order Solution A measure of the validity of St okesâ€™ expansion is whether th e series converges (Dean and Dalrymple, 1991). A measure of convergence R is given by Eq (4.63) 2 3 13cosh2 1 8coshsinhj jkakh R khkh (4.63) For deep water (kh ) the above relation simplifies to Eq (4.64) 3exp2 R khka (4.64) In deep water R is very small because ka is small and is large in shallow water (/10kh ), the hyperbolic functions can be replaced by their asymptotic values, so Eq (4.63) becomes Eq (4.65) 2 332333 864 kaLH R khh (4.65) It is difficult to achieve a small value of R in shallow water. For example, in Gadeâ€™s shallow water experiments, R would be 1.2 based on characteristic valu es of experimental parameters in Table 4.1. Given this limitation, for Stokesâ€™ expans ion to be applicable, a criterion based on the Ursell number Ur must be used. Accordi ng to this criterion, Eq (4.66)can be written as: 2 325 LH Ur h (4.66) Stokesâ€™ expansion is valid for Ur less than 25 because it is a measure of wave steepness over water depth, and wave steepness (slope) is th e perturbing parameter. Small wave steepness ensures that the series would be convergent and can be expande d as a perturbation series. For large wave steepness (large Ur ), in shallow water, the wave form develops anomalous bumps in the trough. Ur in Gadeâ€™s experiments was 258, which i ndicates that the Stokesâ€™ perturbation approach cannot be applied to Gadeâ€™s analysis. PAGE 88 88 4.10 Comparison with Data The described models are now compared usi ng Gadeâ€™s (1958) characteristic parameters given in Table 4.1. Figs 4.2 and 4.3 pl ot dimensionless wave numbers /rrkkgh and dimensionless wave attenuation coefficients /iikkgh from FSM, D&L and Ng, and Gadeâ€™s data. FSM, with inherent boundary laye r, effect is able to predict th e trend in data and the peak in data within 20% error (i n Figure 4.3). D&L does not predict the trend for 2(/)1dd , and the peak is 40% higher than data. Ng does predict the trend but the peak is higher by 40% and is shifted by 0.5d. The reason for the differences between th e models is that Gadeâ€™s experiment was conducted for shallow water conditions with 2223.4cm, which is equal to the depth of sugar solution layer he used as the lower fluid, so bounda ry layer effects in the lower layer become quite important, while these are absent from D&L. FSM is next compared with the flume experime ntal data of Sakakiyama and Bijker (1989) (S&B) using a mixture of kaolinite in water as mud below the water layer. Characteristic parameters are given in Table 4.2 and data fr om S&B. In Figure 4.4, FSM follows the general trend of the data, and predicts the wave dampi ng coefficient within 1-2% error for mud with a density of 1,150 kg/m3. However, as mud density increases to 1,240 kg/m3, the peak is overpredicted by 20%, while for 1,370 kg/m3 the peak is under-predicted by 100%. These trends imply that mud behavior is Newt onian at low densities, but as density increases its rigidity becomes important. Also, with increasing mud dens ity increasing stress is required to shear mud, which is no longer a Newtonian fluid. It behaves as a viscoplastic w ith a yield stress that must be exceeded before it can flow. Figure 4.5 is a plot of the yield stress versus density of a kaolinitic mud tested by S&B. The trend show s that for mud density under 1,150 kg/m3, the yield stress PAGE 89 89 was practically zero. At higher de nsities the yield stress increa sed, which means that mud had a residual shear strength which had to be overcome before it could be sheared. The ratio of interfacial am plitude to surface amplitude /ba is plotted in Figure 4.6. This ratio is 2 times greater for mud D as compare to A, which is 1/15 times less viscous than A. The plot also shows the shift of the peak (of ratio b/a ) towards lower frequencies with more denser and viscous mud. For mud A, the peak in b/a is around 4.8 radians/s, while for D the peak is around 1.5 radians/s. The plot illustrates how mud can heave increasing with decreasing density. A set of flume experiments was conducted by Jiang (1993) using two muds, a clay mixture called AK mud, and natural mud (OK) from Lake Okeechobee in Florida. He conducted several tests with different mud densities and water dept hs. Characteristic experimental parameters are given in Table 4.3. The wave attenuation coefficients from Jiang are plotted in Figure 4.7 for a mud depth of 18 cm. An interesting feature is that in the fre quency range of 3-7 radians/s, the FSM result is lower than data by 8-12%, and the peak is unde r-predicted by 15%. However, in the frequency range of 8-12 radians/s the mode l over-predicts data by 1-5%. This implies that at frequencies greater than the peak frequency, at which re sonance occurs, model agreement with data improves. This could be so because at higher freque ncies mud is in a liquefied state and closer to Newtonian in behavior co mpared to its state at lower freque ncies, when the effect of mudâ€™s rigidity is important. The horizontal acceleration amplitude in mud is plotted in Figure 4.8. FSM shows 30% higher acceleration than data, wh ich is related to the under-pred iction of the wave attenuation peak, as wave dissipation and acceleration are inve rsely related. This rela tionship is due to the following reason. Dense mud is le ss compliant than lighter mud, as exemplified in Figure 4.6 by PAGE 90 90 the ratio of interfacial amplitude to surface amplitude, which is 1/2 times lower for mud A than D. Also the measured dimensionless wave a ttenuation coefficient is 12 times higher for A compared to D in Figure 4.4. So it can be inferre d that the interfacial amplitude is inversely related to wave attenuation. The same is the cas e with acceleration in mu d, which is directly related to the interfacial amplitude. 4.11 Mud Behavior from FSM â€œWave-activeâ€ mud in the marine and lacustrine environments is typically much thinner than the overlying water layer, a nd maximum dissipation occurs between 221.31.5 , i.e., of the order of 2 . Thus it is essential to consider boundary layer effects in, and due to, thin mud layers . FSM predicts the peak and trend of wave attenuation (Figs 4.3, 4.4 and 4.7) over the entire range 0d. Following the testing of FSM with three data sets (Tables 4.1, 4.2 and 4.3), applications of FSM can be considered for a greater insight into wave kinematics. Figure 4.9 is a plot of the horizontal wave orbital velocity amplitude, u against dimensionless depth based on characteristic parameters from the experi mental study of Jiang in Table 4.3 (for T = 1.8 s). For this case, the Ursell number varies from 3 for a wa ter depth of 34 cm (if mud is considered to be a fluid) to 23 for a depth of 16 cm (if mud is not considered). So in th is case the second-order solution is applicable. The second-or der velocity in red is seen to be almost 10% higher at the mud-water interface than the first-order velocity . The dashed black line identifies the interface between mud and water. The velocity is of O (10-2) m/s, and there is an overshoot in velocity at the interface due to matching re quirement of the potential flow solution in water and boundary layer solution in mud. PAGE 91 91 The mud kinematic viscosity is 15,000 times that of water (Table 4.3). However, the requirement that the velocities be continuous a nd that the shear stresses must be equal at the interface leads to a very sharp gradient in water velocity. The vertical wave orbital velocity amplitude w corresponding to u is plotted in Figure 4.10. The second-order velocity is 12.5% higher than first-order. As expe cted, the velocity is continuous across the interface, and at the mud bottom the velocity is almost normal to the abscissa. This behavior indicates that not only the vertical velocity is zero (no-penetration condition), but also that the gradie nt of the vertical ve locity is zero, so th ere is no contribution from the vertical component of velocity to bed shear stress. To highlight second-order effects on wave form , the surface elevation profile is plotted in Figure 4.11 based on parameters in Table 4.4, but for a wave height of 20 cm. Second-order effects, characterized by flattene d troughs and peaked crests, are evident. The second-order wave amplitude at the crest is 20% higher than first-order. The phases of the horizontal orbital velocity (r elative to water surface crest) for muds in Table 4.2 are plotted in Figure 4.12. Mud A with density of 1,370 kg/m3 is out of phase by 38o at the interface and 50o at mud bottom. On the other hand, mud D with density 1,150 kg/m3 is out of phase by 2o at the interface and almost 40o at mud bottom. As the density of the mud increases, mud motion is increasingly out of phase relative to water because the higher the density the lesser the compliance of mud with water motion. Figure 4.13 is a plot of u(horizontal orbital velocity amplitude) for different mud densities and viscosities. As viscosity increases from 5x10-4 m2/s to 1.5x10-2 m2/s, i.e., by a factor of 30 (at a constant wave frequency) , the Stokesâ€™ boundary layer thickness 222 increases by a factor of 5.8, equal to the square -root of the ratio of th e two viscosities. The PAGE 92 92 boundary layer thickness (marked by solid black lines ) is representative of the length over which molecular diffusion is dominant. For low-viscosity (5x10-4 m2/s) mud (Curve-4) the diffusion length in mud is only 1.5 cm, wher eas at the highest viscosity (1.5x10-2 m2/s) the diffusion length is 9 cm. An over-shoot occurs in water velocity close to the interface due to the requirement of matching the inviscid flow solution outside the boundary layer with viscous solution within mud. As mud viscosity increases from 5x10-4 m2/s to 1.5x10-2 m2/s, the position of the over-shoot shifts upward from 5 cm above the mud bottom (Curve-4) to the mud-water interface (Curve-1). Correspondingly, the velocity gradient changes from 1.3 to 0.3 s-1. So there is a 77% decrease in the gradient from Curve-4 to Curve-1, but almo st 30-fold increase in viscosity, resulting in a lower shear stress in the lighter fluid (in Curve-1) compared to th e more dense fluid (Curve-4). 4.12 Mass Transport Velocity 4.12.1 Mass Transport Phenomenon Mass transport in mud is caused by an inter action between water and mud movements. The mass transport velocity, often called Lagrangian mass transport (velocity), has two components, a steady-state streaming velocity, also known as the mean Eulerian streaming velocity (ESV), and Stokesâ€™ drift (SD) (velocity) . One of the early relevant studi es of mass transport in a twolayered fluid system was by Dore (1970). Naga i et al. (1984) conducted laboratory experiments to investigate wave attenuation and mass transpor t in mud, and based on their experimental data concluded that mud mass transpor t is proportional to the square of the wave height. In studies by Shibayama et al. (1986) and Tsuruya and Nakano (1987), ESV was ignored. Thus, the mechanism of mass transport in mud was not full y described. ESV is usually larger then the Stokesâ€™ drift component (Figure 4.14), and ther efore, in general, must not be ignored. Sakakiyama and Bijker (1989) (S&B) measured the mass transport velocity in muds of different kinematic viscosities. As part of th e experiment they also measured the wave PAGE 93 93 attenuation coefficient (Figure 4.14) for muds (Table 4.2). They considered both ESV and Stokesâ€™ drift, and compared predicted mass tran sport velocities with measurements. Ng (2000) provided an analytical expression for the mass transport velocity in the thin mud; however, his expression is valid only for 2d . The average velocity of the fluid particle over a wave period (mass transport velocity) is defined as in Eq (4.67) LESUuu (4.67) The particle executes an orbital tr ajectory during a wave period, and particle velocity at the crest is highest, and at the trough the smallest and th e particle at the cres t is moving in opposite direction to the particle, which results in a net fo rward motion of a particle due to Stokesâ€™ drift. The Stokesâ€™ drift component can be written as in Eq (4.68) 00tt EE SEEuu uudtwdt xz (4.68) 21 22E SEEu k uuw z (4.69) where the Eulerian velocity E u is the same as wave orbital veloc ity obtained from Eq (4.18), i.e., the sum of the first-order a nd the second-order solutions. ESV is due to the viscosity of fluid. The time-mean Eulerian velocity Eu was deduced by Longuet-Higgins (1953) for constant wave condi tions. S&B considered the governing equations under more general conditions in order to make them applicable to soft mud as follows. The horizontal momentum equation of any fluid system is given by Eq (4.70) 2E EEE x xxzu uuw txzxz (4.70) PAGE 94 94 If forcing of the system is due to waves, the time-mean form of the momentum equation must be considered. From Eq (4.70), the time-mean equation is obtained as in Eq (4.71) 20EE x xzx Euw u x zxz (4.71) The terms 2Eu and EEuw are components of Reynolds stre ss tensor representing momentum fluxes due to wave fluctuations. ESV arises because a mean stress field , x xxy must exist to balance the Reynolds stresses field: 2xxu p x , zxuw zx (4.72) Substituting from Eq (4.72) for the normal and shear stresses in Eq (4.71), we obtain Eq (4.73) 22222 22EE EEEuw uuu xzxz (4.73) Based on an order of magnitude analysis the gradient in x -direction can be shown to be much smaller than in the z -direction. So Eq (4.73) simplifies to Eq (4.74). 22 2EE Euw u zz (4.74) There are two such equations, one for mud and another for water. The subscript E is replaced by 1 for water and 2 for mud we obtain Eqs (4.75) and (4.76). 22 111 1 2uwu zz (4.75) 22 222 2 2uwu zz (4.76) PAGE 95 95 which are solved numerically to obtain the mean Eulerian velocities, 1u , 2u with four boundary conditions. [Eqs (4.77)-(4.80)] 20u at zhd (4.77) 21uu at zh (4.78) 12 12uu zz at zh (4.79) 1 10 u z outside the boundary layer or at the surface. (4.80) 4.12.2 Comparison with Data The steady streaming component 2EU , Stokesâ€™ drift 2Su and total mass transport 2LU along with S&B data for mud of different densities are plotted in Figures 4.14 and 4.15. The velocities are normalized by 0.00111m/s.LU The model predicts mass transport velocity of mud (of 1,150 kg/m3 density) at the interface w ithin 2-5% (Figure 4.15), but over-predicts by 20% at mid-depth. Near the bottom, the predicted velo city is substantially lower (200-400%) than measured. However, for mud with a density of 1,250 kg/m3, the predicted velocity is 40% higher than observed. The reason for this over-prediction may be that at lower densities mud behaves as a Newtonian fluid. However, as the density incr eases the (upper Bingham) yield stress develops (Figure 4.5). So, viscoplastic mud modeled as a Newtonian fl uid (Figure 4.15 and Figure 4.16) would indicate a greater mass transport than in reality. From linear wave theory and Newtonian fl uid assumption, it can be shown that mass transport velocity (a second-order quantity) is pr oportional to the square of surface wave height. However, in S&Bâ€™s analysis of their laborator y experiments, mud mass transport velocity was related to higher than second power of wave he ight. At high mud densiti es this law was not PAGE 96 96 found to be applicable to data. Since the relationship between st ress and shear rate of mud is nonlinear (Herchel-Bulkley, Fig 5.1), shear stress under large waves is sm aller than the value that would be attained if the fluid were Newtonian. The non-Newtonian behavior of soft mud nece ssitates an examination of mud rheology. Accordingly, in Chapter 5 mud is modeled as a single-phase (viscoelastic) material, and in Chapter 6 as a two-phase (poroelastic) material. In Chapter 6 also, domains of applicability of viscoelastic, poroelastic and other models are discussed in re lation to mud grain size and density. 4.13 Application of FSM to Newnans Lake In this section, FSM is applied to Newnans Lake using characteristic parameters for the lake in Table 4.4. The amplitude of th e horizontal wave orbital velocity u is plotted in Figs 4.17. The maximum u is around 1 cm/s at the mud-water interface. The Stokesâ€™ boundary layer thickness is 2 cm. Stokesâ€™ drift co mponent of the mud mass transport velocity is plotted in Figure 4.18. The mass transport velocity is very small, of the order of 10-5 m/s for a wave height of 5 cm, but has a much larger value of 10-2 m/s for a height of 20 cm. PAGE 97 97 Table 4.1 Gadeâ€™s experimental parameters T (s) a (m) h (m) d (m) 1 (m2/s) 2 (m2/s) 1 (kg/m3) 2 (kg/m3) 1.4 0.01 3.8x 10-2 3.4x 10-2 2.42x10-6 2.6x10-3 859.3 1504 Table 4.2 Sakakiyama and Bijkerâ€™s experimental parameters Data set T (s) a (m) h (m) d (m) 1 (m2/s) 2 (m2/s) 1 (kg/m3) 2 (kg/m3) A 1.0 0.016 0.3 0.09 1.0x10-6 1.5x10-2 1000 1370 B 1.0 0.016 0.3 0.09 1.0x10-6 1.0x10-2 1000 1300 C 1.0 0.016 0.3 0.09 1.0x10-6 0.4x10-2 1000 1240 D 1.0 0.016 0.3 0.09 1.0x10-6 0.1x10-2 1000 1150 Table 4.3 .Jiangâ€™s experimental parameters Data set a (m) h (m) d (m) 1 (m2/s) 2 (m2/s) 1 (kg/m3) 2 (kg/m3) T (s) Ur 1 0.01 0.16 0.18 1x10-6 1.5x10-2 1000 1200 1.1 3 2 0.0125 0.14 0.17 1x10-6 1.5x10-2 1000 1200 2.0 Table 4.4 Characteristic para meters for Newnans Lake T (s) a (m) h (m) d (m) 1 (m2/s) 2 (m2/s) 1 (kg/m3) 2 (kg/m3) Ur for a =0.05m Ur for a =0.01m 1.2 0.05 1.2 0.3 1x10-6 1x10-3 1000 1200 0.29 0.6 PAGE 98 98 Figure 4.1 Schematic diagram of water-mud system Figure 4.2 Dimensionless wave number fr om models and Gadeâ€™s (1958) data z x 1 2 2 Fluid Mu d Wate r 1 h d Bed a b Result from full model Result from Dalrymple and Liu model Result from Ng * Gades experiment data Dimensionless wave number, Dimensionless mud depth PAGE 99 99 Figure 4.3 Dimensionless wave atte nuation coefficient from mode ls and Gadeâ€™s (1958) data Figure 4.4 Wave attenuation coeffi cient from FSM and data of Sakakiyama and Bijker (1989) Dimensionless wave atte nuation coefficient Wave attenuation coefficient (m-1) Wave frequency (radians/s) Dimensionless mud depth FSM PAGE 100 100 Figure 4.5 Yield stress versus density of kaolin itic mud (derived from data presented in Sakakiyama and Bijker, 1989) Figure 4.6 Dimensionless amplitude in mud from FSM and data of Sakakiyama and Bijker (1989) Wave frequency (radians/s) Dimensionless wa ve amplitude Yield Stress (Pa) PAGE 101 101 Figure 4.7 Wave attenuation coefficient from FSM and data of Jiang (1993). Figure 4.8 Horizontal accelerati on amplitude for Jiangâ€™s data a =0.5cm T =1 s, h =16 cm and d =18 cm Wave frequency (radians/s) Wave attenuation coefficient (m-1) Dimensionless mud thickness Wave acceleration amplitude (m/s2) Interface PAGE 102 102 Figure 4.9 Horizontal wave orbital velocity amplitude for test case in Table 4.2 ( T = 1.8 s) Figure 4.10 Vertical wave orbi tal velocity amplitude for test case in Table 4.2 ( T =1.8 s) Horizontal wave orbital velocity (m/s) Depth (m) Interface Vertical wave orbital velocity (m/s) Depth (m) Interface PAGE 103 103 Figure 4.11 Surface amplitude profiles (firstand second-order) for a wave height of 20 cm in Newnans Lake Figure 4.12 Dependence of the phase of wave hor izontal velocity on mud properties based on data in Table 4.2 ( T =1 s) Water Mud PAGE 104 104 Figure 4.13 Horizontal velocity va riation with density and viscos ity for test case in Table 4.4 Figure 4.14 Normalized Lagrangian, Stokesâ€™ drift and Eulerian streaming mass transport velocity for test case Table 2.4 (D). Data of Sakakiyama and Bijker (1989). PAGE 105 105 Figure 4.15 Normalized Lagrangian, Stokesâ€™ drift and Eulerian streaming velocities for test case in Table 4.4 (C). Data of Saka kiyama and Bijker and (1989). Figure 4.16 Mass transport velocity pr ofile and Jiangâ€™s data a=3.0 cm, T =1s, h =14cm and d =17cm PAGE 106 106 Figure 4.17 Horizontal wave veloc ity using characteristic paramete rs from Newnans Lake (Table 4.4) PAGE 107 107 Figure 4.18 Mass transport velocity in Newnans Lake using characteristic parameters in Table 4.4 PAGE 108 108 CHAPTER 5 MUD AS A SINGLE-PHASE MEDIUM 5.1 Introduction A wave profile is modified when an intera ction occurs between it and soft mud at the bottom. The wave and mud form a coupled system , and as a result the constitutive behavior of mud plays an important role in governing the na ture of wave profile. A few decades ago, bottom friction, percolation, wave refrac tion and shoaling were recognize d as the principal mechanisms of the wave energy attenuation. It was commonly accepted that the friction coefficient for waveinduced bed shear stress is equal or close to 0.01 for sandy bottoms, and a value greater than that is applicable to muddy bottoms (without further qua lification with respect to the nature of mud). Beginning in the 1940â€™s, interest grew in unders tanding mud behavior from a fluid mechanical point of view, and experimental investigations were initiated. For in stance, Ewing and Press (1949) reported that soft m ud attenuates the overlying wave under intermediate and shallow water conditions. Lhemitte (1958) reported that bo ttom mud can be transported by waves even in the absence of tidal currents. As we now know, processes with in mud are also responsible for wave attenuation, such as seepage flow through the bed pores and fricti on on the walls of thes e pores. To date, two approaches have been followed to formulate th e wave-mud interaction problem. The first is to treat mud as a single-phase medium. For example, Gade (1958) and Dalrymple and Liu (1978) assumed soft marine sediment to behave as a single-phase viscoelastic material. The second approach is to treat mud as a two-phase medium. Based on the assumption of a rigid and permeable sandy seabed, Reid and Kajiura (1957 ) investigated wave damping analytically. Subsequently, this (two-phase system) approach has been extended to more complex cases (e.g., PAGE 109 109 Liu, 1973; Dean and Dalrymple, 1991; Kim et al., 2000). Further de tails on some of these cases are mentioned in Chapter 6. Mitchell et al. (1972) concluded from experi ments that wave action can lead to bottom slope failure and thereby generate fluid m ud. Doyle (1973) showed that bottom sediment movement varies considerably with depth belo w the water-mud interface, and that wave-induced bottom pressure is considerably altered from that expected over a rigid bottom. Bea et al. (1973) and Bea (1983) reported that storm waves can cause massive mud flows and thereby decrease water depth. Forristall and Reece et al. (1980) and Forristall and Reece (1985) measured wave attenuation and bottom motion as waves traveled from deep to relatively shallow water at platform VV off the Mississippi Delta. A rapi d increase in wave attenuation was observed with increase in wave height, which is not a typi cal feature of refraction. Based on data from Louisiana and Surinam, Wells and Kemp (1986) i ndicated that interacti ons between waves and cohesive sediment might govern transport proces ses in areas of erosion and accretion at open coasts, and cause extraordinary high rates of mud transport, even under relatively weak currents. Jiang and Zhao (1989) examined solitary wave motion over soft fluid mud. It was shown that the wave attenuated much fa ster than the rate determined by Keulegan (1948) over a rigid smooth bed. Ross and Mehta (1990) indicated th at wave conditions required for significant erosion of soft mud beds are more moderate th an conditions required for erosion of more rigid particulate beds. From the above review it is evident that , rather than bottom friction, the constitutive properties of bottom mud play a cr itical role in determination of the degree of wave attenuation. PAGE 110 110 In this chapter, relevant constitutive rheo logical models that ch aracterize mud behavior under stress are summarized. Then, a viscoelastic model is selected and applied to simulate surface wave attenuation and mud mass transport velocity. 5.2 Rheological Models â€˜Rheological modelâ€™ implies a certain consti tutive relationship between shear stress and strain (or strain rate). Based on theoretical considerations or assu mptions, the five basic rheological behaviors include: elastic, viscous , viscoplastic, viscoela stic and poroelastic behaviors. Based on experiments, some research ers such as Jiang and Watanabe (1996) and Isobe et al. (1992) have also proposed empirical models. They essentially modify the bottom shear stress to account for change s in water content and other prope rties of the bed due to cyclic loading. As a relevant example of the behavior of a single-phase system, the viscoelastic model is discussed further, and th e two-phase (poroelastic) mode l is described in Chapter 6. 5.3 Single-Phase System: Viscoelastic Model The elastic versus viscous nature of the ma terial can be characterized by the so-called Deborah number /,eeeDT a ratio of characteristic time intrinsic to the material e , to the characteristic time of the deformation process, an extrinsic quantity Te. For a Newtonian liquid 0eDand for purely elastic medium eD. Cyclic forcing time-scales ( Te) in the marine environment can vary from 10-3 s (turbulence) to 101 s (waves) to 104 s (tide). Arbitrarily taking 1e representative of mud, the Deborah number is found to vary from 10-4 to 103. Thus, in general, both viscous as well as elas tic behavior of mud is important. Basically there are two linear viscoelastic elements: Voigt and Maxwell (Figure 5.2). In simple shear flow the constitutive equation for the Voigt element is given by Eq (5.1) G (5.1) PAGE 111 111 and for the Maxwell element G (5.2) where is the shear stress, is the shear strain, the dot indicates derivative with respect to time, is the viscosity and G is the shear modulus of elasticity. The Voigt model is also known as a Kelvin solid , because the material deforms under the app lication of stress a nd regains its initial state after removal of stress. This behavior is unlike that of a Maxwell fluid , which continues to flow even after stress is remove d. The properties of two models ar e further described in the next section. To investigate the viscoelastic behavior of materials, resear chers have applied both Voigt and Maxwell models. Christensen and Wu (1964) and Abdel-Hady and Herrin (1966) applied a model that was a combination of Maxwell and Vo igt models in order to investigate the creep behavior of clays. Carpenter et al. (1973) applied the concept of viscoelasticity to marine sediments by considering that a material is viscoelastic if the total deformation is calculated from the sum of elastic and viscous deformations. St evenson (1973) presente d a practical method for determining the modulus of a linear viscoe lastic model for submarine sediment. Macpherson (1980) considered mud as a Voig t material characterized by an equivalent viscosity(/)(/),eiG where is wave frequency and 1i . The results of his model showed that depending on th e elasticity and the viscosity of the seabed, wave attenuation can be of the same or higher orde r of magnitude than that due to bottom friction or percolation in a permeable bed. For the viscosity-dominated case, rapid rate of attenua tion can occur whereby waves are almost completely damped within few wave lengths. Hsiao and Shemdin (1980) PAGE 112 112 compared their results with field measuremen ts. They found that the observed wave energy dissipation was predicted reasonably well by the visc oelastic mud model. Suhayda (1986) presented a simplified t echnique based on an empirical nonlinear viscoelastic model to predict wa ve attenuation as well as shear st ress and shear strain profiles in the soil. Vertical soil movement under hurri cane waves was predicted to be up to 0.5 m, depending on the soil shear streng th profile. Motion was predicted to occur in wa ter depths of over 30 m. Bottom pressure was shown to be more out of phase relative to surface wave than the rigid bottom case. Horizontal and vertical mud motions also showed large phase shifts relative to the surface waves. Maa and Mehta (1988) experimentally invest igated the dynamic properties of soft mud beds. They argued that the Voigt and the Maxwe ll elements are possible choices. It was however further argued that the Voigt model is a be tter choice than Maxwell. Based on the linear viscoelastic theory, these inve stigators (1986, 1990) applied the Voigt model to a multi-layered bed in order to calculate the wa ve attenuation coefficient. They also noted that the interfacial shear stress (0.25 Pa) was out of phase by 100o-120o compared to the rigid bottom case. Among other investigators who have st udied similar systems include Dalrymple and Liu (1978), Chou (1989), Piedra-Cueva (1 993) and Jiang (1993). In wave-mud interaction studies involving visc oelastic behavior of mud, researchers have typically adopted a viscoe lastic solid model (Voigt or an equivalent Voigt) to determine the degree of wave attenuation. Some researchers have also used the same solid model to calculate mud mass transport. When the Navier-Stokesâ€™ equation is used as the governing equation for wave-mud modeling, the use of an equivalent solid voids the continuum assumption of the governing equation. So for studying such phenomena as mud mass transport, it is necessary to PAGE 113 113 adopt a constitutive model that is a viscoelastic flui d. In that regard, different viscoelastic models and their properties are summarized next. 5.4 Viscoelastic Analogs Based on the linear viscoelastic assumption, the rheology of a ma terial can be studied as a superposition of mechanical analogs. Four such an alogs Voigt, Maxwell, Jefferyâ€™s and Burgers, are described. The general differential constitutive equation fo r linear viscoelasticity can be expressed as in Eq (5.3) 12012....... (5.3) where is the shear stress, is the shear strain, the dots indicat e the derivatives with respect to time, and i and i are material constants taken to be i ndependent of time to preserve linear behavior. In Eq (5.3), if 0 is the only non-zero parameter, th e equation reduces to Hookeâ€™s law for solids. On the other hand, if 1 is the only non-zero parameter the equation describes a Newtonian fluid with 1 as the dynamic viscosity. Eq (5.3) can be represented by a general mechan ical analog made up of a combination of elastic (solid) spring s and viscous (fluid) dashpots. The for ce on a spring is proportional to strain , and on a dashpot to strain rate . By adopting various combinations of spring and dashpot the above four analogs are obtained. 5.4.1 Voigt and Maxwell Models If in Eq (5.3) 0 and 1 are both non-zero, while the othe r parameters are zero, we obtain one of the simplest viscoelastic model, i.e., Vo igt or Kelvin (Figure 5.2A), which results from a combination of a spring and a dashpot in parallel . Both the spring and th e dashpot experience the PAGE 114 114 same deformation (strain), and the total stress is e qual to the sum of the stresses on each element. The resulting constitutive relationship is given by Eq (5.1). As mentione d earlier, the material recovers its initial state after the removal of stress, so the mo del exhibits the behavior of a solid . The Maxwell model is obtained from a combin ation of a spring and a dashpot in series (Figure 5.2B). All the parameters in Eq. (5.3) are zero except 1 and1 . Both the spring and the dashpot elements experience the same stress and the total strain is the sum of the strains in the two elements. Thus the constitutive relationship is given by Eq (5.2). As indicated earlier, the material continues to flow even afte r the stress is removed, so this is a fluid analog. 5.4.2 Jeffreyâ€™s and Burgers Models Jeffreyâ€™s model is an extension of the Voigt and Maxwell models, and is obtained either by adding a dashpot in parallel with the Maxwell model, or a dashpot in series with the Voigt model (Figures-5.3 A & B). The two analogs are equivale nt in terms of their mechanical behavior. The coefficients1 , 1 and2 are the only non-zero parameters in Eq (5.3): 112 (5.4) An equation of this form was theoretically de rived by Oldroyd (1953), when he investigated the viscoelastic properties of emulsions and suspen sions. It was successfully applied by Toms and Strawbridge (1953) to describe the behavior of polymer solutions. The model is capable of continued flow even after removal of applied stress. Burgers model is also an extension of Voigt and Maxwell models, obtained either by series combination of Voigt and Maxwell models or a parallel combination of two Maxwell models. [Figures-5.4 A & B)]. Both analogs are mechan ically equivalent. The only non-zero parameters in Eq (5.3) are 1 , 2 , 1 and2 : 1212 (5.5) PAGE 115 115 Compared to a two-parameter (Voigt or Maxwel l) or a three-parameter (Jeffreyâ€™s) model, in Burgers model there are four unknowns, so one needs to conduc t several rheometric tests in order to determine these coefficients. It is an equivalent Voigt solid , so it regains its initial state after stress is removed. 5.5 Viscoelasticity Characterization Tests Two types of rheometric tests (Figure 5.5), the static test a nd the dynamic test, are applied to determine the coefficients in Eq (5.3). Theoretical relationships corresponding to these tests can be obtained analytically for different mechani cal analogs. The static test in general includes the creep test at constant stress and a relaxation test at constant strain. Th e dynamic test includes the oscillatory test and the shear wave test. Only the creep and the oscill atory tests are described here. Details on the other two te sts can be found in, for instan ce, Barnes et al. (1989) and Wilkinson (1960). 5.5.1 Creep Test In this test a constant stress is applied to the material. Theoretically, the input stress is assumed to be applied instantaneously. In reality, a finite time is require d for stress application due to inertia effects in the measuring system, and delay in transmitting the signal across the test sample. The time required for input stress to reach a steady value must be short compared to the time over which strain output is to be recorded. There are two stages of strain output. The stress loading stage 0ctt, often called the creep curve. The stress unloading stage ctt is referred as creep recovery or the reco il. The input stress is given in Eq (5.6) 0,0 0,c ctt tt (5.6) For the Voigt model, loading and unloading strain responses are, respectively, PAGE 116 116 01expG t G for 0ctt (5.7) 0expcG tt G for ctt (5.8) where G is the retardation time repr esenting the characteristic time-scale for strain creep. It is easy to show that in the first stage, given suffici ent time, the strain will reach its final value, 0G exponentially, and that viscosit y retards the approach to this final value. In the second stage the strain returns ev entually to the initial value (of zero) (Figure 5.6). For the Maxwell model the strain response is linear with time and is given by Eq (5.9) and Eq (5.10). 0t for 0ctt (5.9) 0ct for ctt (5.10) In this case, there is no final steady strain due to viscosity-in duced continuous deformation. In the first stage, the strain incr eases linearly with a slope of 0 , and in the second stage remains constant (Figure 5.7). 5.5.2 Oscillatory Test There are two modes of oscillatory tests depending on the input oscillation. One is controlled-strain test in which the input is oscill ating strain. The other is controlled-stress test in which the input is oscillating stress. In both te sts the stress always exhibits a phase shift, 02 , ahead of strain. When is zero, the material re sponse is purely elastic, 2 represents a pure viscous material, and 02 is the viscoelastic range. In the controlled-strain test , the input strain and output stress are expressed as PAGE 117 117 0expit (5.11) 0exp it (5.12) where 0 is the stress amplitude, 0 is the strain amplitude and is the frequency of oscillation. The input stress and output strain in a controlled-stress test are given by Eqs (5.13) (5.14).and 0expit (5.13) 0exp it (5.14) The relationship between stress and strain (or strain rate) in the oscillatory test can be stated in terms of an equivalent complex shear modulus and an equiva lent complex viscosity, respectively, according to Eqs (5.15) and (5.16) *G (5.15) * (5.16) *GGiG (5.17) The complex shear modulus comprises of a real part G known as the storage modulus, and an imaginary part G called the loss modulus. The complex viscosity* is represented in Eq (5.18) *i (5.18) The real part is the dynamic viscosity of material and the complex part is a measure of the elastic response of the materi al under oscillatory forcing. For the Voigt model, in both type s of tests it can be shown that GG G (5.19) G (5.20) PAGE 118 118 Similarly, for the Maxwell model 22 222G G G (5.21) 2 222G G G (5.22) 2 222G G (5.23) 2 222G G (5.24) It can also be shown that for the Voig t and Maxwell models the two parameters ,G can be determined from the oscillatory test al one. However, for Jeffreyâ€™s model (with three parameters) and the Burgers model (with four parameters), additional conditions are needed; these are provided by static testing of the material. 5.6 Mud Behavior as a Visc oelastic Solid and Fluid 5.6.1 Solid Behavior Whether soft mud is a viscoelastic solid or liquid is a question relevant to wave-mud interaction. In general, mud in the density range of interest can be a so lid or a liquid, depending on the conditions to which it is s ubjected. In the creep test, when the applied stress is small, mud tends to behave approximately as a solid. This is seen in Figure 5.10, in which a Voigt element, a two-Voigt-element (Figure 5.8) a nd Jiangâ€™s (1993) element (combi nation of Voigt element and a spring in series, Figure 5.9) are fitted to Jiangâ€™s (1993) flume data given in Figures 4.4 and 4.5 of his work. The two-Voigt element duplicates th e data, and the other two models also fit reasonably well within an error of 10% . PAGE 119 119 Buscall et al. (1987) studied the rheology of highly flocculated suspensions. Creep tests using an applied-stress rheometer showed that the material exhibited solid-like viscoelastic behavior at sufficiently small stresses . On increasing the stress there was a change from solid-like to fluid-like behavior over a narro w range of applied stress. Details are discussed in next section. 5.6.1 Viscosity Dependence on Shear Rate The viscosity of mud in the fl uid state can be represented as a power-law. On increasing the stress the floc structure breaks due to shea r between flocs, and as a result the viscosity decreases (Figure 5.11), thus repres enting a shear-thinning behavior. Feng (1992) showed that the viscosity of m ud follows the Sisko (1958) power-law given by Eq (5.25) 1n sc (5.25) where is the constant viscos ity at high shear rate, cs is a measure of the consistency of the material, and n indicates whether the ma terial is shear-thinning or shear-thickening. When 1n, material shows shear-thickening behavior , and when n < 1 it is shear-thinning ( n = 1 is a Newtonian fluid). Feng (1992) conducted rheometric tests on diffe rent muds using a Brookfield viscometer. For most of the muds, there was a shear-thinn ing behavior. For example, for AK mud (a 1:1 mixture of an attapulgite and a kaolinite in tap water) with a density of 1,200 kg/m3 the coefficients were4.4 , 0.76sc and 1.083n (Figure 5.11). 5.6.2 Creep-Compliance under Cyclic Loading Creep tests in applied-stress ( ) rheometers have been used to investigate the viscoelastic properties of the cohesive materials under a range of gradually increasing le vels of applied stress (Davis et al. 1968; James et al . 1987; Williams and Williams 1989). PAGE 120 120 Typical test curves are shown in Figure 5.12, in which creep-compliance ()()Jtt , and y is the apparent yield stress related to the strength of the space-filling soil structure. The apparent yield stress, also cal led upper-Bingham yield stress, di ffers from Bingham yield stress, below which no viscous flow occurs. Uppe r-Bingham yield stress is obtained by a linear extrapolation of the shear-thinning flow curve (betw een stress and rate of strain) at high rates of strain. The intercept of the line on the stress axis is taken as .y The quantity Jo represents instantaneous compliance due to purely elastic response. With increasing time the compliance increases, and is retarded due to the viscous effect. Wheny , thixotropic breakdown of material structure occurs, i.e., the ne twork structure collapses with disappearance of elastic response. The material is no longer a solid; in fact, it is a Maxwell fluid, which indicates that mud is lique fied. It should be noted that y based on creep-compliance is independent of any assumptions concerning the rheo logical behavior. Thus it is not equivalent to the upper Bingham yield stress, which is only an approximate measured of yield, due to the qualitative nature of the extrapolatio n method used to obtain its value. James et al. (1987) used a combination of an applied-stress rheometer and miniature vane geometry to measure the static yield properties of illitic suspensions. An advantage of using the vane geometry is that it avoi ds the wall-slip problem, which occurs in rotational measuring devices such as a cylindrical bob. From the tests, the instantaneous compliance Jo, was plotted against applied stress. It was found that under sm all stresses the behavior of the suspension was predominantly elastic, becoming essentially visc ous or, more like a fluid, at higher stresses. Creep-compliance tests similar to those of James et al. (1987) we re conducted on AK mud in a controlled-stress rheometer (J iang, 1993). Based on these tests, Jo has been plotted against PAGE 121 121 applied stress 0 for four samples with diffe rent solids volume fractionsvs (Figure 5.13). The solids volume fraction is defined as /,vss where is the bulk density of mud and s is the particle density. The curve for each vs is divided approximately by the dashed line representing the yield condition (i.e., a pplied stress equal to yield stress). At stresses belowy , Jo increases very gradually with 0 , with the rate of increase decreasing with increasing vs . As also found by James et al., except for the curve for 0.03,vs Jo is almost independent of0 , which indicates a practically linear viscoela stic response. Once yield commences, Jo increases rapidly with increasing0 . In Figure 5.14, which is based on the0y line of Figure 5.13, when vs exceeds about 0.05, the rise in Jo is dramatic due to the development of a fully particlesupported, space-filling sediment network. Thus, ~0.05 is the space-filling value of vs . Migniot (1968) showed that soft mud under going oscillatory moti on due to waves can behave as a fluid. He indicated th at there is an initial rigidity (upper Bingham yield stress), and after the particulate structure of mud is broken it behaves as a fluid. Mehta et al. (1987) used a mi niature device called a Virt ual Gap Rheometer (VGR) to measure the shear modulus of soft kaolinite se diment beds subjected to monochromatic water waves in a flume. The VGR, which is a modified shearometer, was placed in situ , thus allowing the time-evolution of the rigidity modulus G to be measured as waves worked on the bed. In a similar study, Babatope et al . (1999) improved the capability of the VGR and measured the storage modulus G and the loss modulusG . These two studies are descriptive of the way in which fluid mud is generated from a (solid) mud bed. Chou (1989) conducted measurements of the rheol ogical properties of soft mud made of a kaolinite and a montmorillonite, under oscillatory shear in an applied rate of strain rheometer. He PAGE 122 122 found that mud response depended on the amplitu des of wave-induced strain and mud density. Under intermediate strain amplitudes (up to 0.1) mud responded as viscoelastic material. He showed that the storage modulus G and the loss modulus G were not sensitive to the forcing frequency. The induced shear dur ing rate of strain loading 0rtt was steady rather than monotonically increasing, as would be predicte d by the Voigt model, and a residual stress remained at the end of the test. Accordingly, both the Voigt and the Maxwell model were shown to be inapplicable for the selected mud. 5.6.3 Shear Stresses in Mud Figure 5.15 shows the shear stre ss profile of AK mud (Jiang, 1993), for two wave heights. Since the flow was viscous-dominated (wave Re ynolds number was 1800), the shear stress at the mud-water interface can be taken as /.imuz The mean stress for wave height H = 2.5 cm is around 8 Pa and for H = 6 cm around 15 Pa, both of which ar e an order magnitude larger than apparent yield stress of 0.05 Pa (Figure 5.13), be yond which mud flows. This finding suggests that under cyclic loading the shea r stress is typically much higher than the apparent yield stress, and therefore mud does not ha ve a solid-like response. Figure 5.16 shows the variation of phase of shea r stress (relative to surface wave crest) with depth (the black dashed line represents the mud-water in terface). The phase changes from 150o at the interface, to 40o at the depth of 1 cm below the interface, and then to -40o at mud bottom. The frequent change in the phase with de pth is due to the fact that at the interface the viscosity increases more than 1,000fold, while the shear stresses in mud and water are equal (at the interface). So to adjust to this sudden change in the material property the velocity gradient is very large, which in turn is refl ected in a sudden change in phase. PAGE 123 123 Figure 5.18 shows that the shear stress decrease s with increasing water content, which in turn relates to increasingly fluid-like state of mud. 5.7 Mud as a Viscoelastic Fluid From the above presentation it is seen that mud flows when the applied stress is greater than the apparent yield stress. Also, this tran sition typically occurs at low applied stresses, because the yield stress of soft mud is usually low (<~25 Pa) compared to wave-induced stress. This observation in turn dictates the need to model mud as a viscoelastic fluid, as mentioned earlier. In this section a cons titutive model for fluid mud is derived based on the linear viscoelastic assumption. Chou (1989) illustrated the general difficulty in the utility of simple two-parameter mechanical analogs to mud rheology. Here a three-component analog based on Jefferyâ€™s model (Figure 5.3B) is propo sed. The coefficients1 , 1 and2 in Eq (5.5) are obtained from force balance, i.e., total applied st ress is equated to the sum of stresses from each elemental (mechanical) analog of the modeled system (and sim ilarly, total applied strain is equated to the strain contribution from each el ement). Elements in parallel undergo the same strain, so the stresses are additive, while th e vice versa is the case for elements in series. Under these conditions 1212 1112;; GG (5.26) From oscillatory tests the equivalent visc osity and shear modulus can be obtained (as described in Section 5.3.2.2) as f unctions of frequency. In the wa ve-dominated environment, the testing frequency is replaced by wave frequency. Accordingly, 223 112121 * 22 11 i G (5.27) PAGE 124 124 22 112 22 11 G (5.28) 3 121 22 11 G (5.29) 2 121112 * 22 11 i (5.30) 2 121 22 11 (5.31) 112 22 11 (5.32) The wave attenuation coefficient is obtain ed by replacing the equivalent kinematic viscosity * 2e in firstand second-order semi-analy tical solutions for mud motion derived in Chapter 4. The equivalent viscosity is a comple x number, the real part of which is related to the material property and the imaginary pa rt to the dissipative (loss) property. The above â€œJeffreyâ€™s-bâ€ model was calibrated for a mud depth of 18 cm used in Jiangâ€™s (1993) experiments, with creep te st data also from Jiangâ€™s work (Figure 5.19). The coefficients obtained were11.07 , 121 and 221 . After model calibration it was validated for a mud depth of 12 cm (also used in experiments). Both results are plotted in Figure 5.18. The Maxwell model result is also plotted for 21Pa.s and 300Pa. G With reference to wave attenuation, Jeffreyâ€™s-b and Maxwell models under-predict the wave attenuation coefficient within 6% of error. However, the Maxwell model under-predict s the peak by 11.5%, and the Newtonian fluid model is within 13% of error. The models were then used to predict the mud mass transport veloci ty, plotted in Figure 5.20 for Jiangâ€™s data (with charac teristic parameters from Table 4.3). Jeffreyâ€™s-b is +50% off PAGE 125 125 from measurement at the interface, the Maxwell model is off by +120%, and the Newtonian fluid model is off by +85%. Although Jefferyâ€™s-b pr edicts wave attenuation within +10%, mass transport is off by +50%. For better agreement, it will be necessary to obtain accurate values of the viscoelastic coefficients from rheome tric tests (described in Section 5.5). The main purpose of the present exercise has been to show that a viscoelastic fluid model, such as Jefferyâ€™s-b, Maxwell and Newtonian flui d, can simulate mud mass transport velocity, while solid models is not suitab le on physical grounds. 5.8 Empirical Models Due to the dependence of mud properties on wa ve forcing, assuming these properties to be time-independent often introduces significant limitations in modeling mud response. The essential purpose of empi rically defined models is to allow the shear stress to be modified by a physical property such as density or water content, or the sh ape of the backbone curve. For example, Tsuruya and Nakano (1987) modified the viscosity (and therefore the shear stress) due to water content (Figure 5.17). Figure 5.18 illu strates a basic concept for constructing a constitutive equation. Figure 5.18A is a char acteristic hysteresis loop of shear stress versus shear rate obtained from rheometric test s. This loop can be separate d into two parts, namely, a backbone curve shown in Figure 5.18B and a comp ensatory curve (not shown). An example of the determination of shear stress in accordance wi th the backbone curve is given by Isobe et al. (1992). Huynh et al. (1990) studied the rheology of mud under different loading modes in a dynamic rotary shearometer. Shen et al. (1993) also conducted similar experiments. Both groups of researchers concluded that th e relationship between shear stress and shear rate is significantly complicated. Jiang and Watanabe (1996) and Isobe et al. (1992) discussed empirical rheological models of soft mud under oscilla tory loading based on a large number of measurements. Jiang PAGE 126 126 and Watanabe developed a simple model with m odified shear stress (inc luding the effect of water content) in the Navier-Stokesâ€™ equatio ns and developed a twodimensional (vertical) numerical model for wave-mud interaction. The rheological model of (Jiang and Watana be, 1996) was derived from the well known Gibson equation for consolidation (Gibson et al., 1967). The constitutive equation for soft mud subject to waves is given by Eq (5.33) 001 1G (5.33) where 0G is the initial shear modul us at zero shear strain 0 , 0 is the initial viscosity at zero shear rate, 0 and and are coefficients determining th e shape of the backbone curve. The following empirical relations hips were obtained from Eqs (5.34)-(5.37) 3 maxmax1.00.85.010exp0.75tanh4 TT (5.34) 0.21.051.42 2 max0.450.110.015TTT (5.35) 52.8 0 1.71/[(19.87exp(0.87))]7.265601.0tanh0.58 0.5610 81.77071tanh0.47TT GW T (5.36) 52.8 05.61037.51.31ln24.7exp0.311TWT (5.37) where T is cyclic loading (or wave) period, max0.50.4is the shear strain amplitude, and W (%) is the water content of mud. Jiang and Wa tanabe used this modi fied shear stress [Eq (5.35)] to model wave damping and mud mass tran sport velocity. Results are plotted in Figures 5.21 and 5.22. The empirical model over-predicts measured wave damping by 16-100% (for wave heights of 1-3cm), and under-pre dicts the mass transport by 5-10%. PAGE 127 127 5.9 Comments on Resonance An important observation related to wave damping is the consistent presence of a resonance peak corresponding to a maximum in the wave attenuation coefficient curve (Figure 5.19). Resonance has been reported by several researchers (e.g., Ng, 2000; Piedra-Cueva, 1993; Maa and Mehta, 1998). According to Ng (2000) the peak always occurred at 1.55 d in his analysis, wheredis mud depth made non-dimensional by Stokesâ€™ boundary laye r thickness in mud. This was so, because he considered the wave dispersion relationship to be independent of mud depth. This assumption is tenuous because, in general, the wave number and the wave attenuation coefficient depend on mud depth. Maa and Mehta (1990) attributed the resonance peak to the presence of spring in the Voigt model. They observed that the resonance fre quency was a function of mud thickness, with resonance occurring at lower fr equencies in thicker mud. Pied ra-Cueva (1993) also concluded that the resonance frequency depends on mud thic kness and wave frequency. The right part of the curve can also be attributed to the decay of the wave attenuation due to the deep water. None of the above researchers, except for Ng (2000), were ab le to explain the resonance frequency in quantitative terms. In shallow water, Gade (1958) showed that the peak occurs at about 1.3 d (Figure 4.3) and for intermediate to d eep water, Dalrymple and Liu (1978) found the peak to be at 1.55 d. So one can at least infer that in ge neral the resonance frequency occurs in the range2 2(3.54)/ dN 2 2(4.8)/ d. To examine the significance of the resonance frequency,N dimensional analysis can be carried out using the Buckinghamâ€™s -theorem. For that purpose, six pertinent physical variables PAGE 128 128 involving a minimum of three dime nsions are here selected. Thus , there exist three independent (dimensionless) -groups: 22,,,,NrFkdG (5.38) 123, (5.39) and 22 2,N r Nd G Fkd G (5.40) The resonance frequency is obtained from the first group. The development is based on the work of Li (1998), who examined the resonan ce frequency based on extensional rheology. Mud is subjected to extension due to compression and expansion, or due to tangential forces during cyclic action of waves, (Figure 6.7), resulting in change in geometry. Due to extension, mud viscosities are modified by a factor which is a function of type of extension (uniaxial, biaxial and planar). More details on ex tensional rheology can be f ound in Barnes et al. (1989). However, since the analysis as such does not le ad to the desired result, the development is simplified here as follows. The bed is modele d as a mass-spring-dashpot system (Figure 5.23) forced by wave-induced cyclic pressure w p at angular frequency of : 1 0sinsin cosh()wga p ptt kh (5.41) where a is the wave amplitude, k is the wave number and h is the water depth. The dynamic equation of the system can be written as in Eq (5.42) 00,0,0,sineeemtctktpt (5.42) PAGE 129 129 where me is the mass per unit area, ce is the damping coefficient, is the displacement, the dot denotes time-derivative and two dots denote second deriva tive with respect to time. The solution of the above equation is given by Eq (5.43) 2 0 2 2 21sin2cos 0,sincos 12t DD p tt teAtBt k (5.43) where the resonance frequency is Neekm , the dimensionless frequency is N , 2eecm and 21D . The first term on the right hand side in Eq (5.43) represents exponential decay, and the second term is a harmonic response. As t , the decay term vanishes and the system shows harmonic response given by Eq (5.44) 0 1/2 2 2 2sin 0, 12tpt t k (5.44) where 12tan21 , and the amplitude of displacement is 0 max 1/2 2 2 212 p k (5.45) In the expression for the resonance freque ncy, the stiffness (of mud) given by Eq (5.46) nG eGG k dd (5.46) where G depends on the selected extensional rheological model. For axial extension its value is 3, and for planar extension 4 (Barnes et al ., 1989). The mass per unit area is given by Eq (5.47) 22 0d emdzd (5.47) Therefore, the resonance frequency becomes PAGE 130 130 21G NG d (5.48) For Jiangâ€™s (1993) data (in Figure 5.18) with a mud depth of 12 cm and a shear modulus G equal to 300 Pa, we obtain 7.22N radians/s, which is very cl ose to the data, which show a peak at about 7.4 radians/s. If Eq (5.48) is combined with the first and s econd dimensionless groups of Eq. (5.42), the resonance frequency is given by Eq (5.49) 22NrG k (5.49) which is the same as that given by Yamamoto and Takahashi (1985) for their poroelastic model (described in Chapter 6). From Jiangâ€™s (1993) da ta (with characteristic parameters from Table 4.3) for a mud depth of 12 cm and a shear modul us of 300 Pa, the real wave number would be 5.6 m-1 at a mud density of 1,200 kg/m3. These values give a resonance frequency of 4.0 radians/s. The third dimensionless group on the right hand side of Eq. (5.42) is related to the kinematic viscosity of the mud, and th e resonance frequency is given by Eq (5.50) 2 2 2 2222/NG GV fff (5.50) So at a given viscosity, the resonance frequency depends on the shear wave speed 2. VG in all the three cases. PAGE 131 131 Figure 5.1 Flow behavior of different materials. Figure 5.2 Mechanical analogs: A) Vo igt model, and B) Maxwell model. y B Herschel-Bulkley Bingham plastic Pseudo-Bingham plastic Pseudoplastic or shear thinning Newtonian Dilatant or shear thickening G G A B PAGE 132 132 Figure 5.3 Mechanical analogs: A) Jefferyâ€™s -a model, and B) Jeffreyâ€™s-b model. Figure 5.4 Mechanical analogs: A) Burger s-a model, and B) Burgers-b model. 2 1 G1 2 G1 1 A B 2 G 1 1 G2 G1 1 G2 2 A B PAGE 133 133 Figure 5.5 Rheometric tests (Mehta 2006) Figure 5.6 Creep test plot for Voigt model Figure 5.7 Creep test plot for Maxwell fluid PAGE 134 134 Figure 5.8 Two-Voigt-elements model Figure 5.9 Jiangâ€™s model (Jiang, 1993) 2 G2 G1 G G PAGE 135 135 Figure 5.10 Creep test data for AK mud (Jiang, 1993), and best-fit curves for Voigt, Jiangâ€™s and two-element models Figure 5.11 Viscosity variation w ith shear rate plot of Feng (1992) for AK mud with density 1200 kg/m3 Time (s) PAGE 136 136 Figure 5.12 Schematic diagram showing changi ng creep-compliance response with increasing applied stress (from James et al., 1987) Figure 5.13 Initial compliance versus appl ied stress for AK mud (from Jiang, 1993) PAGE 137 137 Figure 5.14 Yield stress versus solids volume fraction for AK sediment (from Jiang, 1993) Figure 5.15 Shear stress profile for AK mud in Jiangâ€™s (1993) experiment (characteristic parameters from Table 4.3) Shear stress (Pa) PAGE 138 138 Figure 5.16 Phase of shear stre ss based on Jiangâ€™s (1993) data Figure 5.17 Variation of shear stress with water content (f rom Tsuruya and Nakano, 1987) PAGE 139 139 Figure 5.18 Definition sketch for stress-strain constitutive behavior from Jiang and Watanabe (1996). A) Hysteresis loop a nd B) Compensatory loop. Figure 5.19 Wave attenuation coefficient versus frequency plot for AK mud, for d = 18 and 12 cm (characteristic parameters from Table 4.3) A B Wave frequency (radians/s) Wave attenuation coefficient (m1) PAGE 140 140 Figure 5.20 Mass transport velocitie s and Jiangâ€™s (1993) data (characteristic parameters from Table 4.3) Figure 5.21 Wave height variation with distance (from Jiang and Watanabe, 1996) 0 2 4 6 8 10 12 Distance x (m) 4 3 2 1 0Wave height H (cm) W = 171% Cal. Exp. Mass transport velocity (cm/s) PAGE 141 141 Figure 5.22 Mud mass transport velocity profiles (from Isobe et al., 1992) Figure 5.23 One-dimensional spring -dashpot-mass harmonic oscillator 0 0.2 0.4 0.6 0.8 1.0 U (mm/s) 0 0.2 0.4 0.6 0.8 1.0 Depth d / d0 ( d0 = 9.03 cm) W = 171% T = 1.01 s H0 = 2.7 cm Exp. Cal. Us UE PAGE 142 142 CHAPTER 6 MUD AS A TWO-PHASE MEDIUM 6.1 Introduction As an alternative to the treat ment of mud rheology as a single-phase medium, it can be modeled as a two-phase medium with a solid and a fluid phase. The solid phase forms an interlocking particulate frame with elastic beha vior governed by Hookeâ€™s law. The fluid phase (liquid and sometimes gas) occupies pore spaces in which flow is assumed to be governed by Darcy's law. This system is also known as a poroelastic medium. The coupling between the two phases is affected by the physical properties of the soil and the ch aracteristics of the disturbance which initiates flow. In a saturated soil (of presen t interest) undergoing periodic movement there are three damping forces: (1) frictional resistance offered by the motion of fluid surrounding the solids, (2) viscous shear and kinetic friction between s liding dry surfaces (Coulomb friction), and (3) internal friction or hysteresis in the body undergoing non-linear elastic deformation (e.g., Meriam, 1959). In this chapter a re view of poroelastic models is presented, following which the poroelastic bed assumption is us ed to quantify wave attenuation over different beds (gravel, coarse sand, fine sand and mud). Finally, a po roelastic model based on Coulomb friction is described, and model results are co mpared with results from a vi scoelastic model in order to explore the domains of applic ability of different models with respect to bed type. 6.2 Literature Review Different solutions are obtained for the poroe lastic system based on assumptions for the solid skeleton (rigid or non-rigid) and the por e fluid (compressible or incompressible). For a rigid and permeable sandy seabed, Bretschneider and Reid (1954) presented solutions in the PAGE 143 143 graphical form for wave attenuation by bottom fr iction and percolation (as well as shoaling and refraction). For rigid and non-deformable porous beds with isotropic permeability and incompressible pore fluid (governed by Darcyâ€™s law), numerous researchers (e.g., Liu, 1973; Massel, 1976; Putnam, 1949; Reid and Kajiura, 1957; Sleath, 1970), have analytically investigated flow induced by waves leading to the Laplace equati on for pore-water pressure. Darcy's law for unsteady flow is given by Eq (6.1). w peq pq Knt (6.1) where w p is the gradient of pore pressure, q is the discharge velocity, is the dynamic viscosity, pK is the intrinsic permeability coefficient representing the characteristics of the porous medium, is the density of fluid, and ne the porosity of the medium. Massel (1976) included non-linear damping and in ertia in the momentum equation in place of Darcy's law. He concluded that permeability has a negligible influe nce on the pressure distribution in both water and seab ed. The result was essentially the same as that obtained from Eq (6.1). In another approach, heat conduction equa tion for the pore pressure was obtained by Nakamura et al. (1973) and Moshagen and Toru m (1975) based on the assumption that water is compressible and the porous bed is non-deform able. A conclusion from this development was that pore pressure response is strongly depende nt on the permeability of the bed material. The pressure attenuated rapidly with depth and had a phase lag in fine soil. Biot (1941) presented a theory which takes into account th e elastic deformation of the porous medium, compressibility of pore fluid, an d Darcy flow in pore fluid. Yamamoto et al. PAGE 144 144 (1978) carried out an analysis of porous bed re sponse based on Biotâ€™s (1941) theory. They examined bed response to waves in terms of a combination of a fluid and a solid mechanical model, which included effects of pore-water flow, bed volume change and bed deformation. They assumed the bed to be a two-dimensional, semi-infinitely deep medium with homogeneous sediment. Biot (1956) evaluated the linear response of fluid-filled porous media under dynamic loading. He found that dissipative poroelastic wave s exist in addition to the usual elastic waves. The presence of such dissipative waves was la ter confirmed experimentally by Plona (1980) and Mayes et al., (1986). Based on Bi otâ€™s poroelastic theory, Mei a nd Foda (1981) re-derived the equations for wave-seabed inte raction with dynamic soil behavi or. To simplify the analytic procedure, they proposed a boundary-layer approxi mation to solve the problem, rather than an exact, close-form solution. Dalrymple and Liu (1982) improved the poroelastic model in which they included both soil and wave dynamics including the effect of inertia. It was concluded that wave energy attenuation was larg ely due to energy losses in the porous medium rather than from boundary layer losses. Another approach to dynamic soil response was derived by Jeng et al. (1999, 2000), who directly solved the governing equations proposed by Biot, unlike the simplified formulation from Mei and Foda. However, the acceleration term considered in their model was only generated from soil particles, excluding the acceleration of pore fluid. Thus, Jeng et al.â€™s solution is not a complete dynamic solution. Jeng (2001) further i nvestigated wave disper sion based on Biotâ€™s static elastic model. PAGE 145 145 Lin (2001) and Lee et al. (2002) extended Yama motoâ€™s (1983) analy tical framework to a finite bed thickness model. Lee et al. (2002) obt ained an analytical solution by including the influence of Coulomb friction and fluid acceleration on soil response. 6.3 Comparison of Poroelastic Models Depending on assumptions concerning the solid skeleton and the pore fluid, there can be four kinds of poroelastic models: 1) Full dynami c model (Z-M), 2) Drained model (D-M), 3) Consolidation model (C-M), and 4) Coulomb damping model (Y-M ). Details on the assumptions are provided in Table 6.1. The dispersion relation for each model is given by the following equations (Jeng and Lin, 2003) 21 tanh00 coshzzgkkdwu akd (6.2) 22tanhtanhtanhpK gkkdigkkdkd g (6.3) 22(0) tanh(0)sechp z wK p gkkdiukd z (6.4) 1212 2 1211 tanh1sec 1ffffssss TTTaacaac gkkdhkd ikaac (6.5) where wave number k has two parts the real part is kr and the complex part is ki and the coefficients 121212,,,,,,,,,, f fssTTfsTfsaaaaaaccc can be found from Lin (2001), and Lee et al. (2002). The symbolzu denotes the displacement of solid matrix, zw is the displacement of pore fluid and w is the unit weight of pore fluid. The above models can be compared for thei r prediction of wave damping over different non-cohesive beds (gravel, coarse sand and fine sand). Due to work done by waves in a viscous PAGE 146 146 fluid, energy is dissipated, and the mean rate of energy diss ipation per unit time, D is given by Eq (6.6). 222 022h Duwuw ds xxzz (6.6) 2 0 h DgudEdE dsC zdtdx (6.7) 2 2 0 02 8ikx i Dg rrde gHk dE CE dxkdxk (6.8) where is the kinematic viscosity of water, Cg is the wave group velocity, E0 is the undamped wave energy and H0 is the un-damped wave height. Jeng and Lin (2003) compared the four models with respect to a dimensionless damping coefficient ikdefined as in Eq (6.9) 02i D i rk k Ek (6.9) Data used by Jeng and Lin are summari zed in Table 2.3. The coefficient ikagainst dimensionless water depth is plotted for gravel, coarse sa nd and fine sand in Figur es 6.1A, B, and C, respectively. Figure 6.1 shows that wave damping decreases with increasing relative water depth. The investigators concluded that for gravel and coarse sand beds , the damping ratios calculated by D-M, C-M and Y-M were comparable, and th e one calculated from Z-M was 200% higher in the shallow water case. Soil type also a ffected wave damping. The coefficient ikwas of the order of 10-1 for gravel due to high percolation losses, and 10-3 for coarse and fine sands. The coefficient ik, is plotted against dimensionless bed thickness in Figure 6.2, which indicates that wave damping increases with increase in bed thickness. There is a significant PAGE 147 147 difference between the models for coarse sand vers us fine sand. For fine sand, Coulomb friction becomes important and Y-M gi ves a reasonable value of ik, whereas the other models do not appear to be applicable to fine sand. Z-M always predicts high ik (for gravel, ik is about 300% higher than values from the ot her models). The coefficient ik obtained from D-M and C-M converges for all depths in gravel. As grain size becomes smaller (gravel to fi ne sand), Yamamotoâ€™s Coulomb friction model (Y-M) predicts ik within a reasonable range of values (of the order of 10-3), whereas the other models predict ik to be an order magnitude lower than YM. In the next section, this model is applied to clays. 6.4 Coulomb Friction and Poroelastic Model When waves propagate over a seabed, forces due to viscosity and pressure gradient in the pores are transmitted as effective stresses to the solid skeleton. These stresses deform the solid frame. As the frame possesses rigidity as well as compressibility, two kinds of stress waves, shear and compressional, occur. Also, since the pore fluid is compressi ble, compressional waves are transmitted through the pore fluid. As a result , wave energy is dissipated by viscous friction and solid-to-solid friction (Coulomb damping) at points of contact between the grains (Lee et al., 2002). Mindlin and Deresfewicz (1953) st udied soil response to oscillat ory stresses and noted that non-linear elasticity and energy dissipation due to Coulomb fri ction are independent of the frequency of oscillation. As with a linear viscoelastic material, the shear modulus of a poroelastic bed is a complex number defined as in Eq (6.10) 1icGGiGGi (6.10) PAGE 148 148 where G is the shear modulus of the soil skeleton, and iG is the linearized expression of nonlinear Coulomb damping. The quantityc is called the Coulomb energy loss parameter. In general, the shear modulus G for clay varies between 105 and 107 Pa, and for sand between 106 and 108 Pa. Based on experimental investig ations, Lee et al. (2002) assumed c = 0.05 for coarse sand, 0.4 for fine sand and 0.8 for clay. As mentioned previously, Yamamoto and Ta kahashiâ€™s (1985) analysis is based on the assumption of infinite bed thickness. The wa ve attenuation coefficient due to percolation ipk is given by Eq (6.11). sinh22wrp ip rrkK k khkh (6.11) where is the water dynamic viscosity and 2 2 2 21 2 1wN mw mw wNm (6.12) 2 21 2 ;wm r N emGk m n (6.13) in which m is the bulk density of the soil, w is the water density, m is the virtual-mass density of the soil, m is an added mass coefficient, pK is the permeability and N is the resonance frequency. For a quasi-static bed 20N . Hazen (see Lambe and Whitman, 1968) proposed the following empirical relation for perm eability as a function of grain diameters of sand and silt 2 10100pKd (6.14) PAGE 149 149 where d10 is the cumulative 10-percentile grain diam eter. Permeability is higher in sand than in silt because the characteristic mean diameter of sa nd (1 mm) is larger than the diameter of silt (0.01 mm). Figure 6.3 is a plot of wave attenuation coefficient ,ipk due to percolation as a function of wave frequency and permeability for wave data given in Table 6.3. Since the larger the permeability the higher the percolation loss, percolation loss is the highest in sand. The wave attenuation coefficient relate d to Coulomb friction is given by Eq (6.15). 22 2 21 4cosh 1wc iC r Ncg k nGkh (6.15) 1 2sinhr rkh n kh (6.16) The coefficientiCk is plotted in Figure 6.4 against the shear modulus for different soils (data from Table 6.3). Except for the peak in ,iCk which is explained later, wave attenuation is maximum in clays and minimum in sand. This can be explaine d by the fact that compressional and shear wave celerity in coarse sand are much greater than water wave celerity . Thus, damping is insignificant. On the other hand, wave celerity in fine sand is sm aller than in water, thus leading to resonating vibration in the bed (Lee et al., 2002). This phenomenon is more obvious in clays. Higher coulomb friction losses in clay can also be explained in terms of particle size. Sand size is around 0.25-10 mm, while clay is less than 1 m. So for given volume and porosity, the number of clay particles is substantially higher than sand particles. The more th e particles the higher would be Coulomb friction losses, because there would be mo re particle-particle interaction. Thus, losses due to Coulomb friction can be expected to be higher in clay than in sand. PAGE 150 150 Wave damping coefficient peaks in Figure 6.4 are due to resonance. Tzang et al. (1992) found that a large increase in excess pore pressure, i.e., pore pr essure above its hydrostatic value, was accompanied by a substantial jump at a certain wave frequency, which suggested liquefaction of the bed. It was fu rther suggested that the peak in wave attenuation was due to a significant amplification in the pore pressure amplitude for given wave frequency, which is related to the cavity structure of the soil. When the frequency approaches the natural frequency of a cavity, internal resonance occurs, which in turn leads to liquefaction. Yamamoto and Takahashi (1985) defined the natu ral frequency as given by Eq (6.13). For a given wave number and bulk density of soil, the resonance frequenc y was higher in sand than in clay, because the shear modulus of sand is hi gher than that of clay. 6.5 Domains of Applicability of Constitutive Models Domains of applicability of the different cons titutive models in relation to bed properties are qualitatively illustrated in Figure 6.6. One may consider a relevant non-dimensional parameter characterizing the bed to be the ratio of the (Stokesâ€™) settling velocity of the particles ws to bed permeability Kp s pw U K (6.17) in which ws is a function of grain size, and permeabil ity also depends on porosity. So the above ratio is measure of grain size to Darcy flow velocity within the pores. This ratio is estimated for different soils in Table 6.4; it is the highest for sand (90) and the lowest fo r clay (1). High values of U % for sand indicate that there are a large differe nces between the settl ing velocity and pore fluid velocity. Since high settling velocity connotes large particle size, a high value of U % is indicative of a two-phase medium . For clays, the low value of U % implies that clay-water mixture PAGE 151 151 behaves as a single-phase medium, as the size of solids is small enough to result in Stokesâ€™ settling velocity to be equivale nt to the pore fluid velocity. In gravel and sand, wave dissipation is ma inly due to percolation and bed friction. Coulomb friction loss is less importa nt in sand, and even less so in gravel. For silt, since the characteristic grain size is 10 m, which is in between sand (100 m) and clay (1 m), losses due to both the percolation and Coulomb friction can be expected to be important. The surface solid volume fraction for clay is higher than the mud. As indicated by Jiang and Wata nabe (1996), waves induce stre sses in two ways, i.e., by pumping and by shaking. In pumping, compression a nd expansion of soil due to wave pressure gradient in the direction normal to the bed occurs. Shaking refers to tangential action by oscillatory bottom shear stresses and wave pressure gradient in the direction of wave propagation (Figure 6.8). Under pumping and shaking, inter-p article bonding may begin to break. When this occurs the effective normal stress v begins to decrease. Li quefaction occurs when v becomes zero: 0vvu (6.18) where v is the total normal stress and u is the excess pore pressure. For a bed to be modeled as a poroelastic medium it must possess effective st ress, as without it there would be no grain-grain contact. Feng (1992) measured both total pressure and pore water pressure in a flume to examine the liquefaction of mud by waves. The rate of liquefaction was show n to vary with wave energy. Also, the duration of mud conso lidation prior to initiation of wave action was an important factor. In general, bed shear stre ngth is reduced with the cyclic loading, and if the effective PAGE 152 152 stresses eventually become zero mud behaves as a fluid and cannot be m odeled as a poroelastic medium. In Figure 6.5, wave attenuation coefficientik from the poroelastic (Eq 6.15) and viscoelastic models, and data fr om the work of Sakakiyama and Bijker (1989) (Table 4.4) are shown. Values of ik from the poroelastic model are within the range of 0.07-0.18 m-1, which is also the data range. Plots are show n for selected ranges of porosity ne (0.4 to 0.8), and shear modulus of elasticity G 500 Pa. The bulk density is a function of porosity and can be obtained if the solid density is known. As noted earlier, YM assumes infinite bed thickness. Results from V-M (for finite bed thickness) from Chapter 4 are also plotted. Y-M pr edicts wave attenuation peaks ranging from 0.07-0.19 m-1, and V-M between 0.02-0.1 m-1. The data show peak values in the range of 0.02-0.125 m-1. Closer comparisons cannot be obtained because the porosity and the shear modulus of soil are unknown. Accordingly, repr esentative ranges of these parameters have been selected to show that both Yâ€“M and VM are able to predict the wave attenuation coefficient within these parametric ranges. Y-M pr edicts a constant wave attenuation coefficient at zero frequency, while the value from V-M r eaches zero indicating that there is no dissipation when there are no waves. The behavior of Y-M in this regard (i.e., non-zero damping) may be due to the fact that it is for infinite bed thickness (as in the case of D&L, which also predicts a constant wave attenuation coefficient as mud th ickness tends to zero because the model assumes a very deep mud layer) and is not app licable for shallow water depths. From these observations it can be conclude d that both Y-M and V-M can predict wave attenuation that is generally co mparable to data. Unfortunately, deviations from data cannot be estimated as exact mud properties are not known PAGE 153 153 Fluid mud has a characteristic density range of 1,050-1,200 kg/m3 (Table 6.5). Depending the sediment (inorganic versus organic), the corresponding concentr ation can be expected to be in the range of 90-350 kg/m3. The ratio of floc volume fraction (volume of flocs divided by total volume) to solids volume fractionvs (volume of the solids divided by total volume) can range from 1.5-3.3 (Mehta, 2006). The floc volume fr action can be obtained from the measurements, but for simplicity of analysis in regard to th e domains of applicability of constitutive models, vs will be used in the following analysis. The solids volume fractionvs is 3 to 12% for the above dens ity range. Since the values are very low compare to the total volume, the part icles (or the solids) ar e not in continuous or contiguous contact. So there can be no significant loss due to Coulomb friction, and the poroelastic model would not be applicable. Also, at such low concentrations mud suspensions can be treated as single-phase media. The domain of applicability of the viscoe lastic model depends on mud viscosity and density (solids volume fraction). Typical variation of yield stress with mud density is plotted in Figure 4.5 from the work of Sakakiyama and B ijker (1989). For density lower than 1,150 kg/m3 the yield stress is zero, indicating that mud is a fluid (and can be modeled as a Newtonian fluid). This inference is supported by the wave attenuati on coefficient plot (Fig ure 4.4) for different density muds modeled as a Newtonian fluid (FSM is able to predict the wave attenuation coefficient within 1-5%). For densities in the range of 1,150 to 1,200 or 1,300 kg/m3 (depending on mud composition), vs is low (3-12%) but because the floc volum e fraction is higher (4.5-36%), there is a measurable yield stress. So it is important to incorporate the appropr iate rheology, and mud must be modeled as a viscoelastic (rather than Newtonian) fluid. At densities greater than 1,200- PAGE 154 154 1,300 kg/m3, mud starts to form a space-filling networ k and the solid (elastic) behavior becomes measurable. The solids volume fraction vs range is 3-36%, so the por oelastic model would be applicable. At densities higher than about 1,300 kg/m3 mud hardens and is no longer compliant to wave action. The utility of a rheological model also depe nds on its suitability to predict mud mass transport. Based on the laboratory observations , Lhemitte (1958) indicated that mud may loose its shear resistance due to surface wave penetration to the bottom, and therefore mud transport may occur by waves even in the absence of a stea dy (e.g., tidal) current. As noted in Section 4.12, researchers have measured mass transport in mud. However, mud with density greater than about 1,300 kg/m3 cannot show mass transport as it posse sses yield stress. In such mud if mass transport is found to occur, one may infer that this behavior is due to cyclic loading. As a result of loading, the floc structure breaks down and the viscosity of th e mud decreases as mud exhibits a shear thinning behavior, even as the de nsity may remain unchanged (Section 5.6.1). The poroelastic model does not predict mud mass transport; because mud is modeled as a two-phase medium with a solid skeleton and a solid cannot have mass transport. The poroelastic model is applicable to predic tion of wave attenuation over the entire range of (solid) bed material from grav el to clay. In gravel and clay the order of dissipation due to percolation is higher than dissipa tion due to Coulomb friction. The application of poroelastic model to fluid mud (with density less than 1,200 kg/m3) is inconsistent with the nature of fluid mud (a single-phase medium), as there are no sustained particle-particle stresses. On the other hand, the viscoelastic model is only applicable to clays. As for sand and silt, the settling velocity to permeab ility ratio is high, indicating that a sand bed must be treated a two-phase medium. PAGE 155 155 Table 6.1 Comparison of poroelastic models Model/ Properties Full dynamic Drained behavior Consolidation Coulomb-damping Model (Z-M) (D-M) (C-M) (Y-M) Investigators Zienkiewicz et al. (1980) (Dean and Dalrymple, 1991, Kim et al., 2000). (Yamamoto et al., 1978; Madsen, 1978; Jeng, 1997). Yamamoto (1983), Lin (2001); Lee et al. (2002). Assumptions Incompressible and irrotational flow in water; linearized free surface boundary condition; small loading wave amplitude Rigid seabed; rigid porous medium; incompressible pore fluid Large timescale; all accelerations negligible Coulomb damping Wave dispersion relationship Eq (6.2) Eq (6.3) Eq (6.4) Eq (6.5) Table 6.2 Characteristic parameters for model comparison Parameter Sand Gravel Coarse Sand Fine Sand Soil permeability ks 10 1 10 2 10 4 Shear modulus G (Pa) 57 107 56 Porosity ne 0.4 Particle density s 2650 Density of pore fluid w 1000 Thickness of bed d 0.1 L (wave length) wave period T (s) 10 water depth h (m) 15 Source: Jeng and Lin (2003) Table 6.3 Characteristic parameters for poroelastic beds with Coulomb friction Parameter Coarse Sand Fine Sand Clay Coulomb friction loss c 0.05 0.4 0.8 Porosity ne 0.4 Particle density s 2650 Density of pore fluid w 1000 Added mass coefficient m 0.25 wave period T (s) 1s water depth h (m) 0.3 Shear modulus G (Pa) 500 PAGE 156 156 Table 6.4 Dimensionless velocity for different soils Soil Grain size (m) Stokesâ€™ settling velocity s w (m/s) Permeability pK (m/s)* s pwK Coarse sand 1.00 -3 9 -1 1.00 -2 90 Fine sand 1.00 -4 9 -3 5.00 -4 18 Clay 1.00 -6 9 -7 1.00 -6 1 .* Kp values are taken from Ta ble-1 of Lee et al. (2002) Table 6.5 Fluid mud density and concentration Investigators Density (kg/m3) Concentration (kg/m3) Inglis and Allen (1957) 1,030-1,300 10-480 Krone (1962) 1,010-1,110a 10-170 Nichols (1985) 1,003-1,200 3-320 Kendrick and Derbyshire (1985) 1,120-1,250a 200-400 Hwang (1989) 1,002-1,065b 4.4-120 Mean range 1,033-1,185 4.6-300 Mean range without Hwang (1989) 1,040-1,215 56-340 Source: Mehta (2006) PAGE 157 157 Figure 6.1 Dimensionless wave attenuation coe fficient as a function of dimensionless water depth A) gravel, B) coarse sand, C) fine sand (from Jeng and Lin, 2003) h/L h/L A B PAGE 158 158 Figure 6.1 Continued Figure 6.2 Dimensionless wave attenuation coe fficient as a function of dimensionless bed thickness A) gravel, B) coarse sand, C) for fine sand (from Jeng and Lin, 2003) h/L d/L C A PAGE 159 159 Figure 6.2 Continued d/L d/L C B PAGE 160 160 Figure 6.3 Percolation loss: wave attenuation coefficient as a f unction of wave frequency and permeability Figure 6.4 Coulomb friction loss: wave attenuation coefficient as a function of elastic modulus G (Pa) Wave attenuation coefficient due to coulomb friction Wave attenuation coefficient due to Wave fre q uenc y ( radians/s ) PAGE 161 161 Figure 6.5 Wave attenuation coeffi cient variation with frequency fo r viscoelastic and poroelastic beds Wave fre q uenc y ( radians/s ) Shallow water limit PAGE 162 162 Figure 6.6 Domains of applicability of cons titutive models (revised from Mehta, 2006) Figure 6.7 Bed liquefaction mechanisms (a dapted from Isobe and Watanabe, 1996) Kp is the permeability coefficient, v is effective stresses 0.1 0.2 0.5 1 2 5 10 20 50 100 Uwk = ws/Kp 0.01 vs =1n 0. 1 0. 7 Viscoelastic solid Viscoelastic fluid Poroelastic solid Viscous fluid Clayey mud Silty mud Sand Viscoplastic Porous static bed Surface solids volume fraction Low porosity static bed â€™v Kp â€™v PAGE 163 163 CHAPTER 7 WAVE-MUD ISSUES IN NEWNANS LAKE 7.1 Wave Damping Parameters influencing wave damping depend on the type of bed. Therefore, in order to characterize the behavior of the wave dissipation coefficient, it is useful to express these parameters in the dimensionless form. Wave damping is characterized byD, the mean rate of energy dissipation per unit time. This quantity can be expressed in terms of wave energy E0 according to Eq (7.1). 2 2 02 8ikx i Dgg rk dEdEgHe CCE dtdxk (7.1) 02i D i rk k Ek (7.2) 0eikxaa (7.3) Given the wave attenuation coefficient ,ik the damped wave amplitude can be calculated from Eq (7.3), where 0a is the un-damped amplitude, and x is the distance traveled by the wave at which it damps to amplitude a . From analysis and comments in Chapters 4, 5 and 6, it can be concluded that ik depends on the type of bed, and therefore on the rheological model (viscoelastic or poroelastic) used. In general, the dependence of ik on factors affecting the wave attenu ation can be stated as in Eq (7.4). ,,,,,,,,,,,,,,,,n irswwmmpsvkfakghkdGGKw (7.4) in which a is wave amplitude, is wave frequency, rk is wave number, g is acceleration due to gravity, h is water depth, ks is bottom roughness height, w is density of water, w is dynamic PAGE 164 164 viscosity of water, d is mud depth, m is mud density, m is mud dynamic viscosity, represents rigidity effect on viscosity, G is loss modulus and G is shear modulus of mud. Subscripts w (water) and m (mud) are used in place of 1 and 2 in Chapter 3. The underlined variables in Eq (7.4) are for mud as a vi scoelastic medium (fluid ), and the double-underlined variables are for mud as a poroe lastic medium (bed with pe rcolation and Coulomb friction losses), pK is permeability, s w is particle settling velocity and v is effective stress. According to the -theorem, there are 17 variables with 3 minimum dimensions are required to describe these variables. So there exist 14 independent dimensionless -groups 12314,,.... (7.5) 2,,,,,,, 2 ,,,,,mms rww wwmm i i r s mp mvmk hdd gkha k k k ghgd w G GKG G (7.6) The first -group is the dimensionless wave attenuati on coefficient defined in Eq 6.9, the second group2/rgkis a measure of wave dispersion, the thir d group is the ratio of water depth to the Stokesâ€™ boundary layer in water 2www , the fourth group is re lative water depth, the fifth group is the ratio of mud thickn ess to Stokesâ€™ boundary layer in mud 2mmm , the sixth group is the specific gravity of mud par ticles, and the eighth group is the relative bed roughness. The shear Mach number for the viscoelastic bed, ,vmMaghG is the ratio of the shallow water wave celerity to shear wave velocity in bed. The ratio /m represents loss of mud viscosity due to elas tic properties of mud (nor malized by viscosity) and /GG is the shear loss modulus normalized by the elastic shear modulus. The quantity p vmMagd is the PAGE 165 165 Mach number for a poroelastic bed. Bed gr ain size and porosity are characterized by ws/ Kp, and /vG is the ratio effective stress to shear modulus. The wave orbital velocity, ub is obtained from wave amplitude a , wave number kr and wave frequency . The, wave Reynolds number, Rew is then obtained from Eq (7.7). 2Rewb w wu (7.7) For a rigid bed with a laminar wave boundary layer, iktakes the form in Eq (7.8). 2i i rw wwk hh k k (7.8) From the work of Dean and Dalrymple ( 1991), the above function is derived in Eq (7.9). 22 2sinh2rw i rrk k khkh (7.9) Due to energy dissipation the wave am plitude diminishes with distance x , as given by Eq (7.3). Combining Eqs (7.9) and (7.3). 2 02 exp 2sinh2rw rrk aax khkh (7.10) For a rigid bed with a tur bulent wave boundary layer, ikcan be written as in Eq (7.11). ,, s s i w wwkk hh k aa (7.11) where the relative roughness height is used to determine the friction factor f . The corresponding wave amplitude has been derived by Dean and Dalrymple (1991) as in Eq (7.12). PAGE 166 166 0 2 021 1 32sinh2sinhr rrra a f kax khkhkh (7.12) It should be noted that the exponent ial function in Eq (7.3) is requi red to be expanded as a Taylor series to obtain the above expression. For a soft bed described by the viscoelastic model, the dimensionless wave attenuation coefficient is given by Eq (7.13). 2,,,,,,,,mm iv rwmwwmdhdG kMa gkhG (7.13) where/()/mmmww incorporates the buoyancy e ffect. For mud modeled as a viscoelastic fluid with typically low rigidity, we may further assume that/m and /GG are both unimportant. The shear Mach number can be taken out for the same reason. Also, since most dissipation occurs in mud, we may ignore /.wh Finally, with water as the upper medium,w can be taken to be constant, and sincem determines,m the ratio/mw may be dropped. Equation (7.13) then reduces to Eq (7.14). 2,,,mm i rwd k gkhd (7.14) The Eq (7.14) can be used as a basis of comp arison for various models (Table 7.4) For a poroelastic bed, based on similar arguments we obtain Eq (7.15). 2,,, s ip rpw d kMa gkhK (7.15) in which Map is obtained by conveniently replacingv by G . Eq (7.15) can then be used to compare various poroelas tic models. (Table 7.5) PAGE 167 167 7.2 Wave Damping in Newnans Lake The wave attenuation coefficient values ar e now calculated from different causes of dissipation including a laminar boundary layer, a turbulent boundary layer, a viscoelastic bottom and a poroelastic bed using parameters character istic of Newnans Lake given in Table 7.1. The calculated wave attenuation coeffi cients are given in Table 7.2. For an assessment of these values, we note that the percent reduction in wave amplitude and wave energy are given by, respectively in Eqs (7.16) and (7.17). 0 0100aa a a (7.16) 22 0 2 0100aa E a (7.17) As an illustration of the order of magnitudes of a and, E they are calculated for Newnans Lake at the tower site, and are compared with measurements. To examine the effect of wave damping at the tower site, the longest fetch ( x ) of 5.5 km and a mean water depth h = 1.5 m (between the northern shore line of the lake to the tower) are selected as characteristic values. With these values, wave dissipation over different beds is estimated using additional characteristic quanti ties given in Table 7.1. We will ignore wave generation and consider only diss ipation to occur over the 5.5 km distance. For a wave period of 1.2 s ( = 5.24 radians/s), a wave height 2 a = 5 cm at the tower would (hypothetically) have a height of 2 a0 = 5.003 cm at 5.5 km distan ce upwind of the tower if dissipation were due to a laminar boundary layer, as given by Eq (7.10). This in turn would mean that there would have been less than 1% decay of wave height over the (assumed) rigid bed. PAGE 168 168 Wave decay due to a turbulent boundary layer is given by Eq (7.12). For the lake bottom, which is covered with vegetation, we will assume a friction factor fw = 0.1. In this case the hypothetical upwind wave height w ould have been only about 1.3%. Now let us consider wave energy dissipati on due to fluid mud. The wave attenuation coefficientik obtained from FSM, described in Chapter 4, as a function of water depth for various fluid mud thicknesses is pl otted in Figure 7.1. Also,ik is plotted against mud depth for different water depths in Figure 7.2. The corresponding wave number rk as a function of water and mud depths is plotted in Figs 7.3 and 7.4, respectiv ely. In Newnans Lake, the 1.5 m mean depth includes the thickness d of fluid mud layer. For a sample ca lculation, this fluid mud is assumed to have a typical thickness of d = 30 cm. So the effective water depth becomes h = 1.2 m. For a fluid mud of density m =1,200 kg/m3 and kinematic viscosity vm = 10-3 m2/s, ik would be 1.5x10-4 m-1, and the 5 cm wave height (=2 a ) at the tower would be 10.5 cm upwind, as calculated from Eq (7.3). This would mean a wave decay a / a0 = 0.53, or 53%, at the tower. The change of wave height against distance from the to wer for rigid and soft beds is plotted in Figure 7.5. Corresponding changes in wave en ergy are plotted in Figure 7.6. Percent reductions in wave amplitude a and wave energy E are given in Table 7.2. Similar to above estimation, ik can be evaluated for an a ssumed poroelastic bed using YM (described in Chapter 6). The resulting values of ik are found to be of the order of 10-6 m-1 due to dissipation over an assumed ri gid bed and of the order of 10-4 m-1 over a soft bed. Wave amplitude decay for a rigid bed would be 0.5-1.3%, whereas over a soft bed would be in the range of 60-90%. Likewise, the re spective wave energy dissipation rates would be about 1% over the rigid bed and 85-95% over soft bed. These calculations strongly emphasize the need to account for the correct description of bed ch aracter in wave height determinations. PAGE 169 169 Dimensionless parameters characteristic of Newnans Lake are given in Table 7.3. The ratio of Stokesâ€™ boundary layer to water depth, /wh is 4x10-4, indicating that boundary layer thickness in water is very small, and may be ignored in wave-mud interaction analysis. In contrast,/md is of the order 1, so the boundary layer in mud must be accounted for. The wave Reynolds number in waterRewis only 0.5 at a low wind speed of 2 m/s and 1.5 for a high wind speed of 8 m/s. This implies that the boundary is laminar, i.e., there are no losses due to turbulence. The Mach number v M a is around 6.8. This means that the speed of waves in water is 7 times greater than that in the fluid mud layer, which is a refl ection of the significance of the role of bottom mud as an energy dissipater. Since the speed of wave in the fluid mud bed is an order of magnitude smaller than th e speed of wave in water, the effect of dissipation due to the wave propagation in the bed cannot be ignored. The ratio /GG (loss modulus divided by shear modulus) is around 0.05, indicating that the loss modulus is sma ll compared to total shear modulus. Values ofG and can be estimated from Eqs 5.34 and 5.31, respectively. The quantity /m (ratio of loss due to elasti city of mud to mud viscosity) is very small (0.004), which implies that the soft mud layer does behave like a fluid rather than a solid. The total pore pressure, wp measured in the lake can be obtained from the Figure 2.22. The excess pore pressure is then found from Eq (7.18). ()wwupghd (7.18) The total stress v is equal to the total force per unit area applied at the level of the pore pressure gage. So v is equal to the hydrostatic pressure and the weight of soil per unit area above it. The pore pressure gage was 20 cm inside the fl uid mud layer with a bulk density of 1,100 kg/m3. So the total stress v = 18.8 kPa, was obtained from Eq (7.19). PAGE 170 170 ()vwmghdgd (7.19) The effective stress is given by Eq (7.20). vvu (7.20) Characteristic effective stress va lues for the lake are plotted as functions of wind speed in Figure 7.7. This plot is obtained from Eq 7.19 and mean pore pressure data in Figure 2.22. As the wind speed increases it exerts higher bottom shear stresses which can break the soil matrix and result in a reduction of effective stress. When the soil structure is completely broken it is no longer able to take any effective stresses and is liquefied (Chapter 6). Li quefaction is seen to occur in the lake at the wind speed higher than 7 m/s. So the bottom cannot be modeled as a poroelastic bed in gene ral. The Mach number p M a for a wind speed of 5 m/s (when effective stress is 0.2 kPa) is 2, indicating that the wave sp eed in the bed is compar able to the shear wave velocity. PAGE 171 171 Table 7.1 Characteristic para meters for Newnans Lake T (s) a (m) h (m) d (m) G (Pa) w (Pa.s) m (Pa.s) w (kg/m3) m (kg/m3) Kp (m/s) c 1.2 0.05 1.2 0.3 300 1x10-3 1x10-3 1000 1200 1x10-8 0.8 Table 7.2 Wave attenuation coefficient and percen tage reductions of amplitude and wave energy Laminar boundary layer Turbulent boundary layer Percolation loss Coulomb friction loss Viscoelastic (Newtonian) Viscoelastic (3-component) ik (m-1) 1.06x10-6 2.64x10-6 5.6x10-5 4.6x10-4 1.5x10-4 2.0x10-4 a (%) 0.5 1.3 25 90 53 63 E (%) 1.1 2.6 43 99 78 87 Table 7.3 Characteristic dimensi onless numbers for Newnans Lake /dh /wh /md ()/mww /mw Rew v M a /GG /m p M a 0.25 0.00041 0.067 0.2 0.001 1.5 6.8 0.05 0.004 2 Table 7.4 Comparison of two-layer models w.r.t. dimensionless numbers for viscoelastic model Models/ Dimensionless numbers 2 rgk dh md mw Gade No dispersion (shallow water) Mud boundary layer effect no included D&L Mud boundary layer effect no included FSM From Eq (7.14) PAGE 172 172 Table 7.5 Comparison of two-layer models w.r.t. dimensionless numbers for poroelastic model Models/ Dimensionless numbers 2 rgk dh p K Reid and Kajiura, (1957) kr only a function of water depth Not included Yamamoto and Takahashiâ€™s (1985) kr only a function of water depth infinite Not included Lee et al. (2002) kr function of both water depth and sea bed thickness Not included From Eq (7.15) PAGE 173 173 Figure 7.1 Wave attenuation coeffici ent as a function of water depth Figure 7.2 Wave attenuation coefficien t as a function of mud thickness PAGE 174 174 Figure 7.3 Wave number as a function of water depth Figure 7.4 Wave number as a function of mud thickness PAGE 175 175 Figure 7.5 Change in wave height (over a rigi d bed and over a soft mud bed) with distance Figure 7.6 Change in wave energy (over a rigid bed and over a soft mud bed) with distance PAGE 176 176 Figure 7.7 Effective stress as a function of wi nd speed, where stars represent the effective stresses calculated from the pore pressure data in Fig 2.22 Liquefaction PAGE 177 177 CHAPTER 8 SEDIMENT ENTRAINMENT IN NEWNANS LAKE 8.1 Introduction In this chapter the mechanism by which muck entrains into the water column due to wind and waves in Newnans Lake is briefly exam ined. Specifically, the cause of the observed oscillations in SSC (e.g., Figure 3.7) is of primary interest. Mo des of entrainment are briefly reviewed, and they are consider ed in relation to Newnans Lake . The applicability of â€œautoentrainmentâ€ is explored, and model results ar e compared with data. Finally, a discussion is presented on results from two dye entrainmen t studies in the lake. One took place during a period of low wind speed and water discharge, and the other when wind and discharge were higher. 8.2 Modes of Entrainment Modes of entrainment are depicted in Figur e 8.1.The vertical stru cture of sediment concentration is characteristically sub-divided into four zones. In the upp er zone of the water column the suspension layer (DSL) is dilute, a nd is characterized by Newtonian behavior. The lower zone is occupied by the so-called benthi c nepheloid layer (BNL), which contains fluid muck. Between DSL and BNL, benthic susp ension layer (BSL) o ccurs, in which the concentration is intermediate between BSL and BNL. Its fluid propertie s are non-Newtonian but the concentration is not high enough for settling to be hinde red. Finally, at the bottom a consolidating bed (CB) occurs. It possesses an eff ective stress but is soft enough (i.e., it is not fully consolidated) for the sediment to be sus ceptible to entrainment when wave forcing is sufficiently strong. In the dynamic sense, the four zones are linked by processes of particle settling, coalescence or deposition of particles-in-fluid parcels, and entr ainment of parcels. In the absence PAGE 178 178 of BNL and BS, dilute suspension is sustained by erosion of particles from CB, and in turn particles settle through the water column and may deposit onto the bed, unless the near-bed fluid stresses are high enough to prevent deposition. Th ese are sometimes called classical erosion and deposition modes associated with CB (Winterwer p et al., 2002). When BNL occurs but BSL is (practically) absent, entrainment of fluid muck into DSL occurs by mixing between the two fluid layers. The settling parcels, if and when they reach BNL, will coalesce into BNL. Exchange between BSL and DSL cannot be ca lled coalescence, since in this case settling flocs from DSL mix with BSL merely by penetrati on, as the inter-particle spaci ng is much higher than in BNL. Finally, exchange processes between BSL, BNL and CB can change their thickness and concentration without pa rticipation of DSL. 8.3 Entrainment in Newnans Lake Entrainment can be modeled by the advecti on and diffusion equation (Winterwerp, 2004) given by Eq (8.1). st s SD CC wC tzzz (8.1) where concentration C is the mass of solids per un it volume of suspension, and s D is the molecular diffusion coefficient given by Eq (8.2) 6Bw s pkT D d (8.2) in which Bk is the Boltzmann constant, wT is the water temperature and pd is the particle size. In Eq (8.1) t is the eddy diffusivity S is the Prandtl-Schmidt number and s w is the settling velocity of the particle. The l.h.s. term in Eq (8.1) is the rate of change of concentration, the first r.h.s term is the gradient of the sett ling flux, the second is the diffusion term. PAGE 179 179 To solve the above equation the eddy diffusivityt must be known. Eddy diffusivity closure can be formulated from one of the flowing approaches (Pope, 2000): Assumed constant or obtained fr om empirical relationships Derived from the mixing length theory Modeled using a k turbulence closure model Modeled using a k model To select the appropriate a pproach, the physical environment of Newnans Lake must be considered. It is a shallow wate r body with 0.3 m of active fluid mud (muck) at the bottom. This mud layer has high organic content (40-60%) w ith very fine particles (size less than 2 m), very low settling velocity of the order of 10-4 m/s, and shear strength as low as 0.01 Pa (Gowland, 2005). Wave heights at the tower were of the or der of few centimeters and the current velocity was of the order of cm/s (Chapter 2), so the combined wave and current shear stress was very small. The current induced Reynolds number is of the order of 103 so the flow can become transitional to turbulent due to currents. To characterize wave-induced movement, the wave Reynolds number Rew is plotted in Figure 8.2. The value of Rewwas less than 2 for all wind speeds and less than 1 most of the time. So the flow regime was close to creeping flow. Thus, turbulent closure schemes and mixing length th eory cannot be applied to obtain the eddy diffusivity. The only choice is to use an empirical relationship for the diffusivity. Using an empirical relationship from Hwang (1982), the diffusivity in wave-dominated environment can be roughly estimated from Eq (8.3) 2 2 2sinh() 2sinhr tw rkhz H kh (8.3) PAGE 180 180 where H is the wave height and w is an empirical proportionality coefficient chosen as 1. The origin of the vertical coordinate z is at the water surface. Equation (8.4) was applied to Newnans Lake and the diffusivity due to waves was calculated using the parameters specified in Table 4.5 and plotted in Figure 8.3. The mean diffusivity due to wave is seen to be of the order of 10-7 m2/s, which is close to the molecular diffusivity (diffusion scales are shown in Figur e 8.4). So it can be inferred that wave-induced local entrainment makes a very small contribu tion to the suspended se diment concentration variation in the water column. The combined wave and shear st ress for different wind speeds is plotted in Figure 8.5. The plot also includes a line for the bed shear strength (critical bed shear stress) line. The bed shear strength is equal to the bed shear stress above which the bed will start to erode. Since the bed shear stress did not exceed the bed shear streng th, bed erosion and deposition are not significant mechanisms for the observed oscillations in SSC. Thus, in order to explain these oscillations, in the following section a simple approach is described and applied to the lake. 8.4 Auto-Entrainment Auto-entrainment is based on the hypothesis th at the benthic nepheloid layer (BNL) acts as a reservoir of sediment in suspension, while se diment exchange between BNL and the bed (CB) is absent (Figure. 8.6). Since the wave-mean concen tration is of interest, the steady state form of Eq (8.3) is considered, and a lumped diffu sion coefficient, designated by the symbol E , is selected and parameterized as a function of wind. It is assumed to be independent of the depth. Since SSC-1 and SSC-2 are low (10-30 mg/L), the particles are assumed to settle independently of each other and the settling velocity s w is assumed to be constant over depth. Under these conditions, Eq (8.3) simplifies to Eq (8.4) PAGE 181 181 0sdC wCE dz (8.4) This equation can be solved for concentration C ( z ) with bottom boundary condition C ( az) = Ca where Ca is a reference concen tration at elevation z = az above bed datum ( z = 0). Given water depth h , the solution of Eq. (8.4)gives Eq (8.5) ()/szwzhaE aC e C (8.5) A selected SSC time-series segment is repr oduced in Figure 8.7 for the period January 14th to January 30th, 2004. The reference concentration Ca is taken as 325 mg/L based on extrapolation (shown in black dotte d line) of measured concentrati on profile (in black solid line). This concentration was assumed to occur at a distance az = 10 cm above the mud-water interface (datum). Diffusivity was parameterized as a function of wind for the period January 14th to 20th. The plot of diffusivity with wind speed is shown in Figure 8.8, in which 95% confidence limit lines are also included. The average diffusivity, E (m2/s) was parameterized as a linear function of wind speed U (m/s) given by Eq (8.6). -5-8=3.32x10 + 8.13x10 E U (8.6) The diffusivity equation (Eq 8.7) was th en used to determine SSC using Eq (8.5) at three levels for the period of January 21st to 28th. The diffusivity in Eq (8.6), is larger than the molecular diffusion which might indicate the eff ect due to the presence of current. Since diffusivity in its time -mean form was parameterized as a function of wind speed, calculated SSC using this diffusivity cannot be co mpared with instantane ous values of SSC in the measured time series (at any elevation). So an analysis of data for January 21st to 28th was done in the frequency domain. Figures 8.9 and 8.10 show spectra of SSC-1 and SSC-3 (for PAGE 182 182 illustrative purposes). To determine the degree of correlation, the cohe rence spectrum between the predicted SSC and data is given in Figur e 8.11. The correlation valu e ranges up to 0.5. Since the settling velocity is very low there is a s uperposition of the response at frequencies greater than about 0.28 hr-1. Figure 8.12 shows the concentration profile obtained by time-averaging SSC calculated from Eq (8.5) for the period January 21st to 28th. As can be expected on account of the method used to obtain Eq (8.7), the calculated pr ofile is almost coincident with data. 8.5 Dye Study From the dye study summarized in Section 2.2, dye concentration series is plotted in Figure 8.13. The upper plot (a) is for the calm peri od (average wind speed of 4 m/s and outflow of 0.23 m3/s from Prairie Creek) and the lower plot (b) corresponds to a stormy period (wind 8.4 m/s and outflow 7.5 m3/s). Dye concentration was 50% higher at the depth of 1.5 m compared to 0.83 m. Thus, the vertical gradient of dye c oncentration was 2238 ppb/m. The dye concentration at 1.5 m depth during the calm period was highe r by 1,500 ppb compared to the stormy period. The mean concentration remained same at both elevations during the st ormy period, indicating high vertical mixing. Figure 8.14 is a plot of dye concentration ag ainst wind speed. The concentration does not correlate with wind, which suggests that the e ffect of wind decayed with the depth and was negligible at the depth of 1.5 m. Figure 8.15 is a plot of dye concentration at both elevations against outflow discharge in Prairie Creek. Dye concentration shows some dependence on outflow, with a correlation coefficient of 0.45 for plot (a) and 0.20 for (b). This observation suggests that advective effects during storm even ts may govern entrainment in the lake, although such was not the case during the calm period. PAGE 183 183 8.6 Chemical and Biological Processes Figure 8.16 is a plot of dissolved oxygen (DO) with depth. DO concentration decreases rapidly in the nepheloid layer, indicating an anaerobic condition near the bottom. The onset of seasonal anoxia in water overlying the sediment has been linked to an increase in phosphorus release from the sediment (Caraco et al., 1991). Low DO favors the production of methane sulfide and ammonium, and release of ferrous iron from sediment. Some of these reactions may contribute to release of the sediment from the bottom. Bio activity (fish and alligators ) can also cause the suspension of the sediments. The lake has a large population of fish ( cat fish and large-mouth bass) a nd alligators, which feed at the bottom. PAGE 184 184 Figure 8.1 Lake sediment concentration zones and entrainment modes Figure 8.2 Wave Reynolds number as a function of wind speed in Newnans Lake Entrainment Particle settling and deposition on bed Particle settling Dilute suspension layer ( DSL ) Particle settling and coalescence with fluid mud Entrainment Erosion Benthic suspension layer (BSL) Benthic nepheloid (fluid muck ) la y er ( BNL ) Settling and coalescence Entrainment Hindered settling Consolidating bed (CB) Erosion Settling and deposition Erosion Creeping flow PAGE 185 185 Figure 8.3 Wave diffusion coefficient as a function of wind speed in Newnans Lake Figure 8.4 Scales of diffusion (Chapra, 1997) DIFFUSION IN POROUS MEDIA (sediments) VERTICAL TURBULENT DIFFUSION LONGITUDINAL DISPERSION 10-8 10-6 10-4 10-2 100 102 104 106 108 MOLECULAR DIFFUSION Estuaries HORIZONTAL TURBULENT DIFFUSION Oceans Lakes Surface layer Deep Layers Streams cm2/s Moleculardiffusion PAGE 186 186 Figure 8.5 Bed shear stress as a function of wind speed in Newnans Lake Figure 8.6 Auto-entrainment due to exchange of suspended sediment between BNL, BSL and DSL, without participation by the bed (CB) Particle settling Dilute suspension layer (DSL) Particle settling and coalescence with fluid mud Entrainment Benthic suspension layer (BSL) Benthic nepheloid (fluid muck) layer (BNL) Settling and coalescence Entrainment Wind Entrainment Critical shear stress line PAGE 187 187 Figure 8.7 Suspended sediment concentration time-series Figure 8.8 Eddy diffusivity as a functi on of wind speed in Newnans Lake PAGE 188 188 Figure 8.9 Comparison of spectra of SSC-1 from data (dashed line) and model in Eq (8.6) (solid line) Figure 8.10 Comparison of spectra of SSC-3 from data (dashed line) and model in Eq (8.6) (solid line) PAGE 189 189 Figure 8.11 Coherence spectra of data and model in Eq (8.6) PAGE 190 190 Figure 8.12 Concentration profile fro m model in Eq (8.6) and data Figure 8.13 Dye concentration time-series fo r: A) calm period, and B) storm period ca a z Dye concentration (ppb) Dye concentration (ppb) Data E q 8.6 A B PAGE 191 191 Figure 8.14 Dye concentration variation with wi nd speed for: A) calm period, and B) storm period Figure 8.15 Dye concentration variat ion with the outflow discharge for: A) calm period, and B) stormy period Dye concentration (ppb) Dye concentration (ppb) Dye concentration (ppb) Dye concentration (ppb) A B A B PAGE 192 192 Figure 8.16 Dissolved oxygen profiles at five locations (Table 2.3) PAGE 193 193 CHAPTER 9 CONCLUDING OBSERVATIONS 9.1 Study Summary Shallow-water lakes laden with fluid mud as a nepheloid layer often have problems associated with high levels of tu rbidity. Physical conditions in such lakes require that the issue of turbidity generation be examined in two parts. Th e first is to identify the mechanism(s) by which fluid mud damps wind-generated waves, and th e second is to understand how these (damped) waves entrain sediment stored in the nepheloid layer. In this work these issues have been examined with reference to Newnans Lake, a shallow, 2,700-hectare hypereutrophic lake in north-central Florida. To examine wave-mud interaction in shallow la kes, a first-order semi-analytical solution is presented for simulating the effect of the nepheloid layer on surface wave attenuation. The utility of the solution is tested by comp aring results with laboratory data. A second-order nonlinear solution is then derived using the well known Stokes' perturbation approach, with mud modeled as a Newtonian fluid. As a test of the applicability of this model, calculated mud mass transport is co mpared with laboratory data, and the issue of correct representation of mud rheology is highlighted. The significance of the rheology of mud as a single-phase medium is considered. Mud is treated as a viscoelastic flui d, and its behavior under stress is compared with data from rheometric tests. A viscoelastic fluid model (Jeffrey's-b) is then incorporated in the second-order solution, and the resulting wave attenuation coefficient and mud mass transport are compared with experimental data. Mud is then modeled as a tw o-phase poroelastic medium as an alternative rheological description. Since sediment grain size and density can vary widel y, the domains of applicability PAGE 194 194 of viscoelastic and poroelastic models are di scussed with reference to bottom sediment composition and density. Wave energy dissipation in Newnans Lake is next treated in a gene ral way by carrying out a dimensional analysis for characterization of th e wave attenuation coefficient. The significance of the dimensionless groups is first discussed with respect to soft mud modeled as a viscoelastic or poroelastic medium. Characteristic values of the groups are then discussed with reference to the wave-mud regime of Newnans Lake. Lake circulation due to wind and water discharge from the creeks is simulated using a three-dimensional numerical hydrodynamic code . Suspended sediment transport is then examined using this model. In order to better e xplain the behavior measured SSC oscillations in the lake, it is hypothesized that th e lake is subject to â€œauto-ent rainmentâ€. In this mode of entrainment, vertical exchange of suspended sediment between the ne pheloid layer and water layer above determines SSC, in the ab sence of erosion and deposition. 9.2 Main Observations The following observations have been made in this study: Newnans Lake is a small, fetch-limited lake in which about 0.3 m thick nepheloid layer of fluid mud dissipates waves. The organic content of bottom sediment is 20-4 0% with a very low shear strength (critical shear stress) of 0.01 Pa. Settli ng velocities of the suspende d organic-rich matter are of order on 10-4 m/s. Due to these low settling velocities , settling lags are of the order of 3-4 hours, which results in a superposition of SSC response at high frequencies. Peak wave energy dissipation char acteristically occurs between221.31.5 , which is of the order of 2 , the Stokesâ€™ boundary layer thickness in mud. Thus it becomes essential to consider mud boundary layer effects in prediction of wave heights. At low stresses mud behaves as a solid. Howe ver at high stresses under cyclic loading in both laboratory and field settlings, mud is easily liquefied and start to flow. Therefore, it needs to be modeled as a fluid. PAGE 195 195 Mud can be assumed to be a Newtonian fluid below densities of about 1,150 kg/m3. In the density range of about 1,150-1,300 kg/m3, it can be modeled as a viscoelastic fluid. Beyond about 1,300 kg/m3 it is not compliant to waves. The commonly used Voigt model cannot be applied to simulate wave-mud interaction based on the governing equations for fluid flow , especially to predict mud mass transport velocity, as the Voigt element represents a solid. Due to mud elasticity a resonance frequency ex ists, at which wave en ergy dissipation has its maximum value. This frequency is a func tion of the square-root of the shear wave velocity in mud and a length scale ba sed on the mud depth or wave length. A poroelastic bed with Coulomb friction assump tion can be applied to the entire range of beds from gravel to clay. However, applica tion to soft mud beds becomes empirical as these beds, with density in the range of 1050-1200 kg/m3, have a solids volume fraction in the range of 0.03-0.12, indicating that the mud (sus pension) is dilute and particle-particle interaction is negligible. In this case th ere are no significant losses due to Coulomb friction. The poroelastic model is based on the existen ce of a solid soil skeleton which may be rigid or non-rigid, but it cannot be used to determine mud mass tr ansport velocity. The domains of applicability of various mech anisms of wave energy dissipation can be quantified on the basis of a proposed dimensionless parameter s pUwK ,which is a measure of rate of solid settli ng in the liquid to the rate of liquid flowing in the pore spaces of the solid. A low ratio indicates that th e medium is single-phase (for which the viscoelastic assumption is applicable), while a large value of the ratio indicates differential settling associated with a twophase medium (which can be modeled as poroelastic bed). Physical parameters affecting wave damping are examined using dimensional analysis. It is shown that, depending on the bed conditi ons, dimensionless groups which influence wave attenuation coefficient can be identified. Based on the above dimensional analysis it is shown that in Newnans Lake liquefaction of bottom mud occurs at wind speeds greater than 7 m/s. The amplitudes of SSC oscillations in the lake are not large. The sta ndard deviation of SSC normalized by mean SSC at 0.3 m depth was 14% and at 1.5 m it was only 3.6%. These values indicate that the effect of wind decays rapidly with water depth. A sediment transport model applied to the lake showed that almost nowhere in the lake did the combined bed shear stress due to current and waves exceed the critical shear stress. therefore, classical erosion and deposition m echanism based on sediment exchange at the bed appears not to apply to this lake, wher e the wind speed is usually below 8 m/s. Advection plays a significant role in mixing in the water column during storm events, but is small under calm (normal) conditions. PAGE 196 196 As bed erosion and deposition are not the domin ant modes of sediment entrainment, to explain the observed amplitude oscillations in SSC, a simple concept called autoentrainment has been applied to the lake. It is shown that there is a repository of sediment in the benthic nepheloid layer which contribut es to sediment in the upper water column and allows SSC to oscillate in response to wind speed variation. 9.3 Recommendations for Future Studies To develop a better understandi ng of domains of applicability of different mud behavior models discussed in Chapter 6, and to explor e the physics behind the dimensional analysis presented in Chapter 7, the following reco mmendations are made for future studies. The FSM approach should be tested against field data. Since the second-order Stokesâ€™ theory is not valid for sha llow water, a fully non-linear wave model should be applied to solve the e quations numerically to better understand the effects of non-linearity of waves on wave orbital and mass transport velocities. Fluid mud in the field should be examined in rheometric tests descri bed in Chapter 5. Such tests should be conducted to cal ibrate the viscoelastic fluid model, which can then be applied to understand wave dissipation phenomena. Pore pressure should also be measured in the field and, based on mud properties, the poroelastic model should be applied to estimate the wave attenuation coefficient, ther eby exploring the domains of applicability of mud models in a more extensive way. To gain a better understanding of fluid mud formation, wave damping and entrainment mechanisms along with liquefaction by waves, th ese processes should be incorporated in a comprehensive model that can respond to in teraction between wave s and change in mud state. To understand the effect of advection on entr ainment in Newnans Lake, a minimum of two data measurement stations should be installed. The auto-entrainment concept should be test ed rigorously in lakes similar to Newnans Lake. PAGE 197 197 APPENDIX FIRST-ORDER ANALYTICAL SOLUTION The first-order solution is obtai ned by asymptotically matching the velocity field at the edge of the boundary layer with the velocity field in the outer invi scid core (potential solution). Asymptotic matching allows determination of the complex eigenvalue of the wave number. Solution variables are functions only of relative mud depth, d ( d normalized with respect to2). All the variables are expanded as23 123 ffffO , () ()eikxtffn . Subscript 1 is omitted for the first-order solution for all the variables. nhzd, 0n defines the mud bottom and the mud-water interface is given by 2. nd Pressure is constant at the l eading order. Substituting Eq ( 4.29) into Eq (4.21) we obtain ordinary differential equations for the horizontal components of velocity in water and mud: 2 1 11 2 1du i uU dnv in dn (A.1) 2 2 22 2 2du i uU dnv in 0nd (A.2) where 121 is the density ratio. These differential equations can be solved for water and mud separately to obtain a gene ral solution of the horizontal wave orbital velocities in mud and water: 1111exp(())uDndU in dn (water) (A.3) 2221coshsinhuGnHnU in 0nd (mud) (A.4) where D , G and H are constants,1,21,2(1)/ i and 1,21,22/. Using the boundary conditions at the bottom and at the interface: 20 u on 0n ; 21uu and 2211(/)(/) dudndudn on nd, the above constants can be determined: PAGE 198 198 2 221cosh coshsinh d D dd (A.5) G (A.6) 2 22 221coshsinh coshsinh dd H dd (A.7) where 2121vv is the ratio of Stokesâ€™ boundary laye r thicknesses in mud and water. The vertical component of velocity is obt ained from the continuity equation (Ng, 2000) 22221 2sinhcosh1 ik wnnHnU (A.8) 1211 1()exp(())1kD wwdikndndU (A.9) Adopting the general solution form of Da lrymple and Liu (1978) and applying the boundary conditions, u1 is obtained as 1 11()cosh()sinh()exp(())uzkhzBkhzDhzU (A.10) 1 11 1()sinh()cosh()exp(())kD wzikhzBkhzhzU (A.11) Asymptotically matching the outer solution (potential solution) a nd inner (boundary layer) solution, (Ng, 2000) gives 2B B kd (A.12) 1 22sinhcosh1 B dHdD (A.13) After some manipulation the vertical ve locity in mud can be written as 211 1()expkD wnikndBndU (A.14) PAGE 199 199 The amplitude of interfa ce displacement is obtained from Eq (4.10) as 1 1U kD bB (A.15) 1sinhcosh a U khBkh (A.16) The eigenvalue of wave number k is obtained from the lin earized dynamic free-surface boundary condition 1exp(()) uikxtg tx on 0z (A.17) 2tanh 1tanhkhB gkBkh (A.18) Since B Ok , we may replace the wave number by its expansion 12... kkk . Clearly the leading order k1 is real and the solution is th e well-known disper sion relationship 2 11tanh gkkh (A.19) One should note that the above dispersion rela tionship is approximate for this two-fluid system, as it merely includes dispersion due to water, and no effects are considered from mud. The second-order solution for k2 (which is known as wave attenuation coefficient ik ) is a complex solution 1 2 111sinhcosh Bk k khkhkh (A.20) 2 1 2 11111Im() Im() sinhcoshsinh22riBB Bk k khkhkhkhkh (A.21) 13 23r i B BB B BB (A.22) PAGE 200 200 22222 1 2 2222221sinhcoshcoshsinh 1coshcossinhsin 21coshsinhcos B dddd dddd ddd (A.23) 22 2221sincos 21sinhcoshsin B dd ddd (A.24) 22 22 3coshsinhcossinhcoshsin B dddddd (A.25) where 2/.dd This analytical solution for ki is compared with the full semi-analytical solution (FSM) and is tested on different da ta sets in Section 4.9. PAGE 201 201 LIST OF REFERENCES Abdel-Hady, M., Herrin, M., 1966. Char acteristics of soil-asphalt as a rate process. 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Jour nal of Fluid Mechanics, 87, 193. Zienkiewicz, O.C., Chang, C.T., Bettess, P ., 1980. Drained, undraine d, consolidating and dynamic behaviour assumptions in soils. Geotechnique, 30 (4), 385. PAGE 210 210 BIOGRAPHICAL SKETCH Mamta Jain was born as the only daughter of Kanta and Mohan Jain, in Kathmandu, Nepal. She did her bachelorâ€™s in civil engine ering from Delhi College of Engineering in 1998. After obtaining her degree she worked for 3 year s as an ocean engineer in the oil-sector consultancy, Engineers India Ltd. Her main area of specialization was design of oil terminals. Craving for more knowledge made her take a brea k from the job and she applied to graduate schools. In the fall of 2001 she was admitted to the Graduate School of the University of Florida. She married Parag Singhal in December 2001, who encouraged and supported her to continue her academic work in the Department of Civil and Coastal Engineering. She obtained her M.S.in coastal and oceanographic engin eering in 2002 and she was blesse d with a son (Shivam Singhal). She decided to continue and complete he r doctoral study in coastal and oceanographic engineering. |