DYNAMICS OF LONG WATER WAVES: WAVE-SEAFLOOR INTERACTIONS, WAVES THROUGH A COASTAL FOREST, AND WAVE RUNUP

Transcription

1 DYNAMICS OF LONG WATER WAVES: WAVE-SEAFLOOR INTERACTIONS, WAVES THROUGH A COASTAL FOREST, AND WAVE RUNUP A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by I-Chi Chan August 2011

2 c 2011 I-Chi Chan ALL RIGHTS RESERVED

3 DYNAMICS OF LONG WATER WAVES: WAVE-SEAFLOOR INTERACTIONS, WAVES THROUGH A COASTAL FOREST, AND WAVE RUNUP I-Chi Chan, Ph.D. Cornell University 2011 This dissertation studies three applied topics concerning long-wave dynamics. Interactions between surface waves and a muddy seabed are first investigated. Under the assumption that a seafloor can be modeled as a layer of viscoelastic sediments, a set of depth-integrated equations is derived to describe the propagation of long waves under the effects of seabed conditions. Dynamic responses of a viscoplastic mud bed subject to a surface solitary wave are also studied. Surface waves can be attenuated considerably due to the presence of a muddy seabed. Features of wave-induced mud motions depend largely on the rheology of bottom sediments. Theoretical predictions are tested against available experimental data. A good agreement is observed. Next, a theory is developed to study the effects of emergent coastal forests on the propagation of long surface waves of small amplitudes. The forest is idealized by a periodic array of rigid cylinders. Parameterized models are employed to simulate turbulence and to represent bed friction. A multi-scale analysis is carried out to deduce the averaged equations on the wavelength-scale, with the effective coefficients calculated by numerically solving the flow problem in a unit cell surrounding one or several cylinders. Analytical and numerical solutions for the wave attenuation are presented. Comparisons with laboratory data show very good agreements for both periodic and transient incident waves. Finally, the last topic concerns mainly the runup of leading tsunami waves.

4 Lagrangian long-wave equations are derived to help accurately track the moving shoreline. A series of numerical experiments reveals that the front-profiles of leading tsunami waves dominate the runup processes while the back-profiles are influential for the rundown flows. For a leading elevation wave, stronger acceleration of the wave front results in higher maximum runup height. As far as the maximum runup height is concerned, it is sufficient to consider only the accelerating phase of the main tsunami wave. It is concluded that solitary wave is not a perfect modeling wave for tsunami research. Directly applying the runup rule of solitary wave to tsunami runup can lead to a very inaccurate estimation.

5 BIOGRAPHICAL SKETCH I was born and raised in Taipei, Taiwan. Before being awarded this Doctor of Philosophy degree, I earned my Bachelor of Engineering degree in Water Resources and Environmental Engineering from Tamkang University, my Master of Science degree in Civil Engineering from National Taiwan University, and another Master of Science degree in Civil and Environmental Engineering from University of Illinois at Urbana-Champaign. iii

6 Dedicated to my parents, my wife, and my two sisters. iv

7 TABLE OF CONTENTS Biographical Sketch iii Dedication iv Acknowledgements v Table of Contents vi List of Tables ix List of Figures x 1 Introduction 1 2 Long water waves over a thin muddy seabed Introduction A simplified two-layer model and assumptions An overview of the mud rheology A generalized model for surface waves interaction with a linear viscoelastic muddy seabed Depth-integrated model for weakly nonlinear and weakly dispersive water waves Model equations for mud flow motions Rheology model for a linear viscoelastic mud Solution forms inside the muddy seabed HD application: evolution of wave height of a surface solitary wave HD application: amplitude variation of a linear progressive wave Explicit solutions for 1HD periodic waves Comparison with laboratory experiments Summary Response of a Bingham-plastic muddy seabed to a surface solitary wave Formulation for wave-induced mud motions inside a thin Bingham-plastic seabed Review of Mei & Liu (1987) Solutions inside a Bingham-plastic mud Extension of the solution technique Numerical examples Wave attenuation caused by a thin layer of mud Summary Conclusions Long water waves through emergent coastal forests Introduction Theoretical formulation vi

8 3.2.1 Governing equations and boundary conditions The linearized problem Depth-integrated equations for the constant eddy viscosity model Estimation of controlling parameters Method of homogenization Macro theory for linear progressive waves Homogenization Numerical solution of the micro-scale cell problem HD application: constant water depth HD application: variable water depth Experiments and numerical simulation for periodic waves Macro theory for transient waves Homogenization Numerical solution for the transient cell problem Numerical model for the macro-scale solutions HD application: tsunami waves through a thick forest Comparison with laboratory experiments Conclusions Long-wave modeling in the Lagrangian description Introduction On the solitary wave paradigm for tsunami waves Characteristics of leading tsunamis and solitary waves Leading waves of the 2004 Indian Ocean tsunamis Leading waves of the 2011 Tohoku tsunamis Lagrangian long-wave equations Numerical model and its validation The role of surface profile on the tsunami runup The role of beach slope on the tsunami runup Discussions Concluding remarks and suggestions for future work 194 A Motions of a bi-viscous muddy seabed under a surface solitary wave 198 A.1 Solutions of mud flows inside a bi-viscous seabed A.2 Approximate bi-viscous model B The Lagrangian long-wave equations 204 B.1 Shallow water equations B.2 Boussinesq equations B.3 A stratified multi-layer model B.4 Solid slide on a plane beach vii

9 Bibliography 222 viii

10 LIST OF TABLES 2.1 Laboratory conditions of periodic waves over a viscoelastic mud bed by Maa & Mehta (1987, 1990) Controlling parameters in the proposed wave-forest model: Values of σ and α under different wave conditions Positions of wave gauges in the experiments at NTU, Singapore Laboratory conditions of periodic waves experiments at NTU: Wave periods range from 0.8 to 3.0 seconds Experimental conditions of NTU study: Periodic waves with a wide range of wave amplitudes Experimental conditions of NTU study: Solitary waves cases Solitary wave characteristics for two different scenarios Ocean bottom tsunami meters (TM1, TM2) and the GPS gauge station (Iwate South) off the northeastern coast of Japan ix

11 LIST OF FIGURES 2.1 Surface waves over a muddy seabed Rheology curves for viscous, elastic, and plastic behaviors Schematic sketch of a Maxwell element and a Kelvin-Voigt element Rheology curves for a Maxwell element and a Kelvin-Voigt element Rheology curve for a viscoplastic mud A surface solitary wave over a viscoelastic mud: Time histories of horizontal mud flow velocity at the water-mud interface, u mi A surface solitary wave over a viscoelastic mud: Time histories of bottom shear stress, τ mb A surface solitary wave over a viscoelastic mud: Profiles of horizontal velocity, u m, inside the mud column at different phases Evolution of a surface solitary wave propagating over a viscoelastic mud: Wave height as a function of time Surface solitary wave propagates over a viscoelastic mud: Effect of mud layer thickness on the evolution of wave height A linear progressive wave over a viscoelastic mud: β i as a function of d Periodic wave over a viscous mud: Comparison with Gade (1958) Solitary wave over a viscous mud: Comparison of horizontal velocity component Viscous mud flow induced by a solitary wave: Comparison with Park, Liu & Clark (2008) Solitary wave over a viscous mud: Interfacial displacement and bottom shear stress Periodic waves over a viscoelastic mud: Velocity profiles Sketches of solitary wave induced Bingham-plastic mud flow velocity: Two-layer scenario Sketches of solitary wave induced Bingham-plastic mud flow velocity: Four-layer scenario Sketches of solitary wave induced Bingham-plastic mud flow velocity: Three-layer scenario Bingham-plastic mud flow solutions of 4-layer scenario (1): Yield surfaces and interfacial velocity Sample solutions of 4-layer scenario (2): Vertical profiles of mud flow velocity Bingham-plastic mud flow solutions of 3-layer scenario (1): Yield surfaces and interfacial velocity Sample solutions of 3-layer scenario (2): Vertical profiles of mud flow velocity x

12 2.24 Bingham-plastic mud flow solutions of 2-layer scenario (1): Yield surfaces and interfacial velocity Sample solutions of 2-layer scenario (2): Vertical profiles of mud flow velocity Bingham-plastic mud problem: Comparison with the theory of Mei & Liu (1987) Strain rate of at the bottom of a Bingham-plastic muddy seabed Effects of viscosity on the flow motion inside a Bingham-plastic, τ 0 /d = Effects of viscosity on the flow motion inside a Bingham-plastic, τ 0 /d = Effects of physical mud layer thickness on the flow motion inside a Bingham-plastic mud Energy dissipation of a surface solitary wave over a thin layer of Bingham-plastic mud Sketch of the wave-forest problem Discretization of a typical unit cell and the spatial distributions of K 11 (x) Hydraulic conductivity as a function of depth-to-wavelength ratio, k 0 h Periodic waves through a semi-infinite forest in a constant water depth region: Reflection coefficient and snapshots of free-surface elevation Periodic waves propagating through a finite patch of forest in a constant water depth: Reflection coefficient and snapshots of free-surface elevation Snapshots of periodic waves propagating through a forest on a plane beach Periodic waves through a finite forest belt: Reflection coefficient Sketch of experimental setup at NTU, Singapore Comparison between theory and experimental data: Reflection coefficient for periodic waves Reflection and transmission coefficients against amplitude-todepth ratio: Comparison between theory and measurements Sample solutions of dynamic permeability, K(t) Effects of the cell geometry on the dynamic permeability Leading waves of a tsunami entering a deep forest in a constant water depth: Theoretical and numerical solutions A transient wave packet crossing a forest: Comparison between theory and measurements Sample record of incident wave for solitary wave experiments at NTU, Singapore xi

13 3.16 Solitary waves through a model forest of finite length (H/h 0 = 0.04, ): Comparison between theory and measurements Solitary waves through a model forest of finite length (H/h 0 = , ): Comparison between theory and measurements Solitary waves through a model forest of finite width (H/h 0 = ): Comparison between theory and measurements Indian Ocean tsunamis: Satellite images Indian Ocean tsunamis: Numerical simulations Tohoku tsunamis: Locations of the gauge stations and the epicenter of the earthquake Tohuku tsunamis: Gauge records Runup of surface waves on an infinite sloping beach Runup of a non-breaking solitary wave on a one-slope beach Runup of a non-breaking solitary wave on a three-slope beach Effects of the horizontal length scale of the initial wave condition on the runup and rundown Effects of the back profile of the initial wave condition on the runup and rundown Effects of the preceding waves on the runup processes Runup and drawdown of model waves on a one-slope beach Effects of the bottom slope on the wave runup over a one-slope beach A.1 Sketches of solitary wave induced bi-viscous mud flow velocity. 200 A.2 Rheology curve of a bi-viscous mud xii

14 CHAPTER 1 INTRODUCTION Ocean surface waves, among the best-known oceanic phenomena, perform an essential role in sustaining life on our planet; in part these wave motions transport energy across the continents and shape the coastlines. Ocean waves occur over a tremendously broad range of wavelengths, from a few centimeters capillary ripple to a tsunami spanning hundreds of kilometers. In particular, surface gravity waves are of the greatest importance since gravity is the main restoring force for wave motions associated with most human activities in the seas. In the well-established linear water wave theory, to specify the wave motions one needs to know the water depth h, wave height H, and wavelength L. While the last two describe the physical dimensions of the wave, the first, from a certain perspective, states the property of the medium a wave travels through. As for the seemingly undetermined wave period T, it can be calculated theoretically from the dispersion relationship, ω 2 = gk tanhkh, where ω = 2π/T is the wave frequency, g the gravitational acceleration, and k = 2π/L the wavenumber. Through the theoretical analysis, it is interesting to see that as a surface wave propagates, the active water particles pass along the wave energy by moving in circular orbits. What is more intriguing is that these circular orbital motions are only considerable within the depth no more than half the wavelength. This influential depth, of course, can be smaller the total water depth. Based on the relative magnitudes of water depth and wavelength, surface waves are classified as long waves if h/l is less than On the other hand, for waves of h/l greater than 0.5 they are called short waves. Of course, waves outside these two categories are given the name intermediate waves. To be more 1

15 precise, long waves are of large wavelengths, long periods, and low frequencies while short waves are just the opposite. Long waves, inherently, have several special features. For instance, the water particle trajectories of a long wave are ellipse-like with the horizontal excursion more or less a constant throughout the depth-wise extent, and a comparably negligible vertical component increases linearly from the bottom to the free surface. In other words, the wave-induced pressure is hydrostatic and the horizontal motions have only a weak dependence on elevation. In addition, long waves are nondispersive, i.e. the wave speed c is solely a function of the water depth, c = gh. All leads to a common ground that the three-dimensional long-wave hydrodynamics can be legitimately approximated by some simplified models involving only two horizontal dimensions. Let us also discuss some properties regarding short waves. Quite differently, motions associated with short waves are strongly three-dimensional as the trajectories of active water particles are circles with the orbital diameters decreasing exponentially with depth. Also, the speed of short waves depends on the wavelength, c = gl/(2π), i.e. the so-called frequency dispersion. Tremendous efforts have been made by scientists and engineers to study long-wave mechanisms as these waves are more prominent in association with many human activities in coastal marine environments. The importance of long waves in the complex web of natural waters can be appreciated from several perspectives. A rather intuitive explanation is that long waves travel faster than short waves, and consequently would reach the beaches much earlier. This corresponds to the fact that for short waves c = gl/(2π) < gh/(4π). One shall also realize that long waves interact strongly with the seabed while the bottom conditions have less impact on short waves. This can be understood by considering the previously discussed influential depth of the wave-induced water 2

16 particle motion, along with the underlying assumption on the limits of h/l that distinguish the long and short waves. The presence of ocean floor makes the physical process involving long waves more complicated. In part, bottom sediments can be eroded and transported by wave motions while wave climates can as well be changed simultaneously. One may also argue the significance of long waves from the viewpoint of the amount of collective energy. As the wavelength of a long wave is usually considerable and the corresponding water particle motions are uniform throughout the entire water column, the energy carried by a long wave is substantial. A good example is to consider tsunamis, the extreme long waves which can have a wavelength of several hundred kilometers in an open sea with a water depth of a few kilometers. Furthermore, long waves lose less energy than short waves as they propagate. First, the frictional energy loss is mainly attributed to the oscillations of water particles. Secondly, water particles under short waves move upwards and downwards much more rapidly as can be seen from the previously discussed particle trajectories. Combining these two facts, the conclusion is drawn. Another point to argue the importance of long waves in consideration of wave energy is that the velocity of energy transport is the same as the wave speed of long waves. Consequently, short waves die out fast as surface waves are sustained by energy. The above arguments, although solely based on the linear wave theory along with ideal conditions, support the significance of long waves in understanding the mechanisms of ocean surface waves. Therefore, the objective of this dissertation is to study the dynamics of long water waves. In particular, three specific topics are investigated: Long water waves over a thin muddy seabed (Chapter 2); Long water waves through emergent coastal forests (Chapter 3); Long-wave modeling in the Lagrangian description (Chapter 4). The first topic 3

17 addresses the significance of seabed conditions on the long-wave propagation; the second one investigates wave dynamics in a wave-forest system; finally, the last essentially studies the runup of tsunami waves. Therefore, this dissertation shall cover some fundamental, yet important, features of long water waves in coastal marine environments. To introduce these three problems, which are to be studied specifically in Chapter 2 to Chapter 4, an overview is provided in the following. Long water waves over a thin muddy seabed Most studies of wave-seabed interactions have focused on the wave propagation over non-cohesive sediments, i.e. a sandy bed (see e.g. Liu 1973). Wave attenuation due to percolation 1 in a sandy bed tends to be relatively minor in comparison with other dissipative mechanisms, such as bottom roughness and wave breaking. On the other hand, it is well known that damping of ocean waves can be considerable, if the seabed consists of cohesive sediments. Gade (1958) reported that there is a location in the Gulf of Mexico, nicknamed the Mud Hole, where the attenuation of surface waves due to the mud bed is so great that fishing boats use it as an emergency harbour during severe storms. Similar muddy seafloors have been reported in many coasts, rivers and estuaries around the world (Healy, Wang & Healy 2002). Cohesive sediments, commonly characterized as mixtures of water and clays, are transported as aggregates. In general, mud in different locales can exhibit diverse rheological properties, partly as a consequence of distinct physico-chemical compositions. Facing the rather complex dynamic behaviour of cohesive sediments, many simplified constitutive models have been suggested, including the viscous fluid 1 This can be observed when ocean waves propagate over a permeable seabed. Wave energy is dissipated by the porous bed due to the friction between water and solid skeleton (see Liu & Dalrymple 1984). 4

18 (Dalrymple & Liu 1978), viscoelastic (MacPherson 1980), viscoplastic (Mei & Liu 1987), and poroelastic models (Yamamoto et al. 1978). Clearly, no single model can describe the entire spectrum of the seabed responses because of the great complexity and variety of mud rheology. Nevertheless, it is worthwhile pursuing a deeper understanding of every model as each has its own range of validity 2, and it is the hope that one could build up a complex model closer to the reality with the knowledge gained from this base. In the present study, the emphasis is on muddy seafloors that can be modeled as either viscoplastic or viscoelastic matter. The proposed models shall cover the basic material behaviour of viscosity, elasticity, and plasticity. It is remarked that all these rheological laws have been employed in the context of wave-seafloor interactions (see e.g., Gade 1958; Mallard & Dalrymple 1977; Hsiao & Shemdin 1980; Mei & Liu 1987). However, most of the past studies considered only waves of small amplitudes, i.e. within the framework of linear periodic wave theory. It is known that in shallow waters, where the seabed effects are expected to be more significant, the wave nonlinearity can be considerable; a nonlinear theory of long waves is therefore needed. In what follows, the immediate objective is to develop a general model describing the interactions between long waves and muddy seafloors. This problem is investigated and presented in Chapter 2. Long water waves through emergent coastal forests It is not surprising that coastal forests could serve as natural barriers to protect coastlines from tides, storm surges and tsunamis. Indeed, the field survey conducted by Danielsen et al. (2005) has shown that vegetated coastal areas suffered less damage from the 2004 Indian Ocean tsunamis. In the event of the 1999 Orissa Super Cyclone that struck the eastern coast of India, it was also seen that 2 Wen & Liu (1998) classified the applicability of these models based on soil properties. 5

19 mangroves shielded the coastline and reduced the death toll (Dasa & Vincent 2009). It is indubitably comprehended that surface waves could lose a substantial amount of energy when propagating through coastal forests. Based on the field observations collected at Cocoa Creek in Australia, Massel, Furukawa & Binkman (1999) were able to demonstrate that at low tides nearly 75% of incident wave enery, with a peak period of rougly 2 seconds, was dissipated when waves propagated through a coastal forest of approximate 100 m in length. Of course, a rough seabed can cause certain frictional loss. However, in a waveforest system the energy dissipation, as can be expected, is mainly due to the turbulence generated through the multiple interactions between waves and the vegetation; this most likely occurs throughout the entire water column. Laboratory studies have been designed to build quantitative understanding of energy dissipation process in wetlands, and to evaluate the efficiency of coastal trees in protecting the shore against tsunami attacks. For instance, Nepf (1999) proposed a parameterized model to describe the turbulence for flow through emergent vegetation. Modeling a coastal forest by an array of rigid cylinders, Irtem et al. (2009) have demonstrated that trees planted on the sloping beach can reduce the runup height of a model tsunami approximately by half. Built on the established knowledge, the goal of this study is to develop a sound, yet simple, theory describing the dynamics of long waves through coastal forests. To make the analysis more tractable, a major simplification is adopted to model tree trunks by a periodic array of rigid cylinders but to neglect the effects of tree roots, branches, and leaves. Note that the typical diameter of tree trunks is of O(0.5) m, while the characteristic wavelength of long waves can easily reach O(100) m. The existence of these two distinct scales grants the use of the homogenization technique, which can be viewed as a rigorous two-scale anal- 6

20 ysis, in developing the theoretical model for the present wave-forest problem. The new theory is capable of dealing with both periodic waves and transient waves, and will be tested against the available experimental data. Several numerical examples are also given to illustrate some important features regarding the wave-forest dynamics. This topic will be discussed in Chapter 3. Long-wave modeling in the Lagrangian description A tsunami, which is usually generated by a submarine earthquake, landslide, or volcanic eruption, is an extremely long wave with a wavelength easily up to several hundred kilometers. The science of tsunami waves has been studied systematically for many decades. It is fair to say that a tremendous advance has been achieved after the devastating 2004 Indian Ocean tsunamis shocked the world. Nevertheless, our knowledge is still quite limited. This is evident after another earthquake-triggered destructive tsunami struck the northeastern coast of Japan in March, 2011 and claimed thousands of lives. In studying water wave theory, contributions can be made towards the understanding of tsunami mechanisms by improving the prediction on the generation, propagation, and runup of tsunamis. Of course, the study of wave-structure interactions is also important. In this study, the particular focus is on the terminal effect of tsunami waves running up shoreline 3. The capability of accurately estimating the maximum runup height, i.e. the largest landward excursion of the waves along the shoreline, is crucial in developing a tsunami evacuation plan. However, the problem is challenging as the moving shoreline makes the problem domain time-varying. It is worth remembering that most of the wave-related studies employ the Eulerian approach, which concentrates on the fluid motions 3 A good reference on the modeling of tsunami generation and propagation is that of Wang (2008). For the study of wave-structure interactions, one can refer to, for example, the threedimensional numerical model developed by Mo (2010). 7

21 at specific spatial locations. Several approximation techniques have been applied to address this well-known moving-boundary issue with various degrees of success. An increasingly popular treatment is the higher order interpolation method developed by Lynett (2002). The present study, on the other hand, approaches the problem by the use of the Lagrangian specification. The moving shoreline becomes a fixed point as in the Lagrangian coordinates one essentially follows the history of each individual particle. Thus, with no additional numerical approximation required one can accurately and directly calculate the time history of the shoreline movement, including the position and the velocity. In many cases, it is desirable to have a quick assessment of tsunami inundation with only limited information. It is because of this engineering interest that many runup formulae have been established by assuming certain idealized conditions. For instance, the best-known work is that of Synolakis (1987), relating the maximum runup height to the incident solitary wave height and the beach slope. The performance of this so-called runup rule is very good, if indeed the tsunamis can be scaled by solitary waves. However, Madsen, Fuhrman & Schaffer (2008) cautioned that solitary waves can not be used to model tsunamis due to the limitation of relevant geophysical scales. It is therefore important to understand the consequence if a solitary wave is still in use to model a tsunami. In all, Chapter 4 will start from the introduction of long-wave equations in the Lagrangian description. A Lagrangian numerical model is then developed to study the runup of tsunami waves. 8

22 CHAPTER 2 LONG WATER WAVES OVER A THIN MUDDY SEABED In this chapter, interactions between surface waves and a thin muddy seabed made of cohesive sediments are studied by the use of a immiscible two-layer water-mud system. Modeling the seafloor as a linear viscoelastic body, a set of Boussinesq-type equations for long waves over a thin layer of mud is derived, as presented in section 2.2. Wave damping rates for both periodic waves and solitary waves are calculated using the newly developed model. In section 2.3, the seabed is assumed to be made of Bingham-plastic mud. The dynamics of mud flow induced by a surface solitary wave are investigated. To examine the performance of the proposed model, theoretical predictions of mud flow velocity, bottom shear stress, vertical displacement at wave-mud interface, amplitude attenuation, and wavenumber shift are all compared with available laboratory measurements for the case of viscoelastic mud. As for the study of a surface solitary wave propagating over a layer of Bingham-plastic mud, comparison is made between the model results and field observations. Good agreements are evident in all examples. It is concluded that a muddy seabed can attenuate surface waves considerably. 2.1 Introduction Understanding the interactive processes between surface water waves and muddy seabeds is one of the intriguing research topics in the fields of coastal engineering and ocean science. On one hand, as waves propagate over a seafloor work is done by the wave-induced pressure force to excite the motion of fluid 9

23 mud. The associated wave energy loss can be considerable. Indeed, significant damping of surface waves caused by a mud bottom has been reported by numerous field observations (see e.g., Gade 1958; Wells & Coleman 1981; Forristall & Reece 1985; Elgar & Raubenheimer 2008). On the other hand, the waveinduced mud motions not only affect the wave climate but also have great impacts on the seabed morphology and biological activities in the benthic boundary layer (Foda 1995). For instance, through the resuspension and deposition processes transport of nutrients is enhanced as well as the remobilization of the buried pollutants. In the long run, coastline change can also be expected (Mei et al. 2010). Muddy seabeds are essential cohesive sediments made up of fine particles with a characteristic size less than 2µm (Chou, Foda & Hunt 1991). In contrast to the non-cohesive deposits where particles move individually, the cohesive sediments flow as aggregates. In general, mud in different locales can exhibit diverse rheological properties, partly as a consequence of distinct physicochemical compositions (Balmforth & Craster 2001). Moreover, the rheology of muddy seabed depends also on the wave climate and sediment concentration (see e.g. Krone 1963; Chou, Foda & Hunt 1991). As a result, the rheology of bottom mud could change dynamically. Owing to the difficulty in modeling the complexity of nature, researchers have approached the problem, as the first step, with different simplified rheology models to examine the wave-mud interactions, namely the response of cohesive sediments to surface waves and the impact of seabeds on wave propagation. The hope is that with the knowledge gained from these basic studies, one could build up a complex model closer to the reality. Some representative rheology models employed in the past studies of wave-seafloor problem are: viscous fluid mud (Gade 1958; Dalrymple & Liu 10

24 1978), elastic bed (Mallard & Dalrymple 1977), viscoelastic model (MacPherson 1980; Piedra-Cueva 1993), and viscoplastic seabed (Mei & Liu 1987; Sakakiyama & Bijker 1989). In fact, these simplified rheology models have been shown to fit fairly well with specified field observations (Krone 1963; Maa & Mehta 1987; Mei et al. 2010). Reviews on the early studies of wave-mud interactions have been well documents by Mehta, Lee & Li (1994), Foda (1995) and Wen & Liu (1998). It is noted that almost all of the above mentioned studies have considered only progressive waves of infinitesimal amplitudes. However, seabed effects become more significant as surface waves propagate into shallow waters where the wave nonlinearity is expected to be important as well. It is then the objective of the present study to investigate the problem in a more general context, i.e. relax the time periodicity assumption on the wave motions, and consider also the effects of wave nonlinearity. Since it is well known that the wave system is better described by Boussinesq equations in a shallow sea (see e.g. Peregrine 1972), waves that are both weakly nonlinear and weakly dispersive shall be of particular interest. For the purpose of better understanding the fundamental physics of wave-seafloor interactions, the following mud rheology models shall be considered: viscous, elastic, viscoelastic, and viscoplastic. In section 2.2, a single model is proposed to describe the interactions between surface waves and a muddy seabed made of Newtonian fluid, elastic mud, or linear viscoelastic materials, as it will be shown later in section that these three can actually be incorporated into a generalized viscoelastic model. In section 2.3, the dynamic response of a viscoplastic seabed to a surface wave is discussed. Before proceeding to the detailed analysis, an overview is first given in sec- 11

25 tions and to clarify the theoretical aspect of the present problem along with several important assumptions A simplified two-layer model and assumptions Inspired by the strong field evidence that surface waves can be damped out significantly at certain locales, a phenomena which can not be explained by the classical water wave theory where a rigid bottom is often assumed, Gade (1958) was perhaps the first to investigate the wave-seafloor interactions both theoretically and experimentally. In his study, a immiscible two-layer model, which consists of a layer of water and a relative heavier seabed lying on a flat solid bottom, was employed. In addition, the mud properties were assumed to be homogeneous. Since then, this two-layer approach has been widely adopted by other researchers (see e.g. Dalrymple & Liu 1978; Hsiao & Shemdin 1980; Mei & Liu 1987; Piedra-Cueva 1993; Ng 2000; Mei et al among others) due to its simplicity and good performance when validating with laboratory experiments and field observations. Without any surprise, the simple two-layer approach has been challenged. For instance, Maa & Mehta (1987, 1990) proposed a multi-layer stratified model since the properties of mud, such as viscosity and elasticity, can also depend on the concentration, which is essentially the mud density. To simulate numerically several laboratory-scale examples, they divided the mud beds into four distinct homogeneous viscoelastic layers, each of which has different values of viscosity and elasticity, according to the measured concentration profiles. It is remarked that in the situation where mud density varies considerably, i.e. 12

26 mud properties are not vertically uniform, the multi-layer stratified model certainly outperforms the two-layer approach. However, as the vertical variation becomes important one may also need to consider the time-varying layer thickness, which is not incorporated in this stratified model. To improve the twolayer treatment, Chou, Foda & Hunt (1991) also suggested another approach: a multi-phase layered model. For example, the entire mud column can be dived into three different layer (from top to bottom): viscous fluid, elastic mud, and a solid bottom. The layer thicknesses, which are determined as part of the solution, are no longer fixed. One can argue that this approach seems to be more realistic for the field applications as the moving interfaces have been considered. Nevertheless, one can also question the appropriateness of the viscouselastic-solid configuration for a muddy seabed. How to assign the properties to each sublayer, namely determine the proper rheology of each layer, remains an issue. In addition, while locations of interfaces between different materials change in time, whether mixing starts to play a role needs to be examined more carefully. From another perspective, Shibayama & An (1993) have proposed to consider the mud rheology as a function of wave forcing. More precisely, they suggested that the fluid mud act as either viscoelastic or viscoplastic material depending on the magnitude of the driven pressure force induced by the surface waves. Their model intends to replicate the complex properties of natural mud, although more field evidence is required to confirm the assumption that the rheology of mud is indeed switching between viscoelasticity and viscoplasticity. As a first step to carefully examine the wave-mud interactions under a general surface wave loading, the two-layer model will be adopted in the present study: we shall consider an inviscid water body on top of a layer of heavier 13

27 mud. Schematic sketch of the wave-seafloor system is given in figure 2.1. ζ z y x h 0 Water ξ d Mud Figure 2.1: Surface waves over a layer of mud. h 0 and d are the water depth and the mud thickness, respectively. ζ and ξ denote the displacements at the free-surface and the water-mud interface. x and y are the horizontal coordinates, and z is the vertical axis. The mud layer is sitting on top of a solid bed An overview of the mud rheology In addition to the two-layer assumption, the seabed will be modeled as either a generalized linear viscoelastic material or a viscoplastic mud. To help understand the complex mud rheology, figure 2.2 gives the schematic sketch of rheology curves for purely viscous, elastic, and plastic behaviors. Basically, stress is proportional to strain rate for viscous fluid; elastic behavior shows the linear relation between the stress and the strain; plastic material displays continuous 14

28 deformation after certain value of critical stress (yield stress) is achieved. (a) (b) (c) High viscous More stiff Yield stress Stress Stress Stress Low viscous Less stiff Strain rate Strain Strain rate Figure 2.2: Schematic diagram of rheology curves for: (a) Viscous mud; (b) Elastic mud; (c) Plastic mud. It can be expected that a viscoelastic material exhibits both viscous and elastic behaviors. Two conceptual rheology models are the Maxwell element and the Kelvin-Voigt element, both consisting of a linear combination of an elastic spring and a viscous damper (dashpot), as have been sketched in figure 2.3 (see e.g., Malvern 1969). Using the information given in figures 2.2 and 2.3, the responses of these two elements under a constant stress or a fixed deformation are illustrated in figure 2.4. This provides a qualitatively understanding of linear viscoelastic media. It is noted that the detailed constitutive equation of a viscoelastic mud is to be discussed in section For an ideal viscoplastic mud, namely a Bingham-plastic material, the rheology curve is demonstrated in figure 2.5. A Bingham-plastic mud behaves like a rigid body when the magnitude of the stress is less than the yield stress (see also figure 2.2), and flows pretty much as a viscous fluid at high stress. The detailed analysis of waves over a viscoplastic seabed is presented in section

29 (a) (b) Figure 2.3: Schematic diagram of linear viscoelastic media: (a) Maxwell element; (b) Kelvin-Voigt element. The stress is the same in the spring and the dashpot for a Maxwell element. The spring and the dashpot exhibit the same amount of deformation for a Kelvin-Voigt element. 2.2 A generalized model for surface waves interaction with a linear viscoelastic muddy seabed Early studies on the interactions between a layer of viscoelastic mud and surface waves relied on the introduction of a complex viscosity (see e.g., Tchen 1956; Hsiao & Shemdin 1980; MacPherson 1980; Maa & Mehta 1990; Piedra-Cueva 1993; Zhang &Ng 2006), ) E m ν e = ν m (1 + i, (2.2.1) ρ m ω 0 ν m where ν m is the kinematic viscosity of mud, E m the shear modulus of elasticity of mud, ρ m the mud density, and ω 0 the wave frequency. In terms of this complex viscosity ν e, the viscoelastic model shares the same governing equations with those of Newtonian fluid-mud case and, of course, the solution forms (Tchen 1956). It is remarked that the problem of surface waves over a viscous fluid-mud seabed has been studied extensively. Some representative references are Gade (1958), Dalrymple & Liu (1978), and Ng (2000). Despite the breakthrough of the complex viscosity concept, Ng & Zhang (2007) reiterated that this approach is 16

30 (a) (b) Stress Maxwell Time Time Strain K-V Time Strain Stress Strain Maxwell K-V Time Stress Time Time Figure 2.4: Viscoelastic behaviors of a Maxwell element and a Kelvin-Voigt element: (a) Under a constant load; (b) Apply a fixed deformation. valid only for the wave system of simple harmonic motions. By examining the field samples taken from the eastern coast of China, Mei et al. (2010) have shown that ν m and E m in (2.2.1) are actually functions of ω 0. This further confirms the limited applicability of the complex viscosity model. To investigate the higher harmonic components of wave motions which relate to the mass transport due to the muddy seabed, Ng & Zhang (2007) formulated the problem in the Lagrangian coordinates without using the complex viscosity approach commonly adopted in the Eulerian description. In fact, the work by Ng & Zhang (2007) can be viewed as the extension of Piedra-Cueva (1995) who has developed a Lagrangian model describing how surface waves interact with a layer of viscous fluid mud. The conservation laws presented by these two studies are, of course, general and valid for any surface wave 17

31 Stress Yield stress Viscous Strain rate Bingham-plastic Figure 2.5: Rheology curves for viscous and viscoplastic (Bingham-plastic) materials. The constitutive equation for a Bingham-plastic mud is given in loadings. However, when deducing the analytical solutions both Piedra-Cueva (1995) and Ng & Zhang (2007) considered only small amplitude waves. Therefore, up to now analytical solution for the viscoelastic or viscous mud flow motions driven by a transient long-wave loading is still not available. This motivates the present study. The effects of a muddy seafloor on surface wave propagation become more significant as waves enter shallow waters where the wave system is better described by Boussinesq equations (see e.g. Peregrine 1972). It is, therefore, the objective of this study to derive a set of Boussinesq-type depth-integrated equations for weakly nonlinear and weakly dispersive waves with the effects of a viscoelastic muddy seabed considered. It follows that the perturbation technique outlined in Mei, Stiassnie & Yue (2005) for deriving common Boussinesq equations, and also in Liu & Orfila (2007) for studying the effects of water viscosity on the evolution of shallow-water waves shall be applied. In the water-mud 18

32 system, an immiscible two-layer approach (Gade 1958), consisting of a water body on top of a heavier muddy seabed modeled as a linear viscoelastic material, is adopted. The mud viscosity is assumed to be several orders of magnitude larger than that of water. As a result, the water layer is treated as a inviscid fluid. Furthermore, the thickness of muddy seabed is taken to be very thin in comparison with the typical wavelength of the surface waves. It is reiterated that these assumed mud properties are in the range of field samples. For instance, during the pilot experiment in the Gulf of Mexico Elgar & Raubenheimer (2008) observed a layer of 30 cm thick yogurt-like bottom mud lying under a water body of 5 m in depth; Holland, Vinzon & Calliari (2009) reported a muddy seabed 0.4 m thick and a viscosity m 2 s 2 offshore of the Cassino Beach, Brazil. In the following, the mathematical model and the scalings for the motions of long waves in the water column are first discussed. Next, a set of depthintegrated equations is derived with a closure problem to be addressed by solving the mud flow problem beneath. After formulating the governing equations along with the proper initial and boundary conditions for the mud motions, a generalized rheology model for linear viscoelastic materials is then introduced to constitute the stress-strain relation for the muddy seabed. Subsequently, solutions for the motions of a thin layer of viscoelastic mud induced by a surface wave loading are obtained. The mathematical problem is formally completed. Using the newly derived equations, several important features, namely the bottom shear stress and velocity of mud flow, and the amplitude evolution of both linear progressive waves and solitary waves, are illustrated. Finally, the proposed model is examined against the available laboratory experiments. A good agreement is observed. 19

33 2.2.1 Depth-integrated model for weakly nonlinear and weakly dispersive water waves Consider a train of surface water waves with a characteristic wavelength L 0 and wave amplitude a 0 propagates in a uniform depth h 0 overlying a thin layer of bottom mud of thickness d. The wave-seafloor system is sketched in figure 2.1. In contrast to the mud column which is made of cohesive sediments, the water body of a constant density ρ w is treated as an inviscid fluid following the usual assumption of classical water wave theory. To ease the mathematical manipulation, the following dimensionless variables are introduced: t (x,y) = x, z = z, t = L 0 h 0 L 0 / gh 0 p = p, ζ = ζ, u = (u,v) = (u,v ) ρga 0 a 0 ǫ, w = gh 0 w (ǫ/µ) gh 0, (2.2.2) where (x,y ) and z denotes the horizontal and vertical references, respectively, t the time coordinate, g the gravitational acceleration, p the total pressure, ζ the free-surface displacement, and (u,v,w ) the velocity components of water particles in (x,y,z )-directions. In addition, two dimensionless parameters ǫ = a 0 h 0 and µ = h 0 L 0 (2.2.3) measure the relative importance of the wave nonlinearity and the frequency dispersion, respectively, and both are considered to be small. Consequently, the dimensionless continuity equation in the water body can be expressed in terms of the velocity potential, Φ = Φ(x,y,z,t), as µ 2 2 Φ + 2 Φ = 0, 1 z ǫζ, (2.2.4) z2 20

34 and the kinematic and dynamic free-surface boundary conditions are ( ) ζ µ 2 + ǫ Φ ζ = Φ, z = ǫζ, (2.2.5) t z ( ) [ Φ µ 2 t + ζ + ǫ ( ) ] 2 Φ µ 2 ( Φ) 2 + = 0, z = ǫζ, (2.2.6) 2 z ( where x, y ) denotes the horizontal gradients. In addition, the pressure field can be evaluated from the Bernoulli s equation as { [ p = z ǫ 1 µ 2 Φ µ 2 t + ǫ ( ) ]} 2 Φ µ 2 ( Φ) 2 +, (2.2.7) 2 z where the first term is the hydrostatic pressure and the rest the hydrodynamic pressure. The velocity potential Φ may be expended in terms of a power series in the vertical coordinate z as (see Chapter 12.1 in Mei, Stiassnie & Yue 2005) Φ(x,y,z,t) = (z + 1) n φ n (x,y,t). (2.2.8) n=0 Therefore, the direct substitution of (2.2.8) into the Laplace equation, i.e. the conservation law of mass (2.2.4), leads to a recursive relation Note that and µ 2 φ n+2 = (n + 1)(n + 2) 2 φ n, n = 0, 1, 2,. (2.2.9) φ 0 = Φ z= 1 = u(x,y,z = 1,t) u b (2.2.10) φ 1 = Φ z = w(x,y, 1,t) w b (2.2.11) z= 1 represent the horizontal and vertical velocity components at the water-mud interface z = 1, which are now defied as u b and w b, respectively. In the case of a horizontal solid sea bottom, the no flux condition requires w b = 0 suggesting that each φ n with odd n vanishes. For the present wave-seabed problem, 21

35 the wave-driven mud motion leads to a non-zero w b. It is necessary to estimate the order of magnitude of this quantity. Since in coastal waters the thickness of mud bed, d, is usually very small and the mud viscosity, ν m, is relatively strong, under long water waves the laminar boundary-layer thickness of mud, δ m, can be comparable to d. Therefore, in this study the focus will be on the following scenario: d δ m νm = αl 0, (2.2.12) gh0 /L 0 where α 2 = ν m L 0 gh0 (2.2.13) is a dimensionless parameter. To give a quantitative example, let us consider a typical case: O(ǫ) O(µ 2 ) 0.1, h 0 5 m, d 0.25 m, ν m 0.01 m 2 s 1, (2.2.14) where both ν m and d are in the range of field data reported by Mei et al. (2010). It follows that the value of α is roughly 0.01, i.e. O(α) O(µ 4 ). (2.2.15) Note that the condition (2.2.15) has also been assumed by Ng (2000), Ng & Zhang (2007) and Mei et al. (2010) to study periodic waves over a viscous or viscoelastic muddy seabed. Furthermore, many field observations (see e.g., Sheremet & Stone 2003; Winterwerp et al. 2007; Holland, Vinzon & Calliari 2009) have supported this argument. In this study, the assumption (2.2.15) will be adopted throughout. It also deserves emphasis that the mud motion considered is in the laminar flow regime. A Reynolds number can be introduced as Re m = ( ǫ gh0 ) d ν m = ǫ α 2 d L 0, (2.2.16) 22

36 where α has been defined in (2.2.13). By the use of (2.2.12) and (2.2.15), we obtain O (Re m ) = O(µ 2 ), (2.2.17) which is a moderate value for the weakly dispersive waves to be discussed herein. In fact, this statement complies with the immersible assumption: a sharp density interface is persistent in the two-layer model, which surpasses the possible turbulence. Through the above argument, the horizontal and vertical components of mud flow velocity are estimated to be O (u m) O (ǫ ) gh 0 and O (w m) O (αǫ ) gh 0, (2.2.18) respectively. By virtue of matching the vertical velocity across the water-mud interface, we obtain O(w b ) O(αµ) O(µ 5 ). (2.2.19) Consequently, from (2.2.8) to (2.2.11) the truncated velocity potential with an error of O(µ 6 ) is Φ = (z + 1)w b + u b µ2 2 (z + 1)2 2 u b + µ4 24 (z + 1)4 2 2 u b + O(µ 6 ), (2.2.20) where (2.2.19) has been evoked as well. Under the Boussinesq assumption, i.e. O(ǫ) O(µ 2 ), the use of (2.2.20) into the free-surface conditions, (2.2.5) and (2.2.6), yields 1 H ǫ t + (Hu b) µ2 6 2 u b w b µ = 2 O(µ4 ), (2.2.21) and u b t + ǫu b u b + 1 µ2 H ǫ 2 t u b = O(µ 4 ), (2.2.22) 23

37 where H = 1 + ǫζ (2.2.23) denotes the total water depth. Equations (2.2.21) and (2.2.22) are the approximate continuity and momentum equations in terms of H and the velocity at the bottom of water body, (u b,w b ). These vertical independent Boussinesq-type equations can also be expressed in the form of the depth-averaged horizontal velocity defined by u = 1 H ǫζ 1 Φdz = u b µ2 6 H2 2 u b + O(µ 4 ). (2.2.24) Substituting the above definition into (2.2.21) and (2.2.22), we obtain and 1 H ǫ t + (Hu) w b µ = 2 O(µ4 ), (2.2.25) u t + ǫu u + 1 µ2 u H ǫ 3 t = O(µ4 ). (2.2.26) Equations (2.2.25) and (2.2.26) constitute the Boussinesq-type depth-averaged equations in terms of the total depth, H, and the depth-averaged horizontal velocity, u. The effects of the underlaid thin mud layer appear in the continuity equation through a nonzero w b term and are of O(µ 3 ). In the absence of the muddy sea bed where the solid bottom is also frictionless, w b = 0 and the above equations reduce to the conventional Boussinesq equations. It is remarked that (2.2.25) and (2.2.26) are underdetermined, as three unknowns (ζ,u,w b ) are involved. Ideally, if w b can be expressed in terms of u and/or ζ the mathematical problem is then complete (of course, proper initial and boundary conditions for both ζ and u are still required). Owing to the continuity of vertical velocity at the water-mud interface, w b essentially describes 24

38 the vertical motion of mud flow at z = 1 as well. Therefore, it sheds some insight on this closure issue that the solution form of w b may be obtained from the flow problem inside the mud layer. Details will be elaborated in the following sections, to It is beneficial to point out that actual velocity components, (u,v,w), and pressure field, p, are realized once ζ and u are solved. The approximate velocity is obtained, by definition, as and (u,v) = Φ = u µ2 2 (z + 1)2 u + O(µ 4 ), (2.2.27) From (2.2.7), the total pressure becomes w = Φ z = µ2 (z + 1) 2 u + O(µ 4 ). (2.2.28) p = z ǫ + ζ + µ2 2 ( z 2 + 2z ) + O(µ 4 ). (2.2.29) Model equations for mud flow motions Since viscous shearing is one of the key factors affecting the mud flow motions, we shall introduce new scalings to describe dynamics inside the muddy seabed. For the mud flow velocity components, (u m,v m,w m), pressure, p m, and the shear stress tensor, τ m, the normalizations are as follow: u m = (u m,v m ) = (u m,v m) ǫ gh 0, w m = p m = p m ρ m ga 0, τ m = τ m αǫρ m gh 0 w m αǫ gh 0, (2.2.30) 25

39 where ρ m is the mud density and recall α defined by (2.2.13). Note that τ m, xx τ m, xy τ m, xz τ m = τ m, yx τ m, yy τ m, yz. (2.2.31) τ m, zx τ m, zy τ m, zz Recalling (2.2.12) that the mud depth d is assumed to be comparable to the laminar boundary-layer thickness δ m αl 0, a new dimensionless vertical coordinate is introduced: η = z + d + h 0 αl 0. (2.2.32) The mud then occupies 0 η d in the stretched coordinate where d = d αl 0 + ǫµ α ζ m a 0 (2.2.33) with ζ m denoting the vertical displacement of the water-mud interface. By the order of magnitude analysis on the mass conservation of both water body and mud column, it can be shown that ζ m is much smaller than the free-surface displacement, ζ, O (ζ m/ζ ) O (d /(d + h 0 )) O(d /h 0 ) O(µ 3 ) 1. (2.2.34) The above statement has been verified by laboratory study of solitary waves propagate over a viscous fluid-mud bed (Park, Liu & Clark 2008) and the examination of field samples of viscoelastic mud subject to a surface periodic wave forcing (Mei et al. 2010). Following (2.2.34), d = d αl 0 + O(µ 2 ). (2.2.35) In terms of the above dimensionless variables, we can now formulate the conservation law of mass for mud flow as u m + w m η = 0, (2.2.36) 26

40 and the momentum equations: u m t α 2 [ wm t + ǫ ( ) u m u m u m + w m η = p m + ( α τ HH m + ( )] w m + ǫ u m w m + w m = p ( m η η + α α τ V m H τ HV m η + τ V m V η ), (2.2.37) ) α ǫµ, (2.2.38) where and τ HH m = τ m, xx τ m, yx τ m, xy τ m, yy, τ HV m = (τ m, xz,τ m, yz ), (2.2.39) τ V H m = (τ m, zx,τ m, zy ), τ V V m = τ m, zz. (2.2.40) Referring again to (2.2.25) and (2.2.26), the long-wave equations that describe the motions of water particles, leading-order solutions of (u m,w m ) and p m are sufficient to satisfy the overall truncation error of O(µ 4 ) as w b = αµw m (x,y,η = d,t). (2.2.41) Therefore, it is reasonable to neglect the displacement of water-mud interface in the present study, i.e., (2.2.35) reduces to d d αl 0. (2.2.42) The significance of the above assumption is that d becomes a constant parameter of O(1). All in all, we shall now work on the linearization of (2.2.37), u m t = p m + τ HV m η, 0 η d. (2.2.43) As for the vertical equation, (2.2.38), it suggests that at the leading-order pressure is vertically uniform inside the mud layer, i.e. p m = p m (x,y,t), 0 η d. (2.2.44) 27

41 In addition, the continuity of normal stress along the water-mud interface reduces to where p m (x,y,t) = γp(x,y,z = 1,t), (2.2.45) γ = ρ w ρ m (2.2.46) is the ratio of water density to mud density. Furthermore, the use of (2.2.26) into (2.2.29) leads to p u b, z = 1 (2.2.47) t at the leading-order. The approximate problem, (2.2.43) to (2.2.47), is similar to that of classic laminar boundary-layer theory, which is expected since d δ m. Evoking (2.2.45) and (2.2.47) into the horizontal momentum equation (2.2.43), we obtain u m t = γ u b t HV τ m + η. (2.2.48) The associated boundary conditions in the vertical coordinate are u m = 0, η = 0, (2.2.49) τ HV m = 0, η = d, (2.2.50) which satisfy the no-slip condition and the inviscid water assumption, respectively. In addition, u m = 0, t = 0 (2.2.51) is imposed as the initial condition. Reviewing (2.2.48) to (2.2.51), the solution of u m can be obtained in terms of u b under the circumstances that shear stress is a linear function of u m, um η their time operations. The understanding of mud rheology is therefore essential, and will be discussed next. and 28

42 2.2.3 Rheology model for a linear viscoelastic mud The muddy seabed, made of cohesive sediments, is modeled as a general linear viscoelastic body. Here, the term general signifies the fact that Newtonian fluids and purely elastic media shall be recovered from the proposed viscoelastic model as two limiting cases. In addition, the linearity refers to the direct proportionality between the shear stress τ and shear strain ε at all time (Barnes, Hutton & Walters 1991). In other words, the effects of successive changes in shear strain are additive. Following the study of Boltzmann (see e.g., Fabrizio & Morro 1992; Lakes 2009), the three-dimensional constitutive equation for a linear viscoelastic material can be expressed in a general form as τ ij(x i,t ) = t 0 R ijkl (x i,t t ) ε kl (x i,t ) t dt, i,j,k,l = 1, 2, 3, (2.2.52) where R is the relaxation function, which describes over time under a fixed level of strain the decreasing of stress from its peak value. Note that both ε and τ are zero at t = 0. The shear-strain relation (2.2.52) can be inverted to obtain ε as a similar time convolution integral of C and the rate of change of τ, where C denotes the creep function describing the change of strain in time subject to a constant stress. In practice, the rheology model (2.2.52) is difficult to apply due to the complexity of R. We shall limit ourselves to a special case of homogeneous materials such that the relaxation function is only a function of time. Now, recall the common strain-displacement relationship (see e.g. Kundu & Cohen 2002), ε kl = 1 ( ) X k + X l, (2.2.53) 2 x l x k where X is the displacement vector. Evoking the assumption of homogeneous material properties and (2.2.53), the constitutive equation (2.2.52) can be recast 29

43 into a differential equation (Malvern 1969; Barnes, Hutton & Walters 1991), P p=0 p τ Q T p ij t p = D q q t q q=0 ( X i x j ) + X j, (2.2.54) x i where P, T p, Q(= P or P + 1), and D q are constant coefficients to be determined experimentally. Note that the finite order of (2.2.54) is equivalent to the limited discrete record of continuous relaxation function, R. Note also that (2.2.54) reduces to the constitutive relation of Newtonian fluids if: P = 0, T 0 = 1, Q = 1, D 0 = 0, D 1 = µ v, (2.2.55) where µ v is the dynamic viscosity. Similarly, for P = 0, T 0 = 1, Q = 0, D 0 = E e, (2.2.56) (2.2.54) recovers the case of purely elastic mediums in which E e denotes the shear modulus of elasticity. Therefore, both the viscous and elastic cases can be viewed as special scenarios of the generalized linear viscoelastic problem. Before applying the generalized viscoelastic rheology model, (2.2.54), to our mud flow problem, we shall discuss two elementary cases, namely Maxwell s model and Kelvin-Voigt s model (see e.g., Malvern 1969), for a better understanding of material behaviors and the associated relevance to the wave-mud studies. Both under a phenomenological concept of a two-component Hookean spring-and-newtonian dashpot system, Maxwell element has a spring and a dashpot in series whereas the Kelvin-Voigt element consists of a spring and a dashpot in parallel (see figure 2.3). It is obviously that in Kelvin-Voigt s model both spring and dashpot are constrained to deform the same amount, and the total stress is the sum of the stresses from these two parts. Alternatively, in the design of the Maxwell element, the spring and dashpot are subjected to the 30

44 same stress while the total strain being the summation from both parts. Therefore, cast in the generalized formulation, (2.2.54), these two conceptual models are associated with the following constant coefficients: Maxwell: P,Q = 1, T 0 = 1, T 1 = µ v /E e, D 0 = 0, D 1 = µ v, Kelvin-Voigt: P = 0, Q = 1, T 0 = 1, D 0 = E e, D 1 = µ v. (2.2.57) Recall that E e is the elastic modulus of the Hookean spring and µ v the dynamic viscosity of the Newtonian dashpot. It is known that Maxwell s model does not predict creep in material accurately, and the Kelvin-Voigt element shows a retarded elastic behavior (Malvern 1969). Despite their utility, quantitative representation of real viscoelastic materials is not always guaranteed by these two simple models. Through laboratory rheology tests on the estuarial mud samples 1, exhibiting both viscous and elastic behaviors, Maa & Mehta (1988) have suggested that Kelvin-Voigt element is a better selection for modeling the mud responses. In fact, the two-parameter Kelvin-Voigt s model has been adopted by many researchers (see e.g., Hsiao & Shemdin 1980; MacPherson 1980; Maa & Mehta 1990; Piedra-Cueva 1993 and others) to study the interactions between small-amplitude surface waves and a viscoelastic seabed. A remarkable observation, first revealed by Tchen (1956), is that a mathematical problem encountered in the study of periodic waves interacting with a Kelvin-Voigt viscoelastic medium is essentially identical to that of viscous fluid-mud case. The reasoning is as follows. By the use of complex variables, the periodicity of wave motions permits a new expression of a constitutive relation from (2.2.54) and (2.2.57): τ ij = ( µ + i E ) ( e u i ω x j ) + u j, (2.2.58) x i where ω is the wave frequency and u i = X i t the velocity field of mud flow. 1 Mud samples were taken from Cedar Key, Florida. See Maa & Mehta (1988). 31

45 Consequently, the introduction of a complex viscosity, µ kv = µ v + i E e ω, (2.2.59) into (2.2.58) draws the conclusion suggested by Tchen (1956). Regardless of a great appreciation for the complex viscosity approach, Ng & Zhang (2007) have re-emphasized that this concept can only be applied to problems involving simple harmonic waves. Furthermore, examining the field mud samples from sites along the eastern coast of China, Mei et al. (2010) reported that, under periodic motions, the values of µ v and E e are actually functions of ω when fitting the results of rheology tests to (2.2.59). Similar time-dependent behavior of presumed constant coefficients was also observed by Jiang & Mehta (1995) who studied the properties of viscoelastic mud samples taken from the southwest coast of India. The above two laboratory tests both support the fact that Kelvin-Voigt element is only an approximate rheology model for viscoelastic materials, as has been mentioned. Reviewing (2.2.54) and (2.2.57), findings of Mei et al. (2010) and Jiang & Mehta (1995) can be interpreted as the insufficient order of the truncated constitutive equation, i.e. P and Q are not large enough to provide the desired accuracy. The present study does not intend to suggest a better rheology model. Instead, the research focuses on the interactions between long waves and a viscoelastic seabed where the constitutive relationship is known a priori. Therefore, the solution methodology of mud flow motions shall be developed based on the generalized shear stress formulation for a linear viscoelastic medium, (2.2.54), where the problem of a Newtonian or elastic mud is also a special case (see the discussion in (2.2.55)). 32

46 2.2.4 Solution forms inside the muddy seabed We shall now discuss the solutions of the mud motions. From (2.2.39), (2.2.48) and (2.2.54), let us first formulate the necessary components of shear stress gradient in the normalized form as N p τ HV m T p t p η p=0 = M q=0 M q=0 ( q 2 X m D q + α 2 ) t q η 2 η Z m D q q t q ( 2 X m η 2 ), (2.2.60) where X m = (X m,y m ) and Z m are the horizontal and vertical displacements of mud, respectively, and (T p, D q ) the dimensionless coefficients. Note that the following new normalizations have been introduced: T p = (X m,y m ) = (X m,y m), Z m = Z m ǫl 0 αǫl 0 T p ( L0 / D ) q p, D q = ( gh 0 ρ m ν m L0 / ) q 1 gh 0. (2.2.61) Clearly, the relationship between τ HV m and X m is implicitly defined through (2.2.60). By taking the Laplace transform of (2.2.60), however, an explicit expression can be deduced in terms of transformed variables, provided the necessary initial conditions are accessible. In other words, τ HV m η = S(s) 2 Xm η 2, (2.2.62) where ( ) denotes the transformed function in s domain as defined by F(s) = 0 e st F(t)dt, (2.2.63) and S = S(s) is a function of s only. Note that the actual function of S is determined by the coefficients p n and q m. For instance, as the two simplest models 33

47 are associated with the following dimensionless coefficients (see (2.2.57) and (2.2.61)): Maxwell: P,Q = 1, T 0 = 1, T 1 = Wi, D 0 = 0, D 1 = 1, Kelvin-Voigt: P = 0, Q = 1, T 0 = 1, D 0 = 1/Wi, D 1 = 1, (2.2.64) we obtain Maxwell: S = s 1 + swi, Kelvin-Voigt: S = s + 1 Wi, (2.2.65) where Wi = µ m/e m L 0 / gh 0 (2.2.66) is the Weissenberg number defining the ratio of the relaxation time to the process time. It is reiterated that E m and µ m are the elastic modulus and dynamic viscosity of the viscoelastic mud, respectively. The above discussion suggests the use of the Laplace transform to solve the initial-boundary-value problem, (2.2.48) to (2.2.51) along with (2.2.61). As a result, in the transformed domain the counterpart of the original problem becomes an ordinary differential equation: s 2 Xm = γs 2 Xb + S(s) 2 Xm η 2, (2.2.67) X m = 0, η = 0, (2.2.68) X m η = 0, η = d, (2.2.69) where X b = X b (x,y,t) denotes the horizontal displacements of water particles at the water-mud interface, z = 1 or η = d. By further introducing a new variable, X = X m γx b, (2.2.70) 34

48 the problem is now: s 2 X = S(s) 2 X η 2, (2.2.71) X = γx b, η = 0, (2.2.72) X η = 0, η = d. (2.2.73) Solution can then be obtained as ( cosh s (d η)/ ) S X = γ X b ( cosh sd/ ) γ X b R(η,s), (2.2.74) S where the inverse Laplace transform of R can be viewed as a response function describing the mud motions induced by the surface wave loadings. Applying the convolution theorem, the inversion of X is X(x,y,η,t) = t 0 c i γx b (x,y,t τ)r(η,τ)dτ. (2.2.75) The function R(η,t), by definition, is ( R(η,t) = 1 c+i e stcosh s(d η)/ ) S ( 2πi cosh sd/ ) ds, (2.2.76) S where the path of integration with respect to s is a vertical line parallel to and on the right of the imaginary axis in the complex s-domain. In practice, the complex integral in (2.2.76) can be evaluated using the Cauchy s residue theorem 2. Afterwards, X m = X + γx b and u m = X m t = X t + γu b, u b = X b t, (2.2.77) are finally deduced in the form of u b. Subsequently, the vertical displacement is calculated from (2.2.36) as η Z m (x,y,η,t) = X m (x,y,η,t)dη, (2.2.78) 0 2 An example based on the two-component Kelvin-Voigt element is demonstrated shortly. 35

49 where the bottom boundary condition, Z m = 0, η = 0, (2.2.79) has been evoked. Recalling the condition (2.2.41) which states the matching of vertical velocity at the water-mud interface, we then obtain the vertical component of water particle velocity at the bottom of water body as d w b (x,y,t) = αµ 0 u m (x,y,η,t)dη. (2.2.80) Reviewing the newly derived Boussinesq-type wave equations, (2.2.21) and (2.2.22), the problem is now officially complete as w b has been indirectly expressed in terms of u b, i.e. w b (x,y,t) = αµi(γ,d,u b ), (2.2.81) where the detail of function I is determined by the actual mud rheology. It is clear from (2.2.75), (2.2.77), and (2.2.81) that derivatives involved in I are u b and ( u t b). The same is also true for model equations (2.2.25) and (2.2.26) since u u b at the leading-order. Special cases: Kelvin-Voigt element, viscous fluid mud, and elastic mud Solution technique for the mud flow problem of a generalized linear viscoelastic seabed has been presented in the above. However, detailed expression of the mud response function R(η, t), i.e. (2.2.76), still needs to be worked out once the rheology model is specified. Without loss of generality, let us consider the two-component Kelvin-Voigt element for the demonstration. Two representative limiting cases, namely Newtonian fluid-mud and purely elastic mud, will also be discussed. 36

50 (1) Simple model for a viscoelastic mud: Kelvin-Voigt element For the two-component Kelvin-Voigt s model, S(s) = s + 1/Wi as has been discussed in (2.2.65). Since the viscoelastic mud is of interest, we shall consider a case where O (1/Wi) = O(1), which can be interpreted as both the viscous and elastic effects are considered equally important in our analysis. Now, substitute the expression of S into (2.2.76). Since cosh (i(n + 1/2) π) = 0 for any given integer n, there are poles at s n = 1 [ ] 2 (2n + 1)π 2 2d 1 ± 1 4 [ ] 2 2d, n = 0, 1, 2,. Wi (2n + 1)π (2.2.82) By the Cauchy s residue theorem, the inversion of R can be evaluated as ( cosh s(d η)/ ) S R(η,t) = n=0 (sd/ cosh ) e st S s s=sn = 2 s 2 n cosh (κ n (d η)) e snt, (2.2.83) d κ n (2s n κ 2 n) sinh (κ n d) where Finally, n=0 κ n = s n sn + 1/Wi. (2.2.84) X m = γx b 2 γ d n=0 s 2 n/κ n 2s n κ 2 n cosh (κ n (d η)) t X b (x,y,t τ)e snτ dτ. (2.2.85) sinh (κ n d) 0 The corresponding horizontal velocity component is u m = γu b + 2 γ d n=0 s n /κ n 2s n κ 2 n cosh (κ n (d η)) t sinh (κ n d) 0 u b (x,y,τ) τ [ 1 e s n(t τ) ] dτ. (2.2.86) 37

51 Recalling (2.2.78), the vertical velocity component becomes w m = γη u b 2 γ d n=0 t s n /κ 2 n 2s n κ 2 n 0 { 1 sinh (κ } n(d η)) sinh (κ n d) u b (x,y,τ) τ [ 1 e s n(t τ) ] dτ. (2.2.87) (2) Newtonian fluid mud If the muddy seabed is made of a viscous fluid, i.e. E m = 0 or Wi, solutions of mud flow velocity, (2.2.86) and (2.2.87), are reduced to u m = γu b 2 γ d n=0 sinh ( sn η ) t sn 0 u b (x,y,τ) τ [ 1 e s n(t τ) ] dτ, (2.2.88) and w m = γη u b 2 γ d 1 cosh ( sn η ) t s n=0 n 0 u b (x,y,τ) τ [ 1 e s n(t τ) ] dτ, (2.2.89) where s n = [(2n + 1)π/ (2d)] 2 is also obtained from (2.2.82). Of course, these new solutions can also be obtained following the same procedure introduced above with S(s) = s. Note that solutions (2.2.88) and (2.2.89) are, in fact, identical to the results reported by Liu & Chan (2007a) who studied only waves over a layer of viscous mud and expressed the solutions in terms of complementary error function. Taking (2.2.88) for example, one can follow Tikhonov & Samarskii (1963) to show that the series is formally interchangeable with u m = γu b γ 2 π n= t ( 1) n (η + 2nd) 0 u b (x,y,τ) (t τ) 3/2 exp [ ] (η + 2nd)2 dτ. 4(t τ) (2.2.90) 38

52 Employing integration by parts, the integral in (2.2.90) becomes Therefore, u m = γu b γ 2 π t η + 2nd 0 n= ( u b (x,y,τ) η + 2nd erfc )dτ. (2.2.91) τ 4(t τ) t ( 1) n which is exact the same as that of Liu & Chan (2007a) 3. 0 ( u b (x,y,τ) η + 2nd erfc )dτ, (2.2.92) τ 4(t τ) (3) Purely elastic mud Solutions for the case of a elastic mud can be obtained by taking proper limits of (2.2.86) and (2.2.87) at Wi 0. Whereas the stretched vertical coordinate η breakdowns at α = 0 (i.e., ν m = 0), we shall set an artificial viscosity as ρ m ν m = E m L 0 / gh 0 for the sake of keeping the same vertical coordinate. Therefore, at the elastic limit (2.2.86) reduces to u m = γu b 2 γ d n=0 sin (s n η) t s n 0 u b (x,y,τ) τ [1 cos (s n (t τ))]dτ, (2.2.93) where s n = (2n + 1)π/(2d). Similarly, (2.2.87) becomes w m = γ u b η + 2 γ d n=0 1 cos(s n η) s 2 n t 0 u b (x,τ) τ [1 cos (s n (t τ))]dτ. (2.2.94) Note that the above results can also be obtained by solving the elastic limit of (2.2.67) to (2.2.69) with S = 1 directly, where the reduced equation is identical to the wave equation and the solution is known in the literatures (see e.g. Tikhonov & Samarskii 1963). 3 See Eq. (2.2.1) in Liu & Chan (2007a). Note that in their study d < η < 0, while in the present formulation 0 < η < d. 39

53 1HD example: a surface solitary wave loading To illustrate the mud motions excited by surface waves, let us consider a case of a solitary wave loading in one horizontal dimension (1HD). This example shall also show that the current study outperforms the early studies (Gade 1958; MacPherson 1980; Ng 2000 and others) as they considered only periodic waves. The horizontal velocity of a water particle at the water-mud interface is assumed to be described by the canonical solitary wave solution, ( ) 3ǫ u b = sech 2 2µ (x x 0 c t), (2.2.95) where c = 1 + ǫ is the dimensionless wave celerity and x 0 denotes the initial location of the wave crest. In this section, ǫ = µ 2 = 0.1 and x 0 = 50 are employed. In order to gain some insights on the effects of both elasticity and viscosity, let us examine the responses of three different types of viscoelastic mud (Wi = 0.5, 1, 2) along with a purely elastic mud (Wi = 0) and a viscous fluidmud (Wi = ) 4. Note that the two-component Kelvin-Voigt element has been adopted as the rheology model for the viscoelastic mud. In all cases, γ = 0.85 and d = 5 are fixed. Figure 2.6 shows the time histories of horizontal velocity at the water-mud interface, u mi = u m η=d. The corresponding bottom shear stress, τ mb = τ m,xz η=d, is plotted in figure 2.7. It is not surprising to observe that both u mi and τ mb oscillate for the cases of elastic and viscoelastic mud. Furthermore, as the relative elasticity becomes stronger, i.e. smaller Wi, the oscillation becomes more violent. For both viscous and viscoelastic mud, i.e. Wi > 0, mud motions eventually vanish due to the viscous damping, although it takes a longer time to settle down for more elastic mud. As for the case of a purely 4 It is reminded that Wi, the Weissenberg number, has been defined in (2.2.66). 40

54 elastic mud (Wi = 0), however, it is evident that the excited mud motions do not attenuate. Through this example, effects of both viscosity and elasticity on the mud responses are clearly illustrated Wi = 0.0 Wi = 0.5 Wi = 1.0 Wi = 2.0 Wi = u mi u b (x x 0 ct) Figure 2.6: A surface solitary wave over a viscoelastic mud: Time histories of horizontal mud flow velocity at the water-mud interface, u mi. Lower corner plots the horizontal velocity of imposed solitary wave, u b. While dashed, solid and dashed-dotted lines denote the cases of different viscoelastic mud with Wi = 0.5, 1, 2 respectively, the gray line plots the result of a purely elastic muddy bed (i.e. Wi = 0) and bold line represents the case of viscous fluid-mud (Wi = ). In this example, x 0 = 50, ǫ = µ 2 = 0.1, γ = 0.85 and d = 5. Figure 2.8 further plots the vertical profiles of horizontal velocity inside the mud column corresponding to the cases shown in figures 2.6 and 2.7. During the acceleration phase of imposed solitary wave (i.e. u b t > 0), the velocity profiles for different mud samples are very similar. However, as the solitary wave starts to decelerate, these velocity profiles behave very differently depending on the 41

55 τ mb Wi = 0.0 Wi = 0.5 Wi = 1.0 Wi = 2.0 Wi = (x x 0 ct) Figure 2.7: A surface solitary wave over a viscoelastic mud: Time histories of bottom shear stress, τ mb, with the insert highlights the details. Dashed, solid and dashed-dotted lines plot the results of different viscoelastic mud with Wi = 0.5, 1, 2 respectively, gray line represents the elastic mud (Wi = 0) and the bold line plots the case of viscous fluid-mud (i.e. Wi = ). In this example, x 0 = 50, ǫ = µ 2 = 0.1, γ = 0.85 and d = 5. mud elasticity. The oscillation and attenuation are of course observed. Note that features of mud responses shown in figures 2.6 to 2.8 are similar to the findings of Park & Liu (2010) who studied experimentally pipe flow motions of a viscoelastic-plastic fluid driven by oscillatory pressure gradients. 42

56 Wi = 0.0; Wi = 0.5; Wi = 1.0; Wi = 2.0; Wi = θ = θ = η d θ = θ = u m u m u m u m η d θ = 5 θ = 15 θ = 25 θ = u m u m u m u m Figure 2.8: A surface solitary wave over a viscoelastic mud: Profiles of horizontal velocity component, u m, inside the entire mud column at several different phases, θ = (x x 0 ct). The upper four panels (θ = 1.521, 0.634, 1.521, 3.456) correspond to u b = (+)0.25, (+)0.75, ( )0.25, ( )0.01 where (+) and ( ) represent the accelerating and decelerating phases of the imposed solitary wave, respectively. For the remaining phases, u b 0. All parameters are same as those in figures 2.6 and

57 HD application: evolution of wave height of a surface solitary wave We shall now examine the effects of muddy seabed on the surface water waves. For simplicity, let us consider again a solitary wave in one horizontal dimension (1HD). The solution procedure presented here follows closely the approach outlined in Mei, Stiassnie & Yue (2005) for studying viscous damping of solitary waves (see also Liu & Orfila 2007). For the 1HD problem, the conservation laws of mass and momentum can be reduced from (2.2.25) and (2.2.26) as and ζ t + x [(1 + ǫζ) u] α µ I = O(µ4 ), (2.2.96) u t + ǫu u x + ζ x µ2 3 u 3 x 2 t = O(µ4 ), (2.2.97) respectively. Recall the function I is defined by (2.2.81), and in this case I = I(γ,d,u). (2.2.98) As suggested by Mei, Stiassnie & Yue (2005), let us first introduce a moving coordinate, σ = x t, (2.2.99) and also a slow time variable, ϕ = ǫt. ( ) The temporal and spatial derivatives then become: t σ + ǫ ϕ, x σ. ( ) 44

58 Folding ( ) into the summation of (2.2.96) and (2.2.97), we obtain ǫ (ζ + u) + ǫ (ζu) + ǫu u ϕ σ σ + µ2 3 u 3 σ α 3 µ I = O(µ4 ). ( ) Recall that O(α) O(µ 4 ). Therefore, with an error of O(µ 2 ) only the leadingorder solutions of ζ and u are required in ( ). The leading-order solution of an undisturbed solitary wave can be expressed in terms of the newly defined coordinates as ζ = a sech 2 ( 3a 2 where a = a(ϕ) is the dimensionless wave height. Moreover, ( σ a 2 ϕ ) ), ( ) u = ζ ( ) is the leading-order approximation. Using ( ), we obtain from ( ) a partial differential equation for ζ, where is also evident. ζ ϕ ζ ζ σ + 1 µ 2 3 ζ 6 ǫ σ α ǫµ I = O(µ2 ), ( ) I = I(γ,d,u) I(γ,d,ζ) ( ) Under the effect of a muddy seabed, the surface solitary wave is expected to be perturbed from its original solution, i.e. ( ). Let us introduce the perturbation solution as follows (see Mei, Stiassnie & Yue 2005): where ζ = δ n ζ n, ζ n = ζ n (ρ,ξ), ( ) n=0 δ = α ǫµ, ξ = δϕ = α µ t, ρ = σ 1 2 ϕ a(ϕ )dϕ = σ 1 ξ a(ξ )dξ. ( ) 2δ 45

59 Notice that σ ρ, ϕ δ ξ a 2 ρ. ( ) Substituting ( ) into ( ) and then collecting terms at different orders, we obtain at O(δ 0 ): and at O(δ): a 2 ζ 0 ρ ζ ζ 0 0 ρ + 1 µ 2 3 ζ 0 6 ǫ ζ0 3 = 0, ( ) ζ 0 ξ a ζ 1 2 ρ ρ (ζ 0ζ 1 ) µ 2 ǫ 3 ζ 1 ζ I 0 = 0, ( ) where I 0 is the leading-order of I and is a function of ζ 0. Following Ott & Sudan (1970) (see also Mei, Stiassnie & Yue 2005), ( ) and ( ) can be recast as and L 1 ζ 1 ρ L 0 ζ 0 ρ [ a ζ ] µ 2 2 ζ 6 ǫ ρ 2 0 = 0, ( ) [ a ζ ] µ 2 2 ζ 6 ǫ ρ 2 1 = ζ 0 ξ I 0, ( ) respectively. The operators L 0 and L 1 satisfy (ζ 0 L 1 ζ 1 ζ 1 L 0 ζ 0 )dρ = 0, ( ) as they are adjoint operators of each other (Mei, Stiassnie & Yue 2005). Therefore, ( ) provides a solvability condition for ζ 1, ( ζ 0 ζ 0 ξ + 1 ) 2 I 0 dρ = 0. ( ) It is remarked that this condition is actually valid for any weakly nonlinear and weak dispersive wave loadings, as long as ζ u is legit at the leading-order. 46

60 Apparently, the leading-order solution, ζ 0, is just the solitary wave solution which can be re-expressed from ( ) as ( ) 3a ζ 0 = a sech 2 2 ρ, a = a(ξ). ( ) Substituting ( ) into ( ), we then obtain an integral equation for the wave height, a(ξ), ( ) ( ζ0 3a a sech 2 2 ρ ξ 1 ) 2 I 0 dρ = 0, ( ) where Since ( ) [ ζ 0 ξ = da ( )] 3a 3a 3a dξ sech2 2 ρ 1 2 ρ tanh 2 ρ. ( ) sech 4 (k) [1 k tanh(k)]dk = 1, ( ) the integral equation ( ) becomes da dξ = 3a 4 ( ) 3a sech 2 2 ρ I 0 dρ. ( ) It is now necessary to discuss the detailed function of I 0, which depends on the actual mud rheology. Let us again use the two-component Kelvin-Voigt element as an example. It is reiterated that both viscous fluid-mud and purely elastic mud are just special cases of Kelvin-Voigt s model (see the discussion in section 2.2.4). Recall (2.2.41), (2.2.87) and (2.2.81), we obtain { w b = αµγ d u x + 2 s n /κ 2 t n 2 u(x,τ) [ ] } 1 e s n(t τ) dτ, ( ) d 2s n κ 2 n 0 x τ and n=0 I = γd ζ x γ 2 d n=0 s n /κ 2 n 2s n κ 2 n t 0 2 ζ(x,τ) x τ [ 1 e s n(t τ) ] dτ, ( ) 47

61 where the leading-order approximation of (2.2.24), ( ) and ( ) are evoked. To satisfy the order of accuracy as stated in ( ), ( ) shall be rewritten as I = γd ζ σ + γ 2 d n=0 s n /κ 2 n 2s n κ 2 n t 0 2 ζ(x,τ) σ 2 [ 1 e s n(t τ) ] dτ. ( ) It is then straightforward to obtain the leading-order term as I 0 = γd ζ 0 ρ + γ 2 d n=0 s n /κ 2 n 2s n κ 2 n t 0 2 ζ 0 ρ 2 [ 1 e s n(t τ) ] dτ. ( ) Derivatives in the above equation can be expressed explicitly as: where ζ 0 ρ = a 3a sech 2 (R) tanh (R), ( ) 2 ζ 0 ρ = a2 sech 4 (R) [2 cosh(2r)], ( ) R = 3a ρ. ( ) 2 The substitution of the above expressions back into ( ) gives I 0 =a 3a [ γd sech 2 (R) tanh(r) ] + a t 3a 0 ( ) 2S sech 4 (ρ + S) [2 cosh(ρ + S)] M ds, ( ) 3a where Since M(τ) = 2 γ d s n /κ 2 n [1 e snτ ]. ( ) 2s n κ 2 n n=0 sech 4 (k) tanh(k)dk = 0, ( ) we finally obtain from ( ) and ( ) an evolution equation for the wave height of a surface solitary wave propagating over a thin layer of viscoelastic 48

62 mud modeled by the two-component Kelvin-Voigt element: da dξ = a 3a ( ) 2S sech 2 (R)sech 4 (R + S) [2 cosh 2(R + S)] M dsdr a ( ) The above equation can be integrated numerically to find the time variation of solitary wave heights for a prescribed set of wave data and mud properties. Numerical examples As an example, figures 2.9 and 2.10 plot a(ξ) under different values of Wi and d. The effect of relative elasticity is first discussed by fixing γ and d while varying Wi, as shown in figure 2.9-(a) and (b). In subplot (a) (i.e. d = 5), the wave height tends to decay faster for the case of smaller Wi (i.e. the mud is more elastic). However, as solitary wave propagates further (i.e. at large ξ) the attenuation caused by mud with bigger Wi can be larger. This behavior is more obvious if we decrease the mud layer thickness to d = 2, as shown in figure 2.9-(b). Again, more elastic mud causes a stronger attenuation only at the beginning stage. The complexity can be attributed to the fact that part of surface wave energy is transfered into the mud layer as elastic energy to sustain the oscillation motion (see figures 2.6 to 2.8 for reference). Since elastic energy is proportional to the square of mud displacement, during the oscillation process the stored elastic energy can be released in which a portion of energy is damped out by the viscous mechanism while the rest recharges the surface wave motion. In figure 2.10, the effect of mud thickness is discussed by fixing Wi but varying d. It is observed that for the case of Wi = 1 and moderate mud layer thickness, the increase of d leads to remarkably faster and stronger attenuation of solitary wave height as shown in subplot (a). However, if the mud layer thick- 49

63 (a) d = 5; Wi = 0.5, 1, 2, (b) d = 2; Wi = 0.5, 1, 2, a 0.10 a Wi = 0.5 Wi = 1.0 Wi = 2.0 Wi = ξ Wi = 0.5 Wi = 1.0 Wi = 2.0 Wi = ξ Figure 2.9: Evolution of a surface solitary wave propagates over a viscoelastic mud: Wave height, a, as a function of time, ξ = (α/µ)t. Effect of relative importance of elasticity are shown by fixing d ((a): d = 5; (b): d = 2) and varying Wi as Wi = 0.5, 1, 2,. In all calculations, γ = ness keeps increasing, the attenuation of wave height becomes less sensitive to d and a complex behavior, similar to that shown in figure 2.9, is displayed. Note that, viscous damping does not grow unbounded with d as the viscous effect is mostly contributed by the so-called boundary layer. Regarding the wave variation caused by the elastic mechanism, as discussed above the releasing of elastic energy plays the role where the oscillation frequency of elastic motion is also a function of d. 50

64 (a) Wi = 1; d = 0.5, 1, 1.5, (b) Wi = 1; d = 2.5, 5, 7.5, 10 d = 2.5 d = 5.0 d = 7.5 d = a a d = 0.5 d = 1.0 d = 1.5 d = ξ ξ Figure 2.10: Wave height, a, as a function of time, ξ = (α/µ)t, for a surface solitary wave propagates over a viscoelastic mud: Effect of mud layer thickness, d. While γ = 0.85 and Wi = 1 are fixed, d = 0.5, 1, 1.5, 2 in subplot (a) and d = 2.5, 5, 7.5, 10 in (b) HD application: amplitude variation of a linear progressive wave We can also examine the effects of bottom mud on a simple harmonic wave as the present theory considers a general surface wave loading. For a linear nondispersive progressive wave, the 1HD equations can be reduced from (2.2.96) and (2.2.97) as ζ t + u x = α I, µ ( ) u t + ζ = 0, x ( ) which are just the linear shallow water equations with the effects of muddy 51

65 seabed as a forcing. Recall the moving coordinate, σ = x t, and the slow time variable, ξ = (α/µ)t, by evoking the leading-order approximation, u ξ ζ ξ, ( ) into the summation of ( ) and ( ) we obtain an evolution equation for the free-surface displacement ζ ξ = 1 I. ( ) 2 Note that in function I the spatial and temporal derivative operators should be replaced by: x σ, t σ. ( ) For a linear progressive wave, the displacement can be formally expressed as ζ = a(ξ)e iσ = [a(0)exp (iβ r ξ) exp ( β i ξ)] e iσ, ( ) where a(ξ) is the wave amplitude. Also, β r and β i account for the wavenumber shift and change in magnitude due to the presence of a muddy seabed, respectively. In the case of a frictionless solid bottom, β r = β i = 0 and a(ξ) = a(0) remains as a constant. For convenience, in the dimensionless manner one can further take a(0) = 1. Due to the periodicity of wave loadings, we can express I(ξ,σ) = I (ξ)ζ. ( ) Therefore, the direct substitution of ( ) and ( ) into ( ) leads to β r + iβ i = i 2 I. ( ) 52

66 For the numerical demonstration, let us consider again the two-component Kelvin-Voigt s model. Following ( ), I = γd ζ t σ 0 = iγdζ + ζ = iγd + 2 ζ(x,τ) M(t τ)dτ σ 2 t 0 0 e i(t τ) M(t τ)dτ e is M(S)dS ζ, ( ) where the function M has been defined in ( ). Substituting ( ) into ( ), we finally obtain β r = 1 2 γd sin(s)m(s)ds, ( ) and β i = 1 2 cos(s)m(s)ds. ( ) 0 For a purely elastic mud, M reduces to M(τ) = 2 γ d [ n=0 2d (2n + 1)π Therefore, ( ) has the asymptotic result ] 2 [ ( )] (2n + 1)π 1 cos τ. ( ) 2d β i = 0, ( ) while β r can be approximated as β r = γd [ 2 γd d 2 [(n + 1/2)π] 2 2d (2n + 1)π ] 2 [ 1 cos ( (2n + 1)π 2 d )]. ( ) Clearly, ( ) suggests that there is zero decay for periodic waves propagating over a elastic seabed, which bas been shown by MacPherson (1980) and Mei et al. (2010). It is reiterated that both past theories are limited to small-amplitude 53

67 waves only, which can be viewed as a special case of the present model. It is also noted that the denominator of ( ) vanishes at ( d = n + 1 ) π, n = 0, 1, 2,, ( ) 2 signifying the resonance due to the elasticity of mud. The above criterion is the same as that given by Mei et al. (2010). It is noted that only the first three peaks, i.e. n 3, are physically significant as d is of O(1). If the seabed is made of viscoelastic mud, we can also obtain the similar condition at which the magnitude of β i will be significantly enhanced. For a non-zero Wi, the asymptotic expression of ( ) is β i = γ s n /kn 2 s n. ( ) d 2s n kn s 2 n n=0 By requiring β i = 0, one gets d ( d = n + 1 ) π 4 + 3Wi 2 1, n = 0, 1, 2,. ( ) 2 Wi The above condition reduces to ( ) as Wi 0. Regrading the case of a Newtonian muddy seabed, i.e. Wi, the rate of amplitude attenuation can be further deduced from ( ) as β i = γ 1 d 1 + [(2n + 1)π/(2d)] 4, ( ) n=0 and the criterion ( ) becomes ( d = 3 1/4 n + 1 ) π 2.067(2n + 1). ( ) 2 Again, only n = 0 and n = 1 are meaningful in ( ). To determine the peak wave damping, it is sufficient to consider n 1 in ( ) alone. Therefore, it is found that a maximum β i occurs at d 2.186, or d 2ν 1.55 m ω0 where ω 0 is the angular frequency of the surface progressive wave. This agrees with the result of Ng (2000). 54

68 Numerical examples In figure 2.11, β i is plotted against d under various values of Wi and γ. Both solutions calculated by ( ) and results from MacPherson (1980) are presented. The occurrence of resonance is also compared with the prediction by ( ). As can be seen, a good agreement is evident. The oscillating feature of β i, due largely to the presence of elasticity, is same as that demonstrated by Mei et al. (2010). However, it is remarked that viscous damping is not responsible for the amplitude attenuation (non-zero β i ) caused by a purely elastic mud bed. Instead, wave energy transfered into the mud layer through pressure work at the water-mud interface is stored by the muddy seabed to sustain the mud flow motions. Furthermore, supported by the observation from figure 2.11-(b) and the mathematical argument by Mei et al. (2010), it is realized that local maxima of β i, which correspond to m = 0, 1, 2, in ( ), decrease sequentially as a mud layer becomes thicker (i.e. larger m). We may expect for the case of purely elastic mud β i 0 as d, which is consistent with the conclusion by Mallard & Dalrymple (1977). Given that the problem of linear waves interacting with a muddy seabed has been studied extensively, comprehensive discussions are available in the literature. Some important references are MacPherson (1980), Piedra-Cueva (1995), Ng & Zhang (2007), and Mei et al. (2010) Explicit solutions for 1HD periodic waves For a linear progressive wave propagating over a viscoelastic seabed modeled by a two-component Kelvin-Voigt element, explicit solution forms can actually 55

69 β i (a) γ = Wi = 0.5 Wi = Wi = 1.0 Wi = Wi = Present model (b) Wi = 0.5 γ = 0.85 γ = γ = 0.65 Present model β i d Figure 2.11: Amplitude attenuation rate (β i ) for a linear progressive wave over a viscoelastic mud: (a) Effect of relative importance of elasticity: Wi = 0.5, 1, 2, ; (b) Effect of mud density, γ. Symbols are the solutions of present model while lines plot the damping rate of MacPherson (1980). The insert in subplot (a) shows the details of viscous case with the dashed line indicates the value of d corresponds to the maximum β i as given by ( ). In addition, the vertical bars in (b) show the theoretical prediction of d values correspond to the first two peaks of β i by ( ). 56

70 be deduced. Consider the free-surface displacement in the following form: ζ(x,t) = e i(kx t), ( ) where k is the dimensionless wavenumber. Substitute the above expression into the momentum equation, ( ), the horizontal velocity component is obtained u(x,t) = ke i(kx t). ( ) Furthermore, the vertical component is w(x,z,t) = ik 2 µ 2 ze i(kx t) + w b. ( ) Regarding the mud flow solutions, from the initial-boundary-value problem, (2.2.48) to (2.2.50), the horizontal velocity is u m (x,η,t) = γ [ 1 ] cosh ( (d η)) u(x, t), ( ) cosh ( d) where = 1 i 2 Wi i + Wi ( ) is a complex parameter. Accordingly, w m (x,η,t) = ikγ [ η 1 ] sinh ( d) sinh ( (d η)) u(x, t), ( ) cosh ( d) which suggests that [ I = ik 2 γd 1 ] tanh ( d). ( ) d The dimensionless wavenumber, k, is still an unknown. By evoking the conservation of mass, ( ), the dispersion relation is obtained 1 = k 2 { 1 + α µ γd [ 1 ]} tanh ( d). ( ) d 57

71 In fact, we can further take into account the wave dispersion. Following the continuity equation, (2.2.4), the water particle velocity components are formulated as u = [k cosh (kµ(z + 1)) + D sinh (kµ(z + 1))] e i(kx t), ( ) w = iµ [k sinh (kµ(z + 1)) + D cosh (kµ(z + 1))] e i(kx t), ( ) where D is an unknown to be determined. Note that the mud flow velocity remains the same, as given in ( ) and ( ). The continuity of the vertical velocity at the water-mud interface yields [ ] D = k 2 tanh ( d) αγd 1. ( ) d Therefore, combing the free-surface conditions, (2.2.5) and (2.2.6), we obtain the dispersion relation [ 1 = k 1 + kαγd µ tanh (kµ) [ 1 + kαγd 1 tanh( d) d 1 tanh( d) d ] coth(kµ) ]. ( ) tanh(kµ) The above solution forms agree with the results reported by Ng (2000) if the water viscosity is ignored 5. In addition, at the nondispersive limit, i.e. µ 0, ( ), ( ) and ( ) reduce to ( ), ( ) and ( ), respectively Comparison with laboratory experiments To examine the performance of the present theory, model predictions are now compared with available laboratory measurements. In this section, three examples will be discussed: linear progressive waves over a layer of either viscous 5 The water viscosity is also considered by Ng (2000), while the water body is treated as a inviscid fluid in the current study. 58

72 mud (Gade 1958) or viscoelastic mud (Maa & Mehta 1987, 1990), and a solitary wave interacting with a viscous fluid-mud seabed (Park, Liu & Clark 2008). All experiments were carried out in flat wave flumes. The pioneering work of Gade (1958) modeled the bottom mud using sugar water, considered as a Newtonian fluid. Experimental conditions were: 2π ω 0 = 8 s, h 0 = 4 ft, ν m = 5 ft 2 s 1, γ = 0.487, d 2νm /ω 0 = 0, 0.28, 0.56, 0.84, 1.12, 1.40, Figure 2.12 plots β r and β i calculated by ( ) and ( ), along with the laboratory results. A reasonable quantitative agreement is observed. Due to the presence of the viscous mud, the amplitude of the surface wave is attenuated while the wavelength increases. As a reminder, β r and β i represent the rates of change of wavenumber and amplitude, respectively. It should be noted that in the experiments, even without the mud, a remarkable wave damping is still evident, i.e., value of β i is significant at d 0. Note that this cannot be solely accounted for by the effect of water viscosity, as a considerable discrepancy is also reported by Dalrymple & Liu (1978) 6 whose theoretical model also includes the water viscosity. Park, Liu & Clark (2008) also studied the interaction between surface waves and a viscous mud, but considered solitary waves of weak to moderate nonlinearity instead. In their experiments, a commercial Newtonian silicone fluid was used as the bottom mud. Employing the particle image velocimetry (PIV), Park, Liu & Clark (2008) reported the velocity measurements in both water body and mud column. Experimental conditions of a specific case to be discussed are: h 0 = 10 cm, a 0 = 1.9 cm, d = 1.7 cm, ν m = m 2 s 1, γ = See FIG. 2 in Dalrymple & Liu (1978). 59

73 β r 0.40 β i d d Figure 2.12: Periodic wave over a viscous mud: Rates of change of dimensionless wavenumber, β r, and wave amplitude, β i, as functions of dimensionless mud layer thickness, d. Dots are the laboratory measurements of Gade (1958) and lines plot the results calculated by ( ) and ( ), respectively. Figure 2.13 shows the time histories of horizontal mud flow velocity, u m, at three fixed vertical levels: η/d = 0.25, 0.5, The theoretical predictions agree very well with the measurements. The vertical profiles of u m at several different phases are further illustrated in figure Again, the agreement is also reasonable. Note that panel (c) demonstrates the transition of flow reversal predicted by the current theory. Unfortunately, the experiment did not capture this feature due to the limited sampling rate at 100 HZ, which corresponds to a phase difference of θ 0.04 with θ = (x x 0 ct). In figure 2.15, records of both bottom shear stress, τ mb, and the vertical displacement at water-mud interface, ζ m, are shown. Note that the model predicted interfacial displacement is recovered by numerically integrating the linear approximation ζ m t = w m, η = d. ( ) 60

74 η d = 0.75 η d = 0.50 η d = 0.25 Measurements u m (x x 0 ct) Figure 2.13: Solitary wave over a viscous mud: Records of horizontal mud flow velocity, u m, at three vertical levels, η/d = 0.25, 0.5, Lines plot the theoretical results and dots are the experimental data of Park, Liu & Clark (2008). The measurements are actually those of PIV products. As can be seen, theoretical results agree well with the experimental data. It should be noted that ζ m is at least two orders of magnitude smaller than the free-surface displacement, which is of O(1) in the dimensionless manner. This justifies the assumption of negligible interfacial displacement (see the discussion in (2.2.34)). The model validation is completed by considering a final example of periodic waves over a viscoelastic muddy seabed. Here, the present solutions are compared with the laboratory tests of Maa & Mehta (1987, 1990). It is noted that actual estuarial mud taken from Cedar Key, Florida was used in the experiments. Rheology tests have shown that the mud samples exhibited both viscous and elastic properties. Figure 2.16 shows the comparison of horizontal velocity component across the entire extent of water body and mud column. The cor- 61

75 η d η d η d η d η d (a) θ = (b) θ = (c) θ = 0.012, 0.014, (d) θ = (e) θ = u m (x, η, t) Figure 2.14: Mud flow induced by a surface solitary wave: Profiles of horizontal velocity component, u m, at several different phases, θ = (x x 0 xt). Dots are the PIV products of Park, Liu & Clark (2008) and lines show the theoretical predictions. In panel (c), the transition of flow reversal is demonstrated, which was not captured by the experiment due to the limited sampling rate. 62

76 ζ m τ mb (x x 0 ct) Figure 2.15: Solitary wave over a viscous mud: Time histories of interfacial displacement, ζ m, and bottom shear stress, τ mb. Lines plot the theoretical predictions and dots show the laboratory results of Park, Liu & Clark (2008). responding parameters are listed in Table 2.1. Notice that in both experiments, the properties of mud, namely ρ m, ν m, and E m, are vertically stratified. To adopt their data to our theory, these parameters have been depth-averaged. As can be seen, the agreement is generally acceptable. It is clear that a sharp velocity gradient is shown at the water-mud interface, due largely to the huge difference in viscosities of water and mud. Table 2.1: Experimental conditions of Maa & Mehta (1987, 1990) for periodic waves over a viscoelastic muddy seabed. 2π ω 0 h 0 a 0 d ρ m ν m E m (s) (cm) (cm) (cm) (g cm 3 ) (m 2 s 1 ) (N m 2 ) M&M M&M

77 Distance to bottom (cm) Distance to bottom (cm) (a) Horizontal velocity (cm/s) 5 (b) Horizontal velocity (cm/s) Figure 2.16: Periodic waves over a viscoelastic mud: Profiles of horizontal velocity components. Lines plot the theoretical results while dots are experimental data of Maa & Mehta (1987) (subplot (a)) and Maa & Mehta (1990) (subplot (b)). In addition, dashed lines denote the free surfaces. The corresponding parameters are given in Table Summary A set of depth-averaged continuity and momentum equations, describing motions of surface long waves over a thin viscoelastic muddy seabed, has been derived. The new theory is capable of modeling weakly nonlinear and weakly dispersive waves. A generalized rheology model for a linear viscoelastic material is adopted with both Newtonian fluid and purely elastic mud being the limiting cases. To demonstrate, amplitude evolution of a solitary wave is investigated which shows a significant attenuation caused by the mud. The case of linear progressive waves is also studied. It is found that the wave attenuation can be enhanced remarkably by the elasticity of mud. The performance of 64

78 the proposed model is examined by comparing the results with the available laboratory measurements. The overall agreement is encouraging. 2.3 Response of a Bingham-plastic muddy seabed to a surface solitary wave Mud in different locales can have different rheological behavior, partly as a consequence of diverse chemical composition (Balmforth & Craster 2001). Examining the mud samples taken from the eastern coast of China, Mei et al. (2010) have shown that these cohesive sediments can be modeled as a viscoelastic material. Krone (1963) performed the viscosimetric tests on field mud samples collected along the coasts of the United States. He reported that mud with a concentration roughly lying between 10 to 100 g L 1 displayed both plastic and viscous-like behavior, depending on the external forcing. This observation suggests that the muddy seafloor can sometimes be referred as a Bingham-plastic material, in which the constitutive equation in a simple two-dimensional case is expressed as u m µ m = z τ m τ o sgn 0, τ m τ o ( ) u m, τ z m > τ o, ( ) where τ o > 0 is the yield stress, and µ m is the Bingham-plastic viscosity. It is remarked that the above rheology model is the leading-order approximation as the contribution from u m x has been ignored (see section for the justification). Detailed discussion on the validation of ( ) has been documented in Balmforth & Craster (1999). It is noted that typical values of the physical parameters for different Bingham-plastic muds can be found in Mei & Liu (1987). A 65

79 useful summary on the relationships among yield stress, Bingham-plastic viscosity, and the concentration for various types of mud is also provided by Mei, Liu & Yuhi (2001). The immense challenge of the Bingham-plastic mud problem is posed by the existence of the yield stress. Yet, Mei & Liu (1987) still managed to investigate the effects of a Bingham-plastic muddy seabed on long-wave propagation and shoaling. Neglecting the water viscosity and assuming a thin mud layer, they illustrated elegantly that the motions of a Bingham-plastic muddy seafloor can be approximately divided into two distinct regions: a plug flow layer moving above a shear flow zone. The plug flow velocity and the thickness of the shear flow zone, or equivalently the location of the yield surface, change in time depending on the magnitude of the pressure gradient imposed by the surface wave, and the properties of the Bingham-plastic mud. Solutions have to be obtained numerically by solving two coupled partial differential equations which govern motions in two regions, respectively. In analyzing the shear flow, Mei & Liu (1987) applied the Kármán momentum integral method and adopted the parabolic profile to describe the vertical distribution of horizontal velocity component inside the shear flow region. They further assumed that the plug flow layer is always much thicker than the shear flow zone. With these two additional simplifications, the plug flow velocity can be obtained explicitly without knowing the shear zone thickness, which has to be solved numerically from the deduced ordinary differential equation. Clearly, their analysis does not allow the flow reversal inside the shear flow region as illustrated by figure 2.8 for the case of surface waves over a viscoelastic mud. Although the finding of viscoelastic problem presented in section 2.2 does not necessarily guarantee the same behavior to be observed in the Bingham-plastic fluid, it is desirable 66

80 to analyze the shear flow region more carefully. In addition, it is anticipated that under certain combinations of yield stress, viscosity and pressure gradient, multiple shear flow layers (or plug flow regions) can develop. Therefore, this section is devoted to studying the response of a Binghamplastic muddy seafloor subject to long water wave loadings without some of the constrains imposed in Mei & Liu (1987). In particular, we shall relax the following two assumptions: the parabolic shear flow velocity profile and the negligible shear layer thickness in computing the plug flow velocity. This will allow us to investigate the evolution of yield surfaces, and the associated velocity profiles throughout the entire mud column. Since analytical solutions are impossible for general wave loadings, we shall focus only on the free surface solitary wave loading. The initial-boundary-value problem governing the wave-induced motions inside the thin layer of Bingham-plastic mud shall be first discussed. A brief review of the approach and assumptions of Mei & Liu (1987) follows. Possible scenarios of mud motions, which could have up to four layers of alternating plug flow and shear flow under a surface solitary wave, are then illustrated. Subsequently, semi-analytical/numerical solutions are presented for the velocities inside the mud column, along with detailed discussion on the mud flow dynamics under different physical parameters. The damping rate for a surface solitary wave is calculated using the energy conservation law. Using some estimated but realistic physical parameters, the predicted damping rate is compared with the field observations. Good qualitative agreement is shown. 67

81 2.3.1 Formulation for wave-induced mud motions inside a thin Bingham-plastic seabed A thin layer of Bingham-plastic mud subject to a transient surface long-wave loading is now studied. For simplicity, let us consider only two-dimensional problem where the x-axis coincides with the direction of wave propagation and the η-axis points upwards, denoting the vertical coordinate. As has been discussed comprehensively in section 2.2.2, the mud flow motions can be described by the following linearized boundary-layer equations (see (2.2.36) and (2.2.48)): where u m t u m x + w m η = 0, 0 η d, ( ) = γ u b t + τ m, 0 η d, ( ) η p m x = u b t, η = d ( ) has been evoked. In addition, from (2.2.49) and (2.2.50) the corresponding boundary conditions in the vertical direction are τ m = 0, η = d, ( ) and u m = 0, η = 0. ( ) The idea is again to first express u m in terms of u b by solving the above initialboundary-value problem. Afterwards, w m (x,η,t) = is calculated from ( ) (see also (2.2.78)). η 0 u m dη ( ) x 68

82 2.3.2 Review of Mei & Liu (1987) Fundamental characteristics of Bingham-plastic mud motion is the existence of plug flow and shear flow: when the magnitude of shear stress is larger than the yield stress the mud will move like a viscous fluid (shear flow), otherwise it behaves as a solid (plug flow). The crucial assumption embedded in the analysis by Mei & Liu (1987) is that under wave loadings the shear stress in the mud column decreases monotonically in the vertical direction with the maximum magnitude at the solid bottom, η = 0. Accordingly, inside the Bingham-plastic mud bed there exists only one shear flow zone (0 < η < η 0 ) with a thickness of η 0 (x,t), if the magnitude of bottom frictional stress, τ mb, is greater than the yield stress, i.e., τ mb > τ o. Above the shear flow layer, there is a plug flow region (η 0 < η < d) of thickness κ 0 = d η 0. Mei & Liu (1987) have pointed out that in the plug flow region the horizontal mud velocity, u m = u p (x,t), is vertically uniform 7. It follows from ( ) that τm η must also be independent of η. Therefore, balancing the forces in the plug flow region one obtains u p t = γ u b t + τ osgn(u p ) d η 0, η 0 η d. ( ) By further assuming that the plug flow region occupies most of the mud column, i.e., O(η 0 ) = 1 and κ 0 η 0, or equivalently d 1, the equation above can be approximated as u p t γ u b t + τ osgn(u p ), η 0 η d. ( ) d Clearly, the plug flow velocity can now be determined without knowing the shear flow layer thickness. Within the shear flow region, u m = u s (x,η,t), Mei 7 In this study, the terminology plug flow is actually referred to the flow region where the horizontal velocity component is uniform in the vertical extent, but not necessarily being invariant laterally. 69

83 & Liu (1987) employed the Kármán momentum integral method by assuming a parabolic velocity profile ( ) 2 u s η = + 2 η, 0 η η 0, ( ) u p η 0 η 0 in which the no-slip condition on the solid bottom and two matching conditions along the interface between plug flow and shear flow regions, u p = u s, u s η = 0, η = η 0, ( ) have been applied. Consequently, the momentum equation inside the shear flow zone becomes ( η0 2 u p t + 6γ u ) b t 4 u p η0 2 12u p = 0, ( ) t which is an ordinary differential equation for η 0, as u p can be obtained by ( ) beforehand. Note that the assumed parabolic shear flow velocity profile, along with the negligible water viscosity, ( ), implies that the shear flow layer must vanish at zero plug flow velocity. Based on the model of Mei & Liu (1987), the anticipated Bingham-plastic mud motions under a surface solitary wave loading are sketched in figure Notice that in this figure the yield surface location, η = η 0, is designated as η 1 when the mud flow moves in the direction of wave propagation (from the left to the right) and η 0 = η 3 when it moves in the opposite direction. Before the arrival of the solitary wave, the entire extent of mud column behaves like a solid and is at rest. As shown in phase (1) of figure 2.17, a shear flow layer begins to develop from the solid bottom at t = t s when τ mb = τ o. Clearly, this incipient moment can be calculated as γ u b t τ o d = 0, t = t s, ( ) 70

84 (1) (2) (3) (4) t = t s t = t 0 η d η 1 η 1 η 1 η = η 1 (5) (6) (7) (8) t = t 1 η 3 t = t e η 3 η 3 η 3 Figure 2.17: Sketches of vertical profiles of horizontal velocity, u m, inside the mud bed under a surface solitary wave loading based on the model proposed by Mei & Liu (1987) (two-layer scenario). All dots represent the locations of yield surface (η 0 = η 1 for the positive mud motion and η 0 = η 3 for the backward mud motion), dashed-dotted lines denote the water-mud interface and dotted vertical lines are the zero velocity reference. A shear flow layer develops when the bottom shear stress reaches the yield stress at t = t s (cf. phase (1)). Both mud velocity and the thickness of shear flow region first grow and then decrease as shown in phases (2) to (4). The mud motion pauses at t = t 0 and restarts to move backwards at t = t 1 (see phases (4) and (5)). If t 1 > t 0, the mud flow is intermittent (i.e., the mud is at rest for a finite time interval t 0 < t < t 1 ), otherwise it moves continuously. Eventually, the mud motion stops at t = t e and a cycle of mud motion under a solitary wave loading is completed. In this example, the velocity profile is always monotonically increasing from zero at the bottom to the plug flow velocity at the mud-water interface. 71

85 representing the balance between the driving pressure gradient, which is also proportional to acceleration of wave motions at the water-mud interface, and the bottom friction that is the same as the yield stress at this moment. Phases (2) to (4) in figure 2.17 suggest that both the mud velocity and the thickness of shear flow layer first grow and then diminish as the magnitude of the driving pressure gradient (or acceleration of wave motions) first increases and then decreases. Eventually, the entire mud column pauses and returns to the solid state at t = t 0 (i.e., phase (4) in figure 2.17). The shear flow starts to move in the opposite direction of the wave propagation when the reversed driving pressure gradient yields the bottom mud again at t = t 1 (see phase (5)). During the backward mud flow motion phases, the characteristics of mud velocity and shear flow layer thickness behave very much like those at the forward mud motion phases. Finally, the mud flow ends at t = t e as shown in phase (8). The transition times t 0, t 1, and t = t e are yet to be determined. Mei & Liu (1987) have suggested that when t 1 = t 0 the mud flows continuously while the mud motion is intermittent if t 1 > t 0 (cf. figure 2.17). For the intermittent mud flow, t 1 can be calculated by γ u b t + τ o d = 0, t = t 1. ( ) The physical representation of the above equation is similar to that of ( ) except the directions of mud flow and driving pressure gradient reverse. Although the analysis of Mei & Liu (1987) is ground breaking, the simplifications employed prevent it from being applied to more complex flow conditions. For instance, the assumption that shear flow layer thickness is much smaller than the total mud bed thickness, d 1, is not always applicable. Consider a solitary wave propagating over a depth h 0 = 10 m with ǫ = µ 2 = 0.1. The Bingham-plastic mud has a thickness of d = 0.5 m and a viscosity three orders of magnitude greater than that of water (i.e., αl 0 = 0.05 m); the dimensionless 72

86 mud thickness is only about d = 10. Moreover, it is well-known that for a Newtonian boundary-layer flow under unfavorable pressure gradient, the strain rate at the bottom can become zero and eventually a flow reversal occurs, implying that the vertical variation of the strain rate is no longer monotonic. This feature has been demonstrated in figure 2.8 for the case of a viscoelastic mud. The parabolic velocity profile is adequate when the driving force is always favorable, e.g., in gravity current or debris flow problems (Liu & Mei 1989; Huang & García 1997). Under a transient wave loading, because of the occurrence of unfavorable pressure gradients a multi-layer flow structure inside the Binghamplastic mud, i.e., alternating layers of plug and shear flow regions, can exist. Differing from the approach of Mei & Liu (1987), it is the objective of the present study to provide a general investigation of the response of a Bingham-plastic muddy sea bed to the surface solitary wave propagation Solutions inside a Bingham-plastic mud Figure 2.18 illustrates the complete mud responses under a surface solitary wave loading. During the accelerating phases of solitary wave, (1) and (2) in figure 2.18, a shear flow region develops from the solid bottom when the pressure gradient generated bottom friction overcomes the yield stress. The corresponding yield surface between the plug flow and the shear flow region is designated as η 1 (x,t). As the solitary wave starts to decelerate, the unfavorable pressure gradient creates zero strain rate at the bottom, which implies that the lower portion of mud is solidified (plug flow) and a second yield surface, η 2 (x,t), appears; e.g., panel (4) in figure The corresponding time instant is denoted as t = t 1. In terms of the constitutive curve, ( ), the development of the bottom plug 73

87 (1) (2) (3) (4) (5) t = t s t = t 0 η d η 1 η 1 η 1 η 1 η = η 1 η 2 η 2 (6) (7) (8) (9) (10) η 1 η 1 η 1 = η 2 t = t y η 2 η 3 η 3 t = t 1 η 3 η 3 t = t e η 2 η 3 Figure 2.18: Sketches of vertical profiles of horizontal velocity, u m, inside the mud bed under a surface solitary wave loading: Fourlayer scenario. All dots represent the locations of yield surfaces (η 1,2,3 ), dashed-dotted lines denote the water-mud interface and dotted lines are the zero velocity reference. The mud is yielded at t = t s when the bottom shear stress reaches the yield stress. During the beginning phases (1) to (3), there is only one yield surface. In panel (4), a second plug flow region develops from the solid bottom in response to the unfavorable pressure gradient and the mud plasticity at t = t 0 and the new plug flow layer grows as the strength of the unfavorable pressure gradient increases (cf. phase (5)). As the driving unfavorable pressure gradient becomes stronger, the mud in the lower plug flow region is yielded again at t = t y in (6). The upper shear layer eventually vanishes, i.e., η 1 and η 2 are merged at t = t 1, and the mud motion returns to a single yield surface (η 3 ) structure. The whole process of mud flows ends at t = t e. 74

88 flow layer represents the transition during which the bottom shear stress decreases from the positive yield stress to the negative yield stress, τ mb = τ o. As the solitary wave keeps propagating forward, the newly developed lower plug flow region grows and the positive (unfavorable) pressure gradient can liquefy the bottom solid mud again when the pressure gradient overcomes the yield stress; i.e., panels (5) and (6) in figure The third interface between plug flow and shear flow region is denoted as η 3 (x,t) and the time of its occurrence is marked as t = t y. Consequently, a four-layer structure inside the mud column is formed and a flow reversal occurs as shown in panel (7). The subsequent phases show that the sandwiched shear layer vanishes, i.e., the upper two yield surfaces, η 1 and η 2, merge at t = t 1 in panel (8) of figure 2.18, since the driving (positive) pressure gradient becomes fully favorable again. The sea bed continues to flow with a single yield surface structure (panel (9), figure 2.18) and eventually the whole mud column returns to its initial resting state at t = t e. In addition to the four-layer and two-layer (also the model of Mei & Liu 1987) scenarios, a three-layer scenario is also possible and is sketched in figure This scenario occurs only if the driving pressure gradient is not strong enough so that the second shear flow region does not develop and the second plug flow region builds up only until the whole mud column is solidified before the backward mud motions take place (cf. (4) to (6) in figure 2.19). With this exception, the three-layer scenario is very similar to the four-layer scenario: the mud is first liquefied at t = t s, a bottom plug flow region begins to develop at t = t 0, the sandwiched shear layer vanishes at t = t 1 and the whole mud motion ends at t = t e. Clearly, the flow reversal does not occur in either the three-layer or two-layer scenarios. We reiterate that the two-layer scenario, as shown in figure 2.17, can only occur when the yield stress is so strong that dur- 75

89 ing the middle phases the entire mud column comes to rest without any bottom plug flow zone develops (cf. (4) to (5) in figure 2.17). In addition, there is no presumed shear flow velocity profile in our two-layer scenario and mud flow has to be intermittent (i.e., no mud flow motion during t 0 t t 1 ). This is very different from the proposal of Mei & Liu (1987). Based on the above physical pictures, we can now formulate the mathematical model describing Bingham-plastic mud motions under a surface solitary wave loading within the following framework: (I) t s t t 0 : A plug flow region is on top of a shear flow region with the single yield surface, η 1 (cf: (1)-(4) in figure 2.17; (1)-(3) in figures 2.18 and 2.19); (II) t 0 t t 1 : There are multiple yield surfaces with alternating plug- shear plug shear flow structure (four-layer scenario: figure 2.18, (4)- (8); three-layer scenario: figure 2.19, (4)-(6)) or no mud motion at all (twolayer scenario: figure 2.17, (4)-(5)); (III) t 1 t t e : Flows return to the plug shear flow structure with a single yield surface, η 3 (cf: (5)-(8) in figure 2.17; (8)-(10) in figure 2.18; (7)-(9) in figure 2.19). Notice that all the time stamps, t s, t e, t 0, and t 1, have been illustrated and described in figures 2.17 to In addition, while for all scenarios t s has a common definition (see ( )), t 0 and t 1 are different for two-layer or three/four-layer scenarios. Both t 0 and t 1 are still parts of the solutions to be determined with the exception that for the two-layer scenario t 1 has been defined in ( ). 76

90 η 3 η 3 (1) (2) (3) (4) (5) t = t s t = t 0 η η = η 1 d η 1 η 1 η 2 η 1 η 2 η 1 (6) (7) (8) (9) η 1 = η 2 t = t 1 η 3 t = t e Figure 2.19: Sketches of vertical profiles of horizontal velocity, u m, inside the mud bed under a surface solitary wave loading: Threelayer scenario. All dots represent the yield surfaces (η 1,2,3 ), dashed-dotted lines denote the water-mud interface and dotted lines are the zero velocity reference. The mud motion is initiated at t = t s and a second plug flow region develops from the solid bottom when t = t 0. At t = t 1 the whole mud column is solidified as the transition between positive plug flow velocity and the backward movement (cf. phase (6)). Thereafter, the mud bed moves backwards with a single yield surface (η 3 ) structure towards the ending instant, t = t e. To be consistent with the definition in section 2.3.3, here the notation η 3 denotes the lowest yield surface after η 1 and η 2 have merged. In this example, there is no second shear flow layer and flow reversal does not occur. 77

91 Despite the possibility of having different multi-layer structures, the momentum equation remains the same in each shear flow region, u s t = γ u b t + 2 u s η 2, ( ) while within the plug flow layer the momentum equation becomes u p t = γ u b t + τ pt τ pb κ p, ( ) where τ pt and τ pb are the shear stresses along the top and bottom of a plug flow region, respectively and κ p is the thickness of this specified layer. However, the boundary and interfacial conditions are not the same for different flow scenarios, which will be described in the following sections. Stage (I): Initial single yield surface (η 1 ) structure During this initial stage (t s t t 0 ), there is only one yield surface, η 1 (x,t), and the vertical velocity gradient inside the shear flow layer is always positive, which indicates that plug flow velocity, u p1 (x,t), is non-negative. Therefore, by integrating ( ) in time we obtain [ ] t u p1 (x,t) = γ u b (x,t) u b (x,t s ) + t s τ o η 1 dt, η 1 η d. ( ) As for the shear flow velocity, let us follow the approach presented in section 2.2 (see the mathematical treatment in (2.2.70)) to introduce a new variable v s1 = u s1 γu b. ( ) Thus, the two-point boundary-value problem (BVP) in this region can be expressed in terms of v s1 as v s1 t = 2 v s1 η 2, 0 η η 1, ( ) 78

92 with the initial condition v s1 = γu b, t = t s, ( ) and the following boundary conditions and v s1 η = 0, η = η 1, ( ) v s1 = γu b, η = 0. ( ) In addition, the continuity of mud flow velocity along the yield surface, η = η 1, needs to be satisfied. Hence, from ( ) and ( ) it is required that v s1 (x,η 1,t) = γu b (x,t s ) t t s τ o d η 1 dt. ( ) The BVP, ( ) to ( ), is similar to that presented in section 2.2 for a linear viscoelastic muddy seabed problem. However, the present problem has a moving boundary, i.e., η 1 = η 1 (x,t), which posts a mathematical difficulty in finding an analytical solution. Nevertheless, by adopting the assumption that the thickness of shear flow layer is slowly varying in time, η 1 (x,t) can be approximated as a constant within a small time interval, t. Therefore, using the Green s function method (Mei 1995) the solution form can be obtained as v s1 (x,η,t) = where G(η,ξ,t) = η 1 0 n= v s1 (x,ξ,t )G(η,ξ, t)dξ γ t 0 u b (x,t + t) G (η, 0, t t)dt, ξ ( ) { ( 1) n 2 exp [ (η ξ + 2nη ] 1) 2 exp [ (η + ξ + 2nη ] } 1) 2, πt 4t 4t ( ) 79

93 and 0 < t = t t 1 in order to satisfy the slowly varying assumption, η 1 = η 1 (x,t) from t to t. When t = t s, the solution becomes v s1 (x,η,t) = γu b (x,t s ) γ 2 π n= 1 n=0 m= 1 ( 1) n (η + 2nη 1 ) t 0 ( ( 1) n+m 1 m 2 ) [ ] η + (2n + m)η1 erfc 4 t [ u b (x,t s + t) exp (η + 2nη ] 1) 2 dt, ( ) ( t t) 3 4( t t) with η 1 = η 1 (x,t s + t). Based on ( ), it is possible to formulate the general expression for v s1 (x,ξ,t ) in ( ), which involves a multiple series. However, there is no obvious computational benefit for doing so since the integrals in ( ) still have to be evaluated numerically. In summary, when the properties of the Bingham-plastic mud and the velocity of water along the water-mud interface, u b, are given, the thickness of the shear flow layer, η 1, can be calculated numerically from ( ). Once η 1 is known, the velocities of the plug flow and shear flow can be obtained by ( ) and ( ), respectively. It is remarked that the current stage ends at t = t 0. For a two-layer scenario, t 0 indicates the moment that mud motion pauses from the forward motion while in the three/four-layer scenario it represents the instant that zero shear strain rate appears at the solid bottom (see figures 2.17 to 2.19). Stage (II): Multiple yield surfaces structure for three/four-layer scenario During the unfavorable pressure gradient phase, multiple yield surface structure is formed when t 0 t t 1. As mentioned earlier, the mud bed is stationary during this time interval in the two-layer scenario. Referring to figure 2.18, the maximum possible number of yield surfaces is three, therefore, the momentum 80

94 equations for these four layers can be formulated as u p1 t u s1 t u p2 t u s2 t = γ u b t + τ o d η 1, η 1 η d, ( ) = γ u b t + 2 u s1 η, η 2 2 η η 1, ( ) = γ u b t + 2τ o η 2 η 3, η 3 η η 2, ( ) = γ u b t + 2 u s2 η 2, 0 η η 3, ( ) where η 1, η 2 and η 3 denote the yield surfaces. The associated interfacial and boundary conditions are u p1 = u s1, η = η 1, ( ) u s1 η = 0, η = η 1 or η = η 2, ( ) u s1 = u p2, η = η 2, ( ) u p2 = u s2, η = η 3, ( ) u s2 η = 0, η = η 3 and u s2 = 0, η = 0. ( ) An additional yielding criterion for the second shear flow zone, 0 η η 3, is η 3 = u s2 = u p2 = 0, t < t y, ( ) where t y is illustrated in phase (6) of figure 2.18 and can be determined by γ u b t + 2τ o η 2 = 0, t = t y. ( ) It is reiterated that since η 2 is still part of the unknown solutions, the above criterion has to be checked at every time step. For the three-layer scenario, η 3 is always zero in this stage. As for the four-layer scenario, η 3 > 0 when t t y. In both scenarios, the mud motion returns to a single yield surface setup at t = t 1 with η 1 = η 2 when the wave-induced pressure gradient becomes truly favorable again (see (8) of figure 2.18 and (6) of figure 2.19). 81

95 Following the same solution method as shown in the previous section, the plug flow velocities can be obtained as and [ ] t u p1 (x,t) = u p1 (x,t 0 ) + γ u b (x,t) u b (x,t 0 ) + [ ] t u p2 (x,t) = γ u b (x,t) u b (x,t y ) + t y t 0 τ o d η 1 dt, ( ) 2τ o η 2 η 3 dt, t > t y. ( ) is For the upper shear flow zone, solution form of BVP, ( )) with ( ), v s1 (x,η,t) = 1 2 π t η 1 η 2 0 v s1 (x,ξ + η 2,t t)g 1 (η,ξ)dξ, t > t 0, ( ) where G 1 (η,ξ) = n= m=1 [ 2 ( ) ] η + ( 1) m 2 ξ + 2n(η 1 η 2 ) exp 2, ( ) t and v s1 = u s1 γu b. Note that t should be small in order to satisfy the assumption of slowly varying yield surfaces. In addition, the initial condition, v s1 = v s1 (x,η,t 0 ), has to be computed from ( ). Similarly, for the second shear flow layer, i.e., ( ) with ( ), we obtain v s2 (x,η,t = t + t) = γ t 0 η 3 0 v s2 (x,ξ,t )G 2 (η,ξ, t)dξ u b (x,t + t) G 2 ξ (η, 0, t t)dt, t t y, ( ) where G 2 (η,ξ,t) is same as G given in ( ), except η 1 being replaced by η 3. Recall the initial condition for this region should be u s2 = γu b + v s2 = 0, t = t y. ( ) 82

96 So far, the thicknesses of each layer remain unknown. Three interfacial conditions, ( ), ( ) and ( ), are applied to obtain these variables. Therefore, at every instant we need to solve numerically a nonlinear system that involves three unknowns. Stage (III): Single yield surface (η 3 ) structure with a negative value of plug flow velocity During this final period (t 1 t t e ), the solutions are very similar to those in stage (I). Therefore, we can easily obtain and [ ] t u p1 (x,t) = u p1 (x,t 1 ) + γ u b (x,t) u b (x,t 1 ) + v s2 (x,η,t = t + t) = η 3 t 1 v s2 (x,ξ,t )G 2 (η,ξ, t)dξ τ o d η 3 dt, ( ) γ t 0 0 u b (x,t + t) G 2 ξ (η, 0, t t)dt, t t 1. ( ) Reminded that t 1 is part of the solutions from the previous stages and the location of yield surface, η 3, can be obtained by requiring v s2 (x,η 3,t) = u p1 (x,t 1 ) γu b (x,t 1 ) + t t 1 τ o d η 3 dt. ( ) All solutions need to be carried out until u p1 vanishing at t = t e, which completes the process of Bingham-plastic mud response under a surface solitary wave loading Extension of the solution technique The above solution technique can be extended to study surface waves over another yield-stress fluid-mud, namely the bi-viscous mud: a material tends to 83

97 resist motion at low stress, but flows readily when the yield stress is exceeded. In other words, a bis-viscous mud has two distinct viscosities of finite values and the viscosity is much higher when the magnitude of the applied stress is less than the yield stress. As the solution approach for the bi-viscous problem follows closely the methodology presented in section 2.3.3, the detailed analysis is documented in appendix A instead Numerical examples For illustration, numerical solutions of Bingham-plastic mud motion subject to a surface solitary wave shall be presented. Several different scenarios will be considered. The prescribed water velocity along the water-mud interface is assumed to be the undisturbed solitary wave given as ( ) 3ǫ u b (x,t) = sech 2 2µ (x x 0 c t). ( ) It is reiterated that x 0 is the initial position of the wave crest, and c = 1 + ǫ the dimensionless celerity. In addition, ǫ and µ measure the wave nonlinearity and frequency dispersion, respectively. For all cases presented here, the following wave parameters are used x = 0, x 0 = 50, ǫ = µ 2 = 0.1. As for other physical parameters, let us consider d = 10, γ = 0.7, α = , τ o = 0.2. In terms of dimensional values: h 0 = 10 m, a 0 = 1 m, λ 0 = 200 m, 84

98 p m x γ = 0.7, τ o = 0.2, d = 10 (1) (a) θ = (b) θ = (c) θ = (d) θ = (e) θ = (f) θ = (g) θ = η j d (2) η 1 η 2 η Horizontal velocity (3) θ = (x x 0 ct) u b u p1 Figure 2.20: Muddy sea bed responses under a surface solitary wave loading (4-layer scenario) at different phases: (a) θ = (x x 0 ct) = ; (b) 0.629; (c) 0.760; (d) 1.019; (e) 1.511; (f) 1.851; (g) (1) The pressure gradient (dashed-dotted line indicates the yield stress, τ o /(γd)); (2) Locations of yield surfaces, η j, j = 1, 2, 3; (3) Water-mud interfacial plug flow velocity, u p1 (dashed-dotted line is the water particle velocity at the water-mud interface, u b ). The corresponding velocity profiles are illustrated in figure A second plug flow region develops after phase (b) which is yielded again at (d). The mud flow motion returns to a single yield surface (η 3 ) structure as η 1 and η 2 are merged at phase (f). 85

99 η d η d η d η d η d η d η d (a) η η (a ) (b) η η (b ) (c) η η (c ) (d) η η (d ) (e) η η (e ) (f) η η (f ) (g) η η (g ) u m u m max u m Figure 2.21: Muddy sea bed responses under a surface solitary wave loading (4-layer scenario) vertical profiles of horizontal velocity component, u m, at different phases: (a) θ = (x x 0 ct) = ; (b) 0.629; (c) 0.760; (d) 1.019; (e) 1.511; (f) 1.851; (g) Left panels, (a) (g), show the velocities throughout the entire mud column while the right panels, (a ) (g ), are expanded for 0 < η < η 1 or 0 < η < η 3 at the same phases. In each plot, the dotted line represents the zero velocity reference line. Clearly, a second plug shear flow pair is formed from the solid bottom during the deceleration phase of the surface solitary wave and the flow reversal occurs (cf. phase (e)). 86

100 d = 0.95 m, ρ m = 1.43 g cm 3, ν m = m 2 s 1, τ o = 8.67 N m 2, where λ 0 = 2πL 0 has been defined as the effective wavelength. As mentioned in Mei, Liu & Yuhi (2001), the properties of mud vary widely, depending on the chemical composition, sediment concentration, salinity and other factors. For instance, the mud found in Yunan Province, China has a viscosity three orders of magnitude greater than that of water and the yield stress reaches O(100) N m 2. On the other hand, Krone (1963) reported that the mud in San Francisco Bay, USA has a viscosity which is in the same order of magnitude as the water and the yield stress is much smaller compared to the mud observed in China. The parameter set employed here is within the range of Provins clay data collected by Mei & Liu (1987). In figures (four-layer scenario), (three-layer scenario), and (two-layer scenario), the effects of the yield stress on the resulting mud motions are demonstrated. Three different values of yield stress, τ o = 0.2, 2.0, 4.0, are used while all other parameters remain the same. In the case of a relatively small yield stress (τ o = 0.2), i.e., figures 2.20 and 2.21, the fourlayer scenario inside the mud column results. From ( ) and ( ), it is clear that the dimensionless parameter τo measures the relative ease of the mud γd to be mobilized under a given incident wave. As the yield stress is weak relative to the wave loading ( τo γd = 0.029, see plate (1) in figure 2.20), the Binghamplastic mud is quickly liquefied and a shear flow layer starts to develop from the solid bottom when the friction due to the yield stress is balanced by the pressure force. Note that because of the viscous shear, the plug flow velocity at the water-mud interface, u p1, is not in phase with the velocity of the solitary wave and the mud flow can move in the opposite direction of wave propagation (see (3) in figure 2.20). During the initial period (θ < 0.032, phase (a)) both the plug 87

101 flow velocity, u p1, and thickness of the viscous shear layer, η 1 /d, grow in time. The velocity profile at the phase (θ = 0.032) of maximum plug flow velocity is illustrated in panel (a) of figure As the crest of solitary wave passes, the unfavorable pressure gradient eventually slows down the forward motion in the mud column as shown in (3) of figure However, the corresponding shear layer thickness, η 1 /d, is still increasing until the phase θ = (see (2) in figure 2.20). At the phase θ = (t = t 0 ), i.e., plate (b) of figure 2.21, the shear strain rate vanishes at the bottom of muddy bed and the lowermost Binghamplastic mud returns to its plastic state (plug flow). Once the mud is solidified, the friction between the bottom of the mud layer and solid bed prevents this portion of mud from moving. The material plasticity resists the viscous force. As the unfavorable pressure gradient continues to push the mud column backwards, the thickness of the second plug flow region (with zero velocity), η 2 /d, increases and the shear flow layer thickness, (η 1 η 2 )/d, shrinks (cf. phases (c) to (d) in figures 2.20 and 2.21). Since the yield stress is relatively small in this case, as the unfavorable pressure gradient persists, the bottom plug flow region is eventually yielded again. A new shear layer is formed at θ = (i.e., phase (d), t = t y ) and continues to grow (see (d) to (e) in figures 2.20 and 2.21). At this point, there are two plug flow regions (η 1 < η < d; η 3 < η < η 2 ) and two shear flow layers (η 2 < η < η 1 ; η < η 3 ); the two plug flow regions move in the opposite direction and flow reversal occurs (cf. (e), figure 2.21). The process continues as the lower shear flow layer grows and the middle shear flow layer shrinks. Finally, the sandwiched shear layer vanishes at θ = (i.e., phase (f), t = t 1 ) and the mud motion returns to a single yield surface structure moving towards the end of the event at θ = (t = t e ). Notice that when the wave crest already propagated far away, i.e., u b 0 or pressure gradient vanishes (see 88

102 (3) in figure 2.20), it is actually the inertia of mud drives its motion. For the case with a larger yield stress (τ o = 2.0, figures 2.22 and 2.23), it requires a stronger driving pressure gradient to yield the mud and to create the first shear flow layer (cf. plate (1) in figure 2.22). The shear flow layer thickness is also relatively thinner than that in the previous case. During the unfavorable (positive) pressure gradient period, for instance, phase (c), the strong plasticity suppresses the viscous force and the pressure gradient. As a result, the bottom solid layer (plug flow) builds up and eventually the mud motion pauses (cf. phase (d), figure 2.23). From phase (b) to (d), i.e., stage (II): t 0 t t 1, there is only one shear flow layer being sandwiched by two plug flow regions. Immediately after the zero motion moment, a new shear flow layer develops from the solid bottom and continues to grow as the positive pressure gradient increases, see phases (e) and (f) in figures The mud flow structure now returns to a single yield surface structure progressing towards the end of the whole process. We reiterate that there is no flow reversal in this case and mud flow motion is continuous. Figure 2.24 and 2.25 show a case where the mud has an even stronger yield stress, i.e., τ o = 4.0. The sea bed is barely liquefied and the mud flow motion is relatively small with a single yield surface structure throughout the entire process. Obviously, a flow reversal is impossible in this case. The mud flow moves intermittently with no motion during θ (t 0 t t 1, t 1 given in ( )). While it has been demonstrated that the present results can have very different features from the approach of Mei & Liu (1987) for low yield stress situations (cf. figures and ), the solutions of this high yield stress case (τ o = 4.0) are indeed similar to those presented in Mei 89

103 (1) γ = 0.7, τ o = 2, d = 10 p m x 0.0 (a) θ = (d) θ = (b) θ = (e) θ = (c) θ = (f) θ = η j d (2) η 1 η 2 η (3) u p θ = (x x 0 ct) Figure 2.22: Muddy sea bed responses under a surface solitary wave loading (3-layer scenario) at different phases: (a) θ = (x x 0 ct) = ; (b) 0.363; (c) 0.430; (d) 0.479; (e) 0.594; (f) (1) The pressure gradient (dashed-dotted line indicates the yield stress, τ o /(γd)); (2) Locations of yield surfaces, η j, j = 1, 2, 3 ; (3) Water-mud interfacial plug flow velocity, u p1. The corresponding velocity profiles are illustrated in figure A second plug flow region develops at phase (b) but no flow reversal appears (i.e., at each instant the maximum possible number of yield surface(s) is two). The entire mud column pauses at phase (d) and immediately continues the backward motion as a single yield surface structure. 90

104 η d 0.2 (a) (b) η 0.1 d (c) η 0.1 d (d) η 0.1 d (e) η 0.1 d (f) η 0.1 d η η 1 η η 1 η η 1 η η 1 η η (a ) (b ) (c ) (d ) (e ) u m η η u m max u m (f ) Figure 2.23: Muddy sea bed responses under a surface solitary wave loading (3-layer scenario) profiles of horizontal velocity component, u m, at phases: (a) θ = (x x 0 ct) = ; (b) 0.363; (c) 0.430; (d) 0.479; (e) 0.594; (f) Left panels show the velocities throughout the entire mud column while the right ones give the corresponding details. Dotted lines indicate the zero velocity reference. Due to a large yield stress, a plug flow region builds up from the solid bottom and eventually pauses the mud at the transition between forward and backward mud motion (see (b) to (d)). As a result, it is impossible for the flow reversal to develop. When the sea bed begins to move in the opposite direction to the solitary wave propagation direction, the mud column can be described again by a single yield surface structure. 91

105 & Liu (1987). Figure 2.26 shows the locations of yield surfaces and the plug flow velocity from both studies. Two models give similar results with some differences. The discrepancy can be mainly attributed to one of assumptions by Mei & Liu (1987) that the shear flow layer thickness is small and negligible when computing the plug flow velocity (see ( )). Apparently, this assumption becomes invalid as d decreases. Figure 2.27 shows the shear strain rate along the bottom of muddy bed (η =, which is proportional to the bottom shear stress, τ mb. In all three cases mb 0), um η there exists a time interval where zero velocity gradient appears along the solid bottom, i.e., τ mb τ o. As has been discussed previously, for the large yield stress case, τ o = 4.0 (figures 2.24 and 2.25), within this period the entire mud column is solidified and stays at rest, while for other two cases the upper portion of mud column keeps moving. Therefore, there is no clear trend describing the length of the zero strain rate interval as the physical processes are quite different for the examples shown in figure The mud movement appears to start and end more gradually for the case of smaller yield stress, i.e., τ o = 0.2. When the yield stress is very low, the mud behaves closely to a viscous fluid. However, it is remarked that for a purely viscous fluid mud, the zero bottom strain rate occurs only at one moment. The effect of Bingham-plastic viscosity on the mud flow motion has also been investigated. Figures 2.28 and 2.29 demonstrate the plug flow velocity and locations of yield surfaces for various dimensionless mud layer thickness, d = 1, 5 and 10, with the same initiation parameter: τ o /γd = 0.029, 0.29 in figures 2.28 and 2.29, respectively. Since τ o γd = τ o d 1 ǫµρ w g, ( ) 92

106 (1) γ = 0.7, τ o = 4, d = 10 p m x 0.0 (a) θ = (d) θ = (b) θ = (e) θ = (c) θ = (f) θ = (2) η 1 η 3 η j d (3) u p θ = (x x 0 ct) Figure 2.24: Muddy sea bed responses under a surface solitary wave loading (2-layer scenario) at different phases: (a) θ = (x x 0 ct) = ; (b) ; (c) ; (d) 0.784; (e) 1.141; (f) (1) The pressure gradient (dashed-dotted line indicates the yield stress, τ o /(γd)); (2) Locations of yield surfaces, η j, j = 1, 3; (3) Water-mud interfacial plug flow velocity, u p1. The corresponding velocity profiles are illustrated in figure In this case, the shear flow region is relatively small due to the large yield stress. The mud flow motion pauses for a long interval before it starts to move backwards. Only a single yield surface structure appears throughout the whole process. 93

107 η d 0.10 (a) (b) η 0.05 d (c) η 0.05 d (d) η 0.05 d (e) η 0.05 d (f) η 0.05 d η η 1 η η 1 η η 1 η η 3 η η (a ) (b ) (c ) (d ) (e ) u m η η u m max u m (f ) Figure 2.25: Muddy sea bed responses under a surface solitary wave loading (2-layer scenario) profiles of horizontal velocity component, u m, at different phases: (a) θ = (x x 0 ct) = ; (b) ; (c) ; (d) 0.784; (e) 1.141; (f) Left panels, (a)- (f), show the velocities throughout the entire mud column while the right panels, (a ) (f ), are the detailed features at the same instants. In each plot, the dotted line indicates the zero velocity reference. The velocity profiles vary monotonically in all phases, which is similar to those presented in Mei & Liu (1987). 94

108 M&L η 1 η 3 γ = 0.7, τ o = 4, d = 10 η j d u p (x x 0 ct) Figure 2.26: Comparison of results from Mei & Liu (1987) and current study for the case of large yield stress, τ o = 4.0. The upper panel displays the locations of yield surfaces (M&L: solid line; Current study: dashed-dotted line = η 1, dashed line = η 3 ) and the lower panel is the water-mud interfacial plug flow velocity. in each figure a different value of d (= d /(αl 0 ), α 2 ν m ) can be interpreted as the results of changing viscosity (i.e., treat τ o, ρ m and d as constants). Despite the fact that the mud flow motion is initiated at the same instant with a fixed τ o (γd), it does not guarantee that the subsequent mud flow motions will be the same. For instance, all three cases display a four-layer scenario when τ o /d = 0.02 (figure 2.28) but behave differently for τ o /d = 0.2 (figure 2.29). As can be seen, low viscosity mud (bigger d) can move faster in the forward direction and the duration of mud flow motion lasts longer. In addition, the time interval within which the multiple yield surfaces appear tends to shorten as the viscosity increases (d decreases). However, this does not imply that the single yield surface 95

109 γ = 0.7, d = 10 τ o = 0.2 τ o = 2.0 τ o = 4.0 u m η mb (x x 0 ct) Figure 2.27: The strain rate of various types of mud bed at the bottom. The mud is assumed to have different yield stresses with dashed line: τ o = 0.2, solid line: τ o = 2.0 and dashed-dotted line: τ o = 4.0. In all cases, γ = 0.7 and d = 10. model of Mei & Liu (1987) is adequate when the multi-layer interval becomes small (e.g., d = 1 in figure 2.28) as the mud flow behaves very differently, i.e., flow reversal occurs, within this period. Next, the effects of actual mud layer thickness, d, is examined. In figure 2.30, the mud column is thicker for bigger d since all other physical parameters are kept the same (constant τ o and γ are equivalent to fix τ o and ν m ). It is seen that the thin mud layer case, d = 1, has much smaller plug flow velocity as the relative yield stress, τ o /d, is stronger. Referring to figures 2.20 to 2.25 (various yield stress, τ o ), we can conclude that the mud bed thickness and strength of yield stress show similar effects on the mud flow motion: low τ o /d cases are easier to be initiated and tend to have multi-layer mud structure, stronger plug flow velocity and thicker shear flow region. 96

110 u p d = 1 (τ o = 0.02, 4L) 1.0 γ = 0.7, τ o /d = 0.02 d = 5 (τ o = 0.1, 4L) 1.0 d = 1 d = 5 d = 10 d = 10 (τ o = 0.2, 4L) η j d (x x 0 ct) Figure 2.28: Effects of viscosity on the mud flow motion with τ 0 /d = 0.02: fixed τ 0 /d represents same τ o and d ; small d stands for high viscosity mud. The upper panel shows the water-mud interfacial plug flow velocity, u p1, with dashed line: d = 10, solid line: 5, dashed-dotted line: 1 and dotted line the zero velocity reference. Lower plates are the the locations of the yield surfaces, η j /d, j = 1, 2, 3, with dotted line: η 1, solid line: η 2 and dashed line: η 3. All three cases display a four-layer scenario and the mud is initiated at the same instant. Low viscosity mud (bigger d) tends to have faster forward plug flow velocity and the overall mud flow duration last longer. 97

111 γ = 0.7, τ o /d = 0.2 u p d = d = 1 (τ o = 0.2, 2L) 0.4 d = 5 (τ o = 1, 3L) 0.2 d = 1 d = 5 d = 10 d = 10 (τ o = 2, 3L) η j d (x x 0 ct) Figure 2.29: Effects of viscosity on the mud flow motion with τ 0 /d = 0.2: τ o and d are fixed while a small d corresponds to a high viscosity mud. The upper panel shows the water-mud interfacial plug flow velocity, u p1, with dashed line: d = 10, solid line: d = 5, dashed-dotted line: d = 1. Lower panels are the the locations of the yield surfaces, η j /d, j = 1, 2, 3, with dotted line: η 1, solid line: η 2 and dashed line: η 3. While d = 1 (high viscosity mud) shows a two-layer scenario, the other two are three-layer scenario Wave attenuation caused by a thin layer of mud Estimating wave energy dissipation in the muddy seabed and the corresponding wave damping rate is one of the key objectives in studying the interaction between waves and seafloor. Referring to Dalrymple & Liu (1978) and Mei & Liu (1987), in a moving coordinate following the wave propagation the balance 98

112 u p d = d = 1 (τ o /d = 0.2, 2L) 0.6 γ = 0.7, τ o = 0.2 d = 5 (τ o /d = 0.04, 4L) 0.6 d = 1 d = 5 d = 10 d = 10 (τ o /d = 0.02, 4L) η j d (x x 0 ct) Figure 2.30: Effects of physical mud layer thickness, d, on the mud flow motion: A fixed value of τ o represents the same viscosity and yield stress (see the normalization introduced in (2.2.30)). The upper panel shows the water-mud interfacial plug flow velocity, u p1, with dashed line: d = 10, solid line: d = 5, dasheddotted line: d = 1 and dotted line the zero velocity reference. Lower plates are the the locations of the yield surfaces, η j /d, j = 1, 2, 3, with dotted line: η 1, solid line: η 2 and dashed line: η 3. For a thinner sea bed, d = 1, the mud flow motion shows a two-layer scenario while thicker mud cases are four-layer scenario. 99

113 of wave energy requires de dt = D m, ( ) where E and D m represent the wave energy and the energy dissipation in the muddy seabed, respectively. The dissipation in ( ) can be calculated by D m = 0 d τ m u m dz dx. ( ) z For a solitary wave, the dimensionless free-surface profile, ζ(x, t), can be expressed in the form ζ = ζ a 0 = a sech 2 [ 3ǫa 2µ ( x x ǫa t) ]. ( ) It is reminded that a = a /a 0 1 is the dimensionless wave height. Therefore, the total wave energy for a solitary wave is obtained as where and E p = E k = E = E p + E k, ( ) 1 2 ρ wgζ 2 dx = ρ wg (a h 0 ) 3/2 ( ) 1 2 ρ w(h 0 + ζ )u 2 b dx = ( ǫa)E p ( ) are the potential and kinetic energies, respectively. If the wave nonlinearity is weak, i.e., ǫ is small, we can assume E k E p. In addition, for long waves the celerity is roughly equal to the group velocity, c c g. These two approximations lead to de dt = c de g c de dx dx. ( ) Substituting ( ) and ( ) to ( ) into ( ), we derive the evolution equation of the dimensionless wave height ( da dx = α ) ǫ 3 1 d ( u m τ γµ 2 m 4 a η dηdx = α ǫ γµ 2 ) 3 4 F D a, ( )

114 where F D, the dissipation function, represents the double-integral term and has to be calculated numerically. As an example, figure 2.31 plots the dissipation function F D for several different scenarios calculated by the current model and the theory of Mei & Liu (1987). The parameter sets being used are same as those in figures (four-layer scenario), (three-layer scenario), and (two-layer scenario), respectively. The corresponding evolution of surface wave height is also shown in panel (II) of the same figure. For the case of larger yield stress (τ o = 4.0, two-layer scenario), the present results fit well with those of Mei & Liu (1987) as expected. However, the discrepancy becomes obvious as strength of yield stress decreases. Note that it is not possible to compare the results for τ o = 0.2 (four-layer scenario) since an unbounded shear layer thickness occurs in Mei & Liu (1987) (see their Fig. 4 for more details). In addition, in panel (I) we observe that there is no clear relationship between the values of F D and strength of yield stress. For a larger τ o, the corresponding strain rate is weaker. However, the product of strain rate and shear stress, which actually accounts for the energy dissipation, is not necessary smaller (i.e., F D can be larger). Referring to panel (II), we find that the wave height can be damped out severely by the presence of Bingham-plastic mud. For instance, the case of τ o = 0.2 shows that wave height could be reduced by 50% after propagates over x /h Moreover, as can be seen in panel (II) both the present solutions and those of Mei & Liu (1987) approach asymptotic values (or equivalent to F D approaches zero in panel (I)), which means that the attenuated surface solitary wave can no longer move the Bingham-plastic mud for the diminished wave pressure gradient becomes too weak to yield the mud, i.e., u b < τo. Since mud with weaker yield stress has t γd less ability to resist the viscous shearing than that with stronger yield stress, it 101

115 can eventually dissipate more wave energy (I): Dissipation function, F D τ o = 0.2 τ o = 2 τ o = 4 τ o = 2 (ML87) τ o = 4 (ML87) (II): Dimensionless wave height, a τ o = 4 F D a a τ o = 0.2 τ o = a = a /a Present ML87 ER08 WC Distance travelled, x /h 0 Figure 2.31: (I): Dissipation function, F D (see ( )), for γ = 0.7, d = 10 and τ o = 0.2, 2, 4. Solid lines are the current results while dashed-dotted and dashed lines (ML87) plot the solutions of Mei & Liu (1987). (II): Evolution of dimensionless wave height (a = a /a 0 ) with respect to the dimensionless traveling distance. Solid lines represent the current model results (correspond to figures 2.20 to 2.25, respectively) and dashed lines are the solutions of Mei & Liu (1987). All symbols (WC81) are the field observations of Wells & Coleman (1981) and the dashed-dotted line (ER08) shows the calculation using the measured dissipation rate at 4.5 m deep water by Elgar & Raubenheimer (2008). A constant water depth of h 0 = 10 m is used in the present model calculations while for WC81 the depth ranges from 7.1 to 8.7 m. In figure 2.31 the field observations by Wells & Coleman (1981) (WC81: circles) and Elgar & Raubenheimer (2008) (ER08: dashed-dotted line) are also plot- 102

116 ted in panel (II). To present the data collected by Wells & Coleman (1981), the water depth at the first station in the field experiment has been used (i.e., h 1 in their TABLE 1; h 1 = m). As for the damping curve of Elgar & Raubenheimer (2008), a constant depth of 4.5 m is adopted (see their Figure 2). It is reminded that in the model calculations x /h 0 = x/µ with µ = 0.1 and h 0 = 10 m. As shown in panel (II), one of model predicted wave height curves (τ o = 0.2) is close to WC81 and ER08. In fact if we increase the value of τ o slightly, the results will fit WC81 very well. However, the present model is not expected to fully explain the field observations as the wave conditions and mud properties in both WC81 and ER08 are incomplete. For instance, the mud in WC81 is inhomogeneous with density, ρ m = g cm 3, and viscosity, µ m = kg m 1 s 1. In addition, the mud layer thickness is about half meter. Although no yield stress data is available in Wells & Coleman (1981), it has been mentioned that the mud exhibited very low strength. For the mud property in Elgar & Raubenheimer (2008), the seabed has been described as a layer of 0.3 m thick yogurt-like mud above a harder clay bottom. The mud has a density, ρ m = 1.3 g cm 3, and can resist shear. Despite the fact that the physical parameters in the field studies and the current numerical examples are not perfectly matched, the comparison of wave height attenuation does suggest that the muddy seabeds mentioned in these two sites behave more closely to Bingham-plastic mud with weaker yield stress where the model of Mei & Liu (1987) is not adequate to describe the mud flow motion as the multi-layer scenario occurs. 103

117 2.3.7 Summary Response of Bingham-plastic muddy seafloor under a surface solitary wave loading has been investigated. A semi-analytical/numerical approach is used to obtain solutions inside the mud bed. The present analyses suggest that layered flow structures can occur, depending on the magnitudes of yield stress and the viscosity of the mud, the thickness of the mud bed, and the strength of the solitary wave. Four alternating plug flow and shear flow layers are possible. Detailed mud motions driven by a surface solitary wave have been successfully demonstrated for the possible scenarios. Wave damping rate for the solitary wave is also estimated and there are indications that they agree qualitatively with available field data. 2.4 Conclusions Considering the water-mud system as a two-layer setting, a depth-integrated model has been developed to describe the dynamic interaction between weakly nonlinear and weakly dispersive surface waves and a thin layer of viscoelastic mud. Response of a Bingham-plastic seabed to a surface solitary wave is also studied. Model predictions are examined against the available laboratory measurements and field observations. The overall agreement is reasonably well. In the present study, water viscosity is neglected. This can be improved by installing a viscous boundary layer right above the water-mud interface. However, the correction is expected to be small due to the fact that the water viscosity is much smaller than the typical viscosity of mud, as has been suggested by 104

118 field samples. The assumption of a flat solid bottom beneath the mud layer has also been made in the current analysis. This certainly limits the application of the proposed theory only to the case where bottom slope is negligible. Further investigation is required to examine the wave-mud interaction on an inclined beach. Finally, a true challenge comes from the assumption that thickness of the muddy seabed is fixed in our consideration, i.e. vertical displacement at the water-mud interface is neglected. Although this has been justified both theoretically and experimentally, it is still not satisfactory. Significant interfacial movement can be expected, in particular, when a sloping bottom is considered. For instance, Traykovski et al. (2000) reported a strong field evidence showing a considerable interfacial waves in the wave-mud system on a roughly 1-on-150 slope. As has been pointed out by Mei et al. (2010), predicting the depth is an immense challenge. Of course, in the fluid dynamics problem of wave-mud interactions there are still many physical processes being neglected in the present study, such as resuspension and deposition of cohesive sediments. Nevertheless, supported by the good performance of the theoretical predictions, it is fair to say that the present model provides a better understanding on the change of wave climate caused by the muddy seabed, and the dynamics of wave-induced mud flow. Some of the results presented in this chapter have been published in Liu & Chan (2007a,b) and Chan & Liu (2009). 105

119 CHAPTER 3 LONG WATER WAVES THROUGH EMERGENT COASTAL FORESTS This chapter discuss the effects of emergent coastal forests on the propagation of long surface waves of small amplitudes. While the forest is idealized by a periodic array of vertical cylinders, a two-parameter model is employed to represent bed friction and to simulate turbulence generated by flow through the tree trunks. A multi-scale (homogenization) analysis is carried out to deduce the effective equation on the wavelength-scale with the effective coefficients calculated by numerically solving the flow problem in a unit cell surrounding one or several cylinders. Analytical and numerical solutions for amplitude attenuation of periodic waves for different bathymetries are presented. In addition, results for the damping of a leading tsunami wave are discussed to demonstrate the effects of forests on transient waves. It is seen that strong reflection and energy dissipation can occur when surface waves propagate through a coastal forest. The proposed theory is compared with a series of laboratory data for periodic and transient incident waves. Good agreement is observed. 3.1 Introduction The hydrodynamics of tidal flows through mangrove swamps have been widely studied for understanding the health of coastal ecosystems (see e.g., Wolanski, Jones & Bunt 1980; Wolanski 1992; Mazda, Kobashi & Okada 2005). For inland waters, Nepf (1999) has investigated flow and diffusion of nutrients and solvents in a steady current. It has also been noted that coastal forests can serve as barriers against tides, storm surges and tsunami waves (Kerr & Baird 2007). 106

120 Historical evidence suggests that mangroves shielded the eastern coast of India and reduced the number of deaths in the 1999 cyclone attack (Dasa & Vincent 2009). Records of the 2004 Indian Ocean tsunamis have given strong support to the hypothesis of shore protection by mangroves and trees (Danielsen et al. 2005; Tanaka et al. 2007). Field experiments conducted in Australia and Japan also demonstrated that during high tides only 50% of incident wave energy is transmitted through forests over a distance of 200 m (Massel, Furukawa & Binkman 1999). Indeed, this evidence has motivated suggestions and laboratory studies for planting a strip of trees along the shores. For instance, Hiraishi & Harada (2003) have proposed the Green Belt with trees planted in water to guard against tusnami attacks. Through a seires of experimental studies, Irtem et al. (2009) demonstrated that trees planted on the landward side of the shore can reduce the maximum run-up of a model tsunami by as much as 45%. Augustin, Irish & Lynett (2009) and Thuy et al. (2009) have both reported laboratory studies of wave damping by emergent cylinders along with numerical simulations employing parameterized drag models. In tidal swamps, part or most of the vegetation can be constantly immersed in water. For effective protection against tsunamis, the needed thickness of the green forest can be hundreds of meters. Hence an emerging plantation would likely be a preferred option along well populated shores. It follows that the understanding of the dissipation process of long waves through emergent vegetation is then essential. As can be expected, the dissipation of wave energy is dominated by turbulence generated between the tree trunks, branches and leaves throughout the entire sea depth, and by bed friction (Massel, Furukawa & Binkman 1999). In general, numerical simulations shall provide the most detailed and accurate predictions. Mo & Liu (2009), for instance, developed a 107

121 three-dimensional numerical model to study a solitary wave interacting with a group of cylinders. Similarly, the large eddy simulation for open channel flows through submerged vegetations has also been performed by Stoesser et al. (2009). Both studies showed very good agreements when compared with laboratory measurements. However, it is noted that in Stoesser et al. (2009) the largest dimension in the computational domain is only about forty times of the cylinder diameter. For the study by Mo & Liu (2009), about one and a half million numerical cells are already required for the case of three cylinders. The application to real problems can be computationally expensive. In practice, several simplified mathematical models, all based on the common parameterized drag force concept (e.g. Massel, Furukawa & Binkman 1999; Mazda, Kobashi & Okada 2005; Teh et al. 2009), have been proposed to describe the most important impact of coastal forests on surface wave propagation, namely the wave damping, without the demand of massive computation. In other words, instead of resolving the detailed flow, effects of individual tree trunks are represented by a bulk drag force term. The required model coefficient, i.e. the so-called drag coefficient, is usually obtained by fitting the simulated free-surface profiles with either laboratory measurements (e.g. Thuy et al. 2009) or field observations (e.g. Mazda, Kobashi & Okada 2005). Although easy to implement, this kind of empirical drag force model does not explain well how the detailed structure (i.e. flow problem around tree trunks) affects the global behavior (i.e. the surface wave transformation). One important goal of the present study is to systematically develop a theoretical model for describing the propagation and dissipation process of long waves through emergent coastal forests. Through the substantive effort of mathematical work, it is the hope to better address the effective impact of the tree trunks than the existing drag force approach. 108

122 For the wave-forest problem being considered, the typical wavelength can be of O(100) m while the diameter of tree trunks is O(0.5) m. Therefore, the physical problem can be viewed as a micro-scale structure, i.e. the coastal trees, subjected to a macro-scale forcing, i.e. the surface waves. The goal is then to obtain a macro equation associated with the effective property of micro-scale material in question. In other words, it is to find the representative macroscopic property through multiple-scale analysis. To enable an analysis of the macroscale phenomenon from the micro-scale upwards, a number of simplifications are made. The first is to consider only long waves of small amplitude so that linearized approximation applies. The second is to model tree trunks by rigid cylinders in a periodic array but neglect the effects of tree branches, roots, and leaves. Turbulence generated between tree trunks is then described by the constant eddy viscosity model. Finally, bottom friction is represented by a linear law. With these simplifications, the two-scale method of homogenization is carried out to derive the mean-field equations on the macro-scale. The effect of tree trunks on the mean flows appears in macro equations through an effective hydraulic conductivity, which needs to be calculated from the solution of certain initial-boundary-value problem on the micro-scale. Several macro-scale problems will be discussed, both analytically and numerically. For possible application to wind waves, the marco theory for linear progressive waves is first presented. Of interest to the protection against tsunamis, the transient problem is then considered. Numerical examples will be demonstrated under different incident wave conditions and coastal forest configurations. The present theory is also examined by comparing with a series of laboratory experiments for both periodic and transient incident waves. Comparison shows a very encouraging agreement. 109

123 3.2 Theoretical formulation Consider a train of long water waves entering a thick coastal forest from the open sea. Spanning a large horizontal area, vertical cylinders are erected in a periodic array of uniform spacing to simulate emergent and rigid tree trunks. An illustration of a coastal forest is provided in figure 3.1. The tree spacing l and the typical water depth h 0 are assumed to be comparable. Since long waves are considered, both l and h 0 are much smaller than the typical wavelength L 0, i.e., O(l/L 0 ) = O(h 0 /L 0 ) 1. The variation of water depth is assumed to be appreciable only over a distance scale comparable to a wavelength. (a) (b) l Figure 3.1: Problem sketch. (a) Illustration of a coastal forest. (b) Proposed model: trees are modeled by emergent cylinders of a uniform spacing l. Only tree trunks are considered Governing equations and boundary conditions For clarity, the horizontal and vertical quantities are separated: u = (u 1,u 2 ) and w denote the horizontal and vertical velocity components in x = (x 1,x 2 ) and z coordinates, respectively. 110