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Numerical modeling of submarine landslides and surface water waves which they generate Jiang, Lin 1993

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NUMERICAL MODELING OF SUBMARINE LANDSLIDESAND SURFACE WATER WAVES WHICH THEY GENERATEbyLIN JIANGB.Sc., Tianjin University, China, 1984M.A.Sc., Tianjin University, China, 1987A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Department of Oceanography)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJanuary 1993© Lin Jiang, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission. (Signature)Department of  06atUrtPh8The University of British ColumbiaVancouver, CanadaDate^Ma OP&^fiver3DE-6 (2/88)liAbstractUnderwater landslides are a common source of tsunamis in coastal areas. In this thesis,three numerical models are developed to simulate the coupling of submarine landslides andthe surface waves which they generate. Three different rheologies of the submarinesediments are considered: a viscous fluid, a Bingham-plastic fluid, and a Coulombfrictional material. Formulations of the dynamics are presented for the slides on a gentleuniform slope reactively coupled with the surface waves which they generate. Thegoverning equations are solved by a finite-difference method.In the first model, the submarine slide is treated as a laminar flow of incompressibleviscous fluid. Three major waves are generated: the first wave is a crest which propagatesinto deeper water, this crest is followed by a trough in the form of a forced wave; the thirdwave is a small trough which travels shoreward. Two major parameters dominate theinteraction between the slide and the waves: the density of sliding material and the depth ofwater at the slide site.The landslide is treated as an incompressible Bingham-plastic flow in the second model.Because of the yield stress (or plasticity), the slides stop on the slope when the shear stressexerted on the bottom becomes smaller than the yield stress. The Bingham-plastic behaviorof the mud significantly reduces the run-out distance and the speed of the slide, and alsoreduces the magnitude of the surface waves and the water current generated.In the third model, the submarine sediment is treated as a Coulomb frictional material. Africtional material model can well describe the slide of soft sandy sediment, in which theCoulomb friction law applies to the basal friction on the bottom of the slide. It is found thatthe friction angle between the slide material and the slope has the most significant effect onthe slide dynamics and wve generation.iiiThe numerical results obtained in this thesis indicate that the rheology of the sediment(i.e., the constitutive relation) dominates the slide dynamics, and hence the surface wavegeneration. The greater the mobility of the sediment, the greater the amplitudes of thewaves generated.ivTable of ContentspageAbstract ^ iiList of Figures^ viiiAcknowledgement ^ xvi^1 General Introduction   11.1 Large Water Waves Generated by Sea Bottom Motions^ 11.2 Previous Studies on Submarine Slides and Waves Induced by Seafloor Motion••31.2.1 Modeling of Submarine Landslides^ 31.2.1.1^Viscous Fluid Models 71.2.1.2^Bingham Fluid Models^ 81.2.1.3^Frictional Material Models  . 91.2.2 Modeling of Water Waves Induced by Sea Bottom Motion^ 101.2.2.1^Waves Generated by Earthquake-induced Bottom Motion-101.2.2.2^Waves Generated by Submarine Slides^ 121.3 Objectives 141.3.1 On Surface Waves Generated by Submarine Landslides^ 141.3.2 On the Interaction between Submarine Slides and Waves^ 152 A Viscous Slide and Surface Wave Generation^ 172.1 Introduction^ 172.2 Governing Equations for a Viscous Slide^ 182.3 Governing Equations for the Waves 242.4 One-way Coupling and Fully Coupled Models^ 252.5 Numerical Results for the Slide with an Initial Parabolic Profile^262.5.1 Slide Profiles^ 2 9V2.5.2 Horizontal Velocity of the Slide^ 3 52.5.3 The Frontal Speed of the Slide 382.5.4 Elevations of the Surface Waves^ 432.5.5 Horizontal Velocity of the Water Motion 492.5.6 Energy Transfer from the Slide to the Waves ^522.5.7 Possibility of Resonance between the Slide and the Waves ^552.6 Numerical Results for the Slide with an Initial Triangular Profile ^572.6.1 Numerical Results for a Triangular Slide^ 572.6.2 Comparison with Huppert's Solution 632.6.3 Wave Runup at Shore^ 6 52.7 Numerical Predictions for the ADFEX Experiment^672.7.1 The Slide Surface Variation^ 692.7.2 The Slide Velocity Distribution 692.7.3 The Surface Waves ^ 7 42.7.4 The Velocity Distribution of Water Motion^772.8 Numerical Results for Large-Scale Slides 802.8.1 Slide Profiles^ 8 02.8.2 Horizontal Velocity of the Slide^ 8 42.8.3 The Frontal Speed of the Slide 872.8.4 Elevations of the Surface Waves^ 892.8.5 Waves Generated by a Rigid Triangular Block^922.9 Effects of Turbulence at Slide-Water Interface 1002.9.1 Kelvin-Helmholtz Instability and Turbulence ^ 1002.9.2 Effects of Interfacial Turbulent Shear on Water Motion^ 1012.9.3 Effects of Interfacial Turbulent Shear on the Slide 1052.10 Summary and Conclusions^ 107vi3 A Bingham-Plastic Fluid Model and Surface Wave Generation^ 1103.1 Introduction ^ 1 103.2 Governing Equations for a Bingham-Plastic Slide^ 1113.3 The Viscous Model - A Special Case of the Bingham Model^ 1223.4 Governing Equations for the Waves^ 1233.5 One-way Coupling and Fully Coupled Models^ 1243.6 Numerical Results and Discussions 1253.6.1 Surface Variation of the Slides^ 1273.6.2 Distribution of the Plug Velocity of the Slide^ 1353.6.3 Variation of the Slide Frontal Speed^ 1403.6.4 The Thickness of the Yield Layer ^ 1403.6.5 Elevations of the Surface Waves.^ 1443.6.6 Distribution of the Water Particle Velocity^ 1473.6.7 Energy Transfer from the Slide to the Waves 1523.7 Comparison with a Snow Flow Test ^ 1533.8 Summary and Conclusions^ 1604 A Frictional Material Model and Wave Generation ^ 1624.1 Introduction^ 1624.2 Governing Equations of a Cohesionless Frictional Submarine Slide^ 1644.3 Governing Equations of Surface Waves ^ 1744.4 Numerical Results and Discussions 1744.4.1 Surface Variation of the Slide ^ 1764.4.2 The Slide Velocity Distribution^ 1814.4.3 The Surface Water Elevations ^ 1864.4.4 The Velocity Distribution of Water Motion 1864.5 Summary and Conclusions^ 195vii5 Summary and Conclusions^ 1966 Bibliography^ 202viiiList of Figurespage1.1^Hypothetical stages in the evolution of a submarine slide.^41.2^Conceptual process of mudflow on a slope.^ 61.3 Two kinds of underwater mudflows. 72.1^Definition sketch of an underwater viscous slide and surface waves. ^ 192.2^Definition sketch of a parabolic initial mud surface. ^262.3^Profiles of a viscous slide on a slope with 8=4°, D o=4 m, Lo=100 m, r=1.4,idp2=0.2 m2/s and r=0.44. Solid line: two-way coupling; dotted line: one-waycoupling. ^ 312.4^Profiles of a viscous slide on a slope with the same conditions as specified inFigure 2.3 except r=2.0. Solid line: two-way coupling; dotted line: one-waycoupling.  ^322.5^Profiles of a viscous slide on a slope with the same conditions as specified inFigure 2.3 except K=2/11. Solid line: two-way coupling; dotted line: one-waycoupling. ^2.6^Profiles of a viscous slide on a slope with the same conditions as specified inFigure 2.5 except r=2.0. Solid line: two-way coupling; dotted line: one-waycoupling. ^2.7^Successive variation of the horizontal velocity of the viscous slide, under the sameconditions as specified in Figure 2.3. The maximum velocity reached (0.247[U])corresponds to 4.1 m/s. ^ 362.8^Successive variation of the horizontal velocity of the viscous slide, under the sameconditions as specified in Figure 2.4. The maximum velocity reached (0.246[U])corresponds to 5.43 m/s.^ 3733342.9^Successive variation of the frontal speed of the slide, under the same conditions asspecified in Figure 2.3.^ 42ix2.10 Successive variation of the frontal speed of the slide, under the same conditions asspecified in Figure 2.4.^ 422.11 Successive variation of the frontal Froude number for slide (1) and slide (2)respectively.^ 432.12 Evolution of surface waves by the viscous slide with the same conditions as inFigure 2.3. Solid line: two-way coupling; dotted line: one-way coupling.^452.13 Evolution of surface waves by the viscous slide with the same conditions as inFigure 2.4. Solid line: two-way coupling; dotted line: one-way coupling.^462.14 Evolution of surface waves by the viscous slide with the same conditions as inFigure 2.5. Solid line: two-way coupling; dotted line: one-way coupling.^472.15 Evolution of surface waves by the viscous slide with the same conditions as inFigure 2.6. Solid line: two-way coupling; dotted line: one-way coupling.^482.16 Successive variation of the horizontal velocity of the water motion, under the sameconditions as specified in Figure 2.3. The maximum velocity reached (0.03M)corresponds to 0.501 m/s (shoreward).^ 502.17 Successive variation of the horizontal velocity of the mud flow along the slope,under the same conditions as specified in Figure 2.4. The maximum velocityreached (0.025[U]) corresponds to 0.55 m/s (shoreward).^512.18 The variation of the energy transfer ratio with time and mud properties. Solid lineshows the case as specified in Figure 2.3; dotted line, the case as specified inFigure 2.4.^ 542.16 The variation of the energy transfer ratio with time and mud properties. Solid lineshows the case as specified in Figure 2.5; dotted line, the case as specified inFigure 2.6.^ 542.20 The variation of the Froude number versus time for slide (2) on four slopes. ^562.21 Definition sketch of an initial triangular slide surface.^572.22 The surface variation of the slide with a triangular initial surface on a slope with0=4°, D o=4 m, Lo= 100 m, p2=1400 kg/m3 , p/p2 .2 m2/s and x=0.44. Solid line:two-way coupling; dotted line: one-way coupling.^59x2.23 Successive variation of the horizontal velocity of the slide with an initial triangularsurface, under the same conditions as specified in Figure 2.22. The maximumvelocity reached (0.205[U]) corresponds to 3.42 m/s. Solid line: two-waycoupling; dotted line: one-way coupling.^ 602.24 Evolution of surface waves generated by the slide with an initial triangular surface,with the same conditions as in Figure 2.22. Solid line: two-way coupling; dottedline: one-way coupling.^ 612.25 Successive variation of the horizontal velocity of the water motion induced by antriangular slide, under the same conditions as specified in Figure 2.22. Themaximum velocity reached (0.022[U]) corresponds to 0.37 m/s (shoreward).••••622.26(a) Comparison of the present numerical solution and Huppert's solution forthe slide front positions, for the case of a triangular initial shape.^642.26(b) Comparison of the present numerical solution and Huppert's solution for the slidefront positions, for the case of a parabolic initial shape.^642.27 A definition sketch for wave runup on a uniform slope. ^652.28 Bathymetry of the ADFEX experiment area.^ 682.29 Profiles for the slide with p 2=1.5 g/cm 3 and p/p2=0.2 m2/s.^702.30 Profiles for the slide with p 2=2.5 g/cm3 and p/p2=0.3 m2/s.^712.31 The velocity distribution for the slide specified in Figure 2.29.^722.32 The velocity distribution for the slide specified in Figure 2.30.^732.33 The surface wave evolution for the case specified in Figure 2.29.^752.34 The surface wave evolution for the case specified in Figure 2.30.^762.35 The water particle velocity distribution for the case specified in Figure 2.29.^782.36 The water particle velocity distribution for the case specified in Figure 2.30.^79xi2.37 Profiles of a viscous slide on a slope with 0.4 0 , D o=24 m, Lo=686 m, r=1.2,pip2=0.2 m2/s and tc=0.40. Solid line: two-way coupling; dotted line: one-waycoupling.^  822.38 Profiles of a viscous slide on a slope with the same conditions as specified inFigure 2.37 except r=2.0. Solid line: two-way coupling; dotted line: one-waycoupling. 832.39 Successive variation of the horizontal velocity of slide (a). The maximum velocityreached (0.36[U]) corresponds to 11.9 m/s.^ 852.40 Successive variation of the horizontal velocity of slide (b). The maximum velocityreached (0.44[U]) corresponds to 25.5 m/s.^ 862.41 Successive variation of the frontal speed of the slide (a).^882.42 Successive variation of the frontal speed of the slide (b). 882.43 Successive variation of the frontal Froude number for slide (a) and slide (b)respectively.^ 892.44 Surface waves induced by slide (a). Solid line: two-way coupling; dotted line: one-way coupling.^ 902.45 Surface waves induced by slide (b). Solid line: two-way coupling; dotted line: one-way coupling.^ 912.46 The positions of a rigid triangular block (solid line) with density p 2=1.2g/cm3 , andfriction coefficients f=0.02, sliding on a 4° slope, in comparison with the result ofthe triangular viscous slide (dotted line). 942.47 The positions of a rigid triangular block (solid line) with density p 2=1.2g/cm3 , andfriction coefficients f=0.03, sliding on a 4° slope, in comparison with the result ofthe triangular viscous slide (dotted line). 952.48 Surface waves generated by the rigid triangular block as specified in Figure 2.46(solid line) in comparison with the result of the triangular viscous slide (dottedline).^ 96)di2.49 Surface waves generated by the rigid triangular block as specified in Figure 2.47(solid line) in comparison with the result of the triangular viscous slide (dottedline).^ 972.50 Distribution of the velocity of the water motion induced by the rigid triangular blockas specified in Figure 2.46 (solid line) in comparison with the result of thetriangular viscous slide (dotted line).^ 982.51 Distribution of the velocity of the water motion induced by the rigid triangular blockas specified in Figure 2.47 (solid line) in comparison with the result of thetriangular viscous slide (dotted line).^ 992.52 Kelvin-Helmholtz instability and turbulence in a two layer sheared flow. ^ 1022.53 Variation of Fr=Uf/(gh) 112 versus time for the small-scale slide (as specified inFigure 2.4) and the large-scale slide (as specified in Figure 2.38).^ 1043.1^Constitutional laws of a Newtonian fluid and a Bingham fluid. ^ 1123.2^Definition sketch of an underwater Bingham slide and surface waves.^ 1133.3^Velocity distribution of a one-dimensional Bingham fluid flow.^ 1153.4^Definition sketch of an initial parabolic mud surface.^ 1263.5^Successive shapes of the mud surface, for K=0 (viscous model).^ 1293.6^Successive shapes of the mud surface, for K=0.05.^ 1303.7^Successive shapes of the mud surface, for K=0.2. 1313.8^Successive shapes of the mud surface, for K=0.35.^ 1323.9^The positions of the mud surface for the slides with different yield stressesat time t=8.^ 1333.10 The positions of the mud surface for the slides with different yield stressesat time t= 10 .^ 133xiii3.11 The positions of the mud surface for the slides with different yield stressesat time t= 12 .^ 1343.12 The positions of the mud surface for the slides with different yield stressesat time t=14.^ 1343.13 Transient variation of the slide particle velocity, for K=0 (viscous model). ^ 1363.14 Transient variation^of the plug velocity, for K=0.05.^ 1373.15 Transient variation of the plug velocity, for K=0.2. 1383.16 Transient variation of the plug velocity, for K=0.35.^ 1393. 17 The variation of the frontal speed of the slides with different yield stresses.^ 1413.18 The variation of the thickness of the yield layer, for K=0.05. The solid lineshows the slide surface and the dotted line shows the yield interface.^ 1423.19 The variation of the thickness of the yield layer, for K=0.2. The solid lineshows the slide surface and the dotted line shows the yield interface.^ 1433.20 Evolution of surface waves, for K=0 (viscous model).^ 1453.21 Evolution of surface waves, for K=0.05.^ 1463.22 Evolution of surface waves, for K =0.2. 1473.23 Evolution of surface waves, for K=0.35.^ 1483.24 The variation of the water particle velocity, for K=0 (viscous model).^ 1503.25 The variation of the water particle velocity, for K=0.2.^ 1513.26 The variation of the energy transfer ratio with time for the slides withdifferent yield stresses, K=0, 0.1, 0.2, 0.3.  152xiv3.27 The initial snow profile (t=0), the final snow profile (dots) and the theoreticalpredictions of the fmal snow profile (solid line) with parameters(a) Ty =660 N/m2, it/p2=0.0001 m 2/s. ^ 1573.28 The initial snow profile (t=0), the final snow profile (dots) and the theoreticalpredictions of the fmal snow profile (solid line) with parameters(b) Ty =330 N/m2, u/p2=0.001 m2/s. ^ 1573.29 Variation of the flow speed with time, for snow with parameters(a) Ty =660 N/m2, plp2=0.0001 m 2/s. ^ 1583.30 Variation of the flow speed with time, for snow with parameters(a) ry =330 N/m2 , ,u/p2=0.001 m2/s.  ^ 1594.1^Definition sketch of a frictional slide and the waves which it generates.^ 1654.2^Stresses on a plane soil element. ^ 1664.3^Relation between Mohr circles and Mohr-Coulomb failure criterion. ^ 1704.4^Mohr diagram showing the Coulomb yield criterion and stress states.^ 1714.5^The slide surface variation of a frictional slide with 8=1°, 0=4°. ^ 1774.6^The slide surface variation of a frictional slide with 8=2°, 0=4°.  ^ 1784.7^The slide surface variation of a frictional slide with 8=2°, 0=6°. ^ 1794.8^The slide surface variation of a frictional slide with 8=4°, 0=6°. ^ 1804.9^The slide velocity distribution of a frictional slide with 61°,0=4°. ^ 1824.10 The slide velocity distribution of a frictional slide with 62°, 0=4°. ^ 1834.11 The slide velocity distribution of a frictional slide with 8 =2°, 0 =6°.^ 184xv4.12 The slide velocity distribution of a frictional slide with 8 =4°, 0 =6°.^ 1854.13 The water surface elevations generated by the slide specified in Figure 4.5. 1874.14 The water surface elevations generated by the slide specified in Figure 4.6. 1884.15 The water surface elevations generated by the slide specified in Figure 4.7. 1894.16 The water surface elevations generated by the slide specified in Figure 4.8. 1904.17 The distribution of the water particle velocity under the same conditionsas specified in Figure 4.5.^ 191 4.18 The distribution of the water particle velocity under the same conditionsas specified in Figure 4.6.^ 19 24.19 The distribution of the water particle velocity under the same conditionsas^specified^in Figure^4.7. 1934.20 The distribution of the water particle velocity under the same conditionsas^specified^in Figure^4.8. 194xviAcknowledgmentI sincerely thank my research supervisor, Dr. Paul H. LeBlond, first for providing methe wonderful opportunity to pursue my doctorate degree in such a beautiful country asCanada; and for his support, encouragement, stimulating discussions and helpfulsuggestions during the last two years of research; and for his patience in reviewing variousdrafts of this thesis.My sincere thanks also go to the members of my thesis supervisory committee: Dr. TadMurty, Dr. Susan Allen, and Dr. Gregory Lawrence, for their encouragement, helpfulsuggestions and their input into this thesis.I am also indebted to the staff at Department of Oceanography, Denis Laplante, WarrenLee, and Liusen Xie, for their generous help in computer programming. The suggestionsand assistance from my fellow students are also highly valued.Finally, I thank my wife, Min Luo, and my parents, Wenxiu Jiang and Xiuhua Fang,for their understanding and lasting support during the last three years of overseas studies,without which this thesis would not have been successful.1Chapter 1General Introduction1.1 Large Water Waves Generated by Sea Bottom MotionsUnderwater ground motions can generate surface water waves. A tsunami, a kind ofsurface wave, is formed in the sea following a large-scale, short-duration disturbance of thewater surface. The disturbance is generally produced by two kinds of ground motions. Oneis the earthquake-induced rigid ground motion ending with a permanent bottomdisplacement; the other is a landslide which is usually induced by slope failure or bottomfailure. Many small scale tsunamis have resulted from underwater landslides with sufficientdisplaced volume. This kind of wave, though much more localized than a "true" tsunami,can still produce large run-up heights at the coast, especially when the energy is trapped ina basin (e.g., a fjord) or along the coast. There have been many instances of very largewater waves which were generated by underwater landslides.A major submarine landslide [Murry, 1979] which occurred April 27, 1975, in KitimatInlet, British Columbia, Canada, generated water waves with an estimated range of heightof up to 8.2 m (a 4.6-m crest and a 3.6-m trough). The volume of the material involved inthis slide was estimated to be approximately 2.6x10 7 m3 . The duration of the slide wasestimated to be 0.5-2.0 minutes. The slide occurred at the time of low tide and wasattributed to the shear failure in a soft clay slope.Karlsrud and Edgers [1980] reported that an underwater slide which was triggered by aminor blasting inside a ship wreck in the fjord near Sandnessjoen, Norway, generated aflood wave of 4-7 m in height and broke the telecables across the fjord 2.0 km away fromthe slide site.Miloh and Striem [1978] studied the extreme changes in the sea level along the coast2of Levant, Israel, and found that a recession of the sea occurred more often than aflooding of the shore. Such events may have been caused by mud slumping on thecontinental slope.Recently discovered evidence of slumping at the foot of the Fraser Delta, BritishColumbia, Canada, has also created some apprehension about the possibility of tsunamigeneration in the Strait of Georgia [Hamilton and Wigen, 1987].On a large scale, the Grand Banks earthquake of 1929 produced a major debris flow thatinitiated a major turbidity current and also a tsunami which caused damage toNewfoundland's south shore [Kirwan, 1986; Locat et al., 1990]. Therefore, a betterknowledge of the phenomenon is of great significance to the development of offshoreresources exploration and protection, and also to reservoir management.The Geological Survey of Canada has initiated project ADFEX (Arctic Delta FailureEXperiment) to trigger, for the first time, a submarine landslide under controlled conditionson the Kenamu River delta in Melville lake, Canada. This project presented a rareopportunity to obtain real-time field data on dynamic changes in the physical parameters ofthe sliding mass, and also information on the surface wave generation as a result of asubmarine slide event [Locat et al., 1990]. The analytical and numerical models developedin this thesis aimed towards explaining the results to be obtained in the ADFEXexperiment.Unfortunately the initiation of an underwater landslide at the delta front failed, due tosome unknown technical problems. So there is not any field data from ADFEX experimentto compare with the theoretical predictions of the present models. Comparisons shallhowever be made between my results and those calculated for an underwater sliding rigidobstacle, and between my results and Huppert's [1982] asymptotic solution for the flow ofa finite mass of viscous fluid on a slope, as well as for a field experiment of snowavalanche [Dent, 1980, 1982].31.2 Previous Studies on Submarine Slides and Waves Induced by Seafloor Motion1.2.1 Previous Models of Submarine SlidesThe development of offshore engineering has raised great interest in the investigation ofsubmarine landslides. As more and more offshore engineering structures will be built onthe seafloor, the risks associated with submarine landslides must be taken intoconsideration. Such an evaluation must include answers to a number of very difficultquestions, some of which are:(1) what is the extent of a possible slide, and how does the slide develop with time?(2) what mechanisms govern the movements of the debris downslope, and what will bethe velocity, height, and run-out distance of the masses?(3) what forces might the moving mass exert on structures of different kinds as itmoves downslope, and to what extent will it erode sediments as it moves along? and(4) what kind of surface water waves will be generated, and how large will thesewaves be and where will they propagate?There has been little field observation data on large submarine landslides due to theinaccessibility to submarine environments. The available information of underwater slideshas been obtained from investigation of post-slide seabed topography, the sediment depositdistribution, the break-down of undersea telecables due to slides, as well as limitedlaboratory experiments [Locat, et al., 1990]. Masses involved in a slide might behave in avariety of ways after failure, ranging from slow creep movements, to rapid debris flows,fluidized sediment flows and diluted turbidity currents. Middleton and Hampton [1976]discussed the hypothetical development of different types of flows, and suggested aclassification as shown in Figure 1.1. The governing factors for the transition of sedimentinto different types of flowing material are the soil characteristics (chemical composition,grain size, concentration, etc.) of the original material as well as the environmentalDepositDEBRIS FLOWUpwardFLUIDIZED SEDIMENT FLOW^flow of porele- fluid^ Grain interactionHIGH CONC. TURBIDITY (Dispersive pressure)CURRENTGRAIN FLOWSlide^Remoldingslump liquefactionTurbulenceLOW CONC. TURBIDITY CURRENT4conditions. Soft sensitive clays and loose sand and silts might normally turn into more orless liquid material and flow downslope with considerable velocity and run great distanceson very gentle slopes. As it moves downslope, water might be mixed into the flowingmass, making it more liquid in behavior. Some material might also in the process be "tornloose" from the main body of the flow and turn into dilute turbidity currents.Karlsrud and Edgers [1980] argued that only minor parts of the main body of a flowingmass will normally turn into very low density turbidity currents unless the flowing massmeets some obstructions, or there are very abrupt changes in slope geometry, possiblycreating a "hydraulic jump". This might also be dependent on the grain size distribution ofthe masses involved. Hampton [1972] obtained the same conclusion from his laboratoryinvestigation on the generation of turbidity currents.iTIME AND / OR SPACEFigure 1.1: Hypothetical stages in the evolution of a submarine slide (Middleton &Hampton 1976)5Edgers and Karlsrud [1982] have also reviewed case histories of submarine slides andthe associated soil flows. They collected data from a number of the best documented slides,ranging in size from small coastal slides which ran out a few hundred meters, to largedeepwater slides which ran out hundreds of kilometers. Their examination of the datarevealed the following points:(1) submarine slides may be triggered on very flat slopes, even as low as 1 degree; thevolumes and distances from back-scarp to toe are enormous in comparison to mostterrestrial slides;(2) the predominant soil types having large run-out distances were silts and fine sand;(3) a major factor in the development of all these slides was the availability of weak andunstable sediments and the presence of a triggering mechanism, such as overloading,earthquakes and human activities along the coastline.The failure of the slide initiation in ADFEX experiment is probably due to the lack of aweak and easy-to-fail delta front. So even with a triggering factor (an undersea explosion)the slide of sediment could not be initiated.Conceptually, there are three sequential phases in the mudflow process on a slope asshown in Figure 1.2 [Kirwan, et al., 1986]. Phase I encompasses the initiation of the flowand the acceleration to some terminal velocity, and it is an unsteady state. Phase II can be aperiod of constant velocity if the slope is uniform and long enough for such a constant stateto be developed; it is a relatively steady state. Because the velocity is nearly constant (notexactly constant) during phase II it is the easiest to deal with theoretically. Phase III is aperiod of deceleration. Due to the unsteadiness of phase I and phase III the vast majority oftheoretical and experimental research focuses on phase II. Up to now, there has been littlestudy on the mudflow in phases I and III, especially coupled with surface waves.As to their flow regimes, underwater slides can be classified into two classes: a laminarflow and a turbulent flow as shown in Figure 1.3. A mudslide on a slope will become 6A Phase I^Phase II Phase III^1.1-0^TimeFigure 1.2: Conceptual process of mudflow on a slopeturbulent if it acquires strong enough shear at its surface, either through downslopeacceleration or through encounters with obstructions or abrupt changes in slope. For aturbulent slide, the flow is dominated by the conditions at the head. There will be aturbidity current flowing over a lower dense flow as shown in Figure 1.3(1). When theReynolds number exceeds 1000, the flow structure at the head does not change with theReynolds number [Simpson, 1987]. The dominating factors for the case of a turbulent slideare: (1) turbulence, (2) ambient water flow (water entrainment), (3) viscous force, and (4)buoyancy. Potential development of interfacial instability (Kelvin-Helmholtz type) andonset of turbulence in a stratified shear flow will be briefly discussed in Section 2.9.1.There are also many large-scale examples such as lava flow, debris current andmudslide, in which viscous forces play a large part in the dynamics of the flow [Simpson,1987]. There will be no significant mixing at the slide interface. The flow looks likeFigure 1.3(2). The dominating forces in such flows are (1) inertial force, (2) buoyancy,and (3) viscous force. To solve the complicated problem of underwater slides (soil-watermixture), one has to simplify the problem and attack it step by step. Most theoretical studiesturbidity current.--^..4,,.....,-*°'.- "N•N"..N.Nr"---,--,....,--,-, r^- .- 'Th'dense flow(1) Turbulent(2) LaminarFigure 1.3: Two kinds of underwater mudflows.-), -1I''' ,......... ".% ^..%7focused on the laminar gravity currents [Edgers, 1981; Huppert, 1982; Trunk et al., 1986;Hocking, 1990; Liu and Mei, 1989. Mei and Liu, 1990].Various theoretical models have been proposed to simulate the phenomena of submarinelandslides, which focused upon the laminar case of submarine debris flow or fluidizedsediment flow (the lower dense part of the flowing mass). Most of these studies are one-dimensional and steady-state solutions, and they can be classified into three classes: (1)viscous models [Edgers, 1981; Huppert, 1982; Trunk et al., 1986; Hocking, 1990], (2)visco-plastic models [Johnson, 1970; Bubayada and Prior, 1978; Liu and Mei, 1989. Meiand Liu, 1990], and (3) frictional models [Morgenstern, 1967; De Matos, 1988; Locat andNorem, 1989 ].1.2.1.1 Viscous Fluid ModelsThe viscous approach is based on the assumption that a submarine sediment-watermixture behaves like a viscous fluid, which is true for the mixture of water and fine soil8particles (sometimes called mud or muck)[Krone, 1962; Parker, 1986; Verreet andBerlamont, 1987; O'Brien, 1988, ]. In a viscous model the coefficient of viscosity and theslope angle are the predominant parameters.Edgers [1981] has reviewed previous viscous models and proposed a simple viscousapproach that relates the flow velocity with the displacement of submarine flows. Thismodel was applied to a sample problem of flow within the Norwegian Trench, whichdemonstrated the importance of the detailed slope geometry, the input soil rheologicalparameters, the initial thickness of the flow and non-steady state conditions in modeling theflow.Trunk, et al, [1986] presented a computer-based analysis of the two-dimensionaldynamics of the Madison Canyon rockslide of August 17, 1959, in which a singleviscosity model and a biviscosity model were used to represent the flow. The classicNavier-Stokes equations were used in the numerical form in the modeling. The numericalresults of the two models suggest a mechanism of an active high-shear-stress layer in thebasal zone and the bulk of the slide moves with smaller internal deformations.Huppert [1982] and Hocking [1990] studied the flow and instability of a viscous fluiddrop released on a sloping surface, using the approximations of lubrication theory (laminarflow with inertia effect neglected, and with capillary effect included). It was found, by theirtheory, that the flow evolves into a thin and long fluid sheet. The flow depth profile isindependent of the initial conditions some time after the initiation of the flow.1.2.1.2 Bingham Fluid ModelsJohnson [1970] analyzed one-dimensional subaerial debris flows by means of aBingham model (i.e., visco-plastic model (a laminar model) with a yield resistance, zy , ,and a linear viscosity, au ). This solution predicts a zone within which the shear is9sufficiently low so that it moves as a rigid plug with no internal deformation. Thisprediction was verified by the observations of laboratory slurry experiments and of anatural debris flow in California. Johnson's model [1970], however, only describes steadyuniform debris flows.Liu and Mei [1989, 1990] presented a set of governing equations for a thin sheet ofBingham-plastic fluid spreading slowly on an inclined plane. Their theory was developedon the basis of shallow-water approximation which is similar to the theory of lubrication.Because of the yield stress, static profiles of the fluid with non-uniform depth are possibleon a sloping bed. Transient flows due to either a steady upstream discharge or to thesudden release of a finite fluid mass on another fluid layer were studied by solving a set ofnonlinear partial differential equations.1.2.1.3 Frictional Material ModelsFrictional models [Morgenstern, 1967; De Matos, 1988] were also proposed to studysubmarine landslides. A frictional model is particularly suitable for the slide of coarsesandy sediment, in which the Coulomb friction law can well describe the bottom frictionalresistance.Morgenstern [1967] investigated the generation of a turbidity current from a viewpointof limit equilibrium. He proposed a general model to study the initiation of a slide on aninfinite slope which is set to move by both the action of gravity and earthquakeacceleration, and he derived a useful formula to evaluate the risk of failure for very gentleslopes. De Matos 's [1988] model includes a frictional term in the slide dynamics and wasable to reproduce the cable break data of the Grand Banks Slide of 1929.Suhayda and Prior [1978] extended Johnson's [1970] solution by including a Coulombfriction component of soil resistance which depends on the effective stress. They calculated10the plug velocity for the chute flow in the Gulf of Mexico, on the basis of some simpleassumptions regarding the viscous properties of the chute materials. As an effective stressmethod of analysis, it requires an evaluation of the pore water pressures during the flow,which are very difficult to evaluate.Locat and Norem [1989] examined the previous physical models of submarine slideswhich are considered to behave as granular material. In their model, Coulomb friction andviscoplastic behavior are assumed to be predominant in the slide dynamics.Savage and Hutter [1989] presented a numerical model to simulate a finite mass ofgranular material flowing down a rough slope. Their assumption for the constitutiveequations are mainly based on the experiments made by Hungr and Morgenstern [1984],which indicate that the slow flow of dry granular material is well described by the Coulombfriction law.According to the review by Locat et al. [1990] much less work has been done regardingthe mechanical characteristics and behavior of the sediment and also the tsunamis triggeredby the slide. Little is known on the post failure behavior of the sediments and its effects onthe seabed and on the engineering structures along the flow path. A better knowledge of thephenomenon will impact the development of offshore resources exploitation and helpexplain a fundamental sedimentary process.1.2.2 Modeling of Water Waves Induced by Sea Bottom Motion1.2.2.1 Waves Generated by Earthquake-induced Bottom MotionMost previous studies on surface wave generation by bottom motion have beenconcerned with the surface elevations induced by the impulsive motion of an impermeablerigid bottom resulting from an undersea earthquake. For earthquake-induced waves, oneneed not consider the interaction between the tectonic motion and the surface waves,11because the duration of the earthquake is usually very short (several seconds). To study thewaves caused by an earthquake, one usually treats it as the classical Cauchy-Poissonproblem: the initial value problem for the generation and propagation of surface waterwaves, i.e., assuming the initial surface elevation is the same as the displacement of thebottom and then converting the problem into the initial value problem of wave propagation.For surface waves generated by underwater landslides, one has to consider the couplingof the landslide and the waves which it generates, because the slide duration is relativelylong and the interaction between the slide and the waves will affect the characteristics ofboth. All the previous studies on submarine landslides were based upon the assumption ofa rigid lid at the water surface or under the deep water assumption (the slide can not be feltby the free surface). There has been little investigation of the interactions between anunderwater landslide and the waves which it generates.Tuck and Hwang [1972] presented analytical solutions and a quantitative discussion forthe case when the bottom motion occurs at a place where the bottom has an uniform slopeand the resulting waves then propagate into deep water. Dispersion is neglected in thegeneration region, where the typical horizontal length scale is supposed to be much greaterthan the local water depth. It was shown that dispersion might still be neglected somedistance away from the generation region, even though the depth is continually increasing.This is because the significant wavelength of the generated waves remain long compared tothe water depth up to a distance of order bla2 , where b is the length scale of the groundmotion and a is the bottom slope. Beyond this distance dispersion is undoubtedlysignificant. In the region b,oc«bla2, an asymptotic solution for the wave elevation wasderived. The amplitude of waves in the intermediate far-field zone is characterized by arelatively simple relationship with the space and time history of the ground motion. Withthis relationship, the effects of a presumed transient ground deformation was investigated,and a connection between the time scale of the transient deformation and the amplitude of12generated waves was established. The surface waves generated by a particular class oftransient bottom motion (D(x,0=art 2e -1te -x/b, D(x,t) is the bottom deformation whichbegins slowly from zero initial displacement and velocity, reaches a maximum of 0.54a att=2/y, and then decays exponentially to its original level) are calculated and discussed. Tuckand Hwang's theory, however, can be used only in the case of earthquake-induced waveswith specially presumed bottom motion; it can hardly be used to study landslide-inducedwaves because the bottom motion (i.e., the slide surface variation) is a. priori  unknown and it is also affected by the coupling between the slide and the waves.Liu and Earickson [1983] studied tsunami generation due to transient seabeddeformation by means of shallow-wave approximation and numerical simulations. Aperturbation technique was used to derive systematically a set of governing equations ofnonlinear shallow-water waves. Different regimes of approximation in terms of the order ofthe magnitude of ground motion were presented. Numerical examples of two-dimensionalcase were given and compared favorably with experimental data.1.2.2.2 Waves Generated by Submarine SlidesUp to now there has been little theoretical and experimental study on the surface wavegeneration by a submarine slide, due to difficulties in modeling and observing submarinelandslides and the associated water waves.Miloh and Striem [1978] proposed an empirical evaluation method to study the wavesinduced by underwater landslides. They assume that the strain energy, or the slide potentialenergy is partially converted into surface wave energy. Their main parameter is the energytransfer ratio, which is assumed to be less than 2%. In addition, they postulated that amajor solitary wave, followed by a few small waves that can be neglected, is generated.Equating the wave energy to the transferred part of landslide energy yields the desired13result of wave height. In this method, the energy estimation relies on an almost arbitraryvalue of the energy transfer ratio and also on the assumption of a solitary surface waveprofile. Other factors, such as the viscous energy dissipation and the energy dissipation bythe turbidity current following the slide should be taken into account. Besides, trappedwaves and other secondary phenomena may occur during the slide [Slatkin, 1971].Wiegel [1955] conducted two-dimensional experiments in which the effect of a fallingslide in a wave tank was analyzed. In the experiments, rigid bodies of several shapes,sizes, and weights were allowed to drop vertically or to slide down inclines of severalangles (24°-90°), in water of various depths, from several heights above the bottom, butalways below the water surface. The surface time histories were recorded and sourcemotion pictures were taken. It was found that a crest was always formed first downwardfrom the slope, followed by a trough from one to three times the amplitude of the first crest(depending primarily on the slope of the inclines), the trough is followed by a crest withabout the same amplitude as the trough. The amplitudes of the waves depended primarilyupon the submerged weight of the body but also upon the depth of submergence, the waterdepth, and other characteristics of generation. The transfer ratio, relating the net potentialenergy release to the wave energy, was estimated to be of the order of 1-2%. The slidingbodies used in Wiegel's experiments were rigid blocks; they cannot adequately simulateunderwater slides whose profiles are constantly changing while flowing down the slope.Recently Heinrich [1992] simulated numerically and experimentally the two-dimensionalproblem of waves generated by a rigid body with a triangular cross-section sliding on a 45°slope. The generation and propagation of the waves was numerically examined by solvingthe nonlinear Navier-Stokes equations with a finite-difference method, in which dispersionand nonlinearity of waves was included. Experimental results were compared withnumerical predictions, which showed close agreement except when turbulence occurred inthe flow field. Heinrich's result may give the correct solution for waves produced by an14underwater rockslide, and also may give maximum height of the waves that can beproduced by a deformable landslide.1.3 Objectives1.3.1 On Surface Waves Induced by Underwater LandslidesThe aim of this research is to develop analytical and numerical models to study thecoupling of a laminar flow of an underwater landslide and the surface waves which itgenerates. The coupling includes: (1) surface waves being generated by a downslopemudslide, and (2) at the same time the surface waves reacting on the mudslide. Thisproblem has so far been poorly investigated due to the complexity of submarine sedimentproperties and heavy numerical computation required for calculating surface wave motions.The mechanism which may initiate the landslides (e.g., rapid sedimentation, earthquakeloading, wave and tide pressure, etc.) will tiot be examined in this research. For simplicityin the theoretical analysis, it is assumed that a finite mass of sediment body suddenlyliquefies and reaches a well-mixed phase without appreciable downslope movement, or thata finite mass of sediment suddenly loses its static equilibrium and then begins to flow downthe slope.The following basic assumptions will be adopted in the analysis for the slides:(1) the submarine sediments (soil-water mixture) will be treated as a one-phasehomogeneous isotropic continuum;(2) the turbulent diffusion (mixing) that may occur along the interface between waterand slide flow will be neglected; but its effects and the conditions under which it becomesimportant will be discussed in Section 2.9;(3) an impermeable rigid slope will be considered, and the erosion from the sea bottomcaused by the slide flow will be neglected.15Formulations of the dynamics of the coupling between the slides of three kinds ofrheologies (viscous, Bingham and frictional) and the waves which they generate will bedeveloped. The governing equations for the slide and the waves will be solved numerically.The effects of sediment properties (the densities, viscosities, and plasticities), the flow path(slope angles), the initial slide dimensions (volumes) and shapes (parabolic, triangular, orarbitrary) upon the characteristics of the slide dynamics and the induced water waves willbe investigated. More specifically, the questions to be answered for the waves produced bya submarine slide are:(1) what kind of surface water waves will be generated?(2) how large these waves will be?(3) to which direction will the waves propagate?(4) how much wave energy is transferred from the slide to the waves?The various models developed here were to be used for the calculation of the landslideand the surface waves generated under the conditions of the ADFEX experiment. In theabsence of experimental data only theoretical predictions will be presented. Comparisons ofthe viscous slide solution with Huppert's asymptotic solution of a viscous flow on a slopewill be presented. Snow avalanche experiment results [Dent, 1982] will also be comparedwith the Bingham slide solution.1.3.2 On the Interactions Between Submarine Slides and WavesThrough numerical modeling, I shall examine the differences between the slidedynamics in three cases: the case of a rigid free surface; in presence of surface waves butneglecting reaction of the surface pressure on the slide; and with full consideration of theinteraction between the slide and surface waves. The following characteristics of the slideswill be studied:16(1) the variation of the slide surface with time and with the distance traveled;(2) the particle velocity distributions of the slide flow;(3) the front speeds of the slides;(4) the importance of reaction of waves on the slide;(5) the effects of possible turbulent interfacial mixing.I shall focus on the two-dimensional problem of water waves generated by underwaterlandslides on a gentle uniform slope in shallow water. The formulations of the dynamics ofsubmarine slides with three kinds of rheological properties, viscous fluid, Bingham-plasticfluid, and Coulomb frictional material are to be developed. Water is assumed to beincompressible, and wave motion is assumed to be irrotational. The long-waveapproximation will be employed for both water waves and the mudslide, which implies thatthe characteristic length scale of wave motion is much larger than the local water depth andthat the mud thickness is much smaller than the characteristic length scale of the mudslidealong the slope. The long-wave approximation is valid only for a small slope. Small slopesare very common in submarine slide environment [Edgers and Karlsrud, 1982]. So a slopewith angle of 1 0-10° will be considered. It is assumed that water and mud are initially at restand that the mud starts flowing down the slope at t=0.The slide and the surface waves produced by mudslides with initial parabolic andtriangular shapes are to be calculated by a finite-difference method. The energy exchangebetween the slide and the waves are to be investigated. Finally the effects of the potentialdevelopment of turbulence at the water-slide interface upon the slide dynamics and surfacewave generation will be examined. An empirical expression of turbulent shear stress will beused to assess the significance of the turbulent shear. Conditions under which turbulencebecomes important will be examined. Modeling of underwater turbulent slides and surfacewave generation, however, has to wait for another doctorate thesis.17Chapter 2A Viscous Slide and Surface Wave Generation2.1 IntroductionIn this chapter, a numerical model is presented to investigate water waves generated bythe laminar flow of a viscous slide on a gentle uniform slope in shallow water. Themutual effects between the slide and the waves which it generates are the focus of thischapter. Part of this chapter has been published in a paper entitled " The coupling of asubmarine slide and the surface waves which it generates" in the Journal of GeophysicalResearch [Jiang and LeBlond, 1992a]. The landslide material is assumed to be anincompressible, isotropic viscous fluid. The sea water is assumed to be incompressibleand inviscid, and the wave motion is assumed to be irrotational. For simplicity it isassumed that there exists a sharp interface with negligible mixing. Conditions underwhich the interfacial mixing becomes significant and the effects of interfacial turbulencewill be examined in Section 2.9. The long-wave approximation is employed for both thewater waves and the slide. A formulation of the governing equations of the laminarviscous slide is obtained by using the Navier-Stokes equations and the von Karmanintegral method. Long-wave equations with bottom deformation are also presented.Surface waves produced by a slide with a parabolic and a triangular initial shapes arecalculated by a finite-difference method.The numerical results for the slide surface variations, the slide velocity distributionsand the surface wave evolutions will be presented, which contrast the behavior of themud flow under a fixed surface (rigid lid at the water surface), in the presence of one-waycoupling (bottom deformations affect the free surface), and with full coupling (surfacepressure gradients react on the mud flow). The ratio of energy transfer from the slide to18the waves will be calculated, and the energy exchange will be discussed. The possibilityof a resonance between the slide and the waves will be examined. This model will beused to provide theoretical predictions for the ADFEX experiment. Finally, the effects ofthe potential turbulent shear at water-mud interface will be discussed.2.2 Governing Equations for a Viscous SlideSubmarine slides are assumed to separate into a dense flow close to the bed and aturbidity current above. Here I shall concentrate on the flow of the dense part which isconsidered to behave as a viscous fluid. The focus of the chapter is the two-dimensionalproblem of the coupling of a submarine viscous slide and the surface water waves whichit generates. I shall consider the case where the density difference is large, i.e., the case ofhigh-density flow ((p2 p,)? 0.2 g/cm 3), and I consider the case where the mixing at thewater-mud interface is negligible such that the slide material is not significantly dilutedwhile flowing downslope[Lava/ et al., 1988].Consider a layer of viscous mud flowing down a rigid impermeable slope inclined at asmall angle 0 with respect to the horizontal direction (see Figure 2.1 ). Let the x axis startfrom the upper margin of the slope, coinciding with the still water level (SWL) and bedirected to the deep sea. The z axis is positive upward. The free surface is designated asz=11(x,t), and the sloping bottom as z=-14(x). By the long-wave approximation, thevelocity is essentially in the x axis direction and the vertical momentum equation reducesto a vertical hydrostatic pressure distribution within the mud layer, i.e.,p(x,z,t) = p ig(n + h)— p2g(z + h),^—hs(x) 5_ z —h(x,t),^(2.1)where p(x,z,t) is the pressure in the mud layer; p i and p2 are densities of the water and theslide, respectively, g is the gravitational acceleration, and h(x,t) is the undisturbed depthhP1Xa, ...................".../........°' NriverSWLV 019of the water.Here, I employ the nonlinear Navier-Stokes equation for the horizontal momentumconservation of the viscous slide, which is assumed to be laminar, i.e.,d^dU^dUP2 (—U„,-- + Urn II + Wm ---" ) = (P2 — PI )g sin 0 - (.13 + 0( d2Um + d2 Um , (2.2)at^ax^az^ax ' ax 2^az2 ) where Um(x,z,t) and W„,(x,z,t) are the horizontal and vertical particle velocities in the mudlayer respectively, ii is the dynamic coefficient of viscosity of the slide. I will returnbelow to a discussion of the appropriateness of a laminar flow model.Figure 2.1: Definition sketch of an underwater viscous slide and surface waves.I neglect the tangential stress on the water-mud interface in the following theoreticalanalysis, because the viscosity of water is much smaller than that of the mud, and thebasal shear of the mudflow is much greater than the interfacial shear in the case oflaminar flow [Mei and Liu, 1987]. The conditions that turbulent mixing at the water-mudinterface becomes important will be addressed in Section 2.9. The corresponding20boundary conditions for the slide are: (1) the zero-shear condition on the water-mudinterface, and (2) the no-slip condition on the bottom, i.e.,aU„, . 0 ,'^z = —h(x,t),^ (2.3)Uni = 0,^z = -k (x), (2.4a)W. = 0,^z = —k(x).^ (2.4b)For large scale laminar viscous mud flow and debris flow, there will be two flowregimes: the inertial regime and the viscous regime [Huppert and Simpson, 1980;Simpson, 1987]. In the inertial regime, the dominating factors are the buoyancy (the firstterm in the right-hand side of equation (2.2)) and the inertial force (the left-hand sideterms in equation (2.2)). This is the stage corresponding to phase I in mud flow processeson a slope - the accelerating phase. As the flow accelerates, the resistance to the flowincreases and the acceleration decreases. The inertial force becomes less important andthe viscous force (the last two terms in the right-hand side of equation (2.2)) becomesdominant, and this is the viscous regime. The viscous regime corresponds to phases IIand III in the mud flow - the relatively steady state and the deceleration state. As will beseen in Section 2.5.3, the acceleration stage in a transient slide is a relatively short period,while the viscous regime (phases II and III) is a much longer processes.In the viscous flow regime, the inertial force becomes negligible and the buoyancybalances the viscous force. Thus the left-hand terms of (2.2) vanish and the verticaldistribution of horizontal velocity is found to be parabolic. Here, I assume that themudslide rapidly reaches its equilibrium velocity [see also Edgers, 1981; Mei and Liu,1987; Liu and Mei, 1989] so that a vertical parabolic approximation may be used for thehorizontal velocity, Um(x,z,t), for the whole slide process, i.e.,21)^.4-  yunjx, z, 0 = U(x,t)^D[2(z ± k (z Dk 1(2.5))^)which satisfies the boundary conditions on the water-mud interface and on the bottom,(2.3) and (2.4a, b). U(x,t) is the horizontal velocity of the mudflow at the water-mudinterface, z.-h(x,t), and D(x,t) is the thickness of the mud layer. In the relatively shortinertial regime, the velocity quickly increases and approaches a maximum velocity and avertical parabolic distribution. The viscous force also increases quickly as the velocityprofile approaches its final parabolic profile. Employing the parabolic velocity profile forthe slide in the inertial regime will change the process of acceleration only slightly (seeSection 2.53). Because the inertial stage is relatively short period, this approximation maynot lead to serious errors.The vertical velocity of the slide W.(x,z,t) can be obtained from the continuity equationdUn, + dW„, .... 0.dx^dz(2.6)Integrating the continuity equation (2.6) with respect to z from z=-I4(x) to z=z yieldsthe vertical velocity of the mudflowW,,,(x,z,t). — ---Lc1 zI dUdx. _D(x,o dU(x,t)[(  z + ky 1 r  z + hs ydx^D )^D )].By applying the von Karman momentum integral method [Bachelor, 1967; Mei andLiu, 1987], I integrate (2.2) with respect to z from z.-hs(x) to z=-h(x,t) and substitute(2.1), (2.5), and (2.7). Then I have the governing equation for the mudflow in terms of theunknowns D(x,t) and U(x,t),-h,(2.7)2 p2D dU - 1- p2U —dD + 3p2DU dU3^at 3^at 3^dx=D[ (p2 _ pogra _ dDl_ p1 g dill _ if 2 yi 2 d2U)dx ) dx i D 3 dx2where a= sine, and the water surface gradient term on the right-hand side of the equationrepresents the reaction of the waves on the sliding mud.Conservation of mass in the entire mud layer requiresdD dq _ ndt +^- - 'dx(2.9)where q(x,t) is the volume flux of the mudflow:-h(x,t)q(x,t) = f U .(x,z,t)dz = -2 D(x,t)U(x,t).3-h (x)(2.10)I choose the initial maximum mud thickness, D o, as the vertical length scale, [H]; theinitial mud length, Lo, as the horizontal length scale, [L]; and the horizontal velocity scale[U]=(gILDia. I adopt the following dimensionless variables(x* , z* ,t* ) = W -1 x,[1-1] -1 z, A I g' /[L]t),^(2.11a)^(n* ,D s ,h * ,hss ) =[11] -1 (n,D,h,h,),^(2.11b)(U.,^,^= [U] -1 (Um , Wn„ U), (2.11c)where the variables with an asterisk are dimensionless, and g' is the reduced gravity,defined as22(2.8)f P2 - A g =^g.P2(2.12)23With (2.11a), (2.11b), and (2.11c), equations (2.8) and (2.9) take on the form, with theasterisks being omitted,2^dU^1^dD^2^dUdD^e dry—D----U—+—DU—=D a - E— — 2 ( U 1 d2U( ),3^dt^3^at^3^ax^ax^r —1 ax ) eR —D 3 axe(213).wheredD dqd + dx = " , (2.14)[H]E =^,^r = P2^R= p2[H]Vg'[L] ^[L] A #The factor of the surface gradient term in (2.13) indicates that the reaction of waves onthe mudslide is inversely proportional to (r-1), i.e., the nearer the density ratio approachesunity, the greater the reaction of waves on the mudslide. For small density ratio, thereaction of the waves on the slide is significant; however, the action of the slide on thewater motion will be small because the slide flows more slowly, due to the fact that theactive driving force on the slide, the reduced gravity, is smaller.I have assumed a laminar flow regime in the mud with appropriate values of viscosities.A Reynolds number is defined asRe = p2UD^(2.15)allto characterize the mudflow. Laboratory experiments on gravity currents on a horizontalplane indicate that the flow structure is independent of the Reynolds number when it isgreater than 1000 [Simpson, 1987]. The critical Reynolds number for the onset ofturbulence for a two-layer sheared flow will be much smaller than this. Withoutexperiment data on the critical number for the cases considered here I choose the criticalReynolds number to be approximately Re=50 ( For Re>50, turbulence plays a non-24negligible role in the slide dynamics, see discussion in Section 2.9). It will be seen, insection 2.9.3, that the relative significance of the interfacial turbulent shear stress (at thewater-slide interface) and the bottom shear stress (on the slide bottom) depends on theReynolds number defined in equation 2.15.A more appropriate dimensionless number to examine the flow regime in a two layershear flow may be the Keulegan number [see Turner, 1973], which uses the scale of theinterface instabilities, 8=0(AU2)Ig' (AU=U-u, the velocity difference between the slideand the water), as the length scale the above definition of the Reynolds number.Substituting this length scale into equation (2.15) yields the Keulegan number,Ke=(AU)31 g'v. (2.16)Turner [1973] states that the critical value of the Keulegan number is approximately180 for the interface of the two-layer shear flow to remain laminar. The Keulegannumbers and the Reynolds numbers associated with the cases considered in this thesiswill be calculated and discussed. In section 2.9, I shall discuss the relative significance ofthe interface shear and the bottom shear stress for "the worst case scenario": where youhave significant turbulence at the water-slide interface.2.3 Governing Equations for the WavesFor waves on a gentle slope in shallow water, I adopt the long-wave approximation,i.e., the wave length is much greater than the depth of the water. For shallow waterwaves, the water motion is essentially in the x axis direction and the pressure distributionin the water can be assumed hydrostatic. The dynamic equations for nonlinear shallow-water waves induced by the general motions of an impermeable seabed are (Tuck andHwang, 1972)25d(n + h) + at^dx[u(n + h)] = 0,^(2.17)ud^du d-a-t-+u—+g-11 = 0ax^dx^'(2.18)where u(x,t) is the depth averaged horizontal particle velocity of the water motion. Notethat nonlinear term is included and dispersion of waves is neglected. Tuck and Hwang[1972] showed that dispersion can be neglected up to a distance of L o/(tan0)2.I adopt the same length scales, velocity scale and the time scale as those used for themudslide in (2.11a), (2.11b), and (2.11c), and remembering h(x,t)=14(x)-D(x,t), equations(2.17) and (2.18) in dimensionless variables read, with the asterisks being omitted,dn dD a^a^a= - —(uk)- -(u71)± at Tt dx^70x^dxdu - Er dn dudt^r -1 dx -u dx,(2.19)(2.20)where the first and fourth terms in the right-hand side of (2.19) represent the effects of themudslide on the surface waves, and the factor of the first term in the right-hand side of(2.20) also indicates that the interaction between the waves and the mudslide is greaterfor smaller density ratio, r (r> 1).2.4 One-Way Coupling and Fully Coupled ModelsThere are many factors that affect the coupling of a mudslide and the waves which itgenerates. The important ones are the density and viscosity of the mud, the initiationdepth of the mudflow, its initial shape and dimensions, and the steepness of the slope. Itcan be seen, from the surface pressure term in equation (2.13), that the effect of thesurface pressure gradient on the mudslide is small for large mud density. Mud with large26viscosity flows more slowly, and its effect on the surface of the water is small. For ashallow initiation depth of the mudslide, the slide is felt more strongly by the surfacewater than for a deeper slide initiation depth, and the interaction between the slide and thewaves is also more significant.Consider first the one-way coupling between the mudslide and the waves, i.e., onlyconsider the action of the mudslide on the water and neglect the reaction of the surfacewaves on the mudflow: the water surface gradient term in the governing equation of themudslide vanishes, and equation (2.13) reads_2 D dU 1 aD 2 dU+ DU—3 dt^dt 3^ax = D(a—e— —^(2.21)apj 2 (U 1 d2Uax eR^3 axeTo calculate waves in the one-way coupling model, I first solve (2.14) and (2.21) forthe mudslide to find D(x,t) and U(x,t), then I solve (2.19) and (2.20) to find n(x,t) andu(x,t) for the wave motion.In the complete model, I consider the full coupling between the mudslide and thewaves which it generates, i.e., the mudslide generates surface waves and at the same timesurface pressure gradients react on the mud flow. To calculate the mudflow and thewaves by the two-way coupling model, I need to solve (2.13) and (2.14) for the mudslideand (2.19) and (2.20) for the surface waves simultaneously.2.5^Numerical Results for the Slide with an Initial Parabolic ProfileAssume the mud is initially at rest and has a parabolic surface (see Figure 2.2), i.e.,D(x,0) = Do ll — [2(x — I)1 Lo]2 ) ,^(2.22)where, x is the initial x coordinate of the center of mud, I --.(x 1 +x2)/2; x 1 and x2 are the27initial x coordinates of the rear and front margins of the mud; Lo=x2-x1 , is the initial lengthof the mud, and D o is the initial maximum mud thickness. The reason I choose a parabolicinitial slide profile is that this profile has a continuous surface gradient, and I can avoidunnecessary singularity of surface gradient at the front and rear faces of the slide, whichmay occur if I choose a rectangular initial profile. Later, I shall also examine the case of atriangular initial profile to examine the effects of initial slide profiles on the interactionsbetween the slide and the waves.L0DoDMUM/X2Figure 2.2: Definition sketch of a parabolic initial mud surface.Equation (2.22) in dimensionless variables reads,D(x ,0) = 1 — 4(x — ic)2 .^ (2.23)Given the slope angle and the typical parameters and the initial positions of themudflow, I calculated n(x,t) and u(x,t) for the waves and D(x,t) and U(x,t) for the slide bynumerically solving the governing equations with a finite-difference method, in which theforward difference in time derivatives and the backward difference in space derivativesare used in the discretized equations. The governing equations for the slide and the waves(2.13), (2.14), (2.19), and (2.20) in the finite-difference form read28(D7+1 – 2D7 + D7_1),^(2.24)–U7_1)]711:1-1) ]3UeR(D:1)2(2.25)StDr' = D7 – –2 —St(U7D7 – U: iD:)+3 x^—^(8x)2U 7+ 1 = U7 + –1 U: St "(D – D7 1 ) – U7 St2 D7 Sx^-^Sx+32 [a – e —88xt(D7– D',_,) (r –1) Sx(rig!'e^St+ L ai + 1 ) St ,_+1– (u^ 2U: + U:_ 1 ),Vi.^eR)(&)2 're+1 = Tr --St (4k. –u:L ik i_ i)--St (terr –127, i'. 0Sx^'^Sx '^' + St (u7D7 – ur_ iDil i )+ ita2  St 2 (iii7, 1 — 2n: + 71i-1),^(2.26)Sx (5x)4+1 =Er^St (re – 117_1)– St (le – 1 )(r –1) Sx^Sx +µa2  St(8x)2 (u7+1 –^+ u7_ 1 ), (2.27)where the superscripts "n" and "n+1" represent the values at time t=nSt and t=(n+1)8t,respectively. Note that two small artificial viscosities p a, and pa, (of the form itadu1 xwith ita the artificial viscosity) are introduced in the above discretized equations for theslide and for the waves to avoid numerical instabilities, although the CFL condition wassatisfied. The reason for such a introduction in the mudflow equations is that sometimesthe real viscosity term is too small to smooth out horizontal small-scale instability. It isfound that minimum values of gal =0.002-0.006 for the mudslide, and ga =0.025 for thewaves are sufficient to keep the solutions stable.In the following numerical calculations, I consider a small-scale mudslide with aninitial parabolic surface of length L 0=100 m and maximum thickness D o=4 m, starting ona gentle uniform slope with inclination 0=-4°. Then I shall consider a large-scale slide inSection 2.8 and compare the slide evolution processes and the surface wave generation,29and also discuss the potential development of turbulence at water-slide interface and theeffects of turbulence on the slide and the wave generation in Section 2.9.The following typical parameter ranges are employed for the numerical calculation: thedensity ratio of mud and water r=1.2-2.5 [Mei and Liu, 1987]; the kinematic viscosityfor river and coastal mud v=µ/p2 .002-0.2 m2/s [Mei and Liu, 1987], for mudslide anddebris flows v=i2/p2=0.2-0.6 m2/s [Beaty, 1963; Simpson, 1987]. I consider a river (atx<O) with depth h 1 =0.5D0=2 m to avoid a mathematical singularity at the shore, and also adeep sea with depth h2=19.5D 0=78 m to make the model more practically consistent. Thetotal length of the calculation area is about L =3500 m (including the length of the river,1000 m, the length of slope of 1200 m, and the length of deep water area of 1300 m).The initial position of the slide is indicated by the ratio of the initial maximum slidethickness and the water depth at the slide center, K=D o/ho, where, ho=hs(1). Themaximum Reynolds numbers are to be calculated, and they are to be kept smaller than thecritical value at which the flow becomes turbulent. I shall present below the numericalresults for two different initial mud positions: K=0.44 (x =75 m), 0.36 (1=125 m) (fromshallower to deeper water), and two slide densities and viscosities (1) p 2=1400 kg/m3 andv=iilp2=0.2 m2/s, which will be later referred as slide (1); and (2) p2=2000 kg/m3 andv=p/p2=0.3 m2/s (viscosity increases with density), which will be later referred as slide(2). Two kinds of initial slide profiles will be considered: parabolic and triangular.2.5.1 Slide ProfilesFigures 2.3 and 2.4 show the successive positions of slide (1) and slide (2)respectively, and a shallow initiation depth of the mudslide, K=0.44, or 1=0.754=75 tn.Figures 2.5 and 2.6 show the results for a deeper initiation depth of the mudslide, K=0.36,or 1=1.254=125 m. The solid lines represent the two-way coupling results, and the30dotted lines are the one-way coupling results. In Figures 2.3-2.6 the depth of the slide isplotted on a scale exaggerated about 70 times over the horizontal distance scale. The timescales are [T]=(4/gr2=5.98 sec for slide (1) ( for which g'=2.8 m/s 2 ), and [T]=4.52 secfor slide (2) ( for which g'=4.9 m/s2 )The calculated results indicate that the slide quickly flattens and forms a thin and longflood wave. The reaction of waves on the slide slows down the flow speed of the slideonly slightly. The mud flows faster in the cases of one-way coupling model (with nosurface wave reaction on the slide). As expected, the slide with larger density flowsfaster, and also, the larger the mud viscosity, the more slowly the mud flows. Therelatively light slide specified in Figure 2.3 reaches x=3.424=342 m at t=12[7]=71.7 sec,while the denser slide with a larger viscosity as specified in Figure 2.4 reachesx=3.34=330 m at t=12[T]=54.2 sec which is faster than the slide in Figure 2.3. Anotherfeature of a viscous slide on a slope is that there is amplitude dispersion: the highestdepth transfers from the middle part of the slide to the front face.The differences between the one-way coupling results and the fully coupled resultsindicate the significance of the reaction of the waves on the slide. The numerical resultsindicate that the effect of the two-way coupling is more significant for smaller muddensity and shallower slide initiation depth as in Figure 2.3, and it is small for larger muddensity and deeper slide initiation depth as in Figure 2.6. After some time (t>6,approximately), the two solutions evolve into almost the same profile, with the slideunder a uncoupled free surface moving slightly more quickly. Thus it may be concludedthat for most cases the reaction of the surface waves on the slide is weak and may usuallybe neglected. If one wants to simplify the calculation of the slide dynamics one may justignore the surface wave reaction, especially for the case of a denser slide in deep water,where the reaction is much less significant.1.00.80.60.40.20.0D1.00.80.60.40.20.0t=0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=1.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=2.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=4.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=6.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0311.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=1 0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=12.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Figure 2.3: Profiles of a viscous slide on a slope with 0=4', D o=4 m, Lo= 100 m, r=1.4,p/p2- .2 m2/s and ic=0.44. Solid line: two-way coupling; dotted line: one-way coupling.t=0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=1.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.01.00.80.60.40.20.01.00.80.60.40.20.0321.00.80.60.40.20.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=4.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=6.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=8.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.00.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0t=12.00.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0xD1.00.80.60.40.20.0Figure 2.4: Profiles of a viscous slide on a slope with the same conditions as specified inFigure 2.3 except r=2.0. Solid line: two-way coupling; dotted line: one-way coupling.331.00.80.60.40.20.00.0^0.5^1.0^1.5^2.0^2.5^3.0t=0.03.5^4.0^4.5^5.01.00.80.60.40.20.00.0^0.5^1.0^1.5^2.0^2.5^3.0 3.5^4.0^4.5^5.0t=2.07'N\1.00.80.60.40.20.00.0 0.5^1.0^1.5^2.0^2.5^3.0^3.5 4.0^4.5^5.01.00.80.60.40.20.00.0 0.5^1.0^1.5^2.0^2.5 5.04.53.0 4.03.51.00.80.60.40.20.00.0^0.5^1.0^1.5^2.0^2.5 3.0^3.5^4.0^4.5^5.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0• t=8.0•••0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.01.00.80.60.40.20.01.00.80.60.40.20.0t=12.00.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0xD1.00.80.60.40.20.0Figure 2.5: Profiles of a viscous slide on a slope with the same conditions as specified inFigure 2.3 except ic=2/11. Solid line: two-way coupling; dotted line: one-way coupling.t=0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=1.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=2.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=4.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=6.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=8.0..■■•■•■■•••■.........J....■•■■•■•■•■■■•■•■•■0.0^0.5^1 . 0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0341 .00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.00.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0t=12.00.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0xD1.00.80.60.40.20.0Figure 2.6: Profiles of a viscous slide on a slope with the same conditions as specified inFigure 2.5 except r=2.0. Solid line: two-way coupling; dotted line: one-way coupling.352.5.2 Horizontal Velocity of the SlideI present, in Figures 2.7 and 2.8, the transient variation of the horizontal velocity of themud particles, U(x,t), with nondimensional time t, under the same conditions as specifiedin Figures 2.3 and 2.4, respectively. The velocity scales are [U]=(g'L 0) 1 /2=16.7 m/s forslide (1) (for which g'=2.8 m/s 2 ), and [U]=22.1 m/s for slide (2) (for which g'=4.9 m/s2 )Again it is found that the slide speed at the front face in the one-way coupling model islarger than the fully coupled results, i.e., the reaction of the surface gradient on the slideslows down the slide speed, which is consistent with the results for the slide surfaceprofiles. The slide velocity starts increasing from zero at t=0, approaches its maximumvalue at time t=4-6, and then it starts to decrease. Denser slide flows more rapidly (thecase r=2.0 as in Figure 2.8). Because of the parabolic flow profile equation (2.5), themudflow exhibits velocity amplitude dispersion: the largest velocities are found near thefront of the slide. The distribution of the slide speed behaves similarly to that of the slidethickness: large speeds occur at large thickness.The maximum velocity for the slide specified in Figure 2.3 reached 0.247[U] att=4[T]=23.9 sec near the front with D(x,t)=0. 41 [H]=1.64 m, which corresponds to 4.1m/s. The maximum Reynolds number is Re=33.6 and the maximum Keulegan number isKe=123.1. The maximum velocity for the slide specified in Figure 2.4 reached 0.246[U]at t=4[T]=18.1 sec near the front with D(x,t)=0.42[H]=1.68 m, which corresponds to 5.43m/s, with a maximum Reynolds number being Re=30.4, and the maximum Keulegannumber is Ke=108.9.The numerical results indicate that the Reynolds numbers and the Keulegan numbersare much smaller then their critical value. The two criteria agree fairly well with eachother. Thus the turbulence at water-mud interface may not develop and the flow regimesare laminar for the small-scale slides calculated above.0.400.300.200.100.00.00.400.300.200.100.00.00.400.300.200.100.00.00.400.300.200.100.00.00.400.300.200.100.00.00.400.300.200.100.00.00.400.300.200.100.00.00.400.30U 0.200.100.00.00.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.00.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.00.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.00.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.00.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.00.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.00.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.036t=0.0t=6.00.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0xFigure 2.7: Successive variation of the horizontal velocity of the viscous slide, under thesame conditions as specified in Figure 2.3. The maximum velocity reached (0.247[U])corresponds to 4.1 m/s.t=0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=1.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=4.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=6.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0370.400.300.200.100.00.400.300.200.100.00.400.300.200.100.00.400.300.200.100.00.400.300.200.100.00.400.300.200.100.00.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.00.400.300.200.100.00.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.00.400.30U 0.200.100.00.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0xFigure 2.8: Successive variation of the horizontal velocity of the viscous slide, under thesame conditions as specified in Figure 2.4. The maximum velocity reached (0.246[U])corresponds to 5.43 m/s.382.5.3 The Frontal Speed of the SlideI show, in Figures 2.9 and 2.10, the variation of the frontal speed of the mudflow, Uf=dxf/dt, with nondimensional time t, xf being the x coordinate of the front margin (whichis defined as the position where D(x,t)=0.01) of the mudflow, under the same conditionsas specified in Figures 2.3 and 2.4 respectively. As to the coupling between the slide andthe waves, I obtained the same result as in the previous sections. The fully coupled resultis locally 5-20% smaller in magnitude than in the one-way coupling model for the case inFigure 2.9, where the interaction is strongest. The difference between the one-waycoupling result and the full coupling result for the case in Figure 2.10, however, is muchsmaller, because in this case the reaction of the waves on the mudflow is less significant.From the variation of the slide frontal speeds, it is found that the slide process can beroughly divided into three phases: the acceleration phase t=1-2, the relatively steadyphase t=2-9 (actually can not be really steady!) and the deceleration phase t>9. Densermud moves more rapidly (as in Figure 2.10). The maximum value (the result of the fullycoupled model) of the frontal speed for the mudflow in Figure 2.3 is 0.241[U]corresponding to 4.0 m/s, and for the mudflow in Figure 2.4 is 0.242[U] corresponding5.35 m/s (note the velocity scales for the two cases are different). The calculated resultsof other cases also show that the frontal speed of the slide increases with the increase ofthe initial volume of the slide, and increases with the density difference (p 2 p,). Thefrontal speed also increases with the increase of the inclination of the slope.In 2.2, a parabolic velocity distribution is employed for the viscous mud flow. In thefollowing analysis I shall examine the flow adjustment process at the beginning of theflow and discuss the consequence of this assumption. To analyze the transient flowprocess of a finite mass of viscous fluid released on a slope, I use the following simplified39slide equation, with the advective terms and surface wave reaction being neglected (witha rigid lid at water surface),adtU,„^a2u"`= go sin 9 + vaz2 'Inertial^ViscousBuoyancyforce force(2.28)In the inertial regime, the dominating factors are the buoyancy and the inertial force,the slide accelerates as if it were in free fall, and the slide speed can be written asU. .--. g' sin 0 . t.^ (2.29)As the flow accelerates, the resistance to the flow increases and the accelerationdecreases. The inertial force becomes less important and the viscous force becomesdominant, and this is the viscous regime. The viscous regime corresponds to phases IIand III of the slide - the relatively steady state and the slowing decelerating state. In theviscous regime the viscous force balances the buoyancy, i.e.,0 =gisine+v d2Um.dz 2 (2.30)Solving the above equation with the boundary conditions at the slide bottom and theslide surfaceU„, = 0,^at^z = -k, (2.31)all,. _ 0 at^z = --k + D, (2.32)dz^—^'yields a parabolic vertical distribution of the slide velocity in the viscous regime:U _ g' s2 z ^hs ( z ^hs )21(2.33)in^2 v D ) D40With the above two approximate solutions of the slide velocity in the inertial andviscous regimes, (2.29) and (2.33), one is able to estimate the transient adjustment timefor the velocity profile to reach equilibrium and examine the consequences of theparabolic velocity approximation on the numerical results. From the above equation, themaximum speed (which occurs at the slide surface, z=-h,s+D) that could be reached at theend of the acceleration phase is Umax=0.5g'sineD 2Iv. So the time of the acceleration periodis approximately (using the expression of the velocity in the inertial regime)Ta.U.J(g'sin0)=0.5D 2/v. For the slide specified in Figure 2.3, substituting the numericalresult of the slide front height at the time when there is a maximum speed, D=(0.4-0.5)[H]=1.6-2.0 m and the viscosity of the slide v=0.2 m 2/s yields the estimation of thetime of acceleration T,,=6.4-10 sec ,  (1.1-1.7)[T]. For the slide specified in Figure 2.4, theestimation of the time of acceleration is Ta=4.3-6.7 sec----; (1.0-1.5)[T]. It is found that timefor the adjustment of the velocity to a parabolic profile is almost the same as the timeinterval for the acceleration phase obtained with numerical models. This indicates that theuse of a vertical parabolic velocity profile for the inertial regime [as used in Edgers,1980; Mei and Liu, 1987, Liu and Mei, 1989] will not induce serious errors.There have been many experimental and analytical studies on the frontal speed ofdensity currents [Middelton, 1966a,b, 1967; Middelton and Hampton, 1976; Pantin1979; Huppert and Simpson, 1980; Taira, 1985] with low densities. For flow of a lowdensity suspension (i.e., turbidity current - which has a small density contrast withambient water) on relatively flat slopes, the suggested equation for the average velocity ofthe head of the flow is of the form [ Karlsrud & Edgers, 1982] :Upk(gDf)la^ (2.34)where Df is the height of flow at the head and k is a constant. Suggested values for k liein the range 0.7-1.4. The above range of k is for turbulent, low density contrast and low41viscosity turbidity currents, for which the turbulent mixing and ambient water flow at thehead of the flow are the dominating factors. Because of upward motion of the mud at thefront due to turbulence, the pressure distribution in the flow is no longer hydrostatic, andthe long-wave approximation is not valid. The present solution does not apply. It shouldalso be emphasized, however, that there is only limited experimental evidence on thevalidity of this formula available, and most experiments have been carried out onmaterials with density less than 1400 kg/m 3 [Middelton, 1966a,b, 1967; Middelton andHampton, 1976; Pantin ,1979; Taira, 1985]. Hampton [1973] pointed out that thedensities of turbidity currents lie between 1030-1120 kg/m 3 , and for submarine debrisflow the densities range from 1500-2500 kg/m 3 . It can be seen that there is no experimentdata for the frontal speed of submarine debris flows to compare with the present modelingresults.Figure 2.11 shows the variation of k =Uf/(g'Df) ,a(Froude number of the slide front)versus time for slide (1) and slide (2) respectively. It can be seen that k increasesdrastically with the acceleration of the slide to a maximum value (maximum k=1.84 forslide (1), and 1.76 for slide (2)), and then starts a gradual decrease. The average frontalFroude number is about 1.5. The reasons that k is a little bit larger than 1.4 are notunknown. This might be due to the omission of the effects of turbulence at the head of theslide, or some orther mechanism is functioning. In previous experiments [Middelton,1966a,b, 1967; Simpson and Britter, 1980; Simpson, 1987], it is found that the foremostpoint of the head of a gravity current is raised about 1/8 of the total thickness from thebottom due to the effcet of friction. This may slow down the flow. Interfacial turbulencewill increase energy dissipation and change the flow structure at the slide front. The slidemay then be slowed down after turbulence develops. For most real world underwaterlandslide the flow is most likely to be turbulent, expect for some small-scale mudslideswith very large viscosity in the initial phase.a)PCO0-ocpa)CLCOTaCOLL420^5^10^15Time tFigure 2.9: Successive variation of the frontal speed of the slide, under the sameconditions as specified in Figure 2.3.0^5^10^15Time tFigure 2.10: Successive variation of the frontal speed of the slide, under the sameconditions as specified in Figure 2.4.3.02.52.01.51.00.50.00 151 0543Time tFigure 2.11: Successive variation of the frontal Froude number for slide (1) and slide (2)respectively.2.5.4 Elevations of the Surface WavesFigures 2.12-2.15 illustrate the evolution of the surface waves generated by theunderwater mudslides under the conditions specified in Figures 2.3-2.6 respectively. InFigures 2.12-15, the surface elevations are plotted about 70 times exaggerated over thehorizontal distance.Three major waves are produced: the first wave is a large crest which propagates intodeeper water from the slide site; the second wave is a trough with amplitude of the sameorder as the crest, following the crest as a forced wave and propagating with the speed ofthe slide front to deep water. The third wave is a small trough that propagates shoreward.Following the trough a small crest gradually forms and propagates shoreward. This crest44may cause sea-level increase at the shore some time after the slide initiation. The fact thata trough propagates toward the shore coincides qualitatively with the observation [Milanand Striem, 1978] that a recession of the sea occurs at an early stage as a result of mudslumping on the continental slope.Two major parameters dominate the interactions between the slide and the waves: thedensity of sliding material and the water depth of initiation of the mudslide. The two-wayinteractions are significant for the cases of a smaller mud density and shallower water, asin Figures 2.12 and 2.13. For larger mud density (slide (2)) and shallower water, as inFigure 2.13, the two-way interactions are small, but the amplitudes of the wavesgenerated are very large.The maximum height (the result of the fully coupled model) of the surface wavesgenerated by slide (1) in a shallow position c=0.44 is 0.12[H] corresponding to 0.48 m.The maximum wave height for the waves generated by slide (2) in a shallow positionic=0.44 is 0.17[H] corresponding 0.68 m. The viscosity of mud has only a weak effect onthe interaction between the slide and the waves; mudflow with larger viscosity flowsmore slowly and produces smaller waves. For a mudslide starting in deeper wateric=0.36, as in Figures 2.14 and 2.15, the interactions between the slide and the waves areweak, as expected, and the waves are also smaller than those produced by the same slidein shallower water.Therefore I may conclude that, though the reaction of the waves on the slide isgenerally weak and negligible, the action of the slide on the surface water is significantand cannot be neglected.6^8^10^12 14 16 18 20-4^-2^0^2^4^6^8^10 12 14 16 18 20-4 -2 0 2 40.100.050.0-0.05-0.10 45t-0.00.100.050.0-0.05-0.10-4 -2^0^2^4^6^8^10 12 14 16 18 20t=1.00.100.050.0-0.05-0.100.100.050.0-0.05-0.10^-4^-2^0^2^4^6^8^10 12 14 16 18 20-4^-2^0^2^4^6^8^10 12 14 16 18 20-4^-2^0^2^4^6^8^10^12^14^16 18 201.10.00.060.02/ -0.02-0.06-4^-2^0^2^4^6^8^10^12 14^16 18 20t=12.00.060.02-0.02-0.060.060.02-0.02-0.060.060.02-0.02-0.06-4^-2^0^2^4^6^8^10 12 14 16 18 20xFigure 2.12: Evolution of surface waves by the viscous slide with the same conditions asin Figure 2.3. Solid line: two-way coupling; dotted line: one-way coupling.-4^-2^0^2^4^6^8^10 12 14 16 18 200.060.02-0.02-0.06t=0.0-4 -2 0 2 4 6 8 10 12 14 16t=1.0-4 -2 0 2 4 6 8 10 12 14 16t=2.0-4 -2 0 2 4 6 8 10 12 14 16t=4.0-4 -2 0 2 4 6 8 10 12 14 160.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.104618 2018 2018 200.060.02-0.02-0.0618 20^- 4^-2^0^2^4^6^8^10^12 14^16 18 20^-4^-2^0^2^4^6^8^10^12 14^16 18 200.060.02-0.02-0.060.060.02-0.02-0.06-4^-2^0^2^4^6^8^10 12 14 16 18 20xFigure 2.13: Evolution of surface waves by the viscous slide with the same conditions asin Figure 2.4. Solid line: two-way coupling; dotted line: one-way coupling.-4^-2^0^2^4^6^8^10 12 14 16 18 20xt=0.04 6 8 10 12 14 16 18 20t=1.04 6 8 10 12 14 16 18 201=2.04 6 8 10 12 14 16 18 201=4.04 6 8 10 12 14 16 18 20t=6.04 6 8 10 12 14 16 18 20t=8.04 6 8 10 12 14 16 18 200.100.050.0-0.05-0.10-4^-2^0^20.100.050.0-0.05-0.10-4^-2^0^2Nf0.100.050.0-0.05-0.10-4^-2^0^20.100.050.0-0.05-0.10-4^-2^0^2f0.060.02-0.02-0.06-4^-2^0^20.060.02-0.02-0.06-4^-2^0^20.060.02-0.02-0.060.06ri 0.02-0.02-0.06472^4^6^8^10 12 14 16 18 200-4^-2Figure 2.14: Evolution of surface waves by the viscous slide with the same conditions asin Figure 2.5. Solid line: two-way coupling; dotted line: one-way coupling.0.060.02-0.02-0.06-4^-2 0 10^12^14^16^18 208642770.060.02-0.02-0.060.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.060.02-0.02-0.060.060.02-0.02-0.06-4 -2^0 2 4^6^8^10 12 14 16 18 20-4^-2^0^2^4^6^8^10 12 14 16 18 20x48Figure 2.15: Evolution of surface waves by the viscous slide with the same conditions asin Figure 2.6. Solid line: two-way coupling; dotted line: one-way coupling.492.5.5 Horizontal Velocity of the Water MotionFigures 2.16 and 2.17 illustrate the transient variation of the horizontal velocity of thewater motion for the cases specified in Figures 2.3 and 2.4, respectively. The positivevalue of u(x,t) indicates that the water flows towards the deep water, and the negativevalue indicates water flowing shoreward.It is seen that the water ahead of the front face of the slide is pushed away and movestowards deep water, which is consistent with the result that a crest propagates into deepwater. The water just above the slide flows shoreward, and there appears a water flowdivergence at the front face of the slide. The water near the rear face of the slide moves todeep water, and there is a convergence of water flow at the rear face of the slide. Thewater particle velocity at the front face of the slide quickly decreases as the crestpropagates into deeper water, and the shoreward velocity also decreases quickly as theslide moves downslope. The magnitude of the water velocity near the rear face of theslide does not change very much, because the waves propagate into a river with constantdepth.It is noted that the particle velocity of water is at least one order of magnitude smallerthan that of the slide. The maximum water velocity generated by the slide as specified inFigure 2.3 reached 0.030[U] which corresponds to 0.501 m/s (shoreward), at time t=4.The maximum water velocity produced by the slide specified in Figure 2.4 reached0.025[U] which corresponds to 0.553 m/s (shoreward), at time t=4.Considering that the largest tidal current speed is of the order of tens of cm/s, I mayconclude that the local water current induced by a small-scale underwater slide iscomparable with other local currents .t=0.00.030.01-0.01-0.03-4^-2 0 2^4^6^8^10 12 14 16 18 20f t=1.0^\ t=2.00 2^4^6^8^10 12 14 16 18 20-4^-2-4^-2^0^2^4^6^8^10 12 14 16 18 20-4^-2^0^2^4^6^8^10 12 14 16 18 200.030.01-0.01-0.030.030.01-0.01-0.03-4^-2 0 2^4^6^8^10^12 14^16 18 20-4^-2^0^2^4^6^8^10^12 14^16 18 20-4^-2^0^2^4^6^8^10 12 14 16 18 20xFigure 2.16: Successive variation of the horizontal velocity of the water motion, underthe same conditions as specified in Figure 2.3. The maximum velocity reached (0.03[U)corresponds to 0.501 m/s (shoreward).500.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.03t=4.0-4^-2^0^2^4^6^8^10 12 14 16 18 202^4^6^8^10 12 14 16 18 20-4^-2 00.030.01-0.01-0.0351t=0.0-4 -2 0^2 4 6 8 10 12 14 16 18 20`-t=1 .0-4 -2 0^2 4 6 8 10 12 14 16 18 20t.2.0-L\f-4 -2 0^2 4 6 8 10 12 14 16 18 20t=4.0-4 -2 0^2 4 6 8 10 12 14 16 18 20^-4^-2^0^2^4^6^8^10 12 14 16 18 20^-4^-2^0^2^4^6^8^10^12 14^16^18 20t=12.02^4^6^8^10 12 14 16 18 20xFigure 2.17: Successive variation of the horizontal velocity of the mud flow along theslope, under the same conditions as specified in Figure 2.4. The maximum velocityreached (0.025[1/1) corresponds to 0.55 m/s (shoreward).0.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.03-4^-2^0522.5.6 Energy Transfer From the Slide to the WavesIn this section I shall calculate the ratio of energy transfer from the slide to the waves.The energy source in this problem is the potential energy of the slide due to gravitation.During sliding the potential energy of the slide is converted into three parts: the surfacewave energy, the kinetic energy of the mudflow and viscous energy dissipation. Theenergy transfer ratio is defined as the ratio of the surface wave energy to the releasedpotential energy of mudflow. The surface wave energy per unit width, E,,, can beexpressed as1E), = f[-2pi (h+ n)u2 +—pan 2 _ dx. (2.35)The potential energy of the slide, E„, can be expressed, choosing the still water level asthe reference level, asEp = - .1 (p2 — p)gD(h+ Ddx.^(2.36)Then the energy transfer ratio, X, can be calculated by2^Ew^1 ^f[(h+ n)u2 + gre}dx. =^.AEp^2(r —1) jgD(h + D 1 2)dx(2.37)Figure 2.18 illustrates the variation of the energy transfer ratio with time and mudproperties; the solid line indicates the result for the less dense slide as specified in Figure2.3, and the dotted line indicates the result for the denser slide as specified in Figure 2.4.The calculated results indicate that the energy transfer ratio from the slide to the waves isnot constant. It is zero at t=0, approaches its maximum value of 5.3% at t=2-3, and then53starts decreasing. At the initial stage (t=0-4), the slide is shallow and the waves gainenergy quickly from the slide, thus the energy transfer ratio increases drastically. As theslide moves into deeper water the slide continues to release its potential energy (which isconverted into the kinetic energy and the viscous dissipation) but the waves start todecrease, so the energy transfer ratio continues to decrease with time. At the beginningstage(t=0-5), the energy transfer from the slide to the waves is significant in the case of asmaller mud density (the solid line), where the energy transfer from the slide to the wavesand the reaction of the waves on the slide are significant and the two-way coupling isstrong. This may be seen from equation (2.37) that the energy transfer ratio is inverselyproportional to the density ratio r. The energy transfer is weak and nearly constant forlarger mud density, where the interaction between the slide and the waves is weak.Figure 2.19 shows the variation of the energy transfer ratio with time and mudproperties for the cases with deeper slide initiation depth; the solid line indicates theresult for the case as in Figure 2.5, and the dotted line indicates the result for the case asspecified in Figure 2.6. As expected, the energy transfer from the slide to the waves isless significant in deeper water (as in Figure 2.6), because the free surface cannot feel theslide strongly. It can be seen that the energy transfer is more significant for the case of aless dense slide (as in Figure 2.5), for which case the maximum value of energy transferratio is 4.4%.It is noted that Miloh and Striem [1978] used a constant energy transfer ratio and also asolitary surface profile in their model. Their theory is an empirical one. It cannot give thecorrect solution of the wave evolution and wave height. It is only a rough method to yieldan approximate solution of wave magnitude for the waves produced by submarinelandslides.54O00 105 1503000 5 10 15c000CO00NrO0q0Time tFigure 2.18: The variation of the energy transfer ratio with time and mud properties.Solid line shows the case as specified in Figure 2.3; dotted line, the case as specified inFigure 2.4.Time tFigure 2.16: The variation of the energy transfer ratio with time and mud properties.Solid line shows the case as specified in Figure 2.5; dotted line, the case as specified inFigure 2.6.552.5.7 Possibility of Resonance between the Slide and the WavesThe above numerical results show that the two-way coupling between the slide and thewaves which it generates is weak for realistic conditions: surface gravity waves travelmuch faster than the slide can move, and there is no possibility of resonance. Resonancecould occur if the slide initially moved faster than the waves, i.e., if it began in extremelyshallow water with a sufficiently steep slope such that Uf, the speed of the mudslide front,exceeded the speed of the local long waves (gh) 1a. Such a situation could not hold verylong, since the wave speed would rapidly increase with the water depth: soon (gh) 1I2would catch up with Uf, the interaction would be most effective, and a short period ofresonance would occur. After (gh)112 >Uf, one is back to the conditions examined here.What are the conditions under which the frontal speed Uf >.(gh )1n ? I have calculatedthe Froude number Fr=U1 1(gh)Ir2 for the slides (the ratio of the slide frontal speed withthe local long-water-wave speed) with the same initial profile and volume as specified inFigure 2.4 on four slopes 0=2°, 4°, 6°, and 8°. As shown in Figure 2.20, the Froudenumber quickly reaches a maximum as the mudflow accelerates, gradually decreasingafterward as the depth increases. Higher Froude numbers are reached in the larger slope0=8° because of the relatively faster increase in slide speed.Should the Froude number approach unity, the situation would become analogous tothat of flow over a weir and a hydraulic jump would appear in the lee of the moving mudobstacle. This kind of interaction would be transitory, occurring near the peak of theFroude number Fr (t) curve of Figure 2.20, in very shallow water. While of some interestfrom a hydraulic point of view, and worthy of future study, the calculations have shownthat this phenomenon is not to be expected in many practical cases. Full consideration ofthe highly nonlinear large Froude number regime would require a reformulation of theproblem which will not be considered here.560 5 10 151.00.80.60.20.0Time tFigure 2.20: The variation of the Froude number versus time for slide (2) on four slopes.x l x x2 /MIMI572.6^Numerical Results for the Slide with an Initial Triangular Profile2.6.1. Numerical Results for a Triangular SlideTo examine the effects of the initial slide profile, I also calculated a slide with an initialtriangular surface and the waves which it generates. Here I consider an initial triangularprofile (see Figure 2.21) which can be expressed as D0 [1— 2(7,—x)/L0 ],^xl xD0 [1— 2(x —^L0 ],^x5x5x2,D(x,0) = (2.38)where, x is the initial x coordinate of the center of mud, x =(x 1 +x2 )/2; xi and x2 are theinitial x coordinates of the rear and front margins of the mud; Lo=x2 -xi , is the initial lengthof the mud, and D o is the initial maximum mud thickness.Figure 2.21: Definition sketch of an initial triangular slide surface.Figures 2.22-2.25 show the results of the slide surface variation, the slide velocitydistribution, the surface elevation and the water motion velocity for a triangular initialslide profile under the conditions: 0=4*, D 0=4 m, Lo=100 m, p2=1400 kg/m3 , p/p2=0.2m2/s, and ic---0.44.58The numerical results indicate that whatever the initial profile of the slide is, iteventually evolves into a flood-wave-like flow which has a higher amplitude at thefrontal face and an elongated thin tail. The slide surface profile, the slide velocitydistribution, the surface wave evolution and the water particle velocity distribution aresimilar to those of the slide with an initial parabolic profile. The surface waves (themaximum wave height reaches 0.09[H]=0.36 m, the maximum water velocity is0.022[U]=0.37 m/s) and the slide velocity (the maximum speed is 0.205[U]=3.42 m/s),however, are smaller than those for the parabolic profile case, as a result of the volume ofthe triangular slide being smaller. Therefore I conclude that some time after the initiationof the slide, no matter what the initial slide profile, the slide surface takes a similar form -a long and thin sheet of fluid with a higher thickness and steeper surface at the front. Thisis also the conclusion of an asymptotic solution of Huppert's [1982]. In the followingsection I shall use Huppert's formula to calculate the slide depth at large time, andcompare the result with the above numerical solutions.59t=1.00.0 0.5 1.0 1.5 3.0^3.5^4.0^4.5^5.02.0^2.50.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.01.00.80.60.40.20.01 .00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.0 1.00.80.60.40.20.0D0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=4.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=6.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=8.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=10.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=1 2. 00.0 0.5 1.0 1.5 2.0 2.5x3.0 3.5 4.0 4.5 5.0Figure 2.22: The surface variation of the slide with a triangular initial surface on a slopewith 0=4', Do=4 m, Lo=100 m, p2=1400 kg/m3 , Et/p2=0.2 m 2/s and K=0.44. Solid line:two-way coupling; dotted line: one-way coupling.0.5^1.0^1.5^2.0fti0.5^1.0^1.5^2.00.5^1.0^1.5^2.00.00.400.300.200.100.00.00.400.300.200.100.00.00.400.300.200.100.02.5 3.0 3.5 4.0 4.5 5.0t=4.02.5 3.0 3.5 4.0 4.5 5.0t=6.02.5 3.0 3.5 4.0 4.5 5.00.400.300.200.100.060t=0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0t=1.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.400.300.200.100.00.400.300.200.100.00.400.300.200.100.00.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.00.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.00.400.30U 0.200.100.0 t=12.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Figure 2.23: Successive variation of the horizontal velocity of the slide with an initialtriangular surface, under the same conditions as specified in Figure 2.22. The maximumvelocity reached (0.205[U]) corresponds to 3.42 m/s. Solid line: two-way coupling;dotted line: one-way coupling.t=1.0\IL^f0.040.020.0-0.02-0.04-4^-2^0^2 4 6^8^10^12 14 16 18 20-4^-2^0^2 4 6^8^10 12 14 16 18 20t=6.00.040.020.0-0.02-0.04-4 -2^0^2^4^6^8^10 12 14 16 18 20-4^-2^0^2^4^6^8^10 12 14 16 18 200.040.020.0 --0.02-0.040.040.020.0-0.02-0.04\?'^0.040.020.0-0.02-0.040.040.0277 0.0-0.02-0.04-4^-2^0^2^4^6^8^10 12 14 16 18 20xFigure 2.24: Evolution of surface waves generated by the slide with an initial triangularsurface, with the same conditions as in Figure 2.22. Solid line: two-way coupling; dottedline: one-way coupling.61-4^-2^0^2^4^6^8^10 12 14 16 18 20t=8.0-4^-2^0^2^4^6^8^10^12 14 16 18 20t=10.0-4^-2^0^2^4^6^8^10^12^14 16 18 20t=2.0t=4.00.040.020.0-0.02-0.040.040.020.0-0.02-0.04t=12.0t=0.0u8 10^12 14 16^18 2062I tag0.0t-4 -2^0^2^4^6^8^10 12 14 16 18 20t-1.0if0.030.01-0.01-0.032 4-4^-2^0-4 -2^0^2^4^6t0.030.01-0.01-0.03t0.030.01-0.01-0.030.030.01-0.01-0.03-4^-2^0^2^4^6^8^10 12 14 16 18 200.030.01-0.01-0.03-4^-2^0^2^46^8^10 12 14 16 18 208^10^12 14 16 18 206t.8.0----.____---..^-4^-2^0^2^4^6^8^10^12^14^16^18 201. 10.0---...e' ■---.....-7....:---......).., ..+■••■-•■■•■•■■•^40.030.01-0.01-0.03 t-4^-2^0^2 6^8^10^12^14^16^18 20t.12.0L\r^0.030.01-0.01-0.030.030.01-0.01-0.03N2.0-4^-2^0^2^4^6^8^10^12 14 16 18 20xFigure 2.25: Successive variation of the horizontal velocity of the water motion inducedby an triangular slide, under the same conditions as specified in Figure 2.22. Themaximum velocity reached (0.022[U]) corresponds to 0.37 m/s (shoreward).632.6.2 Comparison with Huppert's SolutionHuppert [1982] presented an analytical solution for the flow of a viscous fluid drop ona slope. The approximations of lubrication theory (inertia of the flow neglected) wereadopted and the capillary effect was discussed. Huppert's theory shows that the flowdepth and front positions of a viscous flow on a slope a certain time after initiation areD(x,t) =1(^ 2 ^1p^) 1-^  (X -- X0 ) 2 t 2 ,^ (2.39)(peg sin 01x^(9A2p2g' sin 0  j 3- --f — xo =^t3,4p (2.40)where A is the area of the slide. Huppert's asymptotic solution (for x»xo , xo is the initialslide front position) suggests that the solution at large time is only related to the density,viscosity and area of the slide, and that the initial stage of motion will not have anylasting effect on the subsequent motion.Figure 2.26(a) and (b) illustrates the comparison of the present numerical results andHuppert's solution for the slide front positions in the cases of a triangular and a parabolicinitial slide shapes, respectively. It is seen that the discrepancy between the two solutionsis great for the initial stage (t=1-20, approximately). It is expected that the two solutionswill approach each other very close for large time. Thus one can not use Huppert'ssolution for the initial phase of the slide, because x o in his solution is non-negligible atthis stage. It is also noted that Huppert's theory can not be used to calculate the waveswhich the slide generates, since it does not include the presence of a free surface. Even ifa free surface is included, the solution still can not give the correct wave evolutionbecause it is only valid for large times, and wave generation in the initial stage is the mostimportant.640Triangle SlideNumerical solutionHuppert's solutionParabolic SlideNumerical solutionHuppert's solution0^2^4^6^8^10^12^14Time tFigure 2.26(a): Comparison of the present numerical solution and Huppert's solution forthe slide front positions, for the case of a triangular initial shape.0^2^4^6^8^10^12^14Time tFigure 2.26(b): Comparison of the present numerical solution and Huppert's solution forthe slide front positions, for the case of a parabolic initial shape.wave run-upincidentwavehl/ ,-/ ,/ // ,TR652.6.3 Wave Run-up at ShoreCan a landslide produce a wave runup at shore (see Figure 2.27)? This is the questionof great public concern. From the evolution of the triangular-slide-induced waves (Figure2.24) it can seen that as the slide moves downslope into deep water and the waves spreadaway from the slide initiation site, a crest with increasing amplitude forms and propagatesshoreward following the small trough. The shoreward trough can not produce a runup:instead, it causes a decline of the water level. The shoreward crest will generate a runupat the shore. The most dangerous wave is the large outgoing crest if there exists anopposite shore. The seaward crest will definitely cause a large wave runup because of itslarge size. More seriously this wave may reflect at the opposite bank and cause wateroscillation in a closed or semi-closed area that might possibly induce resonance.Figure 2.27: A definition sketch for wave runup on a uniform slope.In the previous calculations I have introduced a long river with constant depth and along deep water area, and I focused on the wave generation near the slide initiation site.66Thus the present solution can not be directly used to calculate wave runup at the shore.To calculate wave runup at shore, complicated computation of the motion of the waterline(at which water depth is zero, i.e., 7)4-11,1) is required, and this will not be done in thisstudy. Instead I shall adopt Synolakis' s solution [1987] of the runup of a solitary wave ona plane beach to roughly estimate the magnitude of the runup generated by the shorewardcrest at the shore and the outgoing crest at the opposite shore. Synolakis [1987] showedthat the maximum runup of a solitary wave on a plane beach is (see Figure 2.27)54= 2.8311/cot 0^ H,h(2.41)where R is the maximum wave runup, 01 is the inclination of the beach, 77, is the incidentwave height.For the shoreward, substituting the amplitude of the crest rt f=0.015[D]----0.06 m, thewater depth h 1 =2 m, and the slope angle 01 =4° into (2.41) yields the maximum runupR=0.17/1 1 =0.34 m, which is much larger than the amplitude of the incoming crest. Iassume the opposite shore is the same as the near one. The wave runup produced by theoutgoing crest with 77,=0.02[D]- - .0.08 m will be R=0.23h 1 =0.47 m. It is found that therunup induced by a small-scale slide is small. A large scale slide might generate largerwaves and hence produce larger wave runup. This will be examined in Section 2.8.672.7 Numerical Predictions for the ADFEX ExperimentIn this section, I shall use the viscous slide model to supply numerical predictions forthe slide dynamics and the surface waves for the ADFEX experiment, although slideinitiation in ADFEX failed. The Kenamu River delta in Lake Melville, Labrador waschosen as the site of the ADFEX project. The axial geometry of the experimental area isshown in Figure 2.28. The slope angles in the experimental area are (from river mouth todeep water): 9.7°, 1.1°, 11.3°, 4.0°, 1.5°, and 0.95°, which are all in the range of smallslopes. The slide to be initiated was a slab of Lo=170 m long and D o=12 m thick. Theriver is h 1 =3 m deep at the river mouth. Due to the fact that no data for the slide densityand viscosity are available, I shall presume two sets of possible values for the slide.According to the results of previous laboratory experiments of submarine and coastalsediments [Krone, 1962; Beaty, 1963; Mei and Liu, 1987; Simpson, 1987; O'Brien, 1988;Locat and Demers, 1988; Laval et al., 1988], the viscosity of the mud is nonlinearlyproportional to its bulk density. To simulate the slide in the ADFEX experiment I shallemploy two sets of densities and viscosities for the slides: (1) p 2=1.5 g/cm 3 , it/p2=0.2m2/s; and (2) P2=2.5 g/cm3; it/p2=0.3 m2/s)I use the same finite-difference scheme as I used in previous sections. The artificialviscosity is chosen as pai =0.0025, for the slide; and p a2=0.05, for the waves. The initialslide is divided into 34 cells, each cell of 8r=0.029L 0=5 m long. The slide is initially atrest at t=0, and then it starts flowing down slope. The time step is chosen as 8t=0.005[7].The total length of the experimental area is 2800 m, divided into 560 cells. I shall presentthe one-way coupling results and fully coupled results for the slide profiles, the slidevelocity distribution, the surface waves produced by the slide, and the water particlevelocity distributions.SWLAXIAL DISTANCE (M)^ 1500^1800Figure 2.28 Bathymetry of the ADFEX experiment area.692.7.1 The Slide ProfilesFigure 2.29 shows the slide profiles for the slide with p 2=1.5 g/cm3 and it/p2412 m2/s.For this case, the reduced gravitational acceleration, g'=g/3; the time scale, [T]=(Ligr2sec; and the velocity scale [U]=(g'L 0) 112,---24 m/s. Figure 2.30 shows the slide profilesfor the slide with p2=2.5 g/cm3 and Wp2=0.3 m2/s for which the reduced gravitationalacceleration, g'=3g/5; the time scale, [T]=(L o/g') 112--.5.4 sec; and the velocity scale[U]=(g'L 0) 1 r2-..32 m/s. Again, I find that the reaction of the surface waves on the slide(indicated by the difference between the one-way coupling and the fully-coupled results)is more significant for smaller slide density (Figure 2.29). The reaction of the wavesslows down the slide. The slide specified in Figure 2.29 reaches x.--3[L0]=510 m int=9[T]=64.8 sec, while the slide specified in Figure 2.30, which is denser, reachesx--3[Lo]=510 m in t=9[T]=48.6 sec.2.7.2 The Slide Velocity DistributionFigure 2.31 and 2.32 illustrate the slide velocity distribution for the slides specified inFigure 2.29 and Figure 2.30 respectively. Denser mud flows more rapidly (Figure 2.32).The maximum velocity for the slide specified in Figure 2.29 reached 0.306[U] whichcorresponds to 7.34 m/s at t=3[T]=21.6 sec near the slide front with D=0.3[H]=3.6 m,with maximum Reynolds and Keulegan numbers being Re=132.1 and Ke=593.2. Themaximum velocity for the slide specified in Figure 2.30 reached 0.312[U] whichcorresponds to 9.98 m/s at t=3[T]=16.2 sec near the slide front with D=0.31[H]=3.72 m„with maximum Reynolds and Keulegan numbers being Re=123.7 and Ke=552.2. It isfound the Keulegan numbers exceed the critical value, so interfacial turbulence willdevelop, though it may not be fully developed. The effect of turbulence can be evaluatedas in Section 2.9.700.80.40.0t=0.0-6^-4 -2 0 2 4 6 8 100.8t=1.00.40.0 A I-6^-4 -2 0 2 4 6 8 100.8t=2.00.40.0 A-6^-4^-2^0^2^4^6^8^100.80.40.0EN;c-in0.80.40.00.80.40.00.80.40.0/----Nt.0-6 -4 -2 0 2 4 6 8^10t=7.0-6 -4 -2 0 2 4 6 8^10t=9.0-6 -4 -2 0 2 4 6 8^100.80.40.0 ,-----7\t=10.0 (x7.2sec)-6 -4 -2 0 2 4 6 8^10X (x170m)Figure 2.29: Profiles for the slide with p2=1.5 g/cm3 and p/p2=0.2 m2/s.t=13.0 (x5.4sec),-------\.■■•■■■■...■.•0.80.40.0-6 -4 -2 0 26 8 10-6^-4^-2^0^2^4X (x170m)Figure 2.30: Profiles for the slide with p2=2.5 g/cm3 and p/p2=0.3 m2/s.t.04 6 8^1 0t=7.04 6 8^10t=9.04 6 8^100.80.40.00.80.40.0II tr).0-6^-4 -2 0 2 4 6 8 1071It=1.0-6 -4 -2 0 2 4 6 8 10A0-6 2-4^-2‘,.........-0-6 2-4 -20.80.40.00.80.40.0E 0.8N;c- 0.400.0-6 -4 -2 0 2 4 6 8 10-6 -4 -2 0 2 4 6 8 10-6 -4 -2 0 2 4 6 8 10-6^-4^-2^0^2^4^6^8^10t=2.0A-6^-4^-2^0^2^4^6^8^10t=5.0r f A0.60.40.20.00.60.40.20.00.60.40.20.00.60.40.20.00.60.4020.00.60.40.20.00.60.40.20.0re 0.6s 0.4V 0.2m 0.0-6 -4 -2 0 2 4 6 8^10t=9.0-6 -4 -2 0 2 4 6 8^10t=10.0 (x7.2sec)-6 -4 -2 0 2 4 6 8^1072X (x170m)Figure 2.31: The velocity distribution for the slide specified in Figure 2.29.-6 -4 -2 0 2 4 6 8 10-6 -4 -2 0 2 4 6 8 10-4-6 2 4 6 8 100.60.40.20.0-2 0-680.60.40.20.0106420-2-4-4^-2 0 2 4 6 8 1080 2 4 6-2-410-60.6  0.40.20.0-6t=13.0 (x5.4sec)...N....P.-^....N.t=9.0,-,x•--N0.60.40.20.00.60.40.20.0A t=2.0ri■t=3.0t=7.0y--r--- \0.60.40.20.0-6 -4 -20.60.40.20.0-6^-4^-2^0^2^4^6^8^10X (x170m)Figure 2.32: The velocity distribution for the slide specified in Figure 2.30. 73t=0.0 I0 2 4 6 8 10t=1.0742.7.3 The Surface WavesFigure 2.33 and 2.34 show the surface waves generated by the slides specified in Figure2.29 and Figure 2.30, respectively. The solid line shows the fully-coupled results, and thedotted line shows the one-way coupling results. It is found again that three main wavesare produced: a large crest propagating into deeper water, a trough in the form of a forcedwave and propagating into deep water, and a small trough which propagates shoreward;and that the interaction between the waves and the slide is stronger for the slide withsmaller density and viscosity.For the case in Figure 2.33 (surface waves generated by the slide with small densityand viscosity), the maximum crest amplitude, ri,=0.0478[D]=0.57 m, occurs att=2.0[T]=14.4 sec, and x-1.5[L]=255 m. The maximum trough amplitude (the forcedtrough), n_=-0.0803[H]=-0.964 m, occurs at t----3.0[T]=21.6 sec, near the river mouth x---0.The crest reaches the boundary of the calculation area at about P-40[7]=72 sec.For the case in Figure 2.36 (surface waves generated by the slide with large density andviscosity), the maximum crest amplitude, rii.=0.076[H]=0.91 m, occurs at t--3.0[7]=16.2sec, and x--.1.6[L]=272 m. The maximum trough amplitude, ri_=-0.104[11]=-1.25 m,occurs at t---3.0[T]=16.2 sec, near the river mouth x----0. The crest reaches the boundary ofthe calculation area at about t---.13[7]=70.2 sec.From the above numerical results it is found that the density has the most significanteffect on the magnitude of surface waves: the denser slide (as in Figure 2.36) produceslarger waves. Also, I find that the viscosity has only a weak effect on the slide and thesurface wave generation. The theoretical predictions on wave amplitude show that a largewave with height up to 2 m may be produced near the front of the slope, which may belarge enough to damage small ships and equipment moorings.t=0.0-6 -4 -2 0 2 4 6 8^10t=1.0-6 -4 -2 0 2 4 6 8^10t=2.0-6 -4 -2 0 2 4 6 8^10-6 -4 -2 0 2 4 6 8^10t=5.0-6 -4 -2 0 2 4 6 8^10t=7.0-6 -4 -2 0 2 4 6 8^100-6 -4 -2 0 2 4 6 8^10t=10.0 (x7.2sec)-6 -4 -2 0 2 4 6 8^10X (x170 m)750.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.10E 0.050.0-0.05-0.10Figure 2.33: The surface wave evolution for the case specified in Figure 2.29.t=0.0-6 -4 -2 0 2 4 6 8^10t=1.0-6 -4 -2 0 2 4 6 8^10760.150.05-0.05-0.150.150.05-0.05-0.150.150.05-0.05-0.15-6^-4^-2^0^2^4^6^8^100.150.05-0.05-0.15-6 -4 -2 0 2 4 6 8 10-6^-4^-2^0^2^4 6 8 10___________,...........,.....„7 - t=7.0-6^-4^-2^0^2^4 6 8 10-6 -4 -2 0 2 4^6^8^10t=13.0 (x5.4sec)_________.....er--6 -4 -2 0 2 4^6^8^10X (x170m)0.150.05-0.05-0.150.150.05-0.05-0.150.150.05-0.05-0.150.15ci 0.0574 -0.05=•- -0.15Figure 2.34: The surface wave evolution for the case specified in Figure 2.30.772.7.4 The Velocity Distribution of the Water MotionFigures 2.35 and 2.36 show the transient variations of the horizontal velocities of thewater motion under the same conditions as specified in Figures 2.29 and 2.30,respectively. The water ahead of the front face of the slide moves towards deep water.The water just above the slide flows towards the river (u(x,t) has a negative value), andthere is a water flow divergence at the front face of the slide. The water near the rear faceof the slide moves to deep water, and there is a convergence of water flow at the rear faceof the slide. Again I find that the water velocity at the front face of the slide quicklydecreases as the depth of the water increases.The particle velocity of water produced by the slides is at least one order of magnitudesmaller than the slide velocity. The maximum shoreward velocity generated by the slidespecified in Figure 2.29 reached -0.04[U], corresponding to -0.96 m/s, at timet-..5.0[T]=36.0 sec and x----[L]=170 m. The maximum seaward velocity (for the shore wardtrough) reached 0.033[U], corresponding to 0.79 m/s, at time t-=-5.0[T]=36.0 sec and x----0.5[L]=-85 m at the river side.The maximum shoreward velocity generated by the slide specified in Figure 2.30reached -0.035[U], corresponding to -1.12 m/s, at time t---5.0[T]=27.0 sec, and x---[L]=170m. The maximum seaward velocity (for the shoreward trough) reached 0.0257[U],corresponding to 0.82 m/s, at time t--5.0[T]=27.0 sec, and x,,,-0.5[L]=-85 m.The numerical predictions indicate that significant local current may be producedaround the slope front area; outside this area the current decreases quickly as the wavespropagate away from the slide site. Therefore, precautions should be taken for small shipsand equipment moorings set up for an experiment near the river mouth.6 8 10-6^-4^-2 0 2 4 6 8 100.040.0-0.040.040.0-0.040.040.0-0.040.040.0-0.04Tif 0.04gi^0.0X; -0.04-6^-4^-2^0^2^4X (x170m)0.040.0-0.040.040.0-0.040.040.0-0.04t:1.0-6 -4 -2 o 2 4 6 8^10t=1.0..c.vc..,-6 -4 -2 0 2 4 6 8^10t=2.0 I-6 -4 -2 0 2 4 6 8^1078Figure 2.35: The water particle velocity distribution for the case specified in Figure 2.29.0.040.0-0.04ti 0.040.0xD -0.04-6 -4 -2 0 2 4 6 8 100 2 4 6 8-6^-4^-2^0^2^4X (x170m)6 8 10-610-4 -20.040.0-0.0479W3.0Figure 2.36: The water particle velocity distribution for the case specified in Figure 2.30.802.8 Numerical Results for Large-Scale SlidesI have presented the numerical results on the dynamics of small-scale slides and thesurface wave generated in the pervious sections. It is expected that large-scale slides(with large initial volume) would gain larger speeds and produce larger surface waves.Here I shall discuss the flow of a large-scale slide and the surface wave generation. Theinitial dimensions of a parabolic slide profile is chosen to be D o=24 m and L0=686 m. Theriver depth is h i=0.5Do=12 m and the deep water with h 2=588 m, the slope inclination is4°. Two muds with different densities and viscosities are considered: (a) p 2=1200 kg/m 3 ,v=p/p2=0.2 m2/s, g'=1.6 m/s2; and (b) p2=2000 kg/m3, v=,u/p2=0.2 m2/s. g'=4.9 m/s2;.Theinitial position of the mud center is chosen to be at I =0.5L 0 and the initial positionindicator K=D o/W)=0.40. I shall present here the numerical results for the slide surfacevariation, the slide horizontal velocity distribution, the slide frontal speed, the surfacewave generation and the velocity of water motion distribution.2.8.1 Slide ProfilesFigures 2.37 and 2.38 show the successive positions of slide (a) and slide (b)respectively, and a shallow initiation depth of the mudslide, tc=0.40. The solid linesrepresent the two-way coupling results, and the dotted lines are the one-way couplingresults. In Figures 2.37 and 2.38 the depth of the slide is plotted on a scale exaggeratedabout 70 times over the horizontal distance scale. The time scales for slide (a) are[f].(Lo/ gr2=20.5 sec, and for slide (b) [7]=(L0/ 0' a=11.8 sec.Again it can be seen that the slide quickly flattens into a thin and long flow. Thereaction of waves on the slide slows down the flow speed of the slide only slightly. Asexpected, the slide with larger density (slide (b)) flows faster. The relatively light slide81specified in Figure 2.37 reaches x=3.42L 0=342 m at t=12[7]=71.7 sec, while the denserslide with a larger viscosity as specified in Figure 2.38 reaches x=3.3L 0=330 m att=12[T]=54.2 sec which is much faster than the slide in Figure 2.37. The numericalresults again indicate that the effect of the two-way coupling is more significant forsmaller mud density and shallower slide initiation depth as in Figure 2.37, and it is smallfor larger mud density and deeper slide initiation depth as in Figure 2.38. After sometime (t>6, approximately), the two solutions evolve into almost the same profile, with theslide under a uncoupled free surface moving slightly more quickly.1.00.80.60.40.20.00t=0.01 2 3 4 5 6 7 81.00.80.60.40.20.00 1 2 3 4 5 6 7 8D1.00.80.60.40.21 2 3 4 5 6 7^8t=6.01 2 3 4 5 6 7^8t=8.01 2 3 4 5 6 7^8t=10.01 2 3 4 5 6 7^8t=12.01 2 3 4 5 6 7^8t=14.02 3 4x5 6 7^81.00.80.60.40.20.000.0 ---0 11.00.80.60.40.20.001.00.80.6 -0.40.20.0 --01.00.80.60.40.20.001.00.80.60.40.20.0082Figure 2.37: Profiles of a viscous slide on a slope with 0=4', D o=24 m, Lo=686 m, r= 1.2,m2/s and tc=0.40. Solid line: two-way coupling; dotted line: one-way coupling.1.00.80.60.4020.0ta0.05 6 7 8to2.05 6 7 8t=4.05 6 7 811.6.05 6 7 8t=8.05 6 7 82 430 11.00.80.60.4020.03 41 201.00.80.60.40.20.041 2 301.00.80.60.40.20.00^1^2^3^41.00.80.60.40.2 :_____.--------\2^3^41083/1.00.80.60.40.20.00100^10^12 3 4 5 6 7^8-.-.-----t=12.02 3 4 5 6 7^81.14.02 3 4x5 6 7^81.00.80.60.40.20.01.00.80.60.40.20.0Figure 2.38: Profiles of a viscous slide on a slope with the same conditions as specifiedin Figure 2.37 except r=2.0. Solid line: two-way coupling; dotted line: one-way coupling.842.8.2 Horizontal Velocity of the SlideFigures 2.39 and 2.40 show the transient variation of the horizontal velocities of themud particles, U(x,t), with nondimensional time t, for slide (a) and slide (b) respectively.The solid lines are the results of the fully coupled model, and the dotted lines are the one-way coupling results. The velocity scales for slide (a) are [U]=(g'L 0) 1 /2=33.1 m/s; and forslide (b) [U]=(g 7,0)12=57.9 m/s.The maximum velocity for the slide specified in Figure 2.37 reached 0.36[U] att=8[T]=164 sec near the front with D(x,t)=0.35[H]=8.43 m, which corresponds to 11.9m/s,. The maximum Reynolds number is Re=502.6, and the maximum Keulegan numberis approximately Ke=5526.1. The maximum velocity for the slide specified in Figure 2.38reached 0.44[U] at t=6[T]=70.8 sec near the front with D(x,t)=0.30[H]=7.20 m, whichcorresponds to 25.5 m/s, with a maximum Reynolds number being Re=918.0 and amaximum Keulegan number being Ke=16919.7. It can be seen the Reynolds numbers arenear 1000 for these slide and the Keulegan number are far greater than the critical value.It is likely that turbulence will play a important role in the large slide with low viscosity.The effects of interface turbulence on the slide and the wave generation will be discussedin Section 2.9.0.60.40.20.0t-0.00 1 2 3 4 5 6 7 80.60.40.20.00 1 2 3 4 5 6 7 80.60.40.20.00 1 2 3 4 5 6 7 80.60.40.20.00 1 2 3 5 6 7 8850 1 2 3 40 15^6^7^81.8.02^3^4^5^6^7^8t=10.0U0.60.40.20.00 10^1^2^3^4^5^6^7^8t=12.02^3^4^5^6^7^8t=14.00.60.40.20.04xFigure 2.39: Successive variation of the horizontal velocity of slide (a). The maximumvelocity reached (0.36M) corresponds to 11.9 m/s.0.60.40.20.00.60.40.20.00.60.40.20.00.60.40.20.00.60.40.20.00.60.40.20.00.60.40.20.00.60.40.20.00.60.40.20.0863 43 43 43 43 43 43 4t-0.05 6 7^8t=2.05 6 7^8t.4.05 6 7^8tig6.05 6 7^8tv.8.05 6 7^8t=10.05 6 7^81=12.05 6 7^8__-'-'---\0 1 20 1 2•0 1 20 1 20 1 2•0 1 2.^. ___-------0 1 2U0.60.40.20.00^1^2^3^4^5^6^7^8xFigure 2.40: Successive variation of the horizontal velocity of slide (b). The maximumvelocity reached (0.44[0) corresponds to 25.5 m/s.872.8.3 The Frontal Speed of the SlideFigures 2.41 and 2.42 illustrate the variation of the frontal speed of the mudflow, Uf=dxfidt, with nondimensional time t, for the slides specified in Figures 2.37 and 2.38respectively. The solid lines indicate the results of the fully coupled model, the dottedlines indicate the results of the one-way coupling model. The maximum value (the resultof the fully coupled model) of the frontal speed for the mudflow in Figure 2.37 is 0.30[U]corresponding to 10.0 m/s, and for the mudflow in Figure 2.38 is 0.34[U] corresponding19.5 m/s. So large-scale slides would obtain larger frontal speeds than the small ones Iconsidered in previous sections. But at such high speed, there would be strong shear atthe slide-water interface that might lead to interfacial mixing. The effects of interfacialturbulent shear will be discussed in Section 2.9. Here I first examine the Frontal Froudenumber of the slides.Figure 2.43 shows the variation of k =Ufl(g'Df) 112 (Froude number of the slide front)versus time for slides (a) and slide (b) respectively. It can be found that for large-scaleslide the frontal Froude number increases drastically with the acceleration of the slide to amaximum value (maximum k=2.78 for slide (2), and k=3.11 for slide (b)), and then startsdecreasing. It is doubted that the slides could travel laminarly with such high Froudenumber (k>1.3) at the slide front. What would most likely happen in real world is thatturbulence will develop at that stage and change the flow pattern at the head and slowdown the flow, for which case the laminarity approximation of the slide will not be valid.With the slide speed being overestimated, the magnitude of the surface wave may also beoverestimated. As stated in Section 2.5.3, using a laminar model for a turbulent case,however, could give the upper limit for the slide speed, which may be useful to evaluatethe slide run-out distance and the impact force on a submarine structure for engineeringpurposes.0 2 4 6 8 10 12^140C 088Cl0C4C00^2^4^6^8^10^12^14Time tFigure 2.41: Successive variation of the frontal speed of the slide (a).Time tFigure 2.42: Successive variation of the frontal speed of the slide (b).89c....43Pco 2-......100^5^10^15Time tFigure 2.43: Successive variation of the frontal Froude number for slide (a) and slide (b)respectively.2.5.4 Elevations of the Surface WavesFigures 2.44 and 2.45 illustrate the surface waves generated by slide (a) and slide (b)respectively. The maximum height (the result of the fully coupled model) of the surfacewaves generated by slide (a) is 0.11[H] corresponding to 2.64 m. The maximum waveheight for the waves generated by slide (b) is 0.23[H] corresponding 5.52 m. Wave runupat the shore caused by the shoreward crest (with amplitude of 0.0081[11)=0.19 m) isestimated to be (using equation 2.33) R=0.94 m, whilst wave runup caused by theoutgoing crest (with amplitude of 0.02[11]4.48 m) at the opposite shore (with the samecondition as the near one) is estimated to be R=2.8 m. The wave runups are much largerthan the incident waves, and they are of great public concern. As mentioned early, thelaminar model may overestimate the slide speed and the surface wave amplitude if usedfor a turbulent case. More of this effect will be discussed in Section 2.9.3.-10^-5^0,N--10^-5^0t-0.05 10 15 20 25t=2 .05 10 15 20 25900.060.02-0.02-0.060.060.02-0.02-0.060.060.02-0.02-0.06-10 -5 0 5 10 15 20 250.060.02-0.02-0.06-10 -5 0 5 10 15 20 250.060.02-0.02-0.06-10 -5 0 5 10 15 20 250.060.02-0.02-0.06-10 -5 0 5 10 15 20 250.060.02-0.02-0.06-10 -5 0 5 10 15 20 25t=14.0-10 -5 0 5 10 15 20 25x0.0617 0.02-0.02-0.06Figure 2.44: Surface waves induced by slide (a). Solid line: two-way coupling; dottedline: one-way coupling.Figure 2.45: Surface waves induced by slide (b). Solid line: two-way coupling; dottedline: one-way coupling.0 5-5 1510 20 25-1010.150.05-0.05-0.150.150.05-0.05-0.15-100.150.05-0.05-0.15-100.150.05-0.05-0.15-100.150.05-0.05-0.15-100.150.05-0.05-0.15-100.150.05-0.05-0.15-100.150.05-0.05-0.15-10-5 0 5 10 15 20 25-5 0 5 10 15 20 25-5 50 10 15 20 25-5 0 5 10 15 20 25-5 0 5 10 15 20 25-5 50 10 15 20 25-5^0^5^10x15 20 2591t.0.0922.8.5 Waves Generated by a Rigid Triangular BlockIn this section I shall examine the water waves generated by a rigid block with atriangular cross-section, sliding on a gentle smooth slope. The purpose of this section isnot to determine exactly the waves generated by a slide block, but to examine theinfluence of the deformability of the slide on surface wave generation. For a very longand thin body attached to a wall, the form drag due to the pressure difference between thefrontal and rear faces of the body is very small, because separation of flow at the rear facemight be delayed to the very end of the body or there might even be no separation[Hoerner, 1958; Tritton, 1984]. The drag force exerted on the body is due to skin friction,a shear force on the surface of the body due to viscous effects. The frictional force on thesurface of the thin sliding block can be evaluated with the boundary layer theory for flowover a flat plate [Norem and Locat, 1991]. For turbulent flow over a flat rough plate theshear stress at point x along the plate is given by [Schlichting, 1968]2 = 0.5[2.87 +1.58log(-)"2.5^(2.42)AUwhere k is a roughness length. Averaged over the slide length, the averaged shear stresscan be expressed as^= 0.5[1.89 +1.62log(—L )""AU2(2.43)For the cases calculated here, the length of the block, L=686 m, the slide speed isapproximately U=10 m/s, and I consider a roughness length of, k=0.01 m. With thesedata the averaged shear stress will be z=0.0017p iU2 =170 N/m2, and the total shear forceon the block surface (per unit width) is z t=zs=170(N/m2)x686x1(m2)=1.37x105 N (s- thearea of the block surface per unit width) The bottom friction F r=fmg 'cos0=(0 .02-930.03)x1200(kg/m 3)x8232(m 3)x1.67(m/s2)x0.997=(3.3-4.9)x10 5 N. It is noted that theabove frictional drag force is the maximum one (at the maximum slide speed) the blockcan experience. With the acceleration (t>14), the sliding speed increases, so does the dragforce on the block. So the solid block will obtain a constant speed at a later time. At thattime the block moves into much deeper water and its wave-generating effect will be muchsmaller. In the following calculation, the shear stress on the sliding block is neglected forsimplicity. For the sliding motion, the net slope-parallel force F acting on the block isF = mg' sin 0 — fmg' cos 8, (2.44)where m is the mass of the block, g' is the reduced gravity, f is the bottom frictioncoefficient, and s is the drag area (the surface area of the block) per unit width. ApplyingNewton's second law, the acceleration of the block a can be expressed as,a = g' (sin 0 — f cos 8). (2.45)I present, in Figures 2.46 and 2.47, the positions of a rigid triangular block (solid line)with density p2=1.2 g/cm3 , and two bottom friction coefficients f=0.02 and 0.03, slidingon a 4° slope, in comparison with the result of the triangular viscous slide (dotted line,with D o=24 m, Lo=686 m, r=1.2, and c=0.40). Figures 2.48 and 2.49 show the surfacewaves (solid line) generated by the blocks as specified in Figures 2.46 and 2.47,respectively, with those produced by the triangular viscous slide (dotted line). Themagnitudes of the offshore crest and the shoreward trough are of the same order as thosein the case of the viscous slide. The amplitude of the forced trough, however, is 2-5 timeslarger than that produced by the viscous slide. Figures 2.50 and 2.51 show the velocity ofthe water motion (solid line) generated by the blocks as specified in Figures 2.46 and2.47, respectively, in comparison with those produced by the triangular viscous slide(dotted line). It is found that the water motion is similar to that of a viscous slide, exceptthat the solid block generates a shoreward current 2-5 times larger than the viscous slide.1.00.80.60.40.20.0-2 -1^0^1^2^3^4^5^6^7^8^9 10D1.00.80.60.4020.0-2 -1^0^1^2^3^4^5^6^7^8^9 10-2 -1^0^1^2^3^4^5^6^7^8^9 101.00.80.60.4020.0-2 -1 0 1....--2 -1 0 1-2 -1 0 1-2 -1 0 1- . 1-2 -1 0 12 3 4 5 6A2 3 4 5 6A2 3 4 5^6A2 3 4 5^6A2 3 4 5 6x7 8 9 10t=8.07 8 9 101=10.07 8 9 10t=12.07 8 9 10t=14.07 8 9 10-........ : ---- :............ -...^•Figure 2.46: The positions of a rigid triangular block (solid line) with densityp2=1.2g/cm3 , and friction coefficients f=0.02, sliding on a 4' slope, in comparison withthe result of the triangular viscous slide (dotted line).1.00.80.60.40.20.0941.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.0A^t=0 .0A 1..0.0A 1-14.01.00.80.60.40.20.0-2 -1^0^1^2^3^4^5^6^7^8^9 101.00.80.60.40.20.0-2 -1^0^1^2^3^4^5^6^7^8^9^10-2 -1 0 1. --2 -1 0 1-2 -1 0 1.. -.--2 -1 0 14 5 6 7 8 9 10t=6.04 5 6 7 8 9 101-8.04 5 6 7 8 9 101= 1 0.04 5 6 7 8 9 102 3A2 32 3- - • -. A2 31.00.80.60.40.20.0-2 -1^0^1^2^3^4^5^6^7^8^9^101.00.80.60.40.20.0-2 -1^0^1^2^3^4^5^6^7^8^9^10xFigure 2.47: The positions of a rigid triangular block (solid line) with densityp2=1.2g/cm3 , and friction coefficients f=0.03, sliding on a 4* slope, in comparison withthe result of the triangular viscous slide (dotted line).D951.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.0t=0.0-10 -5 0 5 10 15 20..;z7.t=2.0-10 -5 0 5 10 15 20960.100.0-0.10-0.20-0.300.100.0-0.10-0.20-0.300.100.0-0.10-0.20-0.300.100.0-0.10-0.20-0.30-10 -5 0 5 10 15 20-10 -5 0 5 10 15 200.100.0-0.10-0.20-0.300.10.0-0.1-0.2-030.10.077 -0.1-0.2-0.3-0.4-10 -5 0 5 10 15 20.,....■.^.....■.....t=10.0-10 -5 0 5 10 15 20,.........w...........orlia,, ■■•■■•■t=12.0..•-10 -5 0 5x10 15 20Figure 2.48: Surface waves generated by the rigid triangular block as specified in Figure2.46 (solid line) in comparison with the result of the triangular viscous slide (dotted line).t=0.010 15 20t=2.010 15 20t=4.010 15 200.100.0-0.10-0.20-0.305-5 0-100.100.0-0.10-0.20-0.30-10 50-50.100.0-0.10-0.20-0.30-10 50-597-,f-----0.100.0-0.10-0.20-0.30-10^-5^0^5^10^15^200.100.0-0.10-0.20-0.30-10^-5^0^5^10^15^200.100.0-0.10-0.20-0.30-10^-5^0^5^1 0^15^200.100.077 -0.10-0.20-0.30-10^-5^0^5^10^15^20xFigure 2.49: Surface waves generated by the rigid triangular block as specified in Figure2.47 (solid line) in comparison with the result of the triangular viscous slide (dotted line).0-10 -50-5-100.02-0.02-0.06-0.100 5-5 1510 20-100.02-0.02-0.06-0.100 5-5 1510 20-100.02-0.02-0.06-0.10•t-0.05 10 15 201-2.05 10 15 20-10 -5 0 5 10 15 20t-10.0-10 -5 0 5 10 15 20t -12.0••-10 -5 0 5x10 15 20Figure 2.50: Distribution of the velocity of the water motion induced by the rigidtriangular block as specified in Figure 2.46 (solid line) in comparison with the result ofthe triangular viscous slide (dotted line).980.02-0.02-0.06-0.100.02-0.02-0.06-0.100.02-0.02-0.06-0.100.0u -0.04-0.08-0.120 5-5 1510 20-100.02-0.02-0.06-0.100 5-5 151 0 20-100.02-0.02-0.06-0.100 5-5 10 15 20-100.02-0.02-0.06-0.10-10 0-5 5^10^15^20N0.0-10 -5 0 5 10 15 20t-2.0-10 -5 0 5 10 15 20-10 -5 0 5 10 15 20xFigure 2.51: Distribution of the velocity of the water motion induced by the rigidtriangular block as specified in Figure 2.47 (solid line) in comparison with the result ofthe triangular viscous slide (dotted line).990.02-0.02-0.06-0.100.02-0.02-0.06-0.100.02-0.02-0.06-0.100.02-0.02-0.06-0.101002.9^Effects of Turbulence at Slide-Water Interface2.9.1 Kelvin-Helmholtz Instability and TurbulenceThe purpose of this section is to briefly discuss, following Thorpe [1973], the Kelvin-Helmholtz type instability and the onset of turbulence at the interface of two-layer shearflow. The mechanism of the instability development is out of the scope of this thesis. Formore information on Kelvin-Helmholtz instability, one may refer to the reviews byThorpe [1973], Maxworthy and Browand [1975], and Sherman, et a/[1978].Kelvin-Helmholtz instability is the instability at the interface of a stratified shearedflow. For a two-layer shear flow with a pre-existing sharp interface (Figure 2.52 (a)), theinterface will first be wrinkled by some wavelike disturbances that grow quickly (Figure2.52 (b)). These waves then become sharp crested and roll up to form billows in a fashionas shown in Figure 2.52 (c). Turbulence occurs when the billows acquire enough heightand collapse. The turbulent region rapidly spreads to fill the billow and begin to entrainits surroundings. The fluid entrained from the lighter upper layer is rapidly carried intothe bottom of the billows, and the heavy fluid entrained from the lower layer is carriedinto the top (Figure 2.52 (d)). This process of entrainment is very efficient at destroyingthe density gradient.The vertical spread of the layer continues until the gradient Richardson number (seeg dp f dzaU)2Turner, 1976; Ri = —Paz—/ — ) reaches a value of 0.25 everywhere (see Figure 2.52(e)). Experimental results and theoretical analysis show that Ri>0.25 is sufficient for theflow to be stable [Miles and Howard, 1964; Thorpe, 1973]. Sufficiently large densitydifferences will increase the Richardson number and thus stabilize the shear flow. Thefinal velocity and density profiles are shown in Figure 2.52 (f). Three main conclusionsmay be drawn about Kelvin-Helmholtz instabilities [Sherman, et al, 1978]:101(1) they require a pre-existing sharp interface to get started,(2) they convert mean flow kinetic energy to potential energy rather inefficiently,(3) they are self-limiting, an interface once thickened by a Kelvin-Helmholtz episodewill not support subsequent instabilities of the same kind unless it becomes more stronglysheared, or unless it is followed by some process that sharpens the interface again.2.9.2 Effects of Interfacial Turbulent Shear on Water MotionIn the previous sections I discussed the results of a laminar slide model and the surfacewaves generated. In this section I shall examine the effects of the possible turbulent shearat the slide-water interface. It is noted that interfacial turbulent shear between the water-slide interface acts only over the length of the moving slide. Elsewhere turbulent shear onrigid impermeable slope is usually negligible. So I shall analyze only the forces on thewater body just above the moving slide. With the interfacial turbulent shear stressincluded, the momentum equation of water motion can be written asdu du_—+u--= —g ^,dt^dx^dx pih (2.46)where Ti is the interfacial stress due to the turbulent boundary layer, and h is the waterdepth. To asses the importance of which is due to turbulence near the interface, I adoptthe empirical formula:= Cpp,(U — u)2 CDpiU2, (2.47)where CD is an empirical drag coefficient which is usually taken as (1-2.5)10 3 [Murty,1984]; U and u are the velocities of the slide and the water at the interface respectively; uis much smaller than U so for convenience let U-u—U.102(c)(b)(e)(d)P1132Density Disribution^UVelocity Distribution(a)U)Velocity DistributionP2Density DisributionFigure 2.52: Kelvin-Helmholtz instability and turbulence in a two layer sheared flow.103The ratio of the interfacial turbulent shear to the pressure gradient is/ h^CDU2  =  CD U2 _  C (Fr)2pigan I dx ghat)! dx an 1 dx gh dn I (2.48)where Fr.Ufl(gh) 112 is a Froude number defined as the ratio of the slide speed with thelocal long-water-wave speed. I shall consider the case (the place and the time) wherethere might be the most significant turbulent shear. Fr has its maximum magnitude at theslide front, where there is the strongest shear. Therefore the effect of potential turbulentmixing at the front on the water motion will be examined.Figure 2.53 shows the variation of Fr.U1/(gh) 1/2 versus time for the small-scale slide(as specified in Figure 2.4) and the large-scale slide (as specified in Figure 2.38). Themaximum value of Fr reaches 0.55 for the large-scale slide and 0.47 for the small-scaleslide. The average trough slope (the wave just above the slide) is about 0.001 for thecases calculated here. Thus the ratio of the maximum turbulent shear and the pressuregradient force in the area of the forced trough lies between 0.2-0.8, depending upon thesignificance of the turbulence - the value of the drag coefficient. This indicates that thepossible turbulent shear exerted on the water in this part of the slide might be significantcompared with other forces on the water body. The potential turbulent shear will beadditional downslope force on the water, it will tend to push the water forward andincrease the acceleration of the water.It should be also emphasized that the turbulent shear stress considered above is the mostsignificant at the slide front at the end of the acceleration period; at other places in theslide the velocity is smaller and so is the shear. The interfacial shear also decreases as astratified layer of turbidity water forms at interface ( strong stratification may dampenout turbulence -increase Ri), and it also decreases with the slide speed as the slide moves1.00.80.60.4020 .01040^5^10^15Time tFigure 2.53: Variation of Fr=19(gh)w2 versus time for the small-scale slide (as specifiedin Figure 2.4) and the large-scale slide (specified in Figure 2.38).into the deceleration period. After analyzing some observations of submarine slidedeposits Karlsrud and Edgers [1982] argued that only a small amount of a flowing masswill become low density turbidity currents, flowing above a dense lower flow, unless theflowing mass encounters some obstructions on the flow path or there are very abruptchanges in slope geometry. I feel that the significance of turbidity current might also bedependent on the slide volume. With other parameters (the viscosity, density, slope angle)being fixed, a large-scale slide (thicker and longer) may acquire a larger speed at the endof the acceleration phase (which means a larger Reynolds number and a smallerRichardson number). Hence there will be more significant interfacial shear for a large-scale slide than a small-scale one.1052.9.2 Effects of Interfacial Turbulent Shear on the SlideNow I shall examine the effects of the interfacial shear on the slide dynamics. Thelaminar model assumes that the interfacial shear is small compared with the bottomshear, and neglects the interfacial shear. When will the turbulent shear become importantif turbulence occurs? To answer this question I compare the turbulent shear stress withthe bottom shear stress.The bottom shear stress of the slide can be expressed asdU„, I1.11^--h^2#U = cp2U ^crpiLl2 ,dz ' D (2.49)where c is the laminar friction coefficient, c=2/Re, Re=p2UDI . The ratio of theinterface shear stress and the bottom shear stress is^.CDPP2^= CDRezb rcp1U2 rc^2r (2.50)When can I neglect turbulence effects on the slide? Let me set the condition to be thatVTb<0.01. Then the requirement for Reynolds number is Re<0.2r/CD . Substituting r=1.2-2.0 and CD=0.001-0.0025, I have the condition for neglecting turbulent shear: Re<(8-40)which depends on the magnitude of the turbulent drag coefficient and the Reynoldsnumber of the flow. The turbulent drag coefficient is determined by the structure of theturbulent boundary layer between the slide and the water. Thus it can be seen that forlarge-scale landslide as calculated in Sections 2.8, where the maximum ratio of theinterfacial shear stress and bottom shear stress may reach zi/Tb=0.2-0.9, turbulence plays asignificant role at the head of the slide when the slide accelerates to its maximum speed106and turbulence develops. For the small-scale slides as calculated in Section 2.5, 2.6 and2.7, the slide speeds are small and the Reynolds numbers are less than 40 (where themaximum ratio ilr6=0.02-0.05), turbulence plays a minor role in the slide dynamics. Theturbulent shear exerted on the surface of the slide, though small, will slow down the flowat the head and the non-hydrostatic effects at the head associated with turbulent mixingwill also slow down the slide speed. The laminar model can give relatively more accuratesolutions for small-scale slides than for large-scale ones, and the laminar model willoverestimate the slide speed and wave height due to the omission of effect of turbulence.It is useful to emphasize that the turbulent shear decreases with the slide speed in thecourse of flowing. It has the largest value at the slide front where there might be the mostsignificant shear, and it is small at the rear part of the slide where the velocity is small.Turbulence weakens as the slide speed decreases or the interfacial shear is weakened bystratification.What is the flow picture if turbulence dominated the flow? What would happen to theflow speed and the surface wave height? In the case of a turbulent underwater flow, thereis usually a turbidity current flowing over a dense flow (as shown in Figure 1.3(1)). Thethickness of the dense flow will be smaller than that predicted with the laminar model. Inthe lower dense flow there are usually a head at the front, a body in the middle and a tailat the end. The frontal speed is mainly governed by the flow condition at the head:turbulent mixing, ambient water flow, etc., and it is usually calculated with equation(2.34). In the present laminar model, energy dissipation due to turbulence is neglected, sothe slide speed might be overestimated. Also due to the formation of turbidity wateroverlaying the lower dense flow, the slide height in a turbulent case will be smaller thanthe laminar solution. These factors will decrease the frontal speed of the slide and also theheight of the surface waves. The effects of the turbidity current overlaying the dense flowon the surface wave generation needs more study.107To sum up this section, the present laminar model will give the upper limit of the slidespeed and wave height for the case of large-scale underwater landslides which mayeventually evolve into a turbulence-dominated turbidity current. For small-scale slideswith large enough viscosity the flow may remain laminar if the acceleration does notbring the flow into turbulence, and the laminar model will give more accurate results. Forengineering purposes, people are interested in catastrophic effects of underwaterlandslide. The laminar model, neglecting energy dissipation into turbulence, will give theupper limit of the slide speed and wave height that might be useful for engineers toevaluate the maximum slide speed and wave amplitude.2.10 Summary and ConclusionsI have presented a formulation of the dynamics of a submarine mudslide coupled withthe surface waves which it generates, flowing on a gentle uniform slope. The mud istreated as a viscous fluid, and the water motion is assumed irrotational. The shallow waveapproximation is adopted for both wave motion and mudflow. Dispersion of the waves isneglected. I solved the resulting equations by a finite-difference method.The calculations show that three major waves are generated. The first wave is a crestwhich propagates into deeper water; it is followed by a trough which is a forced wavepropagating with the speed of the slide front into deep water. The third wave is a smalltrough which propagates shoreward. Two major parameters determine the interactionbetween the slide and the waves: the density of sliding material and the depth of initiationof the mudslide. The two-way interactions are significant for the cases of a smaller muddensity and shallower waters. For the cases of denser mud sliding in shallower water thetwo-way interactions are small; however, the amplitudes of waves generated are large. If108the mudslide occurs in deeper water, as expected, the interactions are weak and the wavesare small.The amplitudes of the waves depend primarily on the density of mud, the initiationdepth of the mudslide, the volume of the landslide, and the viscosity of mud. The slidewith larger density and volume, and smaller viscosity in shallower water generates largerwaves. In the calculated examples, a mudslide with a initial length of 686 m and an initialmaximum thickness of 24 m on a 4° slope can generate waves with maximum crestamplitude of up to 3 m. The amplitude of the forced trough associated with the mudslidefront is of the same order as that of the crest, thus the height of the wave can reachapproximately 6 m. The ratio of the energy transfer from the slide to the waves is not aconstant; it has a maximum value at the beginning stage and decreases to 2-4% when thewaves propagate away from the slide site. The energy exchange is significant for the casewhere the interaction between the slide and the waves is strongest.The effect of the configuration of the initial slide profile is also examined. A slide witha triangular initial profile and the waves which it produces are calculated and compared tothose produced by the initial parabolic profile. The numerical results indicate thatwhatever the initial profiles of the slide are, they finally all evolve into a similar profile.Slides with different initial profiles and same volume behave almost identically. It is alsofound that the waves generated by a viscous slide are usually smaller than those producedby a sliding solid block. I examined the occurrence of possible resonance between anunderwater slide and the waves which it generates. The calculated results on the Froudenumbers for various situations indicate that the possibility of such a resonance is verysmall in practical cases.As to the wave runup at the shore, it is found that during a underwater landslide eventa decline of the water level at the shore occurs first as a result of the shoreward trough,soon after a wave runup will be generated by the shoreward crest following the trough. A109rough estimation indicates that the runup induced by a triangular slide is four times largerthan the amplitude of the incident wave.I also presented theoretical predictions for the ADFEX experiment, using the actualbathymetry of the experimental area and two sets of presumed parameters for therheological properties of the submarine sediments. The calculated results indicate that thewaves produced by the proposed slide are large (up to 2 m in height) near the rivermouth, and the local water current can reach significant values (up to 1.0 m/s) which maycause damage to the nearby ships and equipment moorings. With the increase of thedepth of the water, the wave amplitude decreases rapidly.Finally I examined the influence of turbulence on the slide dynamics. As to the effect ofturbulent shear on the slide, for large-scale landslides as calculated in Sections 2.8,turbulence plays a non-negligible as the slide accelerates to great enough speed. Aturbulent slide on a slope is much different from a laminar one. The importance andaccurate evaluation of the interfacial turbulent shear still needs further investigation boththeoretically and experimentally. For the small-scale slides as calculated in Section 2.5and 2.6, the slide speeds are small and the Reynolds numbers are less than 40, turbulencemight play a minor part in the slide dynamics. Therefore the laminar model can giverelatively accurate solutions for small scale slides, and it can also give the upper limit ofthe slide speed and wave height that might be useful for engineers to evaluate themaximum slide speed and wave amplitude for engineering purposes.110Chapter 3A Bingham-Plastic Fluid Model and Surface Wave Generation3.1 IntroductionSubmarine sediments in different places may have different rheological behavior,mostly as a consequence of their chemical composition, grain size distribution, clayconcentration, etc.. Many laboratory experiments on the theological properties of coastaland river sediments [Krone, 1963; Verreet and Berlamont, 1987; O'Brien, 1988; Lavaland Demers, 1988, Nguyen and Boger, 1992] indicate that the Bingham fluid model is agood fit to measured rheograms. To understand surface wave generation due to aBingham plastic slide, I pursue an analysis similar to that in Chapter 2.In this chapter I present a numerical model, similar to the viscous fluid model inChapter 2, to examine water waves generated by an underwater Bingham-plasticlandslide on a gentle uniform slope. The effects of the Bingham-plastic rheology on theslide dynamics and on the wave generation will be the focus of this chapter. Aformulation of the governing equations of the Bingham-plastic slide and the surfacewaves is presented. Surface waves produced by a mudslide with a initial parabolic shapeare calculated with a finite-difference method similar to that I used in Chapter 2.I present here the numerical results for successive profiles of the slide surface, thehorizontal velocity distributions of the slide, the evolution of the surface elevations andthe distributions of the particle velocity of the water motion. The ratio of energy transferfrom the Bingham plastic slides to the waves will be calculated, and the energy exchangewill be discussed. I also present a comparison of the numerical predictions of theBingham slide model with a field snow test of Dent [1980, 1982] which involves thedeceleration of snow flow from a slope onto a flat plane of packed snow.1113.2 Governing Equations for A Bingham-Plastic SlideA Bingham plastic fluid (also called visco-plastic fluid) is one in which nodeformation takes place until a specified shear stress, which is called the yield stress orthe Bingham stress, is applied to the fluid, after which the deformation is driven by theexcess of the stress beyond the yield stress. Many materials such as toothpaste, snow,volcanic lava, submarine and river sediments behave approximately as a Bingham plasticfluid.For one-dimensional shear flow of a Bingham fluid, the stress-strain relation [Duvautand Lions, 1976] is nonlinear (see Figure 3.1):du n,li ^Udz= if ITi< Ty ,^(3.1a)au—du = T — zsgn(—daz ),^ if Iii TY'^ (3.1b)a  where 'I" is the shear stress; T y is the Bingham yield stress; and au is the coefficient ofdynamic viscosity. The sgn(dul az) notation in (3.1b) is the standard form in which therheological relation for a Bingham fluid is expressed, so that correct signs may beobtained on both sides of a pipe flow, for example. For coastal and river mud, both theyield stress and the viscosity increase with the bulk density or the clay concentration[Krone, 1963; O'Brien, 1988]. The constitutive equation (3.1) can be used only if the flowis laminar. I will return below to a discussion of the appropriateness of a laminarBingham fluid flow model.I consider a layer of visco-plastic mud which starts flowing from rest down a rigidimpermeable slope inclined at a small angle 0 with respect to the horizontal direction,and employ the same coordinate system as that in Chapter 2(see Figure 3.2).112shear stress^BinghamTr Newtonian ,..,- shear rateBinghamFigure 3.1: Constitutive laws of a Newtonian fluid and a Bingham fluid.I assume that both the water and the mud are homogeneous with distinct densities, andthat there exists a sharp interface between water and mud. I shall consider a dense slideand neglect the interfacial mixing. As to the effects of interface turbulent mixing one maydo similar discussion as in Section 2.9. I adopt the long-wave approximation for both theslide and the waves, i.e., the horizontal length scale is much greater than the verticallength scale, and the velocity is essentially in the x axis direction. The verticalmomentum conservation reduces to a vertical hydrostatic pressure distribution in the mudlayer, i.e.p(x, z,t) = pig(n + h)— p2g(z + h), —hs (x) 5_ z —h(x,t),^(3.2)where p(x,z,t) denotes the pressure in the mud layer; p 1 and p2 denote the densities ofwater and mud, respectively; and h(x,t) is the undisturbed depth of the water.The longitudinal momentum equation for the entire mud layer isp2(dU  mUr m U , d'r--at +—ax + wm a—dz )-= (P2 — )g sin dx dz—– k(x) z 5_ –h(x,t),where Um(x,z,t) and W„,(x,z,t) are the horizontal and vertical particle velocities in the mudrespectively, and T(x,z,t) is the shear stress in the mud. If the value of r is positive, itmeans the shear stress is in the direction of the positive x axis, if it is negative; the shearstress is in the direction of the negative x axis.113(3.3) 0river^h,,/ ■/////Figure 3.2: Definition sketch of an underwater Bingham slide and the surface waves.For an underwater laminar mudflow on a gentle uniform slope, I may neglect thetangential stress acting on the water-mud interface, because the viscosity of water ismuch smaller than that of the mud, and the basal shear of the mudflow is much greaterthan the interfacial shear [see Section 2.9 and Liu and Mei, 1987]. The corresponding114boundary conditions for the problem are: (1) a zero-shear condition at the water-mudinterface, and (2) a no-slip condition on the slide bottom, i.e.,s.(x,z,t) = 0,^z = —h(x,t),^(3.4)Um (x,z,t) = 0,^z = —h3 (x).^(3.5a)W.(x,z,t) = 0,^z = -k (x).^(3.5b)Integrating (3.3) with respect to z from the mud surface z= -h(x,t) to z= z, andsubstituting (3.2) and (3.4) yields the shear stress distribution through the entire mudlayerT = (Z + h) plig—dn — (P2 — Pi)g(oc d— D j]dx^x(du,n^+ w+ Um dun,^dun,),z — k(x) z — h(x, t),^(3.6)-dx -h(x,t)where a = shit).At the slide bottom, z= - k(x), the shear stress can be expressed asdi^an^)g(Dl^= —D[Pig dx 2 1-kr P2(x)J (aU m + dxdU w dUdz )-h(x,t)(3.7)For an underwater Bingham mudslide starting from rest, it is obvious that I 11, I mustexceed the yield stress Ty for the slide to initiate, i.e., the fluid moves downslope ifaD)(P2 — Pi)g(a — —dx — Pig dx 7),(3.8)and the fluid moves upslope if(P2 -Pi)g(a --P--)-Pig--1-la a -'1'dx^D The slide does not initiate if I SI 1< Ty, oraD)^anTy ^—,rY > D[(P2 - Pi )g(a - Tx  -Pig ax ^y. (3.10)The nonlinear constitutive relation (3.1) leads to two distinct zones in the flow, a shearzone and a plug zone, as shown in Figure 3.3. In the shear flow, the shear stress exceedsthe yield stress, and the velocity varies in z. In the plug flow, the stress is smaller thanthe yield stress, and the horizontal velocity must be uniform in z, i.e.,U nt (X,Z,t) = U p (X,t),^— (hs (x) — Dy (X,t)) 5- z —h(x,t),^(3.11)where Up(x,t) represents the plug velocity in the plug zone, D(x,t) is the total thickness ofthe mud flow, and Dy(x,t) is the thickness of the shear zone.Figure 3.3: Velocity distribution of a one-dimensional Bingham fluid flow.The vertical velocity in the plug zone can be obtained from the continuity equation115(3.9)dU ax ' m+^  = 0.dx^az(3.12)116Integrating (3.12) with respect to z from z=- (125-Dy) to z=z yields^Wp (X,Z,t) = Wm (X,^+DY dU P (Z k - Dy ).^(3.13)At the yield interface in the slide, the shear stress is equal to the yield stress, i.e.,^= 1-ysgn(U p ),^at z = -(hs (x)- Dy (X,t)).^(3.14)The horizontal momentum equation in the plug zone can be obtained from (3.3), bysubstituting (3.2),P2dU^dU p(P2 - Pi adD , as-± (3.15)P UP+at^dx = ax )-^-Pig ax dz-(hs (x)- Dy (x,t)) z -h(x,t).Integrating (3.15) with respect to z from the mud surface z= - h(x,t) to z=z, andsubstituting boundary condition (3.4) yields the linear distribution of the shear stressthrough the plug zone= (z h)P2rLIP u p dU p ) + pig an _ (132 _ fog( a _dx )dt^dx^dx-(hs (x)- Dy (x,t)) 5_ z -h(x,t).(3.16)Substituting boundary condition of the shear stress at the yield interface (3.14), (3.16)readsdU^dU^aD)^an ysgn(U p ) P2(dtP + UP axl = (P2 Pi)g(a^Pigdx^dx D- Dy (3.17)which may be taken as the governing equation linking Up, D, Dy for the slide.117For the shear flow, the horizontal velocity is dependent on z, i.e.,U ,,i (x,z,t) =U s (x,z,t),^— hs (x) z 5_ —(hs (x)— Dy (x,t)),^(3.18)where Us(x,z,t) represents the horizontal velocity in the shear zone which satisfies thefollowing boundary conditions: (1) the horizontal and vertical velocities are continuousacross the yield interface; (2) the horizontal velocity gradient across the yield interface iszero; and (3) the no-slip condition at the slide bottom; i.e.,Us (x,z,t) = Up (x,t),^z = —(hs (x)— Dy(x,t)),^(3.19)aU s = _'u^z = —(hs (x)— Dy (X,t)),^(3.20)dz Us (x, z,t) = 0,^z = —hs (x),^(3.21a)Ws (x, z, t) = 0,^z = —hs (x).^(3.21b)where W,(x,z,t) denotes the vertical velocity in the shear layer.In the shear flow, the horizontal momentum equation of mudflow (3.3) reads,,,( dudts +us idus +ws dus) . (,,,,_ pog(a _ dD )dx^dz )^dxan + if  a2us  ± a2us-PAS ^)ax^dx2 az 2 Y(3.22)which has a similar form to the Navier-Stokes equation.For a steady Bingham-plastic flow on an uniform slope, the left-hand terms of (3.22)vanish, and the vertical distribution of the horizontal velocity in the shear zone isparabolic. Here, I assume that the mudslide rapidly reaches its equilibrium velocity [see118Edgers, 1981; Mei and Liu, 1987; Liu and Mei, 1989] so that I may use a verticalparabolic distribution for the shear velocity, U3(x,z,t), without committing serious errors,i.e.,(z+hs)2Us (x,z,t)= Up(x,t) z +11[2(--Dy^Dy(3.23)which satisfies all of the boundary conditions (3.19)-(3.21a,b).The vertical velocity in the shear zone Ws(x,z,t) may be obtained by integrating thecontinuity equation (3.12) in the shear layer,dUp [(z + k^1  +Ws (x,z,t)=—Dy -- -^Dy^3  Dy (3.24)By applying the von Karman momentum integral method [Bachelor, 1967; Mei andLiu, 1987], I integrate (3.22) with respect to z from the mud bottom z=-hs(x) to the yieldinterface z= -(k(x) -Dy(x,t)), and substitute (3.23) and (3.24) into (3.22). I obtain thegoverning equation of the shear flow in terms of Up, D, Dy , i^2 ^dUp =yP2 2(Dy—dU, 1 Up dD Up Dy3^dt 3^dt 3^dxU 2 a2U[(P2 — )g(a -(2-D-) - g^y - ,u(2 --e-a^ax Dy 3 axeConservation of mass in the entire mud layer requiresdD dq=dt dx(3.25)(3.26)where q(x,t) is the total volume flux in the two regions of the slide:119-(h,-Dy ) 1q(x,t) = Up (D — Dy ) + f Us (x,z,t)dz = —3 U,(3D — Dy ).-h,(3.27)Substituting (3.13), (3.18), (3.23) and (3.24) into the integral in (3.7), I have thebottom shear stress rb(x,t) in terms of n(x,t), U,(x,t), D(x,t), and Dy(x,t),an ,^aD)rb . _or pl x_ (A — po(a — ax )]L 1 ( dU „^aU,) 1 u aDy .— P2^3[(D — —DY ) ,t(--, + UP ax ) 3 P at(3.28)If I "rb I becomes smaller than the yield stress T y during the flow, the mudflow willstop on the slope. Equations (3.17), (3.25), (3.26) and (3.28) are the governing equationsfor the three unknowns Up(x,t), D(x,t), and Dy(x,t) for the mudflow with the reaction ofwater surface elevation on mudflow, the ari/ax term, being included.For a uniform mud layer on an underwater slope with no elevation at the water surface,the threshold for downward flow is Tb < Ty. Using the expression of the shear stress at thebottom, (3.7), I obtain the critical thickness at which the mud can stay stationary on aslope with an inclination of 8,D, — 2"y (3.29)(P2 - A )g sin 8Thus a uniform mud layer can remain stationary on a slope if its thickness, D, issmaller than D,. In contrast, a Newtonian fluid can achieve a state of static equilibriumonly when the bed is horizontal and the depth constant, because a viscous fluid can notresist shear stress in static state.120In the following analysis I shall use the same advantageous nondimensional variablesas I did in Chapter 2. I choose the initial maximum mud thickness, D 0, as the verticallength scale, [H]; the initial mud length, 4 as the horizontal length scale, [L]; and thehorizontal velocity scale as [U]=V[L])1/2. I adopt the following dimensionless variables:(x* , z* ,t* ) =([LI 1 x,[H]-1 z, 11,g' /[L]t),^(3.30a)(n * ,D s ,D y* D: „hs * ) = [li]-1 (n,D,Dy,Doh,k),^(3.30b)(U no tip^=[U]-1(UpoUp,U„W.,Ws),^(3.30c)where the variables with an asterisk are dimensionless and g' is the reduced gravity,defined asP2 - Pi g' =^g.P2(3.31)With (3.30a), (3.30b), and (3.30c), the governing equations for the slide (3.17), (3.25),and (3.26) take on the form, with the asterisks being omitted,at^P dx ^dx r -1 dx D - Dy sgn(Up ),^(3.32)2 dU 1 dp 2^dU^(^dD E an)-D^U —1--F -DU -2- = D a - E —3 Y dt 3 P at 3 Y P ax^Y^dx r — 1 dx )_ 2 (Up _ 1 a2U p )ER Dy 3 ax e f(3.33)aUp± u du p _ a _ e ar) _ e ark  aKdD dq _ ndt ± dx - ‘'' (3.34)121whereDo =^ 2P Do I gi K = , R -£ = - , r = — 74^Do The region of no initiation of slide is defined from (3.10) byaDe (970 > –a Dce–D ^ax r -1 dx )(3.35)I have assumed that the flow remains laminar during the slide. The flow regimedepends on the Reynolds numberRe = p2UDand on the parameter called the Hedstrom number2He =^p2 T D ,au 2(3.36)(3.37)which measures the relative importance of the yield stress and the viscosity. From Meiand Liu [1987] I have refereed that for Bingham mudflow in laboratory experiments thetransition from laminarity to turbulence takes place at the critical Reynolds number Re,which is related to related to He byRe, = 14.0He .", He > 104, (3.38a)Re, -a: 2100, He <104 . (3.38b)It can be seen that the critical Reynolds number increases with the Herdstrom number.Therefore the yield stress enhances the resistance of a Bingham-plastic flow againsttransfer from laminarity to turbulence. The stability of a Bingham flow increases with itsyield strength. I choose a the critical Reynolds number as 1000 for the case He<10 4 forturbulence to become dominant. I use equation (3.38a) to calculate the critical Reynolds122number for the case where He>104. Appropriate values of the viscosity and yield stress ofthe mud will keep the flow laminar. As to the effect of interfacial turbulence, similardiscussion can be done as in Chapter 2.3.3 The Viscous Model - a Special Case of the Bingham ModelComparing the constitutive relations of a viscous fluid and a Bingham fluid, it is foundthat they bear superficial resemblance to some extent. If one lets the yield stress be zero,the Bingham fluid model reduces to the viscous fluid model. Therefore one should beable to derive the solution of a viscous fluid model from the present solution of theBingham model. To get the solution of the viscous fluid model, let the yield stress be zeroand the thickness of the plug zone be zero, i.e., Dy(x,t)=D(x,t); one then can rewrite thegoverning equations of the slide (3.25) and (3.26) in the nondimensional form as follows:2 ()Up 1 dD 2^aUp^(^dD e an)-D— U —+-DUp = D a- e 7^/Ixr -1 dxd3 dt 3 P at 3^dx2 (Up 1 d2Up— eR D 3 dx2 IdD dqdt dx(3.39)(3.40)where Up(x,t) now represents the velocity at the mud surface, z=D(x,t)=Dy(x,t), and q(x,t)is given by- h(x,t)q(x,t) = J Upi (x,z,t)dz = -2 D(x,t)U p (x,t).- h, (z)(3.41)The governing equations of the mudflow (3.39), (3.40), and (3.41) are the same as123those I obtained from the viscous fluid model in Chapter 2. Thus the viscous slide modelcan be taken as a special case of the Bingham slide model.3.4 Governing Equations for the WavesFor waves on a gentle slope in shallow water, I adopt the long-wave approximation,i.e., the vertical length scale is much smaller than the horizontal length scale. Thus thewater motion is essentially in the x axis direction and the pressure distribution in thewater can be assumed hydrostatic. The depth-averaged dynamic equations for nonlinearshallow-water waves induced by the general motions of an impermeable seabed area(ndt h) + ax [u(n+ h)]= 0,du du an.—+u----+ g-13,at dx^dx(3.42)(3.43)where u(x,t) denotes the horizontal particle velocity of the water motion.Using the same length scales and velocity scale as those used for the mudslide in(3.30a), (3.30b), and (3.30c), and remembering h(x,t)=11,(x)-D(x,t), (3.42) and (3.43) indimensionless variables read, with the asterisks being omitted,an . dB, a^a^aat^- dTc(uhs) - Tx(u11)± (liD) ' (3.44)du_ ^di duat  -^- u—.r-1 ax^ax (3.45)For the coupling of a Bingham-plastic slide and the waves which it generates, there arefive unknowns, ri(x,t) and u(x,t) for the wave motion, and D(x,t), Dy(x,t), and Up(x,t) for124the slide, which are all independent of z. Correspondingly there are five governingequations (3.17), (3.25), (3.26), (3.42), and (3.43) and two constraints (3.8) and (3.9) tosolve them.3.5 One-way Coupling and Fully Coupled ModelsThere are many factors that affect the coupling of a mudslide and the waves which itgenerates. The important ones are the density, the viscosity and the yield stress of themud, the initiation depth of the mudflow, its initial shape and dimensions, and theinclination of the slope. The presence of the surface gradient term argax in (3.32), (3.33),and (3.45) indicates that the reaction of waves on the mudslide is inversely proportionalto (r-1), i.e., the nearer the density ratio approaches unity, the greater the reaction ofwaves on the mudslide. Mud with large viscosity flows more slowly, and its effect on thesurface of the water will be small. For a shallow initiation depth of the mudslide, the slideis felt more strongly by the surface water than for a deeper slide initiation depth, and theinteraction between the slide and the waves is also more significant. The parameter K andthe Reynolds number Re are two important parameters that indicate the effects of theyield stress and the viscosity, respectively, which will be discussed in the followingsections.First, I consider the one-way coupling between the mudslide and the waves, i.e., I onlyconsider the action of the mudslide on water and neglect the reaction of surface waves onthe mudflow: then the surface gradient term di1/dx in the governing equations of themudslide vanishes, and (3.32) and (3.33) read,dUtl±u ai, =a e dD aK sgn(U ),dt^' dx^dx D—Dy^' (3.46)1252 dU 1 dB, 2^dUp . D a _ c(^ap)._^p^p2 (U 1 d2U—Dy — —Up +—DyUp --3^at 3^at 3^ax^3'^dx ) eR Dy 3 ax e ) (3 '47)To calculate the mudslide and the waves, first I check the initiation conditions of theslide, equations (3.8) and (3.9), at the front face and at the rear face. Equations (3.8) or(3.9) must be satisfied for the slide to initiate. During the flow the bottom shear stress 'rbmust be examined from time to time to determine whether the flow will stop or continueto move. To calculate waves in the one-way coupling model, I solve (3.34), (3.46) and(3.47) to find D(x,t), D y(x,t), and Up(x,t) for the mudslide, then solve (3.44) and (3.45) tofind rgx,t) and u(x,t) for the waves.In the complete model, I consider the full coupling between the mudslide and thewaves which it generates, i.e., the mudslide generates surface waves, and at the same timesurface pressure gradients react on the mudflow. To calculate the mudflow and the wavesby the two-way coupling model I need to solve (3.32), (3.33), and (3.34) for the mudflow,and solve (3.44) and (3.45) for the surface waves simultaneously.3.6 Numerical Results and DiscussionsAssume that the mud is initially at rest and has a parabolic surface (see Figure 3.4),i.e.,D(x,0) = D0 11 — [2(x — I)/ Lo ] 2 ),^(3.48)where, I is the initial x coordinate of the center of the mud, I = (x 1 + x2 ) / 2 , x 1 and x2are the initial x coordinates of the rear and front margins of the mud; L o = x2 — xl , is theinitial length of the mud, and Do is the initial maximum mud thickness.126Equation (3.48) in dimensionless variables reads,D(x,0) =1— 4(x —x) 2 .^ (3.49)Given the slope angle and the typical parameters (the density, the viscosity and theyield stress) and the initial positions and dimensions of the mudflow, I calculated ii(x,t)and u(x,t) for the waves and D(x,t), Dy(x,t), and Up(x,t) for the mudslide by numericallysolving the governing equations with a finite-difference method, in which a forwarddifference in time and a backward difference in space are used in the discretizedequations. In the numerical calculations, the distance interval is chosen as 8x=0.05 andthe time-step is chosen as Ot=0.002 such that the maximum distance traveled by any partof the fluid remained less than one-fortieth of a cell length. Although the CFL conditionand the viscous diffusion stability condition [see Dent, 1982] were satisfied, I stillencountered instabilities in the numerical calculations. So I introduced two small artificialviscosities (of the for auad2u/dx2, with Pa the artificial viscosity; for the mudslide,i.ta=0.001-0.005; for the waves, jua=0.025) in the discretized equations, for the watermotion and for the mudslide to avoid numerical instabilities.LoFigure 3.4: Definition sketch of an initial parabolic mud surface.In the following numerical calculations, I consider a initial parabolic mud surface withthe same dimensions as I used in Section 2.8 for the parabolic profile, starting on a gentle127uniform slope with an inclination 9=4°. I employ the following typical parametersranges: the density ratio of mud and water r=1.2-2.0; the Reynolds number is kept in therange under 1000 so that the flows remain laminar. As I did in Chapter 2, I consider ariver (at x<O) with depth h1 =0.5D 0, and also a deep sea with depth /1 2.24.5Do . The initialposition of the mudflow is indicated by the parameter K.-Do/ha, where ho=h,(1).I presenthere the results for the cases where w=2/5, r=2.0, R=5000 (corresponding to a mudviscosity of p1p2 =0.278 m2/s), and K=0, 0.05, 0.1, 0.2, 0.3, 0.35, 0.5.3.6.1 Surface Variation of the SlidesThe successive positions of the mud surface for the slides with different yield stresses:K=0, 0.05, 0.2, 0.35 are shown in Figures 3.5, 3.6, 3.7, and 3.8, respectively. The solidlines represent the two-way coupling results, and the dotted lines are the one-waycoupling results. The vertical dimension (the depth of mud) is plotted on a scaleexaggerated by a factor of 70 over the horizontal scale.The results with K=0 are those calculated from the viscous fluid model. Due to theeffect of the yield stress, the Bingham-plastic slide can travel only a finite distance; inclear contrast with the flow pattern of a viscous flow, which can spread very far awayfrom its starting place until other forces, such as capillarity, balance the gravitationalforce. A Bingham-plastic flow will stop at its equilibrium thickness D, on a uniformslope when the shear stress exerted on the bottom of the flow becomes smaller than theyield stress. The rear part of the slide will stop first because the pressure force induced bythe slide surface acts against the driving gravitational force at the rear part of the slideand the bottom shear decreases more quickly below the yield stress. At the front face thepressure force works with the gravity, and the front part will stop later on the slope whenthe shear stress exerted on the slide bottom becomes smaller than the yield stress. If other128parameters are kept fixed, the slide with larger K (i.e., larger yield stress, or largerresistance against shear) flows much more slowly than that with smaller K. When Kapproaches zero the yield stress approaches zero, and the results approach that of theviscous fluid model. A slide on a steep slope will spread farther than one on a gradualslope, because the same yield stress can hold a thinner layer of mud in static equilibriumon a steep slope than on a gentle one.The calculated results also indicate that the interaction between the slide and thewaves is not significant for the calculated cases. The reaction of the waves on the slideslows down the slide only slightly. The one-way coupling and two-way coupling resultsbehave in a very similar manner. These findings are similar to those I already obtained inthe viscous fluid model. Therefore, I shall discuss only the two-way coupling results inthe following sections.The effect of the Bingham plasticity on the slide run-out distance can be seen moreclearly in Figures 3.9, 3.10, 3.11, and 3.12, which illustrate the successive profiles (two-way coupling results) of the mud surface for the slides with increasing yield stresses K =0.0, 0.05, 0.1, 0.2, 0.3, 0.35, 0.5, at different times t= 8, 10, 12, 14, respectively. In thecalculated cases, there was no uphill movement of the initial mud profile at the rear face.It can be seen that the yield stress of the mud has significantly retarded the spreadingspeed and run-out of the slide. The slide with K=0.5 almost stops moving at t=8-10, whilethe slides with smaller yield stresses still moves forward. Another distinct feature of aBingham plastic flow is that the frontal part of the slide tends to pile up due to the frontalresistance induced by the plasticity of mud, while the frontal piling up of a viscous fluidflow (K=0) is less significant.I 2 3 4 5 6t.0.01.00.80.60.40.20.001.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.8D 0 .60.40.20.00 1 2 3 4 5^6t=10.00 1 2 3 4 5^6....--t=1 2.00 1 2 3 4 5^6....---'‘t=14.00 1 2 3x4 5^6Figure 3.5: Successive shapes of the mud surface, for K=0 (viscous model).1290 1 20 1 21.00.80.60.40.20.00^1^23 4 5 6t.4.03 4 5 61.6.03 4 5 6t.8.01.00.80.60.40.20.01.00.80.60.40.20.0Figure 3.6: Successive shapes of the mud surface, for K4105.t-0.01 2 3 4 5 61 2 3 4 5 62 3 4 5 61.00.80.60.40.20.001.00.80.60.40.20.001.00.80.60.40.20.001.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.8D 0 .60.40.20.00 1 2 3 4 5^6t=10.00 1 2 3 4 5^6t=12.00 1 2 3 4 5^6t=14.00 1 2 3x4 5^61301.0 ^0.8 •0.60.4 •0.2 •0.0  0 1 2 3 4 5t=6.063 4 5 61=8.03 4 5 6t=10.03 4 5 6t=12.03 4 5 6t=14.03x 4 5 60^1^20^1^20^ 20^1^20^1^21.00.8D 0 .60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.0I1.00.80.60.40.20.0t=0.00^1^2^3^4^5^61.00.80.60.40.20.00^1^2^3^4^5^61.00.80.60.40.20.00^1^2^3^4^5^61.00.80.60.40.20.0131Figure 3.7: Successive shapes of the mud surface, for K 3.2.Figure 3.8: Successive shapes of the mud surface, for K.35.1 2 3 4 5 61 2 3 4 5 6t=0.01 2 3 4 5 61.00.80.60.4020.001.00.80.60.40.20.001.00.80.60.40.20.003 4 5 61=8.03 4 5 6t=10.03 4 5 61=12.03 4 5 61=14.03x4 5 61.00.80.60.40.20.00 1 21.00.80.60.40.20.00 1 21.00.80.60.40.20.00/1^20^1^20^1^21321.00.80.60.40.20.01.00.8D 0 .60.40.20.0n 1=2.0\•1331.21.00.8D 0.60.4020.00^1^2^3^4^5^6xFigure 3. 9: The positions of the mud surface for the slides with different yield stresses attime t=8.1.21.00.8D 0.60.4020.0 0^1^2^3^4^5^6xFigure 3. 10: The positions of the mud surface for the slides with different yield stressesat time t=10.1341.21.00.8D 0.60.4020.00^1^2^3^4^5^6xFigure 3. 11: The positions of the mud surface for the slides with different yield stressesat time t=12.1.21.00.8D 0.60.40.20.00^I^2^3^4^5^6xFigure 3. 12: The positions of the mud surface for the slides with different yield stressesat time t=14.1353.6.2 Distribution of the Plug Velocity of the SlideTo compare the velocity for a viscous slide and the Bingham slides, I present, inFigures 3.13, 3.14, 3.15, and 3.16, the variation of the plug velocity of the slide, Up(x,t),for the cases specified in Figures 3.5, 3.6, 3.7, and 3.8, respectively. The plug velocity ofa Bingham slide behaves similarly to the velocity of a viscous slide, it starts increasingfrom zero at t=0, reaches its maximum value at t=6-8, and then starts to decrease.The magnitude of the Bingham plug velocity decreases with the increase of the yieldstress. For a slide with a smaller yield stress as in Figure 3.14 (K=0.05) the resultapproaches that of the viscous fluid model as in Figure 3.13 (K=0). The maximumvelocity of the viscous slide is U=23.1 m/s which occurs at The maximum Reynoldsnumber for the viscous slide is Re=755.4. The maximum Reynolds numbers for theBingham slides will be smaller than that of the viscous flow due to the effect of Binghamplasticity and a smaller velocity. For the Bingham slide with a larger yield stress K=0.2 inFigure 3.15, the plug velocity reaches Up=10.9 m/s and the maximum Reynolds numberis Re=568.2. It is also seen in Figure 3.15 that the slide speed gradually decreases to zerofrom the rear margin and the no-motion point gradually moves forward. Only the frontalpart of the mud is moving at time t=12, 14; leaving a uniform layer of mud at itsequilibrium depth. This result is in excellent agreement with the slide profiles.For the slide with larger yield stress as in Figure 3.16 (K=0.35), the maximum plugvelocity reaches Up=6.3 m/s and the maximum Reynolds number is Re=373.5. It can beseen that the mobility of the mud is significantly reduced, and that the whole mud almoststops flowing at time t=14, with a very limited run-out distance. It is also found thatbecause of the parabolic flow profile equation (3.21) and the Bingham-plastic behavior ofmud, the mudflow exhibits amplitude dispersion: the largest velocities are found at thelarge depth of mud near the front of the mudflow.0.60.4t-0.00.20.00^1^2^3^4^5^61360 1 2 3 4 5^6t-4.00 1 2 3 4 5^6t-6.00 1 2 3 4 5^6t.8.00 1 2 3 4 5^6t.10.00 1 2 3 4 5^60.60.40.20.00.60.4020.00.60.4020.00.60.40.20.00.60.4020.00.60.40.20.00.60.4Up020.00^1^2^3^4^5^60^1^2^3^4^5^6xFigure 3.13: Transient variation of the slide particle velocity, for K=O (viscous model).1.2.01.12.02^3^4^5^60^11.14.0--/-\0.60.4.2 1-0.00.00^1^2^3^4^5^61370.60.40.20.00^I^2^3^4^5^61.4.00^I^2^3^4^5^61.6.00^1^2^3^4^5^61.8.00^1^2^3^4^5^6:_-______-,------"..--------\1.10.00^1^2^3^4^5^60.60.40.20.00.60.40.20.00.60.40.20.00.60.40.20.00.60.40.20.00.60.4Up020.00^1^2^3^4^5^6xFigure 3.14: Transient variation of the slide plug velocity, for K3.05.t.4.01.6.0t..8.0N10.00.60.4.2 1.0.00.00^I^2^3^4^5^61380.60.40.20.00.60.4020.0t-2.0----.'"-N.0^1^2^3^4^5^60^1^2^3^4^5^60^1^2^3^4^5^60^1^2^3^4^5^60^1^2^3^4^5^6N12.0- _ _.--/Th .0.60.40.20.00.60.40.20.00.60.40.20.00.60.4020.00^1^2^3^4^5^60.60.4Up 0.2^ 1.14.00.00^1^2^3^4^5^6xFigure 3.15: Transient variation of the slide plug velocity, for K4).2.0^1^2 ...--- ■0^1 2------ N0^1^2:-.--------\0^1^20 1 2:--.----......-.---0 1 20.60.40.20.00.60.40.20.00.60.40.20.00.60.40.20.00.60.4Up020.01-0.03 4 5 61.2.03 4 5 61.4.03 4 5 6tx.6.03 4 5 61.8.03 4 5 61-10.03 4 5 6",...1-12.03 4 5 61390.60.4020.00.60.4020.00.60.40.20.00^1^23x0^I^2 4^5^6Figure 3.16: Transient variation of the slide plug velocity, for K4.35.1403.6.3 Variation of the Slide Frontal SpeedTo examine the effect of the Bingham plasticity on the slide, I also calculated thefrontal velocities for mud with different yield stresses. Figure 3.17 illustrates thevariations of the frontal speed of the slide, Uf=dxj/dt, xf being the x coordinate of the frontmargin of the slide(defined as where D(x,t)=0.01), for K = 0.0, 0.05, 0.1, 0.2, 0.3, 0.35,0.5. It is seen that the slide quickly accelerates to its maximum speed at the initial state,continuing with a relatively steady speed for the cases with smaller yield stresses. Finally,the Bingham-plastic slides will stop at a finite run-out distance.The frontal speed is the largest for the viscous slide (K=0.0), with maximummagnitude of 0.285[U], corresponding to 16.5 m/s, and it decreases with the increase ofthe yield stress of mud. The frontal speed of a Bingham slide approaches its maximumvalue at time t=6-10 and then starts to decrease. Because of the yield stress, the Binghamslides travel more slowly than a viscous one, and the flow regimes are likely to remainlaminar. For the Bingham slide with K=0.2, the maximum Reynolds number is Re=605.3(Up=10.2 m/s, D=16.5 m), and the maximum frontal Froude number is Fr=1.13. The slidewith the largest yield stress, K=0.5, has the smallest frontal speed, with maximummagnitude of 0.07[U], corresponding to 4.1 m/s, and its front stops moving at about t=10.These results show excellent consistency with the results on the slide profiles and on thevariation of the slide plug velocity.3.6.4 The Thickness of the Yield LayerThe distinctive feature of a Bingham fluid flow is that there will be a plug zone, withinwhich there is no deformation, moving above the shear zone within which the shear stressexceeds the yield stress. To examine the Bingham effect on the extents of the two flowregions I calculated the thickness of the yield layer for the slides with different yield141stresses. I present, in Figures 3.18 and 3.19, the variation of the thickness of the yieldlayer (the dotted line) in comparison with the total depth of the mudflow (the solid line)for the cases K.05, 0.2, respectively. For the viscous fluid model, the flow is shear flowthroughout the mud layer. For a Bingham fluid flow, only the lower layer of the mud willyield and the upper part of the mud moves as a plug. For less plastic mud K.05 (Figure3.18), the yield stress is small and the shear stress in the mud exceeds the yield limiteasily. Thus the shear flow thickness is large (but always smaller than the total thicknessof mud) and its variation approaches that of the viscous flow. For more plastic mud(Figure 3.19), the mud does not easily yield, and the shear flow is confuted withinthe very lower part of the mud near the bottom. Some time after the initiation, there isshear flow only at the front part of the slide, the yield layer vanishes and the plug velocityreduces to zero at the rear part of the slide.0.40.3Uf 020.10 .00^5^1 0^15Time tFigure 3. 17: The variation of the frontal speeds of the slides with different yield stresses.A t3.00^1^2^3^4^5^60.80.40.00^1^2^3^4^5^60.80.40.00^1^2^3^4^5^6t=6.00^1^2^3^4^5^60.80.40.00^1^2^3^4^5^60.80.40.00^1^2^3^4^5^60.80.40.00^1^2^3^4^5^60.8DDy 0.40.0 t=14.00^1^2^3^4^5^6x1420.80.40.00.80.40.0Figure 3.18: The variation of the thickness of the yield layer, for K=0.05. The solid lineshows the slide surface and the dotted line shows the yield interface.001122334455661.00.80.60.40.20.01.00.80.60.4020.01.00.80.60.40.20.00 1 2 30^1 2 3/1.00.80.60.40.20.04 5 6t=8.04 5 6t=10.04 5 6t=12.04 5 6t=14.04 5 60^1^2^31.00.8D 0.6Dy 0.40.20.00^1^2^31.00.80.60.40.20.01.00.80.60.40.20.010^1^2^3x1.00.80.60.40.20.0 143n t=0.0Figure 3.19: The variation of the thickness of the yield layer, for K:12. The solid lineshows the slide surface and the dotted line shows the yield interface.1443.6.5 Elevations of the Surface WavesFigures 3.20, 3.21, 3.22, and 3.23 illustrate the transient evolution of the surfacewaves generated by the slides under the conditions specified in Figures 3.5, 3.6, 3.7 and3.8, respectively. Three main waves are produced: the first wave is a large crest whichpropagates into deeper water from the slide site; the second wave is a trough followingthe crest as a forced wave propagating with the speed of the slide front. The third wave isa small trough which propagates shoreward. Thus a recession of the sea at the shorelinewill occur first as a result of the mud slumping on the continental slope. Following thistrough, there is a shoreward crest, which will generate a wave runup at the shore.The evolution of the surface waves generated by Bingham-plastic slides (Figure 3.21-3.23) is similar to that of the waves generated by a viscous slide as in Figure 3.20. Thewave amplitude and the water particle velocity, however, are significantly affected by theyield stress of the mud. The smaller the yield stress of the mud, the greater the amplitudeof the surface waves. The slide with larger yield stress moves much more slowly andgenerates much smaller waves. The viscous slide generates the largest surface waves,with maximum amplitude of n=0.132D 0. The amplitudes of the waves decrease with theincrease of the yield stresses, and the Bingham-plastic slide with K=0.35 only produces awave with a maximum amplitude of 11=0.07D 0 .It is found that three major parameters dominate the interaction between a Bingham-plastic slide and the waves: the Bingham yield stress, the density of mud and the depth ofinitiation of the mudslide. A slide with smaller Bingham yield stress, a larger density anda shallower initiation water depth will generate larger waves.145xFigure 3.20: Evolution of surface waves, for K=0 (viscous model).0.200.100.0-0.10-0.200.200.100.0-0.10-0200.200.100.0-0.10-0.20-100.200.100.0-0.10-0.20-100.200.100.0-0.10-0.20-100.200.100.0-0.10-0.20-100.200.100.0-0.10-0.20-100.200.1077 0.0-0.10-0.20-10t.0.0-10-10-5-5-5-5-5-5-5-5000000005555555510101010101010101515151515151515-10-10t-10t-0.0-5 0 5 10 151=2.0-5 0 5 10 15\7't=4.0-5 0 5 10 150.200.100.0-0.10-0.200.200.100.0-0.10-0.200.200.100.0-0.10-0.200.200.1077^0.0-0.10-0.20xFigure 3.21: Evolution of surface waves, for K41.05.i0.200.100.0-0.10-0.201460.200.100.0-0.10-0.200.200.100.0-0.10-0200.200.100.0-0.10-0.20-10-10-107  0.0-0.10-0.20t=6.0-5 0 5 10 15t=8.0-5 0 5 10 15t=10.0-5 0 5 10 15t=0.0-5 0 5 10 15t=2.0-5 0 5 10 15t=4.0-5 0 5 10 150.200.100.0-0.10-0.200.200.100.0-0.10-0.200.200.100.0-0.10-0.200.200.100.0-0.10-0.200.200.100.0-0.10-0.200.200.100.0-0.10-0.20-10-10-100.200.100.0-0.10-0.200.200.10xFigure 3.22: Evolution of surface waves, for K3.2.147-10 -5 0 5 10 150.150.05-0.05-0.15-10 -5 0 5 10 15-10 -5 0 5 10 15-10 -5 0 5 10 15-10 -5 0 5 10 150.150.05-0.05-0.150.150.05-0.05-0.150.150.05-0.05-0.150.1577 0.05-0.05-0.15-10 -5 0^5 10 151=12.0-=----'..•*--.,........„,------10 -5 0^5 1 0 151=14.0-10 -5 0^5 10 15xFigure 3.23: Evolution of surface waves, for K.35.1480.150.05-0.05-0.150.150.05-0.05-0.150.150.05-0.05-0.15t-0.0^\+-^1-4.01=6.01=8.01493.6.6 Distribution of the Water Particle VelocityFigures 3.24 and 3.25 show the transient variation of the horizontal velocity of thewater particles for the cases K=0.0, 0.2, respectively. Similar to the results obtained inChapter 2, the water particle velocity is one-order of magnitude smaller than the slideparticle velocity. The water ahead of the front face of the slide moves to deep water.Water just above the slide flows shoreward, and water near the rear face of the slidemoves to deep water.The distribution of water particle velocity induced by a Bingham slide behavessimilarly to that of a viscous slide. The magnitude of the velocity, however, issignificantly reduced due to the reduced mobility of the slide. As shown in Figure 3.24and 3.25, the shoreward velocity and seaward velocity of the water induced by a Binghamslide (K=0.2) are smaller than those induced by a viscous slide (K=0.0).0.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.03-10 -5 0 5 10 15-10 -5 0 5 10 15t.4.0150••t-0.0t-2.0•-10^-5^0^5^10^150.030.01-0.01-0.03-10^-5^0^5^10^150.030.01-0.01-0.03-10^-5^0^5^10^150.030.01-0.01-0.03-10^-5^0^5^10^150.030.01-0.01-0.03-10^-5^0^5^10^15u0.030.01-0.01-0.03-10^-5^0^5^10^15xFigure 3.24: The variation of the water particle velocity, for K4:0 (viscous model).-10 -50 50 50 50 50 5••u0.030.01-0.01-0.030 5 10 15t-.14.0-10^-5 0 5 10 150.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.030.030.01-0.01-0.03-10-10-10-10-10-5-5-5-5-510101010101515151515x0.030.01 t t=0.0-0.01-0.03 I0.03-10 -5 0 5 10^150.01 h=2.0-0.01 4c7°.-0.03151Figure 3.25: The variation of the water particle velocity, for K,o.2.1523.6.7 Energy Transfer from the Slide to the WavesTo examine the Bingham effect on the energy transfer from the slides to the waves, Icalculated the ratio of energy transfer from the slides of different yield stresses to thesurface waves, with equations (2.25), (2.26), and (2.27). The variations of the energytransfer ratios with nondimensional time t are illustrated in Figure 3.26, for four cases,K4, 0.1, 0.2, 0.3.Again it is found that the energy transfer ratio is not constant during the slide. It iszero at t4, and increases to a maximum value at then starts decreasing as the slidestops moving on the slope. The energy exchange is the most significant for the case of aviscous slide because it has the largest mobility amongst the Bingham slides. The energytransfer from the slide to the waves becomes less significant for slides with the increasingyield stresses and decreasing mobility.00o000^5^10^15^20Time tFigure 3.26: The variation of the energy transfer ratio with time for the slides withdifferent yield stresses, K4, 0.1, 0.2, 0.3.1533.7 Comparison with a Snow Flow TestUntil now, there has been no observational data on submarine landslides and thesurface waves which they generate. Laboratory experiments on turbidity currents (whichoccur when the density difference is small) can hardly be used to verify the presentmudslide model, because the turbulent mixing becomes important at the interface, and thedensity decreases as the slide travels due to the interfacial mixing. Here, I shall use theBingham-plastic fluid model to model the snow flow test of Dent's experiment [1980,1982].In the test the snow, with a density of 300 kg,/m3 and volume of 2.7 m 3 , descended a30° slope in an almost friction-free polyethylene-lined, 60-meter-long track, and exitedonto a 28-meter-long, flat run-out area of packed snow. The measurements of the snowfront velocity and the final snow depth in the deceleration phase constituted the snowflow test. The snow stopped after running about 26 m in 2 seconds. The initial inflowconditions for the snow profile and the flow velocity were determined from a film takenof the test. The initial velocity of the snow inflow was taken as 17.0 m/s. The test resultsindicated a high shear zone near the bottom and a low shear zone in the upper part of theflow within which there is little deformation, which indicates that a Bingham fluid modelwill properly describe the snow flow.For the flow of a mass of snow with a initial velocity on a flat plane (0=0), I have thefollowing governing equations^dU7^dU^dD Tysgn(U p )^P2 at^dx(--1-U pH= --P2g dx (D — Dy)(3.50)_dU^apax= 2 g^P2(-23 DY ddU: UpP y + pddD^ar ^p - ()X D ti 2Y^76_14y 2 da2 xU2p ), (3.50154aD aqat dxand the condition of no motion, I ;1 < Ty, oraD[^dt1 (aUp^.9(4)ax^31 aDyTy> p2 gDax+Dy )^ +Up^Up dt..' > "Tye3( -- (3.52)(3.53)For snow flow on a flat plane, I choose the initial maximum snow thickness, D., as thevertical length scale, [II]; the initial mud length, L., as the horizontal length scale, [L];(gL0) 1t2 as the horizontal velocity scale [U]; and (Lo/g) 1 /2 as the time scale. The governingequations (3.50)-(3.53) in nondimensional variables read,^(Alp +U aUp^D,dx D-Dy sgnwp),P^=at^a  (3.54)2 dU 1 dD 2^dUP^aD 2 (Up 1 a2Up-D^U3 Y at 3 P at 3 DY P dx Y dx ER Dy 3 ax e ) (3.55)dD aq+—= 0,dt dxwhererDh =  Y^R- p2D.VgL.E = DoTv,P2gDo 1-1^•The snow will stop moving ifDh aU U _DhD[E dD ±(i_lplaUax^3 Dpat^ax ) 3DdDy d >at^D(3.56)(3.57)The numerical modeling then commenced with the snow flowing off the frictionlessslope onto a flat surface of packed snow employing a no-slip boundary condition. Thecomputational area was divided into 140 cells of 0.2 m length each. The maximum initial155snow height is 0.25 m. The time interval and the distance interval for the numericalcalculation were chosen as St=0.001 and &=0.025. An artificial viscosity of p a=0.008was used to avoid numerical instabilities. The two modeling parameters of Bingham fluidflow, Ty and ,u had to be adjusted in order to satisfy the flow criteria obtained from thesnow test. The terminal snow profiles were obtained from the computation when the plugvelocity became zero.The rheogram of the test snow was not measured in Dent's experiment and Dent usedpresumed parameters according to some experimental studies in his numerical modeling.According to Nishimura's [1989] experimental results, the kinematic viscosity v=p/p2 offluidized snow with density p2=200-400 kg/m3 varies from 10-s to 10-4 m2/s, but the yieldstress was not available. Here I have used two sets of parameters of Ty and p , (a) Ty =660N/m2, p2 =300 kg/m 3, p/p2=0.0001 m2/s, (b) ry =330 N/m2 , p2 =300 kg/m 3, 1-up2 =0.001m2/s. This is to show that different combination of the yield stress and viscosity may givealmost identical results. It is also realized that without enough field data of the parametersof snow flows it is hard to use this model to predict the snow flow dynamics. It shouldalso be mentioned that this Bingham fluid model can not be used for powder snowavalanche, because turbulence plays a large role in the flow dynamics of powder snow.Figure 3.27 and 3.28 shows the initial snow profile, the final profile of the snow flowand the predictions from the Bingham fluid model with the above two sets of parameters.It was found that these two combinations of ry and pu produced almost the same run-outdistance which is close to the experimental data (the snow stops at a run-out of 26 m in 2sec), and that the snow flow stops at approximately t=2.1 sec (with U(x,t)---0), which isvery close to the experimental result, t=2.0 sec. Considering that (1) the configuration andthe velocity of the snow flow at the entrance of the flat run-out area could not beaccurately evaluated from the film taken of the test, and (2) that the rheological propertiesof the snow were not measured so that the parameters had to be assumed according to156previous experiments, the general features of the snow deceleration are well predicted bythe numerical solutions. To examine the flow regime in a Bingham plastic flow, one hasto first calculate the Hedstrom number using (3.37), and then calculate the criticalReynolds number using (3.38). For the parameter set (a), He=1.375x107, the criticalReynolds number is then calculated as Re c=14He°33=8.5x104 ; and for the parameter set(b) He=6.87x104, and Rec=5127. The Reynolds numbers (the maximum values at t=0) forthe two cases are Re max=42500 (for (a)), and Remax=4250 (for (b)), respectively. Thesenumbers are smaller than the critical values, it is expected turbulence does not develop.Figure 3.29 and 3.30 show the variation of the snow flow speed with time. The speedof the rear part of the flow becomes zero first and the rear part stops first, which agreeswith Dent's numerical and experimental results [Dent, 1980]. Dent simulated thedeceleration of snow flow on a horizontal plane with a two-viscosity fluid model:"The terminal debris velocity profile were found from the computermodel by freezing the flow when the velocity became zero. It was foundthat the rear portions of the flow stopped first, while the leading edgecontinued to move, which was also observed in the snow tests".Unfortunately, Dent [1980,1982] has not given any measurement of the velocityvariation with time but only the final snow profiles. The reason that the rear part stopsfirst is the pressure force due to the slope of the snow surface acts against the inertialforce (backward) at the rear part, and at the front face the pressure force works with theinertial force (forward). So the bottom shear stress decreases more quickly at the rear partthan at the frontal part, which leads to first stopping at the rear. To use the presentBingham-plastic fluid model for snow avalanche, one needs more information on theTheological properties of snow, and also more careful investigation on the boundaryconditions on the snow flow path.157 (a)t=0^ t=2.1s0^2^4^6^8 10 12 14 16 18 20 22 24 26 28 302.01.5D 1.00.50.0x (m)Figure 3.27: The initial snow profile(t=0), the final snow profile (dots) and thetheoretical predictions (solid lines) of the final snow profiles with parameters (a)zy =660N/m2, p/p2=0.0001 m2/s.(b)t=0^ t=2.1s2.01.5D 1.00.50.00^2^4^6^8 10 12 14 16 18 20 22 24 26 28 30x (m)Figure 3.28: The initial snow profile(9), the final snow profile (dots) and the theoreticalpredictions (solid line) of the final snow profiles with parameters (b)s; =330 N/m 2 ,Wp24).001 m2/s.1612840t=0.0s0 2 4 6 8 10 12^14 16 18 20 22 24^26 281612 1=0.4s8400 2 4 6 8 10 12^14 16 18 20 22 24^26 2886 t=0.8s4200 2 4 6 8 10 12^14 16 18 20 22 24^26 284.03.0 t=1.2s2.01.00.00 2 4 6 8 10 12^14 16 18 20 22 24^26 284.03.0 t=1.6s2.01.00.00 2 4 6 8 10 12^14 16 18 20 22 24^26 281.00.8 t=2.00.60.40.20.00 2 4 6 8 10 12^14 16 18 20 22 24^26 281.0(7) 0.8 t=2.20.6o. 0.40.20.0 .0 2 4 6 8 10 12^14^16 18 20 22 24^26 28x (m)158Figure 3.29: Variation of the flow speed with time, for snow flow with parameters (a).ry::•60N/m2, plp24).0001 m2/s.1612840864204.03.02.01.00.04.03.02.01.00.01.00.80.60.40.20.01.0.7; 0.80.6o. 0.40.20.0t=1.2st=1.6s-7\20 22 24 26 28t=0.8s20 22 24 26 2820 22 24 26 2820 22 24 26 2816128400 2 4 6 8 10 12 14 16 18 20 22 24 26 28159t=0.0s0 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12^14^16 18 20 22 24^26 28t=2.20 2 4 6 8 10 12^14^16 18 20 22 24^26 28x (m)Figure 3.30: Variation of the flow speed with time, for snow flow with parameters (b)s;=330 N/m2, ///p2=0.001 m2/s.1603.8 Summary and ConclusionsA formulation is presented for the dynamics of a submarine Bingham-plastic mudslidecoupled with the surface waves which it generates, flowing on a gentle uniform slope inshallow water. The long wave approximation is adopted for both water waves and themudslide. The resulting differential equations for the Bingham slide and the waves aresolved by the same finite-difference method. The numerical results for the Binghamplastic slides with various yield stresses indicate that the plasticity of the mud willsignificantly reduce the mobility of the slide and the magnitude of the surface wavesgenerated. It is also verified that the solution of the viscous fluid model can be derived asa special case of the Bingham-plastic solution.A Bingham plastic slide can travel only a finite distance; in clear contrast with theflow pattern of a viscous flow, which can spread very far away from its starting place. Itwill stop at its equilibrium thickness D c on the slope when the shear stress exerted on thebottom of the flow becomes smaller than the yield stress. The slide with a larger yieldstress flows much more slowly than that with a smaller one.The evolution of the surface waves generated by Bingham-plastic slides is similar tothat of the waves generated by a viscous slide. The water current speed and the waveamplitude, however, are much smaller. The smaller the yield stress of the mud, thegreater the surface waves it generates. The slide with a larger yield stress moves muchmore slowly and generates much smaller waves. The major parameters that dominate theinteraction between a Bingham-plastic slide and the waves are: the Bingham yield stress,the density of mud and the depth of initiation of the mudslide.The energy transfer from the Bingham slides to surface waves varies with time and theBingham yield stresses of the slides. It has a maximum value at time t--4-8, and it161decreases to approximately 1-3% at time t=20. The magnitude of the energy transfer ratiois larger for smaller yield stress, and it is the largest for a viscous slide.Finally, I present a comparison of the solution of the Bingham-plastic mudslide modelwith a snow flow test. The predictions of the final snow profile by the Bingham fluidmodel are compared favorably with the experimental result. To use the Bingham-plasticfluid model for snow avalanche, however, one needs more information on the Theologicalproperties of snow flow, and also on the boundary conditions on the snow flow path.162Chapter 4A Frictional Material Model and Wave Generation4.1 IntroductionIn this chapter I shall study surface waves generated by a submarine slide of granularmaterial starting from rest on a gentle uniform slope. There have been many studies onwhen and how the sediment liquefies and starts flowing - the initiation of sediment flow,however, there are few investigations on the post-failure behavior of the flow [Middletonand Hampton, 1976; Locat, et al., 1990]. It has generally been held that one of the maincauses of slumping and sliding is earthquakes, the best known example being the GrandBanks earthquake, slump and turbidity current of 1929. Not all slumps and slides aretriggered by earthquakes. In many cases slumping is probably due mainly to rapiddeposition of sediment on a slope, leading to underconsolidation of muds, formation oflarge pore water pressure, and consequent low shear strength. Other reason for theinitiation of such a slide may be the liquefaction of submarine sediment which usuallyresults from cyclic loading of waves and tides.First I briefly discuss, following Middleton and Hampton [1976], the initiation ofsediment flow due to liquefaction. Sediments subject to liquefaction are sands that areloosely packed. If a sand is in very dense condition, the only way that shearing can occuris for grains to move apart. Thus densely packed sands resist liquefaction because it isnecessary to expand the soil skeleton (increase the porosity, - the ratio of the volume ofvoid and the total volume) in order to shear the sand. If a sand is in very loose condition,the grains tend to move together when they are sheared. So loosely packed sands arefavorable to breakdown of the fabric. Once the fabric is destroyed, the grains no longer163form a rigid framework to the deposit, but must be supported at least in part by the porewater: consequently porewater pressures rise much above normal hydrostatic pressure(excess pore pressure). The ultimate porewater pressure, pu , is reached when theporewater pressure becomes equal to the effective overburden pressure exerted on thegrains. At this point the sands are said to be liquefied. So long as the grains are supportedby the pore water, the sand has little shear strength and behaves like a fluid that can flowrapidly down a relatively gentle slope (1 0-10°). However, because of the dilation effect, assoon as deformation or sediment flow begins, the porewater pressure drops and porewateris lost upward [see Luternauer and Finn, 1983]. The sediment will regain some of itsstrength and the sediment flow can continue only if the active driving force exceeds theresidual strength of the liquefied sediment.Unfortunately there is little known about the porewater expulsion and the variation ofthe excess porewater pressure during the whole slide event. Norem and Locat[1989]assumed a constant porewater pressure, p„, 60% higher than the hydrostatic pressure forthe whole process of sediment flow and applied their model to the Sokkelvik submarineslide[Braend, 1961]. Other studies show the value of the excess porewater pressure willprobably depend on both the shear rates and the particle size or permeability of theflowing sediment [Sassa, 1988; Hutchinson, 1986]. It can be seen that the excessporewater pressure establishment for the whole slide process is not well understood.To study underwater granular flow I would rather simplify the complicated problem oftwo-phase motion of a soil-water mixture and ignore the excess porewater pressure (butusing properly chosen parameters for the friction coefficients) than to introduce anarbitrary law for it for the whole slide process. I shall adopt the frictional granular flowmodel of Savage and Hutter [1989], and derive the governing equation of the frictionalflow with the surface water waves included. The focus of this chapter will be on theeffects of a frictional bottom stress on the slide dynamics and the surface wave164generation. The consequence of the omission of the excess porewater pressure will beexamined.In the following analysis the submarine granular material will be treated as a frictionalCoulomb-like incompressible continuum with a Coulomb basal friction law, whichrelates the bottom shear resistance to the normal stress by a friction angle between thesoil and the slope. This bottom friction law can well describe the shear stress at thebottom of a sand slide, a rock slide and a submarine slide consisting of loose sandysediment [Savage and Hurter, 1989; Norem and Locat , 1989].A cohesionless frictional slide will be considered to investigate the effects of the basalCoulomb friction upon the coupling between the slide and the waves. The slide is longand shallow and the underwater slope is gentle, so the long-wave approximation can beemployed for both the slide and the waves. The depth-averaged nonlinear equations ofmomentum and mass conservation will be derived for the frictional slide. Numericalsolutions of the governing equations for an initial parabolic slide profile are sought,which show the surface evolution of the slide and the waves, the distributions ofhorizontal velocities of the slide and the water motion. The interactions between the slideand the waves will also be discussed.4.2 Governing Equations of A Cohesionless Frictional Submarine SlideI consider a finite mass of granular material sliding down an underwater rigidimpermeable slope inclined at a small angle 9 with respect to the horizontal (see Figure4.1). I use the same coordinate system and notations as those in Chapters 2 and 3. Iassume that both the water and the slide material are an incompressible homogeneouscontinuum. Then the equations of mass and momentum conservation in the horizontaland vertical directions can be written as:aU dlY Adx + dz = ‘-''dU dU^dU^.^do dr1,2(—dt +U—dx+W Tz ) = (P2 — P1).g sin co — 'dx^dz 'do-zz= -P2g,where U and W are the velocities of the slide in direction of the x axis and z axis,respectively; p i and p2 are the densities of the water and of the slide; axx, a and 2zz arethe normal and shear stresses acting on a plane soil element as shown in Figure 4.2. Notethat I used the long wave approximation, so the vertical momentum balance (4.3) ishydrostatic.dz(4.3)0jriver h,// //////Figure 4.1: Definition sketch of a frictional slide and the waves which it generates.The kinematic boundary conditions at the slide surface and the slide bottom are: (1) theslide surface is a material surface, and (2) the normal velocity on the slope is zero, i.e.,166—dh +U—dh +W = 0,^at z = –h(x,t),^(4.4)dt^dxdhsU^+ W = 0,^at z = –k(x).^(4.5)dxt z^oZ +^z dzxdzZxz"rxz dz dd'rxz zdux(Tx x^dx dxd'txzz +^dxdxFigure 4.2: Stresses on a plane soil element.It is advantageous to non-dimensionalize the equations. Thus I choose the initialmaximum mud thickness, D o, as the vertical length scale, [H]; the initial mud length, Lo,as the horizontal length scale, [L]; the horizontal velocity scale [1/]=-(gEo) 42; the verticalvelocity scale as [I/]=[t/DA° and the stress scale [o]=p2gDocos0. I adopt the followingdimensionless variables:(x * , z * , ,D * , h* , k * ) = {[Lo ]- ' x,[Do ] -1 (z,D,h,k)},(U* ,W * ,t* ) = ff-ar0 1 -1 U^W ,[-Ao I g' ] -1,(Y z ,Txz* )= [p2g' Do cos Or (oc , o-zz , cot erxz ),(4.6a)(4.6b)(4.6c)where the variables with an asterisk are dimensionless; the quantities in square brackets167are typical values for the variables and g' represents the reduced gravity, defined as1 P2 - P1 g = ^g.P2(4.7)With (4.6a), (4.6b) and (4.6c), equations (4.1)-(4.3) take on the form(?U dWax n+ az = s''Ud^a^a^ar+u u +w u i. sn 0(1– =z--) E cos e a - ,at^a xaz az^axao-zz - ^raz^(r –1)cos 0'in which E=Do/Lo; the asterisks have been omitted for brevity, and subsequently this willalways be done.Integrating (4.10) and using boundary condition: azz.(n+h)1(r-1) at z=-h, I haveazz =--- (17 + h) —r(z + h),r –1^r –1– hs (x) < z < –h(x,t), (4.11)or in dimensional formCT zz = pig(n A- h)cos 0 – p2g(z + h)cos 0,^– hs (x) < z < –h(x,t).^(4.12)Melosh [1986] has reviewed field data from very large landslides and concluded thatthe evidence suggested sliding on a thin basal layer. The basal zone is the active zonewhere the shear rates are high and nearly all the shear takes place. Thus the velocityprofile for these slides is expected to be quite abrupt and the depth-averaged velocityprofile is quite close to the actual velocity everywhere except at the very base. Therefore168I shall be satisfied with the results obtained by using the von Karman momentum integralmethod. Integrating (4.9) with respect to z from z= -hs(x) to z= -h(x,t), and making use ofthe continuity equation and of Leibnitz's rule while interchanging differentiations andintegrations, I havea -fUCI a I 2Z ^U dz + U + ah +w)] +[u(u'±-, +qat^ax at ax^ax-h, -h, z=-h^z.-h,[= sin 61D — 2- zz l z=_CF" r xz I z=_h, j — E COS e —a -1 axxdz^+ a xxax _h,z^_ a z.ah^I^ak^(4.13)aX^xx"' ax * I now define the depth averages of the variables as follows-^ -UD = f h Udz,^U2 D = $ h U2dz a 02D,-h, -h,(4.14a)a zzD = _h  a zzdz,fh^- h adz.^(4.14b)o-xxD = h, xxAs I did in the previous chapters, the shear stress on the slide-water interface isassumed to be zero. For a Coulomb frictional material, the bed shear stress can be relatedto the normal stress at the bed by a constant basal friction angle 8, as1-xz I z=-h, = —sgn(U)D cot 9 tan 8,^(4.15)or in dimensional formr I^ = —sgn(U)(p2 — p,)gD cos 0 tan 8.^(4.16)Substituting the kinematic boundary conditions (4.4) and (4.5) and (4.14)-(4.15) into(4.13), the terms in the square brackets in the left-hand part of the equation vanish, and Iobtain the governing equation in terms of the depth-averaged variables,a —at^ax--(DU)+ a—(DU2)[a= D sin 0 — sgn(U)D cos° tan 8 — ecos 0 --(D.) + a.ax er169ah^1^ah— a Hz(4.17)ax^' axThe stresses at a point in the soil can be described by a Mohr circle. Each point on theMohr circle represents a stress state at a plane in the soil cell. The Coulomb failure theorystates that failure will occur along that plane for which the ratio of shear stress to normalstress reaches a critical limiting value [see Perloff and Baron, 1976], i.e.,1'f = of tan 0.where TT is the shear stress on the failure plane at failure, af is the normal stress on thefailure plane, and is the angle of shear resistance (the slope of failure line). The shearstress on the failure plane at failure , I-1 is called shear strength. Failure is said to occurwhen the shear stress on the potential failure plane equals the shear strength.The development of stress conditions during the conduct of a direct shear test is shownin Figure 4.3. In a direct shear test of a soil sample, a normal force, P, and a shear force,T, are applied on a soil specimen. The Mohr circle corresponding to a small shear stress isshown as the smallest circle in Figure 4.3. Under these circumstances the soil is in stablecondition. If the shear force T is increased (the shear stress is increased), the Mohr circlewill become larger. This is shown as the second circle which also represents a stablecondition. If the shear force T is increased sufficiently, the Mohr circle will just touch thefailure line (with a slope angle of 0). The state of stress at this point of tangencyrepresents the limiting condition: the stress on the failure plane equals to the strength. Atthat point, failure occurs.tf oc<0T^failure line aqfailure circleP-normal forceT-shear forceA direct shear test170Figure 4.3: Relation between Mohr circles and Mohr-Coulomb failure criterion.The slide is assumed to behave as a cohesionless Mohr-Coulomb type material, so thatthe yielding will occur at a point on a plane element when the Mohr circle touches thefailure line (with slope angle of 0) as shown in Figure 4.4 (it has been found that there isan error in Figure 3 of Savage and Hutter [1989], the line with larger inclination shouldbe the failure line of the material, and the lower line should be the failure line for bottommotion; 0 and S were inverted in Savage and Hutter), i.e.,(N( = S tan 0, (4.18)where S and N are the shear and normal stresses acting on the element, respectively, 0 isa constant internal friction angle. At the state of yielding the normal stresses on thebottom can be found by the interception of the bottom failure line with the Mohr circle.There are two kinds of stress states at the bottom: the active and passive states dependingon whether an element of material is being stretched or being compressed. The twobottom stress states and the Mohr-Coulomb yield criterion are shown in Figure 4.4. Thusshearstress failure line0xx ' t xz )passive'txz )activenormalstress^si....._171the bottom stresses parallel and normal to the slope, az. and an can be related through anearth pressure coefficient, k, [ see Savage and Hurter, 1989]a. = kart ,^ (4.19)where k can be found out, after some geometry work, from Figure 4.4,{2[1 — (1 — (1 + tan 2 8)cos2 CI cos2 0-1, for ar1 I ax 0,k=2[1+ (1— (1+ tan 2 8)cos 2 CI cos 2 0-1, for ari1 ax 5. 0, (4.20)in which a ii/ax?_0 and ar//ax5.0 correspond to the active and passive states of stressrespectively.Figure 4.4: Mohr diagram showing the Coulomb yield criterion andstress states [Savage and Hutter, 1989, an error is corrected -0 and 8 inverted].Integrating the continuity equation (4.1) over the depth and using the kinematicboundary conditions (4.4) and (4.5), I obtain the equation of mass conservation, withoverbars omitted,overbars omitted,172L.D + —a (DU) = O.at ax (4.21)Substituting (4.14), (4.15) and (4.17) into (4.19) and neglecting terms of higher orders0(e2), I have the governing equations for the slide, with the overbars omitted forsimplicity,au + U jj- = sin 0 – sgn(U) cos 0 tan 8 – ek cos 0[-1 an aD^(4.22)at^ax r – 1 ax + ax] .From the surface pressure term in (4.22) (the first term in the bracket) I can see that thereaction of the surface gradient on the slide is great for small slide density (r>1), and it issmall for a denser flow which has greater inertia.Equation (4.22) in dimensional form readsau aUP2^ + U Tai--). (132 – pi )g sin 0 – sgn(U)(p2 – pi )g cos 0 tan 8– k cos e[pig —an + (p2 – pi)g —aD 1ax^ax (4.23)Now I analyze the physical meanings of the terms in the right-hand side of equation(4.23). The first term comes from the reduced gravity acceleration, the second term fromthe basal friction resistance and the terms in the square brackets are the forces due to thevariations of the free surface and the slide surface.Comparing (4.23) with the governing equation of the viscous slide (2.6), I find that theforces from the water surface and slide surface gradients are almost the same for the twokinds of slides except that there is a factor k for a frictional slide. I will note later thatk--1 for most submarine slide material. The basal friction resistance for a laminar viscous173slide (the last term of the right-hand side of (2.6)) is inversely proportional to the localReynolds number and proportional to the square of the velocity at the slide surface, whichmay be expressed by a laminar friction coefficient f (function of local Reynolds number,Re) as follows,where2,uD U^21.1) I viscous^ = fP2U2f = Re Re = p2UD(4.24)(4.25)Thus the basal friction for a viscous flow will increase quickly with the increase of thevelocity, while the basal shear for the cohesionless frictional slide (equation (4.16)) isproportional to the soil weight acting on the bed, and it has no direct relation with theslide velocity. The basal friction laws are the major difference between the viscous fluidmodel and a frictional model, which will change the flow patterns of these slides.Now I shall discuss a method to compensate the omission of the excess porewaterpressure by using a pseudo-friction angle or apparent friction angle (which is smaller thanthe real friction angle) for the sliding material. As discussed in section 4.1, failure due toliquefaction in a soil specimen occurs when the shear stress exerted on the specimenbecomes greater than its shear strength. The shear strength of the soil is reduced when thepore water carries part of the burden on the soil skeleton. Because the excess porepressure is unknown (very hard to measure), it is impossible to evaluate the decrease ofthe effective stress on the soil skeleton. It is possible, however, to evaluate the shearstrength by using the original effective stress and a smaller friction angle-which is calledthe pseudo-friction angle or the apparent friction angle. This pseudo-friction angle maydu^er an au-- = -at^r -1 dx - u—ax(4.32)174be back-calculated from field observation or experiment [Henkel, 1970]. Thus I shall usean apparent friction angle for submarine granular slides in the following calculations in acompensation for the omission of the excess pore-water pressure.43 Governing Equations of Surface WavesAs in the previous chapters, I employ the long-wave approximation, which means thehorizontal length scale is much larger than the vertical length scale, and the water motionis almost one dimensional. I have the following dynamic equations for nonlinear shallow-water waves induced by the general seabed motions asd(n d + h) + [u(ri + h)]= 0,t^dx^du du^,itH-u±g-11= 0^dx^ax^'(4.29)(4.30)where u(x,t) is the horizontal particle velocity of the water motion.I adopt the same length scales, velocity scale, and time scale as those used for theslide; (4.29) and (4.30) in dimensionless variables read, with the asterisks being omitted,an aD –a--_- (uhs) - —a (un)+ —d (uD),at at dx^ax^ax(4.31)4.4 Numerical Results and Discussions175In the frictional material model the main parameter affecting the slide dynamics is thefriction angle. Edgers and Karlsrud [1982] reviewed several submarine landslides andfound that the average slope angle varies between 0.3° and 5.7°. This angle reflects themaximum apparent friction angle of the slide, if the frictional model applies to the slides.This angle also includes the effect of the excess porewater pressure on the dynamics ofthe flowing masses. Henkel [1970] calculated the sediment properties according to theobserved slide information at the Mississippi Delta and suggested an apparent frictionangle of 2.25°. Laboratory experiments [Savage and Hutter, 1989] indicate that theinternal friction angle is close to the basal friction angle. Thus in the following numericalcalculations, I consider gentle slopes with angles 0=1%10° and the material with thefriction angles of the same order as the apparent friction of submarine sediment flows: 80 =1%10°. For the flow to start the bottom friction angle must be smaller than the slopeangle. With 85.010° I find, from equation (4.20), that k --.1, so o- „—o-zz . Thus I shall let k=1 in the following calculations without committing serious errors. This means for mostsubmarine landslides the friction angle and the slope angle are very small, and thestresses in the slide follow the constitutive law of the ideal fluid. The only difference isthat in the governing equation of the frictional slide there is a Coulomb friction term.To solve the governing equations of the slide and the waves (4.21), (4.22), (4.31) and(4.32), an explicit finite-difference scheme is used, in which the values at new timet=(n+1)8t are calculated from the values at previous time t=n8t by using one-side(upwind) difference to approximate the first derivatives. I introduced two small artificialviscosities pai and ii..2 in the above discretized equations for the slide and for the waves toavoid numerical instabilities. It is found that minimum values of pal and pa2 of 0.05 forthe slides and the waves are sufficient to keep the solutions stable.As I did in the previous models, I considered an initial parabolic slide surface which isthe same as I used in the viscous model (see Figures 2.3). I also considered a river (at176x<0) with depth h 1 =0.5D0 and a deep sea. The initial position of the mudflow is indicatedby the parameter ic=D o/ho, where ho=hs(I). The distance interval and the time intervalfor the numerical computation are chosen as Sx =0.05 and St =0.005. The density of wateris taken as p 1 =1.0 g/cm3 , and the density of the slide p2 is taken as 2.0 g/cm 3 . The smallparameter e , the ratio of initial slide thickness and length, is chosen to be 0.017, E mustbe small enough (1/20) that the long wave approximation is valid.4.4.1 Surface Variation of the SlideFigures 4.5-4.8 show the profiles for the slides with three friction angles 8 =1° , 2°and 4° on slopes with two small slope angles 0 =4°, 6°. For the calculated cases, I find theone-way coupling results are very close to the fully coupled results, which means thereaction of the surface waves on the slide is generally weak. This finding is consistentwith the results I found in the previous two models. So I only present the results of thefully coupled model.As to the slide profile, It can be seen that slide decreases drastically and spreadsquickly into a thin layer on the slope. There is no apparent slide amplitude dispersion, as Ifound in the cases of viscous slide and Bingham slide. Higher amplitudes are in themiddle part of the slide. This is due to the fact that the slide is actually dominated by thegravity acceleration, the motion of the slide follows approximately the law of free fall.The slide with smaller friction angle (i.e., smaller friction coefficient) flows faster (Figure4.5 compared with Figure 4.6, and Figure 4.7 compared with Figure 4.8); and the sameslide flows faster on a steep slope than on a gentle one (Figure 4.6 compared with Figure4.7).xt-0.0-4^-2^0^2^4^6^8^10^121771.00.80.60.40.20.0-4 -2 0 2^4 6 8^10 12rt--4 4..........t-4.0-4 -2 0 2^4 6 8^1 0 12t-6.0-4 -2 0 2^4 6 8^10 120t-8.0....da1111111111Now....._2^4^6^8^10 12-4t-10.0------______------4 -2 0 2^4 6 8^10 12------t-12.0---____-4 -2 0 2^4 6 8^10 121.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.4020.01.00.80.60.4020.01.00.8D 0.60.4020.0Figure 4.5: The slide surface profile of a friction slide with &I' and 664*. Solid line:two-way coupling; dashed line: one-way coupling.1.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.4020.01.00.80.60.4020.01.00.80.60.4020.01.00.80.60.40.20.0178t-0.02 4 6 8^10 12t-2.02 4 6 8^1 0 12t-4.02 4 6 8^10 12t-6.02 4 6 8^10 12t-8.02 4 6 8^10 12t-10 .02 4 6 8^10 12I \-4 -2 0-4 -2 0-4 -2 0-4 -2 0-4 -2 0-4 -2 00.40.20.0D 0.61.00.8-4^-2^0^2^4^6^8^10^12xFigure 4.6: The slide surface profile of a friction slide with 82° and 664°. Solid line:two-way coupling; dashed line: one-way coupling.-4 -2 0 2 4 6 8^10 12t-2.0-4 -2 0 2 4 6 8^1 0 12t-4.0-4 -2 0 2 4 6 8^10 12t-6.0-4 -2 0 2 4 6 8^10 121.00.80.60.40.20.01.00.80.60.4020.01.00.80.60.40.20.01.00.80.60.40.20.01791.00.80.60.40.20.0-4^-2^0^2^4^6^8^1 0^121.00.80.60.4020.0-4^-2^0^2^4^6^8^10^121.00.8D 0 .0.4020.0-4^-2^0^2^4^6^8^10^12xFigure 4.7: The slide surface profile of a friction slide with 82° and (;6°. Solid line:two-way coupling; dashed line: one-way coupling.-4^-2^0^2^4^6^8^10^12xI \ t-0.0-4 -2 0^2^4 6 8 10 12t-2.0-4 -2 0^2^4 6 8 10 12,,,,,--•-.,,......_t-4.0-4 -2 0^2^4 6 8 10 12.........",..■"----........t-6.0-4 -2 0^2^4 6 8 10 12t-8.0..---.---4 0^2^4 6 8 10 12t-10.0___......-••••"".......-4 0^2^4 6 8 10 121801.00.80.60.4020.01.00.80.60.4020.01.00.80.60.40.20.01.00.80.60.4020.01.00.80.60.4020.01.00.80.60.4020.01.00.80.4020.0D 0.6Figure 4.8: The slide surface profile of a friction slide with &4° and 666°. Solid line:two-way coupling, dashed line: one-way coupling.1814.4.2 The Slide Velocity DistributionFigures 4.9-4.12 show the slide velocity distributions for the frictional slides specifiedin Figures 4.5-4.8, respectively. The velocity is zero at t=0, it starts increasing withincrease of time. The distinct feature of the velocity distribution for the cohesionlessfrictional slides on an uniform slope is that the velocity keeps increasing all the time. Thisis because the surface reaction is weak and the gravity acceleration dominates the flowover the frictional resistance, so that the slide behaves as if it were in free fall. Also I findthat there is no velocity dispersion on the frontal face of the slide, as I found in the resultsof the viscous model. The maximum slide velocity is near the middle part of the slide,which is consistent with the results for the profiles. The velocity is large for the slide withsmaller friction angle (Figure 4.9) and larger slope angle (Figure 4.11).t-8.0t-6.0-4^-2^0^2^4^6^8^10^12t-10.0t-12.0-4^-2^0^2^4^6^8^10^12-4^-2^0^2^4^6^8^10^12-4^-2^0^2^4^6^8^10^12x1.00.80.60.4020.01.00.8U 0.60.4020.01.00.80.60.40.20.0t-0.0-4^-2^0^2^4^6^8^10^12t-2.0-4^-2^0^2^4^6^8^10^12t-4.0-4^-2^0^2^4^6^8^10^121821.00.80.60.40.20.01.00.80.60.40.20.01.00.80.60.4020.01.00.80.60.4020.0Figure 4.9: The slide velocity distribution of a frictional slide with &I° and 6,4*.1.00.80.60.4020.0t-0.0-4 -2 0^2 4 6 8 10 121.0t-2.00.80.60.4020.0-4 -2 0^2 4 6 8 10 121.0t-4.00.80.60.4020.0-4 -2 0^2 4 6 8 10 121.0t-6.00.80.60.4020.0-,_-4 -2 0^2 4 6 8 10 121.0t-8.00.80.60.40.2 .^....................... ....._..._.-..---------.........s...s,0.0-4 -2 0^2 4 6 8 10 121.0t-10.00.80.60.4020.0 .^---"7"---4 -2 0^2 4 6 8 10 121.0t-12.00.8U 0.60.4020.0 .^./..----4 -2 0^2 4 6 8 10 12x183Figure 4.10: The slide velocity distribution of a frictional slide with Em2' and 664'.t-0.01841.00.80.60.40.20.0t-6.0-4 -2 0 2 4 6 8 1 0 12t-2.01.00.80.60.4020.0-4^-2^0^2^4^6^8^1 0^121.00.80.60.4020.0 t-4.0-4^-2^0^2^4^6^8^10^121.00.80.60.4020.0-4^-2^0^2^4^6^8^10^12t-8.0-4^-2^0^2^4^6^8^10^12t=10.0. -4^-2^0^2^4^6^8^10^121.00.80.60.4020.01.00.80.60.4020.01.00.8U 0.60.40.20.07.--4^-2^0^2^4^6^8^10^12xFigure 4.11: The slide velocity distribution of a frictional slide with S =2° and 9U.0.01851.00.80.60.4020.0-4 -2 0 2 4 6 8 10 12-4 -2 0 2 4 6 8 10 121.00.8U 0 .60.4020.0-4^-2^0^2^4^6^8^10^121.00.80.60.4020.0-4^-2^0^2^4^6^8^10^121.00.80.60.4020.01.00.80.60.4020.0t-12.0"...................■-■,..................t-10 .0,....................-------------.....................-4^-2^0^2^4^6^8^10^121.00.80.60.4020.0 t-2.0-4 -2 0 2 4 6 8 10 121.00.80.60.4020.0t-4.0-4^-2^0^2^4^6^8^10^12xFigure 4.12: The slide velocity distribution of a frictional slide with 8=4* and 8 °.1864.4.3 The Surface Water ElevationsFigures 4.13-4.16 show the surface water elevations which are generated by thefrictional slides specified in Figures 4.5-4.8, respectively. I find the waves produced by africtional slide are similar to those generated by a viscous slide. Three main waves areproduced, a crest propagating toward deep water, a trough going shoreward and a weaktrough going slowly to deep water. The reason the forced trough is weak is that theforcing at the front face is very weak, while in the viscous slide model I found a strongforcing at the frontal face of the slide. As to the amplitudes of the waves, I can see for thesame initial slide position and same slope angle the slide with smaller friction angle(Figure 4.13) generates larger waves than the slide with larger one (Figure 4.14).Compared with the surface waves generated by the viscous slide with the same densityand volume (the slide with r=2.0, as specified in Figure 2.4), the waves produced by thefrictional cohesionless slides are usually smaller. Because the frictional slides quicklyflatten and spread, while a viscous slide has a front hence a strong forcing factorassociated with the flow. This could reverse if the viscosity of the viscous slide becomesvery large, or the friction coefficient of the frictional slide becomes very small.4.4.4 The Velocity Distribution of Water MotionFigures 4.17-4.20 show the distribution of water particle velocity for the surface waveswhich are generated by the frictional slides specified in Figures 4.5-4.8. The water aheadof the front face of the slide is pushed away and moves to deep water. Water just abovethe slide flows shoreward and water near the rear face of the slide moves to deep water.The water particle velocity is one-order of magnitude smaller than the slide velocity.Compared with the water motion due to a viscous slide, the water velocity induced by africtional slide is usually small.t-0.0-10 -5 0 5 10 15N2.0....7 ---10 -5 0 5 10 15^20N4.0-10 -5 0 5 10 15^201870.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.10-10 -5 0 5 10 15 200.100.050.0-0.05-0.10-10 -5 0 5 10 15 200.100.050.0-0.05-0.10-10 -5 0 5 10 15 20x0.100.0511 0.0-0.05-0.10-10 -5 0 5 10 15 20Figure 4.13: The water surface elevations generated by the slide specified in Figure 4.5.0.100.050.0-0.05-0.10-10 -5 0 5 10 15 20x0.100.0511 0.0-0.05-0.10-10 -5 0 5 10 15 200.100.050.0-0.05-0.10t-0.0-10 -5 0 5 10 15^20t-2.0-10 -5 0 5 10 15^20t-4.0-10 -5 0 5 10 15^20-10 -5 0 5 10 15^20t-8.0-10 -5 0 5 10 15^201880.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.10Figure 4.14: The water surface elevations generated by the slide specified in Figure 4.6.0.100.050.0-0.05-0.10-10 -5 0 50.100.050.0-0.05-0.10-10...-....."---•-5 0 50.100.050.0-0.05-0.10-100.100.050.0-0.05-0.10-10-5 0 5t-0.010 15 20t-2.010 15 20t-4.010 15 20t-6.010 15 20-5 0 51890.100.050.0-0.05-0.10-10 -5 0 5 10 15 200.100.050.0-0.05-0.10-10 -5 0 5 10 15 200.100.0577 0.0-0.05-0.10-10^-5^0^5^10^15^20xFigure 4.15: The water surface elevations generated by the slide specified in Figure 4.7.t-0.00.100.050.0-0.05-0.10-10190-5^0^5^10^15^200.100.050.0-0.05-0.10-10 -5 0 5 10 15^20t-4.0-10 -5 0 5 10 15^20t-6.0-10 -5 0 5 10 15^20t-8.0-10 -5 0 5 10 15^20t-10.0-10 -5 0 5 10 15^20t-12.0-10 -5 0 5x10 15^200.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.050.0-0.05-0.100.100.0577 0.0-0.05-0.10Figure 4.16: The water surface elevations generated by the slide specified in Figure 4.8.0.0100.0050.0-0.005-0.010t-0.0-10 -5 0^5 10 15 20t-2.0—CT-10 -5 0^5 10 15 20t-4.0Ct.s........----10 -5 0^5 10 15 20t-6.0..le"..'"...\\................,..--10 -5 0^5 10 15 20t-8.0..C... \..._...7.G------..-10 -5 0^5 10 15 201910.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.010-10 -5 0 5 10 15 200.0100.005u 0.0-0.005-0.010-10^-5^0^5^10^15^20xFigure 4.17: The distribution of the water particle velocity under the same condition asspecified in Figure 4.5.0.0100.0050.0-0.005-0.010t-0.0-10 -5 0 5 10 15^200.010 t-2.00.0050.0-0.005-0.010-10 -5 0 5 10 15^200.0100.005 t-4.00.0 , 111••"-0.005-0.010-10 -5 0 5 10 15^200.010t-6.00.0050.0-0.005-0.010-10 -5 0 5 10 15^200.010t-8.00.0050.0-0.005-0.010-10 -5 0 5 10 15^200.010 t-10.00.0050.0-0.005-0.010-10 -5 0 5 10 15^200.0100.005u^0.0-0.005-0.010-10 -5 0 5 10 15^20x192Figure 4.18: The distribution of the water particle velocity under the same condition asspecified in Figure 4.6.x193t-0.0-10 -5 0 5 10 15^20t-2.0-10 -5 0 5 10 15^200.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.010-10 -5 0 5 10 15 20-10 -5 0^5^10 15 20t-8.0-..---__ .,-,,- --'-10 -5 0^5^10 15 20t-10.0--•••••—••••■ ••••°'Ths\,:::„.-..-. .....:„. .g ,...-10 -5 0^5^10 15 20t-12.0-......'"'-'-* ,\.s................ e'•'—'-'---•--..-10 -5 0^5^10 15 200.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.0100.0100.005u 0.0-0.005-0.010Figure 4.19 The distribution of the water particle velocity under the same condition asspecified in Figure 4.7.194t-0.0-10 -5 0 5 10 15^20-0N=...-t-2.0-10 -5 0 5 10 15^20-.Z‘1 ..„....t-4.0-10 -5 0 5 10 15^20t-6.0-------10 -5 0 5 10 15^200.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.0100.0100.0050.0-0.005-0.010-100.0100.0050.0-0.005-0.010-10-5^0^5^10^15^20-5^0^5^10^15^200.0100.005u 0.0-0.005-0.010t-12.0^-V-\\.--_ ---...7-----10^-5^0^5^10^15^20xFigure 4.20: The distribution of the water particle velocity under the same condition asspecified in Figure 4.8.1954.5 Summary and ConclusionsI presented a formulation of the dynamics of an underwater frictional slide coupledwith the surface waves which it generates, flowing on a gentle uniform slope in shallowwater. I adopted the long-wave approximation for both water waves and the frictionalslide and I solved the resulting differential equations by a finite-difference method.Again, I found that the one-way coupling results are very close to the fully coupledresults, which means the reaction of the surface waves on the slide is generally weak. Theslide surface variations, however are quite different from the results of the viscous model.The slide decreases drastically and spreads quickly into a thin layer, and there is noapparent amplitude dispersion. As expected, the slide with smaller friction angle flowsfaster. Another difference between the viscous slide and the frictional slide is that theslide velocity for the cohesionless frictional slides keeps increasing all the time in thetime domain, due to the fact that the surface reaction is weak and the gravity accelerationdominates the slide. This is true only for the slide with depth larger than several particlediameters. When the slide becomes rollings of separated particles, the present theory doesnot apply. Because these particles can not be treated as a continuum. In that case theparticles will not accelerate forever on the slope, the form drag on each individual particlewill balance the gravity as the particle acquire enough speed. A final constant speed willbe achieved.The evolution of the surface waves generated by frictional slides is similar to that ofthe waves generated by a viscous slide, except that the forced trough is small as a resultof the weak forcing at the frontal surface of the slide. For the same initial slide positionand same slope angle the slide with smaller friction angle generates larger waves than theslide with larger one.196Chapter 5Summary and ConclusionsTo study the phenomenon of submarine landslides and the associated tsunamigeneration, I have developed three numerical models, in which three kind of rheologies ofsubmarine sediments have been considered. Two aspects of this problem are investigated.As to the dynamics of the underwater debris flow, I derived the formulations of atransient landslide problem by using a fluid dynamics method. Most previous studies aresteady state solutions which are inadequate to simulate the slide. I examined the influenceof the rheologies of the sediments and the free surface wave reaction on the slide. As tothe tsunami generation, I adopted the long-wave approximation and developed a coupledmodel to examine the surface water waves and currents which are induced by the slide. Icalculated the surface waves produced by the slides with the above three kinds ofrheologies on various slopes, at various initiation water depths.In the first model for submarine debris flows, the slide is treated as a viscous fluidflow, and water motion is assumed irrotational. The shallow wave approximation isadopted for both the surface water waves and for the debris flow. The resulting equationsof the flow and the waves are solved by a finite-difference method for the slide withinitial parabolic and triangle shapes sliding on uniform gentle slopes. It is found that threemain waves are generated by a viscous underwater slide. The first wave is a crest whichpropagates into deeper water; it is followed by a trough in the form of a forced wavewhich propagates with the speed of the slide front to deep water. The third wave is asmall trough which propagates shoreward. It is also found that a small but long crestforms and propagates shoreward. As to the magnitudes of the waves generated by a largescale landslide, it is found that they are of order of several meters.197Another concern on a event of an underwater landslide is whether a wave runup at theshore will occur. The numerical results indicate that there will be first a depression ofwater level at the shore because of the shoreward trough, and then there will be a waverunup induced by the crest following the trough. The calculated result shows that thewave runup is much larger than the amplitude of the incoming crest.I focused on the interaction between the slide and the surface waves in the first model.To examine the significance of the coupling, two aspects of the interaction were studied.First I examined the influence of an underwater slide on the free water surface, andsecondly I investigated the significance of the free surface coupling on the slide. It isfound that the density of the flow (which determines the active driving force on the slide)and the water depth at the slide initiation site have the most important influence upon theinteraction between the slide and the waves. Generally the reaction of the free surface onthe slide is weak and less important for the slide dynamics. One may neglect the surfacecoupling if one wants an approximate estimation of the slide behavior. The influence ofthe slide on the free surface, however, is significant and not negligible if one wants tofind out the tsunamis produced by a submarine slide. The two-way coupling is moresignificant for the case of a smaller slide density and shallower water depth at the slideinitiation site. For the cases of a denser slide in shallower water, the reaction of the freesurface coupling is weak; however, the amplitudes of waves which it generates are verylarge. The amplitudes of the waves depend primarily on the density of slide, the waterdepth at the slide initiation site, the volume and the viscosity of the slide. The slide withlarger density and larger volume, and with smaller viscosity generates larger waves inshallower waters.I examined the energy exchange between the slide and the surface waves. It is foundthat the ratio of the energy transfer from the slide to the waves is not a constant; it has amaximum value at the beginning stage, and decreases to 2-4% when the waves propagate198away from the slide site. The energy exchange is most significant for the case where theinteraction between the slide and the waves is strongest.To examine the effects of the initial shapes of the slide I calculated the slide and thewaves for the case where the slide has an initial triangular profile. The numerical resultsindicate that whatever the initial profiles of the slide are, they finally all evolve into asimilar profile: a long and thin sheet of fluid. The slides with different initial profiles butthe same volume behave almost the same a short time after the initiation of the slide, andthe waves also behave similarly. I also calculated the conditions of possible resonancebetween an underwater slide and the waves which it generates. The calculated results ofthe Froude numbers for various situations indicate that the possibility of such a resonanceis very small in practical cases.The influence of turbulence on the slide dynamics and on the waves is brieflydiscussed. As to the effect of turbulent shear on the slide, for large-scale landslides ascalculated in Sections 2.8, turbulence plays an important role as the slide accelerates togreat enough speed. A turbulent slide on a slope is much different from a laminar one.The present laminar model does not apply. The importance and accurate evaluation of theinterfacial turbulent shear still needs further investigation both theoretically andexperimentally. For the small-scale slides as calculated in Section 2.5 and 2.6, the slidespeeds are small and the Reynolds numbers are less than 50, turbulence might play aminor part in the slide dynamics. Therefore the laminar model can give relativelyaccurate solutions for small scale slides, and it can also give the upper limit of the slidespeed and wave height that might be useful for engineers to evaluate the maximum slidespeed and wave amplitude for engineering purposes.In the second model, I have presented a formulation of the dynamics of a submarineBingham-plastic mudslide coupled with the surface waves which it generates, flowing ona gentle uniform slope in shallow water. I put emphasis in this model on the influence of199Bingham plasticity on the slide dynamics and wave generation. Due to the Bingham-plastic behavior of the mud, the slide can travel only a finite distance, and the slide willstop at its equilibrium thickness D, on the slope when the shear stress exerted on thebottom of the flow becomes smaller than the yield stress. The larger the yield stress of theslide, the smaller the mobility of the slide. The surface water waves generated byBingham-plastic slides behave similarly to those generated by a viscous slide. The waveamplitudes and the water current velocity, however, are significantly reduced due to theBingham plastic effect. The smaller the yield stress of the slide, the greater the surfacewaves it generates. The slide with a larger yield stress moves much more slowly andgenerates much smaller waves. The major parameters that dominate the interactionbetween a Bingham-plastic slide and the waves are: the Bingham yield stress, the densityof mud and the depth of initiation of the slide. Besides, it is found that the solution of theviscous fluid model can also be derived as a special case of the Bingham plastic solution.In the third model of the submarine landslide, I investigated an underwater frictionalslide coupled with the surface waves which it generates. Again, it is found that the one-way coupling results are very close to the fully coupled results, which means the reactionof the surface waves on the slide is generally weak. I have obtained a quite differentpicture, however, for the slide surface variations compared with the viscous solution. Theslide decreases drastically and spreads quickly into a thin layer, and no apparentamplitude dispersion is seen. The slide with a smaller friction angle (thus smaller bottomfriction) flows faster. Another difference between the viscous slide and the frictional slideis that the slide velocity for the cohesionless frictional slides keeps increasing all the timeif there is no slope changes or obstacles on the slide path. This is due to the fact that thesurface reaction is weak and that the gravity acceleration dominates the slide.The surface waves generated by frictional slides are similar to those generated by aviscous slide, but the forced trough is small as a result of the weak forcing at the frontal200surface of the slide. For the same initial slide position and same slope angle the slide withsmaller friction angle generates larger waves. The major parameters that dominate themagnitude of the waves are: the density of the slide, the friction angle between the slideand the slope, and the water depth at the slide initiation site.This study is the first one which addresses the details of underwater landslidedynamics (for sediments with different rheologies) reactively coupled with surface waterwaves which they generate. This study provides more realistic models for transientunderwater landslide as well as for the surface waves which they generate. The numericalmodels developed here can be used to supply theoretical predictions for slide ofsediments with various rheologies on various slopes (even arbitrary, as long as the longwave approximation is satisfied), and also for the surface wave generation by these slides.It is noted that what has been examined here is a two-dimensional problem of anunderwater slide and the waves which it generates. In practical cases the slide and thewaves are expected to be three dimensional, where the slide dynamics and the wavegeneration are subject to the influence of not only the longitudinal spreading but alsotransversal spreading of the slide. For future study on the three dimensional problem Isuggest the following aspects of research: the secondary flow in the slide; the three-dimensional effects of the slide on the waves generation; the pattern of near shore watercirculation induced by the slide. Also it is noted that the present solution is valid only fora slide and waves on a gentle slope, where the long-wave approximation is satisfied andthe slide and water motion are almost horizontal. For an underwater slide on a steep slope(which may happen at the banks of rivers and reservoirs), the long-wave approximation isno longer valid. The vertical acceleration in the slide and the water motion must be takeninto consideration.Another point of interest is wave runup generated by an underwater landslide at theshore. There have been many studies on the wave runup (uprush and downrush) induced201by (1) periodic sinusoidal waves, (2) a solitary wave (or cnoidal waves), (3) a uniformbore, (4) random waves. Wave runup produced by a underwater slide will be unique, asit is due to two waves: a trough and a crest following the trough. It does not fall in any ofthe above categories. The wave runup can be predicted from the computed movement ofthe instantaneous waterline (where water depth is zero) on the slope. The wave runupproblem is of great public concern in a underwater landslide event, and is definitelyworth of further research.As was mentioned already in the introduction, because of the great variety ofsubmarine sediments and the strong dependence of the constitutive relation onconcentration and stress state, differing theoretical models may have their own realms ofapplicability, and laboratory simulations of slide motion in a given natural state aredifficult. 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