Modeling Wave Generation, Evolution, and ... - Semantic Scholar

Modeling Wave Generation, Evolution, and Interaction with Depth-Integrated, Dispersive Wave Equations COULWAVE Code Manual Cornell University Long and Intermediate Wave Modeling Package Patrick J. Lynett∗ & Philip L.-F. Liu†

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Theoretical Background

The governing equations employed in this numerical model allow for the evolution of fully nonlinear ( wave amplitude to water depth ratio = O(1) ) and dispersive waves over variable bathymetry. Additionally, the generation of water waves by movement of the sea floor can be examined. The governing equations are given in the Appendix, which is a summation of the authors’ analytical and numerical work concerning depth-integrated wave theory, submarine landslide modeling, and runup modeling. At this point in the development, modeling of submarine landslides cannot be performed by the user without some guidance by the author.

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Numerical Background

The numerical model uses a predictor-corrector scheme to march forward in time, and uses finite differences to approximate spatial derivatives. The model is formally accurate to ∆t4 in time and ∆x4 in space. The corrector segment of the procedure is implicit in time, and uses iteration to arrive at a solution. Details of the numerical model are given in the Appendix. ∗ †

Assistant Professor, Department of Civil Engineering, Texas A&M University, College Station, TX, USA Professor, School of Civil and Environ. Engrg, Cornell University, Ithaca, NY, USA

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Files

The following files are included in this package: coulwave.f - the source code of the numerical program. plot out.m - a Matlab script file that will load and plot the output of a numerical simulation. bath ss.m - a Matlab script file that loads and saves bathymetry data from the Smith and Sandwell database. The saved data can then be used with a numerical simulation. bath loc.m - a Matlab script file that loads bathymetry data from specified local files, or creates a bathymetry based on user input, and saves that data in a form that the numerical program can read. mygrid.m - a Matlab function file used by ”bath ss.m.” lat2m.m - a Matlab script file used by ”bath ss.m.” spectrum 1d.m - a Matlab script to create a 1HD input spectrum for COULWAVE spectrum 2d.m - a Matlab script to create a 2HD input spectrum for COULWAVE There are numerous example setup files in the /examples/ directory.

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Using the Numerical Code

The code in written in Fortran, and has been compiled using DEC(x86 with Visual Fortran, DEC UNIX) and Microsoft (x86 with Fortran Powerstation) compilers. The DEC compiler, which is currently marketed by Compaq, is strongly recommended. Others compilers, such as Lahey, have not been tested thoroughly. There are a couple simulation options which require the user to recompile the source. These will be described first: 1) The number of layers to be used. At line 192 exists the parameter ”num levels”. This integer should be set to the number of layers required, i.e. 1, 2, 3, or 4. Keep in mind that the memory and CPU time requirements increase linearly with the number of layers. The multi-layer theory is described in the Appendix. 2) The matrix sizes. If the matrix sizes are too small, an error message will appear with instructions on which parameter needs to be changed. These parameters which sometimes need to be altered are all located on line 193. A primitive user interface has been incorporated into the model, which allows the user to set the parameters of the numerical simulation without having to edit the source code and recompile. Therefore, any simulation can be run using the same compiled executable. The parameters/options governing the simulation are described here, in the order in which they appear when using the text interface. It is recommended that for the first few times running the program, the user follow along with this manual while running the executable. The user interface has been constructed such that all of the information contained in the user manual below is also given, in the form of long comments and instructions, in the user interface as well. The first option menu is: 2

****************************************************************************** ID=1:Load simulation parameters from data file, sim set.dat; you will be able to change most of the numerical parameters through menu choices. Also with this choice, you will can load already setup examples, such as wave runup on a conical island, and other more basic examples. ID=2:Input simulation parameters with user interface; this should be run the first time you try to set up a problem - after that you can use ID=1 to modify certain parameters - like wave height - resolution - etc ID=3:Use simulation parameters edited in source code - you will need to re-compile the source Input ID: ****************************************************************************** Choosing ”1” will load the simulation parameters from the data file ”sim set.dat”. Anytime a simulation is run, all of the parameters will be stored to this file - therefore option ”1” can only be chosen after a simulation has been run using option ”2”. However, option ”1” will not force you to run a simulation with the parameters as the are in ”sim set.dat”. When choosing option ”1”, all of the parameter choices will be displayed in a menu format. The user will then be able to change any of the parameter choices. For example, if you have previously run a simulation with the parameters stored in ”sim set.dat”, but you want to change the resolution, or time step, or even the number of dimensions, these parameters can be altered when choosing ”1”. Choosing ”2” will start a text-menu interface where all of the simulation parameters must be inputted by the user. This option should be used when setting up a simulation for the first time. After running ”2” one time, option ”1” can be run subsequently to change any of the inputted parameters. ———————————————————————————————————————The user interface when choosing option ”2” will be described in this section. The next option menu is: ***************************** Simulation Type ****************************** ID=1:Surface wave evolution ID=2:Wave generation by submarine landslide Input ID ****************************************************************************** Choosing ”1” will create a simulation examining surface wave evolution (solitary, cnoidal, or sine) with a water depth profile to be created later. Choosing ”2” will create a simulation examining surface waves created by movement of the sea floor bottom. This option is not ready for general use. ———————————————————————————————————————The next option menu is:

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******************** Number of Horizontal Dimensions ************************ ID=1: 1D simulation ID=2: 2D simulation Input ID ****************************************************************************** Choosing ”1” creates a simulation with only one horizontal dimension. Choosing ”2” creates a simulation with two horizontal dimensions. ———————————————————————————————————————The next option menu is: ***************** Degree of Nonlinearity of Simulation ********************** ID=0: Linear Simulation ID=1: Weakly Nonlinear Simulation ID=2: Fully Nonlinear Simulation Input ID ****************************************************************************** Choosing ”0” will make use of a linear, dispersive set of governing equations. The linear assumption implies that wave amplitude/water depth ¿ 0.1. Choosing ”1” will make use of a weakly nonlinear, dispersive set of governing equations. The weakly nonlinear assumption implies that wave amplitude/water depth ¿ 1. This approximation will significantly reduce CPU time (typically 15-25 % less for 2D), but may lead to large errors in the prediction of large amplitude waves. For example, if prediction of wave shoaling - wave height grows as water depth decreases - is desired, especially over mild slopes ( < 1/20 )the weakly nonlinear assumption should not be used as it tends to overestimate the wave height. Choosing ”2” will make use of a fully nonlinear, dispersive set of governing equations. The fully nonlinear assumption implies that wave amplitude/water depth = O(1). ———————————————————————————————————————The next option menu is: *********************** Dispersion Properties ******************************* ID=1: Use arbitrary level approximation ID=2: Use depth averaged approximation ID=3: Use shallow water equations Input ID ****************************************************************************** Choosing ”1” will make use of a set of governing equations based on evaluation of horizontal velocity at an arbitrary depth, given as z = −βh, where the optimum β = 0.531 for the one-layer (Boussinesq) model. The advantage of using the zβ method is that the wave and group velocities of higher wave numbers (h/λ > .25) are more accurately described. The 4

disadvantage is increased computational cost. When using more than one-layer, this option must be chosen. The evaluation levels for the two and more layer models are given in the Appendix. Choosing ”2” will make use of a set of depth-averaged governing equations. This option will have decreased CPU time, as compared to using option ”1”, but intermediate-depth waves may have significant phase and group speed errors. Therefore, a depth-averaged simulation should be utilized if the user is fairly certain that all wave numbers will be small, i.e h/λ < .2, or if computational speed is important (typically 5-15 % less for 2D). Choosing ”3” will employ the shallow water wave equations. These equations are nondispersive, and are only accurate for very long waves. Note that there are numerical packages in existence that will solve the shallow water equations many times faster than this program. ———————————————————————————————————————Only if performing a ”Surface wave evolution” simulation, the next option menu is: ************************* Type of Input Wave ******************************** ID=1: Solitary Wave ID=2: Solitary Wave with a/h=0.58 ID=3: Solitary Wave with a/h=0.42 ID=4: Cnoidal Waves ID=5: Sine Waves ID=6: Wave Spectrum - created using spectrum.m in Matlab Input ID ****************************************************************************** Choosing ”1” will input a solitary wave (a positive elevation wave of permanent form). This profile is the analytic solution to the weakly nonlinear equations. Therefore, larger amplitude waves will not initially be of permanent form when using this option. Choosing ”2” will input a highly nonlinear solitary wave. This solitary wave is the numerically permanent solution to the fully nonlinear equations. This wave will only work with a water depth of h=0.45 m, and the program will force this water depth to be used. Choosing ”3” will input a highly nonlinear solitary wave. This solitary wave is the numerically permanent solution to the fully nonlinear equations. This wave will only work with a water depth of h=0.45 m, and the program will force this water depth to be used. Choosing ”4” will input a cnoidal wave (oscillatory wave of permanent form) train. Choosing ”5” will input a simple sine wave train consisting of up to two frequencies. Sine waves can be created using an internal source function. The option to use the internal source will be given a little later. Choosing ”6” will input a spectrum of amplitudes. The input data files when implementing a spectrum must be generated by the ”spectrum.m” Matlab file included. Using this script file is discussed in the later section ”Creating a Wave Spectrum in Matlab.”. ———————————————————————————————————————For all ”Surface wave evolution” simulations the following lines will appear 5

Input wave height:(in meters) Input initial/characteristic water depth:(in meters) [this is the water depth at the initial location of the wave or the location of the internal source] ———————————————————————————————————————Only if performing a ”Surface wave evolution”, with ”Solitary Wave” simulation, the user interface will prompt the user to input the following: Input initial location of crest of soliton (meters) [this is the length from the right-side boundary to the crest of the solitary wave] ———————————————————————————————————————If performing a ”Surface wave evolution” with ”Sine Waves”, the interface will display the following: Sine waves can be effectively modeled using an internal source wavemaker. With this option, waves are created inside the numerical domain. Internal source generation, coupled with sponge layer absorbers along the outer domain boundaries allow for the development of a quasi-steady state wave field. ID=0: Do not use internal source wavemaker ID=1: Use internal source wavemaker Entering ”1” will create an internal source function. If the user has chosen to use the internal source, next will be a prompt to enter the location of the centerline of the internal source. For 1D simulations, this is simply the x-coordinate of the internal source centerline. For 2D cases, this the normal distance from the origin. It recommended to always use the internal source wavemaker when generating sine waves. ———————————————————————————————————————Only if performing a ”Surface wave evolution”, with ”Cnoidal Waves” or ”Sine Waves” (with no internal source) simulation, the user interface will prompt the user to input the following: Enter number of wavelengths to be in initial domain - must be at least one [the user can have as many waves in the initial domain as is desired, 2-3 recommended] Enter total number of wavelengths to be created during entire simulation Enter total number of wavelengths to be ramped to eliminate free surface discontinuity - 1 recommended [the initial wave height and velocity, in the domain initially, must be ramped gradually down to zero in order to eliminate a free surface and velocity gradient discontinuity] ———————————————————————————————————————Only if performing a ”Surface wave evolution”, with ”Cnoidal Waves” simulation, the user interface will prompt the user to input the following: For cnoidal waves - enter the modulus of the elliptic integral of the first kind’ [this value theoretically varies from 0 (sine waves) to 1 (solitary waves) but realistically can only be varied from .4-1] 6

———————————————————————————————————————Only if performing a ”Surface wave evolution”, with ”Sine Waves” simulation, the user interface will prompt the user to input the following: For sine waves - enter the wavelength (m) If using an internal source, there is an option to additionally create a second harmonic. If this is desired, the user should input the second harmonic wave height and wavelength when prompted. ———————————————————————————————————————Only if performing a ”Surface wave evolution”, ”2D simulation”, the interface will prompt the user to input the following: Enter incident angle of waves (degrees) - ( 0 = waves traveling in positive x-direction ) ———————————————————————————————————————For all simulations, the following option menu will be presented: ************************* Water Depth Profile *************************** ID=1: Specify Profile by Giving Location/Depth Nodes. This option will allow you to create a piecewise linear topography, where the nodes, in [x-location, water depth] coordinates, indicate the intersections of the piecewise linear segments. ID=2: Load Topographical Data from Matlab Created Files Input ID ************************************************************************* Choosing ”1” will allow the user to input nodes (minimum 2, maximum 7) describing the water depth profile. A node represents the location where the slope of water depth changes. This option does not create a 2HD variable bathymetry. Choosing ”2” will tell the program to load topography data from files. These topography files must be created using the included Matlab files. The Matlab script ”bath ss” will create topography data from the Smith and Sandwell 2-minute database (see the file ”bath ss.m” for details). The Matlab script ”bath loc” will create topography data from local files (again, see the file ”bath loc.m” for details). The files created by these scripts are: ”x topo.dat, y topo.dat, f topo.dat, size topo.dat”, and must be located in the same directory as the executable when running the program. For more information on how to create a topography using the Matlab scripts, see the later section ”Creating a Bathymetry in Matlab.” ———————————————————————————————————————If ”Specify Profile by Giving Location/Depth Nodes” was chosen, the following input is required: 7

Enter the number of nodes required to create bottom profile - including first and last (min 2, max 7): First x-node is x=0 meters [x-location of first node is forced to 0] Enter depth of first node (m): Enter x-location of second node (m): Enter depth of second node (m): ......... ———————————————————————————————————————If a ”2D Simulation” with ”Specify Profile by Giving Location/Depth Nodes” was chosen, the following input is required: Enter domain width (y direction) (m): ———————————————————————————————————————For all types of simulations, the following input is required Enter the physical simulation time in seconds: [total physical time of numerical simulation] Enter the time increment at which spatial snapshots of free surface should be written to file (s): [The time increment to write free surface, depth, and velocity to file] Write time series of zeta,u, and v at specific locations to file? Enter 0 for no, or enter the number of locations you would like time series for(60 max). [ If you choose to write time series to file, you will next be prompted to input the locations of each time series. The time series output form the model can be loaded with the Matlab file ”load ts.m”.] Enter the number of grid points per wavelength: [this is the resolution, a value of 30-50 is typically enough to ensure numerical convergence] Enter the courant number = dx/(dt*co): [this will determine the time step used in the model, for weakly nonlinear simulations, and value of 0.5 will typically yield stability and convergence, but for simulations with highly nonlinear waves, a value as low as 0.1 may be required for stability - this is unfortunately trial and error at this point. However, for nearly all cases a value of 0.5 will be stable.] ———————————————————————————————————————For all types of simulations, the types of boundary conditions must be specified ************************ Boundary Conditions **************************** ID=1: Solid-Reflective Wall ID=2: Input Wave - sending a wave through boundary ************************************************************************* The user will be prompted to specify a value for each of the boundaries (there are 2 for a 1D simulation, and 4 for a 2D simulation). Choosing ”1” will enforce a solid, completely reflecting wall. 8

Choosing ”2” will allow for waves to be sent through the boundary. This is not a radiation boundary condition, and will not allow for waves for arbitrary shape to exit the numerical domain. This boundary is required along the left wall when using cnoidal or sine waves. ———————————————————————————————————————For all types of simulations, sponge layers, which act as open, absorbing boundaries, can be applied along any boundary: ************************ Sponge Layer Absorber ************************* Sponge layers are excellent wave absorber ID=0: Do Not Use Sponge Layer ID=1: Use Sponge Layer ************************************************************************* Adding a sponge layer along any boundary will add a length in the normal direction equal to one wavelength. The sponge layer used here absorbs both mass and energy, and has shown to be an excellent absorber of waves of all types, with negligible reflection. ———————————————————————————————————————************************ Wave Breaking ************************* An eddy viscosity type wave breaking parameterization can be utilized. This breaking model has been empirically tuned to give adequate results for various breaking events. ID=0: Do Not Use Wave Breaking Model ID=1: Use Wave Breaking Model ************************************************************************* For practical simulations, the model should always be used. If the wave transforms to a situation near breaking, and the breaking model is not turned on, the simulation will frequently overflow. ———————————————————————————————————————************************ Bottom Friction ************************* Bottom friction effects can be modeled, using the quadratic friction law ID=0: Do Not Include Bottom Friction ID=1: Include Bottom Friction ************************************************************************* For real coastline problems, the bottom friction model should be used. This is especially the case with breaking waves, where, if the bottom friction model is not used, the thin film rundown can reach extremely large velocities, causing numerical stability problems. If you select to use the Bottom Friction model, you will be prompted: The bottom friction model is partly empirical, and a bottom friction coefficient must be inputted. This value is

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typically in the 10-4 to 10-2 range. Enter coefficient: [For real shoreline problems, where the coefficient is not well known, a value of 0.005-0.01 is recommended] ———————————————————————————————————————For all types of simulations, the following option is given: **************************** Numerical Display *************************** ID=1: Display output data to screen ID=2: Do not display Input ID ************************************************************************* Choosing ”1” will have the program print program information to screen. The information to be displayed is: Current time step out of total time steps, and number of iterations required for convergence in the corrector loop of the last time step. Choosing ”2” will have the program display no information while it is running. If the user has chosen to display data to screen, the following prompt will be given: Time step interval to print select information to screen [i.e. if 10 is inputted, information will be displayed on screen every tenth time step] ———————————————————————————————————————For all types of simulations, the following option is given: ************************ Numerical Filtering **************************** ID=1: Filter entire domain using 9-point filter ID=2: Do not filter entire domain Input ID ************************************************************************* The 9-point filter that can be used is an effective approach to eliminating short wave instabilities. Note that although this filter has a steep response function (i.e. it totally eliminates the 2∆x waves, and removes very little energy from the longer wavelengths) it should be used sparingly. The reason for this is that it does remove a small amount of long wave energy, and thus may effect the final solution in a non-physical manner. If filtering is chosen, the user will be prompted: Number of times to filter entire domain per wave period : [lower values will effect the solution less - values less than 3 are recommended if filtering is needed.] ———————————————————————————————————————The above are all the options that can be inputted when choosing not to load the numerical parameters from file. Additionally, the following parameters are given the default values: 10

Maximum # of iterations allowed in corrector loop = 20 (default) Minimum # of iterations for corrector loop=5 (default) Iteration after which over-relation should be used = 15 (default) Over-relaxation coefficient = 0.5 (default) [Using over relaxation will sometimes make the corrector loop converge more rapidly.] Width of the sponge layer, in wavelengths = 1.0 (default) [ From numerical tests, it was found that using a sponge layer width of one wavelength yielded excellent results for most waves. For highly nonlinear waves, one may want to increase this value. ] These default values are changeable when loading the numerical parameters from the data file ”sim set.dat”. ———————————————————————————————————————-

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Error Messages and Descriptions

Error: Channel width is not large enough, must be at least two grid points wide. Reason: A 2D simulation was selected, but the width of the numerical domain is not large enough to compute derivatives in the y-direction. Solution: Either use a 1D simulation, or expand the width of the domain. Error: nx is not large enough, stop program and change Reason: ”nx” is an integer that sets the maximum number of x-grid points. Its value is set in the ”parameter” section near the top of the source file. Solution: Set ”nx” to a value equal to or greater than ”endx”. The value of ”endx” is printed to the computer screen in the error message. To do this, you will need to edit the source code (parameter section), and recompile. Error: ny is not large enough, stop program and change Reason: ”ny” is an integer that sets the maximum number of y-grid points. Its value is set in the ”parameter” section near the top of the source file. Solution: Set ”ny” to a value equal to or greater than ”endy”. The value of ”endy” is printed to the computer screen in the error message. To do this, you will need to edit the source code (parameter section), and recompile. Error: tt is not large enough, stop program and change Reason: ”tt” is an integer that sets the maximum number of time-grid points. Its value is set in the ”parameter” section near the top of the source file. Solution: Set ”tt” to a value equal to or greater than ”endt”. The value of ”endt” is printed to the computer screen in the error message. To do this, you will need to edit the source code (parameter section), and recompile. Error: Maximum number of iterations reached - corrector did not converge Reason: The iterative corrector loop did not converge to a solution within the specified 11

maximum number of iterations allowed. Solution: This is a tricky one. Most likely, this error is telling the user that there is some instability in the solution. The check this, plot the output in Matlab, and examine the free surface for short wave instabilities. Filtering can be used to reduce the instabilities, letting the simulation proceed. If there is apparently no cause for either the instabilities or the high number of iterations, contact the author my email for assistance at [email protected]. Error: no wave in domain, overflow possible Reason: There is nothing occurring in the simulation. The likely cause is that the simulation overflowed (a physical value was assigned a value of ”infinity”), whereby everything in the simulation is forced equal to zero. This error is compiler specific - some compilers will force everything equal to zero, some will display an error ”overflow” and stop the simulation. An overflow is usually caused by an instability in the somewhere in the domain. Solution: Plot the output that you have in Matlab, and look for any locations that show instabilities. Look for locations with very steep slopes. Steep slopes or corners should be slightly smoothed to eliminate the large first and second derivatives of water depth. Filtering may prevent the simulation from overflowing. If there is apparently no cause for the overflow contact the author my email for assistance at [email protected]. ———————————————————————————————————————-

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Known Problems and Deficiencies

The largest problem to date is that of breaking during the rundown of waves. The author has yet to find a physical dissipation mechanism that can be implemented and keep the simulation from overflowing. The current patchwork solution is to add a large amount of bottom friction during the rundown phase, such that this rundown breaking is minimized or prevented. The simple reason that this type of breaking is troublesome seems to be that the rundown breaking is usually represented by a stationary hydraulic jump. The breaking model, which is keyed into the transients of free surface to determine when a breaking event has started, will not detect a stationary hydraulic jump. A parameterization to describe this breaking has not yet been developed, and the ad-hoc addition of large bottom friction should prevent this phenomenon from causing problems. If the simulation does still overflow on account of the rundown breaking, usually lowering the courant number will solve the problem. This problem is most likely to occur with steep slopes. The moving boundary algorithm is not yet proven to properly model overtopping and reentry. Additionally, reentry, or overtopping water entering into quiescent water, tends to create large instabilities. Some sort of special treatment may be needed from these situations, and none has yet been done. ———————————————————————————————————————-

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Creating a Spectrum in Matlab

The included Matlab scripts ”spectrum 1D.m” and ”spectrum 2D.m” can be used to create an input spectrum to be used in the numerical simulation. The ”spectrum 1D.m” file will create a spectrum in one-horizontal dimension. This file can create a JONSWAP spectrum, a spectrum in the Toba form, with coefficients from Donelan et al(1985), or a Davidan spectrum which is characteristic of swell waves. The user must edit the first few lines of ”spectrum 1D.m” to choose one of these spectrums, as well as input the peak frequency and the wind speed at 10 meters. The output from this script file will be a data file called ”spectrum.dat”, which must be located in the same direction as the executable in order to be loaded by the simulation. The file ”spectrum 2D.m” creates a two-horizontal dimension spectrum. The two-dimensionality is setup as: S(f, θ) = S(f )h(θ)

(1)

where the h(θ) directional function used in the Matlab file is a simple cos2 spreading function. Additionally, the user can load a local energy density spectrum data file using the given Matlab files. ———————————————————————————————————————-

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Creating a Bathymetry in Matlab

The included Matlab scripts ”bath ss.m” and ”bath loc.m” can be used to create a bathymetry profile to be used in the numerical simulation. The first file ”bath ss.m” searches the 30 minute global database, based on latitude and longitude limits that must be manually edited in the script, and creates the appropriate data files to be read by ”coulwave.exe.” Note that the 130 MB database file ”topo.img” must be locally located. Its location directory should be edited into the script file. The location where the 130 MB file can be downloaded from can be found on the model web page : www.cee.cornell.edu/pjl2/research web/COULWAVE/. The file ”bath loc.m” can be used to create a bathymetry profile from other local data files, or by arbitrary creation by the used. This procedure is most easily learned by examining the script file and the comments contained therein.

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a)

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Figure 1: Sine wave runup on a planar beach, a) Numerical free surface at various times, analytic free surface is shown by the dashed line (−−), and is only compared for the maximum and minimum shoreline movement profiles. b) Comparison between analytical (−−) and numerical (−) shoreline movement.

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Examples 1HD: Non-breaking, long wave runup and rundown

A monochromatic wave train is let to runup and rundown a plane beach. This situation has an analytic solution derived by Carrier and Greenspan (1958). Their derivation makes use of the NLSW equations, and thus for consistency the dispersive (µ2 ) terms should be ignored in the numerical simulations for this comparison. The wave and slope parameters for this test case are identical to those used by Madsen et al. (1997) and Kennedy et al. (2000). A wave train with height 0.006 m and period of 10 s travels in a one dimensional channel with a depth of 0.5 m and a slope of 1:25. The results of the numerical simulation are shown in Figure 1. Figure 1a) shows the numerical free surface at various times, along with two profiles of the analytic free surface.

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The comparison between analytic and numerical horizontal shoreline movement is shown as Figure 1b). LOCATION OF INPUT FILES: /examples/Carrier Greenspan FILES NEEDED TO RUN: ”sim set.dat”

9.2

1HD: Regular wave evolution over a submerged shoal

The particular case is a good example of the benefits of using the two-layer model over the one-layer (Boussinesq) model. The setup is taken from the experiments presented by Dingemans (1994), who recorded free surface time series at numerous locations in front of and behind the obstacle. The orientation of the bar is shown in the top subplot of Figure 2. The wave, as it approaches the bar, is nearly a long wave, with a kh=0.7 (wavelength of 7.7 m in 0.86 m of water). This incident wave corresponds to Case A in Dingemans (1994). As the wave shoals, it steepens and nonlinear transfers create superharmonics. The superharmonics, while still shallow or intermediate water waves on top of the bar, become deep water waves as they enter the deeper water behind. As discussed in Woo & Liu (2001), significant wave energy (about 75% of the peak spectral amplitude) is present at kh ≈ 4 in the region behind the bar. For this reason, Boussinesq-type models (one-layer O(µ2o ) models), whose linear dispersion accuracy limit is near kh ≈ 3, do not correctly predict the wave field behind the bar. Time series are taken at the four locations depicted in the top subplot, and both the one- and two-layer models are compared with experimental data. The column on the left shows the one-layer results, the column on the right, the two-layer. On top of the bar, at location #1, both models are in agreement, and the two-layer model shows no benefit. This is expected, as all of the dominant wave components at this location have kh values less than 2.0. However as the wave components progress into deeper water, the one-layer model becomes inaccurate. This is evident at locations #2-#4, where the one-layer model deviates from the experimental results. The two-layer model, on the other hand, shows its strength and predicts the wave field excellently. LOCATION OF INPUT FILES: /examples/Dingemans Step FILES NEEDED TO RUN: ”sim set.dat,x topo.dat, y topo.dat, f topo.dat, and size topo.dat”

9.3

1HD: Breaking solitary wave runup and rundown

Solitary wave runup and rundown was investigated experimentally by Synolakis (1986,1987). In his work, dozens of experimental trials were performed, encompassing two orders of magnitude of solitary wave height. The beach slope was kept constant at 1:19.85. Many researchers have used this data set to validate numerical models (e.g., Zelt, 1991; Lin et al., 1999). Synolakis (1986) also photographed the waves during runup and rundown. One set of these snapshots, for ε = 0.28, was digitized and compared with the numerical prediction, shown in Figure 3. The numerical simulation shown in this figure uses f = 10−3 . The wave begins to break between Figs. 3a) and 3b), and the runup/rundown process is shown in Figs. 3c)d). In Fig 3c), numerical snapshots from three other models are plotted. The comparisons indicate a significant improvement over weakly nonlinear Boussinesq equation results of Zelt (1991) and the NLSW results of Titov and Synolakis (1995). Additionally, the numerical 15

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16

t/T

Figure 2: Comparison between numerical (solid lines) and experimental (dots) free surface displacements for Case A of Dingeman (1994), where the experimental setup and gauge locations are shown in the top subplot. The column on the left shows the numerical results from the one-layer model, the right column shows the two-layer results. Time series locations are indicated in the upper right of each subplot, corresponding to the gauge locations shown in the top subplot. 16

0.6 0.4 0.2 a)

0 −0.2 −20

−15

−10

−5

0

5

10

0.6 0.4 0.2 b)

0 −0.2 −20

−15

−10

−5

0

5

10

0.6

z/h

0.4 0.2 c)

0 −0.2 −20

−15

−10

−5

0

5

10

0.6 0.4 0.2 d)

0 −0.2 −20

−15

−10

−5

0

5

10

Figure 3: Breaking solitary wave runup and rundown on a planar beach at t(g/h)1/2 = a) 15, b) 20, c) 25, d) 45. The solid line represents the numerical results and the points experimental data. In c) the dashed line represents numerical results by Lin et al. (1999) (closest to experiment and numerical results presented in this paper), the dotted line represents results by Zelt (1991), and the dashed-dotted line results by Titov and Synolakis (1995). 17

Wave Height (cm)

10 8 6 4 2 0

0

0.05

0.1

0.15

0.2 Depth (m)

0.25

0.3

0.35

0

0.05

0.1

0.15

0.2 Depth (m)

0.25

0.3

0.35

Mean Free Surface (cm)

0.8 0.6 0.4 0.2 0 −0.2

Figure 4: Experimental (dots) and numerical (line) wave height and mean free surface for Hansen and Svendsen case 041041. results by COULWAVE compare favorably to the two dimensional (vertical plane) results of Lin et al. (1999), which makes use of a complex turbulence model. LOCATION OF INPUT FILES: /examples/Breaking Solitary Wave FILES NEEDED TO RUN: ”sim set.dat”

9.4

1HD: Breaking regular wave runup and rundown

Hansen and Svendsen (1979) performed a number of regular wave tests on plane slopes. One of these experiments, trail 041041, is recreated numerically by COULWAVE. The waves were generated in 0.36 m of water, and shoaled up a 1:34.26 slope. Time series were taken at numerous locations along the wave flume; wave height and mean free surface elevation will be compared in Figure 4. LOCATION OF INPUT FILES: /examples/Regular Breaking Wave FILES NEEDED TO RUN: ”sim set.dat”

18

9.5

1HD: Nonlinear, deep-water wave groups

Stansberg (1993) presented a number of deep water wave experiments, including regular, bichromatic, and irregular wave trials. The experiments were performed in a 270 m in length wave channel at MARINTEK, Norway. The channel is 10 m deep for the initial 80 m of length, and 5 m deep elsewhere. The two depths are connected with a vertical step. The two-layer model will be compared with the experimental run ”00060”. This run employed a bichromatic wave group composed of waves with periods = 1.9 and 2.1 seconds, or equivalently kh = 5.5 and 4.6 in the 5 m depth water (kh = 11 and 9.1 in the 10 m depth water). Each component had an height of 0.16 m. Time series were recorded at 9.3, 40, 80, 120, 160, and 200 m from the wavemaker.Figures 5-10 show the experimental-numerical comparisons at the 6 experimental measurement locations. In each of these figures, the top subplot shows the time series comparison of the free surface displacement, the middle and bottom subplots give the results of FFT’s on both numerics and experiments, where the middle plot looks at the main frequency components and the bottom the high frequency components. As a reminder, ω1 =1/2.1s=0.476 Hz and ω2 =1/1.904s=0.525 Hz. Closest to the wavemaker, in Fig. 5, numerics and experiment match up very well. Note that although the model appears to be predicting the high frequency components well, this is not expected. These components have kh values well out of the range of the two-layer model, for example the 1 Hz components has a kh ≈ 10. These high frequency components will have very large phase errors in the two-layer model. This point becomes increasingly important when we look at the comparisons further downstream, where the sidebands quickly grow. While the individual side bands have kh values in the realm of accuracy of the two-layer model, they are generated through the interaction of the primary components, ω1 and ω2 , with the secondary (high frequency) components. If the physics of the high frequency components is not being captured by the two-layer model, which they are not, it is unfair to expect the model to reproduce correctly the children of the high frequency waves, the side bands. Therefore, the only model that will properly recreate these experiments of Stansberg is one that has good linear and nonlinear accuracy to kh ≈ 12. The three-layer model will likely have these traits, and will be applied to this problem in the near future. LOCATION OF INPUT FILES: /examples/Wave Groups FILES NEEDED TO RUN: ”sim set.dat”

9.6

2HD: Solitary wave runup and rundown on a conical island

Briggs et al. (1994) presented a set of experimental data for solitary wave interaction around a conical island. The slope of the island is 1:4 and the water depth is 0.32m. Three cases were simulated, corresponding to solitary wave heights of 0.013 m (ε = 0.04) , 0.028 m (ε = 0.09), and 0.058 m (ε = 0.18). In addition to recording free surface elevation at a half dozen locations, maximum wave runup around the entire island was measured. This data set has been used by several researchers to validate numerical runup models (e.g., Liu et al., 1995; Titov and Synolakis, 1998; Chen et al., 2000). In this paper, free surface elevation is compared at the locations shown in Figure 11. Gages #6 and #9 are located near the front face of the island, with #9 situated very near the initial shoreline position. Gages #16 and #22 are also located at the initial shoreline, where #16 is on the side of the island and #22 19

Time Series Comparison at x=9.3 m 0.3

Free Surface (m)

0.2 0.1 0 −0.1 −0.2 200

Amplitude (m)

0.08

205

210

215

ω1

2ω1−ω2

220

225 230 Time (s) Main Frequency Components

ω2

2ω2−ω1

235

240

3ω2−2ω1

4ω2−3ω1

0.6 0.65 Frequency (Hz) High Frequency Components

0.7

245

250

5ω2−4ω1

0.06 0.04 0.02 0 0.4

0.45

0.5

0.55

0.75

0.8

0.012

Amplitude (m)

0.01

4ω −2ω 1

2

3ω −ω 1

2

ω +ω

2ω

1

1

2

2ω

2

3ω −ω 2

1

4ω −2ω 2

1

0.008 0.006 0.004 0.002 0 0.8

0.85

0.9

0.95

1 Frequency (Hz)

1.05

1.1

1.15

1.2

Figure 5: Deep water wave group comparisons 9.3 m from wavemaker, where the solid lines are the numerical results, and the dashed line in the top subplot and the stars in the lower two subplots are the MARINTEK data. The top plot shows free surface, the middle plot shows spectral amplitudes for the primary components, and the bottom plot gives amplitudes for the higher frequency components. The ∗’s represent the experimental spectrum in the lower two plots. The thin dashed vertical lines in the bottom two plots indicate expected locations of spectral peaks, where the ω combinations are shown to the right of the lines.

20

Time Series Comparison at x=40 m 0.3

Free Surface (m)

0.2 0.1 0 −0.1 −0.2 200

Amplitude (m)

0.08

205

210

215

ω1

2ω1−ω2

220


ω2

2ω2−ω1

235

240

245

3ω2−2ω1

4ω2−3ω1

5ω2−4ω1


0.7

0.75

250

0.06 0.04 0.02 0 0.4

0.45

0.5

0.55

0.8

0.012

Amplitude (m)

0.01

4ω −2ω 1

2

3ω −ω 1

2

ω +ω

2ω

1

1

2

2ω

2

3ω −ω 2

1

4ω −2ω 2

1

0.008 0.006 0.004 0.002 0 0.8

0.85

0.9

0.95

1 Frequency (Hz)

1.05

1.1

1.15

1.2

Figure 6: Deep water wave group comparisons 40 m from wavemaker, where the solid lines are the numerical results, and the dashed line in the top subplot and the stars in the lower two subplots are the MARINTEK experimental data. Figure setup is the same as with Fig. 5.

21


Free Surface (m)

0.2 0.1 0 −0.1 −0.2 200

Amplitude (m)

0.08

205

210

215

ω1

2ω1−ω2

220


ω2

2ω2−ω1

235

240

245

3ω2−2ω1

4ω2−3ω1

5ω2−4ω1


0.7

0.75

250

0.06 0.04 0.02 0 0.4

0.45

0.5

0.55

0.8

0.012

Amplitude (m)

0.01

4ω −2ω 1

2

3ω −ω 1

2

ω +ω

2ω

1

1

2

2ω

2

3ω −ω 2

1

4ω −2ω 2

1

0.008 0.006 0.004 0.002 0 0.8

0.85

0.9

0.95

1 Frequency (Hz)

1.05

1.1

1.15

1.2


22


Free Surface (m)

0.2 0.1 0 −0.1 −0.2 200

Amplitude (m)

0.08

205

210

215

ω1

2ω1−ω2

220


ω2

2ω2−ω1

235

240

245

3ω2−2ω1

4ω2−3ω1

5ω2−4ω1


0.7

0.75

250

0.06 0.04 0.02 0 0.4

0.45

0.5

0.55

0.8

0.012

Amplitude (m)

0.01

4ω −2ω 1

2

3ω −ω 1

2

ω +ω

2ω

1

1

2

2ω

2

3ω −ω 2

1

4ω −2ω 2

1

0.008 0.006 0.004 0.002 0 0.8

0.85

0.9

0.95

1 Frequency (Hz)

1.05

1.1

1.15

1.2


23


Free Surface (m)

0.2 0.1 0 −0.1 −0.2 200

Amplitude (m)

0.08

205

210

215

ω1

2ω1−ω2

220


ω2

2ω2−ω1

235

240

245

3ω2−2ω1

4ω2−3ω1

5ω2−4ω1


0.7

0.75

250

0.06 0.04 0.02 0 0.4

0.45

0.5

0.55

0.8

0.012

Amplitude (m)

0.01

4ω −2ω 1

2

3ω −ω 1

2

ω +ω

2ω

1

1

2

2ω

2

3ω −ω 2

1

4ω −2ω 2

1

0.008 0.006 0.004 0.002 0 0.8

0.85

0.9

0.95

1 Frequency (Hz)

1.05

1.1

1.15

1.2


24


Free Surface (m)

0.2 0.1 0 −0.1 −0.2 200

Amplitude (m)

0.08

205

210

215

ω1

2ω1−ω2

220


ω2

2ω2−ω1

235

240

245

3ω2−2ω1

4ω2−3ω1

5ω2−4ω1


0.7

0.75

250

0.06 0.04 0.02 0 0.4

0.45

0.5

0.55

0.8

0.012

Amplitude (m)

0.01

4ω −2ω 1

2

3ω −ω 1

2

ω +ω

2ω

1

1

2

2ω

2

3ω −ω 2

1

4ω −2ω 2

1

0.008 0.006 0.004 0.002 0 0.8

0.85

0.9

0.95

1 Frequency (Hz)

1.05

1.1

1.15

1.2


25

Figure 11: Conical island setup. The gage locations are shown by the dots, and the wave approaches the island from the left.

26

120

90 1

60

0.5

150

a)

120

30

180

330 240

270

300

60

0.5

150 0

210

90 1

b)

120

30

180

330 240

270

300

60

0.5

150 0

210

90 1

c) 30

180

0

210

330 240

270

300

Figure 12: Maximum horizontal runup, scaled by the initial shoreline radius, for case A a), case B b), and case C c). Experimental values are shown by the stars and the numerical results by the solid line. on the back face. As mentioned, maximum runup was experimentally recorded. The vertical runup heights are converted to horizontal runups, scaled by the initial shoreline radius, and plotted on Figure 12. The crosshairs represents the experimental data, where Fig. 12a) is for Case 1., Fig. 12b) is for Case 2, and Fig. 12c) is for Case 3. The numerical maximum inundation is also plotted, given by the solid line. The agreement for all cases is very good. This particular simulation is a good example of how the Courant number needs to be lower for ”fully-nonlinear” simulations where the seafloor slope is large. For this case, the slope of the islands sides is roughly 0.25. These steep slopes lead to very rapidly moving water during the rundown phase. The Courant number is calculated based on the maximum velocity of the incident wave in deep water. However, the rundown velocity greatly exceeds this deep water velocity, and therefore the stability requirement is no longer satisfied during the rundown. To eliminate this instability, the time step must be lowered. LOCATION OF INPUT FILES: /examples/Conical Island FILES NEEDED TO RUN: ”sim set.dat,x topo.dat, y topo.dat, f topo.dat, and size topo.dat”

9.7

2HD: Regular wave evolution over a submerged shoal

One of the most frequently studied 2HD problems is that of wave interaction with a submerged elliptic shoal. Experiments by Berkoff et al. (1982), Vincent & Briggs (1989), and others have been used repeatedly to validate mild-slope equations models (e.g. Liu et al., 1985) as well as Boussinesq-type models (e.g. Chen et al., 2000). The submerged shoal is a particularly desirable 2HD validation problem because the wave field behind the shoal can vary greatly in both the along-channel and cross-channel directions, indicating the 2HD effects are very important. One of the experiments of Vincent & Briggs (1989) will be numerically simulated. The elliptic shoal is 6.1 m long in the x direction and 7.92 m wide in the y direction, with a maximum height of 30.5 cm in 45.7 cm of water. The precise mathematical description of 27

Figure 13: Numerical snapshot from a two-layer, 2HD shoal simulation, where the location of the shoal is denoted by the dashed contours. The snapshot is taken 32 s into the simulation, or roughly 24.6 wave periods.

28

the shoal can be found in Vincent & Briggs. A large variety of incident wave conditions were studied, ranging from non-breaking monochromatic waves to breaking directional spectra. A non-breaking monochromatic incident condition is examined in this example. The incident wave has a height of 4.8 cm and a period of 1.3 s (kh=1.27). The incident wave height was chosen based on agreement with experimental data taken nearest to the wavemaker, which varied significantly from the experimental target wave height of 5.5 cm. A snapshot of the quasi-steady state free surface, taken 32 s into the simulation, is shown in Figure 13. LOCATION OF INPUT FILES: /examples/Vincent Briggs Shoal FILES NEEDED TO RUN: ”sim set.dat,x topo.dat, y topo.dat, f topo.dat, and size topo.dat”

9.8

2HD: Directional wave spectra evolving in Wiamea Bay, HI

This example is a simulation setup put together by Greg Wong at the University of Hawaii. The input wave condition is a directional spectrum. The spectrum data is taken from the field, and represents a typical winter storm condition. The area modeled is along the north shore of Oahu, where the waves are known to be extremely large during the winter months. Shown in Figure 14 is snapshot of the wave field, showing the moderate irregularity of the incoming wave condition. Also shown are the locations of the breaking waves, in Figure 15, where the breaking fronts are indicated by the white. LOCATION OF INPUT FILES: /examples/Wiamea FILES NEEDED TO RUN: ”sim set.dat, spectrum.dat, x topo.dat, y topo.dat, f topo.dat, and size topo.dat”

29

Figure 14: Instantaneous snapshot of sea surface elevation for Wiamea Bay simulation.

30

Figure 15: Instantaneous snapshot of breaking locations for Wiamea Bay simulation.

31

10

Appendix

The appendix consists of a a section detailing the properties of the model equations solved by COULWAVE, a section describing the numerical details such as wave breaking, and lastly a collection of papers describing the development of COULWAVE. The first paper, ”A multilayer modeling approach to wave modeling”, gives the mathematical details regarding the derivation of the model equations solved by COULWAVE. The second paper, ”Modeling wave runup with depth-integrated equations”, described the moving boundary algorithm employed by COULWAVE. A third paper, ”A numerical study of submarine landslide generated waves and runup”, shows how the model can be applied to the problem of waves generated by seafloor disturbances. The paper references are: Lynett, P. and Liu, P. L.-F. 2002 A Multi-Layer Approach to Water Wave Modeling. Journal of Fluid Mechanics submitted. Lynett, P., Wu, T.-R., and Liu, P. L.-F. 2002 Modeling Wave Runup with DepthIntegrated Equations. Coast. Engrg., 46(2), 89-107. Lynett, P. and Liu, P. L.-F. 2002 A Numerical Study of Submarine Landslide Generated Waves and Runup. Royal Society of London A in press.

32

11 11.1

Appendix A. Analysis of Multi-Layer Models One-Layer Equation Model

For the one-layer model, the horizontal velocity vector is given as (

U 1 = u1 −

µ2o

)

z12 − κ21 ∇S1 + (z1 − κ1 )∇T1 + O(µ4o ) 2

(2)

where

1 ∂h (3) εo ∂t The exact continuity equation can be rewritten approximately in terms of ζ and u1 as: S 1 = ∇ · u1 ,

T1 = ∇ · (hu1 ) +

1 ∂h ∂ζ + + ∇ · [(εo ζ + h) u1 ] εo ∂t ∂t ("

−µ2o ∇

·

#

ε3o ζ 3 + h3 (εo ζ + h)κ21 − ∇S1 6 2

"

#

)

ε2 ζ 2 − h2 + o − (εo ζ + h)κ1 ∇T1 = O(µ4o ) 2

(4)

Equation (4) is one of two governing equations for ζ and u1 . The momentum equation for u1 is ( ) κ21 ∂u1 2 ∂ + εo u1 · ∇u1 + ∇ζ + µo ∇S1 + κ1 ∇T1 ∂t ∂t 2 "

+εo µ2o

κ2 (u1 · ∇κ1 )∇T1 + κ1 ∇ (u1 · ∇T1 ) + κ1 (u1 · ∇κ1 )∇S1 + 1 ∇ (u1 · ∇S1 ) 2 "

+εo µ2o

Ã

∂T1 T1 ∇T1 − ∇ ζ ∂t "

+ε3o µ2o ∇

!#

Ã

+

ε2o µ2o ∇

ζ 2 ∂S1 ζS1 T1 − − ζu1 · ∇T1 2 ∂t

# ´ ζ2 ³ 2 S1 − u1 · ∇S1 = O(µ4o ) 2

#

!

(5)

This one-layer model, often referred to as the ”fully nonlinear, extended Boussinesq equations” in the literature (e.g. Wei & Kirby, 1995), has been examined and applied to a significant extent. The weakly nonlinear version of (4) and (5) (i.e. assuming O(εo ) = O(µ2o ), thereby neglecting all nonlinear dispersive terms) was first derived by Nwogu (1993). Nwogu, through linear and first-order nonlinear analysis of the equation model, recommended that z1 = −0.531h, and that value has been, for the most part, adopted by other researchers using these equations. Nwogu’s model was extended to ”full nonlinearity” by Liu (1994) and Wei & Kirby (1995). There are some discrepancies between Liu’s and Wei & Kirby’s derived equations, which can be attributed to a neglect of some nonlinear dispersive terms in Wei & Kirby (Hsaio & Liu, 2002). The above, one-layer model equations (4) and (5) are identical to those derived by Liu (1994). The one-layer model has been used to study a number of 2HD real world phenomenon, including rip currents (Chen et al., 1999), longshore currents (Chen et al., 2002), and a 33

variety of harbor problems (e.g. Shi et al., 2002). The numerical scheme employed for these simulations is adopted here for the two-layer model, and will be described in detail in Chapter 4.

11.2

Two-Layer Equation Model

For the two-layer model, we can define the horizontal velocity vectors as (

U 2 = u2 −

µ22 (

U 1 = u1 − where S2 =

µ21

)

z22 − κ22 ∇S2 + (z2 − κ2 )∇T2 + O(µ42 ) 2

(6)

)

z12 − κ21 ∇S1 + (z1 − κ1 )∇T1 + O(µ41 , µ21 µ22 ) 2

d2 ∇ · u2 , ho

T2 = ∇ · (hu2 ) + Ã

d1 S 1 = ∇ · u1 , ho

(7)

1 ∂h εo ∂t !

d1 T1 = η S2 − S1 + T2 d2

(8)

The exact continuity equation can be rewritten approximately in terms of ζ, u1 , and u2 as: "

Ã

!

ho ∂h ho ∂ζ ho + + ∇ · (ε1 ζ − η) u1 + η + h u2 d1 εo ∂t d1 ∂t d1

#

  3 3 3 d1 3 ho  d1 ho 2  η + h (η + h d2 )κ2  d2 d32  d3 −µ22 ∇ ·  2 − d2  ∇S2 d1 6 2  

d2

2

2

2

η 2 1 − h2 hd2o  d2

+

2 ("

−µ21 ∇

·

  



− (η

d1 ho + h )κ2   ∇T2  d2 d2  #

ε31 ζ 3 − η 3 (ε1 ζ − η)κ21 − ∇S1 6 2

"

#

)

ε21 ζ 2 − η 2 + − (ε1 ζ − η)κ1 ∇T1 = O(µ41 , µ21 µ22 , µ42 ) 2

(9)

Equation (9) is one of three governing equations for ζ and un . The governing, momentum equation for u1 is ∂u1 ∂ + εo u1 · ∇u1 + ∇ζ + µ21 ∂t ∂t "

+εo µ21

(

κ21 ∇S1 + κ1 ∇T1 2

)

κ2 (u1 · ∇κ1 )∇T1 + κ1 ∇ (u1 · ∇T1 ) + κ1 (u1 · ∇κ1 )∇S1 + 1 ∇ (u1 · ∇S1 ) 2 "

+εo µ2o

Ã

∂T1 T1 ∇T1 − ∇ ζ ∂t

!#

Ã

+

ε2o µ2o ∇ 34

ho ζ 2 ∂S1 ζS1 T1 − − ζu1 · ∇T1 d1 2 ∂t

!

#

"

+ε2o ε1 µ2o ∇

Ã

ζ2 ho S12 − u1 · ∇S1 2 d1

!#

= O(µ41 , µ20 µ21 , µ21 µ22 )

(10)

Determination of u2 does not require solving an additional momentum equation. With interfacial boundary condition (continuous velocity) and the known velocity profiles (6) and (7), u2 can be explicitly given as a function of u1 :   Ã ! d2    κ22 − d12 η 2  d1 2 2 u 2 + µ2  ∇S2 + κ2 − η ∇T2  = 2 d2   (

u1 +

µ21

)

κ21 − η 2 ∇S1 + (κ1 − η) ∇T1 + O(µ41 , µ21 µ22 , µ42 ) 2

(11)

Thus, the lower layer velocity can be directly calculated with knowledge of the upper layer velocity. Equations (9), (10), and (11) are the coupled governing equations for the two-layer system. 11.2.1

Analysis of Model Equations

In this section, the properties of the two-layer model will be scrutinized and optimized. First, it is shown that the two-layer model will reduce to the well-studied, ”extended” Boussinesq model derived by Ngowu (1993). With the use of O(µ2n ) substitutions, namely: u2 = u1 + O(µ2n ),

(12)

we can eliminate one of the unknowns from our equation system. Rewriting (9) in terms of u1 only, assigning d1 = ho , κ2 = − hd2o h, η = −h, and examining the weakly nonlinear form of the equations, gives 1 ∂h ∂ζ + + ∇ · [(εo ζ + h) u1 ] εo ∂t ∂t ("

−µ2o ∇

·

#

"

#

)

h3 hκ21 h2 − ∇S1∗ − + hκ1 ∇T1∗ = O(εo µ2o , µ4o ) 6 2 2

where S1∗ = ∇u1 ,

T1∗ = ∇ · (hu1 ) +

1 ∂h εo ∂t

(13)

(14)

The momentum equation, (10), becomes ∂u1 ∂ + εo u1 · ∇u1 + ∇ζ + µ2o ∂t ∂t

(

)

κ21 ∇S1∗ + κ1 ∇T1∗ = O(εo µ2o , µ4o ) 2

(15)

This system for ζ and u1 is identical to the model derived by Ngowu. Additionally, the nonlinear dispersive terms, which have been truncated for the sake of brevity in (13) and (15), are identical to those derived by Liu (1994). For the rest of this paper, the ”extended” Boussinesq model including all the nonlinear dispersive terms up to O(µ2o ), as given by Liu (1994), will be referred to as the one-layer model. 35

For the rest of this section, the focus will be on analysis of the three-unknown, (ζ, u1 , and u2 ) two-layer system. Additionally for the rest of this section, all quantities discussed are in dimensional form, with asterisks no longer applied. With the weak rotationality assumption, the momentum equation, (10), can be simplified, in dimensional form, to (see Hsiao and Liu, 2002) ∂u1 1 ∂ + ∇(u1 · u1 ) + g∇ζ + ∂t 2 ∂t

(

Ã

!

)

κ21 ζ2 ∇S1 + κ1 ∇T1 − ∇ S1 − ∇ (ζT1 ) 2 2

(

´ ∂ζ 1³ 2 (T1 + ζS1 ) + (κ1 − ζ) (u1 · ∇) T1 + κ1 − ζ 2 (u1 · ∇) S1 ∂t 2 ¾ i 1h + (T1 + ζS1 )2 = 0 (16) 2 This is the momentum equation that will be analyzed and numerically solved in this paper. Before solving the system, the linear and nonlinear dispersion properties are examined. Let us define the arbitrary evaluation levels and the boundary between the two layers as:

+∇

κ1 = α1 h + β1 ζ,

η = α2 h + β2 ζ,

κ 2 = α 3 h + β3 ζ

(17)

where the coefficients α and β are arbitrary and user-defined. The one-horizontal dimension, constant water depth, two-layer equations are rewritten in dimensional form, keeping, for brevity, only linear terms. These equations are 3 ∂ζ ∂u1 ∂u2 ∂ 3 u1 3 ∂ u2 + δ1 h + δ2 h + δ3 h3 + δ h =0 4 ∂t ∂x ∂x ∂x3 ∂x3 3 3 ∂u1 ∂ζ 2 ∂ u1 2 ∂ u2 +g + δ5 h + δ6 h =0 ∂t ∂x ∂x2 t ∂x2 t 2 ∂ 2 u1 2 ∂ u2 u1 − u2 − δ7 h2 − δ h =0 8 ∂x2 ∂x2

(18) (19) (20)

where

−2α23 + 6α1 α22 − 3α12 α2 , 6 2α3 − 6α1 α22 − 6α1 α2 + 3α32 α2 + 6α3 α2 + 3α32 + 6α3 + 2 δ4 = 2 , 6 α2 α2 + α22 δ5 = 1 − α1 α2 , δ6 = α1 α2 + α1 , δ7 = − 1 + α1 α2 2 2 α2 + α32 δ8 = 2 − α1 α2 + α3 − α1 2 The assumed dimensional solution form δ1 = −α2 ,

δ2 = 1 + α2 ,

δ3 =

(21)

ζ = ²ζ (0) eiθ + ²2 ζ (1) e2iθ + .... (0)

(1)

(0)

u1 = ²u1 eiθ + ²2 u1 e2iθ + ....

(1)

u2 = ²u2 eiθ + ²2 u2 e2iθ + ....

(22)

where θ = kx−wt, k is the wavenumber, w is the wave frequency, and ² is simply an ordering parameter, are substituted into the derived equations. 36

11.2.2

Linear Dispersion Relation

The first order (in ²) system yields the linear dispersion relation: c2 =

w2 gh [1 + N1 (kh)2 + N2 (kh)4 ] = k2 1 + D1 (kh)2 + D2 (kh)4

(23)

where c is the wave celerity and the coefficients N1 , N2 , D1 , and D2 are given in Appendix A.1 and are solely functions of α1 , α2 , and α3 . The above dispersion relation will be compared with both the [4,4] Pade approximation c2 =

w2 gh [1 + 1/9(kh)2 + 1/945(kh)4 ] = k2 1 + 4/9(kh)2 + 1/63(kh)4

(24)

and the [6,6] Pade approximation c2 =

w2 gh [1 + 5/39(kh)2 + 2/715(kh)4 + 1/135135(kh)6 ] = k2 1 + 6/13(kh)2 + 10/429(kh)4 + 4/19305(kh)6

(25)

of the exact linear dispersion relation: w2 g = tanh(kh) 2 k k

c2e =

(26)

The Pade approximates utilized here are approximations of the hyperbolic tangent function, where the numbers in the brackets represent the highest polynomial order of kh in the numerator and denominator. Group velocity of the two-layer model equations, cg , can be determined straightfowardly by taking the derivative of (23) with respect to k. 11.2.3

Vertical Velocity Profile

Let us define the function f1 (z) as the horizontal velocity, with constant water depth, normalized by its value at z = 0. This function is composed of two quadratic polynomial elements, given by: f1 (z) =

1 + (kh)2

f1 (z) = f1 (η)

h

1 2

(0)

i

(0)

(z 2 /h2 − α12 ) + α2 (α1 − z/h) + u2 /u1 (α2 + 1)(z/h − α1 ) 1 − (kh)2

1 + (kh)2

h

h

1 2 α 2 1

(0)

(0)

− α2 α1 + u2 /u1 (α2 + 1)α1

for z ≥ η = α2 h 1 2

1 + (kh)2

1 2

i

(α22 − α32 ) + (α2 − α3 )

, (27)

i

(z 2 /h2 − α32 ) + (z/h − α3 )

h

i

, for z < η = α2 h

(28)

From the linear equation system we know that, (0)

u1 = (0)

u2 =

gζ (0) [kh − δ8 (kh)3 ] hw [1 + D1 (kh)2 + D2 (kh)4 ]

(29)

gζ (0) [kh + δ7 (kh)3 ] hw [1 + D1 (kh)2 + D2 (kh)4 ]

(30)

37

(0)

(0)

and thus the ratio u2 /u1 present in (27) can be evaluated. Similarly, the vertical velocity profile, normalized by the velocity at the still water level, is given by f2 (z): (0)

f2 (z) =

(0)

z/h − α2 + u2 /u1 (α2 + 1) (0)

(0)

−α2 + u2 /u1 (α2 + 1) f2 (z) = f2 (η)

, for z ≥ η = α2 h

z/h + 1 , for z < η = α2 h α2 + 1

(31)

(32)

which is a piecewise linear function. 11.2.4

Linear Shoaling Properties

Based on linear theory, the exact shoaling gradient is given as: aex hx [1 − khtanh(kh)][1 − tanh2 (kh)] hx = Aex = −khtanh(kh) a h {tanh(kh) + kh[1 − tanh2 (kh)]}2 h

(33)

The linear shoaling properties of the two layer model are determined using the constancy of energy flux concept, i.e ax 1 (Cg )x =− (34) a 2 Cg where Cg is the wave group velocity. First, the derivative of (23) is taken with respect to k, giving: w (kh)S1 cg = (35) g S22 where cg is the wave group velocity, and S1 = D2 N2 (kh)8 + 2D1 N2 (kh)6 + (3N2 + D1 N1 − D2 ) (kh)4 + 2N1 (kh)2 + 1 S2 = D2 (kh)4 + D1 (kh)2 + 1

(36) (37)

Taking the derivative of (35) with respect to x, noting that dw/dx=0, we have w S3 (cg )x = (kh)x 3 g S2

(38)

where S3 = D22 N2 (kh)12 + 3D1 D2 N2 (kh)10 + (6D12 N2 − 3D1 D2 N1 + 3D22 )(kh)8 +(17D1 N2 − 10D2 N1 + D12 N1 − D1 D2 )(kh)6 + (15N2 + 3D1 N1 − 12D2 )(kh)4 giving the ratio

+(6N1 − 3D1 )(kh)2 + 1

(39)

(cg )x (kh)x S3 = cg kh S1 S2

(40)

38

Taking the derivative of the dispersion relation (23), with respect to x, gives kx 1 S4 hx =− k 2 S1 h where S4 = D2 N2 (kh)8 + (3D1 N2 − D2 N1 )(kh)6 +(5N2 + D1 N1 − 3D2 )(kh)4 + (3N1 − D1 )(kh)2 + 1

(41)

(42)

Finally, the linear shoaling gradient of the two-layer model can be given: ¶ µ ax S4 S3 hx 1 =− − (43) a 2 4S1 S1 S2 h Note that this solution form is valid in any system for which the dispersion relation can be expressed in the form of (23). 11.2.5

Second Order, Nonlinear Interactions: Steady Waves

Now we find the nonlinear corrections to the linear problem. The two-layer equations must now be truncated to include quadratic nonlinear terms, as well as linear terms. Collecting the O(²2 ) terms from the substitution of the assumed steady wave, (22), into the nonlinear equation system will yield an equation system in the general form: 









(1) b11 b12 b13  ζ R1    (1)      b21 b22 b23   u1  =  R2  (1) 0 b32 b33 R3 u2

where b11 , ..., b33 are functions of the linear δ coefficients, and R1 , .., R3 are tedious functions of the α and β parameters. This approximate expression can be compared to the secondorder solution: 2 kζ (0) (1) ζStokes = [3coth3 (kh) − coth(kh)] (44) 4 which is derived from Stokes theory. 11.2.6

Second Order, Nonlinear Interactions: Bichromatic Interactions

Examining a two-wave group, the free surface can be written as (0)

(0)

(1)

(1)

ζ = ²ζ1 ei(k1 x−w1 t) + ²ζ2 ei(k2 x−w2 t) + ²2 ζ1 e2i(k1 x−w1 t) + ²2 ζ2 e2i(k2 x−w2 t) +²2 ζ + ei(k+ x−w+ t) + ²2 ζ − ei(k− x−w− t)

(45)

where ζ+ , ζ− are the sum and difference components of the two first order wave frequencies, k∓ = k1 ∓ k2 , and w∓ = w1 ∓ w2 . Similar expressions can be given for un . To find the sub- and super-harmonic amplitudes for the bichromatic wave group problem, the procedure is the same as described above for the steady wave (single, first-order harmonic) problem. The assumed solution (45) is substitued into the two-layer equation system. For each of the forced second-order solutions, [(k1 − k2 )x − (w1 − w2 )t] and [(k1 + k2 )x + (w1 − w2 )t], the matrix system is written in the same form as for the steady wave problem. The sum and ∓ difference free surface components can be compared with those from Stokes theory, ζStokes , which can be found in Shaffer (1996). 39

Table 1: α values Ω (kh) 3 5 7.5 10 [4,4] Pade :

11.2.7

from linear optimization for two-layer model. α1 α2 α3 ∆Linear -0.225 -0.420 -0.713 0.00008 -0.204 -0.383 -0.685 0.002 -0.175 -0.331 -0.646 0.009 -0.155 -0.294 -0.620 0.020 -0.248 -0.459 -0.741 ——

Choice of Arbitrary Levels: Linear Optimization

Through examination of linear and nonlinear properties, the most accurate set of arbitrary levels will be chosen in this section. First, the linear properties of the two-layer model will be optimized, independent of nonlinearity. In the linear sense, the three levels are given as κ1 = α1 h, η = α2 h, and κ2 = α3 h, where κ1 and κ2 are the levels at which horizontal velocities are evaluated in the upper and lower layers, and η is the location of the interface between the layers. Of course, possible values are bounded by 0 ≥ α1 ≥ α2 ≥ α3 ≥ −1. Defining a model accuracy, or model error, can be difficult and often can depend on the specific physical problem being examined. For this analysis, a representation of the overall error, including errors in wave speed, group velocity, and shoaling, is sought. The error will be given by the minimization parameter ∆Linear :  

∆Linear

Ω X |ceg − cg |

Ω X |ce − c|

1  kh=0.1 kh =  Ω X 3 |ce |  kh=0.1 kh

+

kh=0.1 Ω X

kh

|ceg | kh=0.1 kh

Ω X |Aex − Ax |

+

kh=0.1 Ω X

kh

|Aex | kh=0.1 kh

      

(46)

where ce , ceg , and Aex are the exact linear phase speed, group velocity, and shoaling gradient, whereas c, cg , and Ax are the approximate values taken from the two-layer model derived here. The right hand side is divided by three, so as to normalize the total error created by the three different sources. All of the summations are divided by kh so that errors at low wave numbers are more important than high wave number errors. The reason for this weighting is a peculiarity of the optimization: it was possible to sacrifice low wavenumber accuracy (kh 7, where hc is the water depth above the centerpoint of the slide mass. This accuracy limitation was found through comparison of the one-layer model with a fully nonlinear potential flow calculation. One of the comparisons presented in the paper by Lynett & Liu (2002) (specifically, ”section 7. Limitations of the Depth-Integrated Model”) is re-examined here with the present two-layer model. The scenario to be recreated in this section involves a non-deforming mass translating down a planar slope in solid body motion. The mathematical description of the slide evolution can be found in Lynett & Liu (2002) (see Figure 4 in referenced paper), and will not be repeated here. This particular problem, already simulated using potential flow theory and a one-layer model, is calculated again with the two-layer equations, and the results are shown in Figure 11. The top subplot of this figure shows the location of the slide mass at four different times, which correspond to the free surface snapshots shown in the lower four subplots. Given on each of these four plots is the ls /hc ratio at the time the snapshot is taken. Note that ls is constant in time, but as the slide moves into deeper water, hc increases. For the times t1 and t2 , all three models agree, and the depth-integrated models are still in the range of accuracy. As can be seen by time t3 , the one-layer model begins to diverge from the potential flow results, indicating that the slide is in water too deep for this model to handle accurately. At this time, the two-layer

Journal of Fluid Mechanics

27

model is still in excellent agreement with potential theory. By time t4 , the two-layer model is beginning to differ from potential theory. Although the rigorous determination of the accuracy limit for the one-layer model done in Lynett & Liu will not be repeated here for the two-layer model, it is clear that the two-layer model gives accurate results into deeper water than the one-layer model does. 4.4. 2HD Wave Evolution over a Shoal One of the most frequently studied 2HD problems is that of wave interaction with a submerged elliptic shoal. Experiments by Berkoff et al. (1982), Vincent & Briggs (1989), and others have been used repeatedly to validate mild-slope equations models (e.g. Liu et al., 1985) as well as Boussinesq-type models (e.g. Chen et al., 2000). The submerged shoal is a particularly desirable 2HD validation problem because the wave field behind the shoal can vary greatly in both the along-channel and cross-channel directions, indicating the 2HD effects are very important. In this paper, one of the experiments of Vincent & Briggs (1989) will be numerically simulated. The elliptic shoal is 6.1 m long in the x direction and 7.92 m wide in the y direction, with a maximum height of 30.5 cm in 45.7 cm of water. The precise mathematical description of the shoal can be found in Vincent & Briggs. A large variety of incident wave conditions were studied, ranging from non-breaking monochromatic waves to breaking directional spectra. A non-breaking monochromatic incident condition is examined in this paper. The incident wave has a height of 4.8 cm and a period of 1.3 s (kh=1.27). The incident wave height was chosen based on agreement with experimental data taken nearest to the wavemaker, which varied significantly from the experimental target wave height of 5.5 cm. The numerical parameters of the simulations are ∆x=0.056m and ∆t=0.026s. A snapshot of the quasi-steady state free surface, taken 32 s into the simulation, is shown in Figure 12. From this image, the processes of wave transformation can be explained.

28

A multi-layer approach to wave modeling

The waves approach from the left, and in passing over the shoal, the wave front slows in the shallower water, and the wave crest narrows and steepens. On the lee of the shoal, refracting wave fronts meet, creating a free surface elevation maximum. Oblique wave interactions dominate the wave field behind the shoal, creating an irregular sea surface. Both one- and two-layer simulations were performed, and shown in Figure 13 is the difference in the free surface elevation predicted by the models at time=32 s. This instantaneous difference shows that the wave fields predicted by the models behind the shoal do not agree particularly well. Differences in the models are in the range of ∓1.25 cm, in an area where the wave has an amplitude of 4-5 cm. As in the one-horizontal dimension problem of section 4.2, it is expected that the higher frequency waves generated due to nonlinear transfers on the shoal are not accurately predicted by the one-layer model behind the shoal. Figure 14 proves this expectation. First, however, given in Fig. 14b) is the experimental wave height along the centerline of the channel (y=0) plotted with the one- and two-layer values. The trends of the experimental and numerical data are very similar, with small but clear differences between the one- and two-layer models behind the shoal. Figs. 14c) and 14d) are the first and second harmonic amplitudes, respectively, from the one- and two-layer simulations. The two models agree closely on first harmonic amplitudes, but do exhibit 0.25 cm differences behind the shoal. Figs 14c) shows very clearly the focusing of wave energy behind the shoal. The refracting wave fronts meet and create a peak height near x=10 m, after which the wave energy spreads laterally, decreasing the first harmonic amplitude. Looking at the second harmonic amplitudes, the results of the two models diverge quickly behind the shoal. In the deep water behind the shoal, this second harmonic has a kh=4.35, and therefore the one-layer model will not be able to properly capture the physical properties of this wave component. This fact also explains


29

the differences shown in Fig. 13, where the errors in the one-layer model are directly attributable to the incorrect phase speed prediction of the second harmonic. Therefore for this type of problem, the accuracy of the one-layer model correlates directly to the magnitude of energy transfer into the second harmonic. The two-layer model will capture the second harmonic properly, although if energy transfer into the third harmonic (kh=9.8 in the water behind the shoal) is significant, even this model will have errors. The amplitude of the third harmonic for the case analyzed above is less than 0.2 cm in the region behind the shoal.

5. Conclusions A model for the transformation of highly nonlinear and dispersive waves is derived. The model utilizes two quadratic polynomials to approximate the vertical flow field, matched along an interface. Through linear and nonlinear optimization of the interface and velocity evaluation locations, it is shown that the two-layer model exhibits accurate linear characteristics up to a kh ≈ 8 and nonlinear accuracy to kh ≈ 6. This is a greater than two-fold extension to higher kh over existing O(µ2o ) Boussinesq-type models, while maintaining the maximum order of differentiation at three. Owing to this maximum order of differentiation, a tractable numerical algorithm is developed for the general 2HD problem, employing a well-studied predictor-corrector scheme. A number of 1HD and 2HD simulations are given, and the presented model shows its strength by accurately capturing the propagation of highly nonlinear and deep-water waves. It is possible that for certain situations, the two-layer model would not be adequate in regard to simulating a correct picture of the wave field. One example of this would be for the 2HD shoal problem discussed in section 4.4. If the incident wave had a greater amplitude, perhaps even breaking, the third harmonic amplitude (kh =9.8 behind the

30


shoal) would be significant. For these types of scenarios, an optimized three-layer equation model with a wave breaking scheme could be an option, and the authors are currently developing this model. The research reported here is partially supported by Grants from National Science Foundation (CMS-9528013, CTS-9808542, and CMS 9908392).


31

Appendix A. Linear Dispersion Properties The linearized one-dimensional equations in a constant depth can be written in the following dimensional form: 3 ∂u1 ∂u2 ∂ζ ∂ 3 u1 3 ∂ u2 + δ1 h + δ2 h + δ 3 h3 + δ h =0 4 ∂t ∂x ∂x ∂x3 ∂x3

(A 1)

∂u1 ∂ζ ∂ 3 u1 ∂ 3 u2 +g + δ5 h2 2 + δ6 h2 2 = 0 ∂t ∂x ∂x t ∂x t

(A 2)

u1 − u2 − δ7 h2

2 ∂ 2 u1 2 ∂ u2 − δ h =0 8 ∂x2 ∂x2

(A 3)

where δ1 = −α2 , δ4 = δ5 =

δ2 = 1 + α2 ,

δ3 =

−2α23 + 6α1 α22 − 3α12 α2 , 6

2α23 − 6α1 α22 − 6α1 α2 + 3α32 α2 + 6α3 α2 + 3α32 + 6α3 + 2 , 6

α12 − α1 α2 , 2

δ6 = α1 α2 + α1 , δ8 =

δ7 = −

α12 + α22 + α1 α2 2

α22 + α32 − α1 α2 + α3 − α1 2

(A 4)

A.1. Linear Dispersion Relation Substituting the solution form (3.6) into (A 1)-(A 3) yields the linear dispersion relation: £ ¤ k 2 gh 1 + (kh)2 N1 + (kh)4 N2 2 w = (A 5) 1 + (kh)2 D1 + (kh)4 D2 where N1 = δ2 δ7 − δ1 δ8 − δ3 − δ4 D1 = −δ8 − δ5 − δ6

N2 = δ3 δ8 − δ4 δ7 D2 = δ5 δ8 − δ6 δ7

(A 6)

A.2. Derivation of Linear Shoaling Gradient First, the derivative of (A 5) is taken with respect to k, giving: (kh)L1 w cg = g L22

(A 7)

32


where cg is the wave group velocity, and L1 = D2 N2 (kh)8 + 2D1 N2 (kh)6 + (3N2 + D1 N1 − D2 ) (kh)4 + 2N1 (kh)2 + 1

(A 8)

L2 = D2 (kh)4 + D1 (kh)2 + 1

(A 9)

Taking the derivative of (A 7) with respect to x, noting that dw/dx=0, we have L3 w (cg )x = (kh)x 3 g L2

(A 10)

where L3 = D22 N2 (kh)12 + 3D1 D2 N2 (kh)10 + (6D12 N2 − 3D1 D2 N1 + 3D22 )(kh)8 +(17D1 N2 − 10D2 N1 + D12 N1 − D1 D2 )(kh)6 + (15N2 + 3D1 N1 − 12D2 )(kh)4 +(6N1 − 3D1 )(kh)2 + 1

(A 11)

(kh)x L3 (cg )x = cg kh L1 L2

(A 12)

giving the ratio

Taking the derivative of the dispersion relation (A 5), with respect to x, gives kx 1 L4 hx =− k 2 L1 h

(A 13)

where L4 = D2 N2 (kh)8 + (3D1 N2 − D2 N1 )(kh)6 +(5N2 + D1 N1 − 3D2 )(kh)4 + (3N1 − D1 )(kh)2 + 1

(A 14)

Finally, the linear shoaling gradient of the two-layer model can be given: ax =− a

µ

1 L4 − 2 4L1

¶

L3 hx L1 L2 h

(A 15)

Note that this solution form is valid in any system for which the dispersion relation can be expressed in the form of (A 5).


33

A.3. Second-Order Nonlinear Interactions Now we find the nonlinear corrections to the linear problem. The two-layer equations must now be truncated to include quadratic nonlinear terms, as well as linear terms. Collecting the O(²2 ) terms from the substitution of the assumed steady wave, (3.6), into the nonlinear equation system will yield an equation system in the general form:       b11    b  21   0

b12 b22 b32

b13   ζ (1)    (1) b23    u1   (1) b33 u2

  R1        = R    2        R3

where b11 , ..., b33 are functions of the linear δ coefficients, and R1 , .., R3 are tedious functions of the α and β parameters. To find the sub- and super-harmonic amplitudes for the bichromatic wave group problem, the procedure is the same as described above for the steady wave (single, firstorder harmonic) problem. The assumed solution (3.22) is substituted into the two-layer equation system. For each of the forced second-order solutions, [(k1 − k2 )x − (w1 − w2 )t] and [(k1 + k2 )x + (w1 − w2 )t], the matrix system is written in the same form as for the steady wave problem. The expressions for the nonlinear harmonics are too tedious to include in the published manuscript, but can be obtained through the first author at ’[email protected]’.

Appendix B. Heuristic Analysis of Truncation Error In this section, the truncation error of the two-layer model equations is examined, and compared to those of existing one-layer depth-integrated model equations. Additionally, minimization of the two-layer truncation error yields an estimate of the layer interface location, and this value can be compared with the interface location determined in section 3.6. It is noted that the form of the truncation error in the present model equations is a

34


direct result of the scaling used to non-dimensionalize the Eulers equations. From (2.32) -(2.34) the overall accuracy of the present model equations, or the largest truncation errors of the three equations, can be expressed as O(µ2o µ21 , µ42 ). The heuristic approach taken here is to look at this truncation error as a finite value, not an order, and use this value to estimate the accuracy of the model compared to the O(µ2o ) Boussinesq equations (i.e., Ngowu, 1993) and the high-order, O(µ4o ) Boussinesq-type equations (i.e., Gobbi et al., 2000). The truncation error of the present two-layer model equations has an upper bound of O(µ4o ) when either d1 or d2 is equal to ho , which corresponds to the truncation error of a traditional (one-layer) Boussinesq model. The lower bound occurs when µ2o µ21 = µ42 , or d1 ho = d22 , which gives d1 = 0.38ho , d2 = 0.62ho . Note that the recommended boundary level from section 3.6, based on agreement with known analytical properties of water waves, is remarkably close to this estimate of d1 = 0.38ho . We can expect that the twolayer model will yield a more accurate result than the one-layer model, due to the fact that the error of approximation for the two-layer model is smaller. The truncation error values for these two models, as well as the high-order Boussinesq-type equations that have a truncation error of O(µ6o ), are shown in Figure 15. This plot indicates that over the range µo < 0.6, the high-order Boussinesq model should yield slightly more accurate results than the two-layer model. Over the range µo > 0.6, however, the two-layer model should be significantly more accurate than the high-order model. In fact, compared to a single layer model of any order, O(µno ), the two-layer model should achieve higher accuracy as µo approaches 1. In the present two-layer model, the truncation error tells us that µ2o µ21 and µ42 should be small compared to the included terms, which, due to the appearance of both O(µ2o ) and O(µ21 ) in the momentum equations, still requires that both O(µ2o ) and O(µ21 ) ¿ 1.


35

The restriction of O(µ2o ) ¿ 1 could be avoided by including O(µ2o µ21 ) terms, thereby making the truncation error of the model O(µ2o µ41 , µ41 , µ42 ). However, inclusion of O(µ2o µ21 ) terms yields a model with fifth-order in space derivatives. This is unacceptable for this particular derivation, whose primary goal is to create a high-order accurate model without requiring high-order derivatives.

36

A multi-layer approach to wave modeling REFERENCES

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Boussinesq modeling of wave transformation, breaking, and runup. Part I: 2D. Journal of Waterway, Port, Coastal and Ocean Engng. 126(1), 57-62. Dingemans, M. 1994 Comparison of computations with Boussinesq-like models and laboratory measurements. Mast-G8M note, H1684, Delft Hydraulics 32 pp. Gobbi, M. F. and Kirby, J. T. 1999 Wave evolution over submerged sills: Tests of a high-order Boussinesq model Coast. Engng. 37, 57-96. Gobbi, M. F., Kirby, J. T., and Wei, G. 2000 A fully nonlinear Boussinesq model for surface waves. Part II. Extension to O(kh)4 ) . J. Fluid Mech. 405, 182-210. Green, A. E. and Naghdi, P.M. 1976 A derivation of equations for wave propagation in water of varaible depth. J. Fluid Mech. 78, 237-346. Hsiao, S.-C. and Liu, P. L.-F. 2002 Permeable effects on nonlinear water waves. Royal Society of London A in press. Israeli, M. and Orszag, S. A. 1981 Approximation of radiation boundary conditions. J. Comp. Phys. 41, 113-115. Kanayama, S., Tanaka, H., and Shuto, N. 1998 A multi-level model for nonlinear dispersive water waves. In Coastal Engineering 1998 (ed. Billy L. Edge), vol. 1, pp.576-588. ASCE. Kennedy, A. B., Kirby, J. T., Chen, Q. and Dalrymple, R. A. 2001 Boussinesq-type equations with improved nonlinear behaviour. Wave Motion 33, 225-243. Kirby, J.T., Wei, G., Chen, Q., Kennedy, A.B. and Dalrymple, R.A. 1998 FUNWAVE 1.0: Fully Nonlinear Boussinesq Wave Model Documentation and User’s Manual. Center for Applied Coastal Research, University of Delaware. 80 pgs. Nwogu, O. 1993 Alternative form of Boussinesq equations for nearshore wave propagation. Journal of Waterway, Port, Coastal and Ocean Engng. 119(6), 618-638.


37

Liu, P. L.-F. 1994 Model equations for wave propagation from deep to shallow water. In Advances in Coastal Engineering (ed. P. L.-F. Liu), vol. 1, pp.125-157. World Scientific. Liu, P.L.-F., Yoon, S.B. and Kirby, J.T. 1985 Nonlinear refraction-diffraction of waves in shallow water. J. Fluid Mech. 153, 184-201. Lynett, P. 2002 A multi-layer approach to modeling nonlinear, dispersive waves from deep water to the shore. PhD thesis, Cornell University. Lynett, P. and Liu, P. L.-F. 2002 A numerical study of submarine landslide generated waves and runup. Royal Society of London A in press. Madsen, P. A., and Sorensen, O. R. 1992 A new form of the Boussinesq equations with improved linear dispersion characteristics. Part II: A slowly varying bathymetry. Coast. Engng. 18, 183-204. Madsen, P. A., and Schaffer, H. A. 1998 Higher order Boussinesq-type equations for surface gravity waves - Derivation and analysis. Royal Society of London A 356, 1-60. Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815-827. Schaffer, H. A. 1996 Second-order wavemaker theory for irregular wave. Ocean Engineering 23(1), 47-88. Tanaka, M. 1986 The stability of solitary waves. Phys. Fluids 29, 650-655. Vincent, C. L. and Briggs, M. J. 1989 Refraction-diffraction of irregular waves over a mound. Journal of Waterway, Port, Coastal and Ocean Engng. 115, 269-284. Wei, G. and Kirby, 1995 A time-dependent numerical code for extended Boussinesq equations. Journal of Waterway, Port, Coastal and Ocean Engng. 120, 251-261. Wei, G., Kirby, J. T., Grilli, S. T., and Subramanya, R. 1995 A fully nonlinear Boussinesq model for surface waves. Part I. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 7192. Woo, S-B. and Liu, P. L.-F. 2001 A Petrov-Galerkin finite element model for one-dimensional fully non-linear and weakly dispersive wave propagation. International Journal for Numerical Methods in Engineering 37, 541-575.

38


Figure 1. Problem setup.


39

1.2

C / Ce

1.15

a)

1.1 1.05 1 0.95

1

2

3

4

5

6

7

8

9

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Cg / Ceg

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b)

1.1

1.05 1 0.95

0.3 c)

Ax

0.2 0.1 0 −0.1

kh

Figure 2. Properties of two-layer model with α1 = −0.204, α2 = −0.383, and α3 = −0.685 (∆Linear =0.002). Comparison of wave speed and group velocity of the two-layer model (dashed line) with the exact linear relation (solid line); the dotted line is the [4,4] Pade approximation, and the dashed-dotted line is the [6,6] Pade approximation. The linear shoaling factor is shown in c), where the [6,6] Pade is not shown.

40

A multi-layer approach to wave modeling 1.2

C / Ce

1.15

a)

1.1 1.05 1 0.95

2

4

6

8

10

12

2

4

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10

12

2

4

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8

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12

1.2

Cg / Ceg

1.15

b)

1.1

1.05 1 0.95

0.3 c)

Ax

0.2 0.1 0 −0.1

kh

Figure 3. Properties of two-layer model with α1 = −0.155, α2 = −0.294, and α3 = −0.620 (∆Linear =0.020). Comparison of wave speed and group velocity of the two-layer model (dashed line) with the exact linear relation (solid line); the dotted line is the [4,4] Pade approximation, and the dashed-dotted line is the [6,6] Pade approximation. The linear shoaling factor is shown in c), where the [6,6] Pade is not shown.

Journal of Fluid Mechanics kh = 3

41

kh = 6

kh = 9

0

0

−0.2

−0.2

−0.2

−0.4

−0.4

−0.4

−0.6

−0.6

−0.6

−0.8

−0.8

−0.8

z/h

0

−1

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−1

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0.5

−1

1

0

0.5 u/umax

1

0

0.5 w/wmax

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u/umax 0

0

−0.2

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−0.8

z/h

0

−1

0

0.5 w/wmax

1

−1

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1

w/wmax

−1

Figure 4. Vertical profile of horizontal velocity (top row) and vertical velocity (bottom row) under the crest of a sine wave for three different kh values, as given by linear theory (solid line), the high-order model of Gobbi et al. (2000) (dotted line), and the 2-layer model presented in this paper employing the ∆Linear =0.002 coefficients (dashed line).

42

A multi-layer approach to wave modeling 1.4 1.35 1.3 1.25

ζ(1) / ζ(1)

Stokes

1.2 1.15 1.1 1.05 1 0.95 0.9

0

1

2

3

4 kh

5

6

7

Figure 5. Second-order free surface correction, ζ (1) , relative to the Stokes solution, with no nonlinear optimization (β1 = β2 = β3 =0) shown by the dash-dotted line, the ∆N onlinear =0.013 results by the dotted line, and the ∆N onlinear =0.036 results by the dashed line.

7

4 1.

1.3

7

3 1.

7

1.2

0 0.3

4 3 2 1

5

10) during runup and rundown. The results of the numerical simulation are shown in Fig. 4. Fig. 4a shows the numerical free surface at various times, along with two profiles of the analytic free surface. The comparison between analytic and

numerical horizontal shoreline movement is shown as Fig. 4b. The agreement is good. Also, as a check on the convergence properties of d, an additional simulation with d = a 0/5000 was run. A comparison between the y = ao/50 shows little difference, and is not given in this paper. The maximum deviation in shoreline at any time between the two d runs is on the order of 0.01% of the maximum excursion. 4.2. Nonbreaking and breaking solitary wave runup Solitary wave runup and rundown was investigated experimentally by Synolakis (1986, 1987). In his work, dozens of experimental trials were performed, encompassing two orders of magnitude of solitary wave height. The beach slope was kept constant at 1:19.85. Many researchers have used this data set to validate numerical models (e.g., Zelt, 1991; Lin et al., 1999). To compare with this data, solitary waves with heights in the range of 0.005 < e < 0.5 are made to

Fig. 5. Nondimensional maximum runup of solitary waves on a 1:09.85 beach versus nondimensional wave height. The points experimental data taken from Synolakis (1986), the dotted line is the numerical result with no bottom friction, the solid line is the numerical result with a bottom friction coefficient, f, of 10 3, and the dashed line with f = 10 2.

P.J. Lynett et al. / Coastal Engineering 46 (2002) 89–107

runup and rundown a slope and the maximum vertical runup is calculated. Note that this range includes both nonbreaking and breaking pffiffiffiffiffiffiffiffi waves. For all simulations, Dx/h = 0.3 and Dt g=h ¼ 0:03 . As a test of the sensitively of wave runup to bottom friction, three sets of simulations were undertaken with different bottom friction coefficients, f. Set 1 was run with no

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bottom friction, Set 2 with f = 10 3, and Set 3 with f = 10 2. The numerical results are compared with the experimental data in Fig. 5, where maximum vertical runup is scaled by the water depth. For the smallest solitary waves (e < 0.01) bottom friction does not affect the runup, as maximum runup is identical for

Fig. 6. Breaking solitary wave runup and rundown on a planar beach at t( g/h)1/2=(a) 15, (b) 20, (c) 25, (d) 45. The solid line represents the numerical results and the points represent the experimental data. In (c), the dashed line represents numerical results by Lin et al. (1999) (closest to experiment and numerical results presented in this paper), the dotted line represents results by Zelt (1991), and the dashed – dotted line results by Titov and Synolakis (1995).

100


all three numerical sets. This is consistent with previous research (e.g., Liu et al., 1995), where it is shown that bottom friction effects are minor for nonbreaking waves, and will typically alter the runup by < 0.5% of the maximum. For larger wave heights, breaking is initiated, both experimentally and numerically, near e = 0.04. It is at this point that the numerical runup for Set 1 and Set 2 begins to diverge. Note that due to the log – log scale used in Fig. 5, the deviation in maximum runup may not be apparent. As an example, for e = 0.3, scaled runup with no bottom friction is 1.21, with f = 10 3 runup is 0.73, and with f = 10 2 is 0.45, which are significantly different. Use of f = 5 10 3 yields the best agreement with experimental data for this particular case. It would seem that inclusion of an accurate bottom friction parameterization becomes increasingly important with increasing degree of wave breaking. The probable reason is that as a broken wave runs up a mild slope, it travels up the slope as a fairly thin layer of water. As can be seen from Eq. (5), the smaller the total water depth, the more important bottom friction becomes. Synolakis (1986) also photographed the waves during runup and rundown. One set of these snap-

shots, for e = 0.28, was digitized and compared with the numerical prediction, shown in Fig. 6. The numerical simulation shown in this figure uses f = 10 3. The wave begins to break between Fig. 6c and b, and the runup/rundown process is shown in Fig. 6c– d. In Fig 6c, numerical snapshots from three other models are plotted. The comparisons indicate a significant improvement over weakly nonlinear Boussinesq equation results of Zelt (1991) and the NLSW results of Titov and Synolakis (1995). Additionally, the numerical results presented in this paper compare favorably to the two dimensional (vertical plane) results of Lin et al. (1999), which makes use of a complex turbulence model.

5. Validation in two horizontal dimensions 5.1. Long wave resonance in a parabolic basin Analytic solutions exist for few nonlinear, two horizontal dimension problems. One such solution is that for a long wave resonating in a circular parabolic basin. Thacker (1981) presented a solution to the

Fig. 7. Initial free surface and depth profile for parabolic basin test.


NLSW equations, where the initial free surface displacement is given as: fðr, t ¼ 0Þ " ¼ h0

ð1 A2 Þ1=2 r2 1 2 1A a

(

1 A2 ð1 A2 Þ2

)# 1 ð21Þ

and the basin shape is given by: r2 hðrÞ ¼ h0 1 2 a

ð22Þ

101

h0 is the center point water depth, r is the distance from the center point, a is the distance from the center point to the zero elevation on the shoreline, and r0 is the distance from the center point to the point where the total water depth is initially zero. The numerical values used for this test are: h0 = 1.0 m, r0 = 2000 m, and a = 2500 m. The centerline initial condition and depth profile is shown in Fig. 7. Thacker showed the solution to this problem to be: "

ð1 A2 Þ1=2 1 1 Acoswt ( )# r2 1A 2 1 a ð1 AcoswtÞ2

fðr, tÞ ¼ h0 where A¼

a4 r04 , a4 þ r04

ð23Þ

ð24Þ

Fig. 8. Centerline free surface profiles for numerical ( – – ) and analytical (: : :) bowl oscillation solutions at t=(a) 5T, (b) 5 1/6T, (c) 5 1/3T, (d) 5 1/2T, where T is the oscillation period.

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where 1 w ¼ ð8gh0 Þ1=2 a


ð25Þ

and g is gravity. Cho (1995) also used this solution as a test for his NLSW moving boundary model. Cho’s model, an explicit leap-frog finite-difference scheme which includes numerical frequency dispersion, reproduced the analytical solution very well for roughly one-half of an oscillation, but began to deviate soon after. A simulation using the extrapolation boundary technique presented in this paper was undertaken, truncating the dispersive terms in Eqs. (2) and (3) to be consistent with the NLSW solution, and using Dx = 28 m and Dt = 0.9 s. Bottom friction is not included and the wave does not break. The comparison between the numerical and analytic results is shown in Fig. 8. The numerical free surfaces shown in Fig. 8a –d are from the fifth oscillation, and show excellent agreement with the analytic solution. Additionally, a test using the full equations (Eqs. (2) and (3)), with dispersive terms, was performed. Interestingly, the wave field in this situation becomes chaotic after the first oscillation, and shows a similar pattern of divergence from the analytical solution as Cho’s results. Therefore, this parabolic basin comparison would appear to be an ideal test for NLSW models, as the effects of numerical dispersion or dissipation become evident rapidly. 5.2. Runup on a conical island Briggs et al. (1994) presented a set of experimental data for solitary wave interaction around a conical island. The slope of the island is 1:4 and the water depth is 0.32 m. Three cases were simulated, corresponding to solitary wave heights of 0.013 m (e = 0.04), 0.028 m (e = 0.09), and 0.058 m (e = 0.18). In addition to recording free surface elevation at a half dozen locations, maximum wave runup around the entire island was measured. This data set has been used by several researchers to validate numerical runup models (e.g., Liu et al., 1995; Titov and Synolakis, 1998; Chen et al., 2000). In this paper, free surface elevation is compared at the locations shown in Fig. 9. Gages #6 and #9 are located near the front face of the island, with #9 situated very near the initial shoreline position. Gages

Fig. 9. Conical island setup. The gage locations are shown by the dots, and the wave approaches the island from the left.

#16 and #22 are also located at the initial shoreline, where #16 is on the side of the island and #22 on the back face. Simulations were performed using Dx = 0.15 m and Dt = 0.02 s; bottom friction is neglected for these numerical tests. A soliton is placed in the numerical domain, as an initial condition. Numerical –experimental time series comparisons are shown in Fig. 10. Fig. 10a –d is for Case A (e = 0.04), Fig. 10e – h is for Case B (e = 0.09), and Fig. 10i – l is for Case C (e = 0.18). The gage number is shown in the upper left of each subplot. For all comparisons, the lead wave height and shape is predicted very well with the current model. Also, for all comparisons, the secondary depression wave is not predicted well. The numerical results show less of a depression following the main wave than in the experiments. This deviation is consistent with other runup model tests (e.g., Liu et al., 1995; Chen et al., 2000). The agreement of Case C is especially notable, as the soliton breaks along the backside of the island as the trapped waves intersect. This breaking occurs both experimentally, as discussed in Liu et al. (1995), and numerically. As mentioned, maximum runup was also experimentally recorded. The vertical runup heights are converted to horizontal runups, scaled by the initial shoreline radius, and plotted in Fig. 11. The crosshairs represents the experimental data, where Fig. 11a is for


Case A, Fig. 11b is for Case B, and Fig. 11c is for Case C. The numerical maximum inundation is also plotted, given by the solid line. The agreement for all cases is very good. 5.3. Soliton evolution in a trapezoidal channel Peregrine (1969) presented laboratory experiments wherein solitary waves propagated through a trapezoidal channel. To experimentally create the solitons,

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a piston wavemaker was cut to fit the channel and could slide horizontally along the trapezoidal channel. In the numerical simulations, as it is difficult to implement a piston wavemaker in a trapezoidal channel, the solutions of solitary waves in rectangular channels are used as an initial condition everywhere in the channel. Once a solitary wave enters a trapezoidal channel, it deforms. Eventually, in certain channels, the leading wave will reach a quasi steady state, and the wave-

Fig. 10. Experimental ( – – ) and numerical ( – ) time series for solitary wave interaction with a conical island. (a – d) are for Case A, (e – h) are for Case B, and (i – l) are for Case C. The gage number is shown in the upper left.

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Fig. 11. Maximum horizontal runup, scaled by the initial shoreline radius, for Case A (a), Case B (b), and Case C (c). Experimental values are shown by the stars and the numerical results are shown by the solid line.

form will not change in time. After reaching this quasi steady state, numerical results of the lead wave height are compared with Peregrine’s experimental results. The comparisons are shown in Fig. 12. For this comparison, a trapezoidal half-channel (one vertical wall and one sloping side wall) with constant depth width of 1.5h0, where h0 is the depth at the nonsloping part the channel, and a side-wall slope of 1:1

is employed. Three different amplitude solitary waves (a = 0.08h0, a = 0.12h0, and a = 0.18h0) are simulated and compared with experimental results. The numerical results show reasonably good agreement with laboratory data, although there is a clear trend of under prediction of wave height near the shoreline. An interesting property of wave evolution in certain trapezoidal channels is the successive regenera-

Fig. 12. The transverse profile of a solitary wave in a trapezoidal channel. The continuous line shows the numerical result; the crosses indicate the measured profile digitized from Peregrine’s (1969) paper.


tion of the wave front. When the channel is wide enough, with respect to the wavelength, and the sidewall slope is gradual enough, the wave energy that is reflected off the side walls does not resituate in the original wave. This occurs in the Peregrine (1969) experiments discussed above, but forms a distinct wave behind the original wave front. Wave energy is continually transferred from the original wave front into the new wave behind, until the original wave front virtually disappears. The new front has a smaller height, and a slightly longer wavelength than the original. One example of the phenomenon is discussed in this section. A half channel is created (one vertical wall at y = 0, one sloping side wall), with a constant water depth width of 9h0 and a length of 250h0, where h0 is the constant water depth along the center of the

105

channel. The side wall is sloped at 1:5. A solitary wave, with wave height 0.1h0 is placed in the channel as an initial condition. The wave does not break, and bottom friction is not included. For this simulation, ffi pffiffiffiffiffiffiffiffiffi Dx/h0 = 0.14 and Dt g=h0 ¼ 0:05 are used. Fig. 13 shows four snapshots, in plan view, of the wave propagating through the part of channel. The dashed line plotted across the channel is the x pctffiffiffiffiffiffiffi =0 line, where c is the linear long wave speed, gh0 . Seafloor elevation contours are also shown on each plot. Fig. 13a shows the wave soon after the simulation has begun, and the front is beginning to arc, due to slower movement in the shallower water. By the time shown in Fig. 13b, wave energy has reflected off the slope, and has formed a second, trailing, wave crest behind the original wave. As this slope-reflected wave crest interacts with the vertical wall (or center-

Fig. 13. Evolution of a solitary wave in a trapezoidal channel (half channel shown), at t( g/h)1/2=(a) 7.5, (b) 35, (c) 65, (d) 93. Seafloor elevation contours are shown at increments of 0.5h0, by the solid lines. The line of x ct = 0 is shown by the dashed line.

106


line of channel), a Mach stem forms at the vertical wall, and virtually no wave energy is reflected off the vertical wall. Also at this time, an oscillatory train, trailing the leading wave, forms along the slope. At time = 65, shown in Fig. 13c, most of the wave energy has transferred from the original wave front, to the secondary crest. In the last plot, Fig. 13d, the process has started to repeat itself, evidenced by the lobe growing behind the second front, near a depth of 0.9. This process can be examined from a different perspective with Fig. 14. This figure shows numerous time series, taken along the centerline of the channel ( y = 0). Also shown are three characteristic lines. Following the first characteristic, we can see that the lead wave as nearly disappeared by x = 140h 0 , whereas the secondary wave is clearly defined by this point. The process repeats; at x = 230h0, the secondary wave is vanishing, and a third wave front is beginning to take shape. The phenomenon shown in Figs. 13 and 14 is an interesting one, although not wholly unexpected, and is a demonstration of the interaction between nonlinearity and refraction.

6. Conclusion A moving boundary algorithm is developed for use with depth integrated equations. Used here in conjunction with a fixed grid finite difference model, the moving boundary algorithm could also be employed by a finite element scheme. Founded around the restrictions of the high-order numerical wave propagation model, the moving boundary scheme employs linear extrapolation of free surface and velocity through the wet –dry boundary, into the dry region. The linear extrapolation is simple to implement and can be straightforwardly incorporated into a numerical model. The technique is numerically stable, does not require any sort of additional dissipative mechanisms or filtering, and conserves mass. The moving boundary is tested for accuracy using one- and two-dimensional analytical solutions and experimental data sets. Nonbreaking and breaking solitary wave runup is accurately predicted, yielding a validation of both the eddy viscosity breaking parameterization and the runup model. For strongly

Fig. 14. Time series along the centerline of the channel ( y = 0); location of each time series is note along the right border of the figure. Characteristics are shown by the dashed – dotted lines.


breaking waves, the proper numerical estimation of bottom friction is shown to be important. Two-dimensional wave runup in a parabolic basin and around a conical island is investigated, and comparisons with published data show excellent agreement. Also, solitary wave evolution in a trapezoidal channel is simulated, and an interesting phenomenon is examined. Acknowledgements This research has been supported in part by National Science Foundation grants to Cornell University (CMS-9908392, CTS-9808542). References Balzano, A., 1998. Evaluation of methods for numerical simulation of wetting and drying in shallow water flow models. Coast. Eng. 34, 83 – 107. Briggs, M.J., Synolakis, C.E., Harkins, G.S., 1994. Tsunami runup on a conical island. Proc. Waves—Physical and Numerical Modeling. University of British Columbia, Vancouver, Canada, pp. 446 – 455. Carrier, G.F., Greenspan, H.P., 1958. Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 97 – 109. Chen, Q., Kirby, J.T., Dalrymple, R.A., Kennedy, A.B., Chawla, A., 2000. Boussinesq modeling of wave transformation, breaking, and runup: Part I. 2D. J. Waterw. Port Coast. Ocean Eng. 126 (1), 57 – 62. Cho, Y.-S., 1995. Numerical simulations of tsunami propagation and run-up. PhD Thesis, Cornell University, 264 pp. Gopalakrishnan, T.C., 1989. A moving boundary circulation model for regions with large tidal flats. Int. J. Numer. Methods Eng. 28, 245 – 260. Hsiao, S.-C., Liu, P.L.-F., 2002. Permeable effects on nonlinear water waves. Proc. R. Soc. Lond. in press. Kennedy, A.B., Chen, Q., Kirby, J.T., Dalrymple, R.A., 2000. Boussinesq modeling of wave transformation, breaking, and runup. Part I: 1D. J. Waterw. Port Coast. Ocean Eng. 126 (1), 39 – 47. Kirby, J.T., Wei, G., Chen, Q., Kennedy, A.B., Dalrymple, R.A., 1998. ‘‘FUNWAVE 1.0. Fully nonlinear Boussinesq wave model. Documentation and user’s manual.’’ Report CACR-98-06, Center for Applied Coastal Research, Department of Civil and Environment Engineering, University of Delaware. Kowalik, Z., Bang, I., 1987. Numerical computation of tsunami runup by the upstream derivative method. Sci. Tsunami Hazards 5 (2), 77 – 84. Leendertse, J.J., 1987. ‘‘Aspects of SIMSYS2D, a system for twodimensional flow computation.’’ R-3752-USGS. Rand, Santa Monica, CA, 80. Lin, P., Chang, K.-A., Liu, P.L.-F., 1999. Runup and rundown of solitary waves on sloping beaches. J. Waterw. Port Coast. Ocean Eng. 125 (5), 247 – 255.

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Liu, P.L.-F., 1994. Model equations for wave propagation from deep to shallow water. In: Liu, P.L.-F. (Ed.), Advances in Coastal Engineering, vol. 1, pp. 125 – 157. Liu, P.L.-F., Cho, Y.-S., Briggs, M.J., Kanoglu, U., Synolakis, C.E., 1995. Runup of solitary waves on a circular island. J. Fluid Mech. 302, 259 – 285. Lynett, P., Liu, P.L.-F., 2002a. A numerical study of submarine landslide generated waves and runup. Proc. R. Soc. Lond. in press. Lynett, P., Liu, P.L.-F., 2002b. A two-dimensional, depth-integrated model for internal wave propagation. Wave Motion in press. Lynett, P., Liu, P.L.-F., Losada, I., Vidal, C., 2000. Solitary wave interaction with porous breakwaters. J. Waterw. Port Coast. Ocean Eng. 126 (6), 314 – 322. Madsen, P.A., Sorensen, O.R., 1992. A new form of the Boussinesq equations with improved linear dispersion characteristics: Part II. A slowly varying bathymetry. Coast. Eng. 18, 183 – 204. Madsen, P.A., Sorensen, O.R., Schaffer, H.A., 1997. Surf zone dynamics simulated by a Boussinesq-type model: Part I. Model description and cross-shore motion of regular waves. Coast. Eng. 32, 255 – 287. Nwogu, O., 1993. Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port Coast. Ocean Eng. 119 (6), 618 – 638. Peregrine, D.H., 1967. Long waves on a beach. J. Fluid Mech. 27, 815 – 827. Peregrine, D.H., 1969. Solitary waves in trapezoidal channels. J. Fluid Mech. 35, 1 – 6. Petera, J., Nassehi, V., 1996. A new two-dimensional finite element model for the shallow water equations using a Lagrangian framework constructed along fluid particle trajectories. Int. J. Numer. Methods Eng. 39, 4159 – 4182. Sielecki, A., Wurtele, M.G., 1970. The numerical integration of the nonlinear shallow-water equations with sloping boundaries. J. Comput. Phys. 6, 219 – 236. Synolakis, C.E., 1986. The runup of long waves. PhD thesis, California Institute of Technology, Pasadena, CA. Synolakis, C.E., 1987. The runup of solitary waves. J. Fluid Mech. 185, 523 – 545. Tao, J., 1983. Computation of wave runup and wave breaking. Internal Report, Danish Hydraulics Institute, Denmark. Tao, J., 1984. Numerical modeling of wave runup and breaking on the beach. Acta Oceanol. Sin. 6 (5), 692 – 700 (in Chinese). Thacker, W.C., 1981. Some exact solutions to the nonlinear shallow water wave equations. J. Fluid Mech. 107, 499 – 508. Titov, V.V., Synolakis, C.E., 1995. Modeling of breaking and nonbreaking long wave evolution and runup using VTCS-2. J. Waterw. Port Coast. Ocean Eng. 121, 308 – 316. Titov, V.V., Synolakis, C.E., 1998. Numerical modeling of tidal wave runup. J. Waterw. Port Coast. Ocean Eng. 124 (4), 157 – 171. Wei, G., Kirby, J.T., 1995. A time-dependent numerical code for extended Boussinesq equations. J. Waterw. Port Coast. Ocean Eng. 120, 251 – 261. Wei, G., Kirby, J.T., Grilli, S.T., Subramanya, R., 1995. A fully nonlinear Boussinesq model for surface waves: Part I. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 71 – 92. Zelt, J.A., 1991. The runup of nonbreaking and breaking solitary waves. Coast. Eng. 15, 205 – 246.

A numerical study of submarine landslide generated waves and runup By Patrick Lynett and Philip L.-F. Liu School of Civil and Environmental Engineering Cornell University, Ithaca, NY 14853 A mathematical model is derived to describe the generation and propagation of water waves by a submarine landslide. The model consists of depth-integrated continuity equation and momentum equations, in which the ground movement is the forcing function. These equations include full nonlinear, but weak frequency dispersion effects. The model is capable of describing wave propagation from relatively deep water to shallow water. Simplified models for waves generated by small seafloor displacement or creeping ground movement are also presented. A numerical algorithm is developed for the general fully nonlinear model. Comparisons are made with a Boundary Integral Equation Method model, and a deep water limit for the depth-integrated model is determined in terms of a characteristic side length of the submarine mass. The importance of nonlinearity and frequency dispersion in the wave generation region and on the shoreline movement is discussed. Keywords: landslide tsunamis, Boussinesq equations, wave runup

1. Introduction In recent years, significant advances have been made in developing mathematical models to describe the entire process of generation, propagation and run-up of a tsunami event (e.g. Yeh et al . 1996; Geist 1998). These models are based primarily on the shallow-water wave equations and are adequate for tsunamis generated by seismic seafloor deformation. Since the duration of the seismic seafloor deformation is very short, the water surface response is almost instantaneous and the initial water surface profile mimics the final seafloor deformation. The typical wavelength of this type of tsunami ranges from 20 km to 100 km. Therefore, frequency dispersion can be ignored in the generation region. The nonlinearity is also usually not important in the generation region, because the initial wave amplitude is relatively small compared to the wavelength and the water depth. However, the frequency dispersion becomes important when a tsunami propagates for a long distance. Nonlinearity could also dominate as a tsunami enters the runup phase. Consequently, a complete model that can describe the entire process of tsunami generation, evolution, and runup needs to consider both frequency dispersion and nonlinearity. Tsunamis are also generated by other mechanisms. For example, submarine landslides have been documented as one of possible sources for several destructive tsunamis (Moore & Moore 1984, von Huene et al . 1989, Jiang & LeBlond 1992, Tappin et al . 1999, Keating & McGuire 1999). On November 29, 1975, a landslide was triggered by a 7.2 magnitude earthquake along the southeast coast of Hawaii. Article submitted to Royal Society

TEX Paper

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Patrick Lynett and Philip L.-F. Liu

A 60 km stretch of Kilauea’s south coast subsided 3.5 m and moved seaward 8 m. This landslide generated a local tsunami with a maximum runup height of 16 m at Keauhou (Cox & Morgan 1977). More recently, the devastating Papua New Guinea tsunami in 1998 is thought to be caused by a submarine landslide (Tappin et al . 1999, Keating & McGuire 1999, Tappin et al . 2001). In terms of tsunami generation mechanisms, two significant differences exist between submarine landslide and coseismic seafloor deformation. First, the duration of a landslide is much longer and is in the order of magnitude of several minutes. Hence the time history of the seafloor movement will affect the characteristics of the generated wave and needs to be included in the model. Secondly, the effective size of the landslide region is usually much smaller than the coseismic seafloor deformation zone. Consequently, the typical wavelength of the tsunamis generated by a submarine landslide is also shorter, i.e., about 1 to 10 km. Therefore, the frequency dispersion could be important in the wave generation region. The existing numerical models based on shallow-water wave equations may not be suitable for modeling the entire process of submarine landslide generated tsunami (e.g., Raney & Butler 1976, Harbitz et al . 1993). In this paper, we shall present a new model describing the generation and propagation of tsunamis by a submarine landslide. In this general model only the assumption of weak frequency dispersion is employed, i.e., the ratio of water depth to wavelength is small or O(µ2 ) ¿ 1. Until the past decade, weakly dispersive models were formulated in terms of a depth-averaged velocity (e.g. Peregrine, 1967). Recent work has clearly demonstrated that modifications to the frequency dispersion terms (Madsen et al ., 1992) or expression of the model equations in terms of an arbitrary-level velocity (Ngowu, 1993; Liu, 1994) can extend the validity of the linear dispersion properties into deeper water. The general guideline for dispersive properties is that the ”extended” versions of the depth-integrated equations are valid for wavelengths greater than two water depths, whereas the depth-averaged model is valid for lengths greater than five water depths (e.g., Nwogu, 1993). Moreover, in the model presented in this paper, the full nonlinear effect is included, i.e., the ratio of wave amplitude to water depth is of order one or ε = O(1). Therefore, this new model is more general than that developed by Liu & Earickson (1983), in which the Boussinesq approximation, i.e., O(µ2 ) = O(ε) ¿ 1 was used. In the special case where the seafloor is stationary, the new model reduces to the model for fully nonlinear and weakly dispersive waves propagating over a varying water depth (e.g. Liu 1994, Madsen & Schäffer 1998). The model is applicable for both the impulsive slide movement and creeping slide movement. In the latter case the time duration for the slide is much longer than the characteristic wave period. This paper is organized in the following manner. Governing equations and boundary conditions for flow motions generated by a ground movement are summarized in the next section. The derivation of approximate two-dimensional depthintegrated governing equations follows. The general model equations are then simplified for special cases. A numerical algorithm is presented to solve the general mathematical model. The numerical model is tested using available experimental data (e.g., Hammack 1973) for one-dimensional situations. Employing a Boundary Integral Equation Model, which solves for potential flow in the vertical plane, a deep water limit for waves generated by submarine slides is determined for the depth-integrated model. The importance of nonlinearity and frequency dispersion

Article submitted to Royal Society

A numerical study of submarine landslide generated waves and runup

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is inferred through numerical simulation of a large number of different physical setups.

2. Governing Equations and Boundary Conditions As shown in Figure 1, ζ 0 (x0 , y 0 , t0 ) denotes the free surface displacement of a wave train propagating in the water depth h0 (x0 , y 0 , t0 ). Introducing the characteristic water depth h0 as the vertical length scale, the characteristic length of the subma√ rine slide region `0 as the horizontal length scale, `0 / gh0 as the time scale, and the characteristic wave amplitude a0 as the scale of wave motion, we can define the following dimensionless variables: (x, y) = (x0 , y 0 )/`0 , z = z 0 /h0 , t =

p

gh0 t0 /`0

h = h0 /h0 , ζ = ζ 0 /a0 , p = p0 /ρga0 · ¸ ³ p ´ εp 0 (u, v) = (u , v )/ ε gh0 , w = w / gh0 µ 0

0

(2.1)

in which (u, v) represent the horizontal velocity components, w the vertical velocity component, p the pressure. Two dimensionless parameters have been introduced in (2.1), which are ε = a0 /h0 , µ = h0 /`0

(2.2)

Assuming that the viscous effects are insignificant, the wave motion can be described by the continuity equation and the Euler’s equations, i.e., µ2 ∇ · u + wz = 0 ut + εu · ∇u + εwt + ε2 u · ∇w +

ε wuz = −∇p µ2

ε2 wwz = −εpz − 1 µ2

(2.3) (2.4) (2.5)

where u = (u, v) denotes the horizontal velocity vector, ∇ = (∂/∂x, ∂/∂y) the horizontal gradient vector, and the subscript the partial derivative. On the free surface, z = εζ(x, y, t), the usual kinematic and dynamic boundary conditions apply: w = µ2 (ζt + εu · ∇ζ), on z = εζ (2.6a) p=0

(2.6b)

Along the seafloor, z = −h, the kinematic boundary condition requires w + µ2 u · ∇h + Article submitted to Royal Society

µ2 ht = 0, on z = −h ε

(2.7)

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For later use, we note here that the depth-integrated continuity equation can be obtained by integrating (2.3) from z = −h to z = εζ. After applying the boundary conditions (2.6a) and (2.7), the resulting equation reads "Z # εζ 1 ∇· udz + Ht = 0 (2.8) ε −h where H = εζ + h

(2.9)

We remark here that (2.8) is exact.

3. Approximate Two-Dimensional Governing Equations The three-dimensional boundary-value problem described in the previous section will be approximated and projected onto a two-dimensional horizontal plane. In this section, the nonlinearity is assumed to be of O(1). However, the frequency dispersion is assumed to be weak, i.e., O(µ2 ) ¿ 1

(3.1)

Using µ2 as the small parameter, a perturbation analysis is performed on the primitive governing equations. The complete derivation is given in Appendix A. The resulting approximate continuity equation is ½ ·µ ¶ 1 1 2 2 1 ht + ζt + ∇ · (Huα ) − µ2 ∇ · H (ε ζ − εζh + h2 ) − zα2 ∇(∇ · uα ) ε 6 2 µ ¶ µ ¶¸¾ 1 ht + (εζ − h) − zα ∇ ∇ · (huα ) + = O(µ4 ) (3.2) 2 ε Equation (3.2) is one of three governing equations for ζ and uα . The other two equations come from the horizontal momentum equation, (2.4), and are given in vector form as ½ · ¸¾ ∂ 1 2 ht uαt + εuα · ∇uα + ∇ζ + µ z ∇(∇ · uα ) + zα ∇ ∇ · (huα ) + ∂t 2 α ε ½· ¸ · ¸ ht ht +εµ2 ∇ · (huα ) + ∇ ∇ · (huα ) + ε ε · µ ¶¸ · ¸ htt ht −∇ ζ ∇ · (huα )t + + (uα · ∇zα )∇ ∇ · (huα ) + ε ε ¾ · µ ¶¸ z2 ht + zα (uα · ∇zα )∇(∇ · uα ) + α ∇ [uα · ∇(∇ · uα )] +zα ∇ uα · ∇ ∇ · (huα ) + ε 2 ½ ¸ · ¸ ¾ · 2 ζ ht ht +ε2 µ2 ∇ − ∇ · uαt − ζuα · ∇ ∇ · (huα ) + + ζ ∇ · (huα ) + ∇ · uα 2 ε ε ¾ ½ 2 ¤ ζ £ (3.3) +ε3 µ2 ∇ (∇ · uα )2 − uα · ∇(∇ · uα ) = O(µ4 ) 2 2



5

Equations (3.2) and (3.3) are the coupled governing equations, written in terms of uα and ζ, for fully nonlinear, weakly dispersive waves generated by a seafloor movement. We reiterate here that uα is evaluated at z = zα (x, y, t), which is a function of time. The choice of zα is made based on the linear frequency dispersion characteristics of the governing equations (e.g., Nwogu 1993, Chen & Liu 1995). Assuming a stationary seafloor, in order to extend the applicability of the governing equations to relatively deep water (or a short wave), zα is recommended to be evaluated as zα = −0.531h. In the following analysis, the same relationship is employed. These model equations will be referred to as FNL-EXT, for fully-nonlinear, ”extended” equations. (a) Creeping ground movements Up to this point the time scale of the seafloor movement is assumed to be in the same √ order of magnitude as the typical period of generated water wave, tw = `0 / gh0 as given in (2.1). When the ground movement is creeping in nature, the time scale of seafloor movement, tc , could be larger than tw . The only scaling parameter that is directly affected by the time scale of the seafloor movement is the characteristic amplitude of the wave motion. After introducing the time scale tc into the time derivatives of h in the continuity equation, (3.2), along with a characteristic change in water depth ∆h, the coefficient in front of ht becomes δ tw ε tc where δ = ∆h/ho . To maintain the conservation of mass, the above parameter must be of order one. Thus, ε=δ

tw δlo = √ tc tc gho

(3.4)

The above relationship can be interpreted in the following way: During the creeping ground movement, over the time period t < tc the generated wave has propagated √ a distance t gho . The total volume of the seafloor displacement, normalized by ho , is δlo ttc , which should be the same as the volume of water underneath the generated √ wave crest, i.e., εt gho . Therefore, over the ground movement period, t < tc , the wave amplitude can be estimated by (3.4). Consequently, nonlinear effects become important only if ε defined in (3.4) is O(1). Since, by definition of a creeping slide, the value t √logh is always less than one, fully nonlinear effects will be important for c o only the largest slides. The same conclusion was reached by Hammack (1973), using a different approach. The importance of the fully nonlinear effect when modeling creeping ground movements will be tested in a following section.

4. Limiting Cases In this section the general model is further simplified for different physical conditions. Article submitted to Royal Society

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Patrick Lynett and Philip L.-F. Liu (a) Weakly nonlinear waves

In many situations the seafloor displacement is relatively small in comparison with the local depth, and the seafloor movement can be approximated as h(x, y, t) = h0 (x, y) + δh(x, y, t),

(4.1)

in which δ is considered to be small. In other words, the maximum seafloor displacement is much smaller than the characteristic water depth. Since the free surface displacement is directly proportional to the seafloor displacement, i.e. O(εζ) = O(δh) or much less than the seafloor displacement in the case of creeping ground movements, we can further simplify the governing equations derived in the previous section by allowing O(ε) = O(δ) = O(µ2 ) ¿ 1 (4.2) which is the Boussinesq approximation. Thus, the continuity equation, (3.2) can be reduced to ¶ ½ ·µ δ 1 2 1 2 2 ζt + ∇ · (Huα ) + ht − µ ∇ · h0 h − z ∇(∇ · uα ) ε 6 0 2 α µ ¶ µ ¶¸¾ 1 δ − h0 + zα ∇ ∇ · (h0 uα ) + ht = O(µ4 , µ2 ε, δµ2 ) (4.3) 2 ε The momentum equation becomes ¾ ½ ∂ 1 2 δ uαt + εuα · ∇uα + ∇ζ + µ2 zα ∇(∇ · uα ) + zα ∇[∇ · (h0 uα ) + ht ] ∂t 2 ε = O(µ4 , εµ2 , δµ2 )

(4.4)

These model equations will be referred to as WNL-EXT, for weakly-nonlinear, ”extended” equations. The linear version of the above will also be utilized in the following analysis, and will be referred to as L-EXT, for linear, ”extended” equations. It is also possible to express the approximate continuity and momentum equations in terms of a depth-averaged velocity. The depth-averaged equations can be derived using the same method presented in Appendix A. One version of the depthaveraged equations will be employed in future sections, which is subject to the restraint (4.2), and is given as δ ζt + ∇ · (Hu) + ht = 0 ε ½ 2 ¾ ∂ h0 h0 δ ut + εu · ∇u + ∇ζ + µ2 ∇(∇ · u) − ∇[∇ · (h0 u) + ht ] ∂t 2 6 ε = O(µ4 , εµ2 , δµ2 ) where the depth-averaged velocity is defined as Z εζ 1 u(x, y, z, t) dz u(x, y, t) = h + εζ h

(4.5)

(4.6)

(4.7)

This set of model equations, (4.5) and (4.6), will be referred to as WNL-DA, for weakly-nonlinear, depth-averaged equations. Article submitted to Royal Society


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(b) Nonlinear shallow-water waves In the case that the water depth is very shallow or the wavelength is very long, the governing equations, (3.2) and (3.3), can be truncated at O(µ2 ). These resulting equations are the well-known nonlinear shallow water equations in which the seafloor movement is the forcing term for wave generation. This set of equations will be referred to as NL-SW, for nonlinear, shallow-water equations.

5. Numerical Model In this paper, a finite difference algorithm is presented for the general model equations, FNL-EXT. This model has the robustness of enabling slide-generated surface waves, although initially linear or weakly nonlinear in nature, to propagate into shallow water, where fully nonlinear effects may become important. The algorithm is developed for the general two-horizontal dimension problem, however in this paper only one-horizontal dimension examples are examined. The structure of the present numerical model is similar to those of Wei & Kirby (1995) and Wei et al. (1995). Differences between the model presented here and that of Wei et al. exist in the added terms due to a time-dependant water depth and the numerical treatment of some nonlinear dispersive terms, which is discussed in more detail in Appendix B. A high-order predictor-corrector scheme is utilized, employing a third order in time explicit Adams-Bashforth predictor step, and a fourth order in time AdamsMoulton implicit corrector step (Press et al., 1989). The implicit corrector step must be iterated until a convergence criterion is satisfied. All spatial derivatives are differenced to fourth order accuracy, yielding a model which is numerically accurate to (∆x)4 , (∆y)4 in space and (∆t)4 in time. The governing equations, (3.2) and (3.3), are dimensionalized for the numerical model, and all variables described in this and following sections will be in the dimensional form. Note that the dimensional equations are equivalent to the non-dimensional ones with ε = µ = 1 and the addition of gravity, g, to the coefficient of the leading order free surface derivative in the momentum equation (i.e., the third term on the left hand side of (3.3)). The predictor-corrector equations are given in Appendix B, along with some additional description of the numerical scheme. Runup and rundown of the waves generated by the submarine disturbance will also be examined. The moving boundary scheme employed here is the technique developed by Lynett et al. (2002). Founded around the restrictions of the high-order numerical wave propagation model, the moving boundary scheme utilizes linear extrapolation of free surface and velocity through the shoreline, into the dry region. This approach allows for the five-point finite difference formulas to be applied at all points, even those neighboring dry points, and thus eliminates the need of conditional statements. In addition to the depth-integrated model numerical results, output from a two dimensional (vertical plane) Boundary Integral Equation Method (BIEM) model will be presented for certain cases. This BIEM model will be primarily used to determine the deep water accuracy limit of the depth-integrated model. The BIEM model solves for inviscid, irrotational flows and converts a boundary value problem into an integral equation along the boundary of a physical domain. Therefore, just as with the depth-integration approach, it reduces the dimension of the problem by one. The BIEM model utilized here solves the Laplace equation in the vertical Article submitted to Royal Society

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plane (x, z), and, of course, is valid in all water depths for all wavelengths. Details of this type of BIEM model, when used to model water wave propagation, can be found in Grilli et al. (1989), Liu et al. (1992), and Grilli (1993) for example. The BIEM model used in this work has reproduced the numerical results presented for landslide generated waves in Grilli & Watts (1999) perfectly.

6. Comparisons with Experiment and Other Models As a first check of the present model, a comparison between Hammack’s (1973) experimental data for an impulsive bottom movement in a constant water depth is made. The bottom movement consists of a length, lo = 24.4 water depths, which is pushed vertically upward. The change in depth for this experiment, δ, is 0.1, so nonlinear effects should play a small role near the source region. Figure 2 shows a comparison between the numerical results using FNL-EXT, experimental data, and the linear theory presented by Hammack. Both the fully nonlinear model and the linear theory agree well with experiment at the edge of the source region (Figure 2a). From Figure 2b, a time series taken at 20 water depths from the edge of the source region, the agreement between all data is again quite good, but the deviation between the linear theory and experiment is slowly growing. The purpose of this comparison is to show that the present numerical model accurately predicts the free surface response to a simple seafloor movement. It would seem that if one was interested in just the wave field very near the source, linear theory is adequate. However, as the magnitude of the bed upthrust, δ, becomes large, linear theory is not capable of accurately predicting the free surface response even very near the source region. One such linear vs. nonlinear comparison is shown in Figure 2 for δ = 0.6. The motion of the bottom movement is the same as in Hammack’s case above. Immediately on the outskirts of the bottom movement, there are substantial differences between linear and nonlinear theory, as shown in Figure 2c. Additionally, as the wave propagates away from the source, errors in linear theory are more evident. A handful of experimental trials and analytic solutions exist for non-impulsive seafloor movements. However, for the previous work that made use of smooth obstacles, such as a semi-circle (e.g., Forbes & Schwartz, 1982) or a semi-ellipse (e.g., Lee et al., 1989), the length of the obstacle is always less than 1.25 water depths, or µ ≥ 0.8. Unfortunately, these objects will create waves too short to be modeled accurately by a depth-integrated model. Watts (1997) performed a set of experiments where he let a triangular block free fall down a planar slope. In all the experiments, the front face (deep water face) of the block was steep, and in some cases vertical. Physically, as the block travels down a slope, water is pushed out horizontally from the vertical front. Numerically, however, using the depth-integrated model, the dominant direction of water motion near the vertical face is vertical. This can be explained as follows. Examining the depth-integrated model equations, starting from the leading order, shallow water wave equations, the only forcing term due to the changing water depth appears in the continuity equation. There is no forcing term in the horizontal momentum equation. Therefore, in the non-dispersive system, any seafloor bottom cannot directly create a horizontal velocity. This concept can be further illuminated by the Article submitted to Royal Society


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equation describing the vertical profile of horizontal velocity: u(x, y, z, t) = uα (x, y, t) + O(µ2 )

(6.1)

Again, the changing seafloor bottom cannot directly create a horizontal velocity component for the non-dispersive system. All of the seafloor movement, whether it is a vertical or translational motion, is interpreted as strictly a vertical motion, which can lead to a very different generated wave pattern. When adding the weakly dispersive terms, the vertical profile of the horizontal velocity becomes: u(x, y, z, t) = uα (x, y, t)− ½ 2 · ¸¾ 2 z − zα ht 2 µ ∇(∇ · uα ) + (z − zα )∇ ∇ · (huα ) + + O(µ4 ) (6.2) 2 ε Now, with the higher-order dispersive formulation, there is the forcing term, ∇ht , which accounts for the effects of a horizontally moving body. Keep in mind, however, that this forcing term is a second-order correction, and therefore should represent only a small correction to the horizontal velocity profile. Thus, with rapid translational motion and/or steep side slopes of a submarine slide, the flow motion is strongly horizontal locally, and the depth-integrated models are not adequate. In slightly different terms, let the slide mass have a characteristic side length, Ls . A side length is defined as the horizontal distance between two points at which ∂h/∂t = 0. This definition of a side length is described graphically in Figure 3. Figure 3a shows a slide mass that is symmetric around its midpoint in the horizontal direction, where the back (shallow water) and front (deep water) side lengths are equal. Figure 3b shows a slide mass whose front side is much shorter that the back. Note that for the slide shown in Fig. 3b, the side lengths, measured in the direction parallel to the slope, are equal, whereas for the slide in Fig. 3a the slide lengths are equal when measured in the horizontal direction. An irregular slide mass will have least two different side lengths. In these cases, the characteristic side length, Ls , is the shortest of all sides. When Ls is small compared to a characteristic water depth, ho , that side is considered steep, or in deep water, and the shallow-water based depth-integrated model will not be accurate. For the vertical face of Watts’ experiments, Ls = 0, and therefore Ls /ho = 0 and the situation resembles that of an infinitely deep ocean. The next section will attempt to determine a limiting value of Ls /ho where the depth-integrated model begins to fail.

7. Limitations of the Depth-Integrated Model Before utilizing the model for practical applications, the limits of accuracy of the depth-integrated model must be determined. As illustrated above, just as there is a short wave accuracy limit (wave should be at least 2 water depths long when applying the ”extended” model), it is expected that there is also a slide length scale limitation. By comparing the outputs of this model to those of the BIEM model, a limiting value of Ls /ho can be inferred. The high degree of BIEM model accuracy in simulating wave propagation is well documented (e.g. Grilli 1989, Grilli et al. 1994). The comparison cases will use a slide mass travelling down a constant slope. The slide mass moves as a solid body, with velocity described following Watts (1997). Article submitted to Royal Society

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This motion is characterized by a decreasing acceleration until a terminal velocity is reached. All of the solid body motion coefficients used in this paper are identical to those employed by Grilli & Watts (1999). Note that all of the submarine landslide simulations presented in this paper are non-breaking. The setup of the slide mass on the slope is shown in Figure 4. The time history of the seafloor is described by · µ ¶¸ · µ ¶¸ ∆h x − xl (t) x − xr (t) h(x, t) = ho (x) − 1 + tanh 1 − tanh (7.1) 2 S S where ∆h is the maximum vertical height of the slide, xl is the location of the tanh inflection point of the left side of the slide, xr is the location of the inflection point on the right side, and S is a shape factor, controlling the steepness of the slide sides. The right and left boundaries, and steepness factor are given by: b xl (t) = xc (t) − cos(θ), 2

b xl (t) = xc (t) + cos(θ), 2

S=

0.5 cos(θ)

where xc is the horizontal location of the center point of the slide, and is determined using the equations governing the solid body motion of the slide. The angle of the slope is given by θ. The thickness of the ”slideless” water column, or the baseline water depth, at the centerpoint of the slide is defined by hc (t) = ho (xc (t)) = ∆h + d(t). With a specified depth above the initial center point of the slide mass, do = d(t = 0), the initial horizontal location of the slide center, xc (t = 0), can be found. The length along the slope between xl and xr is defined as b, and all lengths are scaled by b. For the first comparison, a slide with the parameter set: θ=6◦ , do /b=0.2, and ∆h/b=0.05 is modeled with FNL-EXT and BIEM. With these parameters the characteristic horizontal side length of the slide mass, Ls /b, is 1.7. Ls is defined as in Figure 3, or specifically, the horizontal distance between two points at which ∂h/∂t is less than 1% of the maximum ∂h/∂t value. Note that a 6◦ slope is roughly 1/10. Figure 5 shows four snapshots of the free surface elevation from both models. The lowest panel in the figure shows the initial location of the slide mass, along with the locations corresponding to the four free surface snapshots. Initially, as shown in Figures 5a and b, where Ls /hc =6.1 and 4.5 respectively, the two models agree, and thus are still in the range of acceptable accuracy of the depth-integrated model. In Figure 5c, as the slide moves into deeper water, where Ls /hc =3.1, the two models begin to diverge over the source region, and by Figure 5d, the free surface responses of the two models are quite different. These results indicate that in the vicinity of x/b=5, the depth-integrated model becomes inaccurate. At this location, hc /b=0.5, and Ls /hc =3.4. Numerous additional comparison tests were performed, and all indicated that the depth-integrated model becomes inaccurate when Ls /hc < 3 ∼ 3.5. One more of the comparisons is shown here. Examining a 20◦ slope and a slide mass with a maximum height ∆h/b=0.1, the initial depth of submergence, do /b, will be successively increased from 0.4 to 0.6 to 1.0. The corresponding initial Ls /hc values are 3.4, 2.4, and 1.5, respectively. Time series above the initial centerpoint of the slide masses and vertical shoreline movements are shown in Figure 6. The expectation is that the first case (Ls /hc = 3.4 initially) should show good agreement, the middle Article submitted to Royal Society


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case (Ls /hc = 2.4 initially) marginal agreement, and the last case (Ls /hc = 1.5 initially) bad agreement. The time series above the center, Figures 6a, c, and e, do clearly agree with the stated expectation. Various different zα levels were tested in an attempt to better the agreement with the BIEM model results for the deeper water cases, but zα =-0.531h provided the most accurate output. Rundown, as shown in Figures 6b, d, and f, shows good agreement for all the trials. The explanation is that the wave that creates the rundown is generated from the backface of the slide mass. This wave sees a characteristic water depth that is less than hc , and thus this backface wave remains in the region of accuracy of the depth-integrated model, whereas the wave motion nearer to the front face of the slide is inaccurate. This feature is also clearly shown in Figure 5. Thus, if one was solely interested in the leading wave approaching the shoreline, the characteristic water depth should be interpreted as the average depth along the backface of the slide, instead of hc . The inaccurate elevation waves created by the front face of the moving mass could be absorbed numerically, such as with a sponge layer, so that they do not effect the simulation. A guideline that the depth-integrated ”extended” model will yield accurate results for Ls /hc > 3.5 is accepted. This restriction would seem to be more stringent than the ”extended” model frequency dispersion limitation, which requires that the free surface wave be at least two water depths long. In fact, the slide length scale limitation is more in line with the dispersion limitations of the depth-averaged (conventional) model. The limitations of the various model formulations, i.e. ”extended” and depth-averaged, are discussed in the next section.

8. Importance of Nonlinearity and Frequency Dispersion Another useful guideline would be to know when nonlinear effects begin to play an important role. This can be determined by running numerous numerical trials, employing the FNL-EXT, WNL-EXT, and L-EXT equation models. These three equation sets share identical linear dispersion properties, but have varying levels of nonlinearity. The linear dispersion limit of these ”extended” equations, for the rigid bottom case, is near kh=3, where k is the wavenumber. Nonlinearity, however, is only faithfully captured to near kh=1.0 for the FNL-EXT model, and to an even lesser value for WNL-EXT (Gobbi et al., 2000). The source-generation accuracy limitation of the model is such that the side length of the landslide over the depth must be greater than 3.5. If the slide is symmetric in the horizontal direction, which is the only type of slide examined in this section, then the wavelength of the generated wave will be 2*3.5*h, or roughly kh=1. Thus up to the accuracy limit found in the previous section, nonlinearity is expected to be well captured. The FNL-EXT model will be considered correct, and any difference in output compared to the other models with lesser nonlinearity would indicate that full nonlinear effects are important. The importance of nonlinearity will be tested through examination of various ∆h/do combinations, using the slide mass described in the previous section. The value of ∆h/do can be thought of as an impulsive nonlinearity, as this value represents the magnitude of the free surface response if the slide motion was entirely vertical and instantaneous. The procedure will be to hold the value hco = hc (t = 0) = ∆h + do constant for a given slope angle, while altering ∆h and do . Two Article submitted to Royal Society

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output values will be compared between all the simulations: maximum depression above the initial centerpoint of the slide and maximum rundown. For all simulations p presented in this section ∆x/b = 0.003 and ∆t ghco /b = 0.0003. Set # 1 2 3 4

Slope (degrees) 30 15 5 5

hco /b 0.55 0.55 0.55 0.15

Ls /hco 3.5 3.5 3.5 13

Table 1. Characteristics of the simulations performed for the nonlinearity test. Figure 7 shows the output from four sets of comparisons, whose characteristics are given in Table 1. Figures 7a & b show the depression above the centerpoint and the rundown for Set 1, Figures 7c & d for Set 2, Figures 7e & f for Set 3, and Figures 7g & h for Set 4. Examining the maximum depression plots for Sets 1,2, & 3, it is clear that the trends between the three sets are very similar, with FNL-EXT predicting the largest depression, and L-EXT predicting the smallest. The difference between FNL-EXT and WNL-EXT is solely due to nonlinear dispersive terms, which are of O(εµ2 ), while the difference between WNL-EXT and L-EXT is caused by the nonlinear divergence term in the continuity equation and the convection term in the momentum equation, which are of O(ε). The relative differences in the maximum depression predicted between FNL-EXT and WNLEXT are roughly the same as the differences between WNL-EXT and L-EXT for Sets 1,2, & 3. Therefore, in the source region, for Ls /hco values near the accuracy limit of the ”extended” model (near 3.5), the nonlinear dispersive terms are as necessary to include in the model as the leading order nonlinear terms. As the Ls /hc value is increased, the slide produces an increasingly longer (shallow water) wave. Frequency dispersion plays a lesser role, and thus the nonlinear dispersive terms become expectedly less important. This can be seen in the maximum depression plot for Set 4. For this set, Ls /hco = 13, and the FNL-EXT and WNL-EXT results are nearly indistinguishable. Inspecting the maximum rundown plots for Sets 1, 2, & 3, it seems that the trends between the three different models have changed. Now, WNL-EXT predicts the largest rundown, while L-EXT predicts the smallest. It is hypothesized that the documented over-shoaling of WNL-EXT (Wei et al., 1995) cancels out the lesser wave height generated in the source region compared to FNL-EXT, leading to rundown heights that agree well between the two models. As the slope is decreased, the error in the L-EXT rundown prediction increases. This is attributed to a longer distance of shoaling before the wave reaches the shoreline. As the slope is decreased, while hco is kept constant, the horizontal distance from the shoreline to the initial centerpoint of the slide increases. The slide length is roughly the same for the three sets, therefore the generated wavelength is roughly the same. Thus, with a lesser slope the generated wave shoals for a greater number of wave periods. During this relatively larger distance of shoaling, nonlinear effects, and in particular the leading order nonlinear effects, accumulate and yield large errors in the linear (L-EXT) simulations. This trend is also evident in the rundown plot for Set 4. Also note that in Set 4, where the nonlinear dispersive terms are very small, the FNL-EXT and WNL-EXT rundowns are identical. Article submitted to Royal Society


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A deep water limit has been determined for the ”extended” model (Ls /hc > 3.5), but it would also be interesting to know the limits of applicability of the depthaveraged (WNL-DA) and shallow water (NL-SW) models. The only differences between these three models (the weakly nonlinear ”extended”, weakly nonlinear depth-averaged, and nonlinear shallow-water) are found in the frequency dispersion terms - the nonlinear terms are the same. The testing method to determine the deep water limits of the various model types will be to fix both a slope of 15◦ and a slide mass, with ∆h/b = 0.05 and Ls /b = 1.85, while incrementally increasing the initial water depth above the centerpoint of the slide, d. Figure 8 shows a summary of the comparisons of the three models. Figures 8a & d show the maximum free surface depression measured above the initial centerpoint of the slide and the maximum rundown for various Ls /hco combinations. WNL-EXT solutions are indicated by solid lines, WNL-DA by dashed lines, and NL-SW by the dotted lines. Also shown in Figures 8b & e are the maximum depression and rundown results from WNL-DA and NL-SW relative to the results from WNL-EXT, thereby more clearly depicting the differences between the models. These figures show WNL-EXT and WNL-DA agreeing nearly exactly, while the errors in NL-SW decrease with increasing Ls /hco . The NL-SW results do not converge with the WNL-EXT results until Ls /hco >∼15. Figures 8c & f are time series of the free surface elevation above the initial centerpoint of the slide and the vertical movement of the shoreline for the case of Ls /hco =3.5, respectively. Differences between NL-SW and WNL-EXT are clear, with NL-SW under-predicting the free surface above the slide, but overpredicting the rundown due to over shoaling in the non-dispersive model. The only significant difference between the WNL-EXT and WNL-DA results come after the maximum depression in Figure 8c, where WNL-DA predicts an oscillatory train following the depression. These results indicate that to the deep water limit that WNL-EXT was shown to be accurate, WNL-DA is accurate as well. As mentioned previously, altering the level on which zα is evaluated in the ”extended” model does not increase the deep water accuracy limit for slide generated waves. In summary, the nonlinear dispersive terms are important for slides near the deep water limit (Ls /hc = 3.5) whose heights, or ∆h/do values, are large (> 0.4). For shallow water slides (Ls /hc > 10), the nonlinear dispersive terms are not important near the source, even for the largest slides. The ”extended” formulation of the depth-integrated equations does not appear to offer any benefits over the depthaveraged formulation in regards to modeling the generation of waves in deeper water. The ”extended” model would be useful if one was interested in modeling the propagation of shallow-water, slide-generated waves into deeper water, which is not the focus of this paper. The shallow water wave equations are only valid for slides in very shallow water, where Ls /hco >∼15.

9. Conclusions A model for the creation of fully nonlinear long waves by seafloor movement, and their propagation away from the source region, is presented. The general fully nonlinear model can be truncated, so as to only include weakly nonlinear effects, or model a non-dispersive wave system. Rarely will fully nonlinear effects be important above the landslide region, but the model has the advantage of allowing the slideArticle submitted to Royal Society

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generated waves to become fully nonlinear in nature, without requiring a transition among governing equations. A high-order finite difference model is developed to numerically simulate wave generation by seafloor movement. The numerical generation of waves by both impulsive and creeping movements agrees with experimental data and other numerical models. A deep-water accuracy limit of the model, Ls /hc > 3.5, is adopted. Within this limitation, the ”extended” formulation of the depth-integrated equations shows no benefit over the ”conventional”, depth-averaged approach near the source region. Leading order nonlinear effects were shown to be important for prediction of shoreline movement, and the fully nonlinear terms are important for only the thickest slides with relatively short length scales. Although only one-horizontal dimension problems are examined in this paper, slides in two-horizontal dimensions have been analyzed by the authors, but, due to paper length limitations, will be presented in a future publication. As a final remark, it is noted that prediction of landslide tsunamis in real cases is subject to the large uncertainty inherent in knowing the time-evolution of a landslide. Extensive field research of high-risk sites is paramount to reducing this uncertainty. The research reported here is partially supported by Grants from National Science Foundation (CMS-9528013, CTS-9808542, and CMS 9908392) and a subcontract from University of Puerto Rico. The authors wish to thank Ms. Yin-yu Chen for providing the numerical results based on her BIEM model.

Appendix A. Derivation of Approximate Two-Dimensional Governing Equations In deriving the two-dimensional, depth-integrated governing equations, the frequency dispersion is assumed to be weak, i.e., O(µ2 ) ¿ 1

(A.1) 2

We can expand the dimensionless physical variables as power series of µ . f=

∞ X

µ2n fn ;

(f = ζ, p, u)

(A.2)

n=0

w=

∞ X

µ2n wn ;

(A.3)

n=1

Furthermore, we will adopt the following assumption on the vorticity field. We assume that the vertical vorticity component, (uy −vx ), is of O(1), while the horizontal vorticity components are weaker and satisfy the following conditions ∂ u0 = 0, (A.4) ∂z ∂ u1 = ∇w1 . (A.5) ∂z Consequently, from (A.4), the leading order horizontal velocity components are independent of the vertical coordinate, i.e., u0 = u0 (x, y, t), Article submitted to Royal Society

(A.6)


15

Substituting (A.2) and (A.3) into the continuity equation (2.3) and the boundary condition (2.7), we collect the leading order terms as ∇ · u0 + w1z = 0, −h < z < εζ

(A.7)

ht = 0 on z = −h (A.8) ε Integrating (A.7) with respect to z and using (A.8) to determine the integration constant, we obtain the vertical profile of the vertical velocity components: w1 + u0 · ∇h +

ht (A.9) ε Similarly, integrating (A.5) with respect to z with information from (A.8), we can find the corresponding vertical profiles of the horizontal velocity components: · ¸ z2 ht u1 = − ∇(∇ · u0 ) − z∇ ∇ · (hu0 ) + + C 1 (x, y, t) (A.10) 2 ε w1 = −z∇ · u0 − ∇ · (hu0 ) −

in which C 1 is a unknown function to be determined. Up to O(µ2 ), the horizontal velocity components can be expressed as ½ 2 · ¸ ¾ z ht u = u0 (x, y, t) + µ2 − ∇(∇ · u0 ) − z∇ ∇ · (hu0 ) + + C 1 (x, y, t) 2 ε +O(µ4 ), −h < z < εζ

(A.11)

Now, we can define the horizontal velocity vector, uα (x, y, zα (x, y, t), t) evaluated at z = zα (x, y, t) as ½ 2 · ¸ ¾ zα ht 2 uα = u0 + µ − ∇(∇ · u0 ) − zα ∇ ∇ · (hu0 ) + + C 1 (x, y, t) + O(µ4 ) 2 ε (A.12) Subtracting (A.12) from (A.11) we can express u in terms of uα as ½ 2 · ¸¾ z − zα2 ht u = uα − µ2 ∇(∇ · uα ) + (z − zα )∇ ∇ · (huα ) + + O(µ4 ) (A.13) 2 ε Note that uα = u0 + O(µ2 ) has been used in (A.13). The exact continuity equation (2.8) can be rewritten approximately in terms of ζ and uα . Substituting (A.13) into (2.8), we obtain ½ ·µ ¶ 1 1 2 2 1 Ht + ∇ · (Huα ) − µ2 ∇ · H (ε ζ − εζh + h2 ) − zα2 ∇(∇ · uα ) ε 6 2 µ ¶ µ ¶¸¾ 1 ht + (εζ − h) − zα ∇ ∇ · (huα ) + = O(µ4 ) (A.14) 2 ε in which H = h + εζ. Equation (A.14) is one of three governing equations for ζ and uα . The other two equations come from the horizontal momentum equation, (2.4). However, we Article submitted to Royal Society

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must find the pressure field first. This can be accomplished by approximating the vertical momentum equation (2.5) as εpz = −1 − µ2 (εw1t + ε2 u0 · ∇w1 + ε2 w1 w1z ) + O(µ4 ), −h < z < εζ

(A.15)

We can integrate the equation above with respect to z to find the pressure field as ½ 1 2 htt z 2 (z − ε2 ζ 2 )∇ · u0t + (z − εζ)[∇ · (hu)0t + ] p = (ζ − ) + µ ε 2 ε · ¸ ht ε 2 2 2 ε + (z −ε ζ )u0 ·∇(∇·u0 )+ε(z −εζ)u0 ·∇ ∇ · (hu0 ) + + (ε2 ζ 2 −z 2 )(∇·u0 )2 2 ε 2 ¾ ht +ε(εζ − z)[∇ · (hu0 ) + ]∇ · u0 + O(µ4 ), −h < z < εζ (A.16) ε We remark here that (A.11) has been used in deriving (A.16). To obtain the governing equations for uα , we first substitute (A.13) and (A.16) into (2.4) and obtain the following equation, up to O(µ2 ), ½ · ¸¾ 1 2 htt 2 uαt + εuα · ∇uα + ∇ζ + µ z ∇(∇ · uαt ) + zα ∇ ∇ · (huα )t + 2 α ε ½ · ¸¾ ht +µ2 zαt zα ∇(∇ · uα ) + ∇ ∇ · (huα ) + ε ½· ¸ · ¸ ht ht 2 +εµ ∇ · (huα ) + ∇ ∇ · (huα ) + ε ε · ¸ · ¸ htt ht −∇ ζ(∇ · (huα )t + ) + (uα · ∇zα )∇ ∇ · (huα ) + ε ε · ¸ ¾ ht zα2 +zα ∇ uα · ∇(∇ · (huα ) + ) + zα (uα · ∇zα )∇(∇ · uα ) + ∇ [uα · ∇(∇ · uα )] ε 2 ½ · ¸ · ¸ ¾ ζ2 ht ht 2 2 +ε µ ∇ − ∇ · uαt − ζuα · ∇ ∇ · (huα ) + + ζ ∇ · (huα ) + ∇ · uα 2 ε ε ½ 2 ¾ ¤ ζ £ +ε3 µ2 ∇ (∇ · uα )2 − uα · ∇(∇ · uα ) = O(µ4 ) (A.17) 2 Equations (A.14) and (A.17) are the coupled governing equations, written in terms of uα and ζ, for fully nonlinear, weakly dispersive waves generated by a submarine landslide.

Appendix B. Numerical Scheme To simplify the predictor-corrector equations, the velocity time derivatives in the momentum equations are grouped into the dimensional form: U =u+

zα2 − ζ 2 uxx + (zα − ζ)(hu)xx − ζx [ζux + (hu)x ] 2


(B.1)


17

zα2 − ζ 2 vyy + (zα − ζ)(hv)yy − ζy [ζvy + (hv)y ] (B.2) 2 where subscripts denote partial derivatives. Note that this grouping is different from that given in Wei et al. (1995). The grouping given above in B.1 and B.2 incorporates nonlinear terms, which is not done in Wei et£al.. These nonlinear¤ time htt derivatives arise ³ ´ from the nonlinear dispersion terms ∇ ζ(∇ · (huα )t + ε ) and V =v+

∇

ζ2 2 ∇

· uαt , which can be reformulated using the relation:

· · ¸ · ¸ ¸ ht htt ht ∇ ζ(∇ · (huα )t + ) = ∇ ζ(∇ · (huα ) + ) − ∇ ζt (∇ · (huα ) + ) ε ε t ε µ ∇

ζ2 ∇ · uαt 2

¶

µ =∇

ζ2 ∇ · uα 2

¶ − ∇ (ζζt ∇ · uα ) t

The authors have found that this form is more stable and requires less iterations to converge for highly nonlinear problems, as compared to the Wei et al. formulation. The predictor equations are n+1 n ζi,j = ζi,j +

∆t n−1 n−2 n (23Ei,j − 16Ei,j + 5Ei,j ) 12

(B.3)

n+1 n Ui,j = Ui,j +

∆t n−1 n−2 n−2 n (23Fi,j −16Fi,j +5Fi,j )+2(F1 )ni,j −3(F1 )n−1 (B.4) i,j +(F1 )i,j 12

n+1 n Vi,j = Vi,j +

∆t n−2 n−1 n−2 n (23Gni,j −16Gn−1 (B.5) i,j +5Gi,j )+2(G1 )i,j −3(G1 )i,j +(G1 )i,j 12

where E = −ht − [(ζ + h)u]x − [(ζ + h)v]y ½ ·µ ¶ µ ¶ ¸¾ ¢ 1 1¡ 2 1 + (h + ζ) ζ − ζh + h2 − zα2 Sx + (ζ − h) − zα Tx 6 2 2 x ½ ·µ ¶ µ ¶ ¸¾ ¡ ¢ 1 2 1 1 + (h + ζ) ζ − ζh + h2 − zα2 Sy + (ζ − h) − zα Ty (B.6) 6 2 2 y 1 F = − [(u2 )x + (v 2 )x ] − gζx − zα hxtt − zαt hxt 2 · ¸ ¢ 1¡ 2 2 + (Eht + ζhtt )x − [E(ζS + T )]x − z − ζ (uSx + vSy ) 2 α x h i 1 2 (T + ζS) − [(zα − ζ) (uTx + vTy )]x − 2 x F1 =

ζ 2 − zα2 vxy − (zα − ζ)(hv)xy + ζx [ζvy + (hv)y ] 2


(B.7) (B.8)

18


1 G = − [(u2 )y + (v 2 )y ] − gζy − zα hytt − zαt hyt 2 ¸ · ¢ 1¡ 2 zα − ζ 2 (uSx + vSy ) + (Eht + ζhtt )y − [E(ζS + T )]y − 2 y h i 1 2 − [(zα − ζ) (uTx + vTy )]y − (T + ζS) 2 y G1 =

ζ 2 − zα2 uxy − (zα − ζ)(hu)xy + ζy [ζux + (hu)x ] 2

(B.9) (B.10)

and S = ux + vy

T = (hu)x + (hv)y + ht

(B.11)

All terms are evaluated at the local grid point (i, j), and n represents the current time step, when values of ζ, u and v are known. The above expressions, (B.6) (B.11), are for the fully nonlinear problem; if a weakly nonlinear or non-dispersive system is to be examined, the equations should be truncated accordingly. The fourth-order implicit corrector expressions for the free surface elevation and horizontal velocities are n+1 n ζi,j + = ζi,j

n+1 n Ui,j + = Ui,j

n+1 n Vi,j = Vi,j +

∆t n+1 n−1 n−2 n (9Ei,j − 5Ei,j + 19Ei,j + Ei,j ) 24

(B.12)

∆t n+1 n−1 n−2 n n (9Fi,j − 5Fi,j + 19Fi,j + Fi,j ) + (F1 )n+1 i,j − (F1 )i,j (B.13) 24 ∆t n−1 n−2 n+1 n n (9Gn+1 i,j + 19Gi,j − 5Gi,j + Gi,j ) + (G1 )i,j − (G1 )i,j (B.14) 24

The system is solved by first evaluating the predictor equations, then u and v are solved via (B.1) and (B.2), respectively. Both (B.1) and (B.2) yield a diagonal matrix after finite differencing. The matrices are diagonal, with a bandwidth of five (due to five point finite differencing), and an efficient LU decomposition can be utilized. At this point in the numerical system, we have predictors for ζ, u, and v. Next, the corrector expressions are evaluated, and again u and v are determined from (B.1) and (B.2). The relative errors in each of the physical variables is found, in order to determine if the implicit correctors need to be reiterated. This relative error is given as: wn+1 − w∗n+1 (B.15) wn+1 where w represents ζ, u, and v, and w∗ is the previous iterations value. The correctors are recalculated until all errors are less than 10−4 . Note that inevitably there will be locations in the numerical domain where values of the physical variables are close to zero, and applying the above error calculation to these points may lead to unnecessary iterations in the corrector loop. Thus it is required that u,v | > 10−4 for the corresponding error calculation to proceed, where a is | aζ |, | ²√ gh determined from equation (3.4) for a creeping slide. For the model equations, linear Article submitted to Royal Society


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stability analysis gives that ∆t < ∆x 2c , where c is the wave celerity in the deepest water. Note that when modeling highly nonlinear waves, a smaller ∆t is usually required for stability. In this analysis, ∆t = ∆x 4c produced stable and convergent results for all trails. For the numerical exterior boundaries, two types of conditions are applied: reflective and radiation. The reflective, or no-flux, boundary condition for the Boussinesq equations has been examined by previous researchers (Wei & Kirby, 1995), and their methodology is followed here. For the radiation, or open, boundary condition, a sponge layer is utilized. The sponge layer is applied in the manner recommended by Kirby et al. (1998). Runup and rundown is modeled with the ”extrapolation” moving boundary algorithm described in Lynett et al. (2002).


20


Appendix C. Notation a b c d do g ho h h hc

= = = = = = = = = =

wave amplitude length along the slope between xl and xr for the tanh slide wave celerity depth of water above the centerpoint of the slide, function of time initial depth of water above the centerpoint of the slide, i.e. at t = 0 gravity characteristic water depth or baseline water depth, function of space water depth profile, function of space and time the changing part of the water depth profile = (h − ho )/δ baseline water depth at the centerpoint of the slide = ∆h + d

hc o H lo Ls p S t tc tw u, v, w uα , v α u, v u xc , yc xl , xr zα δ ∆h ∆t ∆x, ∆y ε ∇ ρ

= = = = = = = = = = = = = = = = = = = = = = =

initial baseline water depth at the centerpoint of the slide = ∆h + do total water depth = h + εζ characteristic horizontal lengthscale of the submarine slide characteristic horizontal side length of the submarine slide depth dependant pressure shape factor for tanh slide time time scale of seafloor motion typical period of wave generated by a specified seafloor motion depth dependant components of velocity in x, y, z magnitude of horizontal velocity components, u, v, evaluated on zα depth-averaged horizontal velocity components horizontal velocity vector = (u, v) horizontal coordinates of the midpoint of the seafloor movement locations of the left and right inflection points for the tanh slide profile arbitrary level on which the ”extended” equations are derived scaled characteristic change in water depth due to seafloor motion = ∆h/ho characteristic, or maximum, change in water depth due to seafloor motion time step in numerical model space steps in numerical model nonlinearity parameter, = a/ho horizontal gradient vector density of water

θ = µ =

slope angle frequency dispersion parameter, = ho /lo

ζ

free surface displacement

=


(C 1)


21

References Cox, D. C. & Morgan, J. 1977 ”Local Tsunamis and Possible Local Tsunamis in Hawaii”, Hawaii Institute of Geophysics, University of Hawaii, Report HIG 77-14. Chen, Y., & Liu, P. L.-F., 1995. “Modified Boussinesq Equations and Associated Parabolic Model for Water Wave Propagation.” J. Fluid Mech., 228, 351–381. Forbes, L. K., and Schwartz, L. W., 1982. ”Free-Surface Flow over a Semicircular Obstruction,” J. Fluid Mech. 114, 299-314 Geist, E. L. 1998. ”Local Tsunami and Earthquake Source Parameters” Advances in Geophysics, 39, 117-209. Gobbi, M., Kirby, J.T., & Wei, G., 2000. “A Fully Nonlinear Boussinesq Model for Surface Waves. Part 2. Extension to O(kh4 ),” J. Fluid Mech. 405, 181–210. Grilli, S.T. 1993. ”Modeling of nonlinear wave motion in shallow water.” Chapter 3 in Computational Methods for Free and Moving Boundary Problems in Heat and Fluid Flows (eds. I.C. Wrobel and C.A. Brebbia), pps. 37-65, Computational Mechanics Publication, Elsevier Applied Sciences, London, UK. Grilli, S.T, Skourup, J., and Svendsen, I.A. 1989. ”An Efficient Boundary Element Method for Nonlinear Waves .” Engng. Analysis with Boundary Elements , 6, 97-107. Grilli, S.T, Subramanya, R., Svendsen, I.A., and Veeramony, J. 1995. ”Shoaling of Solitary Waves on Plane Beaches.” Journal of Waterway, Port, Coastal and Ocean Engng., 120, 74-92. Grilli, S.T, and Watts, P. 1999. ”Modeling of Waves Generated by a Moving Submerged Body. Applications to Underwater Landslides” Engng. Analysis with Boundary Elements , 23, 645-656. Harbitz, C. B., Pedersen, G., & Gjevik, B. 1993 ”Numerical Simulation of Large Water Waves due to Landslides”, J. Hydraulic Engineering, ASCE 119, 1325-1342. Hammack, H. L., 1973. “A Note on Tsunamis: Their Generation and Propagation in an Ocean of Uniform Depth,” J. Fluid Mech. 60, 769-799. Jiang, L., Ren, X., Wang, K.-H., & Jin, K.-R. 1996. ”Generalized Boussinesq Model for Periodic Non-Linear Shallow-Water Waves,” Ocean Engng, 23, 309-323. Jiang, L. & LeBlond, P. H., 1992 ”The Coupling of a Submarine Slide and the Surface Waves Which It Generates”, J. Geophys. Res., 97, 12,731-12,744. Kirby, J.T., Wei, G., Chen, Q., Kennedy, A.B. & Dalrymple, R.A., 1998. “FUNWAVE 1.0: Fully Nonlinear Boussinesq Wave Model Documentation and User’s Manual,” Center for Applied Coastal Research, University of Delaware. Keating, B. H. & McGuire, W. J. 1999 ”Island Edifice Failures and Associated Hazards”. In Pure and Applied Geophysics, Special Issue: Landslide and Tsunamis. In press. Kennedy, A. B., Kirby, J. T., Chen, Q. & Dalrymple, R. A., 2001. “Boussinesq-type Equations with Improved Nonlinear Behaviour,” Wave Motion, 33, 225-243. Lee, S.-J., Yates, G. T., and Wu, T. Y., 1989. ”Experiments and analyses of upstreamadvancing solitary waves generated by moving disturbances,” J. Fluid Mech. 199, 569593. Liu, P. L.-F., 1994. “Model Equations for Wave Propagations From Deep to Shallow Water,” in Advances in Coastal and Ocean Engineering, Vol. 1 (ed. P.L.-F. Liu), 125– 158. Liu, P. L.-F., Hsu, H.-W., and Lean, M.H. 1992. ”Applications of Boundary Integral Equation Methods for Two-Dimensional Non-Linear Water Wave Problems,” Int. J. for Numer. Meth. in Fluids, 15, 1119-1141. Liu, P. L.-F. & Earickson, J. 1983 ”A Numerical Model for Tsunami Generation and Propagation,” in Tsunamis: Their Science and Engineering (eds. J. Iida and T. Iwasaki), Terra Science Pub. Co., 227-240. Article submitted to Royal Society

22


Lynett, P., Wu, T-W., & Liu, P. L.-F., 2002. “Modeling Wave Runup with DepthIntegrated Equations,” submitted to Coast. Engng. Madsen, P. A., and Sorensen, O. R., 1992. “A new form of the Boussinesq equations with improved linear dispersion characteristics. Part II: A slowly varying bathymetry,” Coast. Engng., 18, 183-204. Madsen, P.A. & Sch¨ affer, H.A., 1998. “Higher-Order Boussinesq-type Equations for Surface Gravity Waves: Derivation and Analysis,” Phil. Trans. R. Soc. Lond. A, 356, 2123–3184. Moore, G. & Moore, J. G. 1984 ”Deposit from Giant Wave on the Island of Lanai,” Science, 222, 1312-1315. Nwogu, O., 1993 ”Alternative Form of Boussinesq Equations for Nearshore Wave Propagation,” J. Wtrwy, Port, Coast and Ocean Engrg., ASCE, 119(6), 618-638. Peregrine, D. H., 1967. ”Long Waves on a Beach,” J. Fluid Mech. 27, 815-827 Press, W.H., Flannery, B.P., & Teukolsky, S.A. 1989. ”Numerical Recipes,” Cambridge University Press, 569-572. Raney, D. C. & Butler, H. L. 1976 ”Landslide Generated Water Wave Model,” J. Hydraulic Div., ASCE 102, 1269-1282. Tappin, D. R., et al. 1999 ”Sediment Slump Likely Caused 1998 Papua New Guinea Tsunami,” EOS, 80(30), 329. Tappin, D. R., Watts, P., McMurtry, G. M., Lafoy, Y., and Matsumoto, T. 2001 ”The Sissano, Papau New Guinea Tsunami of July 1998 - Offshore Evidence on the Source Mechanism,” Marine Geology, 175, 1-23. von Huene, R. Bourgois, J. Miller, J. & Pautot, G. 1989 ”A Large Tsunamigetic Landslide and Debris Flow Along the Peru Trench,” J. Geophys. Res., 94, 1703-1714. Wang, K-H. 1993. ”Diffraction of Solitary Waves by Breakwaters,” Journal of Waterway, Port, Coastal and Ocean Engng., 119, 49-69. Watts, P. 1997. Water Waves Generated by Underwater Landslides. Ph. D. Thesis, California Institute of Technology, 1997 Wei, G. & Kirby, J. T. 1995. ”A Time-Dependent Numerical Code for Extended Boussinesq Equations,” Journal of Waterway, Port, Coastal and Ocean Engng., 120, 251-261. Wei, G., Kirby, J.T., Grilli, S.T., & Subramanya, R., 1995. “A Fully Nonlinear Boussinesq Model for Surface Waves. Part 1. Highly Nonlinear Unsteady Waves,” J. Fluid Mech. 294, 71–92. Yeh, H., Liu, P. L.-F. & Synolakis, C. (Eds) 1996 ”Long-wave Runup Models,” Proc. 2nd Int. Workshop on Long-wave Runup Models, World Scientific Publishing Co., Singapore.



23

Figure 1. Basic formulational setup.

0.08

0.08 a)

0.06

b) 0.06

δ=0.1 x/h=0 ζ/h

0.04

ζ/h

0.04

δ=0.1 x/h=20

0.02

0.02

0

0

−0.02 −10

0

10

20 0.5

t (g/h)

30

40

0

10

20

30

40

50

t (g/h)0.5 − x/h 0.6

c) δ=0.6 x/h=0

0.2

0

d) δ=0.6 x/h=20

0.4 ζ/h

0.4

−0.2 −10

−0.02 −10

− x/h

0.6

ζ/h

50

0.2

0

0

10

20 0.5

t (g/h)

30 − x/h

40

50

−0.2 −10

0

10

20

30

40

50

t (g/h)0.5 − x/h

Figure 2. Shown in subplots a) and b), a comparison between Hammack’s (1973) experimental data (dots) for an impulsive seafloor upthrust of δ=0.1, FNL-EXT numerical simulation (solid line), and linear theory (dashed line); a) is a time series at x/h=0 and b) is at x/h=20, where x is the distance from the edge of the impulsive movement. Subplots c) and d) show FNL-EXT (solid line) and L-EXT (dashed line) numerical results for Hammack’s setup, except with δ=0.6.


24


a)

b)

0.2

0.2 −∂ h / ∂ t

−∂ h / ∂ t

0

0 L1

L1

L2 Ls = L2 = L1

Ls = L2 < L1

z

−0.2

z

−0.2

L2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1

0

0.2

0.4

0.6 x

0.8

1

−1

0

0.2

0.4

0.6

0.8

1

x

Figure 3. Graphical definition of the characteristic side length of a slide mass. The slide mass at time=to is shown by the solid line, while the profile at some time=t > to is shown by the dashed line. The negative of the change in water depth (or the approximate free surface response in the non-dispersive equation model) during the increment t−to is shown by the thick line plotted on z=0.1.



Figure 4. Setup for submarine landslide comparisons.


25

26


z/d

o

0.02 0 −0.02 −0.04 −0.06

a)

L /h =6.1 s

0

1

2

3

4

5

6

7

z/d

o

0.02 0 −0.02 −0.04 −0.06

c

b)

Ls/hc=4.5 0

1

2

3

4

5

6

7

z/d

o

0.02 0 −0.02 −0.04 −0.06

c)

Ls/hc=3.1 0

0.02 0 −0.02 −0.04 −0.06

1

2

3

4

5

6

7

z/d

o

d) Ls/hc=2.5

∆ h/d

0

1

2

3 t=0

0.2

a)

4

5

b)

6 c)

7 d)

0.1 0

0

1

2

3

4 x/b

5

6

7

Figure 5. Free surface snapshots for BIEM (solid line) and depth-integrated (dashed line) results at t(g/do )1/2 = a) 10.6, b) 21, c) 31.6, and d) 41. The lower subplot shows the location of the slide mass in each of the above four snapshots.



0

0

27

0

−0.005

−0.002 −0.005

−0.01

−0.004

z/d

o

−0.015 −0.01

−0.02

−0.006

−0.025

−0.008 −0.015

−0.03

−0.01 a) Ls/hc =3.4

−0.035

c) Ls/hc =2.4

O

0

5

10

−0.02

e) Ls/hc =1.5

O

0

5

10

−0.012

0

0

−0.01

−0.005

−0.002

−0.02

−0.01

−0.004

Vert. Shoreline Movement / do

0

−0.01

b) Ls/hc =3.4

−0.05

−0.025 d) Ls/hc =2.4

O

0

5

10

−0.008

−0.02

−0.04

5

−0.006

−0.015

−0.03

O

0

10

t(g/do)1/2

−0.03

0

5 t(g/do)1/2

f) Ls/hc =1.5

−0.012

O

10

O

0

5

10

t(g/do)1/2

Figure 6. Time series above the initial centerpoint of the slide (top row) and vertical movement of the shoreline (bottom row) for a 20◦ slope and a slide mass with a maximum height ∆h=0.1. BIEM results are shown by the solid line, depth-integrated results by the dashed line. Subplots a & b are for do /b=0.4, c & d for do /b=0.6, and e & f for do /b=1.0.


28


Max Depression/do

0

0 a)

b) −0.05

Max Rundown/do

−0.05 −0.1

−0.15

−0.15

o

Slope=30 , h /b=0.55 c

−0.2

−0.25

O

L /h =3.5 s

0

−0.1

c

−0.2

s

O

0.2

0.4

0.6 ∆ h/do

0.8

−0.25

1

0

c

O

0.2

0.4

0.6 ∆ h/do

0.8

1

0 c)

d)

Max Rundown/do

−0.05

−0.05

−0.1

−0.15

−0.1 o

Slope=15 , hc /b=0.55 O Ls/hc =3.5

−0.15

Slope=15o, hc /b=0.55

−0.2

0.2

0.4

0.6 ∆ h/do

O

Ls/hc =3.5

−0.25

O

0

O

0.8

1

0

0

0.2

0.4

0.6 ∆ h/do

0.8

1

0 f) Max Rundown/do

Max Depression/do

e) −0.05

−0.02

−0.04 o

Slope=5 , hc /b=0.55 Ls/hc =3.5

−0.06

O

−0.1 Slope=5o, hc /b=0.55 Ls/hc =3.5

−0.15

O

0

0.2

0.4

O

O

0.6 ∆ h/do

0.8

1

0

0

0.2

0.4

0.6 ∆ h/do

0.8

1

0 g)

h)

−0.1

Max Rundown/do

−0.05

Max Depression/do

O

L /h =3.5

c

0 Max Depression/do

o

Slope=30 , h /b=0.55

−0.1

−0.2

−0.15

−0.3

−0.2

−0.4


−0.25

Ls/hc =13

−0.3

0.5

Ls/hc =13

−0.6

O

0


−0.5

O

1 ∆ h/do

1.5

2

O

O

0

0.5

1 ∆ h/do

1.5

2

Figure 7. Maximum depression above the initial centerpoint of the slide mass and maximum rundown for four different trial sets. FNL-EXT results indicated by the solid line, WNL-EXT by the dashed line, and L-EXT by the dotted line.



29

−3

0

0 a)

o

−0.05

c)

−0.1 −2 −0.15 o

Slope=15 ∆ h/b=0.05 Ls/b=1.85

−0.2 −0.25 5

−4

10

15

z/do

Max Depression/d

x 10

20

L /h Max Dep/Max Dep WNL−EXT

s

c

O

−6

b)

1.02 1

Ls/hc =3.5 O

0.98 −8 0.96 0.94 0.92

0 5

10

15 Ls/hc

10

20

30

40

50

time*(g/do)1/2

20

O

0 d) 0.005 −0.1

f)

−0.15 −0.2

Max Rundown/Max Rundown WNL−EXT

0

Slope=15o ∆ h/b=0.05 Ls/b=1.85 5

10

Shoreline Movement/do

Max Rundown/d

o

−0.05

−0.005

15

20

Ls/hc

O

1.3 e)

1.25

−0.01

−0.015

1.2 1.15

−0.02

1.1

Ls/hc =3.5 O

1.05 −0.025

1 5

10

15 Ls/hc

O

20

0

10

20

30

40

50

time*(g/do)1/2

Figure 8. Maximum depression above the initial centerpoint of the slide mass a) and maximum rundown d) for a set of numerical simulations on a 15◦ slope. Shown in b) and e) are the maximum depression and maximum rundown scaled by the corresponding values from the WNL-EXT model. Time series comparisons for Ls /hco = 3.5 showing the free surface elevation above the centerpoint c) and vertical shoreline movement f) are given on the right. WNL-EXT results indicated by the solid line, WNL-DA by the dashed line, and NL-SW by the dotted line.


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