Nonlinear Wave Model Research Articles

Boundary‐fitted non‐linear dispersive wave model for regions of arbitrary geometry

AbstractA vertically integrated non‐linear dispersive wave model is expressed in non‐orthogonal curvilinear co‐ordinate system for simulating shallow or deep water wave motions in regions of arbitrary geometry. Both dependent and independent variables are transformed so that an irregular physical domain is converted into a rectangular computational domain with contravariant velocities. Thus, the wall condition for enclosures surrounding a typical physical domain, such as a channel, port or harbor, is satisfied accurately and easily. The numerical scheme is based on staggered grid finite‐difference approximations, which result in implicit formulations for the momentum equations and semi‐explicit formulation for the continuity equation. Test cases of linear wave propagation in converging, diverging and circular channels are performed to check the reliability of model simulations against the analytical solutions. Cnoidal waves of different steepness values in a circular channel are also considered as examples to non‐linear wave propagation within curved walls. In closing, remarks concerning versatility and practical uses of the numerical model are made. Copyright © 2004 John Wiley & Sons, Ltd.

An efficient and flexible computational model for solving the mild slope equation

The mild slope equation is used to describe wave propagating in the near shore region. In this paper, the finite difference method is used to discretize the governing elliptic equation and the discretized linear equation is solved using GPBiCG ( m, n) method. Two cases of wave propagating are used to test the model, and reasonable agreements have been achieved. It is shown that the present algorithm: GPBiCG ( m, n) with a flexible and diverse form, has a fast and relatively robust convergence rate, and can be efficiently and economically run in either the linear or nonlinear wave model and easily used to a complicated region.

Fully dispersive nonlinear water wave model in curvilinear coordinates

A vertically integrated fully dispersive nonlinear wave model is expressed in curvilinear coordinates with non-orthogonal grids for the simulation of broad-banded nonlinear random water waves in regions of arbitrary geometry. The transformation is performed for both dependent and independent variables, hence an irregular physical domain is converted into a rectangular computational domain with contravariant velocities. Use of contravariant velocity components as dependent variables ensures easy and accurate satisfaction of the wall condition for lateral enclosures surrounding a physical domain, such as a coastal area, channel, or harbor. The numerical scheme is based on finite-difference approximations with staggered grids which results in implicit formulations for the momentum equations and a semi-explicit formulation for the continuity equation. Linear long wave propagation in a channel of varying cross-section and linear random wave propagation in a circular channel are presented as test cases for comparisons with the corresponding analytical solutions. Cnoidal and Stokes waves in a circular channel are also simulated as examples to nonlinear wave propagation within curved walls.

The Nonlinear Alfvn Wave Model for Solar Coronal Heating and Nanoflares

The mechanism of solar coronal heating has been unknown since the discovery that the coronal plasma temperature is a few million degrees. There are two promising mechanisms, the Alfven wave model and the nanoflare-reconnection model. Recent observations favor the nanoflare model since it readily explains the ubi-quitous small-scale brightenings all over the Sun. We have performed magnetohydrodynamic (MHD) simulations of the nonlinear Alfven wave coronal heating model that include both heat conduction and radiative cooling in an emerging flux loop and found that the corona is episodically heated by fast- and slow-mode MHD shocks generated by nonlinear Alfven waves via nonlinear mode-coupling. We also found that the time variation of the simulated extreme-ultraviolet and X-ray intensities of these loops, on the basis of the Alfven wave model, is quite similar to the observed one, which is usually attributed to nanoflare or picoflare heating. This suggests that the observed nanoflares may not be a result of reconnection but in fact may be due to nonlinear Alfven waves, contrary to current widespread opinion.

Traveling Wave Solutions for a Class of One-Dimensional Nonlinear Shallow Water Wave Models

In this paper we consider a class of one-dimensional nonlinear shallow water wave models that support weak solutions. We construct new traveling wave solutions for these models. Moreover, we show that these new traveling wave solutions are stable.

Asymptotic nonlinear wave modeling through the Dirichlet-to-Neumann operator

New nonlinear evolution equations are derived that generalize the system by Matsuno [16] and a terrain-following Boussinesq system by Nachbin [23]. The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. The asymptotic simplification of the nonlinear potential theory equations is performed through a perturbation anaylsis of the Dirichlet-to-Neumann operator on a highly corrugated strip. This is achieved through the use of a curvilinear coordinate system. Rather than doing a long wave expansion for the velocity potential, a Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply-valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. These equations represent a fully dispersive system in the sense that the original (hyperbolic tangent) dispersion relation is not truncated. The formulation is done over a periodically extended domain so that, as an application, it produces efficient Fourier (FFT) solvers. A preliminary communication of this work has been published in the Physical Review Letters [1].

Landslide tsunami case studies using a Boussinesq model and a fully nonlinear tsunami generation model

Abstract. Case studies of landslide tsunamis require integration of marine geology data and interpretations into numerical simulations of tsunami attack. Many landslide tsunami generation and propagation models have been proposed in recent time, further motivated by the 1998 Papua New Guinea event. However, few of these models have proven capable of integrating the best available marine geology data and interpretations into successful case studies that reproduce all available tsunami observations and records. We show that nonlinear and dispersive tsunami propagation models may be necessary for many landslide tsunami case studies. GEOWAVE is a comprehensive tsunami simulation model formed in part by combining the Tsunami Open and Progressive Initial Conditions System (TOPICS) with the fully non-linear Boussinesq water wave model FUNWAVE. TOPICS uses curve fits of numerical results from a fully nonlinear potential flow model to provide approximate landslide tsunami sources for tsunami propagation models, based on marine geology data and interpretations. In this work, we validate GEOWAVE with successful case studies of the 1946 Unimak, Alaska, the 1994 Skagway, Alaska, and the 1998 Papua New Guinea events. GEOWAVE simulates accurate runup and inundation at the same time, with no additional user interference or effort, using a slot technique. Wave breaking, if it occurs during shoaling or runup, is also accounted for with a dissipative breaking model acting on the wave front. The success of our case studies depends on the combination of accurate tsunami sources and an advanced tsunami propagation and inundation model.

Numerical modeling of nonlinear acoustic waves containing shocks

Theoretical models and corresponding numerical algorithms for the description of propagation of nonlinear acoustic waves containing shocks are presented. Plane waves and high intensity acoustic beams are considered. Various time and frequency domain numerical schemes that include numerical or artificial smoothening of the shocks and their specific features and artifacts are discussed. A modified spectral approach for the description of nonlinear waves containing discontinuous shocks is described. The method utilizes an analytical high frequency asymptote of the abrupt shock, which is used in the numerical algorithm to model sawtooth-like waves with a relatively few number of harmonics. Further generalizations of the method are discussed, in particular, to account for the discontinuity in the first derivative at the shock front, typical for the shock pulses; and for the finite thickness of the shock. The results of modeling of nonlinear waves containing shocks are presented: the effect of saturation in focused acoustic beams, double shock formation in a nearfield of a piston source, saturation of nonlinear waves in active media, the effect of various frequency dependent absorption on nonlinear interactions, and stationary and quasi-stationary solutions of the Burgers-type equations with the high order derivative absorption term. [Work supported by ONRIFO, CRDF, and RFBR.]

Field Survey and Numerical Simulations: A Review of the 1998 Papua New Guinea Tsunami

The Papua New Guinea (PNG) tsunami of 1998 is re-examined through a detailed review of the field survey as well as numerous numerical computations. The discussion of the field survey explores a number of possible misinterpretations of the recorded data. The survey data are then employed by a numerical model as a validation tool. A Boussinesq model and a nonlinear shallow water wave (NLSW) model are compared in order to quantify the effect of frequency dispersion on the landslide-generated tsunami. The numerical comparisons indicate that the NLSW model is a poor estimator of offshore wave heights. However, due to what appears to be depth-limited breaking seaward of Sissano spit, both numerical models are in agreement in the prediction of maximum water elevations at the overtopped spit. By comparing three different hot-start initial profiles of the tsunami wave, it is shown that the initial shape and orientation of the tsunami wave is secondary to the initial displaced water mass in regard to prediction of water elevations on the spit. These numerical results indicate that agreement between numerical prediction of runup values with field recorded values at PNG cannot be used to validate either a NLSW tsunami propagation model or a specific landslide tsunami hot-start initial condition. Finally, with the use of traditional tsunami codes, a new interpretation of the PNG runup measurements is presented.

A Comparison of Nonlinear Water Wave Models

We compare the numerical evolution of one-dimensional gravity waves in response to a traveling surface pressure pulse using a highly accurate boundary integral method and two relatively efficient approximate models (West et al. and Benney–Luke). In both water of finite-depth and in the deep-water limit, the steady state effect of the decaying pressure ramp is to create a profile which approximates a Stokes wave. Moreover, the transient surface profile appears to evolve through a series of Stokes waves of time-varying amplitude. Results show all three models yield similar predictions for lower amplitude waves, while the West et al. and boundary integral predictions differ from the Benney–Luke model at higher amplitudes.

High‐frequency internal waves in large stratified lakes

Observations are presented from Lake Biwa and Lake Kinneret showing the ubiquitous and often periodic nature of high‐frequency internal waves in large stratified lakes. In both lakes, high‐frequency wave events were observed within two distinct categories: (1) Vertical mode 1 solitary waves near a steepened Kelvin wave front and vertical mode 2 solitary waves at the head of an intrusive thermocline jet were found to have wavelengths ~64–670 m and ~13–65 m, respectively, and were observed to excite a spectral energy peak near 10-3 Hz. (2) Sinusoidal vertical mode 1 waves on the crests of Kelvin waves (vertically coherent in both phase and frequency) and bordering the thermocline jets in the high shear region trailing the vertical mode 2 solitary waves (vertically incoherent in both phase and frequency) were found to have wavelengths between 28–37 and 9–35 m, respectively, and excited a spectral energy peak just below the local maximum buoyancy frequency near 10−2 Hz. The waves in wave event categories 1 and 2 were reasonably described by nonlinear wave and linear stability models, respectively. Analysis of the energetics of these waves suggests that the waves associated with shear instability will dissipate their energy rapidly within the lake interior and are thus responsible for patchy turbulent events that have been observed within the metalimnion. Conversely, the finite‐amplitude solitary waves, which each contain as much as 1% of the basinscale Kelvin wave energy, will propagate to the lake perimeter where they can shoal, thus contributing to the maintenance of the benthic boundary layer.

Lattice modelling of nonlinear waves in a bi-layer with delamination

A lattice model consisting of two one-dimensional periodic chains with linear links between elements and nonlinear interaction between the chains is suggested to study nonlinear dynamics of a bi-layer. The properties of the model are discussed, and the influence of a delamination zone on the propagation of solitary waves is studied numerically.

An electrostatic model for nonlinear waves in the upper ionosphere

According to observations by the Freja satellite in the upper ionosphere, amplitudes of magnetic disturbances associated with observed density solitons are only several 10 −2 nT or less. These very weak magnetic disturbances can be neglected against the ambient magnetic field of 0.2 Gauss in Freja's orbit region. In this paper we establish an electrostatic model for the nonlinear waves observed on board of Freja. The fluid equations are used and the “Sagdeev potential” is derived to study the nonlinear waves. The results show the existence not only of solitons with a density hump and solitons with a density dip, but also of periodic density waves and density shock waves. The theoretic amplitudes of the solitons may vary between 20% and a factor of 2, which is consistent with the observations on board of Freja. Some waveforms of the solitons with a density dip are very similar to the observed ones.

Study on Nonlinear Coupling Wave Model for Liquid Droplet-Solid Impact

In order to investigate the material corrosion by liquid droplet-solid impact, a nonlinear coupling wave model adopted to analyze the impact between the spherical liquid droplet and an elastic solid plane has been developed. Many usable results such as the dimensionless pressure in the contact plane of liquid-solid and inside the liquid droplet, the equivalent stress distribution inside the solid, the effect of solid elasticity on the impact, and the locations of the maximum equivalent stress in different time steps are presented. It is shown clearly that the model can analyze the impact accurately and provide detailed information of impact, they are usable data to study material corrosion.

A comparison of the performance of profile position and composition estimators for quality control in binary distillation

In this study a number of control strategies have been developed for control of the overhead composition of a binary distillation column. The nonlinear wave model as presented in the literature, has been substantially modified in order to express it in variables that can easily be measured and make it more robust to feed flow and feed composition changes. The new model consists essentially of the equation for wave propagation and a static mass and energy balance across the top section of the column. Taylor series developments are used to relate the temperature on the measurement tray to the temperature and concentration on the tray where the inflection point of the concentration profile is located. The model has been incorporated in control of the overhead quality of a toluene/ o-xylene benchmark column. In addition, a number of partial least squares (PLS) estimators have been developed: a nonlinear estimator for inferring the overhead composition from temperature measurements and a linear and nonlinear estimator for inferring the inflection point of the concentration profile in the column. These estimators are also used in a cascade control strategy and compared with use of the wave propagation model. Finally a control strategy consisting of a simple temperature controller and a composition controller were implemented on the simulated column. The study shows that the inferential control using PLS estimators performs equally well than control using the nonlinear wave model. In all cases the advantage of using inferential controllers is substantial compared with using single tray temperature control or composition control.

Time-domain lifted wavelet collocation method for modeling nonlinear wave propagation

A time-domain adaptive numerical method for modeling nonlinear wave propagation is developed. This method is based on a second-generation wavelet collocation using a lifting scheme and makes use of the multilevel decomposition nature of the scheme to allow for automatic grid refinement according to the magnitude of waveform steepening. The multiplication in the nonlinear term is also easy due to the collocation nature. With thresholding, the solution is compact at every level of resolution and computed only at collocation points associated with the remaining significant wavelet coefficients. The error tolerance and compression ratio of the new method are totally controlled by the threshold value used. This brings substantial savings in computation time when compared to the conventional finite difference scheme on a uniformly fine grid.

A weakly nonlinear water wave model taking into account dispersion of wave phase velocity

This paper presents a weakly nonlinear water wave model using a mild slope equation and a new explicit formulation which takes into account dispersion of wave phase velocity, approximates Hedges' (1987) nonlinear dispersion relationship, and accords well with the original empirical formula. Comparison of the calculating results with those obtained from the experimental data and those obtained from linear wave theory showed that the present water wave model considering the dispersion of phase velocity is rational and in good agreement with experiment data.

Surf and Swash Zone Morphology Evolution Induced by Nonlinear Waves

Nonlinear wave transformation in the surf and swash zone is computed by a nonlinear breaking wave model based on the numerical solution of the Boussinesq-type equations. Instead of classical equations, higher-order equations in dispersion as well as in nonlinearity are derived and used inside and outside the surf zone. The Dibajnia and Watanabe transport rate formula involving unsteady aspects of the sand transport phenomenon is adopted for estimating sheet flow sediment transport rates as well as the bed load and suspended load over ripples. For the calculation of the suspended load induced by wave breaking, the Bailard formula is used after the consideration that the only dissipation mechanism is the wave breaking. Runup and rundown as well as sediment transport in the swash zone are incorporated in the model. The model is able to reproduce beach accretion and erosion without the use of an erosion/accretion criterion. The model is validated against large-scale experimental data.

Localization and Coherent Structures in Wave Dynamics via Multiresolution

We apply variational-wavelet approach for constructing multiscale high-localized eigenmodes expansions in different models of nonlinear waves. We demonstrate appearance of coherent localized structures and stable pattern formation in different collective dynamics models.

Hamiltonian Structure and Linear Stability of Solitary Waves of the Green-Naghdi Equations

We investigate linear stability of solitary waves of a Hamiltonian system. Unlike weakly nonlinear water wave models, the physical system considered here is nonlinearly dispersive, and contains nonlinearity in its highest derivative term. This results in more detailed asymptotic analysis of the eigenvalue problem in presence of a large parameter. Combining the technique of singular perturbation with the Evans function, we show that the problem has no eigenvalues of positive real part and solitary waves of small amplitude are linearly stable.