Nonlinear Wave Model Research Articles

“Choppy wave” model for nonlinear gravity waves

We investigate the statistical properties of a three‐dimensional simple and versatile model for weakly nonlinear gravity waves in infinite depth, referred to as the “choppy wave model” (CWM). This model is analytically tractable, numerically efficient, and robust to the inclusion of high frequencies. It is based on horizontal rather than vertical local displacement of a linear surface and is a priori not restricted to large wavelengths. Under the assumption of space and time stationarity, we establish the complete first‐ and second‐order statistical properties of surface random elevations and slopes for long‐crested as well as fully two‐dimensional surfaces, and we provide some characteristics of the surface variation rate and frequency spectrum. We establish a relationship between the so‐called “dressed spectrum,” which is the enriched wave number spectrum of the nonlinear surface, and the “undressed” one, which is the spectrum of the underlying linear surface. The obtained results compare favorably with other classical analytical nonlinear theories. The slope statistics are further found to exhibit non‐Gaussian peakedness characteristics. Compared to observations, the measured non‐Gaussian omnidirectional slope statistics can only be explained by non‐Gaussian effects and are consistently approached by the CWM.

High-resolution numerical simulation of 2D nonlinear wave structures in electromagnetic fluids with absorbing boundary conditions

Here we show how the full set of governing equations for the dynamics of charged-particle fluids in an electromagnetic field may be solved numerically in order to model nonlinear wave structures propagating in two dimensions. We employ a source-term adaptation and two-fluid extension of the second-order high-resolution central scheme of Balbas et al. (2004) [1]. The model employed is a 2D extension of that used by Baboolal and Bharuthram (2007) [5] in studies of 1D shocks and solitons in a two-fluid plasma under 3D electromagnetic fields. Further, we outline the use of free-flow boundary conditions to obtain stable wave structures over sufficiently long modelling times. As illustrative results, we examine the formation and evolution of shock-like and soliton structures of the magnetosonic mode.

Kelvin-Helmholtz billows and their effects on mean state during gravity wave propagation

Abstract. The Kelvin-Helmholtz (KH) billows which appear in the process of gravity wave (GW) propagation are simulated directly by using a compressible nonlinear two-dimensional gravity wave model. The differences between our model and others include: the background field has no special initial configuration and there is no initial triggering mechanism needed in the mesosphere and lower thermosphere (MLT) region to excite the KH billows. However, the initial triggering mechanism is performed in the lower atmosphere through GW, which then propagate into the MLT region and form billows. The braid structures and overturning of KH billows, caused by nonlinear interactions between GWs and mean flow, can be resolved precisely by the model. These results support the findings in airglow studies that GWs propagating from below into the MLT region are important sources of KH billows. The onset of small scale waves and the wave energy transfer induce the shallower vertical wave number power spectral densities (PSD). However, most of the slopes are steeper than the expected kz−3 power law, which indicates that GWs with 10 km vertical wavelength are still a dominant mode. The results also show that the evolution of mean wind vary substantially between the different processes of GWs propagation. Before the KH billows evolve, the mean wind is accelerated greatly by GWs. By contrast, as the KH billows evolve and mix with mean flow, the mean wind and its peak value decrease.

A numerical method for nonlinear water waves

This article presents a numerical method for modeling nonlinear water waves based on the High Order Spectral (HOS) method proposed by Dommermuth and Yue and West et al., involving Taylor expansion of the Dirichlet problem and the Fast Fourier Transform (FFT) algorithm. The validation and efficiency of the numerical scheme is illustrated by a number of case studies on wave and wave train configuration including the evolution of fifth-order Stokes waves, wave dispersive focusing and the instability of Stokes wave with finite slope. The results agree well with those obtained by other studies.

Generalized solitary and periodic wave solutions to a (2 + 1)-dimensional Zakharov–Kuznetsov equation

In this paper, the Exp-function method is employed to the Zakharov–Kuznetsov equation as a (2 + 1)-dimensional model for nonlinear Rossby waves. The observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient. The obtained results may be useful for better understanding the properties of two-dimensional coherent structures such as atmospheric blocking events.

Simulating and understanding sand wave variation: A case study of the Golden Gate sand waves

In this paper we present a detailed comparison between measured features of the Golden Gate sand wave field and the results of a nonlinear sand wave model. Because the Golden Gate sand waves exhibit large variation in their characteristics and in their environmental physics, this area gives us the opportunity to study sand wave variation between locations, within one well‐measured, large area. The nonlinear model used in this paper is presently the only tool that provides information on the nonlinear evolution of large‐amplitude sand waves. The model is used to increase our understanding of the coupling between the variability in environmental conditions and the sand wave characteristics. Results show that the model is able to describe the variation in the Golden Gate sand waves well when both the local oscillating tidal current and the residual current are taken into account. Current and water depth seem to be the most important factors influencing sand wave characteristics. The simulation results give further confidence in the underlying model hypothesis and assumptions.

Numerical simulations of the Boussinesq equation by He's variational iteration method

In this paper, we present a reliable algorithm to study the well-known model of nonlinear dispersive waves proposed by Boussinesq. We solve the Cauchy problem of Boussinesq equation using variational iteration method (VIM). The numerical results of this method are compared with the exact solution of an artificial model to show the efficiency of the method. The approximate solutions show that VIM is a powerful mathematical tool for solving nonlinear problems.

Inhibiting Shear Instability Induced by Large Amplitude Internal Solitary Waves in Two‐Layer Flows with a Free Surface

We consider a strongly nonlinear long wave model for large amplitude internal waves in two‐layer flows with the top free surface. It is shown that the model suffers from the Kelvin–Helmholtz (KH) instability so that any given shear (even if arbitrarily small) between the layers makes short waves unstable. Because a jump in tangential velocity is induced when the interface is deformed, the applicability of the model to describe the dynamics of internal waves is expected to remain rather limited. To overcome this major difficulty, the model is written in terms of the horizontal velocities at the bottom and the interface, instead of the depth‐averaged velocities, which makes the system linearly stable for perturbations of arbitrary wavelengths as long as the shear does not exceed a certain critical value.

Analysis of water drop erosion on turbine blades based on a nonlinear liquid–solid impact model

Water drop erosion is regarded as one of the most serious reliability concerns in the wet steam stage of a steam turbine. One of the most challenging aspects of this problem involves the fundamental solution of the transient pressure field in the liquid drop and stress field in the metal substrate, which are coupled with each other. In this paper, we first solve the fundamental problem of high-speed liquid–solid impact, both analytically and numerically, based on a nonlinear wave model. The transient pressure distribution in liquid (include shock wave) and transient stress distribution in solid are obtained for representative water drop-1Cr13 impacts, with impact speed varying from 10 m/s to 500 m/s. The relationship between the most important parameters characterizing impact and incident speed is established. With the statistics of water drop impact on the blade, a simple fatigue model is employed in this paper to obtain the lifetime map on a blade surface under typical working conditions, in terms of both impact times and operation hours. The most dangerous water drop erosion regions and operating conditions of the steam turbine blade are deduced. These results are useful for evaluating the water drop erosion mechanisms based on the fundamental solution of liquid–solid impact.

Three geomorphological rainfall–runoff models based on the linear reservoir concept

Abstract Three geomorphological models, based on linear reservoirs cascade, are developed; two of the models are unit hydrograph (UH) models and one contains a non-linear routing approach. In the first UH model, each sub-basin is represented as a cascade of equal linear reservoirs (Nash's model) which directly discharges to the watershed outlet through a linear channel. In the second UH model each sub-basin is represented as a linear reservoir arranged in a sequence based on the drainage network. For both unit hydrograph models, parameters are explicitly derived, which make the models simple and applicable for real world problems. In the third model, output hydrographs of the sub-basins are calculated by the Nash model but with consideration of geomorphological properties of the sub-basins. Then, the obtained hydrographs are routed through the main channel using a non-linear kinematic wave model. Combination of the kinematic wave routing model (a non-linear routing model) and the Nash model (a linear lumped model) makes this third model more appropriate for runoff modeling. Two important properties of these three models are that they consider the effect of watershed geomorphology and include only one parameter which can be estimated using observed rainfall–runoff data. GIS tools are used for determining the watershed geomorphological parameters. The results of these models are compared with the Nash's black box and the geomorphological SCS models for the Amameh watershed, Iran. Although the results show that the proposed models yield good efficiency in rainfall–runoff modeling, the semi-distributed routing ability in the second and third models produces better results.

Four-wave resonant interactions in the classical quadratic Boussinesq equations

We investigate theoretically the irreversibile energy transfer in flat bottom shallow water waves. Starting from the oldest weakly nonlinear dispersive wave model in shallow water, i.e. the original quadratic Boussinesq equations, and by developing a statistical theory (kinetic equation) of the aforementioned equations, we show that the four-wave resonant interactions are naturally part of the shallow water wave dynamics. These interactions are responsible for a constant flux of energy in the wave spectrum, i.e. an energy cascade towards high wavenumbers, leading to a power law in the wave spectrum of the form of k−3/4. The nonlinear time scale of the interaction is found to be of the order of (h/a)4 wave periods, with a the wave amplitude and h the water depth. We also compare the kinetic equation arising from the Boussinesq equations with the arbitrary-depth Hasselmann equation and show that, in the limit of shallow water, the two equations coincide. It is found that in the narrow band case, both in one-dimensional propagation and in the weakly two-dimensional case, there is no irreversible energy transfer because the coupling coefficient in the kinetic equation turns out to be identically zero on the resonant manifold.

An efficient flexible-order model for 3D nonlinear water waves

The flexible-order, finite difference based fully nonlinear potential flow model described in [H.B. Bingham, H. Zhang, On the accuracy of finite difference solutions for nonlinear water waves, J. Eng. Math. 58 (2007) 211–228] is extended to three dimensions (3D). In order to obtain an optimal scaling of the solution effort multigrid is employed to precondition a GMRES iterative solution of the discretized Laplace problem. A robust multigrid method based on Gauss–Seidel smoothing is found to require special treatment of the boundary conditions along solid boundaries, and in particular on the sea bottom. A new discretization scheme using one layer of grid points outside the fluid domain is presented and shown to provide convergent solutions over the full physical and discrete parameter space of interest. Linear analysis of the fundamental properties of the scheme with respect to accuracy, robustness and energy conservation are presented together with demonstrations of grid independent iteration count and optimal scaling of the solution effort. Calculations are made for 3D nonlinear wave problems for steep nonlinear waves and a shoaling problem which show good agreement with experimental measurements and other calculations from the literature.

A three-dimensional coupled numerical model of nonlinear waves in a harbor

A 3-D time-domain numerical coupled model for nonlinear waves acting on a ship in a harbor has been developed in the present study. The whole domain is divided into the inner domain and the outer domain. The inner domain is the area around the ship, where the flow is expressed by the Laplace equation and numerically solved by the finite element method. The other area is the outer domain, where the flow is described by the higher-order Boussinesq equations and numerically solved by the finite difference method. The matching conditions on the interfaces between the inner domain and the outer domain, the procedure of coupled solution, the length of common domain and the mesh generation in the inner domain are discussed in detail. The other coupled model with the flow in the inner domain governed by the simplified linear Euler equations and relevant physical experiment are adopted to validate the present coupled model, and it is shown that the numerical results of the present model agree with the experimental data, so the present model can be used for the study on the effect of nonlinear waves acting on a fixed ship in a large area and provide a reference for the time-domain simulation of nonlinear wave forces on an arbitrary object in a large harbor and the 3-D district computation in the future.

Variation Assimilating of Synthetic Radar Data into a Pseudo-spectral Wave Model

In this paper, a computationally-efficient scheme is developed to assimilate radar images into a pseudo-spectral wave evolution model. The nonlinear wave model uses a fourth-order Runge-Kutta scheme to integrate in time a coupled set of equations for the evolution of the free surface elevation and velocity potential at the free surface. An asymptotic expansion of the model variables in terms of wave steepness parameter is used to close the system of equations and determine the vertical velocity at the free surface from the velocity potential at the free surface. A variational data assimilation scheme is then developed to find an optimal initial wave field that minimizes a cost function defined as the squared difference between model predictions and radar observations over an assimilation interval. The conjugate gradient method is used for the minimization scheme with the adjoint technique used to calculate the gradient of the cost function with respect to the initial condition. Numerical experiments have been conducted with one dimensional pseudo-observations that have been generated by the forward model by adding uncorrelated noise, as well as synthetic radar data to validate the proposed assimilation scheme.

Liquid drop impact on solid surface with application to water drop erosion on turbine blades, Part I: Nonlinear wave model and solution of one-dimensional impact

Water drop erosion is regarded as one of the most serious reliability concerns in the wet steam stage of a steam turbine. The most challenging aspect of this problem involves the fundamental solution of the transient pressure field in the liquid drop and stress field in the metal substrate, which are coupled with each other. We solve the fundamental problem of high-speed liquid–solid impact both analytically and numerically. In Part I of this paper, the governing equations based on a nonlinear wave model for liquid are derived. Analytical and approximate solutions of one-dimensional liquid–solid impact are given for both linear and nonlinear models, which provide critical insights into the water drop erosion problem. Both continuous and pulsant impacts on rigid and elastic substrates are analyzed in detail. During continuous impact, the maximum impact pressure is always higher than the water hammer pressure. Upon pulsant impact and at a particular instant related with the impact duration, the maximum tensile stress appears at a certain depth below the solid surface, which can be readily related with the erosion rate. In Part II of this paper, two-dimensional (axisymmetric) liquid–solid impact is solved numerically, from which the most dangerous impact load/duration time and the most likely crack positions are deduced. Based on our recent solution of the water drop impact statistics (associated with the fluid flow in the blade channel), a comprehensive numerical study of the water drop erosion (fatigue) on a turbine blade is carried out.

Finite element solution of the Boussinesq wave equation

In this work, we investigate a Boussinesq-type flow model for nonlinear dispersive waves by developing a computational model based on the finite element discretisation technique. Hermite interpolation functions were used to interpolate approximation elements. The system is modeled using a time dependent equation. Solution to the model is obtained, through a combination of two different schemes namely: a time approximation scheme (the Newmark Method) and the eigenvalue finite element method. Using this schemes, discrete solutions of the model at different time steps, were obtained. Graphical illustrations of solutions for the transient displacements at the center and right end of the rod are presented. The results obtained are very accurate and the model efficient. JONAMP Vol. 11 2007: pp. 223-238

Modeling fully nonlinear shallow-water waves and their interactions with cylindrical structures

Introduction of a time-accurate stabilized finite-element approximation for the numerical investigation of fully nonlinear shallow-water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the triangular elements by the Galerkin method, the fourth-order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The Streamline-Upwind Petrov–Galerkin (SUPG) method with cross-wind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Numerical results obtained for cases representing propagation of solitary waves, collisions of two solitary waves, and wave-structure interactions show fairly good agreement with experimental measurements and other published numerical solutions. The comparisons between results from present fully nonlinear wave model and weekly nonlinear wave model are presented and discussed.

Wave dissipation by muddy seafloors

Muddy seafloors cause tremendous dissipation of ocean waves. Here, observations and numerical simulations of waves propagating between 5‐ and 2‐m water depths across the muddy Louisiana continental shelf are used to estimate a frequency‐ and depth‐dependent dissipation rate function. Short‐period sea (4 s) and swell (7 s) waves are shown to transfer energy to long‐period (14 s) infragravity waves, where, in contrast with theories for fluid mud, the observed dissipation rates are highest. The nonlinear energy transfers are most rapid in shallow water, consistent with the unexpected strong increase of the dissipation rate with decreasing depth. These new results may explain why the southwest coast of India offers protection for fishing (and for the 15th century Portuguese fleet) only after large waves and strong currents at the start of the monsoon move nearshore mud banks from about 5‐ to 2‐m water depth. When used with a numerical nonlinear wave model, the new dissipation rate function accurately simulates the large reduction in wave energy observed in the Gulf of Mexico.

Computation of Nonlinear Free-Surface Flows by a Meshless Numerical Method

A meshless numerical model for nonlinear free surface water waves is presented in this paper. Using the fundamental solution of the Laplace equation as the radial basis functions and locating the source points outside the computational domain, the problem is solved by collocation of boundary points. The present model is first applied to simulate the generation of periodic finite-amplitude waves with high wave steepness and then is employed to simulate the modulation of monochromatic waves passing over a submerged obstacle. Very good agreements are observed when comparing the present results with an analytical solution, experimental data, and other numerical results.

Drift solitons and shocks in inhomogeneous quantum magnetoplasmas

Linear and nonlinear drift waves are studied in inhomogeneous electron-ion quantum magnetoplasma with neutrals in the background. The Korteweg–de Vries–Burgers equation is derived by using the quantum hydrodynamic model for nonlinear drift waves with quantum corrections. Both soliton and shock solutions are obtained in different limits. It is noticed that the width of the solitary hump is decreased with the increase in the quantum parameter. However this effect is reversed for the solitary dip case. It is also found that oscillatory shock wave is dependent on the quantum parameter. However, the monotonic shock formation is independent of the quantum parameter.