Water Wave Model Research Articles

Spectral boundary element modeling of water waves in Pohang New Harbor and Paradip Port

A novel mathematical model formulation based on spectral boundary element method (SBEM) is presented to examine the wave response in the Pohang New Harbor (PNH), South Korea and Paradip port, Odisha, India. In SBEM, the boundary element method (BEM) is coupled with the spectral element method (SEM) to enhance the numerical accuracy of the present numerical scheme. The numerical solution on each boundary element is obtained by using boundary integral associated with Chebyshev point discretization. The boundary Integrals are transformed using Jacobians and evaluated with Clenshaw Curtis Quadrature rule. Convergence analysis is performed on rectangular domain for present scheme, BEM and hybrid finite element method (HFEM), which shows that the present numerical scheme is better as compared to other traditional numerical schemes. Further, the simulation results are validated with BEM, analytical method and experimental data from Lee (1971) and Ippen and Goda (1963). The wave amplification is obtained at six record stations inside the PNH, South Korea and Paradip port, Odisha, India. In addition, the spectral density is also determined for multidirectional random waves propagating towards the PNH and Paradip port at the same record station. The resonant frequencies are estimated in PNH and Paradip port for the safe navigation of moored ship.

Energy accumulation during the growth of forced wave induced by a moving atmospheric pressure disturbance

ABSTRACTAtmospheric pressure disturbances of high translational speed can excite significant water waves, which are widely believed to be the main cause of destructive meteotsunamis. During the response of sea surface to adisturbance, energy is transferred from the atmosphere to the ocean, while the forced wave induced by the disturbance gradually grows and finally reaches a quasi-steady state. This study aims at quantifying the time required to excite the forced wave. Based on the nonlinear shallow water wave model, the evolution of the forced wave is numerically investigated. Analyses of the accumulation of wave energy are imported to help understand the evolution of the forced wave in different situations. The characteristics of energy accumulation show a great difference in the cases of different Froude number Fr. A parameter named response time is defined to quantify the time required for the evolution of the forced wave. It can be seen that Fr is the principal parameter that controls the wave evolution, while the effects of nonlinearity and bottom friction are significant only when Fr is close to 1. Simple relationships are proposed to roughly estimate the response time for Fr<0.9 and Fr>1.1, which can also provide a lower limit for Fr≈1.

Construction of solitary wave solutions of some nonlinear dynamical system arising in nonlinear water wave models

The higher order of nonlinear partial differential equations in mathematical physics is studied. We used the analytical mathematical methods of the nonlinear (3+1)-dimensional extended Zakharov–Kuznetsov dynamical, modified KdV–Zakharov–Kuznetsov and generalized shallow water wave equations to demonstrate the efficiency and validity of the proposed powerful technique. The shallow water wave models have been applied in tidal waves and weather simulation. Exact wave solutions of these models in various forms such as Kink and anti-Kink solitons, bright–dark soliton, solitary wave and periodic solutions are constructed that have plenty of applications in diverse areas of physics. Graphically, we presented the movement of some obtained solitary wave solutions that aids in understanding the physical phenomena of these models.

Development and assessment of semi-implicit nonhydrostatic models for surface water waves

This paper presents two semi-implicit nonhydrostatic models to simulate surface water waves. Both models utilize the pressure correction method to solve three-dimensional (3D) incompressible Navier–Stokes equations (NSEs), but one is a noniterative model and the other is associated with iterating to solve the Poisson equation for pressure correction terms. To meet the high computational demands, the Poisson equation is solved on the graphics processing unit (GPU) in conjunction with the conjugate gradient method. Both the accuracy in resolving the linear dispersion relation and the serially implemented efficiency of the models are assessed and discussed. Then, both models are validated with several test cases ranging from shallow to deep water. The results of the two models are very similar and show good agreement with experimental data in the shallow-water test cases. The accuracy of both models are also evaluated in deep-water nonlinear sloshing waves and 2D/3D focusing waves. In addition, the acceleration due to GPU parallelization is evaluated in a final 3D test case.

One-Dimensional Symmetry for the Solutions of a Three-Dimensional Water Wave Problem

We prove a one-dimensional symmetry result for a weighted Dirichlet-to-Neumann problem arising in a model for water waves in dimensions 2 and 3. More precisely, we prove that stable solutions in dimension 2 and minimizers and monotone solutions in dimension 3 depend on only one Euclidean variable. Monotone solutions in the 2-dimensional case without weights were studied in de la Llave and Valdinoci (Math Res Lett 16(5):909–918, 2009). In our paper, a crucial ingredient in the proof is given by an energy estimate for minimizers obtained via a comparison argument.

Singularity formation as a wetting mechanism in a dispersionless water wave model

The behavior of a class of solutions of the shallow water Airy system originating from initial data with discontinuous derivatives is considered. Initial data are obtained by splicing together self-similar parabolae with a constant background state. These solutions are shown to develop velocity and surface gradient catastrophes in finite time and the inherent persistence of dry spots is shown to be terminated by the collapse of the parabolic core. All details of the evolution can be obtained in closed form until the collapse time, thanks to formation of simple waves that sandwich the evolving self-similar core. The continuation of solutions asymptotically for short times beyond the collapse is then investigated analytically, in its weak form, with an approach using stretched coordinates inspired by singular perturbation theory. This approach allows to follow the evolution after collapse by implementing a spectrally accurate numerical code, which is developed alongside a classical shock-capturing scheme for accuracy comparison. The codes are validated on special classes of initial data, in increasing order of complexity, to illustrate the evolution of the dry spot initial conditions on longer time scales past collapse.

Two-dimensional surface waves in magnetohydrodynamics

The study of nonlinear waves in water has a long history beginning with the seminal paper by Korteweg &amp; de Vries (Phil. Mag., vol. 39, 1895, p. 240) and more recently for magnetohydrodynamics Danov &amp; Ruderman (Fluid Dyn., vol. 18, 1983, pp. 751–756). The appearance of a Hilbert transform in the nonlinear equation for magnetohydrodynamics (MHD) distinguishes it from the water wave model description. In this paper, we are interested in examining weakly nonlinear interfacial waves in $2+1$ dimensions. First, we determine the wave solution in the linear case. Next, we derive the corresponding generalisation for the Kadomtsev–Petviashvili (KP) equation with the inclusion of an equilibrium magnetic field. The derived governing equation is a generalisation of the Benjamin–Ono (BO) equation called the Benjamin equation first derived in Benjamin (J. Fluid Mech., vol. 245, 1992, pp. 401–411) and in the higher-dimensional context in Kim &amp; Akylas (J. Fluid Mech., vol. 557, 2006, pp. 237–256).

On pressure forcing techniques for numerical modelling of unidirectional nonlinear water waves

A method of incorporating pressure forcing into a nonlinear potential flow wave model is presented. A semi-analytical pseudo-spectral method is used to calculate dynamic response of a water body exposed to evolving local pressure distribution. Surface slope coherent and slope proportional pressure functions are directly applied through a pressure term appearing in the dynamic free-surface boundary condition of a formulated initial boundary-value problem. First, a monochromatic pressure distribution is used to generate steady regular waves of permanent form. The pressure-induced wave motion exhibits stable harmonic structure for deepwater, transitional water and shallow water waves. In the next step, a more complex pressure system is used to initiate multi-component wave propagation. It is demonstrated that the proposed method provides well-posed initial conditions for studying various water wave scenarios within a framework of nonlinear potential flow solutions.

New Wave Solutions of Time-Fractional Coupled Boussinesq–Whitham–Broer–Kaup Equation as A Model of Water Waves

The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.

Transverse linear stability of line periodic traveling waves for water-wave models

We review some recent results on transverse linear stability of line periodic traveling waves for the water-wave problem. A common feature of these results is that they can be obtained from two, rather simple, abstract stability criteria. While the first criterion gives sufficient conditions for linear instability, the second one, which is a counting result for unstable eigenvalues, leads to sufficient conditions for spectral stability. We restrict to waves of small amplitude bifurcating in four different parameter regimes. We focus on the simplest model equations, the Kadomtsev-Petviashvili I and II equations, and refer to existing works for other models, including the full Euler equations.

On sharp global well-posedness and ill-posedness for a fifth-order KdV-BBM type equation

We consider the Cauchy problem associated to the recently derived higher order Hamiltonian model for unidirectional water waves and prove global existence for given data in the Sobolev space Hs, s≥1. We also prove an ill-posedness result by showing that the flow-map is not continuous if the given data has Sobolev regularity s<1. The results obtained in this work are sharp.

Well-posedness and wave breaking for a shallow water wave model with large amplitude

Considered herein is the Cauchy problem of a model for shallow water waves of large amplitude. Using Littlewood–Paley decomposition and transport equation theory, we establish the local well-posedness of the equation in Besov spaces $$B^s_{p,r} $$ with $$1\le p,r \le +\infty $$ and $$s>\max \{1+\frac{1}{p },\frac{3}{2}\}$$ (and also in Sobolev spaces $$H^s=B^s_{2,2}$$ with $$s>3/2$$). Then, the precise blow-up mechanism for the strong solutions is determined in the lowest Sobolev space $$H^s $$ with $$s>3/2$$. Our results improve the corresponding work for this model in Quirchmayr (J Evol Equ 16:539–567, 2016), in which the Sobolev index $$s=3$$ is required. In addition, we also investigate the asymptotic behaviors of the strong solutions to this equation at infinity within its lifespan provided the initial data lie in weighted $$L_{p,\phi }:=L_p(\mathbb {R},\phi ^pdx)$$ spaces.

Non-uniform dependence and well-posedness for the rotation-Camassa-Holm equation on the torus

In this paper, we study the periodic Cauchy problem for a mathematical model of the equatorial water waves propagating mainly in one direction with the weak Coriolis effect due to the Earth's rotation, which reduces to the Camassa-Holm equation as the Coriolis effect vanishes. We first prove that this model is well-posed in the sense of Hadamard when initial data belongs to the Sobolev space Hs(T) with s>3/2. Then we show that the well-posedness is sharp in the sense that the continuity of the data-to-solution map is not better than continuous, whose proof rests upon the method of approximate solutions and the well-posedness estimate. However, if Hs(T) is equipped with a weaker Hr-topology for 0≤r<s, we demonstrate that the data-to-solution map is Hölder continuous with the exponent depending on r and s. Finally, we establish a Cauchy-Kowalevski type theorem for this model.

The effects of surface tension on modulational instability in full-dispersion water-wave models

We study the modulational instability of a shallow water model, with and without surface tension, which generalizes the Whitham equation to include bi-directional propagation. Without surface tension, the small amplitude periodic traveling waves are modulationally unstable if their wave number is greater than a critical wave number predicting a Benjamin–Feir type instability and the result qualitatively agrees with the shallow water model in Hur and Pandey (2016). With surface tension, the result qualitatively agrees with the physical problem except for the large surface tension limit which is accurately predicted by the shallow water model in Hur and Pandey (2016). We also compare the results with the Whitham and full-dispersion Camassa–Holm equations. We conclude that the shallow water model in Hur and Pandey (2016) is a better model than the shallow water model presented here when the effects of surface tension on modulational instability is considered.

Blow-up phenomena for the rotation-two-component Camassa–Holm system

In this paper, we study blow-up phenomena for the rotation-two-component Camassa–Holm (R2CH) system, a model of equatorial water waves that includes the effect of the Coriolis force. We first derive the precise blow-up scenario for R2CH with Then, we present two new blow-up results for strong solutions to the system and then the precise blow-up rate for strong solutions to the system is given.

Nonlinear wave generation by a wavemaker in deep to intermediate water depth

A new analytical method to calculate the nonlinear correction to the wavemaker motion needed for the accurate generation of steep waves is suggested. It is developed taking advantage of the existing models of nonlinear water waves, such as Zakharov, Dysthe, or Schrödinger equations. The limitations of the method in terms of the order of nonlinearity and spectral width are determined by the wave model used, thus allowing its extension to the 3rd and higher orders, beyond the limits of the existing wavemaker theories. Application of the nonlinear water waves models allowed significant simplification of the procedure of determination of the nonlinear correction to the wavemaker driving signal. The suggested method was carefully validated against the theory of Schäffer, fully-nonlinear numerical simulations and wave flume experiments. The advantages of the method were exploited to carefully investigate different regimes of wave generation. An appreciable deviation of the actual value of the linear transfer function from the theoretical predictions was found in agreement with the earlier experimental investigations. The mean current arising at the 2nd order due to finite displacements of the wavemaker surface was suggested as one of the reasons for that deviation.

A New Multilayer Nonhydrostatic Formulation for Surface Water Waves

Wang, G.; Liang, Q.; Zheng, J., and Wan, P., 2019. A new multilayer nonhydrostatic formulation for surface water waves. Journal of Coastal Research, 35(3), 693–710. Coconut Creek (Florida), ISSN 0749-0208.This work presents a new multilayer nonhydrostatic formulation for surface water waves. The new governing equations define velocities and pressure at an arbitrary location of a vertical layer and only contain spatial derivatives of maximum second order. Stoke-type Fourier and shoaling analyses are carried out to scrutinize the mathematical properties of the new formulation, subsequently optimizing the representative interface and the location to define variables in each layer to improve model accuracy. Following the analysis, the one-layer model exhibits accurate linear and nonlinear characteristics up to kd = π, demonstrating similar solution accuracy to the existing second-order Boussinesq-type models. The two-layer model with optimized coefficients can maintain its linear and nonlinear accuracy up to kd = 4π, which boasts of better solution accuracy a larger application range than most existing fourth-order Boussinesq model and two-layer Boussinesq models. The three-layer model presents accurate linear and nonlinear characteristics up to kd = 10π, effectively removing any shallow water limitation. The current multilayer nonhydrostatic water wave model does not predefine the vertical flow structures, and more accurate vertical velocity distributions can be obtained by considering the velocity profiles in coefficient optimization.

A Nonlocal Shallow-Water Model Arising from the Full Water Waves with the Coriolis Effect

In the present study a mathematical model of long-crested water waves propagating mainly in one direction with the effect of Earth's rotation is derived by following the formal asymptotic procedures. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations and has a formal bi-Hamiltonian structure. Its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Our refined analysis is approached by applying the method of characteristics and conserved quantities to the Riccati-type differential inequality.

Rigorous Asymptotic Models of Water Waves

We develop a rigorous asymptotic derivation of two mathematical models of water waves that capture the full nonlinearity of the Euler equations up to quadratic and cubic interactions, respectively. Specifically, letting $$ \epsilon $$ denote an asymptotic parameter denoting the steepness of the water wave, we use a Stokes expansion in $$ \epsilon $$ to derive a set of linear recursion relations for the tangential component of velocity, the stream function, and the water wave parameterization. The solution of the water wave system is obtained as an infinite sum of solutions to linear problems at each $$O( \epsilon ^k)$$ level, and truncation of this series leads to our two asymptotic models, which we call the quadratic and cubic h-models. These models are well posed in spaces of analytic functions. We prove error bounds for the difference between solutions of the h-models and the water wave system. We also show that the Craig–Sulem models of water waves can be obtained from our asymptotic procedure. We then develop a novel numerical algorithm to solve the quadratic and cubic h-models as well as the full water wave system. For three very different examples, we show that the agreement between the model equations and the water wave solution is excellent, even when the wave steepness is quite large. We also present a numerical example of corner formation for water waves.

Numerical Study of the Second-Order Correct Hamiltonian Model for Unidirectional Water Waves

Second-order correct versions of the usual KdV–BBM models for unidirectional propagation of long-crested, surface water waves are considered here. The class of models studied here has a Hamiltonian structure and, in certain circumstances, is globally well posed. A fully discrete, numerical algorithm based on the Fourier-spectral method is developed and its convergence tested. We then use this algorithm to generate solitary-wave solutions to the model. While such waves are known to exist, exact formulas for them are not available. The heart of the paper is a sequence of numerical experiments aimed at understanding the stability of individual solitary waves, their interaction, and whether or not the model exhibits resolution of general initial data into solitary waves. A comparison is made between the first-order correct KdV–BBM models and the associated second-order correct equations. A number of tentative conjectures pertaining to the models are put forward on the basis of these experiments.