Nonlinear Wave Model Research Articles

Riding the Crest: A Tale of Two Wave Experiments

This paper gives a general overview of two ocean wave experiments. The experimental goals of the Surface Wave Processes Program (SWAPP) and of the Surface Wave Dynamics Experiment (SWADE) are quite different but complementary. In general terms, SWAPP is focused on local processes: principally wave breaking, upper mixed layer dynamics, and microwave and acoustic signatures of wave breaking. SWADE, on the other hand, is concerned primarily with the evolution of the directional wave spectrum in both time and space, improved understanding of wind forcing and wave dissipation, the effect of waves on the air-sea coupling mechanisms, and the radar response of the surface. Both programs acknowledge that wave dissipation is the weakest link in our understanding of wave evolution on the ocean. SWAPP takes a closer look at wave dissipation processes directly, while SWADE, with the use of fully non-linear (third generation) wave models and carefully measured wind forcing, provides an opportunity to study the effect of dissipation on spectral evolution. Both programs involve many research platforms festooned with instruments and large teams of scientists and engineers gathering and analyzing huge datasets. The success of SWAPP and SWADE will be measured in the degree to which the results can be integrated into a far more complete picture than we have had heretofore of interfacial physics, wave evolution, and mixed layer dynamics.

4次元データ同化の変分法による解の一意性問題

A model for nonlinear resonant Rossby waves is utilized to obtain analytical solutions. "Identical twin" approach is adopted, in which one specific solution is chosen as the control which serves as pseudo-observation. Then initial condition is varied and thus specified various solutions are compared with the control. The cost function is computed and plot vs. the initial condition. Multiple minima are revealed in the cost function. In the context of variational adjustment approach to four-dimensional data assimilation, which is usually conducted with some minimization scheme based on the gradient information, this multiple-minimum phenomenon in cost function implies nonuniqueness, i. e., the final convergent solution depends upon the first guess for the initial condition. Such nonuniqueness problem becomes more serious due to the increasing complexity of the cost function as the assimilation period gets longer. The relation to nonlinear energy exchange is discussed.

Laboratory measurements of wind‐wave dynamics

Doppler measurements can help to discriminate between waves and bubbles as mechanisms for surface backscattering. However, the analysis depends on knowledge of the near‐surface dynamics. Wind‐wave measurements of wave height versus time in a laboratory flume indicate that conventional linear wave models are not adequate at the higher wave numbers involved. Frequency autospectra have F−3 asymptotic dependence (compared to F−5 for the Pierson‐Moskowitz spectrum) and cross spectra show nearly dispersionless downwind propagation. the speed increasing with wind speed. The F−3 spectral dependence appears to be due to an effective discontinuity of slope at the wave fronts caused by the finite‐amplitude overtaking effect. Backscattering calculations based on a nonlinear wave model similar to that used for convective turbulence are in fair agreement with flume measurements.

Analysis of nonlinear internal waves observed by Landsat thematic mapper

In this work we test the compatibility between the theoretical parameters of a nonlinear wave model and the quantitative information that one can deduce from satellite‐derived data. The theoretical parameters are obtained by applying an inverse problem to the solution of the Cauchy problem for the Korteweg‐de Vries equation. Our results are applied to the case of internal wave patterns elaborated from two different satellite sensors at the south of Messina (the thematic mapper) and at the north of Messina (the synthetic aperture radar).

Conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion

Conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion

Recurrence in truncated Boussinesq models for nonlinear waves in shallow water

The rapid spatial recurrence of weakly nonlinear, weakly dispersive, progressive shallow‐water waves is examined with numerical simulations using a discretized and truncated (i.e., finite number of allowed frequency modes) form of the Boussinesq equations. Laboratory observations of sandbar formation under recurring, mechanically generated monochromatic waves with small Ursell number have motivated others to suggest that recurrence in naturally occurring random waves contributes to the establishment of periodic longshore sandbars on beaches. The present study primarily examines recurrence in wave fields with Ursell number O(1) and characterizes the sensitivity of recurrence to initial spectral shape and number of allowed frequency modes. It is shown that rapid spatial recurrence is not an inherent property of discretized and truncated Boussinesq systems for evolution distances of 10–50 wavelengths. When a small number of Fourier modes are used to represent an initially monochromatic wave field with significant nonlinearity (the Ursell number is O(1)), there is a trend toward recurrence of initial modal amplitudes, consistent with the known periodic solutions for a primary wave and its harmonic. However, for 32 modes or more, numerical simulations indicate only a few cycles of a damped recurrence, followed by disordered evolution of the Fourier amplitudes. For initial conditions similar to ocean field measurements of frequency‐sorted swell with Ursell number O(1) and many (>300) modes in the numerical model, the Fourier coefficients of the wave field do not recur rapidly. Thus in these cases the predictions of many rapid recurrence cycles by few‐mode models is an artifact of severe truncation. On the other hand, even with many allowed modes, pronounced recurrence is predicted when the Ursell number is small and the initial wave field is monochromatic. In this case, few‐ and many‐mode solutions are similar.

Comments on ‘‘The onset of Alfvénic turbulence’’ [Phys. Fluids 3 , 1371 (1987)

The interpretation of the derivative nonlinear Schrödinger model for nonlinear Alfvén waves is questioned. In particular, the damping term used by Ghosh and Papadopoulos [Phys. Fluids 30, 1371 (1987)] is shown to be inconsistent.

Measurements of edge turbulence with Langmuir probes in currentless Heliotron E plasmas

A study of microscopic fluctuations of density ñ and potential Φ̃ in the scrape-off layer has been performed. The frequency spectra of n̄ and Φ̃ show a strongly turbulent character, as observed in many tokamaks. The fluctuation levels of n̄/n and eΦ̃/kTe are consistent with predictions based on non-linear collisional drift wave models. The line average density and heating power dependences of n̄/n and eΦ̃/kTe are studied in two types of currentless plasma, namely electron cyclotron resonance heating plasmas and neutral beam injection heating plasmas. Their parametric dependences are compared with the trends of the energy confinement time. Finite beta effects on eΦ̃f/kTe and n̄/n are also discussed.

Discontinuous solutions of one-dimensional hamiltonian systems

Systems of evolutionary equations in a class having many physical applications as models for nonlinear waves are investigated from the standpoint of hamiltonian theory. Although in most applications these equations are otherwise accountable as hyperbolic systems, they are not necessarily so and the present account bypasses this more familiar approach, gaining several novel advantages. Conservation laws are related through hamiltonian structure to symmetries of the evolutionary system. Then, on the basis of the established facts about conservation laws, various types of discontinuous solutions are classified systematically; and all possible shocks or other discontinuities are shown to have variational characterizations inherited from the hamiltonian description of continuously differentiable solutions. Focusing the variational properties established in general, simple algebraic representations are found for discontinuities of limitingly small strength. Finally, two examples are treated: the system of equations modelling long waves in liquid-filled elastic tubes, applying also to long water waves in horizontal channels of arbitrary cross-section; and the equations of one-dimensional gas dynamics.

Computation of Finite‐Amplitude Dispersive Waves

An explicit finite element model for nonlinear dispersive waves is developed and applied to problems in one and two space dimensions. The proposed method uses linear chapeau functions for interpolation and is very simple to formulate and inexpensive to execute compared to most existing models. The model is sufficiently accurate so that the cumulative effect of numerical errors does not affect the results even at very long times of computation. The method is applied to the computation of solitary waves of various amplitudes and undular bores propagating through two-dimensional channel transitions. The proposed scheme is found to be equally accurate with the best available methods and even superior in eliminating spurious oscillatory tails often encountered near computed nonlinear dispersive waves.

A Model for Unidirectional Water Waves

An explicit finite element model for unidirectional nonlinear dispersive waves is developed and applied to the problems of propagation and interaction of solitary waves. The proposed method uses linear chapeau functions for interpolation and is very simple to formulate and inexpensive to execute compared to most existing models. The model is sufficiently accurate so that the cumulative effect of numerical errors does not affect the results even at very long computation times. A stability and accuracy analysis of the method is presented and the limits of applicability of the method are established both analytically and by means of computational tests. The method is found to be as accurate as the best available methods and even superior in eliminating spurious oscillatory tails often found near computed solitary waves.

Covariances and spectra of the kinematics and dynamics of nonlinear waves

Using the Stokes waves as a model of nonlinear waves and considering the linear component as a narrow‐band Gaussian process, we derive, for points in the vicinity of still water level, the covariances and spectra of velocity and acceleration components and pressure, taking into consideration the effects of free surface fluctuations. The results are compared with those obtained earlier using linear Gaussian waves.

Numerical schemes for a model for nonlinear dispersive waves

A description is given of a number of numerical schemes to solve an evolution equation (Korteweg-deVries) that arises when modelling the propagation of water waves in a channel. The discussion also includes the results of numerical experiments made with each of the schemes. It is suggested, on the basis of these experiments, that one of the schemes may have (discrete) solitary-wave solutions.

A unified model for the evolution nonlinear water waves

Details of a new model of water waves that describes wave propagation over long distances accurately, at low cost, and for a wide variety of physical situations are given. The analysis and numerical methods selected for computer solution are given in some detail. The model uses exact prognostic equations, and a high-order expansion to relate variables at each time step. The accuracy of the model is demonstrated most completely for solitary wave propagation, where model results are compared to exact results. It is found that the model results are much more accurate for high solitary waves than are earlier, Boussinesq-type theories, and give good results for waves so high that they are almost breaking. The capability of the model to treat a variety of situations is demonstrated for colliding solitary waves, nonlinear dispersive wave trains, waves in channels of varying breadth, and undular bores. Formally, the model incorporates nonlinear long wave theory exactly, incorporates enough dispersion to describe linear waves with fourth-order precision, so that both shallow water waves and deep water waves are included, and describes accurately waves for which dispersive and nonlinear effects are both important.

Internal waves in the Gulf of California: Observations from a spaceborne radar

Pronounced signatures of internal waves were detected repeatedly in the Gulf of California by the Seasat synthetic aperture radar (SAR). A series of nine images with exactly repeating ground coverage was used to study the temporal variability of the internal wave field in the area. It was found that the number of observed wave groups was highly correlated with the strength of the local tides: the maximum number occurred during spring tides and the minimum number occurred during neap tides, indicating that the internal waves were tidally forced. Most of the wave activity was found to the north of 28°N where the tides were the strongest in the Gulf. The application of a simple, nonlinear internal wave model to the observations indicated that the peak‐to‐peak amplitude of the observed waves was about 50 m with an uncertainty of a factor of 2. The estimated upper bound for the rate of the loss of tidal energy to internal waves was about 5×1015 erg/s, representing only 10% of the rate of the dissipation of the dominant M2 tide in the Gulf.

The Connection between Partial Differential Equations Soluble by Inverse Scattering and Ordinary Differential Equations of Painlevé Type

A completely integrable partial differential equation is one which has a Lax representation, or, more precisely, can be solved via a linear integral equation of Gel’fand–Levitan type, the classic example being the Korteweg–de Vries equation. An ordinary differential equation is of Painlevé type if the only singularities of its solutions in the complex plane are poles. It is shown that, under certain restrictions, if G is an analytic, regular symmetry group of a completely integrable partial differential equation, then the reduced ordinary differential equation for the G-invariant solutions is necessarily of Painlevé type. This gives a useful necessary condition for complete integrability, which is applied to investigate the integrability of certain generalizations of the Korteweg–de Vries equation, Klein–Gordon equations, some model nonlinear wave equations of Whitham and Benjamin, and the BBM equation.

Diffusion-Wave Flood Routing in Channel Networks

A nonlinear diffusion-wave model is developed for routing of floods in dendritic open-channel networks. The model considers backwater effects from both upstream and downstream ends of a channel. An overlapping-segment technique is adopted to decompose the network. The nonlinear algebraic finite-difference equations are solved through a special formulation of tridiagonal banded matrices. Comparison of the results of the proposed model to those of the dynamic-wave and nonlinear kinematic wave models indicates that the proposed model is nearly as accurate as, and computationally much cheaper, than the dynamic-wave model. It is computationally as inexpensive but more accurate than the nonlinear kinematic-wave model.

Stability of nonlinear ion sound waves and solitons in plasmas

Large amplitude ion acoustic waves and solitons in two component plasmas are investigated for stability. The soliton solutions are found to be stable, while the nonlinear waves are always unstable, though for a significant range of parameters they are only unstable to fully three-dimensional perturbations. The results in one dimension are compared with those obtained from the K. –de V. equation, which gives stability for the nonlinear waves and solitons. Agreement is surprisingly good for Mach numbers less than about 1.5. A three-dimensional generalization of the K. –de V. equation is considered but this leads to stability for all nonlinear solutions and hence is not a good model for nonlinear waves. It is, however, reasonable in the soliton limit.

A New Model for Nonlinear Wind Waves. Part 2. Theoretical Model

A first‐order theory for the development of a wind‐driven wave field is proposed. This takes advantage of the physical interpretation presented by Lake and Yuen (1978) that a nonlinear wind‐wave system can be completely characterized, to first approximation, by a single nonlinear wave train with carrier frequency coinciding with the local dominant frequency. The result is the nonlinear Schrodinger equation for the wave envelope, supplemented by an equation describing the change of dominant‐wave frequency (or wavenumber) with fetch (or duration). The latter equation is a consequence of a form‐drag model proposed by Deardorff (1967) for the input of wind energy into the dominant wave. The extension of the model to a random wave field is briefly discussed.

A new model for nonlinear wind waves. Part 1. Physical model and experimental evidence

A new interpretation of a nonlinear wind-wave system is proposed. It is proposed that, for steady wind blowing in one direction, a nonlinear wind-wave system can be completely characterized, to a good first approximation, by a single nonlinear wave train having a carrier frequency equal to that of the dominant frequency in the wind-wave spectrum. In this model, the spectral components of the wind-wave system are not considered a random collection of free waves, each obeying the usual dispersion relation, but are effectively non-dispersive bound-wave components of a single dominant wave, travelling at the speed of the dominant wave. To first order, the nonlinear wind-wave system is considered to be a coherent bound-wave system which propagates energy only at the group velocity of the dominant wave and is governed by nonlinear self-interactions of the type found in amplitude-modulated wave trains. The role of short free waves in the system is discussed. Results of laboratory experiments performed by the authors and by Ramamonjiarisoa & Coantic (1976) are found to provide evidence supporting the applicability of such a model to wind waves under virtually all laboratory conditions. Preliminary consideration is given to possible application of the model to oceanic wind waves and conditions are identified for which the model would be most likely to apply.