Gerassimos A. Athanassoulis

A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions

Extended mild-slope equations for the propagation of small-amplitude water waves over variable bathymetry regions, recently proposed by Massel (1993) and Porter & Staziker (1995), are shown to exhibit an inconsistency concerning the slopingbottom boundary condition, which renders them non-conservative with respect to wave energy. In the present work, a consistent coupled-mode theory is derived from a variational formulation of the complete linear problem, by representing the vertical distribution of the wave potential as a uniformly convergent series of local vertical modes at each horizontal position. This series consists of the vertical eigenfunctions associated with the propagating and all evanescent modes and, when the slope of the bottom is dierent from zero, an additional mode, carrying information about the bottom slope. The coupled-mode system obtained in this way contains an additional equation, as well as additional interaction terms in all other equations, and reduces to the previous extended mild-slope equations when the additional mode is neglected. Extensive numerical results demonstrate that the present model leads to the exact satisfaction of the bottom boundary condition and, thus, it is energy conservative. Moreover, it is numerically shown that the rate of decay of the modal-amplitude functions is improved from O(n 2 ), where n is the mode number, to O(n 4 ), when the additional sloping-bottom mode is included in the representation. This fact substantially accelerates the convergence of the modal series and ensures the uniform convergence of the velocity eld up to and including the boundaries.

A coupled-mode model for the refraction–diffraction of linear waves over steep three-dimensional bathymetry

Abstract A consistent coupled-mode model recently developed by Athanassoulis and Belibassakis [1] , is generalized in 2+1 dimensions and applied to the diffraction of small-amplitude water waves from localized three-dimensional scatterers lying over a parallel-contour bathymetry. The wave field is decomposed into an incident field, carrying out the effects of the background bathymetry, and a diffraction field, with forcing restricted on the surface of the localized scatterer(s). The vertical distribution of the wave potential is represented by a uniformly convergent local-mode series containing, except of the ususal propagating and evanescent modes, an additional mode, accounting for the sloping bottom boundary condition. By applying a variational principle, the problem is reduced to a coupled-mode system of differential equations in the horizontal space. To treat the unbounded domain, the Berenger perfectly matched layer model is optimized and used as an absorbing boundary condition. Computed results are compared with other simpler models and verified against experimental data. The inclusion of the sloping-bottom mode in the representation substantially accelerates its convergence, and thus, a few modes are enough to obtain accurately the wave potential and velocity up to and including the boundaries, even in steep bathymetry regions. The present method provides high-quality information concerning the pressure and the tangential velocity at the bottom, useful for the study of oscillating bottom boundary layer, sea-bed movement and sediment transport studies.

Extension of second-order Stokes theory to variable bathymetry

In the present work second-order Stokes theory has been extended to the case of a generally shaped bottom profile connecting two half-strips of constant (but possibly different) depths, initiating a method for generalizing the Stokes hierarchy of second-and higher-order wave theory, without the assumption of spatial periodicity. In modelling the wave-bottom interaction three partial problems arise: the first order, the unsteady second order and the steady second order. The three problems are solved by using appropriate extensions of the consistent coupled-mode theory developed by the present authors for the linearized problem. Apart from the Stokes small-amplitude expansibility assumption, no additional asymptotic assumptions have been introduced. Thus, bottom slope and curvature may be arbitrary, provided that the resulting wave dynamics is Stokes-compatible. Accordingly, the present theory can be used for the study of various wave phenomena (propagation, reflection, diffraction) arising from the interaction of weakly nonlinear waves with a general bottom topography, in intermediate water depth. An interesting phenomenon, that is also very naturally resolved, is the net mass flux induced by the depth variation, which is consistently calculated by means of the steady second-order potential. The present method has been validated against experimental results and fully nonlinear numerical solutions. It has been found that it correctly predicts the second-order harmonic generation, the amplitude nonlinearity, and the amplitude variation due to non-resonant first- and-second harmonic interaction, up to the point where the energy transfer to the third and higher harmonics can no longer be neglected. Under the restriction of weak nonlinearity, the present model can be extended to treat obliquely incident waves and the resulting second-order refraction patterns, and to study bichromatic and/or bidirectional wave-wave interactions, with application to the transformation of second-order random seas in variable bathymetry regions.

A nonstationary stochastic model for long-term time series of significant wave height

In this paper an attempt is initiated to analyze long-term time series of wave data and to model them as a nonstationary stochastic process with yearly periodic mean value and standard deviation (periodically correlated or cyclostationary stochastic process). First, an analysis of annual mean values is performed in order to identify overyear trends. It turns out that it is very likely that an increasing trend is present in the examined hindcast data. The detrended time series Y(τ) is then decomposed, using an appropriate seasonal standardization procedure, to a periodic mean value μ(τ) and a residual time series W(τ) multiplied by a periodic standard deviation σ(τ) of Y(τ)=μ(τ)+σ(τ)W(τ). The periodic components μ(τ) and σ(τ) are estimated and represented by means of low-order Fourier series, and the residual time series W(τ) is examined for stationarity. For this purpose, spectral densities of W(τ), obtained from different-season segments, are calculated and compared with each other. It is shown that W(τ) can indeed be considered stationary, and thus Y(τ) can be considered periodically correlated. This analysis has been applied to hindcast wave data from five locations in the North Atlantic Ocean. It turns out that the spectrum of W(τ) is very weakly dependent on the site, a fact that might be useful for the geographic parameterization of wave climate. Finally, applications of this modeling to simulation and extreme-value prediction are discussed.

The truncated Hausdorff moment problem solved by using kernel density functions

In this work, the problem of an efficient representation and its exploitation to the approximate determination of a compactly supported, continuous probability density function (pdf) from a finite number of its moments is addressed. The representation used is a finite superposition of kernel density functions. This representation preserves positivity and can approximate any continuous pdf as closely as it is required. The classical theory of the Hausdorff moment problem is reviewed in order to make clear how the theoretical results as, e.g. the moment bounds, can be exploited in the numerical procedure. Various difficulties arising from the well-known ill-posedness of the numerical moment problem have been identified and solved. The kernel coefficients of the pdf expansion are calculated by solving a constrained, non-negative least-square problem. The consistency, numerical convergence and robustness of the solution algorithm have been illustrated by numerical examples with unimodal and bimodal pdfs. Although this paper is restricted to univariate, compactly supported pdfs, the method can be extended to general pdfs either univariate or multivariate, with finite or infinite support.

Open-ocean deep convection explored in the Mediterranean

Open-ocean deep convection is a littleunderstood process occurring in winter in remote areas under hostile observation conditions, for example, in the Labrador and Greenland Seas and near the Antarctic continent. Deep convection is a crucial link in the “Great Ocean Conveyor Belt” [Broecker, 1991], transforming poleward flowing warm surface waters through atmosphere-oceaninteraction into cold equatorward flowing water masses. Understanding its physics, interannual variations, and role in the global thermohaline circulation is an important objective of climate change research. In convection regions, drastic changes in water mass properties and distribution occur on scales of 10–100 km. These changes occur quickly and are difficult to observe with conventional oceanographic techniques. Apart from observing the development of the deep-mixed patch of homogeneous water itself, processes of interest are convective plumes on scales <1 km and vertical velocities of several cm s−1 [Schott et al., 1994] that quickly mix water masses vertically, and instability processes at the rim of the convection region that expedite horizontal exchanges of convected and background water masses [e.g., Gascard, 1978].

Bivariate distributions with given marginals with an application to wave climate description

Abstract The class of bivariate probability distributions with given (prespecified) marginals is studied, and a special member of this class, the Plackett model, is applied to represent the joint probability distribution of significant wave height ( H s ) and mean zero-upcrossing period ( T 02 ). The distinctive features of the bivariate Plackett model are: (i) it accepts any kind of univariate probability models as marginals, (ii) it can accurately model the degree of correlation between H s and T 02 , and (iii) it permits an easy and reliable estimation of parameters. Applications of the Plackett model to the description of ( H s , T 02 ) statistics for certain sea locations illustrate its overall performance and flexibility. The possibility of extending this approach to the multivariate case is discussed.

A unified methodology for the analysis, completion and simulation of nonstationary time series with missing values, with application to wave data

A new methodology for the analysis, missing-value completion and simulation of an incomplete nonstationary time series of wave data is presented and validated. The method is based on the nonstationary modelling of long-term time series developed by the authors [J. Geophys. Res. 100 (1995) 16,149]. The missing-value completion is performed at the level of the series of the uncorrelated residuals, obtained after having removed both any systematic trend (e.g. monotone and periodic components) and the correlation structure of the remaining stationary part. The incomplete time series of uncorrelated residuals is then completed by means of simulated data from a population with the same probability law. Combining then all estimated components, a new nonstationary time series without missing values is constructed. Any number of time series with the same statistical structure can be obtained by using different populations of uncorrelated residuals. The missing-value completion procedure is applied to an incomplete time series of significant wave height, and validated using two synthetic time series having the typical structure of many-year long time series of significant wave height. The missing-value patterns used for validation have been obtained from existing (measured) wave datasets with 16.5 and 33% missing values, respectively.

An atlas of the wave energy resource in Europe

This paper presents an Atlas of the European offshore wave energy resource that is being developed within the scope of an European project. It will be mainly based on wave estimates produced by the numerical wind-wave model WAM that is in routine operation at the European Centre for Medium-Range Weather Forecasts, Reading, UK. This model was chosen after a preliminary verification of two models again buoy data for a one-year period. Wave measurements will be used for the Norwegian Sea and the North Sea. The Atlas will be produced as a user-friendly software package for MS-DOS microcomputers permitting fast retrieval of information as well as saving and printing of statistics and maps. The Atlas will include annual and seasonal statistics of significant wave height, mean and peak period, mean direction and wave power levels (global values as well as directional distributions). These data will be both presented as tables, graphs and as geographic maps.

Ocean acoustic tomography based on peak arrivals

The recently introduced notion of peak arrivals [Athanassoulis and Skarsoulis, J. Acoust. Soc. Am. 97, 3575–3588 (1995)], defined as the significant local maxima of the arrival pattern, is studied here as a modeling basis for performing ocean tomography. Peak arrivals constitute direct theoretical counterparts of experimentally observed peaks, and offer a complete modeling of experimental observables, even in cases where ray or modal arrivals cannot be resolved. The coefficients of the resulting peak‐inversion system, relating travel‐time with sound‐speed perturbations, are explicitly calculated in the case of range‐independent environments using normal‐mode theory. To apply the peak‐inversion scheme to tomography the peak identification and tracking problem is examined from a statistical viewpoint; maximum‐likelihood and least‐square solutions are derived and discussed. The particular approach adopted treats the identification and tracking problem in close relation to the inversion procedure; all possibi...