Polynomial chaos (PC) expansions are used to propagate parametric uncertainties in ocean global circulation model. The computations focus on short-time, high-resolution simulations of the Gulf of Mexico, using the hybrid coordinate ocean model, with wind stresses corresponding to hurricane Ivan. A sparse quadrature approach is used to determine the PC coefficients which provides a detailed representation of the stochastic model response. The quality of the PC representation is first examined through a systematic refinement of the number of resolution levels. The PC representation of the stochastic model response is then utilized to compute distributions of quantities of interest (QoIs) and to analyze the local and global sensitivity of these QoIs to uncertain parameters. Conclusions are finally drawn regarding limitations of local perturbations and variance-based assessment and concerning potential application of the present methodology to inverse problems and to uncertainty management.

The method of polynomial chaos expansions is illustrated by showing how uncertainties in boundary conditions specifying the flow from the Caribbean Sea into the Gulf of Mexico manifest as uncertainties in a model?s simulation of the Gulf?s surface elevation field. The method, which has been used for a variety of engineering applications, is explained within an oceanographic context and its advantages and disadvantages are discussed. The method?s utility requires that the spatially and temporally varying uncertainties of the inflow be characterized by a small number of independent random variables, which here correspond to amplitudes of spatiotemporal modes inferred from an available boundary climatology.

A preconditioning approach is developed that enables efficient polynomial chaos (PC) representations of uncertain dynamical systems. The approach is based on the definition of an appropriate multiscale stretching of the individual components of the dynamical system which, in particular, enables robust recovery of the unscaled transient dynamics. Efficient PC representations of the stochastic dynamics are then obtained through non-intrusive spectral projections of the stretched measures. Implementation of the present approach is illustrated through application to a chemical system with large uncertainties in the reaction rate constants. Computational experiments show that, despite the large stochastic variability of the stochastic solution, the resulting dynamics can be efficiently represented using sparse low-order PC expansions of the stochastic multiscale preconditioner and of stretched variables. The present experiences are finally used to motivate several strategies that promise to yield further advantages in spectral representations of stochastic dynamics.

In this study, singular vectors (SVs) are calculated for tropical cyclone (TC)-like vortices on an f plane and beta plane using a barotropic model, and the structure and time evolution of the SVs are investigated. In the f-plane study. SVs are calculated for TC-like vortices that do and do not satisfy a necessary condition of barotropic instability of normal modes, in which the vorticity gradient changes sign. It is found that, in the case where the initial vortices do not meet the condition, 1) the SVs are tilted against the shear of the background angular velocity as found earlier by Nolan and Farrell, indicating the growth of SVs through the Orr mechanism; 2) the leading singular value increases with the maximum tangential wind speed V(max) and decreases with the radius of the maximum wind (RMW); and 3) the locations of SVs move outward with increasing RMW, V(max), and the optimization time. In the case where the initial vortex allows for barotropic instability, the SV is initially tilted against the background shear and exhibits transient growth for a limited period. At a certain time during the initial growth, the SV “locks in” to a normal mode structure and remains in that structure so that it may grow exponentially with time. In contrast to the SVs on an f plane, the azimuthal distribution of the SVs on a beta plane becomes more asymmetric, and the extent of the asymmetry increases as the strength of the beta gyres increases. On the beta plane, all first and second SVs calculated in this study have an azimuthal wavenumber-1 structure at the optimization time, regardless of whether the vorticity gradient of initial TC-like vortices changes sign and the TC-like vortices include the beta gyres at initial time. It is found that when the first and second SVs are used as ensemble initial perturbations, the linear combination of the initial first and second SVs can shift the vortex toward any direction at the optimization time. This is true even when SVs with a low horizontal resolution are used as initial perturbations, as in the European Centre for Medium-Range Weather Forecasts (ECMWF) and Japan Meteorological Agency (JMA) ensemble prediction system. Such wavenumber-1 perturbations could be useful for generating sufficient spread among the tropical cyclone tracks in ensemble forecasts.

Previous studies have offered hypotheses for the mechanisms that lead to secondary eyewall formation in tropical cyclones by using two-dimensional incompressible flow. Those studies represented the convection-induced vorticity field as either large but weak vortices that are the same sign as the tropical cyclone core or as purely asymmetric vorticity perturbations that are an order of magnitude weaker than the core. However, both observations and full-physics simulations of tropical cyclones indicate that the convection-induced vorticity field should also include clusters of small vorticity dipoles whose magnitude is comparable to that of the high-vorticity core. Results of numerical simulations indicate that the interaction between the tropical cyclone core vortex and the convection-induced small vorticity dipoles of considerable strength in two-dimensional flow does not lead to coherent concentric vorticity ring formation. The axisymmetrization process under the simplification of two-dimensional incompressible flow appears to be incomplete for describing secondary eyewall formation.

The Deep Water Horizon well blowout on April 20th 2010 discharged between 40,000-1.2 million tons of crude oil into the Gulf of Mexico. In order to understand the fate and impact of the discharged oil, particularly on the environmentally sensitive Florida Keys region, we have implemented a multi-component application which consists of many individual tasks that utilize a distributed set of computational and data management resources. The application consists of two 3D ocean circulation models of the Gulf and South Florida and a 3D oil spill model. The ocean models used here resolve the Gulf at 2 km and the South Florida region at 900 m. This high resolution information on the ocean state is then integrated with the oil model to track the fate of approximately 10 million oil particles. These individual components execute as MPI based parallel applications on a 576 core IBM Power 5 cluster and a 5040 core Linux cluster, both operated by the Center for Computational Science, University of Miami. The data and workflow between is handled by means of a custom distributed software framework built around the Open Project for Networked Data Access Protocol (OPeNDAP). In this paper, we present this application as an example of Many Task Computing, report on the execution characteristics of this application, and discuss the challenges presented by the many task distributed workflow involving heterogeneous components. The application is a typical example from the ocean modeling and forecasting field and imposes soft timeliness and output quality constraints on top of the traditional performance requirements.

Abyssal pathways and volume transports in the Indian Ocean obtained from a 1/12 degrees global data assimilative hindcast for the years 2004-2006 are presented. The known features of bottom water circulation such as boundary currents and inter-basin exchanges are well represented in the hindcast solution. Further, the hindcast solution reveals a basin-wide, anti-cyclonic deep circulation pattern with the main inflow located in the east, in the Perth Basin, and outflow in the west along the African continent and Madagascar. Inter-basin westward transport at 10 degrees S across the Ninetyeast Ridge and along the Mascarene Plateau connect the inflow and outflow locations. The model develops a net bottom water inflow of 10 +/- 8 Sv (mean +/- standard deviation) across 32 degrees S, below approximately 3500 m of which 4+/- 7 Sv are returned south across the same latitude in the deep layers between 2000 and 3500 m. Significant variability on intra-seasonal, semi-annual and annual time scales is present in the abyssal flow fields. The vertical mixing algorithm, based on resolved vertical shear, suggests enhanced diapycnal upwelling in the vicinity of the Central Indian Ridge.

Keywords: Abyssal circulation; Indian Ocean; Ocean modeling; Data assimilation; Meridional overturning

Mixing between overflows and ambient water masses is a critical problem of deep-water mass formation in the downwelling branch of the meridional overturning circulation of the ocean. Modeling approaches that have been tested so far rely either on algebraic parameterizations in hydrostatic ocean circulation models, or on large eddy simulations that resolve most of the mixing using nonhydrostatic models. In this study, we examine the performance of a set of turbulence closures, that have not been tested in comparison to observational data for overflows before. We employ the so-called very large eddy simulation (VLES) technique, which allows the use of k-ε models in nonhydrostatic models. This is done by applying a dynamic spatial filtering to the k-ε equations. To our knowledge, this is the first time that the VLES approach is adopted for an ocean modeling problem. The performance of k-ε and VLES models are evaluated by conducting numerical simulations of the Red Sea overflow and comparing them to observations from the Red Sea Outflow Experiment (REDSOX). The computations are constrained to one of the main channels transporting the overflow, which is narrow enough to permit the use of a two-dimensional (and nonhydrostatic) model. A large set of experiments are conducted using different closure models, Reynolds numbers and spatial resolutions. It is found that, when no turbulence closure is used, the basic structure of the overflow, consisting of a well-mixed bottom layer (BL) and entraining interfacial layer (IL), cannot be reproduced. The k-ε model leads to unrealistic thicknesses for both BL and IL, while VLES results in the most realistic reproduction of the REDSOX observations.

Mixing of overflows released from polar and marginal seas is a key process shaping the structure of the meridional overturning circulation. Ocean general circulation models have difficulty in resolving the overflows, and therefore they must rely on parameterizations. In this study, the performance of a set of turbulence closures in reproducing mixing of an overflow is quantified. We simulate the Red Sea overflow by employing standard k-ε, k-ω and Mellor?Yamada schemes with various stability functions, as well as a modified k-ε model that relies on the prescription of the turbulent Prandtl number rather than on stability functions. The simpler KPP mixing scheme and experiments without turbulent fluxes serve as useful references. To our knowledge, this is the first time that the performance of two-equation turbulence models has been examined so closely using data from an overflow. It is found that without turbulence closures, the hydrodynamic model has difficulty in reproducing the correct three-dimensional pathway of the Red Sea overflow, consisting of a distinct bifurcation into two diverging channels. All turbulence models capture the vertical structure of this overflow consisting of an interfacial layer, characterized by the density gradient, and a well-mixed bottom layer. Mean eddy diffusivity values from most closures are comparable those from observations. But we find that KPP leads to eddy diffusivity values that are too small while those from Mellor?Yamada with Galperin [Galperin, B., Kantha, L.H., Hassid, S., Rosati, A., 1988. A quasi-equilibrium turbulent energy model for geophysical flows. J. Atmos. Sci. 45, 55?62] stability functions are too large. Such high diffusivities lead to excessive mixing in the bottom layer of the overflow, ultimately resulting in a salinity deficit of approximately 1 psu in the product water mass. Salinity deviations between the models and observations are quantified using both data taken along the channels and two sections across the overflow. KPP and Mellor?Yamada with Galperin (1988) stability functions produce the largest deviations from the observations, while the modified k-ε exhibits the smallest deviations. The other four closures fall in between, showing results similar to one another. The performance of the Mellor?Yamada turbulence closure is improved considerably by using the stability functions by Kantha and Clayson [Kantha, L.H., Clayson, C.A., 1994. An improved mixed layer model for geophysical applications. J. Geophys. Res. 99 (December), 25235?25266], which allow for a stationary Richardson number of 0.21. In conclusion, we find that most turbulence closures lead to a satisfactory reproduction of the Red Sea overflow, within the temporal and spatial sampling uncertainties of the REDSOX data, provided that fairly high-resolution regional models are used.

The thin aspect ratio of oceanic basins is simultaneously a complication to contend with when developing ocean models and an opportunity to simplify the equations of motion. Here we discuss these two aspects of this geometric feature in the context of hydrostatic and non-hydrostatic ocean models. A simple analysis shows that the horizontal viscous operator in the hydrostatic primitive equations plays a central role in the specification of boundary conditions on the lateral vertical surfaces bounding the domain. The asymptotic analysis shows that for very thin aspect ratios the leading-order flow cannot be closed unless additional terms in the equations are considered, namely either the horizontal viscous forces or the non-hydrostatic pressure forces. In either case, narrow boundary layers must be resolved in order to close the circulation properly. The computational cost increases substantially when non-hydrostatic effects are taken into account.

A numerical study aimed at investigating the roles of both the stratification and topographic slope in generation of turbulent coherent structures in the lee of capes is presented. We consider a steady barotropic current impinging on an obstacle in a rotating and linearly stratified environment. The obstacle is a triangular prism and represents an idealized headland extending from the coast. Numerical experiments are conducted at constant Rossby number Ro=0.06, varying the Burger number, Bu, and the obstacle slope, α. Flow regime diagrams in the Bu?α space are determined. For Bu<0.1, vertical movement over the obstacle is enhanced and a fully attached regime with pronounced internal waves is established. For 0.1<=Bu<1, fluid parcels flow more around the obstacle than over it. Flow separation occurs and small tip eddies start to shed. For Bu?1, tip eddies merge to form larger eddies in the lee of the cape. We find that previous laboratory results cannot be used for gentler slopes, since bottom flow regimes are strongly dependent on α when Bu?1. The form drag coefficient exerted by the cape is at least two orders of magnitude larger than the one due to skin friction. It increases with increasing Burger numbers and decreasing slopes. When no separation occurs (low Bu), the increase with decreasing slopes is the result of the mixing associated with hydraulic phenomena. For intermediate and high Bu, form drag coefficients reach larger values as a result of the boundary layer mixing associated with flow separation. We put forth an empirical parametrization of form drag in the Bu?α space.

We present the derivation of the discrete Euler?Lagrange equations for an inverse spectral element ocean model based on the shallow water equations. We show that the discrete Euler?Lagrange equations can be obtained from the continuous Euler?Lagrange equations by using a correct combination of the weak and the strong forms of derivatives in the Galerkin integrals, and by changing the order with which elemental assembly and mass averaging are applied in the forward and in the adjoint systems. Our derivation can be extended to obtain an adjoint for any Galerkin finite element and spectral element system. We begin the derivations using a linear wave equation in one dimension. We then apply our technique to a two-dimensional shallow water ocean model and test it on a classic double-gyre problem. The spectral element forward and adjoint ocean models can be used in a variety of inverse applications, ranging from traditional data assimilation and parameter estimation, to the less traditional model sensitivity and stability analyses, and ensemble prediction. Here the Euler?Lagrange equations are solved by an indirect representer algorithm.

The discontinuous Galerkin method is implemented in the spectral element ocean model to replace a continuous Galerkin discretization of the continuity and the tracer evolution equations. The aim is to improve the model?s local conservation properties, and thus its performance in advection-dominated flows. The new model is validated against several oceanic benchmark problems, particularly ones that feature frontal structures and under-resolved features. Comparisons confirm the advantages of the DGM, including enhanced model robustness.

We review and compare advection schemes designed for high-order finite element/finite volume methods. The emphasis is on studying, by numerical examples, the properties of these schemes in terms of accuracy, and monotonicity, and their viability for oceanic applications. The schemes reviewed are classical spectral element, Taylor Galerkin Least Square method, the Discontinuous Galerkin method and high-order finite volume method. The latter two schemes exhibit a definite robustness due to their small, but finite, inherent numerical dissipation. They also prove the most flexible since their discontinuous representation of the solution allows easy implementations of flux limiting or adaptive procedure. Finally, an ad-hoc but simple adaptive procedure is presented to illustrate DGM?s potential; this procedure proved to be extremely effective at controlling Gibbs oscillations in 1D but was too dissipative on the Hecht problem.

A spectral finite-volume (SFV) method is proposed for the numerical solution of the shallow water equations. This is the first phase in the development of a layered (isopycnal) ocean model. Its target applications include, in particular, the simulation of the wind-driven oceanic circulation in geometrically complex basins where layer outcropping and/or isopycnal?bathymetry intersection must be handled explicitly. The present formulation is geometrically flexible and can extend accuracy to arbitrary high order with no change to the basic algorithm. A flux-corrected transport (FCT) algorithm ensures the stability of the computations in regions of vanishing layer thickness and in areas where the flow features are underresolved. The spatial discretization is based on a two-level grid: a globally unstructured elemental grid and a locally structured grid consisting of N × N quadrilateral cells within each element. The numerical solution is continuous within each element but discontinuous across elements; the discontinuity is resolved by upwinding along characteristics. The accuracy and convergence rate of the SFV method are verified on two linearized problems amenable to analytical solution; the SFV solution exhibits a convergence order of N + 1 for smooth solutions. The FCT portion of the model is tested by simulating the formation of an oblique hydraulic jump in a supercritical channel flow. The model is then applied to simulate, in reduced-gravity mode, the double-gyre and wind-driven upper-ocean circulations in a square basin. Finally, the previous experiment is repeated in the North Atlantic basin to illustrate the application of the model in a realistic geometry.

The World Ocean Circulation Experiment Indian Ocean helium isotope data are mapped and features of intermediate and deep circulation are inferred and discussed. The He-3 added to the deep Indian Ocean originates from (1) a strong source on the mid-ocean ridge at about 19degreesS/65degreesE, (2) a source located in the Gulf of Aden in the northwestern Indian Ocean, ( 3) sources located in the convergent margins in the northeastern Indian Ocean, and ( 4) water imported from the Indonesian Seas. The main circulation features inferred from the He-3 distribution include (1) deep (2000-3000 m) eastward flow in the central Indian Ocean, which overflows into the West Australian Basin through saddles in the Ninetyeast Ridge, (2) a deep (2000-3000 m) southwestward flow in the western Indian Ocean, and (3) influx of Banda Sea Intermediate Waters associated with the deep core (1000-1500 m) of the through flow from the Pacific Ocean. The large-scale He-3 distribution is consonant with the known pathways of deep and bottom water circulation in the Indian Ocean.

We present a spectral element model to solve the hydrostatic primitive equations governing large-scale geophysical flows. The highlights of this new model include unstructured grids, dual h?p paths to convergence, and good scalability characteristics on present day parallel computers including Beowulf-class systems. The behavior of the model is assessed on three process-oriented test problems involving wave propagation, gravitational adjustment, and nonlinear flow rectification, respectively. The first of these test problems is a study of the convergence properties of the model when simulating the linear propagation of baroclinic Kelvin waves. The second is an intercomparison of spectral element and finite-difference model solutions to the adjustment of a density front in a straight channel. Finally, the third problem considers the comparison of model results to measurements obtained from a laboratory simulation of flow around a submarine canyon. The aforementioned tests demonstrate the good performance of the model in the idealized/process-oriented limits.

The spectral element method offers distinct advantages for geophysical simulations, including geometric flexibility, accuracy and scalability. Developers of atmospheric and oceanic models are capitalizing on these properties to create new models that can accurately and effectively simulate multi-scale flows in complex geometries.

This work is an attempt to simulate the Mediterranean Sea general circulation with a Spectral Finite Element Model. This numerical technique associates the geometrical flexibility of the finite elements for the proper coastline definition with the precision offered by spectral methods. The model is reduced gravity and we study the wind-driven ocean response in order to explain the large scale sub-basin gyres and their variability. The study period goes from January 1987 to December 1993 and two forcing data sets are used. The effect of wind variability in space and time is analyzed and the relationship between wind stress curl and ocean response is stressed. Some of the main permanent structures of the general circulation (Gulf of Lions cyclonic gyre, Rhodes gyre, Gulf of Syrte anticylone) are shown to be induced by permanent wind stress curl structures. The magnitude and spatial variability of the wind is important in determining the appearance or disappearance of some gyres (Tyrrhenian anticyclonic gyre, Balearic anticyclonic gyre, Ionian cyclonic gyre). An EOF analysis of the seasonal variability indicates that the weakening and strengthening of the Levantine basin boundary currents is a major component of the seasonal cycle in the basin. The important discovery is that seasonal and interannual variability peak at the same spatial scales in the ocean response and that the interannual variability includes the change in amplitude and phase of the seasonal cycle in the sub-basin scale gyres and boundary currents. The Coriolis term in the vorticity balance seems to be responsible for the weakening of anticyclonic structures and their total disappearance when they are close to a boundary. The process of adjustment to winds produces a train of coastally trapped gravity waves which travel around the eastern and western basins, respectively in approximately 6 months. This corresponds to a phase velocity for the wave of about 1 m/s, comparable to an average velocity of an internal Kelvin wave in the area.

Computers today rely heavily on good utilization of their cache memory subsystems. Compilers are optimized for business applications, not scientific computing ones, however. Automatic tiling of complex numerical algorithms for solving partial differential equations is simply not provided by compilers. Thus, absolutely terrible cache performance is a common result. Multigrid algorithms combine several numerical algorithms into a more complicated algorithm. In this paper, an algorithm is derived that allows for data to pass through cache exactly once per multigrid level during a V cycle before the level changes. This is optimal cache usage for large problems that do not fit entirely in cache. The numerical techniques and algorithms discussed in this paper can be easily applied to numerical simulation of fluid flows in porous media.

Keywords: multigrid; cache; threads; sparse matrix; iterative methods; domain decomposition; compiler optimization

The early stages in the adjustment of a mid-latitude abyssal basin with realistic geometry are studied using an inverted one and one-half layer model of the Eastern Mediterranean Sea as a natural test basin. The model is forced with a localized sidewall mass source and a compensating distributed mass sink. A flat bottom basin is investigated for comparison with existing theories on abyssal gyral spin-up, and as a precursor to a study with realistic topography. As in existing theories, the early adjustment is dominated by sub-inertial Kelvin and Rossby waves. Obstacles and the varying coastal geometry do not impede the passage of the Kelvin wave, though the circuit time of the main Kelvin wave signal is reduced by an aggregate 6 basin. The scattering of the Kelvin wave due to small-scale variations in the coastline is also shown not to be significant to the adjustment. The relatively short period of time needed to reach a statistical steady state is attributed to western boundary current formation in response to local Kelvin wave dynamics. Upon cessation of the sidewall forcing, sub-inertial motion controls the spin-down adjustment with basin-scale Rossby waves becoming the most pronounced feature of the flow. Two dynamical issues of particular interest emerge in these simulations: the retardation of Kelvin wave propagation around the abyssal basin and the roles of detrainment and sidewall forcing in the interior vorticity balance. An idealized simulation using an elliptical basin is used to illustrate that the mechanism for Kelvin wave retardation is a geometrically induced dispersion due to large-scale variations in the coastline. A dynamical analysis of the interior circulation shows that detrainment alone does not develop a Sverdrup response. Both the localized sidewall injection and the detrainment are needed to describe the interior dynamics, with both poleward and equatorward flows developing during the adjustment.

A shallow-water spectral element ocean model is implemented on multiple instruction multiple data, distributed memory parallel computers. A communications-minimizing partitioning algorithm for unstructured meshes, based on graph theory, is presented and is shown to improve the efficiency in a limited range of granularities. A domain decomposition implementation with an architecture-independent communications scheme, using message passing, is devised and tested on an nCUBE/2, a Cray T3D, and an IBM SP2. The implementation exhibits high efficiencies over a wide range of granularities. An order of magnitude analysis shows that, to leading order, the efficiency stays constant when KN² grows proportionally to P, where K is the total number of elements, N is the order of the spectral truncation within an element, and P is the number of processors.

The mass transport induced by a small amplitude progressive wave traveling in a rectangular wave tank is investigated. Attention is focused on the three-dimensional mean flow structure generated by the Stokes boundary layers near the side walls. The mass-transport problem is formulated in terms of vorticity and velocity field. A numerical scheme is developed to solve the coupled transport equation for the vorticity and the Poisson equation for the stream function. It is found that the side-wall boundary layers generate mean downstream vorticities. When the Reynolds number is small, the diffusion process dominates. Therefore, the vorticities generated from the boundary layers are diffused into the entire wave tank. On the other hand, when the Reynolds number is much larger than one, the convection process becomes as important as the diffusion process, the steady vorticities are confined within a small area adjacent to the solid boundaries. When the aspect ratio, width divided by depth, is of the order of magnitude of one, a pair of circulation cells appear on the plane perpendicular to the direction of wave propagation. As the width of the tank increases, more cells appear. The spanwise variations of the mass-transport velocity in the wave propagation direction become more significant when the aspect ratio is larger.

A spectral scheme is developed to study the mass transport in three-dimensional water waves where the steady flow is assumed to be periodic in two horizontal directions. The velocity-vorticity formulation is adopted for the numerical solution, and boundary conditions for the vorticity are derived to enforce the no-slip conditions. The numerical scheme is used to calculate the mass transport under two intersecting wave trains; the resulting flow is reminiscent of the Langmuir circulation patterns. The scheme is then applied to study the steady flow in a three-dimensional standing wave.

Mass transport in various kind of two-dimensional water waves is studied. The characteristics of the governing equations for the mass transport depend on the ratio of viscous lengthscale to the amplitude of the free-surface displacement. When this ratio is small, the nonlinearity is important and the mass transport flow acquires a boundary-layer character. Numerical schemes are developed to investigate mass transport in a partially reflected wave and above a hump in the seabed. When the mass transport is periodic in the horizontal direction, a spectral scheme based on a Fourier-Chebyshev expansion, is presented for the solution of the equations. For the case of a hump on the seabed, the flow domain is divided into three regions. Using the spectral scheme, the mass transport in the uniform-depth regions is calculated first, and the results are used to compute the steady flow in the inhomogeneous flow region which encloses the hump on the seabed.

A fully submerged horizontal plate has been suggested as a break-water. With a proper design in the length, thickness, and submerged depth of the plate, most of the incoming short waves can indeed be reflected. The method of eigenfunction expansion is used to solve for the reflection and transmission of the short waves. The present results agreed with earlier experimental data and computations. If the incident wave is bichromatic, a second-order long wave train accompanies the wave groups. This long wave train is also scattered by the horizontal plate. It is shown here that the scattered long waves exist on both sides of the horizontal plate even when the short waves are completely reflected. The long waves may prove to be a bigger hazard than the short waves, because the wave period of the long wave (1-3 min) is closer to the natural frequency of coastal structures.