Response of a harbor with two connected basins to incoming long waves

The general features of the resonant response for a given basin, including the determination of the normal modes and relative amplification of incoming long waves can be easily determined by using a numerical model. However, when the relative role of the different geometric parameters want to be studied, the use of analytical solutions becomes extremely useful. An analytical model is developed to determine the coupled oscillations between two rectangular basins connected through a gap, when only one of them is opened to the ocean. This geometry resembles the topographic features of Ciutadella inlet and its adjacent sub-basin (Cala Busquets) in Menorca, Balearic Islands. The analytical solutions show a good agreement with numerical simulations and capture well the changes induced by some hypothetical modifications of Ciutadella–Cala Busquets system. The agreement shown justifies the use of the analytical model as a basis for the discussion on the influence of the geometrical parameters in the normal mode characteristics of the coupled system.     

间断地形下的亚惯性频率地形波

在纯间断化地形的条件下,从理论分析和数值计算二方面探讨了亚惯性频率地形波的频散关系。指出正压亚惯性频率地形波的波模直接取决于间断点的个数和间断度。连续地形可以看成是纯间断地形的极限情况,因此,连续地形的起伏程度有类似间断度的作用。在对连续地形的频散关系进行数值计算时,水平步长的选取对计算结果有很大的影响。     

Water wave scattering by bottom undulations in an ice-covered two-layer fluid

The scattering of plane surface waves by bottom undulations in an ice-covered ocean modelled as a two-layer fluid consisting of a layer of fresh water of lesser density above a deep layer of salt water, is investigated here by using a simplified perturbation analysis. In such a two-layer fluid there exist waves of two different modes, one with higher mode propagates along the interface and the other with lower mode propagates along the ice-cover. An incident wave of a particular mode gets reflected and transmitted by the bottom undulations into waves of both the modes so that transfer of wave energy from one mode to another takes place. The first-order reflection and transmission coefficients of two different modes are obtained due to incident waves of again two different modes by employing Fourier transform technique in the mathematical analysis. For sinusoidal bottom topography these coefficients are depicted graphically against the wavenumber. These figures show how the transfer of energy from one mode to another takes place.     

A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, steady current over general bottom topography. The vertical distribution of the scattered wave potential is represented by a series of local vertical modes containing the propagating mode and all evanescent modes, plus additional terms accounting for the satisfaction of the free-surface and bottom boundary conditions. Using the above representation, in conjunction with unconstrained variational principle, an improved coupled system of differential equations on the horizontal plane, with respect to the modal amplitudes, is derived. In the case of small-amplitude waves, a linearised version of the above coupled-mode system is obtained, generalizing previous results by Athanassoulis and Belibassakis [J Fluid Mech 1999;389:275–301] for the propagation of small-amplitude water waves over variable bathymetry regions. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to an one-equation model, that is shown to be compatible with mild-slope model concerning wave–current interaction over slowly varying topography, and in the case of no current it exactly reduces to the modified mild-slope equation. The present coupled-mode system is discretized on the horizontal plane by using second-order finite differences and numerically solved by iterations. Results are presented for various representative test cases demonstrating the usefulness of the model, as well as the importance of the first evanescent modes and the additional sloping-bottom mode when the bottom slope is not negligible. The analytical structure of the present model facilitates its extension to fully non-linear waves, and to wave scattering by currents with more general structure.     

Effect of stratification on long period trapped waves on the shelf

The effect of stratification on very long-period waves trapped on a straight continental shelf of constant depth is examined for a two-layer model. There are 4 modes in this system. The characteristics of the mode with the largest phase velocity can be approximated by the barotropic mode. The mode corresponding to the barotropic shelf-wave mode is modified by the baroclinic motions significantly, and in the limit of very narrow shelf width, the mode characteristics are transformed from those of the barotropic shelf-wave to the baroclinic Kelvin wave if the long-shore wave length is larger than the internal deformation radius. In this case, the stratification has an apparent effect of increasing phase velocity of barotropic shelf-waves. The remaining two modes are dominated by baroclinic motions with significant contribution from barotropic motions: among which the one has a shelf-wave characteristics for small values of the shelf width and approaches the mode corresponding to the baroclinic Kelvin wave in shallower water for large shelf width and the other is a stationary mode. If the long-shore wave length is much shorter than the internal deformation radius, the motions in the upper and lower layers are decoupled: the surface and bottom modes analogous to those discussed byRhines (1970) appears.If the interface is deeper than the shelf depth, the stationary mode is absent and the characteristics of the third mode approaches those of the baroclinic double Kelvin wave mode as the shelf width increases.     

A two-mode numerical model with applications to coastal upwelling

A continuously stratified, linear two mode numerical model has been developed. The model incorporates a free surface and finite amplitude topography.The vertical dependence in the equations is removed by applying a Galerkin procedure which uses the normal modes as test functions. The vertical structure is therefore determined by the normal modes.In order to find a suitable efficient numerical scheme to solve the equations a fairly general phase and stability analysis is carried out for the one dimensional gravity wave equations. The A.D.I. scheme was found to be the most suitable scheme.The model is applied to coastal upwelling. A number of two dimensional (x, z) experiments have been carried out. The advantage of the two mode model above the two layer models is that considerable detail of the vertical structure is readily obtained and that no difficulties with the intersection of interfaces with the topography or the seasurface are present. A three dimensional (x, y, z) test run was done for a region along the south western coast of Africa. The results of this experiment are discussed.     

太平洋中的主要起伏模态

对海洋中起伏运动(heaving)信号的时空分布研究能够帮助我们更好地了解气候系统中的年际和年代际变率。文章通过再分析资料和模式对太平洋区域的heaving主要模态进行了研究。研究结果表明: 太平洋区域主要存在两种heaving模态: 第一模态主要表现为赤道东西两侧的温跃层异常信号反位相; 第二模态表现为赤道区域和副热带区域的温跃层异常信号呈现反位相变化的规律。本文对这两个主要heaving模态所涉及的物理过程进行详细讨论, 结果表明: 东西反位相模态主要是受赤道波动调节的结果; 而经向结构模态则主要是由赤道地区的波动和副热带区域的风应力旋度异常作用共同导致。此外, 我们还讨论了heaving模态可以通过海洋波动以及Ekman输送等过程对海盆尺度的热输送(振幅约为5×10¹⁴W)以及海洋热含量(振幅约为1.5×10²⁰J)的再分配起到了关键的调制作用, 进一步表明heaving模态对全球气候变化有着重要的作用。     

On natural modes in moonpools with recesses

The theoretical model of Molin [6] is extended to the case of rectangular moonpools with one or two recesses, as can be found in some drillships. Obtained natural frequencies and modal shapes of the piston and first sloshing modes are compared with experimental results available in literature, with good agreement. An approximation easy to implement is proposed for the natural frequency of the piston mode. Further illustrative results are presented when some geometrical parameters of the moonpool are being varied.     

Transmission and reflection of the Rossby waves in a stratified ocean with bottom topography

Transmission and reflection coefficients are calculated for Rossby waves incident on a bottom topography with constant slope in a continuously stratified ocean. The characteristics of the coefficients are interpreted in terms of the quasigeostrophic waves on the slope. In the parameter range where only the barotropic Rossby waves can propagate in the region outside the slope, the bottom trapped wave plays the same role as the topographic Rossby wave in a homogeneous ocean, and hence the transmission is weak unless phase matching takes place. When both of the barotropic and baroclinic Rossby waves can propagate outside the slope, the total transmission can be strong. The bottom trapped wave affects the transmission and reflection, and it leads to the possibility that the Rossby wave is transmitted as a mode different from the incident mode. When the number of the wavy modes on the slope is smaller than that of the Rossby wave modes outside the slope, strong reflection occurs.The results for an ocean with linear distribution of the squared Brunt-Väisälä frequency are compared to those in a uniformly stratified ocean. The weakening of the stratification near the bottom is almost equivalent to reducing the effect of the slope.     

Stability of a barotropic jet on a sloping bottom

Linear stability of a barotropic jet on a sloping bottom with and without a side boundary is examined. When a sloping bottom and a side boundary are absent, a symmetric jet generally has two unstable modes: a symmetric mode and an antisymmetric mode. In the presence of a sloping bottom or a side boundary, they are modified and lose their symmetry.The presence of a side boundary does not produce substantial change in the stability characteristics, except that it stabilizes the flow to some degree. In the presence of a sloping bottom, the following features are noted; 1) when the direction of the jet is opposite to the propagation direction of topographic Rossby waves, the change of a preferred mode occurs at a certain slope, 2) when the direction of the jet is opposite to 1), with a side boundary, the dispersion relations change from unstable mode type to shelf wave type at a certain slope, accompanied by kissing.     

Weak-mode identification and time-series reconstruction from high-level noisy measured data of offshore structures

The identification of true weak modes buried in high-level, noisy, measured data from offshore structures is a practical but challenging problem because weak modes are typically eliminated as noise and rarely, yield a discrete time series. This study proposes a weak-mode identification and time-series reconstruction method for offshore structures when high-level noise is present. A theoretical development proposed in this study extends the traditional modal analysis to reconstructing the discrete time series of weak modes, thereby removing its previous limitations to only frequencies, damping ratios and mode shapes. Additionally, a second development proposed in this study makes the reconstructed time series not simply a combination of harmonic components from a Fourier transform but rather complex exponentials; the damping of the test structure is thus estimated with a better accuracy. A third theoretical development avoids variations in the results from different original signals by handling multiple signals simultaneously. The proposed approach primarily includes three steps: (1) estimate the poles and corresponding residues of high-level, noisy, measured data by converting high-order difference equations to first-order difference equations; (2) isolate the poles of weak modes by assigning multiple rough-pole windows, and subsequently extract the corresponding residues based on the row number of the isolated pole vector; and (3) identify and reconstruct the time series of the weak modes of interest in the form of complex exponentials. The most primary advantage of the proposed process in engineering applications is that the pole windows can be easily obtained and assigned from the relationship between the frequencies and their poles. Three numerical examples are studied: the first presents the detailed numerical operation of the proposed method, the second extends the proposed method from managing one signal to managing multiple signals, and the third demonstrates the advantage of the approach compared with traditional methods. The numerical results indicate that the original signals can be decomposed into multiple complex exponentials with representative poles and corresponding residues, and that the new signals representing weak modes could be reconstructed by assigning a range of frequencies in terms of their relations with the poles. To study the performance of the proposed method when applied to offshore structures such as offshore platforms and marine risers, the experimental data from the high mode VIV experiments sponsored by the Norwegian Deepwater Programme (NDP) are used firstly. The results show that two dominant frequencies corresponding to the in-line and cross-flow directions can be identified simultaneously even one mode is very weak compared with the other, and the time series of the weak mode could be reconstructed with a rough frequency window. Then sea-test data of two offshore platforms are used: one was collected from the JZ20-2MUQ offshore platform when it was excited by ice, and the other was collected from the WZ11-4D platform when it was excited by waves. The results further demonstrated that a large model order is required to estimate all poles and residues of the original noisy signals, and that the row number corresponding to a weak mode of the isolated pole matrix could be easily determined via finite element analysis or engineering experiences. Therefore, the proposed approach provides not only modal parameters, such as frequencies and damping ratios of true weak modes buried in high-level noise, but also the discrete time series of the weak mode.     

Empirical orthogonal function analysis of shoreline changes behind two different designs of detached breakwaters

This paper compares the shoreline responses immediately shoreward of two adjacent schemes of segmented shore parallel rubble mound breakwaters undergoing the same forcing by waves and tides. Scheme one consists of four longer, emergent breakwaters that have produced tidal tombolos in their lee. Scheme two consists of five shorter, lower breakwaters that are submerged at higher tides with salients in the breakwater's lee. Empirical orthogonal function analysis was used to decompose a video derived shoreline dataset into the dominant modes of shoreline change for both schemes. The two schemes showed similar modes of change. The primary mode of change for both schemes was the cross-shore growth and shrinking of the salients/tombolos. The secondary mode of change was the longshore movement of the salients/tombolos. For all modes of change, the dominant length scale was dictated by the breakwater dimensions and locations. A new manifestation of hydrodynamic parameters is introduced: the cumulative integral of the de-meaned parameters. This parameterisation allowed for meaningful correlation of the temporal EOF components with forcing parameters and identification of the important influence of the tide on observed morphodynamic change. Clear differences were observed between the shoreline responses measured in schemes one and two; including differences in bay erosion/accretion; and the longshore translation of salients/tombolos. The beaches in scheme two showed behavioural patterns similar to unprotected beaches which were not observable in scheme one. It is postulated that these differences are caused both by the different breakwater designs and by variation in longshore sediment supply.     

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

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.     

A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions

A non-linear coupled-mode system of horizontal equations is presented, modelling the evolution of nonlinear water waves in finite depth over a general bottom topography. The vertical structure of the wave field is represented by means of a local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and the local-mode series exhibits fast convergence. Thus, a small number of modes (up to 5–6) are usually enough for precise numerical solution. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady travelling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate depth to shallow-water wave conditions, and its results are compared vs. Stokes and cnoidal wave theories, as well as with fully nonlinear Fourier methods. Furthermore, numerical results are presented for waves propagating over variable bathymetry regions and compared with nonlinear methods based on boundary integral formulation and experimental data, showing good agreement.     

The relation of surface forcing of the Bering Sea to large-scale climate patterns

Climate fluctuations, or modes, are largely manifested in terms of coherent, large-scale (3000 km) patterns of anomalous sea-level pressure or geopotential height at various altitudes. It is worthwhile to investigate how these modes relate to the specific processes associated with atmospheric forcing of the ocean, in this case for the southeast Bering Sea. This approach has been termed “downscaling.” Climate-scale patterns in this study are derived from covariance-based empirical orthogonal functions (EOFs) of low-pass filtered (10-day cut-off) 700-mb geopotential height fields for 1958–1999. By design, this EOF analysis elicits sets of patterns for characterizing the variability in the large-scale atmospheric circulation centered on the Bering Sea. Four modes are considered for each of three periods, January–March, April–May, and June–July. These modes are compared with atmospheric circulation patterns formed by compositing 700-mb height anomalies based on the individual elements constituting the local forcing, i.e. the surface heat and momentum fluxes.In general, different aspects of local forcing are associated with different climate modes. In winter, the modes dominating the forcing of sea-ice include considerable interannual variability, but no discernible long-term trends. A prominent shift did occur around 1977 in the sign of a winter mode resembling the Pacific North American pattern; this mode is most significantly related to the local wind-stress curl. In spring, forcing of currents and stratification are related to the two leading climate modes, one resembling the North Pacific (NP) pattern and one reflecting the strength of the Aleutian low; both exhibit long-term trends with implications for the Bering Sea. In summer, an NP-like mode and a mode featuring a center over the Bering Sea include long-term trends with impacts on surface heating and wind mixing, respectively. Rare events, such as a persistent period of strong high pressure or a major storm, also can dominate the summer Bering Sea forcing in particular years.     

Observation of edge waves trapped on the continental shelf in the vicinity of Makurazaki Harbor, Kyushu, Japan

Sea-bottom pressure gauges were used to measure sea levels at two points on the shelf off the southern coast of Satsuma Peninsula, Kyushu, Japan. Spectral analysis of the observed records and the tide-gauge record of Makurazaki Harbor revealed several predominant common peaks. At the same time, the eigenmodes for the trapped waves on the shelf and inside Makurazaki Bay were obtained numerically using a two-dimensional model, and the periods and the spatial distribution of amplitudes of the proper modes were obtained. A comparison of the calculated modes with the periods and phase patterns of the observed peaks clarified that peaks with periods of 19.5, 16, 13.3, and 12.2 minutes in the shelf region were the modes of standing-edge waves, and the peak with the period of 16 minutes in Makurazaki Harbor was the fundamental mode of the harbor. Among the modes of standing-edge waves, the mode of the period 16 minutes on the shelf had nearly the same period as that of the fundamental mode of Makurazaki Harbor. An analysis of changes of spectral densities of these two modes confirmed that the fundamental mode of the Makurazaki Harbor was induced by this standing-wave mode.     

Internal wave spectrum in shallow water: measurement and comparisonwith the Garrett-Munk model

The covariance matrix of sound-speed variations is determined from yo-yo CTD data collected during the SWARM 95 experiment at a fixed station. The data covered approximately 2 h and were collected during a period when nonlinear solitary internal waves were absent or negligible. The method of empirical orthogonal functions (EOF) is applied to the sound-speed covariance matrix assuming that the internal wave modes are uncorrelated. The first five eigenvectors are found to agree well with the theoretically modeled eigenfunctions based on the measured buoyancy frequency and the internal wave eigenmode equation. The mode amplitudes for the first five modes are estimated from the corresponding eigenvalues. They agree with the Garrett-Munk model if j*=1 is used instead of j*=3. A second method is used to deduce the mode amplitudes and mode frequency spectra by projecting the sound-speed variation (as a function of time) onto the theoretical mode depth functions. The mode amplitudes estimated with this method are in agreement with the EOF results. A modified Garrett-Munk model is proposed to fit the frequency spectrum of linear internal waves in shallow water     

Numerical Simulation on Oscillation-Sliding-Uplift Rock Coupled Motion of Caisson Breakwater Under Wave Excitation

Corresponding to the sliding and the overturning failure, the elementary motion modes of caisson breakwater include the horizontal-rotational oscillation coupled motion, the horizontal sliding-rotational oscillation coupled motion, the horizontal vibrating-uplift rocking coupled motion, and the horizontal sliding-uplift rocking coupled motion. The motion mode of a caisson will transform from one to another depending on the wave forces and the motion behaviors of the caisson. The numerical models of four motion modes of caisson are developed, and the numerical simulation procedure for joint motion process of various modes of caisson breakwater under wave excitation is presented and tested by a physical model experiment . It is concluded that the simulation procedure is reliable and can be applied to the dynamic stability analysis of caisson breakwaters.     

Numerical Simulation on Oscillation-Sliding-Uplift Rock Coupled Motion of Caisson Breakwater Under Wave Excitation

Corresponding to the sliding and the overturning failure, the elementary motion modes of caisson breakwater include the horizontal-rotational oscillation coupled motion, the horizontal sliding-rotational oscillation coupled motion, the horizontal vibrating-uplift rocking coupled motion, and the horizontal sliding- uplift rocking coupled motion. The motion mode of a caisson will transform from one to another depending on the wave forces and the motion behaviors of the caisson. The numerical models of four motion modes of caisson are developed, and the numerical simulation procedure for joint motion process of various modes of caisson breakwater under wave excitation is presented and tested by a physical model experiment. It is concluded that the simulation procedure is reliable and can be applied to the dynamic stability analysis of caisson breakwaters.     

Wind effects on sub-tidal currents in Puget Sound

Wind effects on sub-tidal currents are studied using current meter records obtained at six moorings across the main basin of Puget Sound. High correlations between wind speeds and currents are found near the surface and at mid-depths of about 100 m. Empirical Orthogonal Function analysis applied to the axial currents in 1984 and 1985 shows that mode 1, containing over 60% of the variance, is highly correlated with wind speed even without any near surface current records. When near surface stratification is strong, direct wind effects are limited to the upper 30 m with counter currents in the lower layer indicating a baroclinic response. The transport in the lower layer almost balances the transport in the upper layer. When near surface stratification is weak, direct wind effects on currents can be detected to about 100 m. In this case, there is no clear and consistent depth at which one can separate the upper from the lower layer. Time series show that the acceleration in the surface layer initially increases in the same direction as the wind when the wind starts blowing, but it reaches a maximum, starts decreasing, and eventually changes to the opposite direction (decelerates) while the wind continues to blow in one direction. Results of a continuously stratified normal mode model and estimations from the observations suggest that friction at solid boundaries is a major cause of these phenomena. The model shows that modal currents of normal modes 2 and 3 are as important as mode 1, although the resultant vertical structure of total current shows a two-layer type pattern with only one zero crossing. The effect of the baroclinic pressure gradient is only apparent at low frequencies and among lower modes.