Linear Shallow Water Equations Research Articles

Application of Dispersive Numerical Wave Models to Propagation of Long Distance Tsunami

A series of numerical studies of the December 26, 2004 Indian Ocean tsunami had been carried out using dispersive wave models i. e. weakly and fully nonlinear Boussinesq-type equations and nondispersive wave models i. e. linear and nonlinear shallow water equations. Numerical model results are compared each other or with field measurements. General features of tsunami propagation agree well in all numerical results. Numerical results of water level distribution calculated by dispersive and nondispersive wave models showed important differences especially in development of wave front. Discussion of free surface spatial distribution concluded that the dispersion mechanism is important for tsunami propagation in deep sea region, whereas nonlinearity is important in shallow water region.

Chebyshev generalized finite spectral method for linear and nonlinear waves

A Chebyshev generalized finite spectral method is proposed for 2-D linear and nonlinear waves. To attain high accuracy in time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme was used. The proposed numerical scheme is validated by applications to the linear shallow water equations and extended Boussinesq equations, where analytical solution and experimental data are available for comparison respectively. Numerical results agree very well with the corresponding analytical solution or experimental data in all cases.

Balance regimes for the stability of a jet in an f-plane shallow water system

To study a limit of validity of balanced models, stability of a zonal jet is investigated both linearly and nonlinearly in an f-plane shallow water system for a wide range of parameter. It is shown that quasi-geostrophic approximation gives not only a good estimation of maximum growth rate for high Rossby number, Ro, but also is valid even in the nonlinear phase of instability for high Ro as long as Froude number, Fr, is low. While the maximum growth rate of unstable modes is well estimated by the quasi-geostrophic approximation, dominant balance is different between high and low Ro. In the low Ro regime ( Ro < 5 ), geostrophic balance is dominant in the perturbation field, the ratio ∥ φ ∥ / ∥ ψ ∥ of the amplitudes of divergent flow to that of rotational flow is proportional to Fr 2 / Ro , where φ and ψ are velocity potential and streamfunction, respectively. On the other hand, in the high Ro regime ( Ro > 5 ), cyclostrophic balance with basic shear is dominant, ∥ φ ∥ / ∥ ψ ∥ ∝ Fr 2 . Considering that the barotropic instability is caused by the resonance of neutral Rossby wave modes, we can explain the difference of the ratio in each regime. Using the ratio ∥ φ ∥ / ∥ ψ ∥ being small, different approximation of the linear shallow water equations for each regime is deduced. Properties of the linear unstable modes are explained with these approximations.

Estimating Open-Ocean Barotropic Tidal Dissipation: The Hawaiian Ridge

Abstract The generalized inverse of a regional model is used to estimate barotropic tidal dissipation along the Hawaiian Ridge. The model, based on the linear shallow-water equations, incorporates parameterizations for the dissipation of energy via friction in the bottom boundary layer and form drag due to internal waves generated at topographic slopes. Sea surface height data from 364 orbit cycles of the Ocean Topography Experiment (TOPEX)/Poseidon satellite mission are used to perform inversions at eight diurnal and semidiurnal tidal frequencies. It is estimated that the barotropic M2 tide loses energy at a rate of 19 GW, of which 88% is lost within 250 km of the ridge, presumably via conversion to the internal or baroclinic tide. Uncertainty in the assumed model error and wave drag in the forward model suggest that M2 dissipation values from 18 to 25 GW are consistent with the altimetric observations. Other barotropic tidal constituents are estimated to lose a total of 5.7 GW. The spatial distribution of barotropic dissipation along the ridge is similar to that inferred from three-dimensional primitive equation models, and it is largely insensitive to details of assumed model and data errors. Dissipation at semidiurnal frequencies is most intense at the French Frigate Shoals with lesser, but significant, contributions at other sites. Diurnal tidal dissipation is concentrated to the east of the French Frigate Shoals, at the Gardner Pinnacles. Further work with three-dimensional models will be necessary to determine the fate of the energy that is removed from the barotropic tide.

General Cauchy problem for the linear shallow -water equations on an equatorial beta-plane

Based on the theory of stratification, the well-posedness of the initial value problem for the linear shallow-water equations on an equatorial beta-plane was discussed. The sufficient and necessary conditions of the existence and uniqueness for the local solution of the equations were presented and the existence conditions for formal solutions of the equations were also given. For the Cauchy problem on the hyper-plane {t=0}, the local analytic solution were worked out and a special case was discussed. Finally, an example was used to explain the variety of formal solutions for the ill-posed problem.

A one-dimensional benchmark for the propagation of Poincaré waves

Several numerical methods are employed to solve the linear shallow-water equations describing the propagation of Poincaré waves within a one-dimensional finite domain. An analytical solution to the problem, set off by a discontinuous steplike elevation, is known and allows for assessing the accuracy and robustness of each method and in particular their ability to capture the traveling discontinuities without generating spurious oscillations. The following methods are implemented: the method of characteristics, the Galerkin finite-element method (FEM) and the discontinuous Galerkin FEM with two different ways of computing the numerical fluxes.

Stability Analysis of Geostrophic Adjustment on Hexagonal Grids for Regions with Variable Depth

Abstract Hexagonal grids have been used in a number of numerical studies, and especially in relation to atmospheric models. Recent studies have suggested that ocean circulation models may also benefit from the use of hexagonal grids. These grids tend to induce less systematic errors and have better horizontal isotropy properties than traditional square grid schemes. If hexagonal grids are to be applied in ocean models, a number of features that are characteristic of ocean circulation problems need to be attended to. The topography of the ocean basin is an important feature in most ocean models. Ocean modelers can experience instabilities due to depth variations. In the present paper, analysis of the propagation matrix for the spatially discretized system is used to explain unphysical growth of the numerical solutions of the linear shallow water equations when using hexagonal grids over domains with variable depth. It is shown that a suitable weighting of the Coriolis terms may give an energy-conserving and stable numerical scheme.

동해의 지진해일 처오름 모의

본 연구에서는 지진해일에 의하여 동해안에 발생하는 처오름을 모의하기 위하여 2차 정확도의 풍상차분기법을 사용하였으며, 그 결과는 현장관측값과 1차 정확도의 풍상차분기법의 산정 결과와 비교하였다. 수치해석 모형에서는 지배방정식으로 원해에서는 선형 천수 방정식, 근해에서는 비선형 천수방정식을 사용하였다. 대상 지진해일은 1983년 지진해일과 1993년 지진해일이다. 수치해석 결과 동해안의 처오름높이를 잘 재현하는 것으로 확인되었다. 또한, 지진해일 처오름 모의 결과를 범람도 작성에 이용할 수 있을 것으로 판단된다. A second-order upwind scheme is used to investigate the run-up heights of tsunamis in the East Sea and the predicted results are compared with the field data and results of a first-order upwind scheme. In the numerical model, the governing equations solved by the finite difference scheme are the linear shallow-water equations in deep water and nonlinear shallow-water equations in shallow water. The target events are 1983 Central East Sea Tsunami and 1993 Hokkaido Nansei Oki Tsunami. The predicted results represent reasonably well the run-up heights of tsunamis in the East Sea. And, the results of simulation are used for the design of inundation map.

Computational improvement of gravitational mode calculation of basins discretised by orthogonal curvilinear grids

The present note illustrates a criterion to improve the computational capability of the approaches proposed by Beltrami et al. [Beltrami, G.M., Bargagli, A., Briganti, R., 2003. Gravitational mode calculation of basins discretised by orthogonal curvilinear grids. Ocean Engineering 30, 833–853] for the direct numerical solution of the eigenvalue problem associated to the linear shallow-water equations when adiabatic boundary conditions apply. It is shown that—given the nature of its spatial differential operator—the problem can be solved by the singular value decomposition (SVD) of the real bidiagonal matrix resulting from a previous ad hoc Householder reduction of the operator matrix image. This procedure actually requires 1/8 of the random-access memory (RAM) needed by a standard library routine to compute all the eigenvalues and eigenvectors of the matrix image of the above-mentioned differential operator. Given the intrinsic limitation of a computing-machine RAM, this procedure dramatically improves the computational capability of both the proposed approaches.

The Cauchy-Riemann conditions and localized asymptotic solutions of the linearized shallow-water equations

Singular solutions of the two-dimensional shallow-water equations with algebraic singularities of the “square root” type, which have been studied before [1–4], propagate along the trajectories of the external velocity field, over which this field satisfies the Cauchy-Riemann conditions. In other words, the differential of the phase flow on such a trajectory is proportional to an orthogonal operator. It turns out that in the linear approximation this situation is strongly linked with the “spreading” effect of solutions of the hydrodynamic equations (cf. [5,6]); namely, a localized asymptotic solution of the Cauchy problem for the linearized shallowwater equations maintains its form (i.e. does not spread) if and only if the Cauchy-Riemann conditions hold on the trajectory of the outer flow along which the disturbance is propagating.

Data Assimilation Modeling of the Barotropic Tides in the Korea/Tsushima Strait

During 1999–2000, 13 bottom mounted acoustic Doppler current profilers (ADCPs) and 12 wave/tide gauges were deployed along two lines across the Korea/Tsushima Strait, providing long-term measurements of currents and bottom pressure. Tidally analyzed velocity and pressure data from the moorings are used in conjunction with other moored ADCPs, coastal tide gauge measurements, and altimeter measurements in a linear barotropic data assimilation model. The model fits the vertically averaged data to the linear shallow water equations in a least-squares sense by only adjusting the incoming gravity waves along the boundaries. Model predictions are made for the O1, P1, K1, μ2, N2, M2, S2, and K2 tides. An extensive analysis of the accuracy of the M2 surface-height predictions suggests that for broad regions near the mooring lines and in the Jeju Strait the amplitude prediction errors are less than 0.5 cm. Elsewhere, the analysis suggests that errors range from 1 to 4 cm with the exception of small regions where the tides are not well determined by the dataset. The errors in the model predictions are primarily caused by bias error in the model’s physics, numerics, and/or parameterization as opposed to random errors in the observational data. In the model predictions, the highest ranges in sea level height occur for tidal constituents M2, S2, K1, O1, and N2, with the highest magnitudes of tidal velocities occurring for M2, K1, S2, and O1. The tides exhibit a complex structure in which diurnal constituents have higher currents relative to their sea level height ranges than semi-diurnal constituents.

A stratified dispersive wave model withhigh-order nonreflecting boundary conditions

A layered-model is introduced to approximate the effects of stratification on linearizedshallow water equations. This time-dependent dispersive wave model is appropriate for describing geophysical (e.g., atmospheric or oceanic) dynamics. However, computational models that embrace these very large domains that are global in magnitude can quickly overwhelm computer capabilities. The domain is therefore truncated via artificial boundaries, and nonreflecting boundary conditions (NRBC) devised by Higdon are imposed. A scheme previously proposed by Neta and Givoli that easily discretizes high-order Higdon NRBCs is used. The problem is solved by finite difference (FD) methods. Numerical examples follow the discussion.

An analytical solution of the mild-slope equation for waves around a circular island on a paraboloidal shoal

In this paper, we develop an analytical technique in terms of series expansions to solve the mild-slope equation on an axi-symmetric topography. This technique is applied to study the combined refraction and diffraction of plane monochromatic waves by a circular cylindrical island mounted on a paraboloidal shoal. By using the direct solution for the wave dispersion equation by Hunt [J. Waterw., Port, Coast., Ocean Div. Proc ASCE 4 (1979) 457], the mild-slope equation becomes explicit and it is then solved in terms of combined Fourier series and Taylor series. It is found that, to calculate the wave elevation along a perimeter with a specific radius, more terms in the Taylor series and angular modes in the Fourier series are needed for shorter waves. On the other hand, for the same incident wave, the outer the solutions are sought, more angular modes are needed to obtain the converged result of Fourier series. The comparison with the analytical solution based on the linear shallow-water equation by Homma [Geophys. Mag. 21 (1950) 199] is made for long wave incidence and excellent agreements are obtained. For long waves and waves in intermediate water depth, comparisons are made with other numerical results of the mild-slope equation and an equally good quality of agreement is achieved.

Analytical solutions for forced long waves on a sloping beach

We derive analytic solutions for the forced linear shallow water equation of the following form:&lt;?TeX \partial^2 Y \over \partial t^2}-b{\partial \over \partial x}\left(x {\partial Y \over \partial x}\right)={\partial^2 f\over \partial t^2 ?&gt;for x&gt;0, in which Y(x,t) denotes an unknown variable, f(x,t) a prescribed forcing function and b a positive constant. This equation has been used to describe landslide-generated tsunamis and also long waves induced by moving atmospheric pressure distributions. We discuss particular and general solutions. We then compare our results with numerical solutions of the same equation and with the corresponding solutions of the nonlinear depth-integrated equations and discuss them in terms of landslide-generated tsunamis.

Exact solution of some hyperbolic systems with source terms

In this paper we present a methodology for constructing exact solutions to hyperbolic systems with source or forcing terms. We illustrate the approach as applied to some inhomogeneous systems arising in environmental problems, namely the inhomogeneous linearized shallowwater equations. We solve exactly a generalized Riemann problem that models dambreak problems in channels with variable bottom elevation and variable width. The exact solutions obtained are very valuable in the process of assessing the performance of numerical methods intended for solving morecomplex problems; they can also be useful in providing new ideas to be used in the design of numerical methods for moregeneral inhomogeneous systems.

Research on Atmospheric Motion in Horizontal Discrete Grids

Starting from the linear shallow-water equations and using Arakawa A-E and Z grids, a study is made of inertia-gravitational and Rossby waves in the six grids from the perspectives of frequency and group velocity. Results indicate that grids C, Z, and E thereof give smaller distortion compared to the others.

Time-Reversed Refocusing of Surface Water Waves

A time-reversal mirror is, roughly speaking, a device which is capable of receiving a signal in time, keeping it in memory, and sending it back into the medium in the reversed direction of time. A brief mathematical review of the time-reversal (in reflection) theory is presented in the context of the linear shallow water equations. In particular, an explicit expression is given for the refocused pulse in the simplest time-reversal case. The explicit expression for the power spectral density of the reflection process is used to construct the highpass filter, which controls the refocusing process. Time-reversal numerical experiments in the (effectively) linear regime are used to validate the nonlinear shallow water code. The numerically refocused pulse is compared with the theoretical predicted shape. Further numerical experiments illustrate the robustness of the theory, in particular the time-reversal refocusing with smaller cutoff windows, the self-averaging property, and finally refocusing when the nonli...

Gravitational mode calculation of basins discretized by orthogonal curvilinear grids

This paper presents a simple and straightforward method for carrying out the direct numerical solution of the eigenvalue problem associated to the homogeneous linear shallow-water equations expressed using orthogonal curvilinear coordinates, when ‘adiabatic’ boundary conditions apply. These equations, together with the boundary conditions, define a self-adjoint problem in the continuum. The method presented here, which is thought for calculating the 2-D theoretical gravity modes of both natural and artificial basins, relies on a change of basis of the dependent variable vector. This preliminary transformation makes it, in fact, possible to formulate two different numerical approaches which guarantee the self-adjoint property of the discrete form of the system consisting of the governing equations and the boundary conditions. The method is tested using a square and a fully circular domain, both of which allow comparisons with well-known analytical and numerical solutions. Discretizing the physical domain of a fully circular basin by a cylindrical coordinate grid makes it possible to show the actual efficiency of the method in calculating the theoretical gravity modes of basins discretized by a boundary-following coordinate grid which allows laterally variable resolution.

Linear modeling of infragravity waves during Delilah

The generation of infragravity waves by directionally spread shortwaves incident on an alongshore uniform beach is investigated. Two mechanisms responsible for the generation of infragravity waves are considered: the release of the bound infragravity waves associated with changes in the spatial variation of the incident shortwave energy and the forcing of trapped waves by obliquely incident directionally spread shortwaves. The infragravity wave response is examined using linear shallow water equations, taking into account the presence of bottom friction, setup of the mean water level, rollers, and longshore current. Summing all infragravity contributions due to difference interactions between pairs of shortwave spectral components with the same frequency difference results in infragravity energy density spectra. Calculated spectra are compared with measured spectra obtained during the Delilah field experiment for 11 consecutive days, including calm, mild, and storm conditions. In the comparison, attention is focused on the frequency distribution of the infragravity energy density throughout the surf zone and the transformation of the root mean square infragravity wave height. The measured infragravity response in the frequency range from 0.01 to 0.06 Hz is reproduced by the computations, with relatively broad spectra offshore and a clear nodal structure closer to shore. Typically 80% of the infragravity wave height variability is explained by the model of which 30% or less is due to bound infragravity waves. Differences occur mainly during storm events in which the infragravity energy is underpredicted in the outer surf zone and overpredicted in the inner surf zone.

Energy conservation and physical optics for discrete long wave equations

In a preceding paper [Wave Motion 32 (2000) 79] physical optics was developed for a selection of discrete long wave solutions by application a perturbation expansion. Herein, we approach discrete optics from analysis of discrete energy, in the sense of discrete expressions fulfilling a local energy conservation law. The expressions for energy density and flux are found from finite difference and element discretizations of linear and non-linear shallow water equations as well as linear Boussinesq equations with constant depth. The discrete energies are generally not perfectly positive definite and ambiguity in their definition is encountered and discussed. Still, the discrete energies appear as well behaved, which is confirmed through calculation of averaged energy quantities for harmonics, that reproduces the physical optics from Pedersen [Wave Motion 32 (2000) 79] under the assumption of negligible diffraction. The optics and the derived expressions for the energies are verified through direct solution of discrete long wave equations. Discrete energies are also discussed in relation to Hamilton’s principle. The discrete momentum equation can be obtained from Hamilton’s principle, inserted energy-like quadratures. However, those energy forms are not consistent with the energies that are conserved.