Basin Of Constant Depth Research Articles

Numerical solution of the boundary value problem for inertia-gravity internal waves

This paper presents a numerical calculation of the boundary value problem for the equation of free internal inertia-gravity waves in an unbounded basin of constant depth in the Boussinesq approximation and the presence of a two-dimensional vertically inhomogeneous flow. The boundary value problem for the vertical velocity amplitude has complex coefficients and is solved both numerically and by perturbation theory. Using the example of calculating the decrement of attenuation of internal waves and momentum wave flows, it is shown that the exact numerical calculation gives significantly better estimates in comparison with the perturbation method. In particular, at minimum divergence in the dispersion curves for both calculation methods, the imaginary part of the wave frequency, interpreted as the decrement of attenuation, can differ by two or three orders of magnitude. Vertical wave momentum fluxes are comparable to turbulent ones and may exceed them, with results obtained by the numerical method being almost an order of magnitude smaller than those calculated by the perturbation theory method.

Experimental reproduction of inhomogeneous fjord waves

Waves in coastal areas and fjords can present inhomogeneities that affect the responses of very large floating structures. However, spatial inhomogeneities of waves are usually not included in model testing. In this paper, the feasibility of generating inhomogeneous wave conditions in an Ocean basin is investigated. For that purpose, a fast linear numerical model of a basin of constant depth is applied, and an optimization method is presented, that allows to find the control signal for the multiflap wavemaker. Comparisons with experiments in SINTEF’s Ocean basin demonstrate the validity of the method for generating simple monochromatic synthetic waves, as well as short-crested irregular waves representing realistic conditions in Sulafjorden.

Vertikal'nyy perenos impul'sa inertsionno-gravitatsionnymi vnutrennimi volnami na dvumernom sdvigovom techenii

Purpose. The paper is aimed at investigating the momentum vertical transfer by inertia-gravity internal waves on a two-dimensional flow with a vertical shear of velocity, and also at studying the Stokes drift of liquid particles and the mean current effect on it. Methods and Results. Free internal waves in an infinite basin of constant depth are considered in the Boussinesq approximation with the regard for the Earth rotation. Two components of the mean current velocity depend on the vertical coordinate. The equation for the vertical velocity amplitude has complex coefficients; therefore the eigenfunction and the wave frequency are complex. The corresponding boundary value problem is solved numerically by the implicit Adams scheme of the third order of accuracy. The wave frequency at a fixed wavenumber was found by the shooting method. It was determined that the frequency imaginary part was small and could be either negative or positive depending on a wave number and a mode number. Thus, both weak attenuation and weak amplification of an internal wave are possible. The vertical wave momentum fluxes are nonzero and can exceed the corresponding turbulent fluxes. The Stokes drift velocity, transverse to the wave direction, is nonzero and less than the longitudinal velocity. The vertical component of the Stokes drift velocity is also nonzero and four orders of magnitude less than the longitudinal component. The signs of the vertical component of the Stokes drift velocity for the waves with the frequencies 10 and 16 cycle/h are opposite, since the signs of their frequency imaginary parts are different; and the vertical component of the Stokes drift velocity is proportional to the wave frequency imaginary part. Conclusions. The vertical momentum wave flux of inertia-gravity internal waves differs from zero in the presence of the current whose velocity component, transverse to the wave propagation direction, depends on the vertical coordinate. The component of the Stokes drift velocity, transverse to the wave propagation direction, is nonzero and less than the longitudinal one. The vertical component of the Stokes drift velocity is also nonzero and can contribute to formation of the vertical fine structure

Vertical Transfer of Momentum by Inertia-Gravity Internal Waves on a Two-Dimensional Shear Flow

Purpose. The paper is aimed at investigating the momentum vertical transfer by inertia-gravity internal waves on a two-dimensional flow with a vertical shear of velocity, and also at studying the Stokes drift of liquid particles and the mean current effect on it. Methods and Results. Free internal waves in an infinite basin of constant depth are considered in the Boussinesq approximation with the regard for the Earth rotation. Two components of the mean current velocity depend on the vertical coordinate. The equation for the vertical velocity amplitude has complex coefficients; therefore the eigenfunction and the wave frequency are complex. The corresponding boundary value problem is solved numerically by the implicit Adams scheme of the third order of accuracy. The wave frequency at a fixed wavenumber was found by the shooting method. It was determined that the frequency imaginary part was small and could be either negative or positive depending on a wave number and a mode number. Thus, both weak attenuation and weak amplification of an internal wave are possible. The vertical wave momentum fluxes are nonzero and can exceed the corresponding turbulent fluxes. The Stokes drift velocity, transverse to the wave direction, is nonzero and less than the longitudinal velocity. The vertical component of the Stokes drift velocity is also nonzero and four orders of magnitude less than the longitudinal component. The signs of the vertical component of the Stokes drift velocity for the waves with the frequencies 10 and 16 cph are opposite, since the signs of their frequency imaginary parts are different; and the vertical component of the Stokes drift velocity is proportional to the wave frequency imaginary part. Conclusions. The vertical momentum wave flux of inertia-gravity internal waves differs from zero in the presence of the current whose velocity component, transverse to the wave propagation direction, depends on the vertical coordinate. The component of the Stokes drift velocity, transverse to the wave propagation direction, is nonzero and less than the longitudinal one. The vertical component of the Stokes drift velocity is also nonzero and can contribute to formation of the vertical fine structure.

Seiches in a Basin with an Open Entrance

Two-dimensional seiche oscillations in a rectangular basin of constant depth with an open entrance are considered within the framework of the linear theory of long waves. An analytical solution is obtained for the case where a nodal level line is located at the entrance to the basin. It is shown that transverse seiches in a basin with an open entrance are two-dimensional in contrast to similar modes in a completely closed basin, which are one-dimensional. Seiche parameters are calculated for model basins with dimensions and depth characteristic of a number of Sevastopol bays. Comparison of the calculation results with observations data shows that they are in satisfactory agreement.

Geostrophic adjustment in a closed basin with islands

AbstractWe consider the geostrophic adjustment of a density-stratified fluid in a basin of constant depth on an $f$-plane in the context of linearized theory. For a single vertical mode, the equations are equivalent to those for a linearized shallow-water theory for a homogeneous fluid. Associated with any initial state there is a unique steady geostrophically adjusted component of the flow compatible with the initial conditions. This steady component gives the time average of the flow and is analogous to the adjusted flow in an unbounded domain without islands. The remainder of the response consists of superinertial Poincaré and subinertial Kelvin wave modes and expressions for the energy partition between the modes in arbitrary basins again follow directly from the initial conditions. The solution for an arbitrary initial density distribution released from rest in a circular domain is found in closed form. When the Rossby radius is much smaller than the basin radius, appropriate for the baroclinic modes, the interior adjusted solution is close to that of the initial state, except for small-amplitude trapped Poincaré waves, while Kelvin waves propagate around the boundaries, carrying, without change of form, the deviation of the initial height field from its average.

The structure of basin‐scale internal waves in a stratified lake in response to lake bathymetry and wind spatial and temporal distribution: Lake Iseo, Italy

Field data, theoretical analyses, and numerical simulations were used to investigate the effects of lake basin bathymetry and spatial and temporal structures of the wind field on the basin‐scale internal wave spectrum in Lake Iseo. This deep Italian basin is characterized by the presence of a large island, imparting attributes of an annulus to the lake. During the summer of 2010, the internal wave activity was dominated by a basin‐scale internal wave of vertical and horizontal modes 1 (V1H1), superimposed on which were occasional higher vertical modes (V2H1) and higher horizontal modes (V1H5) trapped by the main island. The occurrence of these motions was interpreted as forcing by the wind components with similar horizontal structures and with energies at frequencies near the natural oscillations of the excited modes. The modifications of the wind field by the topography, in particular, controlled the excitation of an anticyclonic wave trapped around the island, whose features were investigated on the basis of an extension of the circular model to the case of an annular basin of constant depth.

Baroclinic seiches in rotating basins of variable depth in the case of two-layer density stratification

We solve a plane problem of linear baroclinic seiches in closed rotating basins of variable depth with two-layer density stratification. In the long-wave approximation, we get a boundary-value problem for a system of ordinary differential equations and propose a numerical procedure for finding internal seiches. The analytic solutions of the problem are obtained for a basin of constant depth. The numerical analysis of seiches is performed for the distributions of depth corresponding to the zonal and meridional sections of the Black Sea and model basins including the cases of a shelf zone and an underwater ridge. It is shown that the baroclinic seiches become more intense in shallow-water regions and that the intense longshore currents caused by Earth’s rotation are formed in the shelf zones and over the underwater ridges.

Generation of long waves in ice-covered lakes by moving disturbances of atmospheric pressure

The effect of ice cover on the oscillations of water in a two-dimensional constant-depth basin within the linear theory of long waves is considered. The ice cover is treated as thin elastic plate in the presence of compressive force. The ice is fastened hardly to a shore. The eigenfrequencies and eigenfunctions of free oscillations (seishes) are obtained. The forced oscillations of fluid and ice cover under the action of a moving disturbance of atmospheric pressure are investigated. The input data are taken to have values typical of Lake Baikal. The time dependence of total mechanical energy of the fluid is calculated for different velocities of pressure motion. The variations in ice-bending stresses are determined and the causes of shore-ice breaking are explained.

Numerical simulations using momentum source wave-maker applied to RANS equation model

For Navier-Stokes equation model using the VOF scheme, Lin and Liu (Lin, P. and Liu, P.L.-F. (1999). Internal wave-maker for Navier-Stokes equations models. J. Waterw. Port Coast. Ocean Eng., 125 (4), 207–215.) developed an internal wave-maker method for which a mass source function of the continuity equation was used to generate target wave trains. Using this internal wave-maker method, various numerical experiments have been conducted without any problems due to waves reflected by a wave-maker. In this study, an internal wave-maker method using a momentum source function was proposed. Various numerical simulations in two and three dimensions were performed using the momentum source wave-maker applied to the RANS equation model in a CFD code, FLUENT. To verify their applicability in 2 dimensions, the computational results obtained using the momentum source wave-maker in a channel of constant depth were compared with the results obtained by using the mass source wave-maker and with the analytical solutions. And the results of the present numerical simulations of hydraulic experiments, which represent nonlinear waves on a submerged shoal and breaking waves on a plane beach, were compared with measurements. The comparisons showed good agreements between them. To see their applicability into 3-dimensional cases, the present results in a basin of constant depth were compared with the analytical solutions, and they agreed well with each other. In addition, vertical variation of longshore current was presented by using the 3-dimensional simulation results.

Application of asymptotic solutions for the computation of the piston model of tsunami

Conditions are derived under which asymptotic solutions of the Cauchy problem for the wave equation can be applied to the computation of a piston model of tsunami in a basin of constant depth. The derivation is carried out for the initial displacement of the bottom in the form of Gaussian exponentials by comparing the results of numerical integration with computations using asymptotic formulas.

Effect of ice cover on oscillations of fluid in a closed basin

Within the framework of the linear theory of long waves, the problem of the effect of ice cover on seiche oscillations of fluid in a two-dimensional constant-depth basin is solved. The eigenfrequencies and eigenfunctions of seiche oscillations are obtained for different boundary conditions at ice edges: rigid coupling and free edges. The forced oscillations of fluid and ice under the action of a moving disturbance of atmospheric pressure are investigated. The change in the stress of ice bending is considered and it is shown that the coast ice can be broken.

Modeling drift currents in a shallow basin with allowance for variations in tangential stresses induced by wind waves

A coupling model for calculating wind-driven currents and waves in a shallow basin with allowance for current-wave interactions is introduced. The model is constructed on the basis of the three-dimensional σ-coordinate model of currents [3] and the SWAN (Simulating Waves Nearshore) spectral wave model [4]. The effect of waves on currents is taken into account in the coefficients of surface and bottom friction through roughness parameters. Results of combined modeling of stationary fields of currents and waves generated by spatially homogeneous wind are correlated with the corresponding results of separate modeling for a cylindrical basin of constant depth and the water area of Lake Donuzlav (the northwestern coast of the Crimea). The allowance for the effect of waves during calculation of tangential wind stresses in the model of currents is shown to be among major factors intensifying water circulation and forming spatial inhomogeneities of the vortex type. In addition, some cases of local decreases in tangential wind stresses are revealed; they appear when the lake is penetrated from the side of the open sea by relatively long waves, which significantly decrease the roughness of the water surface.

Effect of coastal boundary resolution and mixing upon internal wave generation and propagation in coastal regions

A non-linear two-dimensional vertically stratified cross-sectional model of a constant depth basin without rotation is used to investigate the influence of vertical and horizontal diffusion upon the wind-driven circulation in the basin and the associated temperature field. The influence of horizontal grid resolution, in particular the application of an irregular grid with high resolution in the coastal boundary layer is examined. The calculations show that the initial response to a wind impulse is downwelling at the downwind end of the basin with upwelling and convective mixing at the opposite end. Results from a two-layer analytical model show that the initial response is the excitation of an infinite number of internal seiche modes in order to represent the initial response which is confined to a narrow near coastal region. As time progresses, at the downwind end of the basin a density front propagates away from the boundary, with the intensity of its horizontal gradient and associated vertical velocity determined by both horizontal and vertical viscosity values. Calculations demonstrate the importance of high horizontal grid resolution in resolving this density gradient together with an accurate density advection scheme. The application of an irregular grid in the horizontal with high grid resolution in the nearshore region enables the initial response to be accurately reproduced although physically unrealistic short waves appear as the frontal region propagates onto the coarser grid. Parameterization of horizontal viscosity using a Smagorinsky-type formulation acts as a selective grid size-dependent filter, and removes the short-wave problem although enhanced smoothing can occur if the scaling coefficient in the formulation is too large. Calculations clearly show the advantages of using an irregular grid but also the importance of using a grid size-dependent filter to avoid numerical problems.

The influence of an irregular grid upon internal wave propagation

A cross-sectional non-linear model of wind induced circulation in a stratified basin of constant depth, neglecting rotational effects, is used to examine the influence of grid resolution upon the accuracy of internal wave propagation. In addition the effect upon the accuracy and stability of the solution on an irregular grid of using a scale selective formulation of horizontal diffusion is considered. The initial response to an along-basin wind impulse is coastal downwelling at the downwind end, with upwelling of heavy water and convective mixing at the upwind end. Subsequently a horizontal density front and associated internal waves propagate away from the coast. Calculations show that the intensity of the horizontal and vertical currents associated with the front depends upon vertical and horizontal eddy viscosity, with intensity and hence horizontal gradient increasing as viscosity decreases. Calculations using a range of uniform and variable horizontal grids show the importance of a fine near-coastal grid on the short time scale when the front is formed. With an irregular grid and constant horizontal viscosity physically unrealistic short waves appear as the frontal region propagates onto the coarser grid. However when horizontal viscosity is computed using a Smagorinsky formulation, this acts as a grid size dependent filter and removes the physically unrealistic short internal waves, thereby stabilizing the solution. If the empirical coefficient in the Smagorinsky formulation is too large, excessive smoothing can occur, spoiling the advantages of using an irregular grid. A reduced value of the coefficient reflecting the grid refinement on an irregular grid gave a stable solution without excessive smoothing.

Forced Waves in a Bounded Basin with Two Pycnoclines

The behavior of waves induced in a continuously stratified fluid with two pycnoclines by periodic variations of pressure on the free surface is studied within the framework of the linear theory by taking into account the action of the Coriolis force. The fluid fills a bounded basin of constant depth. We establish the dependences of the amplitudes and velocities of waves on the period of disturbing pressure. The process of transition of generated waves from the barotropic character of propagation to baroclinic with changes in the period of disturbing pressure is studied in detail.

NONLINEAR SURFACE WAVES IN A BASIN WITH FLOATING BROKEN ICE

The method of multiple scales is used to deduce equations for three nonlinear approximations of a wave disturbance in a basin of constant depth covered with broken ice. In deducing these equations, we take into account the space and time variability of the wave profile in the expression for the velocity potential on the basin surface. These equations are used to construct uniformly suitable asymptotic expansions up to quantities of the third order of smallness for the liquid-velocity potential and elevations of the basin surface formed by a periodic running wave of finite amplitude. We analyze the dependence of the amplitude-phase characteristics of elevations of the basin surface on the thickness of ice, nonlinearity of its vertical acceleration, and the amplitude and wavelength of the fundamental harmonic.

Forced Waves in a Bounded Basin with One Thermocline

Within the framework of the linear theory taking into account the action of the Coriolis force, we study waves induced in a continuously stratified fluid with one thermocline by periodic variations of pressure on the free surface. The fluid fills a bounded basin of constant depth. We establish the dependences of the wave amplitudes and wave velocities on the period of disturbing pressure. The sequence of transformations of generated waves from the barotropic character of motion to baroclinic depending on changes in the period of disturbing pressure is studied in detail.

Nonlinear mechanism of tsunami wave generation by atmospheric disturbances

Abstract. The problem of tsunami wave generation by variable meteo-conditions is discussed. The simplified linear and nonlinear shallow water models are derived, and their analytical solutions for a basin of constant depth are discussed. The shallow-water model describes well the properties of the generated tsunami waves for all regimes, except the resonance case. The nonlinear-dispersive model based on the forced Korteweg-de Vries equation is developed to describe the resonant mechanism of the tsunami wave generation by the atmospheric disturbances moving with near-critical speed (long wave speed). Some analytical solutions of the nonlinear dispersive model are obtained. They illustrate the different regimes of soliton generation and the focusing of frequency modulated wave packets.

Generation of Harmonics by Sea Breeze in Nontidal Water Bodies

The sea breeze introduces harmonics of its diurnal cycle into water bodies, which can be notable for strong sea breeze and microtidal and nontidal conditions. The harmonics can be of the same magnitude as those produced by bottom friction and dominate those produced by nonlinear terms in the equations of motion. A closed-form analytic solution of the linearized depth-averaged equations of motion including friction is discussed for the situation of a sea breeze blowing on an idealized one-dimensional basin of constant depth. The solution reveals the generation of odd harmonics introduced by the quadratic wind stress and role of (linearized) friction. Sea-breeze forcing on the idealized basin is numerically modeled, and agreement with the analytic solution is found. The numerical model is then run with quadratic bottom friction and nonlinear terms to compare relative contributions to the generation of harmonics. Harmonics of the water motion are distinguished as forced, or arising from the wind forcing, and as response, or arising from the interactions within the water. The hydrodynamics of Baffin Bay, Texas, are modeled and spatial variation and relative strength of the harmonics investigated. Baffin Bay is a large shallow embayment with a weak connection to the Gulf of Mexico that experiences a strong southeast wind and sea breeze during the summer. The wind induces even and odd forced harmonics through the combined quasi-steady southeast wind and sea breeze. At Baffin Bay, ratios of the semidiurnal to diurnal amplitudes of water level and current speed are found to be comparable to M4/M2 ratios for U.S. Atlantic coast embayments.