Per A. Madsen

A new form of the Boussinesq equations with improved linear dispersion characteristics

Abstract A new form of the Boussinesq equations is introduced in order to improve their dispersion characteristics. It is demonstrated that the depth-limitation of the new equations is much less restrictive than for the classical forms of the Boussinesq equations, and it is now possible to simulate the propagation of irregular wave trains travelling from deep water to shallow water. In deep water, the new equations become effectively linear and phase celerities agree with Stokes first-order theory. In more shallow water, the new equations converge towards the standard Boussinesq equations, which are known to provide good results for waves up to at least 75% of their breaking height. A numerical method for solving the new set of equations in two horizontal dimensions is presented. This method is based on a time-centered implicit finite-difference scheme. Finally, model results for wave propagation and diffraction in relatively deep water are presented.

A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry

Abstract A new form of the Boussinesq equations applicable to irregular wave propagation on a slowly varying bathymetry from deep to shallow water is introduced. The equations incorporate excellent linear dispersion characteristics, and are formulated and solved in two horizontal dimensions. In an earlier paper we concentrated on wave propagation and diffraction on a horizontal bottom in deep water. In this paper these principles are generalized and the Boussinesq equations are extended to include terms proportional to the bottom slope, which are essential for the shoaling properties of the equations. The paper contains a linear shoaling analysis of the new equations and a verification of the numerical model with respect to shoaling and refraction-diffraction in deep and shallow water.

A Boussinesq model for waves breaking in shallow water

Abstract A simple description of wave breaking in shallow water is incorporated in the Boussinesq equations by using the concept of surface rollers. The roller is considered as a volume of water being carried by the wave with the wave celerity. The effect of the roller is included in the vertical distribution of the horizontal velocity, which leads to an additional convective momentum term. The breaking criterion is related to the local slope of water surface and the thickness of the roller is determined from simple geometrical considerations. Although the model is simple, it is capable of representing a variety of processes such as the initiation and cessation of wave breaking, the evolution of wave profiles before, during and after wave breaking, the initial conversion of potential energy into forward momentum flux, and the associated horizontal shift between the break point and the point where the setup in mean water level is initiated. Results are presented for regular and irregular wave trains and comparison with measurements shows good agreement.

Higher–order Boussinesq–type equations for surface gravity waves: derivation and analysis

Boussinesq–type equations of higher order in dispersion as well as in nonlinearity are derived for waves (and wave–current interaction) over an uneven bottom. Formulations are given in terms of various velocity variables such as the depth–averaged velocity and the particle velocity at the still water level, and at an arbitrary vertical location. The equations are enhanced and analysed with emphasis on linear dispersion, shoaling and nonlinear properties for large wave numbers. As a starting point the velocity potential is expanded as a power series in the vertical coordinate measured from the still water level (SWL). Substituting this expansion into the Laplace equation leads to a velocity field expressed in terms of spatial derivatives of the vertical velocity ŵ and the horizontal velocity vector û at the SWL. The series expressions are given to infinite order in the dispersion parameter, &mgr;. Satisfying the kinematic bottom boundary condition defines an implicit relation between ŵ and û, which is recast as an explicit recursive expression for ŵ in terms of û under the assumption that &mgr; ≪ 1. Boussinesq equations are then derived from the dynamic and kinematic boundary conditions at the free surface. In this process the infinite series solutions are truncated at O(&mgr;6), while all orders of the nonlinearity parameter, ε are included to that order in dispersion. This leads to a set of higher–order Boussinesq equations in terms of the surface elevation η and the horizontal velocity vector û at the SWL. The equations are recast in terms of the depth–averaged velocity, U leaving out O(ε2&mgr;4, which corresponds to assuming ε =O(&mgr;). This formulation turns out to include singularities in linear dispersion as well as in nonlinearity. Next, the technique introduced by Madsen and others in 1991and Schäffer & Madsen in 1995 is invoked, and this results in aset of enhanced equations formulated in U and including O(&mgr;4,ε&mgr;4)terms. These equations contain no singularities and the embedded linear and nonlinear properties are shown to be significantly improved. To quantify the accuracy, Stokess third–order theory is used as reference and Fourier analyses of the new equations are carried out to third order (in nonlinearity) for regular waves on a constant depth and to first order for shoaling characteristics. Furthermore, analyses are carried out to second order for bichromatic waves and to first order for waves in ambient currents. These analyses are not restricted to small values of the linear dispersion parameter, &mgr;. In conclusion, the new equations are shown to have linear dispersion characteristics corresponding to a Pade [4,4] expansion in k′h′ (wave number times depth) of the squared celerity according to Stokess linear theory. This corresponds to a quite high accuracy in linear dispersion up to approximately k′h′ = 6. The high quality of dispersion is also achieved for the Doppler shift in connection with wave–current interaction and it allows for a study of wave blocking due to opposing currents. Also, the linear shoaling characteristics are shown to be excellent, and the accuracy of nonlinear transfer of energy to sub– and super–harmonics is found to be superior to previous formulations. The equations are then recast in terms of the particle velocity, ũ, at an arbitrary vertical location including O&mgr;4,ε5&mgr;4)terms. This formulation includes, as special subsets, Boussinesq equations in terms of the bottom velocity or the surface velocity. Furthermore, the arbitrary location of the velocity variable can be used to optimize the embedded linear and nonlinear characteristics. A Fourier analysis is again carried out to third order (in ε) for regular waves. It turns out that Padé [4,4] linear dispersion characteristics can not be achieved for any choice of the location of the velocity variable. However, for an optimized location we achieve fairly good linear characteristics and very good nonlinear characteristics. Finally, the formulation in terms of ũ is modified by introducing the technique of dispersion enhancement while retaining only O(&mgr;4,ε5&mgr;4) terms. Now the resulting set of equations do show Padé [4,4] dispersion characteristics in the case of pure waves as well as in connection with ambient currents, and again the nonlinear properties (such as second– and third–order transfer functions and amplitude dispersion) are shown to be superior to those of existing formulations of Boussinesq–type equations.

Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves

Abstract This is the first of three papers on the modelling of various types of surf zone phenomena. In this first paper, part I, the model is presented and its basic features are studied for the case of regular waves. The model is based on two-dimensional equations of the Boussinesq type and it features improved linear dispersion characteristics, possibility of wave breaking, and a moving boundary at the shoreline. The moving shoreline is treated numerically by replacing the solid beach by a permeable beach characterized by an extremely small porosity. Run-up of nonbreaking waves is verified against the analytical solution for nonlinear shallow water waves. The inclusion of wave breaking is based on the surface roller concept for spilling breakers using a geometrical determination of the instantaneous roller thickness at each point and modelling the effect of wave breaking by an additional convective momentum term. This is a function of the local wave celerity, which is determined interactively. The model is applied to cross-shore motions of regular waves including various types of breaking on plane sloping beaches and over submerged bars. Model results comprise time series of surface elevations and the spatial variation of phase-averaged quantities such as the wave height, the crest and trough elevations, the mean water level, and the depth-averaged undertow. Comparisons with physical experiments are presented. The phaseaveraged balance of the individual terms in the momentum and energy equation is determined by time-integration and quantities such as the cross-sectional roller area, the radiation stress, the energy flux and the energy dissipation are studied and discussed with reference to conventional phase-averaged wave models. The companion papers present cross-shore motions of breaking irregular waves, swash oscillations and surf beats (part II) and nearshore circulations induced by breaking of unidirectional and multidirectional waves (part III).

A new Boussinesq method for fully nonlinear waves from shallow to deep water

A new method valid for highly dispersive and highly nonlinear water waves is presented. It combines a time-stepping of the exact surface boundary conditions with an approximate series expansion solution to the Laplace equation in the interior domain. The starting point is an exact solution to the Laplace equation given in terms of infinite series expansions from an arbitrary z -level. We replace the infinite series operators by finite series (Boussinesq-type) approximations involving up to fifth-derivative operators. The finite series are manipulated to incorporate Pade approximants providing the highest possible accuracy for a given number of terms. As a result, linear and nonlinear wave characteristics become very accurate up to wavenumbers as high as kh = 40, while the vertical variation of the velocity field becomes applicable for kh up to 12. These results represent a major improvement over existing Boussinesq-type formulations in the literature. A numerical model is developed in a single horizontal dimension and it is used to study phenomena such as solitary waves and their impact on vertical walls, modulational instability in deep water involving recurrence or frequency downshift, and shoaling of regular waves up to breaking in shallow water.

Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: Derivation and analysis.

Boussinesq formulations valid for highly dispersive and highly nonlinear water waves are derived with the objective of improving the accuracy of the vertical variation of the velocity field as well as the linear and nonlinear properties. First, an exact solution to the Laplace equation is given in terms of infinite–series expansions from an arbitrary z–level which is a constant fraction of the still–water depth. This defines the fully dispersive and fully nonlinear water–wave problem in terms of five variables: the free–surface elevation and the horizontal and vertical velocities evaluated at the free surface and at the arbitrary z–level. Next, the infinite series operators are replaced by finite–series (Boussinesq–type) approximations. Three different approximations are introduced, each involving up to fifth–derivative operators, and these formulations are analysed with respect to the linear–velocity profile, linear dispersion and linear shoaling. Nonlinear characteristics are investigated by a perturbation analysis to third order for regular waves and to second order for bichromatic waves. Finally, a numerical spectral solution is made for highly nonlinear steady waves in deep and shallow water. It can be concluded that the best of the new formulations (method III) allows an accurate description of dispersive nonlinear waves for kh (wavenumber times water depth) as high as 40, while accurate velocity profiles are restricted to kh < 10. These results represent a major improvement over existing Boussinesq formulations from the literature.

Cerebral desaturation during exercise reversed by O2 supplementation

The combined effects of hyperventilation and arterial desaturation on cerebral oxygenation (ScO2) were determined using near-infrared spectroscopy. Eleven competitive oarsmen were evaluated during a 6-min maximal ergometer row. The study was randomized in a double-blind fashion with an inspired O2 fraction of 0.21 or 0.30 in a crossover design. During exercise with an inspired O2 fraction of 0.21, the arterial CO2 pressure (35 +/- 1 mmHg; mean +/- SE) and O2 pressure (77 +/- 2 mmHg) as well as the hemoglobin saturation (91.9 +/- 0.7%) were reduced (P < 0.05). ScO2 was reduced from 80 +/- 2 to 63 +/- 2% (P < 0.05), and the near-infrared spectroscopy-determined concentration changes in deoxy- (DeltaHb) and oxyhemoglobin (DeltaHbO2) of the vastus lateralis muscle increased 22 +/- 3 microM and decreased 14 +/- 3 microM, respectively (P < 0.05). Increasing the inspired O2 fraction to 0.30 did not affect ventilation (174 +/- 4 l/min), but arterial CO2 pressure (37 +/- 2 mmHg), O2 pressure (165 +/- 5 mmHg), and hemoglobin O2 saturation (99 +/- 0.1%) increased (P < 0. 05). ScO2 remained close to the resting level during exercise (79 +/- 2 vs. 81 +/- 2%), and although the muscle DeltaHb (18 +/- 2 microM) and DeltaHbO2 (-12 +/- 3 microM) were similar to those established without O2 supplementation, work capacity increased from 389 +/- 11 to 413 +/- 10 W (P < 0.05). These results indicate that an elevated inspiratory O2 fraction increases exercise performance related to maintained cerebral oxygenation rather than to an effect on the working muscles.The combined effects of hyperventilation and arterial desaturation on cerebral oxygenation ([Formula: see text]) were determined using near-infrared spectroscopy. Eleven competitive oarsmen were evaluated during a 6-min maximal ergometer row. The study was randomized in a double-blind fashion with an inspired O2 fraction of 0.21 or 0.30 in a crossover design. During exercise with an inspired O2 fraction of 0.21, the arterial CO2 pressure (35 ± 1 mmHg; mean ± SE) and O2 pressure (77 ± 2 mmHg) as well as the hemoglobin saturation (91.9 ± 0.7%) were reduced ( P < 0.05).[Formula: see text] was reduced from 80 ± 2 to 63 ± 2% ( P < 0.05), and the near-infrared spectroscopy-determined concentration changes in deoxy- (ΔHb) and oxyhemoglobin (ΔHbO2) of the vastus lateralis muscle increased 22 ± 3 μM and decreased 14 ± 3 μM, respectively ( P < 0.05). Increasing the inspired O2fraction to 0.30 did not affect ventilation (174 ± 4 l/min), but arterial CO2 pressure (37 ± 2 mmHg), O2 pressure (165 ± 5 mmHg), and hemoglobin O2saturation (99 ± 0.1%) increased ( P < 0.05).[Formula: see text] remained close to the resting level during exercise (79 ± 2 vs. 81 ± 2%), and although the muscle ΔHb (18 ± 2 μM) and ΔHbO2 (-12 ± 3 μM) were similar to those established without O2 supplementation, work capacity increased from 389 ± 11 to 413 ± 10 W ( P < 0.05). These results indicate that an elevated inspiratory O2fraction increases exercise performance related to maintained cerebral oxygenation rather than to an effect on the working muscles.

WAVE CHARACTERISTICS IN THE SURF ZONE

The equations describing conservation of mass, momentum and energy in a turbulent free surface flow are derived for a controle volume extending over the whole depth. The effect of the turbulent surface oscillations are discussed but neglected in the following analysis, where the equations are applied to the energy balance in a surf zone wave motion. This leads to results for the wave height variation and the velocity of propagation. The results cannot be reconciled completely with measurements and the concluding discussion is aimed at revealing how the model can be improved.A three-dimensional morphodynamic model of sequential beach changes Is presented. The model Is based on variations in breaker wave power generating a predictable sequence of beach conditions. The spectrum of beach conditions from fully eroded-dissipatlve to fully accreted reflective is characterised by ten beach-stages. Using the breaker wave power to beach-stage relationship the model Is applied to explain temporal, spatial and global variations In beach morphodynamlcs.The agents of initial damage to the dunes are water, which undermines them, and animals (including man) which damage the protective vegetation by grazing or trampling. Of these, man has recently assumed predominant local importance because of the popularity of sea-side holidays and of the land-falls of certain marine engineering works such as oil and gas pipelines and sewage outfalls. The need is therefore increasing for active dune management programmes to ensure that under these accentuated pressures, the coast retain an equilibrium comparable with that delicately balanced equilibrium which obtains naturally at a particular location.

A new approach to high-order Boussinesq models

An innite-order, Boussinesq-type dierential equation for wave shoaling over variable bathymetry is derived. Dening three scaling parameters { nonlinearity, the dispersion parameter, and the bottom slope { the system is truncated to a nite order. Using Pad e approximants the order in the dispersion parameter is eectively doubled. A derivation is made systematic by separately solving the Laplace equation in the undisturbed fluid domain and then addressing the nonlinear free-surface conditions. We show that the nonlinear interactions are faithfully captured. The shoaling and dispersion components are time independent. Boussinesq-type equations have been widely studied in recent years. The introduction of Pad e approximants greatly improves the dispersion and shoaling characteristics of these equations, making them an attractive tool for general coastal applications. A review of the subject is given by Madsen & Sch¨ (1998, referred to as MS in the following). The methods used most often for deriving high-order Boussinesq equations are based on two techniques: one is the choice of appropriate velocity variables which improves the dispersion characteristics of the resulting equation, and the other is enhancement of the equations by applying appropriate linear operators to the continuity equation and the momentum equation. In MS the two methods were combined leading to an improved dispersion relation. The variables used are normally a velocity variable (the velocity at some fraction of the depth or the mean velocity) and the free-surface elevation. This choice leads to a time-dependent system in which dispersion, nonlinearity and shoaling are all coupled. In water of intermediate depth, the highest order terms are required for representation of dispersion and shoaling. This is successfully achieved by using Pad e approximants. However, in previous studies there appears to be some trade o among the performance regarding nonlinearity, shoaling and dispersion in the dierent sets of equations. Our primary goal is to present an approach which performs well in all three respects. The present approach is based on decoupling the problem into two subproblems. One is the linear part of the problem, which involves solving the Laplace equation in the undisturbed fluid domain, and accounts for dispersion and shoaling. The idea of Boussinesq-type equations is to eliminate the vertical coordinate from the problem. Inx 2, this is achieved by the use of innite power series expansions. The kinematic bottom boundary condition then provides a relation between the horizontal and