Ove Skovgaard

On the origin of meandering and braiding in alluvial streams

The paper describes a hydrodynamic stability analysis of the flow in an alluvial channel in which dunes have developed along the bed. The purpose is to develop a mathematical model describing the three-dimensional flow leading to instability of an originally straight channel. The model offers an explanation of the fact that some channels tend to meander, others to braid.

COMPUTATION OF LONGSHORE CURRENTS

The steady state profile of the longshore current induced by regular, obliquely incident, breaking waves, over a bottom with arbitrary parallel bottom contours, is predicted. A momentum approach is adopted. The wave parameters must be given at a depth outside the surf zone, where the current velocity is very small. The variation of the bottom roughness along the given bottom profile must be prescribed in advance. Depth refraction is included also in the calculation of wave set-down and set-up. Current refraction and rip-currents are excluded. The model includes two new expressions, one for the calculation of the turbulent lateral mixing, and one for the turbulent bottom friction. The term for the bottom friction is non-linear. Rapid convergent numerical algorithms are described for the solution of the governing equations. The predicted current profiles are compared with laboratory experiments and field measurements. For a plane sloping bottom, the influence of different eddy viscosities and constant values of bottom roughness is examined.The calculation of turbulent flow using Naviers equations assumes the introduction of a turbulent viscosity coefficient the value of which is normally constant, conforming with Boussinesqs hypothesis. It was shown that setting aside this hypothesis, a velocity profile quite different to that resulting from the classic theory is obtained in the case of flow induced by wind. This result appears to be confirmed by the tests carried out in the Mediterranean. The advantage of this method is that it gives the vertical turbulent diffusion which is of particular interest to pollution studies.In the numerical method of prediction of wind waves in deep water, Hasselmanns nonlinear interaction theory is applied. This method assumes the energy balance of individual component waves. However, the total energy balance must exist in the transformation of irregular waves in shoaling water. In this investigation, experiments were carried out on the transformations in shoaling water of composite waves having two components and random waves having one or two main peaks. It was found that the elementary component wave height of the composite waves and the elementary peak power of the random waves decrease with decrease in the water depth. This reason can be explained qualitatively by the theory of the elementary component wave height change of finite amplitude waves in shoaling water. The secondary component wave height of the composite waves and the secondary peak power of the random waves increase with decrease in the water depth. This can be explained qualitatively by Hamadas theory of nonlinear interaction in uniform depth.Experiments have been carried out by using non-breaking waves and breaking waves to investigate the wave forces on a vertical circular cell located in the shallow water. Based on the experimental data, the drag coefficient and the inertia coefficient of a circular cylinder and the curling factor of breaking waves are estimated, and the computation methods of wave forces are examined. As a result, it is shown that the phase lag of inertia forces behind the accelerations of water particles should be considered for the estimation of the drag coefficient as well as the inertia coefficient. In addition the previous formula of the maximum breaking wave forces acting on a cell or a pile is revised by introducing the effects of the above-mentioned phase lag and another phase difference, both of which are functions of the ratio of the cell diameter to the wave length. • It is confirmed that the proposed formula is applicable even to the large cell with the diameter comparable to the wave length. INTRODUCTION Many studies have been done on the impulsive pressures acting on a vertical wall, but there has been very little investigation of breaking wave forces on a cell-type structure. The breaking wave forces should be taken into consideration all the same in the design of pile-type or cell-type structures in nearshore area, because breaking waves cause extreme shock pressures on a cell structure asThe air bubble plume induced by the steady release of air into water has been analyzed with an integral technique based on the equations for conservation of mass, momentum and buoyancy. This approach has been widely used to study the behavior of submerged turbulent jets and plumes. The case of air-bubble induced flow, however, includes additional features. In this study the compressibility of the air and the differential velocity between the rising air bubbles ,and the water are introduced as basic propertie s of the air bubble plume in addition to a fundamental coefficient of entrainment and a turbulent Schmidt number characterizing the lateral spreading of the air bubbles. Theoretical solutions for twoand three-dimensional air-bubble systems in homogeneous, stagnant water are presented in both dimensional and normalized form and compared to existing experimental data. The further complication of a stratified environment is briefly discussed since this case is of great practical interest. This paper is to be considered as a progress report, as future experimental verification of various hypotheses is needed.

A mild‐slope wave equation and its application to tsunami calculations

A wave equation correct to first order in wave amplitude and bottom slope is used to calculate the wave field around an island. This is of circular cylindrical shape, and is situated on a paraboloidal shoal in an ocean of constant depth (Figure 1). The sides of the island are assumed fully reflecting. The incident waves are plane, periodic, and of small amplitude. Periods up to 30 min are investigated, and the Coriolis force is neglected. The wave equation is solved analytically, and a great number of numerical computations are carried through. The total wave field over the shoal is presented for two discrete periods in the upper end of the tsunami frequency range. The amplitudes at the middle of the front face of the island, and at the middle of the lee face, are given as functions of the wave period, and the existence of “resonance”; periods is thus demonstrated. Comparison with solutions to the linearized long‐wave equation is made, and the validity range of the shallow water theory is estimated. The g...

Perturbation theories for sine-Gordon soliton dynamics

Abstract Three recent perturbation assumptions for soliton dynamics on Josephson junctions are compared with direct numerical integrations of the perturbed sine-Gordon equation. The McLaughlin-Scott theory yields the better prediction of soliton transmission or pinning at a microshort in presence of loss and bias without or with surface resistance loss.

Refraction of cnoidal waves

Abstract Depth refraction of a first-order cnoidal surface gravity wave in the shallow water zone is calculated for a quasi two-dimensional situation, i.e., a gently sloping bathymetry characterized by straight and parallel sea bed contours (not necessarily a plane bottom), and no phase-independent or phase-averaged wave parameters vary with distance along-shore. The wave system has no energy exchange with the wind or the sea bed. The basic assumption is, that the energy flux is constant between adjacent wave orthogonals. Two non-linear algebraic equations are derived, which govern the refraction of a cnoidal wave train for the quasi two-dimensional situation. The equations are solved numerically. For a deep water angle of incidence α0 = 60° are angles, wave heights and values of the Ursell parameter versus a dimensionless depth given in figures for all practical ranges of the deep water wave steepness. The refraction solution is compared with the linear solution (Airy theory). The solution is also compared with a solution from another non-linear wave theory, viz. Deans stream function wave theory. A detailed numerical cnoidal refraction solution is tabulated as a function of the depth.

WAVE REFRACTION ACROSS A SHEARING CURRENT

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.

CURRENT DEPTH REFRACTION USING FINITE ELEMENTS

A significant portion of the damage by hurricanes is the storm surges. The National Weather Service has developed a dynamical-numerical model to forecast hurricane storm surges. The model is used operationally for prediction, warning, and planning purposes. The model requires fixed oceanographic and real time meteorological input data. The oceanographic data were prepared for the Gulf and East coasts of the U.S. and are stored as an essential part of the program. Meteorological data for any tropical storm are supplied by the forecasters or planners using the model. The model was applied to hurricane Camille 1969. Comparison between the observed and computed surges for Camille was satisfactory for prediction purposes.The main consideration in harbor master planning is to maximize the amount of time that the harbor can be used. The potential level of harbor utilization can be evaluated by analyzing vessel performance during harbor operations in terms of the range of imposed environmental conditions. The harbor utilization level is expressed statistically as the probable amount of time that the harbor can be used as planned.The design and construction of a major ocean outfall and diffuser system for disposal of wastewater effluents is a complex process involving an interplay of requirements originating from various disciplines. These include, among others, considerations of physical oceanography, mixing and dispersion, treatment processes, regulatory requirements, marine geology, economics and construction. The recently completed Sand Island Outfall and the newly designed Barbers Point Outfall are both on the southern coast of the island of Oahu, Hawaii, and are designed for treated sewage effluents from the densely populated portion of the City and County of Honolulu. In this paper, some design considerations of these outfalls will be examined. The emphasis in this paper is on the hydrodynamics, although other design aspects are also discussed briefly.The height of dikes and other coastal structures can only be calculated after determination of the wave run-up. Several formulas for the calculation of wave run-up are developed after model tests as a rule. But the influences of scale effects and natural wind conditions are practically unknown. To clear these questions further investigations and especially field measurements should be carried out. By measuring the markerline of floating trash on the slope of the seadikes the maximum wave run-up could be found out after four storm surges in 1967 and 1973In two graphs it will be shown that on the tidal flats the run-up depends on the waterdepth. The run-up was higher than it could be expected after model tests of 1954. With a newly developed special echo sounder the run-up could be measured in January 1976. The waves and the run-up could be registrated synchronously during two severe storm surges. As shown in Fig. 9 it was found a logarithmic distribution of the wave height, wave period and the higher part of the wave run-up. The found wave run-up is considerably higher than estimated before. The measured 98 % run-up is found about twice the computed value. That is an interesting and important result of the first synchronous recording of wave run-up on sea dikes.In March 1972 the authors firm in association with two Portuguese firms of consulting engineers, Consulmar and Lusotecna, were appointed by the Portuguese Government agency Gabinete da Area de Sines to prepare designs for the construction of a new harbour at Sines on the west coast of Portugal. The location is shown in Figure 1. The main breakwater, which is the subject of this paper, is probably the largest breakwater yet built, being 2 km long and in depths of water of up to 50 m. It is exposed to the North Atlantic and has been designed for a significant wave height of 11 m. Dolos units invented by Merrifield (ref. 1) form the main armour. The project programme required that studies be first made of a wide range of alternative layouts for the harbour. After the client had decided on the layout to be adopted, documents were to be prepared to enable tenders for construction to be invited in January 1973. This allowed little time for the design to be developed and only one series of flume tests, using regular waves, was completed during this period. Further tests in the regular flume were completed during the tender period and a thorough programme of testing with irregular waves was commenced later in the year, continuing until August 1974 when the root of the breakwater was complete and the construction of the main cross-section was about to start. The model tests, which were carried out at the Laboratorio Nacional de Engenharia Civil in Lisbon, were reported by Morals in a paper presented to the 14th International Coastal Engineering Conference in 1974. (ref. 2)Estuaries may be sequentially classified into highly stratified, moderately mixed and vertically homogeneous. An important difference between moderately stratified or vertically homogeneous estuaries, and highly stratified estuaries (salt wedges) is that, in the former, tidal currents are sufficient to cause turbulent mixing of fresh water and sea water over the full depth of the estuary. In the latter, a distinct interface or interfacial layer exists which separates the two nearly homogeneous layers. The vertical advectlon of salt in this two-layer flow is the dominant process in maintaining the salt balance. This paper presents an analytical model describing this process. Experiments have been conducted in the laboratory to compare with the developed theory.

A theoretical and computational study of the soliton laser

Abstract A theoretical and computational study is presented of the soliton laser recently constructed by Mollenauer and Stolen. In particular, the role played by the optical fibre in determining the properties of the optical pulses is analysed in detail.

Correlation exponents for trajectories in the low-dimensional discrete selftrapping equation☆

Abstract The non-linear dynamics of small polyatomic molecules has recently been modeled by the discrete selftrapping equation (DST) for which an alternative formulation is given. In the three-bond case of this hamiltonian system chaos occurs in the intermediate range of the non-linearity to dispersion ratio. Here calculations of the Grassberger-Procaccia correlation exponent exhibit noninteger values.

Chaos in the Beeler-Reuter system for the action potential of ventricular myocardial fibres

Abstract The response of the membrane of mammalian ventricular myocardial fibres, described by the Beeler-Reuter differential equation system, is investigated in the case where the system is subjected to sinusoidal stimulation. For some stimulation frequencies the system exhibits periodic behaviour, frequency locked to the driving cycle. For other values of the stimulation frequency we find chaotic responses clearly distinguished by broad band power spectra. The Lyapunov exponents are computed for this case from which the fractal dimension of the corresponding strange attractor can be estimated.