The infrastructural plans in the Jakarta Bay to reduce risks of flooding in Jakarta city comprise... more The infrastructural plans in the Jakarta Bay to reduce risks of flooding in Jakarta city comprise a large sea dike that encloses a retention lake. Part of the planned dike has the shape of the iconic Garuda bird. This shape is based on NCICD Stage-B Master Plan, where the form shape has not been tested on hydraulic perspective. Therefore, testing of wave run-up has been investiagated to find the optimum form of the Garuda Shape. The simulation of wave run-up uses Hawassi Model, where the model is governed by Boussinesq wave equation with considering wave-wave interaction. This paper shows that if in the future an explosion of Anak Krakatau will occur with strength 1/4th of the original Karkatau 1883 explosion, wave crests of 11m and troughs of 6m may collide against the birds head. As an alternative example, a more optimized design of the dike is constructed that reduces the maximal wave effects considerably.
This dissertation has been approved by: prof. dr. ir. E. W. C. van Groesen prof. dr. ir. O. Bokho... more This dissertation has been approved by: prof. dr. ir. E. W. C. van Groesen prof. dr. ir. O. Bokhove this thesis is dedicated to my mother and the memory of my father to my husband Contents Summary ix Samenvatting xi xii CONTENTS water en in tweede instantie niet-lineaire vergelijkingen voor ondiep water. De lineaire benadering wordt gestart met het eenvoudigste geval, dat is een vlakke bathymetrie met een vaste wand als randvoorwaarde. Verder is een langzaam variërende bathymetrie beschouwd. De analytische oplossing is gebaseerd op lineaire ondiep water theorie en de Wentzel-Kramer-Brillouin benadering, evenals uitbreidingen voor het dispersieve Boussinesq model. Vervolgens worden in de lineaire benadering de waterdiepten aan de kust benaderd door een strand met een constante gemiddelde helling. De oploophoogten aan de kust en de reflectie veroorzaakt door de helling worden vervolgens gemodelleerd op basis van nietlineaire ondiep water theorie over hellende bathymetrie. De koppeling tussen de numerieke en analytische dynamiek in de twee gebieden wordt behandeld met behulp variatieprincipes, wat bij benadering leidt tot behoud van de totale energie in beide gebieden. De numerieke oplossing in het simulatiegebied is gebaseerd op een variationele eindige elementenmethode. Verificaties van de effectieve randvoorwaarde techniek en implementatie worden gedaan in een reeks van numerieke testcases van toenemende complexiteit, waaronder een geval verwant aan tsunami-oploop naar de kustlijn op Atjeh, Sumatra, Indonesie. Bij vergelijking blijkt dat de effectieve randvoorwaarde methode een goede voorspelling geeft van zowel de golf die aankomt bij de kustlijn alsmede van de golfreflectie, terwijl deze methode uitsluitend is gebaseerd op de informatie van het golfsignaal bij dit zeewaartse grenspunt. De rekentijden die nodig zijn in simulaties met behulp van de effectieve randvoorwaarde tonen een reductie ten opzichte van tijden die nodig zijn voor overeenkomstige volledige numerieke simulaties. CHAPTER 1 Introduction 1.1 Motivation The word tsunami is a Japanese word, represented by two characters: tsu, meaning, harbor, and nami, meaning, wave. Tsunami is defined as a set of ocean waves with very long wavelengths (typically hundreds of kilometres) and relatively small amplitude (a metre or so is typical), so that it often passes by ships in the deep ocean without anyone on board even noticing. The cause of a tsunami is any large, sudden disturbance of the sea-surface, such as an underwater earthquake, landslide, or volcanic eruption. More rarely, a tsunami can be generated by a giant meteor impact with the ocean. About 80 percent of tsunamis happen within the Pacific Ocean's "Ring of Fire", which is an active geological area where tectonic shifts make volcanoes and earthquakes common. Historical data of tsunamis record that since 1850 tsunamis have killed more than 420,000 people and caused billions of dollars of damage to coastal structures and habitats [Bernard and Robinson, 2009]. Knowing no international boundaries across the sea, tsunami propagation is a problem with global dimensions and ranks high on the scale of natural disasters. Most of the casualties were caused by local tsunamis that occur about once per year somewhere in the world. The December 26, 2004 Indian Ocean Tsunami (IOT) with a Moment magnitude (Mw) of 9.2 was likely the most devastating tsunami in recorded history, causing over 200,000 fatalities within a few hours in 27 countries across the entire Indian Ocean basin, with tens of thousands reported missing and over one million left homeless [
Physica A: Statistical Mechanics and its Applications, 1988
In this paper we study the long-time behaviour of solutions with a two-dimensional structure of t... more In this paper we study the long-time behaviour of solutions with a two-dimensional structure of the Navier-Stokes equations on a periodic grid. From a rigorous investigation of the decrease of the energy E and the enstrophy W, it follows that the Rayleigh quotient Q = W/E is monotonically decreasing on solutions. This is shown to imply that the spatial structure of any solution tends to a critical point of Q, which is the structure of some planar vortex, and that all these structures are unstable except for the fundamental one which has the longest wavelength. The time dependence of the approach towards this seif-organized state is investigated in some detail. In the spectral plane, the mean-squared wave-numbers of both the spectral energy-and cnstrophy density are shown to decrease as a function of time. For the enstrophy this implies that the normal cascade does not match the discriminating effect of the viscous dissipation.
Different approaches are discussed of variational principles characterizing coherent vortex struc... more Different approaches are discussed of variational principles characterizing coherent vortex structures in two-dimensional flows. Turbulent flows seem to form ordered structures in the large scales of the motion and the self-organization principle predicts asymptotic states realizing an extremal value of the energy or a minimum of enstruphy. On the other hand the small scales take care of the increase of entropy, and asymptotic results can be obtained by applying the theory of equilibrium statistical mechanics.
Two examples from integrated optics are described that motivate the use of explicit variational c... more Two examples from integrated optics are described that motivate the use of explicit variational characterizations for physical parameters that are relevant for the functioning of optical devices. For 1D optical gratings the boundary of the bandgaps, and for 2D square micro-resonators the resonant frequency, are formulated as eigenvalues of the governing Helmholtz equation. This requires to define the set on which the corresponding functional is optimised in an appropriate way, depending on the application
2008 12th International Conference on Mathematical Methods in Electromagnetic Theory, 2008
A Variational Vectorial Mode Solver for 3-D dielectric waveguides with arbitrary 2-D crosssection... more A Variational Vectorial Mode Solver for 3-D dielectric waveguides with arbitrary 2-D crosssections is proposed. It is based on expansion of each component of a mode profile as a superposition of some a priori defined functions defined on one coordinate axis times some unknown continuous coefficient functions, defined on the other axis. By applying a variational restriction procedure the unknown coefficient functions are determined as an optimum approximation of the true vectorial mode profile. A couple of examples illustrate the performance of the method.
This paper describes freak waves by a (pseudo-)maximal wave proposed in [1]. The freak wave is a ... more This paper describes freak waves by a (pseudo-)maximal wave proposed in [1]. The freak wave is a consequence of a group event that is present in a time signal at some position and contains successive high amplitudes with different frequencies. The linear theory predicts the position and time of the maximal amplitude wave quite well by minimizing the variance of the total wave phase of the given initial signal. The formation of the freak wave is shown to be mainly triggered by the local interaction of waves evolving from the group event that already contained large local energy. In the evolution, the phases become more coherent and the local energy is focussed to develop a larger amplitude. We investigate two laboratory experimental signals, a dispersive focussing wave with harmonic background and a scaled New Year wave. Both signals generate a freak wave at the predicted position and time and the freak wave can be described by a pseudo-maximal wave with specific parameters. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Uni-directional wave models are used to study wave groups that appear in wave tanks of hydrodynam... more Uni-directional wave models are used to study wave groups that appear in wave tanks of hydrodynamic laboratories; characteristic for waves in such tanks is that the wave length is rather small, comparable to the depth of the layer. In second-order theory, the resulting Nonlinear Schrodinger (NLS) equation for the envelope of the wave group contains the dispersion of the group velocity multiplying the linear term and a ‘gen-coefficient’ that results from mode generation multiplying the nonlinear term. The signs of these coefficients determine whether experimentally relevant wave groups are possible or not. If the dispersion is modelled in such a way that it is correct for all wave lengths for infinitesimal waves, relevant wave groups are obtained consisting of constituent waves with a certain maximal wave length; other models for the dispersion (such as in the KdV-equation) lead to different results.
Nonlinear Optics for the Information Society, 2001
It is well known that the propagation of optical pulses and beams in material with third order no... more It is well known that the propagation of optical pulses and beams in material with third order nonlinearity can be modelled by the NLS equation. In this paper we present a low-dimensional model for the deformation of bichromatic pulses (beams). Motivated by the cubic nonlinearity in the NLS equation, the model is constructed in a 4D manifold of the complex coefficients of the two most dominan harmonic temporal modulation. By exploiting the conservation of energy and gauge, the model is reduced to a 2D Hamiltonian system. Numerical solutions of the system show that the present model describes the correct qualitative deformations. Furthermore, as long as higher frequency temporal modulations are ngeligible, the description is also quantitavely correct.
Based on the Hamiltonian formulation of water waves, using Hamiltonian consistent modelling metho... more Based on the Hamiltonian formulation of water waves, using Hamiltonian consistent modelling methods, we derive higher order Hamiltonian equations by Taylor expansions of the potential and the vertical velocity around the still water level. The polynomial expansion in wave height is mixed with pseudo-differential operators that preserve the exact dispersion relation. The consistent approximate equations have inherited the Hamiltonian structure and give exact conservation of the approximate energy. In order to deal with breaking waves, we extend the eddy-viscosity model of Kennedy et al. (2000) to be applicable for fully dispersive equations. As breaking trigger mechanism we use a kinematic criterion based on the quotient of horizontal fluid velocity at the crest and the crest speed. The performance is illustrated by comparing simulations with experimental data for an irregular breaking wave with a peak period of 12 s above deep water and for a bathymetry induced periodic wave plunging breaker over a trapezoidal bar. The comparisons show that the higher order models perform quite well; the extension with the breaking wave mechanism improves the simulations significantly.
In a previous paper [1] the inverse problem for wave crests was introduced and a solution strateg... more In a previous paper [1] the inverse problem for wave crests was introduced and a solution strategy for two-wave interactions was given. Here these solutions are actually constructed, in particular for the cases with small interaction angle, moderate phase shifts, and/or symmetric interactions. Two detailed examples are presented and analyzed. The sensitivity of the method is investigated, and conclusions about the practical applicability are given.
The infrastructural plans in the Jakarta Bay to reduce risks of flooding in Jakarta city comprise... more The infrastructural plans in the Jakarta Bay to reduce risks of flooding in Jakarta city comprise a large sea dike that encloses a retention lake. Part of the planned dike has the shape of the iconic Garuda bird. This shape is based on NCICD Stage-B Master Plan, where the form shape has not been tested on hydraulic perspective. Therefore, testing of wave run-up has been investiagated to find the optimum form of the Garuda Shape. The simulation of wave run-up uses Hawassi Model, where the model is governed by Boussinesq wave equation with considering wave-wave interaction. This paper shows that if in the future an explosion of Anak Krakatau will occur with strength 1/4th of the original Karkatau 1883 explosion, wave crests of 11m and troughs of 6m may collide against the birds head. As an alternative example, a more optimized design of the dike is constructed that reduces the maximal wave effects considerably.
This dissertation has been approved by: prof. dr. ir. E. W. C. van Groesen prof. dr. ir. O. Bokho... more This dissertation has been approved by: prof. dr. ir. E. W. C. van Groesen prof. dr. ir. O. Bokhove this thesis is dedicated to my mother and the memory of my father to my husband Contents Summary ix Samenvatting xi xii CONTENTS water en in tweede instantie niet-lineaire vergelijkingen voor ondiep water. De lineaire benadering wordt gestart met het eenvoudigste geval, dat is een vlakke bathymetrie met een vaste wand als randvoorwaarde. Verder is een langzaam variërende bathymetrie beschouwd. De analytische oplossing is gebaseerd op lineaire ondiep water theorie en de Wentzel-Kramer-Brillouin benadering, evenals uitbreidingen voor het dispersieve Boussinesq model. Vervolgens worden in de lineaire benadering de waterdiepten aan de kust benaderd door een strand met een constante gemiddelde helling. De oploophoogten aan de kust en de reflectie veroorzaakt door de helling worden vervolgens gemodelleerd op basis van nietlineaire ondiep water theorie over hellende bathymetrie. De koppeling tussen de numerieke en analytische dynamiek in de twee gebieden wordt behandeld met behulp variatieprincipes, wat bij benadering leidt tot behoud van de totale energie in beide gebieden. De numerieke oplossing in het simulatiegebied is gebaseerd op een variationele eindige elementenmethode. Verificaties van de effectieve randvoorwaarde techniek en implementatie worden gedaan in een reeks van numerieke testcases van toenemende complexiteit, waaronder een geval verwant aan tsunami-oploop naar de kustlijn op Atjeh, Sumatra, Indonesie. Bij vergelijking blijkt dat de effectieve randvoorwaarde methode een goede voorspelling geeft van zowel de golf die aankomt bij de kustlijn alsmede van de golfreflectie, terwijl deze methode uitsluitend is gebaseerd op de informatie van het golfsignaal bij dit zeewaartse grenspunt. De rekentijden die nodig zijn in simulaties met behulp van de effectieve randvoorwaarde tonen een reductie ten opzichte van tijden die nodig zijn voor overeenkomstige volledige numerieke simulaties. CHAPTER 1 Introduction 1.1 Motivation The word tsunami is a Japanese word, represented by two characters: tsu, meaning, harbor, and nami, meaning, wave. Tsunami is defined as a set of ocean waves with very long wavelengths (typically hundreds of kilometres) and relatively small amplitude (a metre or so is typical), so that it often passes by ships in the deep ocean without anyone on board even noticing. The cause of a tsunami is any large, sudden disturbance of the sea-surface, such as an underwater earthquake, landslide, or volcanic eruption. More rarely, a tsunami can be generated by a giant meteor impact with the ocean. About 80 percent of tsunamis happen within the Pacific Ocean's "Ring of Fire", which is an active geological area where tectonic shifts make volcanoes and earthquakes common. Historical data of tsunamis record that since 1850 tsunamis have killed more than 420,000 people and caused billions of dollars of damage to coastal structures and habitats [Bernard and Robinson, 2009]. Knowing no international boundaries across the sea, tsunami propagation is a problem with global dimensions and ranks high on the scale of natural disasters. Most of the casualties were caused by local tsunamis that occur about once per year somewhere in the world. The December 26, 2004 Indian Ocean Tsunami (IOT) with a Moment magnitude (Mw) of 9.2 was likely the most devastating tsunami in recorded history, causing over 200,000 fatalities within a few hours in 27 countries across the entire Indian Ocean basin, with tens of thousands reported missing and over one million left homeless [
Physica A: Statistical Mechanics and its Applications, 1988
In this paper we study the long-time behaviour of solutions with a two-dimensional structure of t... more In this paper we study the long-time behaviour of solutions with a two-dimensional structure of the Navier-Stokes equations on a periodic grid. From a rigorous investigation of the decrease of the energy E and the enstrophy W, it follows that the Rayleigh quotient Q = W/E is monotonically decreasing on solutions. This is shown to imply that the spatial structure of any solution tends to a critical point of Q, which is the structure of some planar vortex, and that all these structures are unstable except for the fundamental one which has the longest wavelength. The time dependence of the approach towards this seif-organized state is investigated in some detail. In the spectral plane, the mean-squared wave-numbers of both the spectral energy-and cnstrophy density are shown to decrease as a function of time. For the enstrophy this implies that the normal cascade does not match the discriminating effect of the viscous dissipation.
Different approaches are discussed of variational principles characterizing coherent vortex struc... more Different approaches are discussed of variational principles characterizing coherent vortex structures in two-dimensional flows. Turbulent flows seem to form ordered structures in the large scales of the motion and the self-organization principle predicts asymptotic states realizing an extremal value of the energy or a minimum of enstruphy. On the other hand the small scales take care of the increase of entropy, and asymptotic results can be obtained by applying the theory of equilibrium statistical mechanics.
Two examples from integrated optics are described that motivate the use of explicit variational c... more Two examples from integrated optics are described that motivate the use of explicit variational characterizations for physical parameters that are relevant for the functioning of optical devices. For 1D optical gratings the boundary of the bandgaps, and for 2D square micro-resonators the resonant frequency, are formulated as eigenvalues of the governing Helmholtz equation. This requires to define the set on which the corresponding functional is optimised in an appropriate way, depending on the application
2008 12th International Conference on Mathematical Methods in Electromagnetic Theory, 2008
A Variational Vectorial Mode Solver for 3-D dielectric waveguides with arbitrary 2-D crosssection... more A Variational Vectorial Mode Solver for 3-D dielectric waveguides with arbitrary 2-D crosssections is proposed. It is based on expansion of each component of a mode profile as a superposition of some a priori defined functions defined on one coordinate axis times some unknown continuous coefficient functions, defined on the other axis. By applying a variational restriction procedure the unknown coefficient functions are determined as an optimum approximation of the true vectorial mode profile. A couple of examples illustrate the performance of the method.
This paper describes freak waves by a (pseudo-)maximal wave proposed in [1]. The freak wave is a ... more This paper describes freak waves by a (pseudo-)maximal wave proposed in [1]. The freak wave is a consequence of a group event that is present in a time signal at some position and contains successive high amplitudes with different frequencies. The linear theory predicts the position and time of the maximal amplitude wave quite well by minimizing the variance of the total wave phase of the given initial signal. The formation of the freak wave is shown to be mainly triggered by the local interaction of waves evolving from the group event that already contained large local energy. In the evolution, the phases become more coherent and the local energy is focussed to develop a larger amplitude. We investigate two laboratory experimental signals, a dispersive focussing wave with harmonic background and a scaled New Year wave. Both signals generate a freak wave at the predicted position and time and the freak wave can be described by a pseudo-maximal wave with specific parameters. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Uni-directional wave models are used to study wave groups that appear in wave tanks of hydrodynam... more Uni-directional wave models are used to study wave groups that appear in wave tanks of hydrodynamic laboratories; characteristic for waves in such tanks is that the wave length is rather small, comparable to the depth of the layer. In second-order theory, the resulting Nonlinear Schrodinger (NLS) equation for the envelope of the wave group contains the dispersion of the group velocity multiplying the linear term and a ‘gen-coefficient’ that results from mode generation multiplying the nonlinear term. The signs of these coefficients determine whether experimentally relevant wave groups are possible or not. If the dispersion is modelled in such a way that it is correct for all wave lengths for infinitesimal waves, relevant wave groups are obtained consisting of constituent waves with a certain maximal wave length; other models for the dispersion (such as in the KdV-equation) lead to different results.
Nonlinear Optics for the Information Society, 2001
It is well known that the propagation of optical pulses and beams in material with third order no... more It is well known that the propagation of optical pulses and beams in material with third order nonlinearity can be modelled by the NLS equation. In this paper we present a low-dimensional model for the deformation of bichromatic pulses (beams). Motivated by the cubic nonlinearity in the NLS equation, the model is constructed in a 4D manifold of the complex coefficients of the two most dominan harmonic temporal modulation. By exploiting the conservation of energy and gauge, the model is reduced to a 2D Hamiltonian system. Numerical solutions of the system show that the present model describes the correct qualitative deformations. Furthermore, as long as higher frequency temporal modulations are ngeligible, the description is also quantitavely correct.
Based on the Hamiltonian formulation of water waves, using Hamiltonian consistent modelling metho... more Based on the Hamiltonian formulation of water waves, using Hamiltonian consistent modelling methods, we derive higher order Hamiltonian equations by Taylor expansions of the potential and the vertical velocity around the still water level. The polynomial expansion in wave height is mixed with pseudo-differential operators that preserve the exact dispersion relation. The consistent approximate equations have inherited the Hamiltonian structure and give exact conservation of the approximate energy. In order to deal with breaking waves, we extend the eddy-viscosity model of Kennedy et al. (2000) to be applicable for fully dispersive equations. As breaking trigger mechanism we use a kinematic criterion based on the quotient of horizontal fluid velocity at the crest and the crest speed. The performance is illustrated by comparing simulations with experimental data for an irregular breaking wave with a peak period of 12 s above deep water and for a bathymetry induced periodic wave plunging breaker over a trapezoidal bar. The comparisons show that the higher order models perform quite well; the extension with the breaking wave mechanism improves the simulations significantly.
In a previous paper [1] the inverse problem for wave crests was introduced and a solution strateg... more In a previous paper [1] the inverse problem for wave crests was introduced and a solution strategy for two-wave interactions was given. Here these solutions are actually constructed, in particular for the cases with small interaction angle, moderate phase shifts, and/or symmetric interactions. Two detailed examples are presented and analyzed. The sensitivity of the method is investigated, and conclusions about the practical applicability are given.
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Papers by E. Van Groesen