arXiv subjects

Henrik Kalisch

Publications and source records attributed to Henrik Kalisch.

At least 19 recordsLinked to original sources

arXiv2026

A Regularized Shallow Water System

The shallow-water system is a standard model for long waves in shallow water. The system is hyperbolic and, for a large class of initial data, solutions develop steep gradients leading to shock formation in finite time. Since such singularities violate the long-wave assumptions underlying the model, their appearance limits the regime of validity of the equations. In this work, we introduce a regularized shallow-water system in which the nonlinear terms are modified by a bounded operator. This regularization removes the standard derivative-steepening mechanism responsible for shock formation in the classical system while remaining consistent with the long-wave regime. We establish local well-posedness and small-data global well-posedness in Sobolev spaces that exclude singularity formation. In addition, numerical simulations indicate that the system admits solitary-wave solutions.

math.AP↗

arXiv2026

Recovery of directional wave spectrum from sparse data with compressed sensing

Compressed sensing provides an efficient framework for reconstructing wave signals from reduced measurements. For multi-channel buoy data, the three displacement components exhibit intrinsic correlations, as wave motion contributes simultaneously to all directions according to linear wave theory. Meanwhile, conventional compressed sensing methods based on $\ell_1$-shrinkage tend to underestimate signal energy when sparsity is not strictly satisfied, leading to biased spectral estimation. This paper introduces a group sparsity constraint to promote physically consistent sparse representations across channels. An energy constraint is proposed in the form of a soft lower bound, enabling an isotropic rescaling of the recovered spectrum while preserving its sparse structure. Considering a large volume of buoy data, we demonstrate through a series of experiments that the proposed approach enables compression by retaining a subset of original measurements.

physics.geo-ph↗

arXiv2024

Listen to the Waves: Using a Neuronal Model of the Human Auditory System to Predict Ocean Waves

Artificial neural networks (ANNs) have evolved from the 1940s primitive models of brain function to become tools for artificial intelligence. They comprise many units, artificial neurons, interlinked through weighted connections. ANNs are trained to perform tasks through learning rules that modify the connection weights. With these rules being in the focus of research, ANNs have become a branch of machine learning developing independently from neuroscience. Although likely required for the development of truly intelligent machines, the integration of neuroscience into ANNs has remained a neglected proposition. Here, we demonstrate that designing an ANN along biological principles results in drastically improved task performance. As a challenging real-world problem, we choose real-time ocean-wave prediction which is essential for various maritime operations. Motivated by the similarity of ocean waves measured at a single location to sound waves arriving at the eardrum, we redesign an echo state network to resemble the brain's auditory system. This yields a powerful predictive tool which is computationally lean, robust with respect to network parameters, and works efficiently across a wide range of sea states. Our results demonstrate the advantages of integrating neuroscience with machine learning and offer a tool for use in the production of green energy from ocean waves.

eess.SP↗

arXiv2023

Mechanical balance laws for two-dimensional Boussinesq systems

Most of the asymptotically derived Boussinesq systems of water wave theory for long waves of small amplitude fail to satisfy exact mechanical conservation laws for mass, momentum and energy. It is thus only fair to consider approximate conservation laws that hold in the context of these systems. Although such approximate mass, momentum and energy conservation laws can be derived, the question of a rigorous mathematical justification still remains unanswered. The aim of this paper is to justify the formally derived mechanical balance laws for weakly nonlinear and weakly dispersive water wave Boussinesq systems. In particular, two asymptotic expansions used for the formal and rigorous derivation of the Boussinesq systems and the same are employed for the derivation and rigorous justification of the balance laws. Numerical validation of the asymptotic orders of approximation is also presented.

math.AP↗

arXiv2023

Infra-gravity Waves and Cross-shore Transport -- A Conceptual Study

Infra-gravity waves are generally known as small-amplitude waves of periods between 25 seconds and 5 minutes. They originate from the presence of wave groups in the open ocean waves and can move freely after being released near the surf zone where they can be further fueled with energy from the spatially varying break point of swell waves . As these waves approach the shore, the relative importance of the infra-gravity wave signal increases, and its impact on the shorter waves gets stronger. In addition, infra-gravity waves drive strong cross-shore currents, which lead to significant back-and-forth motion of the underlying sea water. This strong cross-shore motion has been made visible by recent field studies, where significant cross-shore movement was detected and found to be correlated with the infra-gravity wave signal. In the present work, the connection between infra-gravity waves is explored further using linear wave theory and an established numerical nearshore wave model (BOSZ). It is shown that in all cases, the presence of infra-gravity waves leads to strong cross-shore motion. This behavior can be understood by considering the infra-gravity waves as separate free waves, and then following the fluid particle trajectories excited by these waves. As it is shown, these trajectories have a very large horizontal extent which -- if not separated from the main gravity wave field -- appears as a large, but often not directly visible back-and-forth motion, underlying the more readily observable gravity ocean waves.

physics.geo-ph↗

arXiv2023

Identification of wave breaking from nearshore wave-by-wave records

Using data from a recent field campaign, we evaluate several breaking criteria with the goal of assessing the accuracy of these criteria in wave breaking detection. Two new criteria are also evaluated. An integral parameter is defined in terms of temporal wave trough area, and a differential parameter is defined in terms of maximum steepness of the crest front period. The criteria tested here are based solely on sea surface elevation derived from standard pressure gauge records. They identify breaking and non-breaking waves with an accuracy between 84% and 89% based on the examined field data.

physics.flu-dyn↗

arXiv2023

The superharmonic instability and wave breaking in Whitham equations

The Whitham equation is a model for the evolution of surface waves on shallow water that combines the unidirectional linear dispersion relation of the Euler equations with a weakly nonlinear approximation based on the KdV equation. We show that large-amplitude, periodic, traveling-wave solutions to the Whitham equation and its higher-order generalization, the cubic Whitham equation, are unstable with respect to the superharmonic instability (i.e. a perturbation with the same period as the solution). The threshold between superharmonic stability and instability occurs at the maxima of the Hamiltonian and $\mathcal{L}_2$-norm. We examine the onset of wave breaking in traveling-wave solutions subject to the modulational and superharmonic instabilities. We present new instability results for the Euler equations in finite depth and compare them with the Whitham results. We show that the Whitham equation more accurately approximates the wave steepness threshold for the superharmonic instability of the Euler equations than does the cubic Whitham equation. However, the cubic Whitham equation more accurately approximates the wave steepness threshold for the modulational instability of the Euler equations than does the Whitham equation.

physics.flu-dyn↗

arXiv2023

A novel energy-bounded Boussinesq model and a well balanced and stable numerical discretisation

In this work, a novel Boussinesq system is put forward. The system is naturally nonlinearly entropy/energy-stable, and is designed for problems with sharply varying bathymetric features. The system is flexible and allows tuning of the dispersive parameters to the relevant wavenumber range of the problem at hand. We present a few such parameter sets, including one that tracks the dispersive relation of the underlying Euler equations up to a nondimensional wavenumber of about $30$. In the one-dimensional case, we design a stable finite-volume scheme and demonstrate its robustness and accuracy in a suite of test problems including Dingemans's wave experiment. We generalise the system to the two-dimensional case and sketch how the numerical scheme can be straightforwardly generalised.

math.NA↗

arXiv2022

Breather Solutions to the Cubic Whitham Equation

We are concerned with numerical approximations of breather solutions for the cubic Whitham equation which arises as a water-wave model for interfacial waves. The model combines strong nonlinearity with the non-local character of the water-wave problem. The equation is non-integrable as suggested by the inelastic interaction of solitary waves. As a non local model, it generalizes, in the low frequency limit, the well known modified KdV (mKdV) equation which is a completely-integrable model. The mKdV equation has breather solutions, i.e. periodic in time and localized in space biparametric solutions. It was recently shown that these breather solutions appear naturally as ground states of invariant integrals, suggesting that such structures may also exist in non-integrable models, at least in an approximate sense. In this work, we present numerical evidence that in the non-integrable case of the cubic Whitham equation, breather solutions may also exist.

nlin.PS↗

arXiv2022

The Cubic Vortical Whitham Equation

The cubic-vortical Whitham equation is a model for wave motion on a vertically sheared current of constant vorticity in a shallow inviscid fluid. It generalizes the classical Whitham equation by allowing constant vorticity and by adding a cubic nonlinear term. The inclusion of this extra nonlinear term allows the equation to admit periodic, traveling-wave solutions with larger amplitude than the Whitham equation. Increasing vorticity leads to solutions with larger amplitude as well. The stability of these solutions is examined numerically. All moderate- and large-amplitude solutions, regardless of wavelength, are found to be unstable. A formula for a stability cutoff as a function of vorticity and wavelength for small-amplitude solutions is presented. In the case with zero vorticity, small-amplitude solutions are unstable with respect to the modulational instability if kh > 1.252, where k is the wavenumber and h is the mean fluid depth.

physics.flu-dyn↗

arXiv2022

A regularized shallow-water waves system with slip-wall boundary conditions in a basin: Theory and numerical analysis

The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche's method. Convergence of the numerical method is analyzed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.

math.NA↗

arXiv2021

Fully dispersive Boussinesq models with uneven bathymetry

Three weakly nonlinear but fully dispersive Whitham-Boussinesq systems for uneven bathymetry are studied. The derivation and discretization of one system is presented. The numerical solutions of all three are compared with wave gauge measurements from a series of laboratory experiments conducted by Dingemans. The results show that although the models are mathematically similar, their accuracy varies dramatically.

physics.flu-dyn↗

arXiv2021

A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal Shoaling

In this article we provide formulations of energy flux and radiation stress consistent with the scaling regime of the Korteweg-de Vries (KdV) equation. These quantities can be used to describe the shoaling of cnoidal waves approaching a gently sloping beach. The transformation of these waves along the slope can be described using the shoaling equations, a set of three nonlinear equations in three unknowns: the wave height H, the set-down and the elliptic parameter m. We define a numerical algorithm for the efficient solution of the shoaling equations, and we verify our shoaling formulation by comparing with experimental data from two sets of experiments as well as shoaling curves obtained in previous works.

physics.flu-dyn↗

arXiv2020

Extreme Wave Runup on a Steep Coastal Profile

It is shown that very steep coastal profiles can give rise to unexpectedly large wave events at the coast. We combine insight from exact solutions of a simplified mathematical model with photographs from observations at the Norwegian coast near the city of Haugesund. The results suggest that even under moderate wave conditions, very large run-up can occur at the shore.

physics.ao-ph↗

arXiv2020

The Whitham Equation with Surface Tension

The viability of the Whitham equation as a nonlocal model for capillary-gravity waves at the surface of an inviscid incompressible fluid is under study. A nonlocal Hamiltonian system of model equations is derived using the Hamiltonian structure of the free surface water wave problem and the Dirichlet-Neumann operator. The system features gravitational and capillary effects, and when restricted to one-way propagation, the system reduces to the capillary Whitham equation. It is shown numerically that in various scaling regimes the Whitham equation gives a more accurate approximation of the free-surface problem for the Euler system than other models like the KdV, and Kawahara equation. In the case of relatively strong capillarity considered here, the KdV and Kawahara equations outperform the Whitham equation with surface tension only for very long waves with negative polarity.

physics.flu-dyn↗

arXiv2019

Particle trajectories in nonlinear Schrodinger models

The nonlinear Schrodinger equation is well known as a universal equation in the study of wave motion. In the context of wave motion at the free surface of an incompressible fluid, the equation accurately predicts the evolution of modulated wave trains with low to moderate wave steepness. While there is an abundance of studies investigating the reconstruction of the surface profile $η$, and the fidelity of such profiles provided by the nonlinear Schrodinger equation as predictions of real surface water waves, very few works have focused on the associated flow field in the fluid. In the current work, it is shown that the velocity potential $ϕ$ can be reconstructed in a similar way as the free-surface profile. This observation opens up a range of potential applications since the nonlinear Schrodinger equation features fairly simple closed-form solutions and can be solved numerically with comparatively little effort. In particular, it is shown that particle trajectories in the fluid can be described with relative ease not only in the context of the nonlinear Schrodinger equation, but also in higher-order models such as the Dysthe equation, and in models incorporating certain types of viscous effects.

physics.flu-dyn↗

arXiv2019

A comparative study of bi-directional Whitham systems

In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties. In the current work, we review some of the existing fully dispersive systems. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.

physics.comp-ph↗

arXiv2018

Approximate Conservation Laws in the KdV Equation

The Korteweg-de Vries equation is known to yield a valid description of surface waves for waves of small amplitude and large wavelength. The equation features a number of conserved integrals, but there is no consensus among scientists as to the physical meaning of these integrals. In particular, it is not clear whether these integrals are related to the conservation of momentum or energy, and some researchers have questioned the conservation of energy in the dynamics governed by the equation. In this note it is shown that while exact energy conservation may not hold, if momentum and energy densities and fluxes are defined in an appropriate way, then solutions of the Korteweg-de Vries equation give rise to approximate differential balance laws for momentum and energy.

math-ph↗