# Henrik Kalisch's research while affiliated with University of Bergen and other places

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## Publications (122)

A weak notion of solution for systems of conservation laws in one dimension is put forward. In the framework introduced here, it can be shown that the Cauchy problem for any n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgr...

The paper investigates waves generated by the moving loads on ice plates floating on an incompressible fluid. Two different viscoelastic approximations are considered for the ice cover: A model depending on the strain-relaxation time, and a model including a hereditary delay integral. The problem is formulated in terms of the exact dispersion relat...

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 equa...

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, traveli...

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...

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 appr...

Plain Language Summary
A sea state is defined broadly by an average wave period and an average perceived (significant) wave height, but wave heights and periods of individual waves may differ significantly from these average values. The differences between individual waves are exacerbated as the waves approach the shore and interact with the bottom...

It is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarca...

The cubic-vortical Whitham equation is a two-dimensional 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 p...

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 (Comparison of computations with Boussin...

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 equat...

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 ca...

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 ca...

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 appr...

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 mode...

Consideration is given to the shallow-water equations, a hyperbolic system modeling the propagation of long waves at the surface of an incompressible inviscible fluid of constant depth. It is well known that the solution of the Riemann problem associated to this system may feature dry states for some configurations of the Riemann data. This article...

The nonlinear Schrödinger 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 reco...

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 gravitatio...

Internal solitary waves (ISWs) propagating in a stably stratified two‐layer fluid in which the upper boundary condition changes from open water to ice are studied for grease, level, and nilas ice. The ISW‐induced current at the surface is capable of transporting the ice in the horizontal direction. In the level ice case, the transport speed of, rel...

In this work, a detailed description of the internal flow field in a collapsing bore generated on a slope in a wave flume is given. It is found that in the case at hand, just prior to breaking, the shape of the free surface and the flow field below are dominated by capillary effects. While numerical approximations are able to predict the developmen...

The modulational instability of two-dimensional nonlinear traveling-wave solutions of the Whitham equation in the presence of constant vorticity is considered. It is shown that vorticity has a significant effect on the growth rate of the perturbations and on the range of unstable wavenumbers. Waves with kh greater than a critical value, where k is...

The response of a floating elastic plate to the motion of a moving load is studied using a fully dispersive weakly nonlinear system of equations. The system allows for an accurate description of waves across the whole spectrum of wavelengths and also incorporates nonlinearity, forcing and damping. The flexural–gravity waves described by the system...

A weakly nonlinear fully dispersive model equation is derived which describes the propagation of waves in a thin elastic body overlying an incompressible inviscid fluid. The equation is nonlocal in the linear part, and is similar to the so-called Whitham equation which was proposed as a model for the description of wave motion at the free surface o...

It is well known that most fluid flows feature vorticity. In order to avoid mathematical complexity, the study of surface waves is often carried out in the context of potential flow. In studies where vorticity is taken into account, it usually enters in a standard way, such as a background flow or in the boundary layer.
In the current contribution,...

A long-wave model for the evolution of long waves at the interface of a deep and a shallow fluid is put forward. The model allows for a uniform stream in one of the layers, and the existence of interfacial capillarity. The model can be used to study the dynamics of the interface between liquid CO2 and seawater in the deep ocean, including the evolu...

If a weir is dragged through a wave flume, the upstream flow takes the form of an undular bore propagating ahead of the weir. It was found previously in the work of Wilkinson and Banner ("Undular bores," in 6th Australian Hydraulics and Fluid Mechanics Conference, Adelaide, Australia, 1977) that the leading wave of the undular bore will break if th...

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...

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 t...

We investigate the effect of constant-vorticity background shear on the properties of wavetrains in deep water. Using the methodology of Fokas ( A Unified Approach to Boundary Value Problems , 2008, SIAM), we derive a higher-order nonlinear Schrödinger equation in the presence of shear and surface tension. We show that the presence of shear induces...

In this work, a shallow-water system for interfacial waves in the case of a neutrally buoyant two-layer fluid system is considered. Such a situation arises in the case of large underwater lakes of compressible liquids such as CO in the deep ocean which may happen naturally or may be man-made. Depending on temperature and depth, such deposits may be...

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...

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 reco...

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 t...

Consideration is given to the KdV equation as an approximate model for long waves of small amplitude at the free surface of an inviscid fluid. It is shown that there is an approximate momentum density associated to the KdV equation, and the difference between this density and the physical momentum density derived in the context of the full Euler eq...

In this work, the influence of constant background vorticity on the properties of shock waves in a shallow water system are considered. It is shown that the flow-depth ratio of stationary shocks can be written as a function of two non-dimensional parameters: the Froude number, suitably defined in the presence of the shear flow, and a non-dimensiona...

In this note, we prove local-in-time well-posedness for a fully dispersive Boussinesq system arising in the context of free surface water waves in two and three spatial dimensions. Those systems can be seen as a weak nonlocal dispersive perturbation of the shallow-water system. Our method of proof relies on energy estimates and a compactness argume...

The Brio system is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics (MHD). The system has the form \begin{align*} \partial_t u+\partial_x \Big({\textstyle \frac{u^2+v^2}{2}}\Big)=0,\\ \partial_t v+\partial_x \big(v(u-1)\big)=0. \end{align*} It was found in previous works that the standard theory o...

The KdV equation is a model equation for waves at the surface of an inviscid incompressible fluid, and it is well known that the equation describes the evolution of unidirectional waves of small amplitude and long wavelength fairly accurately if the waves fall into the Boussinesq regime. The KdV equation allows a balance of nonlinear steepening eff...

The paper describes the critical breaking waveheight for long surface water waves on a flow with constant vorticity in the KdV approximation. Given a background linear shear flow, a KdV equation can be found with coefficients depending on the strength of the shear flow. The derivation also shows that the velocity field under the wave can be constru...

Persistence of spatial analyticity is studied for periodic solutions of the dispersion-generalized KdV equation for . For a class of analytic initial data with a uniform radius of analyticity , we obtain an asymptotic lower bound on the uniform radius of analyticity at time , as , where . The proof relies on bilinear estimates in Bourgain spaces an...

Explicit parametric solutions are found for a nonlinear long-wave model describing steady surface waves propagating on an inviscid fluid of finite depth in the presence of a linear shear current. The exact solutions, along with an explicit parametric form of the pressure and streamfunction give a complete description of the shape of the free surfac...

The Whitham equation was proposed as an alternate model equation for the simplified description of unidirectional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitud...

Long waves in shallow water propagating over a background shear flow towards a sloping beach are being investigated. The classical shallow-water equations are extended to incorporate both a background shear flow and a linear beach profile, resulting in a non-reducible hyperbolic system. Nevertheless, it is shown how several changes of variables bas...

The two-dimensional motion of point vortices in an inviscid fluid with a free surface and an impenetrable bed is investigated. The work is based on forming a closed system of equations for surface variables and vortex positions using a variant of the Ablowitz, Fokas, and Musslimani formulation [M. J. Ablowitz, A. S. Fokas, and Z. H. Musslimani, J....

The shallow-water equations for two-dimensional hydrostatic flow over a bottom bathymetry b(x) areht+(uh)x=0,ut+(gh+u2/2+gb)x=0.
It is shown that the combination of discontinuous free-surface solutions and bottom step transitions naturally lead to singular solutions featuring Dirac delta distributions. These singular solutions feature a Rankine–Hug...

The so-called Kaup-Boussinesq system is a model for long waves propagating at the surface of a perfect fluid. In this work, a derivation of approximate local conservation equations associated to the Kaup-Boussinesq system is given. The derivation of the approximate balance laws is based on reconstruction of the velocity field and the pressure in th...

In nonlinear dispersive evolution equations, the competing effects of nonlinearity and dispersion make a number of interesting phenomena possible. In the current work, the focus is on the numerical approximation of traveling-wave solutions of such equations. We describe our efforts to write a dedicated Python code which is able to compute traveling...

The so-called Whitham equation arises in the modeling of free surface water waves, and combines a generic nonlinear quadratic term with the exact linear dispersion relation for gravity waves on the free surface of a fluid with finite depth. In this work, the effect of incorporating capillarity into the Whitham equation is in focus. The capillary Wh...

The object of this article is the comparison of numerical solutions of the so-called Whitham equation to numerical approximations of solutions of the full Euler free-surface water-wave problem. The Whitham equation ηt+32c0h0ηηx+Kh0∗ηx=0
was proposed by Whitham (1967) as an alternative to the KdV equation for the description of wave motion at the su...

Watershed segmentation is useful for a number of image segmentation problems with a wide range of practical applications. Traditionally, the tracking of the immersion front is done by applying a fast sorting algorithm. In this work, we explore a continuous approach based on a geometric description of the immersion front which gives rise to a partia...

The Serre-Green-Naghdi system is a coupled, fully nonlinear system of
dispersive evolution equations which approximates the full water wave problem.
The system is an extension of the well known shallow-water system to the
situation where the waves are long, but not so long that dispersive effects can
be neglected.
In the current work, the focus is...

It is shown how delta shock waves which consist of Dirac delta distributions and classical shocks can be used to construct non-monotone solutions of the Buckley-Leverett equation. These solutions are interpreted using a recent variational definition of delta shock waves in which the Rankine-Hugoniot deficit is explicitly accounted for [6]. The delt...

The time development of an interface separating two immiscible fluids of different densities in heterogeneous two-dimensional porous media is studied. The governing equations are simplified with the help of approximate Green’s functions which allow computation of the shape of the interface directly without resolving the fluid flow in the entire dom...

The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitu...

The Green-Naghdi system is used to model highly nonlinear weakly dispersive
waves propagating at the surface of a shallow layer of a perfect fluid. The
system has three associated conservation laws which describe the conservation
of mass, momentum, and energy due to the surface wave motion. In addition, the
system features a fourth conservation law...

The Korteweg-de Vries (KdV) equation is widely recognized as a simple model for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of fluid. While solutions of the KdV equation describe the shape of the free surface, information about the underlying fluid flow is encoded into the derivation of the equation, and the pr...

Convective mixing of dissolved carbon dioxide (CO2) with formation brine has been shown to be a significant factor for the rate of dissolution of CO2 and thus for determining the viability of geological CO2 storage sites. In most previous convection investigations, a no-flow boundary condition was used to represent the interface between an upper re...

The waveheight change in surface waves with a sufficiently slow variation in depth is examined. Using a new formulation of the energy flux associated to waves modeled by the Korteweg-de Vries equation, a system of three coupled equations is derived for the determination of the local wave properties as waves propagate over gently changing depth. The...

The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. An advantage of the Whitham equation over the KdV equation is that it provides a more faithful description of short waves of small amplitude. Recently, Ehrnström and Kalisch [19] establi...

Consideration is given to slow interfacial dynamics in a two-layer system of viscous fluids of comparable density. The fluid flow is governed by the two-dimensional Navier-Stokes equations where it is assumed that the inertial terms can be disregarded. A numerical methodology is presented which allows the study of the dynamics in the nonlinear regi...

This paper is focused on finding rules for waveheight change in a
solitary wave as it runs up a slowly increasing bottom. A coupled BBM
system is used to describe the solitary waves. Expressions for energy
density and energy flux associated with the BBM system are derived, and
the principle of energy conservation is used to develop an equation
rela...

We prove the existence of a global bifurcation branch of
$2\pi$-periodic, smooth, traveling-wave solutions of the Whitham
equation. It is shown that any subset of solutions in the global branch
contains a sequence which converges uniformly to some solution of
H\"older class $C^{\alpha}$, $\alpha < \frac{1}{2}$. Bifurcation
formulas are given, as we...

The effect of constant background vorticity on the pressure beneath steady long gravity waves at the surface of a fluid is investigated. Using an asymptotic expansion for the streamfunction, we derive a model equation and a formula for the pressure in a flow with constant vorticity. The model equation was previously found by Benjamin (1962), [3], a...

The regularized long-wave equation admits families of positive and negative solitary waves. Interactions of these waves are studied, and it is found that interactions of pairs of positive and pairs of negative solitary waves feature the same phase shift asymptotically as the wave velocities grow large as long as the same amplitude ratio is maintain...

This work presents a derivation of the energy density and energy flux of water waves modeled by
the so-called Kaup system, a variant of the Boussinesq system. The derivation is based on
reconstruction of the velocity field and the pressure in the fluid column below the free surface,
and is an extension of a method recently proposed by the authors....

The KdV equation arises in the framework of the Boussinesq scaling as a model equation for waves at the surface of an inviscid fluid. Encoded in the KdV model are relations that may be used to reconstruct the velocity field in the fluid below a given surface wave. In this paper, velocity fields associated to exact solutions of the KdV equation are...

In geological storage of carbon dioxide (CO2), the buoyant CO2 plume eventually accumulates under the caprock. Due to interfacial tension between the CO2 phase and the water phase, a capillary transition zone develops in the plume. This zone contains supercritical CO2 as well as water with dissolved CO2. Under the plume, a diffusive boundary layer...

In this article, consideration is given to weak bores in free-surface flows. The energy loss in the shallow-water theory for an undular bore is thought to be due to upstream oscillations that carry away the energy lost at the front of the bore. Using a higher-order dispersive model equation, this expectation is confirmed through a quantitative stud...

During geological storage of carbon dioxide (CO2), several mechanisms contribute to safe storage by immobilizing the CO2 in the injection formation. It has been shown that dissolution into resident brine can be one of the major contributors. The injected supercritical CO2 is buoyant, but dissolved CO2 increases brine density and therefore reduces t...

The method of weak asymptotics is used to find singular solutions of the shallow-water system which can contain Dirac-δ distributions (Espinosa & Omel'yanov, 2005). Complex-valued approximations which become real-valued in the distributional limit are shown to extend the range of possible singular solutions. It is shown, in this paper, how this app...

Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and i...

In geological storage of carbon dioxide (CO2 ), the buoyant CO2 plume eventually accumulates under the
caprock. Due to interfacial tension between the CO2 phase and the water phase, a capillary transition zone
develops in the plume. This zone contains supercritical CO2 as well as water with dissolved CO2 . Under
the plume, a diffusive boundary laye...

Two-dimensional inviscid channel flow of an incompressible fluid is considered. It is shown that if the flow is steady and features no horizontal stagnation, then the flow must necessarily be a parallel shear flow.

Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion that govern the evolution of the barycenter of the landslide mass include various dissipative effects due to bottom friction, internal energy dissipation, and viscous drag. The surface waves are studied in t...

Existence and admissibility of $\delta$-shock type solution is discussed for
the following nonconvex strictly hyperbolic system arising in studues of
plasmas: \pa_t u + \pa_x \big(\Sfrac{u^2+v^2}{2} \big) &=0 \pa_t v
+\pa_x(v(u-1))&=0. The system is fully nonlinear, i.e. it is nonlinear with
respect to both variables. The latter system does not adm...

A nonlinear dispersive model equation is used to study the onset of breaking in long waves behind the front of an undular bore. According to experiments conducted by Favre (1935) [1], weak bores have a smooth, but oscillatory structure, with undulations appearing behind the bore front. With increasing bore strength, the amplitude of these oscillati...