Special Session 150: Water Waves and Beyond

A fluid-solid interaction problem in porous media

Diego Alonso Oran Universidad de La Laguna Spain Co-Author(s):    Rafael Granero-Belinchon

Abstract: In this work, we derive asymptotic interface models for an elastic Muskat free boundary problem describing Darcy flow beneath an elastic membrane. In a weakly nonlinear regime of small interface steepness, we obtain nonlocal evolution equations that capture the free-boundary dynamics up to quadratic order. In the long-wave thin-film regime, we rewrite the kinematic condition in flux form, flatten the moving domain, and derive a lubrication-type equation. Moreover, we establish well-posedness for these models in suitable Wiener spaces.

Stability of Hydroelastic Waves in Deep Water

Mark Blyth University of East Anglia England Co-Author(s):    Emilian Parau

Abstract: In this talk we discuss two-dimensional periodic travelling hydroelastic waves on water of infinite depth. We track a bifurcation branch that connects small amplitude periodic waves to a large amplitude state in which the wave is motionless at rest and the fluid is static. The stability of the periodic waves on this branch is computed using a surface-variable formulation, computing growth rates numerically via Floquet theory. The stability spectrum including both superharmonic and subharmonic perturbations will be presented. For superharmonic perturbations the onset of instability via a Tanaka-type collision of eigenvalues at zero is identified. We shed light on the structure of the spectrum as wave amplitude increases and reveal a highly intricate structure.

Symmetric periodic and rotational three-dimensional waves in water of infinite depth

Stefano Boehmer Lund University Sweden Co-Author(s):

Abstract: We prove existence of small, symmetric, doubly periodic and rotational gravity-capillary waves in water of infinite depth in three space dimensions. Following Lortz` ansatz from magnetohydrodynamics [1], we write the vorticity as the cross-product of two gradients and we assume the Bernoulli function to depend on the orbital period of the particles. This gives rise to a coupled elliptic-hyperbolic formulation which we cast as a fixed-point problem. Here we follow the strategy in [2], where the case of finite depth is treated. In our case the domain is not compact, which requires a more careful treatment of both the elliptic and the hyperbolic parts. Finally, we prove existence of small-amplitude solutions to the free-boundary problem with small vorticity using local bifurcation theory. This is ongoing work. [1] D. Lortz, \Uber die Existenz toroidaler magnetohydrostatischer Gleichgewichte ohne Rotationstransformation, Journal of Applied Mathematics and Physics (ZAMP), 1970 [2] D. S. Seth, K. Varholm, and E. Wahl\`en, Symmetric doubly periodic gravity-capillary waves with small vorticity, Advances in Mathematics, 2024

A variational construction of solitary waves for the Babenko equation on finite depth

Mats Ehrnstrom NTNU Norwegian University of Science and Technology Norway Co-Author(s):    Tomas Dohnal, Halle University, and Johanna Marstrander, Norwegian University of Sciences and Technology

Abstract: We construct solitary waves for the two-dimensional steady water wave problem directly from the Babenko formulation in the case of pure gravity waves over finite depth. The proof is straight-forward in the sense that it is built on constrained minimization of a scalar nonlinear and nonlocal equation as in Weinstein (1987), and a limiting procedure from periodic waves as Ehrnstr\om-Groves-Wahl\`en (2012), using concentration--compactness. The novelty lies not the least in the nonlocal cubic terms appearing in the Babenko variational formulation, which carry negative order, and make nonlinear harmonic analysis an important part of the proof. The constructed smooth and localised solutions of the Babenko equation correspond to smooth and localised free-surface waves in the two-dimensional steady water wave problem over a flat bed with gravity. We show that the Babenko solitary waves are approximated by scalings of the classical Korteweg--De Vries solitary waves.

Ocean currents with constant vorticity

Anna Geyer TU Delft Netherlands Co-Author(s):    Ronald Quirchmayr, Calin Martin

Abstract: In this talk I will present two recent results on time-dependent, three-dimensional water flows with constant non-vanishing vorticity. In the first part, I will discuss a rigidity result for flows with continuously depth dependent density. In the second part, I will consider the f-plane approximation for off-equatorial oceanic flows and show that these flows are necessarily steady, zonal and fully explicit, with a free surface structure that is parabolic in the latitudinal coordinate. I will also provide an application to the Antarctic Circumpolar Current (ACC) for which I compare the sea surface height predicted by our constant vorticity model with satellite altimetry measurements available in the literature.

Global bifurcation of doubly periodic water waves in Beltrami flows

Bastian Hilder Technical University of Munich Germany Co-Author(s):    Giang To (Lund University); Erik Wahl\`{e}n (Lund University)

Abstract: In this talk, I will present a global bifurcation result for doubly periodic gravity-capillary water waves with vorticity. Specifically, I consider the class of Beltrami flows, in which the velocity and vorticity fields are collinear. This structure allows us to reformulate the governing equations as a perturbation of the identity by a compact operator, enabling the application of analytic global bifurcation theory to obtain solutions along a global continuum emerging from the laminar flow. The main challenge is the presence of a two-dimensional kernel at the bifurcation point, and I will explain how this can be handled by treating the parameterisation of the local bifurcation curve as a new bifurcation parameter. This is joint work with Giang To and Erik Wahl\`{e}n (both Lund University).

Think Global, Act Local: Inducing Fully Localised Planar Patterns via Spatial Heterogeneity

Dan J Hill University of Oxford England Co-Author(s):    David J.B. Lloyd, Matthew R. Turner

Abstract: The existence of localised two-dimensional patterns has been observed and studied in numerous experiments and simulations: ranging from optical solitons, to patches of desert vegetation, to fluid convection. And yet, our mathematical understanding of these emerging structures remains extremely limited beyond one-dimensional examples. In this talk I will discuss how adding a compact region of spatial heterogeneity to a PDE model can not only induce the emergence of fully localised 2D patterns, but also allows us to rigorously prove and characterise their bifurcation. The idea is inspired by experimental and numerical studies of magnetic fluids, where our compact heterogeneity corresponds to a local spike in the magnetic field of the experiment. In particular, we obtain local bifurcation results for fully localised patterns both with and without radial or dihedral symmetry, and rigorously continue these solutions to large amplitude. Notably, the initial bifurcating solution (which can be stable at bifurcation) varies between a radially-symmetric spot and a `dipole` solution as the width of the spatial heterogeneity increases.

The convergence of time delayed discrete model of traffic flow

Wei Lian NTNU Norway Co-Author(s):    Wei Lian

Abstract: The problem of convergence of particle models to continuum models is fundamental. In the context of traffic flow, the rigorous proof of such convergence has been established recently for the Lighthill-Whitham-Ricchards model. In traffic flow studies, it seems natural to introduce the effect of delay, accounting for drivers` reaction time. We consider a delayed version of Follow-the-Leader model, and aims to show the continuum limit. In this talk, we review the state of the art and present technical difficulties raised by time delay in the microscopic model, and some partial results.

Slope bounds for water waves

Evgeniy Lokharu Lund University Sweden Co-Author(s):

Abstract: In this talk we will discuss some recent results on slope bounds for water waves.

Global bifurcation of hollow vortex streets

Vasileios Oikonomou University of Missouri USA Co-Author(s):    Samuel Walsh

Abstract: In this talk, we will present new results on the existence of periodic configurations of hollow vortices. A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a fluid, whose flow is governed by the 2D Euler equations. We will show that there exists a one-parameter family of steady, periodic hollow vortices that bifurcates from a periodic point vortex configuration and extends up to the onset of certain singularities, which we also classify. Some notable applications are the existence of von K\'{a}rm\'{a}n, and 2P, hollow vortex streets. The technique involves reformulating the problem using complex analysis tools, including conformal mappings and layer potential representations, then carrying out a global bifurcation argument.

Hydroelastic Waves Under Ice Covers: From Moving Loads to Variable Bathymetry

Emilian Parau University of East Anglia England Co-Author(s):    Claudia Tugulan, Olga Trichtchenko

Abstract: Waves propagating at the surface of fluids covered by ice are considered. The waves can be generated by moving loads. The ice is modelled as an elastic or viscoelastic plate. The water is of finite depth and can have variable bathymetry. Some theoretical, numerical and experimental steady and time-dependent result in two and three-dimensions will be presented.

Instability of the halocline at the North Pole

Christian Puntini University of. Vienna Austria Co-Author(s):    Christian Puntini

Abstract: In this paper talk we discuss the instability for the near-inertial Pollard waves, as a model for the halocline in the region of the Arctic Ocean centered around the North Pole, derived in Puntini \emph{Differential and Integral Equations} (2026). Adopting the short-wavelength instability approach, the stability of such flows reduces to study the stability of a system of ODEs along fluid trajectories, leading to the result that, when the steepness of the near-inertial Pollard waves exceeds a specific threshold, those waves are linearly unstable. The explicit dispersion relation of the model allows to easily compute such threshold, knowing the physical properties of the water column.

On large-scale oceanic wind-drift currents

Luigi Roberti Leibniz Universit\\"{a}t Hannover Germany Co-Author(s):    Christian Puntini; Eduard Stefanescu

Abstract: Starting from the Navier--Stokes equations in rotating spherical coordinates with constant density and eddy viscosity varying only with depth, and appropriate, physically motivated boundary conditions, we derive an asymptotic model for the description of non-equatorial wind-generated oceanic drift currents. We do not invoke any tangent-plane approximations, thus allowing for large-scale flows that would not be captured by the classical f-plane approach. The strategy is to identify two small intrinsic scales for the flow (namely, the ratio between the depth of the Ekman layer and the Earth`s radius, and the Rossby number) and, after a careful scaling, perform a double asymptotic expansion with respect to these small parameters. This leads to a system of linear ordinary differential equations with nonlinear boundary conditions for the leading-order dynamics. First, we establish the existence and uniqueness of the solution to the leading-order equations and show that the solution behaves like a classical Ekman spiral for any eddy viscosity profile. Then, we discuss several cases of explicit eddy viscosity profiles (constant, linearly decreasing, linearly increasing, piecewise linear, and exponentially decaying) and compute the surface deflection angle of the wind-drift current. We obtain results that are remarkably consistent with observations.

Stratified Large-Amplitude Steady Periodic Water Waves with General Density

Giang To Lund University Sweden Co-Author(s):    Giang To

Abstract: By means of a conformal mapping and bifurcation theory, we prove the existence of large-amplitude steady stratified periodic water waves, with general density function, which may have critical layers and overhanging profiles. A nodal analysis is carried out for these solutions under certain assumption involving the Bernoulli function and the density.

Hollow desingularization of vortices

Kristoffer Varholm University of Pittsburgh USA Co-Author(s):    Robin Ming Chen, Samuel Walsh, Miles H. Wheeler

Abstract: It is well-known that there exist a number of steady solutions with point vortices to the Euler equations. Both configurations in the plane, and embedded in water waves. In recent years there has been a program to desingularize these steady solutions, by expanding the point vortices into hollow vortices. These are essentially spinning air bubbles. In this talk I will give an overview of this effort, including work in progress on hollow vortex rings.