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Dynamics of motions and deformations of an arbitrary geometry flexural floe in ocean waves

Andrei Ludu

TL;DR

This work develops a comprehensive, linear-hydroelastic framework for sea-ice floes of arbitrary geometry and nonuniform thickness under ocean-wave forcing. By coupling potential-flow theory with a generalized Kirchhoff-Love plate model and a Green-function formulation, the authors decompose the flow into incident, scattered, rigid-motion, and elastic contributions and reduce the coupled problem to Fredholm integral equations of the second kind for surface densities. The approach yields a complete eigenbasis for elastic modes with added-mass corrections, explicit rigid-motion boundary conditions, and a unified treatment that applies to arbitrary floe shapes while capturing resonance, attenuation, and mode coupling. The resulting framework has direct implications for predicting wave attenuation in ice-covered seas, floe breakup dynamics, and climate-relevant wave-ice interactions, with clear avenues for incorporating viscosity, thickness evolution, and three-dimensional effects in future work.

Abstract

This paper develops a comprehensive mathematical framework for modeling the coupled hydroelastic dynamics of sea-ice floes of arbitrary shape and non-uniform thickness under linear ocean wave forcing. We simultaneously incorporate four dominant rigid-body motions (heave, surge, roll, pitch) and the complete spectrum of flexural deformation modes within a unified Green function formulation. The water flow is modeled using potential theory with Laplace's equation, while the floe obeys a generalized Kirchhoff-Love plate equation with spatially varying flexural rigidity. We formulate the coupled fluid-structure interaction problem through kinematic velocity-matching conditions and dynamic pressure-continuity conditions at the ice-water interface. The elastic eigenproblem with free-edge boundary conditions yields a complete orthogonal basis of deformation modes, accounting for added mass effects through modified natural frequencies. By decomposing the velocity potential into partial potentials associated with incident waves, scattered waves, rigid motions, and elastic modes, we reduce the problem to a system of Fredholm integral equations of the second kind for surface density functions on all boundary segments. The solution methodology employs single-layer potential representations with fundamental Green functions for Laplace's equation. We present explicit formulations for all boundary conditions in compact tensor form, provide asymptotic analysis for the spectrum of non-uniform thickness floes, and discuss resonance phenomena arising from the interaction between incident wave frequency and natural vibration modes.

Dynamics of motions and deformations of an arbitrary geometry flexural floe in ocean waves

TL;DR

This work develops a comprehensive, linear-hydroelastic framework for sea-ice floes of arbitrary geometry and nonuniform thickness under ocean-wave forcing. By coupling potential-flow theory with a generalized Kirchhoff-Love plate model and a Green-function formulation, the authors decompose the flow into incident, scattered, rigid-motion, and elastic contributions and reduce the coupled problem to Fredholm integral equations of the second kind for surface densities. The approach yields a complete eigenbasis for elastic modes with added-mass corrections, explicit rigid-motion boundary conditions, and a unified treatment that applies to arbitrary floe shapes while capturing resonance, attenuation, and mode coupling. The resulting framework has direct implications for predicting wave attenuation in ice-covered seas, floe breakup dynamics, and climate-relevant wave-ice interactions, with clear avenues for incorporating viscosity, thickness evolution, and three-dimensional effects in future work.

Abstract

This paper develops a comprehensive mathematical framework for modeling the coupled hydroelastic dynamics of sea-ice floes of arbitrary shape and non-uniform thickness under linear ocean wave forcing. We simultaneously incorporate four dominant rigid-body motions (heave, surge, roll, pitch) and the complete spectrum of flexural deformation modes within a unified Green function formulation. The water flow is modeled using potential theory with Laplace's equation, while the floe obeys a generalized Kirchhoff-Love plate equation with spatially varying flexural rigidity. We formulate the coupled fluid-structure interaction problem through kinematic velocity-matching conditions and dynamic pressure-continuity conditions at the ice-water interface. The elastic eigenproblem with free-edge boundary conditions yields a complete orthogonal basis of deformation modes, accounting for added mass effects through modified natural frequencies. By decomposing the velocity potential into partial potentials associated with incident waves, scattered waves, rigid motions, and elastic modes, we reduce the problem to a system of Fredholm integral equations of the second kind for surface density functions on all boundary segments. The solution methodology employs single-layer potential representations with fundamental Green functions for Laplace's equation. We present explicit formulations for all boundary conditions in compact tensor form, provide asymptotic analysis for the spectrum of non-uniform thickness floes, and discuss resonance phenomena arising from the interaction between incident wave frequency and natural vibration modes.

Paper Structure

This paper contains 16 sections, 76 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. The model equations
  3. Basic hypotheses
  4. Boundary conditions
  5. The elastic model for the floe
  6. Coupling the boundary conditions for the elastic deformation modes on the interface $S_I$
  7. Boundary conditions associated to rigid motions
  8. Heave mode kinematic boundary conditions, $k=1$
  9. Surge mode kinematic boundary conditions, $k=2$
  10. Roll mode kinematic boundary conditions, $k=3$
  11. Pitch mode kinematic boundary conditions, $k=4$
  12. Dynamical boundary conditions for rigid motions
  13. The general solution for the potential and the floe dynamics
  14. Conclusions
  15. Appendix A
  16. ...and 1 more sections

Figures (1)

  • Figure 1: A quasi-planar ice floe region $R_E$ of non-uniform thickness $d$ floating over the water domain $R_F$ which is bounded by the free water surface $S_{F}$, interface with the floe $S_I$, horizontal water bed $S_B$ at $z=h$ and auxiliary cylindrical surface $S_{\sigma}$.