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

Dynamics of motions and deformations of an arbitrary geometry flexural floe in ocean waves A. Ludu∗ Embry-Riddle Aeronautical University, Dept. of Mathematics & Wave Lab Daytona Beach, FL 32124 USA (Dated: December 15, 2025) 1 arXiv:2512.11008v1 [physics.flu-dyn] 11 Dec 2025 Abstract This paper develops a comprehensive mathematical framework for modeling the coupled hydroe- lastic 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 eigen- problem 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 methodol- ogy 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. I. INTRODUCTION The interaction between ocean waves and sea ice represents a fundamental problem in polar ocean dynamics, with significant implications for climate modeling, maritime operations, and understanding the rapidly changing Arctic and Antarctic environments. This report provides a comprehensive review of the mathematical-physical theoretical framework for modeling the dynamics of sea-ice floes under linear plane ocean waves, with particular emphasis on Green function methods. Sea-ice floes in the marginal ice zone exhibit complex hydroelastic behavior when subjected to ocean wave forcing. The floes simultaneously undergo rigid-body ∗ludua@erau.edu 2 motions (heave, surge, sway, roll, pitch, and yaw) and elastic deformations that can lead to fracture. Understanding these coupled dynamics is essential for predicting wave attenuation in ice-covered seas, ice breakup patterns, and the evolution of floe size distributions. This topic attracted much attention in the recent years. In water wave theory the most studied situation, both by analytical and numerical techniques, is the wave interaction with a single or with an array of floating of fixed bodies with axial symmetry. In the complex dynamics of floating systems, the main parameters to take into account are the nature of the wave field, and the geometry and material properties of the body. Clearly, in order to obtain a more realistic model one can work in a larger parameter spaces, including currents, bathymetry, wind interaction, thermodynamics of the sea, wave run-up during unavoidable over-topping events in rough seas, etc. The mathematical modeling of sea-ice floe dynamics under ocean wave forcing has evolved significantly over the past several decades, driven by advances in both computational meth- ods and analytical techniques for fluid-structure interaction problems. The foundational work on floating body dynamics in the naval architecture community, particularly the sem- inal contributions in [1, 2] and [3, 4], established the theoretical framework for analyzing rigid body motions of axisymmetric structures in waves. These early studies focused pri- marily on six degrees of freedom for rigid body motion—heave, surge, sway, roll, pitch, and yaw—treating the floating body as completely rigid and often restricting attention to bodies with circular horizontal cross-sections. The extension of these methods to ice floe modeling began with investigations of wave at- tenuation in ice-covered seas, where researchers recognized that elastic deformation of the floe could play a crucial role. In [5] the authors studied spectral evolution of waves in dispersed ice fields, while in [9] the authors provided a comprehensive review of wave-ice interaction phenomena, highlighting the hydroelastic nature of the problem. The Green function method emerged as a particularly powerful approach for these problems, as demon- strated by a series of influential papers [6–8, 10–14]. These works treated circular ice floes with uniform thickness, solving coupled problems invo

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