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Particle, kinetic and hydrodynamic models for sea ice floes. Part II: Rotating floes with nonlinear contact forces

Quanling Deng, Seung-Yeal Ha, Jaemoon Lee

Abstract

This paper extends the multiscale modeling framework introduced in Part I (Deng and Ha, Physica D: Nonlinear Phenomena 483 (2025) 134951) for sea-ice floe dynamics with non-rotating floes to the case with rotational floes and nonlinear contact interactions. Building on the particle-kinetic-hydrodynamic hierarchy developed for non-rotating floes, we generalize the particle model to describe ice floes as rigid bodies characterized by position, linear velocity, angular velocity, size, and moment of inertia. The interaction rules now include nonlinear contact forces and torques arising from short-range compression, restitution, and tangential friction laws, together with hydrodynamic drag that couples translational and rotational motions. These particle descriptions lead to an enriched Vlasov-type kinetic equation posed on an extended phase space, whose moments yield a hydrodynamic system for mass, momentum, and angular-momentum balances. Compared with Part I, the resulting macroscopic equations feature additional stress contributions, rotational transport, and dissipative mechanisms stemming from nonlinear collisions. The proposed framework provides a more realistic description of sea-ice floe dynamics and offers a systematic pathway toward multiscale modeling of sea-ice rheology under complex environmental forcing.

Particle, kinetic and hydrodynamic models for sea ice floes. Part II: Rotating floes with nonlinear contact forces

Abstract

This paper extends the multiscale modeling framework introduced in Part I (Deng and Ha, Physica D: Nonlinear Phenomena 483 (2025) 134951) for sea-ice floe dynamics with non-rotating floes to the case with rotational floes and nonlinear contact interactions. Building on the particle-kinetic-hydrodynamic hierarchy developed for non-rotating floes, we generalize the particle model to describe ice floes as rigid bodies characterized by position, linear velocity, angular velocity, size, and moment of inertia. The interaction rules now include nonlinear contact forces and torques arising from short-range compression, restitution, and tangential friction laws, together with hydrodynamic drag that couples translational and rotational motions. These particle descriptions lead to an enriched Vlasov-type kinetic equation posed on an extended phase space, whose moments yield a hydrodynamic system for mass, momentum, and angular-momentum balances. Compared with Part I, the resulting macroscopic equations feature additional stress contributions, rotational transport, and dissipative mechanisms stemming from nonlinear collisions. The proposed framework provides a more realistic description of sea-ice floe dynamics and offers a systematic pathway toward multiscale modeling of sea-ice rheology under complex environmental forcing.
Paper Structure (18 sections, 10 theorems, 180 equations, 4 figures)

This paper contains 18 sections, 10 theorems, 180 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Particle description for ice floes
  3. The particle model
  4. Asymptotic behavior
  5. From particle to kinetic description
  6. Kinetic model for ice floe dynamics
  7. Macroscopic behavior of the kinetic model
  8. From kinetic to hydrodynamic description
  9. Hydrodynamic balance laws
  10. Total energy for the closed system
  11. Numerical simulations
  12. Example 1: Floes under constant ocean forcing
  13. Example 2: Consistency test between the particle and hydrodynamic models
  14. Hydrodynamic model configuration and discretization.
  15. Particle-to-continuum observables and comparison.
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $(\boldsymbol{x}^i, \boldsymbol{v}^i, \theta^i, \omega^i)$ be a global solution to the system eq:dem. Then, the following assertions hold.

Figures (4)

  • Figure 1: Example 1 floe trajectories. Arrows represent floe velocities and colors represent floe angular velocity. Ocean velocity $\boldsymbol{u}_o=(0.3, 0)^T$
  • Figure 2: Example 1 floe moments. Time evolution of mean velocities, velocity differences, and angular velocities, total momentum, total angular momentum, and kinetic and normal contact strain energies.
  • Figure 3: Model comparison at $T=1$. Ocean velocity $\boldsymbol{u}_o=(-ys, xs)^T$ with $s= (x^2+y^2-4)e^{(-(x^2+y^2)(x^2 + y^2 -8)/8)}/32$.
  • Figure 4: Model comparison at $T=10$. Ocean velocity $\boldsymbol{u}_o=(-ys, xs)^T$ with $s= (x^2+y^2-4)e^{(-(x^2+y^2)(x^2 + y^2 -8)/8)}/32$.

Theorems & Definitions (22)

  • Lemma 2.1: Total momentum balances
  • proof
  • Remark 2.2
  • Lemma 2.3: Total energy balance
  • proof
  • Lemma 2.4: Total energy lower bound
  • proof
  • Remark 2.5
  • Lemma 2.6: Barbalat's Lemma
  • Theorem 2.7
  • ...and 12 more