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arxiv: 2604.23596 · v1 · submitted 2026-04-26 · 🧮 math.AP

Analysis and numerical simulations of a landfast ice model

Felix Brandt , Carolin Mehlmann This is my paper

Pith reviewed 2026-05-08 05:44 UTC · model grok-4.3

classification 🧮 math.AP
keywords landfast icesea ice dynamicswell-posednessviscous-plastic modelpartial differential equationsstationary equilibriatime-periodic solutionsnumerical simulation
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The pith

A model for landfast ice has local strong solutions, global solutions near constant equilibria without external forces, and time-periodic solutions, with numerics showing stationary states of vanishing velocity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies equations that describe sea ice fixed to coastlines and therefore nearly motionless. It proves that these equations admit unique local-in-time strong solutions from suitable starting data. When external forces are removed and the initial state is close to a uniform equilibrium, the solutions extend to all future times. The same equations also possess solutions that repeat after a fixed period. Numerical runs illustrate the ice velocity dropping to exactly zero and remaining there, matching the analytical guarantees of stability near equilibria.

Core claim

The system of nonlinear partial differential equations for landfast ice thickness, concentration, and velocity possesses strong local solutions for general initial data, strong global solutions when external forces vanish and data lies near constant equilibria, and time-periodic solutions; numerical experiments confirm the formation of stationary equilibria in which ice velocity is identically zero.

What carries the argument

The coupled system of nonlinear partial differential equations that evolves ice thickness, concentration, and velocity under a viscous-plastic constitutive law modified to allow vanishing motion.

If this is right

  • Strong solutions exist at least locally in time for initial data in appropriate Sobolev spaces.
  • Global-in-time strong solutions exist when external forces are absent and initial data is sufficiently close to a constant equilibrium.
  • Time-periodic solutions exist for the forced system.
  • Numerical solutions starting near equilibrium converge to stationary states with exactly zero velocity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The stability near equilibria could be used to construct numerical schemes that exactly preserve zero-velocity states once reached.
  • The same analytical approach may extend to other grounded-ice models whose rheology permits zero motion.
  • Periodic solutions suggest that coastal ice could exhibit repeating cycles under seasonally varying winds or currents.

Load-bearing premise

The ice's resistance to deformation and its interaction with the coast are captured exactly by the chosen viscous-plastic relations and forcing terms.

What would settle it

An explicit initial condition near a constant equilibrium, run without external forces, that develops large velocities or ceases to exist after finite time would contradict the global well-posedness and stability statements.

Figures

Figures reproduced from arXiv: 2604.23596 by Carolin Mehlmann, Felix Brandt.

Figure 1
Figure 1. Figure 1: Comparison of sea-ice concentration, thickness and velocity after 2 days of simulation using the standard viscous–plastic model (I, II) and its landfast ice extension (III, IV), both under dominant rightward wind forcing. While in the standard case (I, II), the ice drifts with the wind, the landfast ice extension (III, IV) leads to the formation of stationary sea-ice along the coast and an opening of sea-i… view at source ↗
Figure 2
Figure 2. Figure 2: Sea-ice evolution without external forcing. The kinetic energy is presented in Panel I. Panel II and Panel III show the sea-ice concentration after 6 and 12 simulated days respectively. In the absence of external forces the sea-ice velocity approaches zero over time, which results in a stable sea-ice cover over time. (compare Panels II and IV in view at source ↗
Figure 3
Figure 3. Figure 3: Sea-ice evolution under constant rightward wind forcing over 27 simulated days. Panels I–IV show the velocity field scaled by the sea-ice concentration. In Panel I, the scaling factor is 1, as the entire domain is initially ice-covered. Over time, the system approaches a stationary state, indicated by near-zero velocities in ice-covered regions (gray plateau in Panels III and IV), while non-zero velocities… view at source ↗
read the original abstract

In this manuscript, we consider a common modeling framework for Arctic landfast ice based on the work of Lemieux et al. [27], which is designed for use in large-scale climate models. This approach extends the classical viscous-plastic sea-ice model introduced by Hibler [18], which remains the most used model for simulating large-scale sea-ice dynamics in climate science. In particular, landfast ice refers to sea-ice that is attached to the coastline or grounded and therefore exhibits nearly vanishing motion. We present a rigorous analytical and numerical study of this landfast ice model. The main analytical contributions are the local strong well-posedness, the global strong well-posedness in the absence of external forces and for initial data close to constant equilibrium solutions, and the existence of time-periodic solutions. Complementing the analysis, we perform numerical simulations that illustrate key qualitative differences between landfast ice and classical viscous-plastic sea-ice models. In particular, the simulations reveal the formation of stationary equilibrium states characterized by vanishing ice velocity. These observations are consistent with the global-in-time existence result close to equilibria established in Theorem 4.1 as well as the time-periodic result in Theorem 5.2. The combined analytical and numerical results provide new insight into the structure, stability, and long-term behavior of landfast ice dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript analyzes a landfast ice model extending the Hibler viscous-plastic sea-ice model with landfast modifications from Lemieux et al. It establishes local strong well-posedness, global strong well-posedness without external forces for initial data near constant equilibria (Theorem 4.1), and existence of time-periodic solutions (Theorem 5.2). Numerical simulations demonstrate the formation of stationary equilibrium states with vanishing ice velocity, consistent with the theoretical results.

Significance. If the well-posedness theorems hold, the paper makes a valuable contribution by providing rigorous mathematical analysis for a model relevant to Arctic climate simulations. The combination of existence results for local, global, and periodic solutions, along with numerical illustrations of key physical behaviors like stationary states, offers new insights into the stability and long-term dynamics of landfast ice. This bridges applied mathematics and geophysical modeling effectively.

minor comments (2)
  1. [Numerical simulations] The numerical illustrations show qualitative agreement with the theorems but lack quantitative error metrics or convergence rates, which would help confirm the accuracy of the simulations.
  2. [Introduction] A more detailed comparison with existing mathematical analyses of the classical Hibler model would contextualize the novelty of the landfast extensions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, which accurately summarizes the local and global strong well-posedness results, the existence of time-periodic solutions, and the supporting numerical simulations for the landfast ice model. We appreciate the recognition of the work's relevance to Arctic climate modeling and the bridging of mathematical analysis with geophysical applications.

Circularity Check

0 steps flagged

No significant circularity detected in analytical claims

full rationale

The paper establishes local strong well-posedness, global strong well-posedness near equilibria without forcing, and existence of time-periodic solutions for the landfast ice extension of the Hibler viscous-plastic system. These are proved using standard fixed-point and continuation arguments for quasilinear parabolic-hyperbolic PDEs once the landfast modifications (vanishing velocity near coast/grounding) are incorporated into the constitutive law. The modeling framework is taken from the external citation to Lemieux et al. [27], but the existence theorems themselves are independent mathematical results with no reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. Numerical illustrations of equilibria are presented as consistent with the theorems rather than as substitutes for them. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard viscous-plastic rheology and landfast ice parameterizations from the cited references; no new free parameters, ad-hoc axioms, or invented physical entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard assumptions on initial data regularity and forcing terms required for strong solutions of the viscous-plastic sea-ice PDE system
    Invoked to obtain local and global well-posedness

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