Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Singular Limit in Hibler's Sea Ice Model

Robert Denk, Franz Gmeineder, Matthias Hieber

TL;DR

The paper addresses the singular limit of Hibler's sea-ice momentum balance by formulating an energy-driven, BD-based notion of solution that accommodates plasticity without stress regularization. It introduces relaxed Hibler energies and proves their lower semicontinuity, together with a bulk approximation mechanism for boundary terms, enabling a robust variational existence theory. The main result guarantees the existence of energy-driven variational solutions on a fixed time interval, derived as limits of regularized problems and attaining the initial data in $L^{2}$. The framework accommodates measure-valued stresses and extends to a broader class of $oldsymbol{C}$-elliptic operators, indicating potential applicability beyond Hibler’s model. Collectively, these contributions provide a rigorous, variationally grounded approach to singular Hibler stresses and lay groundwork for numerical schemes and non-constant-mass generalizations via a weak-variational extension.

Abstract

We establish the existence of energy-driven solutions to the momentum balance equation in Hibler's sea ice model. As a main novelty and different from previous results, we deal with the singular limit and therefore cover the true unregularized Hibler stress. To this end, we introduce an energy-based notion of solution that is able to capture plasticity effects of sea ice. This requires certain relaxations of the Hibler energies and, by the different function space set-up, comes with novel challenges. In particular, we establish a bulk approximation result of the boundary terms in the evolutionary relaxed Hibler energies. This is achieved by developing a novel reduction scheme for nonlinear trace expressions which should be of independent interest. Finally, based on our main results, we classify our findings within a broader concept of solutions that is applicable to the non-constant mass case too.

On the Singular Limit in Hibler's Sea Ice Model

TL;DR

The paper addresses the singular limit of Hibler's sea-ice momentum balance by formulating an energy-driven, BD-based notion of solution that accommodates plasticity without stress regularization. It introduces relaxed Hibler energies and proves their lower semicontinuity, together with a bulk approximation mechanism for boundary terms, enabling a robust variational existence theory. The main result guarantees the existence of energy-driven variational solutions on a fixed time interval, derived as limits of regularized problems and attaining the initial data in . The framework accommodates measure-valued stresses and extends to a broader class of -elliptic operators, indicating potential applicability beyond Hibler’s model. Collectively, these contributions provide a rigorous, variationally grounded approach to singular Hibler stresses and lay groundwork for numerical schemes and non-constant-mass generalizations via a weak-variational extension.

Abstract

We establish the existence of energy-driven solutions to the momentum balance equation in Hibler's sea ice model. As a main novelty and different from previous results, we deal with the singular limit and therefore cover the true unregularized Hibler stress. To this end, we introduce an energy-based notion of solution that is able to capture plasticity effects of sea ice. This requires certain relaxations of the Hibler energies and, by the different function space set-up, comes with novel challenges. In particular, we establish a bulk approximation result of the boundary terms in the evolutionary relaxed Hibler energies. This is achieved by developing a novel reduction scheme for nonlinear trace expressions which should be of independent interest. Finally, based on our main results, we classify our findings within a broader concept of solutions that is applicable to the non-constant mass case too.

Paper Structure

This paper contains 27 sections, 31 theorems, 272 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hibler's model
  3. Main results, notions of solutions and outline of the paper
  4. On the momentum balance equation and modelling of stresses
  5. Hibler deformations, stresses and energies
  6. The momentum balance equation
  7. The true Hibler stress and plasticity effects
  8. Strategy and selected obstructions
  9. Function spaces and auxiliary results
  10. General notation
  11. Functions of bounded deformation
  12. Integrands of linear growth
  13. The coarea formula
  14. Monotone operators and compactness
  15. Relaxations and the bulk approximation of boundary terms
  16. ...and 12 more sections

Key Result

Lemma 3 .2

Let $\Omega\subset\mathbb R^{n}$ be open and bounded with Lipschitz boundary $\partial\Omega$. Then the following hold:

Figures (2)

  • Figure 1: Two variants of stress-strain relations based on the Hibler deformation tensor $\mathbb{T}\mathbf{u}$, see \ref{['eq:hiblerwritedown']}. The light blue line corresponds to the standard Hibler stress, whereas the dashed blue line corresponds to a modified Hibler stress à la Mohr-Coulomb.
  • Figure 2: The construction of the spatial cut-offs in the bulk-approximation of boundary terms.

Theorems & Definitions (71)

  • Remark 3 .1
  • Lemma 3 .2: Weak*-compactness and smooth approximation
  • Lemma 3 .3: Poincaré-Sobolev
  • Lemma 3 .4: Trace operator and extensions
  • Remark 3 .5: On the right-inverse of the trace operator
  • Lemma 3 .6: of Poincaré-type
  • Remark 3 .7
  • Lemma 3 .8
  • proof
  • Lemma 3 .9: Reshetnyak's lower semicontinuity theorem, Reshetnyak
  • ...and 61 more