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On Well-Posedness of the Viscous–Plastic Sea Ice Model

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  • 1 Sandia National Laboratories,* Albuquerque, New Mexico
  • | 2 University of New Mexico, Albuquerque, New Mexico
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Abstract

This study examines the well-posedness of the initial-value problems that arise in common models of sea ice. The model equations describe the balance of linear momentum combined with simplified thermodynamics represented by two continuity equations for effective ice thickness and ice concentration. The constitutive model for sea ice is given by two possible variants of the viscous–plastic model: the viscous–plastic model with pressure replacement and the viscous–plastic model with pressure replacement plus a tensile cutoff. The authors identify regimes of well- and ill-posedness for both models in one and two space dimensions. In one space dimension, the study finds that the viscous–plastic model and viscous–plastic model with pressure replacement behave similarly: there is ill-posedness when the divergent flow rate is larger than a minimum value. On the other hand, the viscous–plastic model with pressure replacement plus a tensile cutoff is ill-posed for all divergent flows. In two space dimensions the analysis is inconclusive for the viscous–plastic model with pressure replacement, but with the tensile cutoff the problem is ill-posed for certain divergent flows. The authors also discuss energy bounds and the difference between ill-posedness and stability of a solution. The study shows by examples that boundedness of solutions does not imply well-posedness and that it is possible for well-posed problems to have unstable solutions. The analysis shows that previous arguments in the literature, which state that a bound on the energy in sea ice models provides control over ill-posedness, are flawed.

Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under Contract DE-AC04-94AL85000.

Corresponding author address: Deborah Sulsky, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131. E-mail: sulsky@math.unm.edu

Abstract

This study examines the well-posedness of the initial-value problems that arise in common models of sea ice. The model equations describe the balance of linear momentum combined with simplified thermodynamics represented by two continuity equations for effective ice thickness and ice concentration. The constitutive model for sea ice is given by two possible variants of the viscous–plastic model: the viscous–plastic model with pressure replacement and the viscous–plastic model with pressure replacement plus a tensile cutoff. The authors identify regimes of well- and ill-posedness for both models in one and two space dimensions. In one space dimension, the study finds that the viscous–plastic model and viscous–plastic model with pressure replacement behave similarly: there is ill-posedness when the divergent flow rate is larger than a minimum value. On the other hand, the viscous–plastic model with pressure replacement plus a tensile cutoff is ill-posed for all divergent flows. In two space dimensions the analysis is inconclusive for the viscous–plastic model with pressure replacement, but with the tensile cutoff the problem is ill-posed for certain divergent flows. The authors also discuss energy bounds and the difference between ill-posedness and stability of a solution. The study shows by examples that boundedness of solutions does not imply well-posedness and that it is possible for well-posed problems to have unstable solutions. The analysis shows that previous arguments in the literature, which state that a bound on the energy in sea ice models provides control over ill-posedness, are flawed.

Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under Contract DE-AC04-94AL85000.

Corresponding author address: Deborah Sulsky, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131. E-mail: sulsky@math.unm.edu
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