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A Model to Describe the Response of Arctic Sea Ice

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Mathematical Approach to Climate Change and its Impacts

Part of the book series: Springer INdAM Series ((SINDAMS,volume 38))

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Abstract

In this paper we develop a model for the flow of Arctic sea ice within the context of the theory of interacting continua that takes into account the change of phase between the two constituents, ice and water. After documenting the general balance laws for mass, linear and angular momentum, energy, the second law of thermodynamics and the volume additivity constraint, we discuss the specific constitutive relations that are to be used for the various quantities that appear in the balance laws. Ice is modeled as a non-Newtonian fluid that is a generalization of the usual model due to Glen to take into account the ability of ice to develop normal stress differences in simple shear flow, while water is modeled as a Navier–Stokes fluid. Constitutive relations are discussed for the change of phase, the interaction forces such as the drag, etc. In order to make the problem amenable to analysis, we simplify the governing equations, keeping the quintessential features of the problem of interest in mind.

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Notes

  1. 1.

    Here, homogenization does not refer to any specific mathematical process of mathematical homogenization procedure but refers to the sense in which we think of a body as a single continuum though clearly at the sub-atomic level the body is really not a three dimensional continuum. Each constituent of the mixture is assumed to be present at each point in the region occupied by the mixture, we do not have regions where we have only one constituent and not the others.

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Acknowledgement

KRR thanks the Office of Naval Research for support of this work.

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Authors and Affiliations

  1. Department of Mathematics, US Naval Academy, Annapolis, MD, USA

    Reza Malek-Madani

  2. Department of Mechanical Engineering, Texas A&M University, College Station, TX, USA

    Kumbakonam R. Rajagopal

Authors
  1. Reza Malek-Madani
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  2. Kumbakonam R. Rajagopal
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Correspondence to Reza Malek-Madani .

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Editors and Affiliations

  1. Department of Mathematics, University of Rome "Tor Vergata", Rome, Italy

    Piermarco Cannarsa

  2. Istituto per le Applicazioni del Calcolo ‘M. Picone’, CNR, Rome, Italy

    Daniela Mansutti

  3. Institute of Geosciences & Earth Resources (IGG), National Research Council (CNR), Pisa, Italy

    Antonello Provenzale

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Malek-Madani, R., Rajagopal, K.R. (2020). A Model to Describe the Response of Arctic Sea Ice. In: Cannarsa, P., Mansutti, D., Provenzale, A. (eds) Mathematical Approach to Climate Change and its Impacts. Springer INdAM Series, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-030-38669-6_5

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