• Gray, J. M. N. T., and P. D. Killworth, 1995: Stability of the viscous-plastic sea ice rheology. J. Phys. Oceanogr.,25, 971–978.

  • Lefschetz, S., 1957: Differential Equations: Geometric Theory. Interscience, 364 pp.

  • Straughan, B., 1992: The Energy Method, Stability, and NonlinearConvection. Springer-Verlag, 242 pp.

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  • 1 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico
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Abstract

No abstract available.

Corresponding author address: Dr. John K. Dukowicz, TheoreticalDivision, Los Alamos National Laboratory, Group T-3, Mail StopB216, Los Alamos, NM 87545.

Abstract

No abstract available.

Corresponding author address: Dr. John K. Dukowicz, TheoreticalDivision, Los Alamos National Laboratory, Group T-3, Mail StopB216, Los Alamos, NM 87545.

Recently, Gray and Killworth (1995, henceforth GK)appeared to show that the viscous–plastic sea ice rheology based on an elliptical yield curve is linearly unstable in divergent flow. They recommended modifyingthe yield curve to avoid this apparent instability. However, as shown below, energy considerations demonstrate that this rheology is necessarily dissipative andtherefore always stabilizing.

For simplicity, following GK, we restrict the analysisto the one-dimensional case; a similar analysis holds inthe two-dimensional case. The momentum equationconsidered by GK for the elliptical yield curve is
i1520-0485-27-3-480-e1
where mi = rih ≥ 0 is the ice mass per unit area, ri isthe density of ice, h ≥ 0 is the ice thickness, α ≥ 0 isa frictional drag coefficient, P ≥ 0 is the ice strength,e is the eccentricity of the elliptical yield curve (typically, e ∼ 2), υ is the ice velocity, D = ∂υ/∂x is thedivergence, and d/dt = ∂/∂t + υ(∂/∂x) is the total timederivative. The continuity equation, rewritten in termsof mi, instead of h, and neglecting source terms, is
i1520-0485-27-3-480-e2
Multiplying (1) by υ, and combining with (2), we obtainan equation for the rate of change of the ice kineticenergy:
i1520-0485-27-3-480-e3
We now integrate this equation over a fixed region ofocean sufficiently large to contain all the ice. The secondterm on the left produces only boundary terms proportional to υmi, and the rheology term on the rhs may beintegrated by parts such that it contributes boundaryterms proportional to υP and a term from the interior.The region of integration is bounded either by landboundaries, where υ = 0, or by ice-free ocean, whereh = 0, and therefore mi = 0 and P = 0, because P isproportional to h. Should the region of integration bebounded by the ice edge, the correct boundary conditionis that the ice stress is zero. In any case, all boundaryterms vanish, and the result is
i1520-0485-27-3-480-e4
where we have moved the time derivative outside theintegral because the region of integration is fixed. Thisresult is sufficient to demonstrate the stability of thedynamics by Liapunov’s direct method (Lefschetz 1957)or the energy method (Straughan 1992). Each of theterms on the rhs is negative definite; in particular, therheology term is negative definite, and so the viscous–plastic rheology is always dissipative. The rheologyterm on the rhs of (1) will therefore always tend to drivethe total kinetic energy (a positive norm of the solution,i.e., a Liapunov functional) to zero, no matter what thevelocity field υ is. That is, the rheology term will alwaysdrive the dynamics toward an equilibrium such that υand/or D are zero (absent forcing). This consequenceimplies that this term will always contribute to stability,rather than detract from it, in the sense of Liapunov.The conclusion, in contradiction to the impression givenin GK, is that viscous–plastic rheology with an ellipticyield curve is always stabilizing.

The preceding, as in GK, is a continuum analysis. Itdemonstrates the usefulness of the dissipative nature ofthe viscous–plastic ice rheology. This property shouldbe preserved as much as possible in the discretization.A discretization typically involves a timewise linearization of the equations. The above considerations implythat the eigenvalues associated with the linear operatorderived from the rheology term would all be real andnegative, provided care is taken in the discretization.

Acknowledgments

This work was performed withsupport from the DOE CHAMMP program.

REFERENCES

  • Gray, J. M. N. T., and P. D. Killworth, 1995: Stability of the viscous-plastic sea ice rheology. J. Phys. Oceanogr.,25, 971–978.

  • Lefschetz, S., 1957: Differential Equations: Geometric Theory. Interscience, 364 pp.

  • Straughan, B., 1992: The Energy Method, Stability, and NonlinearConvection. Springer-Verlag, 242 pp.

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