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Better Physics in Embedded Ice Sheet Models

Better Physics in Embedded Ice Sheet Models

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Better Physics in Embedded Ice Sheet Models

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  1. James L Fastook Aitbala Sargent University of Maine We thank the NSF, which has supported the development of this model over many years through several different grants. Better Physics in Embedded Ice Sheet Models

  2. EMBEDDED MODELS • High-resolution, limited domain • runs inside • Low-resolution, larger domain model. • Modeling the whole ice sheet allows margins to be internally generated. • No need to specify flux or ice thickness along a boundary transecting an ice sheet. • Specification of appropriate Boundary Conditions for limited-domain model, based on spatial and temporal interpolations of larger-domain model.

  3. Shallow Ice Approximation • Only stress allowed is txz, the basal drag. • Assumed linear with depth. • Velocity profile integrated strain rate. • Quasi-2D, with Z integrated out. • 1 degree of freedom per node (3D temperatures). • Good for interior ice sheet and where longitudinal stresses can be neglected. • Probably not very good for ice streams.

  4. Barely Grounded Ice Shelf • A modification of the Morland Equations for an ice shelf pioneered by MacAyeal and Hulbe. • Quasi-2D model (X and Y, with Z integrated out). • 3 degrees of freedom (Ux, Uy, and h) vs 1 (h). • Addition of friction term violates assumptions of the Morland derivation. • Requires specification as to where ice stream occurs.

  5. Full Momentum Equation • No stresses are neglected. • True 3-D model. • Computationally intensive, with 3-D representation of the ice sheet, X and Y nodes as well as layers in the Z dimension. • 3 degrees of freedom per node (Ux, Uy, and Uz) as well as thickness in X and Y. (all three of these require 3-D temperature solutions).

  6. Einstein Notation • The convention is that any repeated subscript implies a summation over its appropriate range. • A comma implies partial differentiation with respect to the appropriate coordinate.

  7. The Full Momentum Equation • Conservation of Momentum: Balance of Forces • Flow Law, relating stress and strain rates. • Effective viscosity, a function of the strain invariant.

  8. The Full Momentum Equation • The strain invariant. • Strain rates and velocity gradients. • The differential equation from combining the conservation law and the flow law.

  9. The Full Momentum Equation • FEM converts differential equation to matrix equation. • Kmn as integral of strain rate term. • Shape functions as linear FEM interpolating functions.

  10. The Full Momentum Equation • Elimination of pressure degree of freedom by Penalty Method. • K'mn as integral of the pressure term. • Load vector, RHS, as integral of the body force term.

  11. The Heat Flow Equation • The strain-heating term, a product of stress and strain rates. • Time-dependent Conservation of energy. • The total derivative as partial and advection term.

  12. The Heat Flow Equation • Heat flow differential equation. • FEM matrix equation with time-step differencing.

  13. The Heat Flow Equation • Capacitance matrix integral for time-step differencing. • Stiffness matrix integral with diffusion term and advection term. • Load vector integral of internal heat sources.

  14. The Continuity Equation • Conservation of mass, time-rate of change of thickness, gradient of flux, and local mass balance. • FEM matrix equation with time-step differencing.

  15. The Continuity Equation • Capacitance matrix same as from heat flow. • Stiffness matrix as integral of the flux term. • Load vector as integral of mass balance.