Better Physics in Embedded Ice Sheet Models 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. 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.

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. ● 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.

Shallow Ice Approximation ● Only stress allowed is  xz, 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. ● Only stress allowed is  xz, 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.

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. ● 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.

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). ● 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).

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. ● 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.

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. ● Conservation of Momentum: Balance of Forces ● Flow Law, relating stress and strain rates. ● Effective viscosity, a function of the strain invariant.

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

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. ● FEM converts differential equation to matrix equation. ● Kmn as integral of strain rate term. ● Shape functions as linear FEM interpolating functions.

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. ● 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.

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. ● The strain-heating term, a product of stress and strain rates. ● Time-dependent Conservation of energy. ● The total derivative as partial and advection term.

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

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. ● Capacitance matrix integral for time-step differencing. ● Stiffness matrix integral with diffusion term and advection term. ● Load vector integral of internal heat sources.

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. ● Conservation of mass, time-rate of change of thickness, gradient of flux, and local mass balance. ● FEM matrix equation with time-step differencing.

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. ● Capacitance matrix same as from heat flow. ● Stiffness matrix as integral of the flux term. ● Load vector as integral of mass balance.

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