Better Physics in Embedded Ice Sheet ModelsBetter Physics in Embedded Ice Sheet Models

James L FastookAitbala 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 FastookAitbala Sargent

University of Maine

We thank the NSF, which has supported the development of this model over many years through several different grants.

EMBEDDED MODELSEMBEDDED 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 ApproximationShallow 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 ShelfBarely 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 EquationFull 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 NotationEinstein 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 EquationThe 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 EquationThe 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 EquationThe 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 EquationThe 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 EquationThe 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 EquationThe 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 EquationThe 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 EquationThe 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 EquationThe 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.

THANK YOUTHANK YOU