better physics in embedded ice sheet models james l fastook aitbala sargent university of maine we...
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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.
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