Better Physics in Embedded Models: Iceberg arcing and Lake-surface profiles Aitbala Sargent, James L Fastook, Ted Scambos. University of Maine We thank the NSF, which has supported the development of this model over many years through several different grants. Aitbala Sargent, James L Fastook, Ted Scambos. University of Maine We thank the NSF, which has supported the development of this model over many years through several different grants.

EMBEDDED MODELS ● Better physics, 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. ● Better physics, 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.

Force Balance longitudinal drag (compression from up/down glacier) direction of flow basal drag lateral drag driving stress

Shallow Ice Approximation ● Only stress allowed is τ xz, the basal drag. ● 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. basal drag driving stress direction of flow

Barely Grounded Ice Shelf ● A modification of the Morland Equations for an ice shelf (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. direction of flow driving stress lateral drag longitudinal drag (compression from up/down glacier)

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). direction of flow basal drag lateral drag driving stress longitudinal drag (compression from up/down glacier)

Field Equations ● Conservation of momentum, ● Conservation of mass, ● Conservation of energy, and ● Constitutive relation. ● Conservation of momentum, ● Conservation of mass, ● Conservation of energy, and ● Constitutive relation.

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

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 Continuity Equation ● Conservation of mass, time-rate of change of thickness, gradient of flux, and local mass balance.

Work in Progress ● Sliding Boundary Conditions: – flat bed: – non-flat bed: ● Grid Resolution: – largest grid solved: ● 25x25x10=6,250 points ● 6,250x4 (u,p)=25,000 variables ● 25,000x81(bandwidth)=2,025,000 matrix size

2-D Applications ● Iceberg Edge Warping: – Toe-up and Toe-down Configurations. ● Lake Vostok – Trench and Trough

Iceberg Edge Warping Past models: toe-down profiles ● Neils Reeh, 1968 theoretical model ● James Fastook, 1984 numerical model Text

Iceberg Ponds, prelude to breakup

Berg Profiles

Berg Concepts

Simulation Results

Lake Vostok Radarsat image of the ice-sheet surface across subglacial Lake Vostok (© RADARSAT) Lake Vostok location map and survey area

● Image: M. Studinger

East-West Transect: Surface and Bed Elevation

Surface Evolution

Velocity Evolution

Velocities Magnitude

X and Y Velocities

Longitudinal Stresses and Strains ●

Vertical Stresses and Strains

Shear Stresses and Strains

Strain Invariant

3D Applications: A beginning

3D Application: A beginning

3D Application: Vertical Transect

THANK YOU