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Research
 
Introduction
The Globe is warming at an alarming rate causing many changes on Earth, including the melting of Earth’s polar caps. If the ice sheets blanketing Greenland and Antarctica were to disappear it will cause the sea level to rise by about 70 meters (nearly 230 feet). This rise will not only cause Memphis, TN to be a seaport and New England to be an island, but it will effect the world as a whole. In order to prepare for these changes scientists are attempting to calculate how ice sheets evolve. This has been done through various approaches, one of which will be addressed here. 
 
Research
A simple first order ice sheet model was derived from simple geometrical consideration of a force balance between gravitational forcing and resistive stress due to basal coupling of the ice sheet frozen to its bed.
The model is used to provide a first order understanding of the thickness of an ice sheet as it spreads across a continental land mass, progressing from Sheet Flow to Stream Flow to Shelf Flow
Sheet Flow: a grounded ice sheet with 100% of its basal surface frozen to its bed
Stream Flow: a mass of ice, flowing toward the sea whose basal surface is partially afloat and partially frozen to its bed.
Shelf Flow: a large floating area of Ice.
Ice Sheet thickness progresses from its thickest state as a grounded ice sheet to its thinnest where it is 100% afloat at its grounding line boundary.
Once afloat the ice shelf continues to thin due to the force of gravity and difference in density between ice and water.
 
Byrd Glacier Data and Model Results

The top graph shows the results of the RADAR data providing ice sheet thickness.
The lower graph shows the results of a model interpretation of the data with regard to the stream’s floating fraction (Pw/PI): a 2nd-3rd order approximation of the floating fraction Phi.

 
Hypothesis
The natural domain of ice lies between air and water due to its intermediate density.
The natural tendency of grounded ice is to seek sea level
A first order approach should provide a reasonable description of the ice sheet & stream thickness as it flows toward its ice shelf grounding line.
Continental ice proceeds from totally grounded sheet flow to partially floating stream flow to totally floating ice flow behavior based on the relative coupling of the ice to its bed
Increases in sea level reduce the coupling and lead to faster stream flow and thinner streams.
The first order model captures the variability of bed coupling through the parameter “Phi” defined as the floating fraction of the ice stream
Phi = 0 for totally grounded ice
Phi = 1 for totally floating ice
The derived first order value for Phi= ho/hI

 
First Order Model
Derived from force balance between gravity (weight of the ice sheet) and basal coupling of the ice to its bed.
The ice sheet frozen to its bed assumes a parabolic shape.
Ice Sheets on land terminate as ice lobes
Partially submerged ice sheets become ice streams as bed coupling diminishes
Ice streams terminate as floating ice shelves
 
The Floating Fraction Phi
As a first order approximation, thickness of the static (force balanced) ice stream must be greater than ho (the thickness at the grounding line) The ratio ho/hI should be proportional to the floating fraction Phi.
Higher order models of the thickness, include side shear stress, mass balance and compression and tension forces.
 
Variation of Phi By Model
Blue curve is 1st Order F derived from RADAR ice thickness.
Pink shows Higher Order Model Phi (Phi=Pw/PI) from RADAR data.
Green is the First Order Phi = ho/hI varying as cosine squared through 3/8 of a period
 
Ice Thickness By Model
Blue curve is ice measured thickness.
Pink shows ice thickness derived from the Higher Order Model Phi (Phi=Pw/PI)
Green is the thickness of ice derived from First Order Phi varying as cos2(3•p•x/(2•L)) for 3/8 of a period