Earth and Space Sciences 533 – Atmospheric Sciences 512
Dynamics of Snow and Ice Masses
Mid-term Study Questions 2016
This is a closed-book test, with a time limit of 80 minutes. You are welcome to use your calculator.
Please answer all 3 questions. Each question has equal value.
Please time yourself
Start time: __________________
Finish time: __________________
There are 6 questions in this study guide. Three of them will be chosen for the mid-term. I will
give you the 3 questions in a sealed envelope in class on Wednesday May 4. Please take 80
minutes at your convenience to answer the 3 questions. Turn in your answers by 5 PM on
Monday May 9.
(1) Mechanics of Glacier Ice, Snow, and Sea Ice
Describe in prose how the mechanical properties of glaciers, snow, and sea ice are related, and how
they are different. You will probably want to discuss controlling processes at the grain scale, and at
much larger scales where these bodies can be treated as continua. Using diagrams to illustrate your
prose is also fine.
(2) Tales of Two Ice Sheets
(a) An isothermal, temperate, and perfectly plastic ice sheet with a span (half-width) of L=400 km sits
on a flat bed. The ice thickness at 100 km from the ice divide is 2000 m.
• What is the plastic yield stress τ0 for this ice sheet?
• What is the surface slope at 100 km from the ice divide?
• How thick is the ice at the ice divide?
• Suppose that this perfectly plastic ice sheet cools by 10oC, and its yield stress does not
change. What will happen to the thickness at its center? Why? How realistic is this?
(b) A circular ice sheet with radius R is frozen to its bed. It is in steady state. The accumulation rate
b!(r ) is a function only of distance r from the ice-sheet center.
• Using conservation of mass and b!(r ) , find an expression
for the total ice flux crossing the circular gate at distance r
from the ice-sheet center.
•
•
Suppose that the accumulation rate b!(r ) = b!0 is actually
uniform within a circle of radius R/2 centered on the ice
divide (dashed line on the map). Simplify your expression
above to evaluate the specific case of total ice flux
crossing the circular gate at R/2.
If the ice thickness is h at radius r=R/2, find an expression
for the depth-averaged velocity at radius r=R/2 in terms of
h, R, and b!0 .
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(3) Momentum-Conservation Equations
Using prose, possibly illustrated by a few equations and diagrams, explain how you can derive the
momentum-conservation equations in tensor form for a continuum such as glacier ice, or snow. Be sure to
write down the momentum-conservation equations that you derive.
(4) Stress Guide
A uniform slab of ice with thickness h sits on a long planar surface with low slope. The origin is at the
ice surface. The x1 axis points down-slope along the ice surface, which has a slope of α (|α|<<1) , and the
x2 direction is downward and normal to the sloping surface. The surface slope and the ice thickness are
independent of x1 and x3. The density of ice is ρ, and gravitational acceleration is g. The slab is also
stretching uniformly in the x1 direction at a longitudinal strain rate ε!11 = ε!0 , which is independent of
depth. A uniform accumulation rate b! of ice on the top surface of the ice prevents the slab from getting
thinner over time. Flow speed does not vary in the x3 direction.
•
Find an expression relating b! and ε!0 .
•
Using Glen’s flow law for ice with exponent n, find an equation relating the second deviatoric stress
invariant τe, to longitudinal strain rate ε!0 , and shear strain rate ε!21.
•
Find an expression for longitudinal deviatoric stress τ11 in terms of Glen flow-law parameters n and
A(T), longitudinal strain rate ε!0 , and shear strain rate ε!21 .
•
Near the upper surface, shear strain rate ε!21 is small. Roughly how do you expect ε!21 to vary with
depth x2 (for example, in the SIA)?
Suppose the ice slab is isothermal at temperature T0. Which part of the ice slab supports the greatest
longitudinal deviatoric stress τ11? Discuss the implications for stress distribution in a valley glacier in
the Cascades.
Suppose the ice slab is colder at the top than at the bottom. What additional effect would this have on
the distribution of longitudinal deviatoric stress? Discuss the implications for stress distribution in the
East Antarctic ice sheet.
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•
(5) Shallow Ice
The Shallow Ice Approximation is widely used in ice-flow models. Without going into all the details,
• Explain the motivation for the SIA.
• Outline how the SIA is derived.
• Explain how the result helps modelers make decisions.
• Discuss the limitations of a model that applies the SIA.
• Identify 3 areas in Antarctica where the SIA is not a good approximation (for different reasons), and
explain why not.
(6) Crystal Physics
• Outline the Weertman explanation for why n=3 in Glen’s Law, and discuss the associated assumptions
and the limitations.
• Discuss the factors that can cause c-axis fabric to develop as a polycrystalline ice aggregate is strained.
• Describe the sequence of processes that cause primary, secondary, and tertiary creep in deforming
polycrystalline ice.
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