The “Shallow Ice Approximation”aoss-research.engin.umich.edu/faculty/bassis/... · AOSS 605...
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The “Shallow Ice Approximation” AOSS 605 (004)
Transcript of The “Shallow Ice Approximation”aoss-research.engin.umich.edu/faculty/bassis/... · AOSS 605...
The “Shallow Ice Approximation”
AOSS 605 (004)
The “Shallow Ice Approximation”
AOSS 605 (004)
The “Shallow Ice Approximation”
AOSS 605 (004)
The “Shallow Ice Approximation”
AOSS 605 (004)
The “Shallow Ice Approximation”
AOSS 605 (004)
The “Shallow Ice Approximation”
AOSS 605 (004)
The “Shallow Ice Approximation”
AOSS 605 (004)
The “Shallow Ice Approximation”
AOSS 605 (004)
The Velocity Field
AOSS 605 (004)
The Velocity Field
AOSS 605 (004)
Non-Newtonian Rheology:
The Velocity Field
AOSS 605 (004)
Non-Newtonian Rheology:
Plastic Rheology:
Shear stress at the bed is constant
Thickness profile of a plastic ice sheet
AOSS 605 (004)
Plastic Rheology:
Shear stress at the bed is constant
Thickness profile of a plastic ice sheet
AOSS 605 (004)
Plastic Rheology:
Shear stress at the bed is constant
What does a plastic ice sheet look like?
Thickness profile of a plastic ice sheet
AOSS 605 (004)
Plastic Rheology:
Shear stress at the bed is constant
What does a plastic ice sheet look like?
Thickness decreases towards the margins
Thickness profile of a plastic ice sheet
AOSS 605 (004)
Plastic Rheology:
Shear stress at the bed is constant
What does a plastic ice sheet look like?
Thickness decreases towards the margins
Implies surface slope must increase
Thickness profile of a plastic ice sheet
AOSS 605 (004)
Plastic Rheology:
On a flat bed:
Thickness profile of a plastic ice sheet
AOSS 605 (004)
Plastic Rheology:
On a flat bed:
Thickness profile of a plastic ice sheet
AOSS 605 (004)
Plastic Rheology:
On a flat bed:
Integrate from x’= L to x’ = x
x = 0 - ice marginx = L - ice divide
Thickness profile of a plastic ice sheet
AOSS 605 (004)
Plastic Rheology:
On a flat bed:
Integrate from x’= L to x’ = x
x = 0 - ice marginx = L - ice divide
Thickness profile of a plastic ice sheet
AOSS 605 (004)
Plastic Rheology:
On a flat bed:
Integrate from x’= L to x’ = x
x = 0 - ice marginx = L - ice divide
If the ice sheet is symmetric we have:
What does it look like?
AOSS 605 (004)
Thickn
ess
Distance
Divide
Margin
Margin
Thickness profile of a plastic ice sheet
AOSS 605 (004)
If the ice sheet is symmetric we have:
Thickness is independent of accumulation and abalation?Thickness only depends on length of the ice sheet/glacier?
Climate and Glaciers Revisited
AOSS 605 (004)
Define ice sheet bed:
Define balance rate:
Climate and Glaciers Revisited
AOSS 605 (004)
Define ice sheet bed:
Define balance rate:
The equilibrium line elevation
Climate and Glaciers Revisited
AOSS 605 (004)
Define ice sheet bed:
Define balance rate:
The equilibrium line elevation
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
E
Climate and Glaciers Revisited
AOSS 605 (004)
Define ice sheet bed:
Define balance rate:
The equilibrium line elevation
Steady-state mass balance dictates:
Climate and Glaciers Revisited
AOSS 605 (004)
Define ice sheet bed:
Define balance rate:
The equilibrium line elevation
Steady-state mass balance dictates:
Climate and Glaciers Revisited
AOSS 605 (004)
Define ice sheet bed:
Define balance rate:
The equilibrium line elevation
Steady-state mass balance dictates:
Climate and Glaciers Revisited
AOSS 605 (004)
Climate and Glaciers Revisited
AOSS 605 (004)
Climate and Glaciers Revisited
AOSS 605 (004)
Climate and Glaciers Revisited
AOSS 605 (004)
Climate and Glaciers Revisited
AOSS 605 (004)
Climate and Glaciers Revisited
AOSS 605 (004)
Assume plastic in a global sense:
Climate and Glaciers Revisited
AOSS 605 (004)
Assume plastic in a global sense:
Climate and Glaciers Revisited
AOSS 605 (004)
0 0.2
120
00.1
60
Slope
Stea
dy-s
tate
leng
th (km
)
z0 - E = 500 mτ
0 = 100 kPa
Climate and Glaciers Revisited
AOSS 605 (004)
0 0.2
120
00.1
60
Slope
Stea
dy-s
tate
leng
th (km
)
z0 - E = 500 mτ
0 = 100 kPa
Small slopes - large glacier?
Climate and Glaciers Revisited
AOSS 605 (004)
Small slopes - large glacier?
Equilibrium line
Small slope means glacier has to grow large to drop beneath the equilibrium line
Climate and Glaciers Revisited
AOSS 605 (004)
Small slopes - large glacier?
Equilibrium line
Large slope means that as the glacier grows it moves beneath the equilibrium line quickly
A first look at climate sensitivity
AOSS 605 (004)
A first look at climate sensitivity
AOSS 605 (004)
A first look at climate sensitivity
AOSS 605 (004)
Glaciers on a bed with a small slope are more sensitive than glaciers on a bed with a large slope
A first look at climate sensitivity
AOSS 605 (004)
Glaciers on a bed with a small slope are more sensitive than glaciers on a bed with a large slope
Fractional change in length:
Climate sensitivity and lapse rate
AOSS 605 (004)
Recall: E = elevation of equilibrium line
Climate sensitivity and lapse rate
AOSS 605 (004)
Recall: E = elevation of equilibrium line (z)
Lapse rate
Climate sensitivity and lapse rate
AOSS 605 (004)
Recall: E = elevation of equilibrium line (z)
Lapse rate
Change in equilibrium line with temp.
Climate sensitivity and lapse rate
AOSS 605 (004)
Recall: E = elevation of equilibrium line (z)
Lapse rate
Change in equilibrium line with temp.
Climate sensitivity and lapse rate
AOSS 605 (004)
0
-10
-5
0.20Slope
dL/d
T a (km
/K)
γ = -4 K/kmγ = -6 K/kmγ = -10 K/km
