On Notation
Thermodynamics of Glaciers
McCarthy Summer School
Andy Aschwanden
Arctic Region Supercomputing Center
University of Alaska Fairbanks, USA
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(hopefully) consistent with Continuum Mechanics (Truffer)
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with lots of input from Luethi & Funk: Physics of Glaciers I lecture at
ETH
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notation following Greve & Blatter: Dynamics of Ice Sheets and Ice
Sheets
June 2012
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Types of Glaciers
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Why we care
The knowledge of the distribution of temperature in glaciers and ice sheets
is of high practical interest
cold glacier
ice below pressure melting point, no liquid water
temperate glacier
ice at pressure melting point, contains liquid water in the
ice matrix
polythermal glacier
cold and temperate parts
Introduction
Introduction
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A temperature profile from a cold glacier contains information on past
climate conditions.
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Ice deformation is strongly dependent on temperature (temperature
dependence of the rate factor A in Glen’s flow law);
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The routing of meltwater through a glacier is affected by ice
temperature. Cold ice is essentially impermeable, except for discrete
cracks and channels.
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If the temperature at the ice-bed contact is at the pressure melting
temperature the glacier can slide over the base.
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Wave velocities of radio and seismic signals are temperature
dependent. This affects the interpretation of ice depth soundings.
Introduction
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Energy balance: depicted
Energy balance: equation
P
∂u
+ v · ∇u = −∇ · q + Q
ρ
∂t
R
QE
fricitional heating
surface energy balance
QH
Noteworthy
latent heat sources/sinks
strain heating
Qgeo
firn + near surface layer
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strictly speaking, internal energy is not a conserved quantity
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only the sum of internal energy and kinetic energy is a conserved
quantity
geothermal heat
Energy balance
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Temperature equation
Energy balance
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Thermal properties
ice is cold if a change in heat content leads to a change in
temperature alone
independent variable: temperature T = c(T )−1 u
ρc(T )
∂T
+ v · ∇T
∂t
Fourier-type sensible heat flux
c(T )
k(T )
Cold Ice
q = qs = −k(T )∇T
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Cold Ice
heat capacity is a
monotonically-increasing
function of temperature
2000
1900
1800
−40 −30 −20 −10
temperature θ [° C]
0
2.6
2.4
2.2
−50
heat capacity
thermal conductivity
Equation
2100
−50
= −∇ · q + Q
k [W m−1 K−1]
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c [J kg−1 K−1]
Temperature equation
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ice density
internal energy
velocity
heat flux
dissipation power
(strain heating)
ρ
u
v
q
Q
−40 −30 −20 −10
temperature θ [° C]
Thermal properties
0
thermal conductivity is a
monotonicallydecreasing function of
temperature
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Flow law
Ice temperatures close to the glacier surface
Viscosity η is a function of effective strain rate de and
temperature T
η = η(T , de ) =
Assumptions
1/2B(T )de(1−n)/n
where B = A(T )−1/n depends exponentially on T
Cold Ice
Flow law
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only the top-most 15 m experience seasonal changes
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heat diffusion is dominant
We then get
∂2T
∂T
=κ
∂t
∂h2
where h is depth below the surface, and κ = k/(ρc) is the
thermal diffusivity of ice
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Ice temperatures close to the glacier surface
Cold Ice
Examples
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Ice temperatures close to the glacier surface
Boundary Conditions
T
T (0, t) = T0 + ∆T0 · sin(ωt) ,
ΔT0
0
t
T (∞, t) = T0 .
T0
∆T0
2π/ω
mean surface temperature
amplitude
frequency
h
Cold Ice
Examples
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Cold Ice
T0
Examples
φ(h)
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Ice temperatures close to the glacier surface
Ice temperatures close to ice divides
Analytical Solution
Assumptions
T (h, t) = T0 + ∆T0 exp −h
∆T (h)
|
{z
∆T (h)
r
r
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ω ω
sin ωt − h
.
2κ
2κ
| {z }
}
We then get
∂2T
∂T
= w (z)
2
∂z
∂z
where w is the vertical velocity
κ
ϕ(h)
amplitude variation with depth
Analytical solution
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Cold Ice
Examples
only vertical advection and diffusion
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Cold Glaciers
Cold Ice
can be obtained
Examples
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Water content equation
Water content equation
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Ice is temperate if a change in heat content leads to a
change in water content alone
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independent variable: water content (aka moisture
content, liquid water fraction) ω = L−1 u
∂ω
+ v · ∇ω = −∇ · q + Q
ρL
∂t
⇒ in temperate ice, water content plays the role of temperature
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Dry Valleys, Antarctica
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(very) high altitudes at lower latitudes
Cold Ice
Examples
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Temperate Ice
Equation
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Flow law
Sources for liquid water in temperate Ice
temperate ice
Flow Law
Viscosity η is a function of effective strain rate de and water
content ω
η = η(ω, de ) = 1/2B(ω)de(1−n)/n
mWS
. .. . . .
................. . .
where B depends linearly on ω
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MWS
but only very few studies (e.g. from Lliboutry and Duval)
bedrock
temperate ice
q = ql =
Fick-type
Darcy-type
water
inclusion
c
cold-dry ice
a
Latent heat flux
(
CTS
b
.. .
. . firn
cold ice
mWS microscoptic water system
MWS macroscoptic water system
1. water trapped in the ice as water-filled pores
⇒ leads to different mixture theories (Class I, Class II, Class III)
2. water entering the glacier through cracks and crevasses at the ice surface in
the ablation area
3. changes in the pressure melting point due to changes in lithostatic pressure
4. melting due energy dissipation by internal friction (strain heating)
Temperate Ice
Flow law
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Temperature and water content of temperate ice
Temperate Ice
Flow law
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Temperate Glaciers
Temperature
Tm = Ttp − γ (p − ptp ) ,
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I
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(1)
Ttp = 273.16 K triple point temperature of water
ptp = 611.73 Pa triple point pressure of water
Temperature follows the pressure field
Water content
Temperate glaciers are widespread, e.g.:
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generally between 0 and 3%
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Alps, Andes, Alaska,
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water contents up to 9% found
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Rocky Mountains, tropical glaciers, Himalaya
Temperate Ice
Flow law
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Temperate Ice
Flow law
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Polythermal glaciers
Scandinavian-type thermal structure
b)
a)
b)
a)
temperate
cold
temperate
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contains both cold and temperate ice
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Scandinavia
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separated by the cold-temperate transition surface (CTS)
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Svalbard
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CTS is an internal free surface of discontinuity where
phase changes may occur
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Rocky Mountains
polythermal glaciers, but not polythermal ice
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Alaska
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Antarctic Peninsula
Polythermal Glaciers
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Scandinavian-type thermal structure
Polythermal Glaciers
cold
Thermal Structures
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Canadian-type thermal structure
Why is the surface layer in the ablation area cold? Isn’t this
counter-intuitive?
b)
a)
temperate
temperate
Polythermal Glaciers
Thermal Structures
cold
firn
cold
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high Arctic latitudes in Canada
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Alaska
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both ice sheets Greenland and Antartica
meltwater
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Polythermal Glaciers
Thermal Structures
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Thermodynamics in ice sheet models
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only few glaciers are completely cold
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most ice sheet models are so-called cold-ice method models
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so far two polythermal ice sheet models
Thermodynamics in ice sheet models
Cold vs Polythermal for Greenland
∂T
+ v · ∇T = ∇ · k∇T + Q
∂t
∂ω
ρL
+ v · ∇ω = Q
∂t
ρc(T )
or
ρ
∂E
+ v · ∇E
∂t
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better conservation of energy
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more realistic basal melt rates well-defined interfaces to the
atmosphere and subglacial hydrology
= ∇ · ν∇E + Q
Ice Sheet Models
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Why polythermal is better: Antarctica
Ice Sheet Models
Why polythermal is better: Greenland
temperature equation
enthalpy equation
temperature equation
enthalpy equation
basal melt rate, meters per year
basal melt rate, meters per year
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conservation of
energy
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more realistic
basal melt rates
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more realistic ice
streams
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conservation of
energy
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more realistic
basal melt rates
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more realistic ice
streams
M. Martin, PIK
Ice Sheet Models
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Ice Sheet Models
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