glacier thermodynamics: ice temperature and heat transfer ...modes of heat transfer 1....
TRANSCRIPT
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Glacier Thermodynamics: Ice Temperature and Heat Transfer Processes
ESS431: Principles of GlaciologyESS505: The Cryosphere
Wednesday, 10/24 – Ben Hills
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Today’s Objectives:
• Why do we care about ice temperature?
• Cover the fundamentals of heat transfer. What are the important time/length scales for heat transfer in ice?
• What is the general thermal structure of a glacier or an ice sheet? What heat transfer processes are important in which regions?
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Importance of Ice Temperature
Background on Heat Transfer
Ice Sheet Thermal Structure
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" = $ % &'
Why Ice Temperature?
1. Ice Deformation (Viscosity)
$ % = $()*+,-.
Joughin et al. (2010)
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Credit: Paul D. Bons, Ilka Weikusat
Why Ice Temperature?
1. Ice Deformation (Viscosity)2. Basal Sliding
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Why Ice Temperature?
1. Ice Deformation (Viscosity)2. Basal Sliding3. Paleoclimate
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Why Ice Temperature?
1. Ice Deformation (Viscosity)2. Basal Sliding3. Paleoclimate4. Drilling
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Why Ice Temperature?
1. Ice Deformation (Viscosity)2. Basal Sliding3. Paleoclimate4. Drilling5. Geophysics
(Radar and Seismic)
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1. Ice Deformation (Viscosity)2. Basal Sliding3. Paleoclimate4. Drilling5. Geophysics
(Radar and Seismic)6. Mass Balance
mm
.w.e
.
Why Ice Temperature?
van den Broeke et al. (2011)
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Importance of Ice Temperature
Background on Heat Transfer
Ice Sheet Thermal Structure
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Laws of Thermodynamics1. Energy Conservation:
• Energy can neither be created nor destroyed; energy can only be transferred or changed from one form to another.
• Δ" = $ −&
2. Entropy• The entropy of any isolated system always increases. Isolated systems spontaneously
evolve towards thermal equilibrium—the state of maximum entropy of the system. • Δ' ≥ 0
Background on Heat Transfer
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Modes of Heat Transfer1. Conduction
- Transfer of heat via direct molecular collision.
Background on Heat Transfer
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Fourier’s Law:
Background on Heat Transfer
Modes of Heat Transfer1. Conduction
- Transfer of heat via direct molecular collision.
! = −$∇&Hot
Cold
Heat Flux
!$ is the thermal conductivity
of the material
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Modes of Heat Transfer1. Conduction
- Transfer of heat via direct molecular collision.
2. Convection- Thermal energy is carried by a moving fluid.
Background on Heat Transfer
dTdt = −&∇(
&
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Modes of Heat Transfer1. Conduction
- Transfer of heat via direct molecular collision.
2. Convection- Thermal energy is carried by a moving fluid.
3. Radiation- The emission of electromagnetic waves carries energy away from a thermal body.
Background on Heat Transfer
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3. Radiation- The emission of electromagnetic waves carries energy away from a thermal body.
Background on Heat Transfer
Stefan-Boltzmann Law:
! = #$%
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!(#$)!& = ( )
*− ( )
*,-*+ /
!(#$)!& = − 0(01 + /
#23040& + u 0401 = −6 0
74017 + /
The Heat Equation
Heat Capacity Definition
$(4) = 234
!#!& 234 + #23
!4!& =
001 6 0401 + / = 0
Total Derivative
!4!& =
040& +
0401
010& =
040& + 9 ⋅ ∇4
040& = < 0
74017 − u 0401 +
/#23
Incompressibility and Total Derivative
/#$(
1 1+dx
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• Diffusion(mostly vertical)
• Advection (all 3 directions)
• Sources(see figure)
The Heat Equation for Ice!"!# = %∇'" − ) ⋅ ∇" + 1
-./( 23 + 4565)
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Characteristic Diffusion Length!"!# = %∇'" − ) ⋅ ∇" + 1
-./( 23 + 4565)
8 = 2 %#
Water 1.43 × 10−7 m2/s
Copper 1.11 × 10−4 m2/sIron 2.3 × 10−5 m2/sAir 1.9 × 10−5 m2/sStainless Steel 4.2 × 10−6 m2/sIce 1.09 × 10−6 m2/s
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Peclet Number!"!# = %∇'" − ) ⋅ ∇" + 1
-./( 23 + 4565)
89 = ):%
Solid earth (mantle convection) Pe~103
Ice Sheet (horizontal) Pe~105
Gulf Stream Pe~1012
Ice Sheet (vertical) Pe~100-101
Ice Sheet Pe~10-1Near-surface
Granite Outcrop Pe<<1
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Importance of Ice Temperature
Background on Heat Transfer
Ice Sheet Thermal Structure
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Thermal Classifications of Glaciers
Temperate
• High Accumulation
• High Melt
• Fast Moving
Franz Josef Glacier
New Zealand Cold
• Low Accumulation
• Low Melt
• Slow Moving
McMurdo Dry Valleys
Antarctica
Polythermal
• Possibly bridges climate zones (i.e.
cold at top and warm at bottom)
• Includes both ice sheets
Grand Pacific Glacier
Glacier Bay, Alaska
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Regions1. Surface2. Bed3. Ice Divide4. Downstream of Divide
5. Fast-Flowing Ice6. Ablation Zone
Boundary Conditions
Advection
Frictional Heating
Latent Heating
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1) Surface!"!# = %∇'" − ) ⋅ ∇" + 1
-./( 23 + 4565)
• Dominated by conduction with the overlying atmosphere.• Melting becomes important in the ablation and
percolation zones.
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1) Surface
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Skin Depth is period dependent
1) Surface
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1) Surface
Hills et al. (2018)
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1) Surface
Hills et al. (2018)
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2) Bed!"!# = %∇'" − ) ⋅ ∇" + 1
-./( 23 + 4565)
• Heat sources from the friction of ice sliding over bedrock and from geothermal heating.
• Basal melting can add to the total mass balance, but is mostly important for its influence on sliding.
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2) Bed
MacGregor et al. (2016)
Pres
sure
Mel
ting
Poin
t
Geothermal Flux
FrozenMelted
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3) Ice Divide!"!# = %∇'" − ) ⋅ ∇" + 1
-./( 23 + 4565)
• Ice flow is purely vertical.• The only source term here is firn compaction which is
small compared to shear strain or latent heating.
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3) Ice DivideAir temperature variations through time
Geothermal flux
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4) Downstream of Divide!"!# = %∇'" − ) ⋅ ∇" + 1
-./( 23 + 4565)
• Ice starts moving downhill, so the surface boundary warms at the lapse rate.
• Horizontal advection is typically faster than diffusion.
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Jakobshavn Isbræ
Luthi et al. (2002), J.Glac.Iken et al. (1993), J.Glac.
4) Downstream of Divide
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5) Fast-Flowing Ice!"!# = %∇'" − ) ⋅ ∇" + 1
-./( 23 + 4565)
• Ice streams represent a sharp boundary in ice velocity, so the ‘shear margins’ at their boundary have very high strain rates.
• Some argue that the strain rate is high enough that a significant portion of the ice column is temperate.
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5) Fast-Flowing IceIce StreamMargin Margin
!"#
!"$
&! = (()
*(+ = ,
(-)
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5) Fast-Flowing Ice
Harrison et al. (1998), JGlac.Engelhardt (2004), Ann. Glac.
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6) Ablation Zone!"!# = %∇'" − ) ⋅ ∇" + 1
-./( 23 + 4565)
• Much liquid water available at the surface. • If water finds a pathway into the ice (through fractures) it
can refreeze and release the latent heat to warm the ice.
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Latent Heat
6) Ablation Zone
Breaking Bonds
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0.04℃/m
0.033℃/m" = $ %&
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0.04℃/m
0.01℃/m 0.033℃/m
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Hypothesis: Differential advection creates horizontal temperature gradients.
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50 years
Crevasse Field
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Summarya
c
d
b
!"!# = %∇'" − u ⋅ ∇" + ,
-.
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• Cuffey and Paterson (2010) – Chapter 9
• van der Veen (2013) – Chapter 6
• Hooke (2005) – Chapter 6
• Zotikov (1986)
• Thermodynamics of Glaciers (Aschwanden, 2014; McCarthy Summer School Notes)
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