Thermodynamics of GlaciersGlaciers 617

Andy Aschwanden

Geophysical InstituteUniversity of Alaska Fairbanks, USA

October 2011

1 / 33

Types of Glaciers

cold glacierice below pressure melting point, no liquid water

temperate glacierice at pressure melting point, contains liquid water in theice matrix

polythermal glaciercold and temperate parts

Introduction 4 / 33

Why we care

The knowledge of the distribution of temperature in glaciers and ice sheetsis of high practical interest

� A temperature profile from a cold glacier contains information on pastclimate conditions.

� Ice deformation is strongly dependent on temperature (temperaturedependence of the rate factor A in Glen’s flow law);

� The routing of meltwater through a glacier is affected by icetemperature. Cold ice is essentially impermeable, except for discretecracks and channels.

� If the temperature at the ice-bed contact is at the pressure meltingtemperature the glacier can slide over the base.

� Wave velocities of radio and seismic signals are temperaturedependent. This affects the interpretation of ice depth soundings.

Introduction 5 / 33

Energy balance: depicted

QE QH

R

Qgeo

P

surface energy balance

strain heating

fricitional heating

geothermal heatfirn + near surface layer

latent heat sources/sinks

Energy balance 7 / 33

Energy balance: equation

ρ�

∂u∂t + v · ∇u

�= −∇ · q + Q

ρ ice densityu internal energyv velocityq heat fluxQ dissipation power

(strain heating)Noteworthy

� strictly speaking, internal energy is not a conserved quantity� only the sum of internal energy and kinetic energy is a conserved

quantity

Energy balance 8 / 33

Temperature equation

Temperature equation� ice is cold if a change in heat content leads to a change in

temperature alone� independent variable: temperature T = c(T )−1u

ρc(T )�

∂T∂t + v · ∇T

�= −∇ · q + Q

Fourier-type sensible heat flux

q = qs = −k(T )∇T

c(T ) heat capacityk(T ) thermal conductivity

Cold Ice Equation 10 / 33

Thermal properties

−50 −40 −30 −20 −10 01800

1900

2000

2100

temperature θ [° C]

c [J

kg−

1 K−1

]heat capacity is amonotonically-increasingfunction of temperature

−50 −40 −30 −20 −10 0

2.2

2.4

2.6

temperature θ [° C]

k [W

m−1

K−1

]

thermal conductivity is amonotonically-decreasing function oftemperature

Cold Ice Thermal properties 11 / 33

Flow law

Viscosity η is a function of effective strain rate de andtemperature T

η = η(T , de) = 1/2B(T )d (1−n)/ne

where B = A(T )−1/n depends exponentially on T

Cold Ice Flow law 12 / 33

Ice temperatures close to the glacier surface

Assumptions� only the top-most 15 m experience seasonal changes� heat diffusion is dominant

We then get∂T∂t = κ

∂2T∂h2

where h is depth below the surface, and κ = k/(ρc) is thethermal diffusivity of ice

Cold Ice Examples 13 / 33

Ice temperatures close to the glacier surface

Boundary Conditions

T (0, t) = T0 +∆T0 · sin(ωt) ,

T (∞, t) = T0 .

T0 mean surface temperature∆T0 amplitude2π/ω frequency

Cold Ice Examples 14 / 33

Ice temperatures close to the glacier surface

h

0

T0

t

T

0

φ(h)

Cold Ice Examples 15 / 33

Ice temperatures close to the glacier surface

Analytical Solution

T (h, t) = T0 +∆T0 exp�

−h�

ω

2κ

�

� �� �∆T (h)

sin�ωt − h

�ω

2κ� �� �ϕ(h)

�.

∆T (h) amplitude variation with depth

Cold Ice Examples 16 / 33

Ice temperatures close to ice divides

Assumptions� only vertical advection and diffusion

We then get

κ∂2T∂z2 = w(z)∂T

∂zwhere w is the vertical velocity

Analytical solution� can be obtained

Cold Ice Examples 17 / 33

Cold Glaciers

� Dry Valleys, Antarctica� (very) high altitudes at lower latitudes

Cold Ice Examples 18 / 33

Water content equation

Water content equation� Ice is temperate if a change in heat content leads to a

change in water content alone� independent variable: water content (aka moisture

content, liquid water fraction) ω = L−1u

ρL�

∂ω

∂t + v · ∇ω�= −∇ · q + Q

⇒ in temperate ice, water content plays the role of temperature

Temperate Ice Equation 20 / 33

Flow law

Flow LawViscosity η is a function of effective strain rate de and watercontent ω

η = η(ω, de) = 1/2B(ω)d (1−n)/ne

where B depends linearly on ω

� but only very few studies (e.g. from Lliboutry and Duval)

Latent heat flux

q = ql =

�Fick-typeDarcy-type

⇒ leads to different mixture theories (Class I, Class II, Class III)

Temperate Ice Flow law 21 / 33

Sources for liquid water in temperate Ice

. .. .. .. .. .. .. . .. .. .. . ... . . . ...

bedrocktemperate ice

firncold ice

. ... . mWS  microscoptic  water  systemMWS  macroscoptic  water  system

mWS

waterinclusionb

temperate ice

cold-­dry icecWS

C SMT

a

1. water trapped in the ice as water-filled pores2. water entering the glacier through cracks and crevasses at the ice surface in

the ablation area3. changes in the pressure melting point due to changes in lithostatic pressure4. melting due energy dissipation by internal friction (strain heating)

Temperate Ice Flow law 22 / 33

Temperature and water content of temperate ice

Temperature

Tm = Ttp − γ (p − ptp) , (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� generally between 0 and 3%� water contents up to 9% found

Temperate Ice Flow law 23 / 33

Temperate Glaciers

Temperate glaciers are widespread, e.g.:� Alps, Andes, Alaska,� Rocky Mountains, tropical glaciers, Himalaya

Temperate Ice Flow law 24 / 33

Polythermal glaciers

temperate cold

a) b)

� contains both cold and temperate ice� separated by the cold-temperate transition surface (CTS)� CTS is an internal free surface of discontinuity where

phase changes may occur� polythermal glaciers, but not polythermal ice

Polythermal Glaciers 26 / 33

Scandinavian-type thermal structure

temperate cold

a) b)

� Scandinavia� Svalbard� Rocky Mountains� Alaska� Antarctic Peninsula

Polythermal Glaciers Thermal Structures 27 / 33

Scandinavian-type thermal structure

Why is the surface layer in the ablation area cold? Isn’t thiscounter-intuitive?

temperate cold firn meltwater

Polythermal Glaciers Thermal Structures 28 / 33

Canadian-type thermal structure

temperate cold

a) b)

� high Arctic latitudes in Canada� Alaska� both ice sheets Greenland and Antartica

Polythermal Glaciers Thermal Structures 29 / 33

Alaskan-type thermal structure

Field work 2011 on KahiltnaGlacier, Denali

Polythermal Glaciers Thermal Structures 30 / 33

Thermodynamics in ice sheet models

� only few glaciers are completely cold� most ice sheet models are so-called cold-ice method models� so far two polythermal ice sheet models

ρc(T )�

∂T∂t + v · ∇T

�= ∇ · k∇T + Q

ρ L�

∂ω

∂t + v · ∇ω�= Q

or

ρ�

∂E∂t + v · ∇E

�= ∇ · ν∇E + Q

Ice Sheet Models 32 / 33

Thermodynamics in ice sheet models

Cold vs Polythermal� better conservation of energy� more realistic basal melt

rates� more realistic ice streams

Ice Sheet Models 33 / 33