On Notation Thermodynamics of Glaciers. Types of Glaciers. Why we care. McCarthy Summer School

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1 On Notation Thermodynamics of Glaciers McCarthy Summer School Andy Aschwanden Geophysical nstitute University of Alaska Fairbanks, USA (hopefully) consistent with Continuum Mechanics (Truffer) with lots of input from Luethi & Funk: Physics of Glaciers lecture at ETH notation following Greve & Blatter: Dynamics of ce Sheets and ce Sheets August ntroduction 3 Types of Glaciers Why we care 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 The knowledge of the distribution of temperature in glaciers and ice sheets is of high practical interest A temperature profile from a cold glacier contains information on past climate conditions. ce deformation is strongly dependent on temperature (temperature dependence of the rate factor A in Glen s flow law). The routing of meltwater through a glacier is affected by ice temperature. Cold ice is essentially impermeable, except for discrete cracks and channels. f the temperature at the ice-bed contact is at the pressure melting temperature the glacier can slide over the base. Wave velocities of radio and seismic signals are temperature dependent. This affects the interpretation of ice depth soundings. ntroduction 5 ntroduction 6

2 Energy balance: depicted Energy balance: equation P QE fricitional heating firn + near surface layer R QH surface energy balance latent heat sources/sinks strain heating Qgeo geothermal heat ρ U t + v } {{ U } = q + Q }{{}}{{} advection diffusion production Noteworthy ρ U v q Q ice density internal energy velocity heat flux dissipation power (strain heating) 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 Energy balance 9 Case study: Colle Gnifetti Cold ice uppermost part of Grenzgletscher, Monte Rosa, at an altitude of 4500 m mean annual air temperature of -13 C amplitude of 7 C T how does a temperature in the first 15 look like? Temperature equation ice is cold if a change in heat content leads to a change in temperature alone independent variable: temperature T = c(t ) 1 u T ρc(t ) t + v T = q + Q Fourier-type sensible heat flux q = q s = k(t ) T z c(t ) k(t ) heat capacity thermal conductivity Energy balance 10 Cold ce Equation 12

3 Thermal properties c [J kg 1 K 1 ] temperature θ [ C] heat capacity is a monotonically-increasing function of temperature Flow law Viscosity η is a function of effective strain rate d e and temperature T η = η(t, d e ) = 1/2B(T )d (1 n)/n e where B = A(T ) 1/n depends exponentially on T k [W m 1 K 1 ] temperature θ [ C] thermal conductivity is a monotonicallydecreasing function of temperature Cold ce Thermal properties 13 Cold ce Flow law 14 ce temperatures close to the glacier surface Assumptions ce temperatures close to the glacier surface Boundary Conditions only the top-most 15 m experience seasonal changes heat diffusion is dominant We then get T t = κ 2 T h 2 where h is depth below the surface, and κ = k/(ρc) is the thermal diffusivity of ice T 0 T 0 2π/ω T (0, t) = T 0 + T 0 sin(ωt), T (, t) = T 0. mean surface temperature amplitude frequency Cold ce Examples 15 Cold ce Examples 16

4 ce temperatures close to the glacier surface T ce temperatures close to the glacier surface Analytical Solution 0 ΔT0 t ( ) ω ( T (h, t) = T 0 + T 0 exp h sin 2κ }{{} T (h) ω ωt h 2κ }{{} ϕ(h) ). T (h) amplitude variation with depth h T0 φ(h) Cold ce Examples 17 Cold ce Examples 18 Case study: Colle Gnifetti ce temperatures close to ice divides depth below surface (m) January February March April May June July August September October November December Assumptions only vertical advection and diffusion We then get κ 2 T z 2 = w(z) T z where w is the vertical velocity Analytical solution can be obtained temperature ( C) Cold ce Examples 19 Cold ce Examples 20

5 Cold Glaciers Temperate ice Water content equation ce 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 1 u ω ρl t + v ω = q + Q in temperate ice, water content plays the role of temperature Dry Valleys, Antarctica (very) high altitudes at lower latitudes Cold ce Examples 21 Temperate ce Equation 23 Flow law Flow Law Viscosity η is a function of effective strain rate d e and water content ω η = η(ω, d e ) = 1/2B(ω)d (1 n)/n e where B depends linearly on ω but only very few studies (e.g. from Lliboutry and Duval) Latent heat flux q = q l = { Fick-type Darcy-type leads to different mixture theories (Class, Class, Class ) Sources for liquid water in temperate ce a bedrock temperate ice... firn cold ice WS CT S 1. water trapped in the ice as water-filled pores M temperate ice mws b water inclusion c cold-dry ice mws microscoptic water system MWS macroscoptic water system 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 ce Flow law 24 Temperate ce Flow law 25

6 Temperature and water content of temperate ice Temperate Glaciers Temperature Tm = Ttp γ (p ptp ), (1) Ttp = K triple point temperature of water ptp = Pa triple point pressure of water Temperature follows the pressure field Water content Temperate glaciers are widespread, e.g.: generally between 0 and 3% Alps, Andes, Alaska, water contents up to 9% found Rocky Mountains, tropical glaciers, Himalaya Temperate ce Flow law 26 Polythermal glaciers Flow law 27 Scandinavian-type thermal structure b) a) temperate b) a) cold temperate cold contains both cold and temperate ice Scandinavia separated by the cold-temperate transition surface (CTS) Svalbard CTS is an internal free surface of discontinuity where phase changes may occur Rocky Mountains Alaska (e.g. McCall Glacier, see exercise) Antarctic Peninsula Polythermal Glaciers Temperate ce polythermal glaciers, but not polythermal ice 29 Polythermal Glaciers Thermal Structures 30

7 Scandinavian-type thermal structure Why is the surface layer in the ablation area cold? sn t this counter-intuitive? Canadian-type thermal structure a) b) temperate cold high Arctic latitudes in Canada Alaska both ice sheets Greenland and Antartica temperate cold firn meltwater Polythermal Glaciers Thermal Structures 31 Polythermal Glaciers Thermal Structures 32 Thermodynamics in ice sheet models Thermodynamics in ice sheet models or only few glaciers are completely cold most ice sheet models are so-called cold-ice method models so far two polythermal ice sheet models T ρc(t ) t + v T = k T + Q ω ρ L t + v ω = Q E ρ t + v E = ν E + Q Cold vs Polythermal for Greenland better conservation of energy more realistic basal melt rates well-defined interfaces to the atmosphere and subglacial hydrology ce Sheet Models 34 ce Sheet Models 35

8 Why polythermal is better: Antarctica temperature equation enthalpy equation basal melt rate, meters per year Why polythermal is better: Greenland temperature equation enthalpy equation basal melt rate, meters per year conservation of energy more realistic basal melt rates more realistic ice streams conservation of energy more realistic basal melt rates more realistic ice streams M. Martin, PK ce Sheet Models 36 ce Sheet Models 37