Six Lectures on Theoretical Glaciology

1. Six Lectures on Theoretical Glaciology Ravello: 11-16 September 2006 Kolumban Hutter, Professor emeritus Bergstr. 5, 3044 Zürich hutter@vaw.baug.ethz.ch c/o Laboratory of Hydraulics, Hydrology and Glaciology ETH Zürich, Gloriastrasse 37, 8092 Zürich

2. Contents 1. Ice and Snow in the Geophysical Context 2. Material Description of Ice in the Geophysical Context 3. The Shallow Ice Approximation for Ice Sheets 4. Induced Anisotropy in Polar Ice and its Role in the Reconstruction of the Past Climate 5. Sea Ice Dynamics 6. An Integrated View of the Role of the Ice in Climate Dynamics

3. 1. Ice and Snow in the geophysical context Contents 1.1 What is snow and ice mechanics ? 1.2 Balance laws for material bodies 1.3 Mass balance 1.4 Linear momentum balance 1.5 Energy conservation 1.6 Entropy balance

4. 1.1 What is snow and ice mechanics ? Definition Snow and ice mechanics are those subfields of physics and engineering, which are concerned with the mechanical and thermodynamical concepts of snow and ice, when such bodies are subjected to external driving agents (forces, atmospheric and terrestrial conditions). Remark: Thermodynamics is a rather important part of it. Goal is the mathematical description of the processes that arise in snow and ice masses under temporal and spatial changes of such conditions. Methods are those of classical physics (mechanics and thermodynamics) with the intention of a deterministic relation between input and output. Principle of Causality Statistical considerations enter often, because input quantities are not well constrained in geophysical systems.

5. Principalassumptions ●Snow is a porous material whichisformedbyicecrystals – grains–whichmoreorless stick toeachotherowingtocontactforces (oftenelectrostaticoradhesive). ● Icein glaciers, icesheets, iceshelvesandicestreamsis a polycrystal, constitutingoforiented hexagonal singlecrystalswithwaterand/orairinclusions, impurities, fracturesandcrevasses. Remark: On differential lengths, which are relevant in the geophysical context, the influence of these effects can be attributed to the meso- and micro structure. Postulate: Snow and ice are treated as continuous media. To each point of the 3D physical space material substance is assigned which is equipped with mass, linear and angular momenta, energy, entropy, etc. The complexity of the microstructure is incorporated in this continuum description via the mathematical complexity of the material behaviour.

6. Ice structure ..... Ice structure free surface Gas inclusions Ice with water and gas inclusion The big ice crystals close to 3000m's depth are clearly seen through a polarization filter. Microscopic photograph of snow crystals

7. Structure of the mathematical description ●Fundamental rules/-lawshavegeneral validity for all continuous media and remain unquestioned. Balances: - Mass - Linear Momentum (Newton´s law) - Angular Momentum (Euler´s law) - Energy (1. law of Thermodynamics) - Entropy (2. Law of Thermodynamics) ● Material dependent statements (constitutive relations) are postulated and must be verified by laboratory experiments and/or field observations.

8. Remark: Each constitutive equation for snow or ice delivers together with the fundamental balance laws a model for particular processes in snow or ice. There is no absolute mathematical description for the thermo-mechanical description of snow and ice, however. Models are developed for particular processes. Consequences: For the derivation of an effective model for snow and ice the following steps must be taken: ●Selection and delineation of the processesfor which a model is to be developed ● Postulation of constitutive relationshaving the potential to describe the selected processes ● Identification of the parameters in the constitutive model by solving typical problems (IBVPs) and comparison with experiments and field observations, to delineate the validity of the model.

9. Time and Space Scale ●Snow - Process episodes are at most of seasonal duration and relate to regions. - Snow can change its microstructure and therefore its physical behaviour within hours or minutes. This shows itself at the macro-level as a change of the constitutive behaviour, often dramatic. - Fracture of a snow cover is usually a matter of micro-seconds and the motion of an avalanche from initiation to run-out is taking place within minutes

10. ●Ice - Process episodes in land-based ice sheets may stretch over ice ages (~ 100000 years) and embrace the entire globe. - Changes in large ice masses (ice sheets, ice shelves, etc.) due to atmospheric climatic variations and the associated sea level rise take place on time scales of thousands and hundreds of years. - In glaciers large variations may occur on time scales of several decades. - The deformation of the ice in hanging glaciers may occur within months or weeks. - Ice avalanches are formed by break-off of ice chunks. The latter have time scales of seconds, the former of minutes. - A water outburst from a glacier (Jökulhlaup) yields a flood with the maximum discharge occuring between 0.5 - 2 days after onset; the total duration is a few days, perhaps 7 – 14 days.

11. Motivation for snow & ice research Historically, the interest in the mathematical-physical behaviour of snow and ice developed because of the wish and concern, to protect land and humans against natural catastrophs. Examples: ● Ice break-off from hanging glaciersand related threat of settlements in the valley below, ● Blocking of valleys by ice-avalanches, formation of ice dammed lakes, which, owing to a possible breaking of the dam, may threaten the population in the valley below by the flood, Example: Gornersee, Jökulhlaup, ● Protection of land and mankind from damage by avalanches, To apply measures against such damages requires understanding of the processes !

12. Technically, scientifically, Snow and ice have always caught the interest of various people from natural sciences and engineering. Engineers attempt to use the potential of snow and ice in applications ●Construction of high-pressure hydro-power plants or the use of melting water (during the winter half year) ● Construction of subglacial water catchment sites (Argentière) ● Controlled discharge of ice-dammed lakes to avoid floods ● Installation of snow bridges to circumvent the formation of avalanches ● Passive and active avalanche protections which control the track of possible avalanche events ● Katastration of the terrain with regard to its probability to be hit by an avalanche ● Estimation of the influence of the anthropogenic Green House Effect in the atmosphere on - retreat of the permaforst in the high Alps - increased melting of ice from glaciers and ice sheets and sea level rise - increased occurrence of avalanches, debris and mud flows

13. Use of snow and ice as structural material ●Cylindrical and hemi-spherical snow roofs for refrigerating halls (Mega-Iglus) ● Artificial ice-islands in shallow water of the polar ocean as drill-islands for oil extraction in the Artic, ● Floating ice on lakes and rivers, used as tracks of transportation, air plane pistes and parking places ● Freezing method in tunnel constructions when propulsing in loose soil

14. Rhonegletcher Rhonegletcher 1907 1979 1912 Variegated Glacier 23 Juni 1983 Variegated Glacier April 1982

16. 1.2 balance laws for material bodies a) General physical quantitydefined per unit volume, e.g. density , specific linear momentum v total amount of g contained in B Question: flux of g through B supply of g into B productionof g in B

17.  Global balance (*) Assumptions: (i) (*) holds for any material body, also infinitesimal volume elements (ii) All field variables are assumed to be differentiable throughout the body, with the exception of possible internal surfaces, across which fields may experience finite jumps → Local field equations → Jump conditions

18.  Local balance laws Consequences are purely mathematical. With the use of Reynolds´ transport theorem and Gauss´ law one obtains ●In points, in which the physical fields are differentiable ● In points on singular surfaces w= velocity of the singular surface n = unit normal vector perpendicular to the singular surface

19. Definition ● A balance law, for which the production vanishes, is called a conservation law ● A physical quantity G (g), for which there exists no production is called a conserved quantity Pg 0 + Remark In a one-constituent continuum, the mass, linear and angular momenta and energy are conserved quantities, but entropy is not.

20. 1.3 Mass balance (conserved) Fact: In a material body the total mass is conserved. No mass can flow through a material boundary, and mass supply and mass production vanish F = Z = P = 0,   mass density → Definition: The quantity (v – w)● n = Mis called mass flux across a singular surface. It is continuous across the singular surface since [[M]] = 0.

21. Definition ● A process, for which /t = 0 for some nonvanishing time interval, is called steady (relative to the density). ●A body is called density preserving, if the density does not change its value along particular trajectories, i.e., d/dt = 0. ●A body is called volume preserving, if it can only suffer unimodular deformations. It follows Continuity equation In a density preserving material the velocity field is solenoidal. Prove that a density preserving body is also volume preserving.

22. A 1st important application Atmosphere Consider the free surface of an ice sheet or glacier. What is the kinematic equation for the motion of this surface, if accumulation/ablation is accounted for ? Equation of the surface w n s z=S(x,y,t) ice z x,y since Fs = 0 for all t

23. Remark: For v, one may substitute the material velocity on the + or – side of the surface With Remarks: 1. In glaciology one substitutes v-, for one is interested in glaciers and ice sheets. 2. = -Ms/- is the accumulation/ablation function  0 for accumulation (snow fall),  0 for ablation (melting) Msis the better variable, because it is easier to measure. 3. Climatologists call mass balance

24. A 2nd important application Atmosphere Consider the boundary between ice/snow and the ground, which may move Ice/snow Rockbed Here we have Interpretation:Mb in a glacier or ice sheet is the mass of basal ice per unit area and unit time that melts on the ice sole. Also, is the freezing rate per surface area and time of the basal ice. In an avalanche, is the entrainment rate of snow for the snow cover.

25. A 3rd important application Consider a glacier/ice sheet or flow avalanche with free surface Fs(x,t) = 0 and basal surface Fb(x,t) = 0 . Assume the material to be density preserving. Then

26. Kinematic (wave) equation Remark: If functional relations of the form are known, then the kinematic wave equation delivers an evolution equation for the thickness of the ice sheet or avalanche. This equation is hyperbolic or parabolic

27. 1.4 Linear momentum (conserved) Cauchy lemma T: Cauchy stress tensor n: unit normal Remark: Ice flow is always Stokes flow Jump condition

28. A. 1st application: Slope parallel flow in a layer of constant depth in the context of Stokes flow , , no y-dependence no x-dependence BC´s: Tzz(z=h)=0, Txz(z=h)=0, Integration yields , , Remarks: ● This result is independent of the material law ●In glaciers, usually   5°. Thus, since sin ~ , cos ~1. ●The shear stress in a glacier or ice sheet at a certain depth is equal to the weight of the ice above the point of consideration times the surface slope Txz=g(h-z) Tzz=g(h-z) →

29. A. 2nd application: Stress boundary condition Example: Powder snow avalanche (Mass) Entrainment of air Case I. Free surface Result: The traction immediately inside the surface equals the traction immediately outside the surface plus the impulse of the mass flowIt = v-Ms through the surface.

30. Case II. Traction boundary condition at the free surface on a glacier or ice sheet with accumulation of snow and/or ablation by melting. Here, the impulse of the accumulated mass is non-zero, but negligibly small. Case III. At the base of an ice sheet at which the ice reaches the melting temperature an analogous situation prevails. The impulse of the melting ice is again negligibly small. Case IV. Snow entrainment or deposit in an ice avalanche also generates an ice impulse and here it is not negligible.

31. 1.5 Energy conservation First Law of Thermodynamics: The time rate of change of the total energy (consisting of kinetic energy and internal energy) is balanced by the sum of the power of working of the external forces plus the non-mechanical energy supplied to the body from outside : specific internal energy ● Power of working of the external forces ● Supplied non-mechanical energy r: specific radiation q: heat flux

32. ● First law Mathematical transformations

33. Application: Thermal boundary condition at the base for ice that is sliding on the base ice sole bed ~ latent heat of melting, L ground (geothermal heat) frictional heat frictional heat

34. 1.6 Entropy balance Second Law of Thermodynamics ● Physically realizable processes are irreversible. The only reversible processes are the equilibrium processes where nothing happens. ● There exists a physical quantity whose production can only be of one sign. If adequately defined this production is non-negative for all processes that are physically realizable. Physical quantities: specific entropy entropy flux entropy supply entropy production for all thermodynamic processes

35. Mathematical transformations [[ . For classical thermodynamics abs. temperature Clausius-Duhem inequality