Simple analytical models of glacier-climate interactions - by Prof. J ...

More documents

Recommendations

Info

Problems: • Find an expression for A* in eq. (7.3). • Suppose that the accumulation rate is not constant but increases with r: b = B R r/R. Find a new expression for the plane-shear solution and compare with the case of constant b. • The flow parameter A depends on ice temperature, approximately following the relation A = A 0 exp(-Q/RT). Find out for the plane-shear solution by how much the maximum ice thickness changes when the characteristic ice temperature drops from 275 K to 285 K. Parameter values: A 0 = 1.14⋅10 10 Pa -3 yr -1 , Q = 100 kJ mol -1 . • We design a schematic mass continuity model for a marine ice sheet. Suppose that the ice sheet rests on a bed that slopes downwards at a constant rate: b = b 0 - s r and that the accumulation rate is constant (a). The ice-sheet edge is in the sea, and the azimuthally averaged ice velocity is proportional to the water depth d (so U R = f * d). Find an expression for the total mass budget of the ice sheet and solve for the icesheet radius R. Apply the model to the West Antarctic ice sheet (set b 0 = 0 for simplicity) and try to fimd out the sensitivity of R to changes in accumulation rate (use R = 600 km, f = 1 yr -1 , a = 0.25 m yr -1 ). REFS Haeberli W. and M. Hoelzle (1995): Application of inventory data for estimating characteristics of and regional climate-change effects on mountain glaciers: a pilot study with the European Alps. Annals of Glaciology 21, 206-212. Jóhannesson T., C.F. Raymond and E.D. Waddington (1989): Time-scale for adjustment of glaciers to changes in mass balance. Journal of Glaciology 35, 355- 369. Oerlemans J. (1981): Some basic experiments with a vertically-integrated ice-sheet model. Tellus 33, 1-11. Vialov S.S. (1958): Regularities of glacial shields movement and the theory of plastic viscous flow. International Association of Hydrology, Scientific Publication 47, 266- 275. Weertman J. (1961): Stability of ice-age ice sheets. Journal of Geophysical Research 66, 3783-3792. 26
Problem HMB-feedback and response time (P 1) Conservation of ice volume V can be expressed as d V d t = d A H m d t = H m d A d t + A dH m d t . (P 1.1) Here A is the glacier area and H m the mean ice thickness. Now we write V = V 0 + V', A = A 0 +A' and H m = H m,0 + H m ', where the reference state is defined as (V 0 , A 0 , H m,0 ). By neglecting higher-order terms we then obtain the perturbation equation d V' d t = H m,0 d A' d t + A 0 dH m ' d t . (P 1.2) Again we assume: A' = w f L' , (P 1.3) Here w f is the characteristic width of the glacier tongue. Now we include the height-mass balance feedback by assuming that the change in mean thickness is proportional to the change in glacier length: H m ' = η L' . (P 1.4) So for the change of ice volume with time we find d V' d t = η A 0 + w f H m,0 d L' d t = amount of mass added . (P 1.5) We now have three contributions to the mass added to or removed from the glacier: change of volume = A 0 B' + A 0 β H m ' + w f B f L' = A 0 B' + A 0 β η' + w f B f L' . (P 1.6) B' is the perturbation of the balance rate (constant over the glacier), β the balance gradient and B f the characteristic balance rate at the glacier front. The first term is simply the amount of ice added or removed by the balance perturbation on the reference glacier area. The second term represents the feedback between balance rate and surface elevation. The third term is the amount of ice lost or gained because the length of the glacier deviates from that of the reference state. Combining the equations yields 27
Page 1 and 2: SIMPLE ANALYTICAL MODELS OF GLACIER
Page 3 and 4: 1. A mass-balance model The process
Page 5 and 6: (iii) A 0 > A 1 (continuous ablatio
Page 7 and 8: 2. Ice deformation: perfect plastic
Page 9 and 10: = ρ i h ρ i - ρ m = -δ h . (2.7
Page 11 and 12: 3. A simple glacier model We consid
Page 13 and 14: d L d T a = ∂ L ∂ E d E d T a =
Page 15 and 16: For L < L ub the solution is given
Page 17 and 18: 5. A volume time scale for valley g
Page 19 and 20: Problem: • The time scale derived
Page 21 and 22: Note that values of L for which N <
Page 23 and 24: 7. Steady state ice-sheet profiles
Page 25: Fig. 7.2 3500 3000 2500 h (m) 2000
Page 29 and 30: Problem ice-sheet profile (P 2) For
Page 31: Problem West Antarcic ice sheet (P

Simple analytical models of glacier-climate interactions - by Prof. J ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?