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

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We assume that the ice-sheet base is flat and replace sin α by dH/dx. With zero sliding velocity we then have: U r = - A* H n+1 d H d r n . (7.6) Now the velocity can be eliminated by combining eqs. (7.5) and (7.6): 1+2/n H dH dr 1/n = - b r 1/n . (7.7) 2 A* Integration with respect to r yields n 2n+2 H 2+2/n - H 0 2+2/n = - n n+1 1/n b 2 A* r 1+1/n , (7.8) or H 2+2/n - H 0 2+2/n = - 2 1/n b 2 A* r 1+1/n . (7.9) Next we have to apply a boundary condition, namely H = 0 at r = R. This leads to a relation between the ice thickness in the centre and the ice-sheet radius: H 0 2+2/n = 2 1/n b 2 A* R 1+1/n . (7.10) The final solution then becomes H(r) = 2 n/(2n+2) b/(2A*) 1/(2n+2) R 1+1/n - r 1+1/n n/(2n+2) . (7.11) The most commonly used exponent in Glen's law is 3. In this case we have H(r) = 2 3/8 b/(2A*) 1/8 R 4/3 - r 4/3 3/8 . (7.12) The solution is shown in Fig. 7.2, together with the profile of a pefectly plastic ice sheet as derived in section 2. The plane shear solution is shown for two values of A*: one for which the height at the centre is equal to that of the pefectly plastic solution, and one in which the mean ice thickness is approximately equal to the mean ice thockness for the perfectly plastic ice sheet. 24
Fig. 7.2 3500 3000 2500 h (m) 2000 1500 1000 500 perfectly plastic plane shear 1 plane shear 2 0 0 200 400 600 800 1000 1200 1400 1600 r (km) A few interesting conclusions can be draw from the plane-shear solution. For n=3 eq. (7.10) reads H 0 = 2 1/8 b 2 A* R 1/2 (7.13) This shows that the dependence of the ice thickness on the accumulation rate is quite weak. Halving the accumulation rate reduces the ice thickness by only 8%. The same applies to the relation between ice thickness and flow parameter. On the other hand, the dependence on ice-sheet radius is larger. Halving the radius reduces the ice thickness by 30%. It is hard to judge which model performs best. It is clear that the perfectly plastic model is not very accurate in the central part of an ice sheet. However, in many applications this does not matter at all. One can try to check the validity of the profiles by comparison with observations. This does not give definite answers as to which profile performs best (Van der Veen, 1999). The less steep slope of the parabolic profile closer to the ice edge seems more realistic in many cases, albeit for the wrong reason (in reality softer ice and sliding over deformable beds leads to reduced ice thickness). It is noteworthy that the dependence of ice thickness on the radius is the same for the plane-shear and perfectly-plastic models. Altogether, when interest is in the large-scale response of ice sheets to environmental change, and when changes in ice volume are expected to be mainly the result of changes in R, then the perfectly plastic model is not a bad choice. 25
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: 7. Steady state ice-sheet profiles
Page 27 and 28: Problem HMB-feedback and response t
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 ...

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