Simple analytical models of glacier-climate interactions - by Prof. J ...
Simple analytical models of glacier-climate interactions - by Prof. J ... Simple analytical models of glacier-climate interactions - by Prof. J ...
L = 2 s τ 0 ρ g s + b 0 - E . (3.5) The solution is shown in Fig. 3.2 (parameter values b 0 -E = 500 m, τ 0 /ρ g = 10 m). For reference the solution for zero ice thickness is also plotted. In this case the intercept of the equilibrium line and bed is simply at x = L/2. For the full solution ice thickness increases with decreasing slope, which implies an upward shift of the equilibrium point (intercept of equilibrium line and glacier surface). There is no solution for a flat bed, unless the equilibrium line is allowed to slope upwards. Fig. 3.2 80 60 L (km) 40 20 full solution 0 H m = 0 -20 0 0.05 0.1 0.15 0.2 0.25 Slope of bed The simple model can be used to make an order-of-magnitude estimate of climate sensitivity. Differentiating Eq. (3.5) with respect to E yields: d L dE = - 2 s . (3.6) So glaciers on a bed with a smaller slope are more sensitive in an absolute sense. It is tempting to use eq. (3.6) to make a first-order estimate of the response of glacier length to a change in free atmospheric temperature Ta. We assume that the equilibrium-line is linked to this temperature, which implies that d E/d T a = -γ -1 . Here γ is the temperature lapse rate in the atmosphere, typically -0.007 K km -1 ). It follows that 12
d L d T a = ∂ L ∂ E d E d T a = 2 γ s . (3.7) So we have arrived at the remarkable result that, for the given simple model, only two parameters are needed to estimate the sensitivity of glacier length to atmospheric temperature, namely, the characteristic bed slope and the temperature lapse rate! Fig. 3.3 shows d L/d T a in dependence of s for the above mentioned valueof the lapse rate. Larger valley glaciers typically have mean slopes between 0.1 and 0.2, implying that a 1 K temperature rise would lead to a 1 to 3 km decrease in glacier length. These figures appear reasonable, and we can conclude that the simple glacier model provides an interesting first-order description of the relation between climate change and glacier response. Fig. 3.3 0 dL/dT a (km K -1 ) -2 -4 -6 -8 large valley glaciers 0.05 0.1 0.15 0.2 0.25 Slope of bed Problems: • The absolute change in glacier length for a given change in E is larger when the slope is smaller. However, does this also apply to the relative change in L? • What are, in your judgement, the largest deficiencies of the model presented above? 13
- 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: 3. A simple glacier model We consid
- 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 and 26: Fig. 7.2 3500 3000 2500 h (m) 2000
- 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
d L<br />
d T a<br />
= ∂ L<br />
∂ E<br />
d E<br />
d T a<br />
= 2<br />
γ s . (3.7)<br />
So we have arrived at the remarkable result that, for the given simple model, only two<br />
parameters are needed to estimate the sensitivity <strong>of</strong> <strong>glacier</strong> length to atmospheric<br />
temperature, namely, the characteristic bed slope and the temperature lapse rate! Fig. 3.3<br />
shows d L/d T a in dependence <strong>of</strong> s for the above mentioned value<strong>of</strong> the lapse rate. Larger<br />
valley <strong>glacier</strong>s typically have mean slopes between 0.1 and 0.2, implying that a 1 K<br />
temperature rise would lead to a 1 to 3 km decrease in <strong>glacier</strong> length. These figures<br />
appear reasonable, and we can conclude that the simple <strong>glacier</strong> model provides an<br />
interesting first-order description <strong>of</strong> the relation between <strong>climate</strong> change and <strong>glacier</strong><br />
response.<br />
Fig. 3.3<br />
0<br />
dL/dT a<br />
(km K -1 )<br />
-2<br />
-4<br />
-6<br />
-8<br />
large valley <strong>glacier</strong>s<br />
0.05 0.1 0.15 0.2 0.25<br />
Slope <strong>of</strong> bed<br />
Problems:<br />
• The absolute change in <strong>glacier</strong> length for a given change in E is larger when the slope<br />
is smaller. However, does this also apply to the relative change in L?<br />
• What are, in your judgement, the largest deficiencies <strong>of</strong> the model presented above?<br />
13
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