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 ...
Fig. 2.2 ice volume • 0 (cold) • E 0 (warm) h mean Problems • The mean ice thickness for a perfectly plastic ice sheet can be written as H = c L 1/2 . Find the constant c. • Suppose that a perfectly plastic ice sheet is axi-symmetrical rather than 'onedimensional'. How would this affect the profile? • An axi-symmetric perfectly plastic ice sheet has radius R. Find an expression for the vertical mean ice velocity u(r). Assume that the specific balance rate b is constant (and positive). There is a problem at r=R. Can you solve it? • For the Greenland ice sheet the mean value of E is about 1200 m, for the Antarctic ice sheet the equilibrium line is 'below sea level'. Can you locate these ice sheets on the solution diagram of Fig. 2.2? 10
3. A simple glacier model We consider a glacier that has a uniform width, rests on a bed with a constant slope s, and behaves perfectly plastically in 'a global sense' (Fig. 3.1). Fig. 3.1 Altitude h = b + H b = b 0 - sx equilibrium line L x The specific balance is written as b = β (h - E) , (3.1) where E is the equilibrium-line altitude and β the balance gradient (assumed to be constant). The glacier is in balance when the total mass budget is zero: L b n dx = β 0 0 L (H + b 0 - s x - E) dx = 0 . (3.2) Solving for glacier length L yields: L = 2 (H m + b 0 - E) s . (3.3) In this expression H m is the mean ice thickness. Note that the solution does not depend on the balance gradient! The next step is to find an equation for H m . We use again the concept of perfect plasticity: H m = τ 0 ρ g s . (3.4) Substituting this in eq. (3.3) yields: 11
- 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: = ρ i h ρ i - ρ m = -δ h . (2.7
- 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 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
Fig. 2.2<br />
ice volume<br />
•<br />
0<br />
(cold)<br />
•<br />
E<br />
0 (warm)<br />
h mean<br />
Problems<br />
• The mean ice thickness for a perfectly plastic ice sheet can be written as H = c L 1/2 .<br />
Find the constant c.<br />
• Suppose that a perfectly plastic ice sheet is axi-symmetrical rather than 'onedimensional'.<br />
How would this affect the pr<strong>of</strong>ile?<br />
• An axi-symmetric perfectly plastic ice sheet has radius R. Find an expression for the<br />
vertical mean ice velocity u(r). Assume that the specific balance rate b is constant (and<br />
positive). There is a problem at r=R. Can you solve it?<br />
• For the Greenland ice sheet the mean value <strong>of</strong> E is about 1200 m, for the Antarctic ice<br />
sheet the equilibrium line is 'below sea level'. Can you locate these ice sheets on the<br />
solution diagram <strong>of</strong> Fig. 2.2?<br />
10
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