I. Glacier Equilibrium Response to a Change in Climate

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Amice Short
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1 EARTH AND SPACE SCIENCE 431 PRINCIPLES OF GLACIOLOGY 505 THE CRYOSPHERE Autun Credits, SLN Credits, SLN Lab Week 6 Glacier Variations (Solutions I. Glacier Equilibriu Response to a Change in Cliate Consider three siple glacier geoetries. Assue they reside in the sae cliate, all with an Equilibriu Line Altitude (ELA of 3000, and ass balance gradient!" of 8 (//k. All!# have constant widths. Glacier 1 starts at an elevation of Z = 4000 and terinates at Its steady state length, L %, is Its characteristic thickness is 80. Glacier 2 starts at an elevation of Z = 3500 and terinates at Its steady state length, L %, is Its characteristic thickness is 160. Glacier 3 starts at an elevation of Z = 3250 and terinates at Its steady state length, L %, is Its characteristic thickness is Find the response ties and equilibriu sensitivities (/db for each glacier. See if you can arrive at a copact sybolic expression for equilibriu sensitivity (valid for these siple geoetries. You ll need it later. We will start to solve this proble by deterining the atheatical expressions needed. For each glacier we know where the equilibriu line is (3000 elevation. Thus we can use the given vertical gradient in ass balance to calculate the glacier s terinus balance relative to the balance at the equilibriu line, which by definition is 0, or: dz z In class on Monday (you will also derive this in hoework #5, we deterined the characteristics response tie: τ = / 0 " 1 where H % is the characteristic thickness and b is the specific ass balance at the terinus at the glacier terinus. The glaciers are initially in balance (no net change in ass or length and have constant widths (so we can neglect the width ter, so any change in balance will results in a change in length of the glacier. We can deterine the change in length through ass flux conservation. If db is the spatial average of the change in specific ass balance

2 9 0 % (db = b 4 (xdx, where b 4 (x is your perturbed ass balance, then the change in ass flux at the glacier s original terinus is db L %. For a positive change db, this flux ust be ablated away over the length the glacier advances, thus: db L % = where is the change in length and is the ablation rate (positive if the ass is being lost, i.e., the glacier s surface is lowering. Rewriting this equation for the change in length for a given change in ass balance (the glacier s equilibriu sensitivity gives: Let s think about what this eans for one inute (we will need this for part 2. For glacier 1, we can calculate this as: = 8000 = / Thus, if the ass balance changes by a spatial average of 0.5 / (ice equivalent, the glacier would advance by: db = db = 1000 db / 0.5 = 500 Conversely, if the if the ass balance changes by a spatial average of -0.5 /, the glacier would retreat by 500. Note that fro the relationships above describe different aspects of the glacier s behavior. The characteristic tie is an e-folding tie describing how long the glacier takes to reach its new equilibriu length. The is the glacier s change in length, but that relationship does not contain inforation on how long that change takes to occur. Now we are in the position to calculate these quantities for each glacier. Glacier 1: Glacier 2: Glacier 3: dz / z = 8 1 k = 8 / k = 80 b 8 = 10 s = / = 1000 / = 1 k / dz / z = k = 4 / k = 160 b 4 = 40 s = / = 2000 / = 2 k / / z = k = 2 / dz k = 75 b 2 = 37.5 s = / = 1000 / = 1 k /

3 2 You are doing fieldwork on glacier #2 and, on your lunch break, ski over a sall pass and find another glacier on the sae ountain. You re at the head of the glacier, which your GPS tells you is at You can see that it has the sae slope, and assue being close by and sae aspect, it has the sae ass balance gradient and ELA. But, it s a bit cloudy down below so you can t see the terinus. a. Estiate the new glacier s length and terinus elevation fro what you know about glacier #2. This glacier has the sae ELA as glacier #2, which is 3000 elevation. Since it is in steady state, the ELA ust be halfway in elevation between the head of the glacier and its terinus, or is at Once we know it has the sae slope as glacier #2, we can calculate its length too. Glacier #2 has a length of 8000 and an elevation range of 1000, so its slope is 1/8. So, for this new glacier, we have fro the definition of slope: # = L%%M = N OPQR!S T V L % = = b. You don t know the thickness of this new glacier, but using what you know about glacier #2, estiate this glacier s response tie and sensitivity. We can estiate the glacier s response tie and sensitivity using the sae process as for nuber 1. New Glacier: dz / z = k = 1.6 / k = 160 b 1.6 = 100 s = / = 2000 / = 2 k / where we have assued the glacier is the sae thickness as glacier #2. This is probably not the best assuption, even if it is the siplest. A uch better assuption ight be to scale the glacier s thickness by the relative area of the glacier, as the area deterines the total flux that uch be conserved (accuulation above the ELA, ablation below the ELA. If we ake this assuption, this new glacier is only 64 thick. And the response tie and sensitivity would be: = 64 b 1.6 = 40 s = / = 2000 / = 2 k / c. Given what you know about ice dynaics, explain why you think this estiate is likely an over or under estiate. You don t have to solve for the thickness, but explain the physical basis for your answer. *hint: think about last hoework on kineatics and dynaics The assuption of the sae thickness as glacier #2 is clearly a vast overestiate, as this glacier has a uch saller accuulation area and the sae surface slopes, so saller balance fluxes would be needed to evacuate the accuulated ass. A better assuption would be to scale by area as we just did, and then it s easy to

4 see that this glacier is ost likely to respond to cliate change in an identical way to glacier #2, despite its different size and terinus ablation rate (the geoetry would copensate for these differences to reach a steady state!! II. Glacier Transient Response to a Change in Cliate The siplest odel for transient response is exponential (as discussed in reading and hoework. The transient length solution for trend in ass balance (b!" starting at t = 0 is: = L % b [t τ\1 e^/_`a where is the ablation rate at the terinus and τ is the glacier s response tie. 1 If glaciers 1 3 are subject to the sae trend of (//, after 100 years how far out of equilibriu are they? (solve for the length difference between transient and equilibriu responses. This first proble is ainly plug and chug. For each glacier, we have (using the values fro part I: Glacier 1: The total change in length (total length change in response to this balance trend is: =!9 db = 1000 M b 0.005!" The change in length after 100 years is L Y (100 s = b ef! 100 s = 500 g100 s 10 s 4% efi jk = 450 so this glacier is coitted to 50 additional retreat 100 years after the trend in ass balance started before a new equilibriu (steady state is reached. Glacier 2: The total change in length (total length change in response to this balance trend is: =!9 db = 2000 M b 0.005!" The change in length after 100 years is L Y (100 s = b ef 100 s = 1000 g100 s 40 s L% efi jk = 630 so this glacier is coitted to 370 additional retreat 100 years after the trend in ass balance started before a new equilibriu (steady state is reached.

5 Glacier 3: The total change in length (total length change in response to this balance trend is: =!9 db = 1000 M b 0.005!" The change in length after 100 years is L Y (100 s = b ef 100 s = 500 g100 s 37.5 s l.n efi jk = 325 so this glacier is coitted to 175 additional retreat 100 years after the trend in ass balance started before a new equilibriu (steady state is reached. 2 Is the difference due to their different response ties or different sensitivity? To find out, odify your expression for equilibriu sensitivity fro part I so that it expresses the equilibriu response to a trend as a function of tie. Then, divide the transient response (above by the equilibriu response to get an expression for fractional adjustent. To find the fractional length adjustent, we siplify divide the transient length solution by the total length adjustent (or the total length change, which is you note the calculation above is sybolically given by: = db b t = L % b t Dividing the transient length solution by this ter gives: L % = a b [t τ\1 e^/_`a % = 1 τ \1 e^/_` L % a b t t % as the fractional adjustent of a glacier at any given tie after the start of a trend. 3 What does it depend on? What is it for each glacier, after 100 years? This expression depends only on the tie since the trend in ass balance started and the characteristic response tie of the glacier (so there is an iplicit dependence on the characteristic thickness and terinus ass balance. For each glacier, we have: Glacier 1: = 1 τ t Glacier 2: Glacier 3: = 1 τ t 10 s 100 s 40 s 100 s 4% efi j = 0.9 L% efi j = 0.63 = 1 τ 37.5 s l.n efi j = 0.65 t 100 s Note that these values are the sae as those you would get fro using your solution fro proble 2 above dividing the change in length after 100 years by the total length change.

6 4 Now, consider two ore glaciers. We don t know their ass balance gradients or sensitivities. But we do know their response ties (10 and 50 years, and have observed both retreat 800 in the last 100 years. If we again assue a ass balance trend started 100 years ago, estiate how far each of these glaciers would retreat if the ass balance trend stopped today. We can calculate their fractional retreats. For the first glacier: and then = 1 τ t 10 s 100 s 4% efi j = 0.9 L Qr = = LY (t 0.9 = = 890 so an additional 90 of retreat would occur if the forcing stopped today. For the second glacier: and then = 1 τ t 50 s 100 s n% efi j = 0.57 L Qr = = LY (t 0.57 = = 1400 so an additional 600 of retreat would occur if the forcing stopped today. These glaciers have characteristic ties scales that are siilar to the local glaciers in the Pacific Northwest. The first glacier (with the shorter characteristic tiescale ight represent a glacier on a big volcano (like Mt. Rainer or Mt. Baker, and the second glacier (with the longer characteristic tiescale ight represent a valley/cirque glacier in the North Cascades. So the glaciers on the Cascade volcanoes are likely uch ore closely adjusted to their cliate due to their ability to respond rapidly to a cliate change. The glaciers in the North Cascades are likely to be ore out of equilibriu and have ore coitted retreat. Finally, please note that this is the siplest odel, which has soe known issues; naely, the odel assues the terinus starts responding iediately (exponential decay and that ass balance rate at the terinus is fixed in tie, when it would really change as the glacier s length changes. More coplex odels show a spin-up phase of retreat, that is due to the tie delay between the forcing and the thickness changes that are necessary to drive flux changes and advance/retreat. This would slow the response of the glacier, increasing the lag between the glacier s current length and its new equilibriu length, and thus also increasing disequilibriu.

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