DEPTH-AGE PROFILES NEAR THE ICE DIVIDE IN CENTRAL GREENLAND

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1 Annals / G/aci%gy 12 International Glaciological Society TWO-DIMENSIONAL THERMO-MECHANICAL MODELLING OF FLOW AND DEPTH-AGE PROFILES NEAR THE ICE DIVIDE IN CENTRAL GREENLAND by Dor Dahl-Jensen (Department of Glaciology, Geophysical Institute, University of Copenhagen, Haraldsgade 6, DK -22 Copenhagen N, Denmark) ABSTRACT The ice divide on an ice sheet is a special place on ice sheet, where velocity and stress distributions are different from those found just two ice thicknesses from ice divide. A two- dimensional rmo-mechanical model has been used to model flow near ice divide in central Greenland. The results show that surface strain-rates are increased by 5% of values found 15 km down-stream and that basa l temperatures rise by 3 Qc. Estimates of a depth-age distribution in vicinity of Crete show that isochrones rise at dome. The model calculations predict 16 m of Holocene ice, 1 m of Wisconsin ice, and 4 m of ice older than lis kyear. 15 km down-stream from dome, thickness of ice older than lis kyear will be reduced by 5%. In se calculations it has been ass umed that ice thickness at dome is time-invariant. The existence of old ice depends on basal temperature. The estimates of basal temperature depend strongly on past accumulation rates and on geormal heat flux but ice has most likely been below pressure-melting point throughout las t glacial period. INTRODUCTION The ice divide on an ice sheet has been shown to be a location where temperature, velocity, and stress distributions in ice are special. Many ice-sheet flow models have been made to describe flow of ice along an entire flow line or locally at some distance from ice divide. In order to simplify very complex set of rmo-mechanical equations, those special requirements essential to a good ice-divide so lution have often been omitted. Since ice-divide region is narrow, poor so lutions here will not seriously affect model calculations of surface elevation, temperature distribution, stress, and velocity distribution away from ice divide. However, ice-divide region, though it is a complicated region to model, is of great interest as a deep ice-core drilling site. There is no horizontal ice motion, so re is no need for corrections for advective transport. If position of ice divide has been unchanged with time, annual layer thickness, depth-age relationship, isotopic composition, concentrations of impurities, etc. in ice cores must be explained by temporal changes of precipitation rates, surface temperatures, and atmospheric conditions at ice divide. Hence, we do not have to make up-flow corrections along an entire flow line from ice divide to drill location. In recent years, ice-divide conditions have been modelled by Raymond (1983), Reeh (1988), and Dahl-Jensen (in press). In this paper, Dahl-Jensen's fully coupled rmo-mechanical flow model is used to discuss situation at Crete, central Greenland, where an ice core will hopefully be drilled to bedrock in near future. THE ICE-DIVIDE MODEL The model used to calculate an ice-divide solution has been described by Dahl-Jensen (in press). It is a steady-state two-dimensional rmo-mechanical model in which free surface, velocity, stress, and temperature distributions are calculated along a flow line from ice divide to ice-sheet terminus. The model calculations have been done by finite differences, and an iteration procedure has been used in order to de-couple energy-balance equation from rest of field equations. This is essential in order to obtain a good ice-divide so lution. The shallow-ice approximation (Hutter, 1982, 1983; Moriand, 1984) is used to obtain a set of leadorder equations. The inputs to model are: bedrock topography and geormal heat flux along flow line, ice thickness at ice divide, and as functions of unknown surface elevation, surface temperature and accumulation/ ablation rates. Glen's (1955) flow law is used as constitutive relationship with a flow-law exponent n = 3. (i.j = x,z) where E ij are strain-rates, a;. are stress deviators, and T e is second invariant of' stress-deviator tensor 71.2 Z B h--.--'-r.-,--,,-h--,---,-h,--,--,-!--::,--,--,-t- 71.B km EAST 2 JO o, I,, I!,!,, Fig. I. Site-location map of central Greenland in vicinity of Crete (CR). Sites A-H were established during post-gisp cam,f'aign. The crosses in a west-east direction around 71 N show E.G.LG. sites in area. The solid line on E.G.LG. line is flow line used in model calculations. The numbers on surface-contour lines show elevation in m a.s.l. The solid lines give areal distribution of accumulation rates (in mm of ice equivalent per year). (From Clausen and ors, 1988.) B

2 32 Dahl-Jensen: Modelling of flow in celllral Greenland TABLE Ice thickness (m) 3125 Malzer, 1964 Surface elevation (m) 3172 Clausen and ors, 1988 Accumulation rate (m/year).29 Clausen and ors, 1988 Surface temperature (C) -3.4 Clausen and ors, 1988 Geormal heat flux (mw/m2) 4 Lee and Uyeda, 1965 (Paterson, 1981, p. 31). B(T) is temperature-dependent part of flow-law parameter, and E, enhancement factor, depends on or ice properties. B(T) is an Arrhenius function of temperature and with E = I, B(-2 C) is 5.5 x 1-3 bar- 3 year- 1 (Paterson, 1981, p. 39). The model is used for conditions found at Crete (71.12 on, W) in central Greenland. Crete is located on main northouth ice divide of Greenland ice sheet and is close to summit point of east-west E.G.I.G. line, which in west terminates north of Jakobshavns Isbrre (Malzer, 1964; Reeh, 1983; Clausen and ors, 1988). The velocity data from this area (Hofmann, 1986) suggest a northward velocity which must be doubted, because surface slopes towards south (Fig. I) (Clausen and ors, 1988, fig. I). Table I lists data input used in model calculations in order to determine flow between Crete and 3 km down-stream along E.G.I.G. line. The topography along E.G.I.G. flow line is shown in Figure 2a. The observations show a flat bedrock in Crete area, so mean slope of bedrock for first 3 km is set to, and this mean bed is used in calculations. The model results for E = I are shown in Figure 2b and c. It is seen that surface longitudinal strain-rate at ice divide increases by 5% of strain-rate 3 km away. The basal temperature rises in a narrow area near ice divide, with 3 C creating a "hot spot" here (Paterson and Waddington, 1986). Figure 3 shows stress, velocity, and strain-rate profiles over depth at ice divide and five ice thicknesses away. The horizontal velocity and shear stress are at ice divide. The profiles of vertical velocity and longitudinal stress deviator at divide are different from similar profiles five ice thicknesses away. Compared to off-ice-divide locations, vertical velocities found near base at divide are smaller than those found furr down-stream. The 35 3 t 25 '-="" 2 C 15 += 1 > W 5-5 IS? 16. 1' 'L >- '--" x 1' a.) Surface - Bedrock ) I I I -8 U -1 '--" f- -12 ( Surface strain-rate Basal temperature -14 I I I I I Fig. 2. Steady-state model calculations for first 3 km of flow line from Crete (x = km) along E.G.I.G. line. The dashed lines in (a) are bedrock and surface elevations. The solid line at bed is mean bedrock for first 3 km of E.G.I.G. line. This is bedrock used in model calculations. The solid surface elevation is calculated surface. The surface strain-rates in (b) and basal temperatures in (c) demonstrate special conditions found near an ice divide. longitudinal stress deviator at ice divide has a very strong vertical gradient close to bedrock and corresponding horizontal strain-rate is concave with an inflection point 8 m above bedrock (Dahl-Jensen, in press). These ice-divide results are similar to results obtained by Raymond (1983), Reeh (1988), and Reeh and Paterson (1988).." 3 CJ x I U QU/OX 25 \:\ t 2 '-="" c... Q }:, > /."" '. w ' po' _ stress (bar) Velocity (m/yr) Strain-rate (yr -1) *1-5 Fig. 3. Depth profiles of shear stress T xz' longitudinal stress deviator a: p horizontal velocity u, vertical velocity w, and horizontal strain-rate au/ax at Crete (solid curves, x = km) and at 15 km distance from Crete along E.G.I.G. flow line (dashed curves). At ice divide, shear stress T xz and horizontal velocity u are zero..'

3 Dahl - ] ensen. Mod elling of flow in celltral Greenland DEPTH-AGE PROFILE AT CRETE 3 In order to discuss advantages/ disadvantages of drilling a deep ice core on an ice divide, calculations of a depth-age profile at Crete have been made. Results from existing ice cores and bore holes in Greenland and norrn Canada show that large ice sheets are not in a steady state. There are several reasons for this: (b) The temperatures in central parts of large ice sheets are still cooled by cold climate of Wisconsin period. At Dye 3, this cooling is calculated to be 5 C (Oahl-Jensen and Johnsen, 1986). (c) The ice sheets are not in mass balance. It has not yet been possible to measure wher cen tral parts are thinning or thickening (Paterson, 1981, p ; Reeh, 1985). have investigated what effect (a) and (b) would have on model solution in vicinity of ice divide, with assumption that ice thickness at Crete is kept constant at its present value. The values from Table I are used unless orwise noted. Figure 4 shows vertical velocity at ice divide and surface slope in vicinity of ice divide has been calculated for each of following cases: (a) Enhancement factor E = I steady-state Holocene situation. for z H. This is a (b) Enhancement factor E = 3 for z H. This is still climatically a Holocene situation but enhancement factor has been raised to 3 as for Wisconsin ice. (c) Enhancement factor E = I for z.5h. 3 for.5h z H. This enhancement-factor distribution is close tq that expected at Crete at present, where transition between Holocene and Wisconsin ice is believed to be 15 m above bedrock. (d) Enhancement factor E = I, as in (a), but geormal heat flux has been lowered from 4 mw / m2, as used in (a)-(c), to 3 mw/ m2. This lowers basal temperatures by 4.5 C. (e) Enhancement factor E = I, as in (a), geormal heat flux 5 mw / m 2. This increases basal temperatures by 4.5 C. (f) Enhancement factor E = and geormal heat flux 4 m W/ m2 as in (a), but surface temperature has been lowered by 12 C from -3.4 c, as used in (a)-(e), to c. This is cooling expected during glaciation. (g) Enhancement factor E = 3, geormal heat flux, and surface temperature as in (f), but accumulation rate has been changed from.29 m/year, as used in (a)-(f), to.1 m/ year. These are conditions which could be found in a teady-state Wisconsin ice sheet. It can be seen from Figure 4a that vertical velocity profiles (a)-(f), in which accumulation rate has been kept constant, are nearly identical. They are not seriously affected by changes of enhancement-factor distribution «a)-(c», nor by changes of temperature distribution «a), (d), (e». However, increase of surface slope with distance from ice divide does change in order to maintain mass balance (Fig. 4b). Close to ice divide, where horizontal velocity is small, depth-age profile of ice is controlled by ;- 2. b) 25 "' 1: & 2... C.Q ' (a) The ice deposited on ice sheet during Wisconsin period, last glaciation, deforms more easily than ice from Holocene (Reeh, 1985; Fisher and Koerner, 1986; Dahl-Jensen and Gundestrup, 1987). The boundary between Holocene and Wisconsin ice moves downward in ice with time and this changes rheology of ice sheet. As a first-order approximation, this phenomenon can be included in Glen's flow law by an enhancement-factor value of I in Holocene ice and of 3 in Wisconsin ice. Vi > u.g W 1 :J V1... b 5 ==:: :=-...:.: r Velocity (m/yr) Fig. 4. Vertical velocity-depth profiles for cases (a)-(g) in Figure 4a. In Figure 4b, surface slope from ice divide (x = km) and 5km down-stream from cases (a)-(g) are shown. vertical velocity. It is refore seen that depth-age profile close to an ice divide cannot be seriously affected by distribution of enhancement factor or by temperature through ice. This is an astonishing result which is due to special situation at ice divide. The stress configuration here is dominated by longitudinal stress deviator, which can be interpreted as drag of ice sheet on ice divide. The shear stress is small in this area. Away from ice divide, where shear stress dominates, flow will depend on temperature and enhancement-factor distribution. During Wisconsin period, surface-accumulation rate on Greenland ice sheet was lower than at present. Most derivations of average Wisconsin accumulation rate suggest values of one-half to one-third of present (Nye, 1963; Hammer and ors, 1978; Beer and ors, 1984; Paterson and Waddington, 1984; Dahl-Jensen and Johnsen, 1986). Although temperature and enhancement-factor distributions at ice divide have little influence on vertical velocity profile, accumulation rate has a large influence when ice thickness at ice divide is kept constant. In Figure 4a, vertical velocity profile with an accumulation rate of.1 m/ year, which is one-third of present, is shown. The velocity profile (g) is seen to have same shape as (a)-(f) but scaled with.33 as accumulation rate. In area close to ice divide, it is possible to derive a "non-steady-state" model by combining a steady-state Holocene model and a steady-state Wisconsin model in order to calculate a depth-age profile here. In Wisconsin model, surface temperature is -12 C colder than present temperature (-42.4 C), enhancement factor is 3, and accumulation rate is reduced to 33% (>'gl =.1 m/ year) or to 5% (>'gl =. 15 m/ year) of present accumulation rate. In Holocene model, present surface temperature (-3.4 C), accumulation rate (.29 m/ year), and enhancement factor E = I are used. The fundamental and critical assumption is that ice thickness at ice divide has been unchanged with time. How ice thickness at Crete has actually changed is unknown (Reeh, 1985). Lowering of surface temperature will produce a cold wave that will reach bedrock after 1-2 kyear. This will make ice more difficult to deform and reby increase ice thickness at Crete. The ice deposited during Wisconsin period, however, deforms more easily than ice from Holocene period. After 5 years of glaciation, most of ice will be of Wisconsin type and this softer ice will result in a decrease of ice thicknesses at Crete. Decreasing accumulation rate will both change mass balance of ice sheet and increase basal temperatures. To obtain a complete answer to this question involves non-steady-state modelling (Budd and Young, 1983). In this reconstruction ice thickness at dome is kept constant. 33

4 Dahl-Jensen: Modelling of flolv in central Greenland The "non-steady-state" model is created by repeating a glacial cycle with 1 kyear of Wisconsin conditions ( steady-state Wisconsin model) and 1 kyear of Holocene conditions ( steady-state Holocene model), assuming constant ice thickness at ice divide. 2.5 kyear are allowed between each step in which temperature and accumulation rate are shifted. This attempt to make a "non-steady" model near an ice divide is justified by fact that changing temperatures and enhancement factor in ice has little effect on vertical velocities. The vertical velocity, however, is sensitive to mass balance and some time must be expected before ice sheet has adjusted to "new" kinematic conditions, so steady-state models give reasonable predictions of velocities. Isochrones are constructed by following particle paths with time. In Figure 5, isochrones marking transitions between Wisconsin and Holocene conditions are -3.4n-n O.29n-n Temperature (OC) Accumulation rate (m/yr) -42.4A 12 1 BO r2 1 Years S.P : Years S.P. 1>=. 1O. 1>,,=.15 i:i\i:! li l mlrr mlrr 2 c o 1 15fllR!.11 w 1 5 O o lilt (deg) Fig. 5. Model calculations of depth-age distribution in vicinity of an ice divide. The climatic inputs used in calculations are surface temperatures and accumulation rates through ails kyear glacial-interglacial c ycle as shown at top of figure. Two cases are shown, )..gl =.1 m/ year and )..gl =. 15 m/ year. The isochrones are those from sfart and end of glacial periods, i.e. 1 kyear B.P., 115 kyear B.P., 125 kyear B.P., 235 kyear B.P. The shaded areas represent intermediate glacial periods. The isochrone bump at ice divide will result in a tilted annual layer, which is shown in last frame of Figure 5. presented. At ice divide it is seen that isochrones rise, creating an isochrone bump. At ice divide re are 16 m of Holocene ice, 1 m of Wisco nsin ice, and 5 m of ice older than 115 kyear when kgl =.1 m/year. For kgl =.15 m/ year, re are 16 m of Holocene ice, 11 m of Wisconsin ice, and 4 m of ice older than 115 kyear. For both examples, it can be seen that five ice thicknesses from ice divide (15 km) thickness of ice older than 115 kyear is reduced to half that at ice divide. The model predicts 6-8 m of Eemian ice at ice divide and 3-5 m of Eemian ice at a 15 km distance from ice divide. The isochrone bump at ice divide is only one to two ice thicknesses wide, so isochrones will have a slope because of this bump. Annual layers in a core drilled at an ice divide will refore be tilted. The slope of annual layers between ice d ivide and one-half ice thickness away are shown in Figure 5. The maximum slope is m above bedrock. Increasing layer tilting from 5 at 2 m depth to a maximum of 13 at 15 m 34 depth have been observed in two surface to bedrock cores (total ice thickness 13 m) taken within 3 m of top of Agassiz ice cap, Ellesmere Island, Canada. These have been attributed to proximity to top of ice cap (personal communication from R.M. Koerner, D.A. Fisher, and N. Reeh). BASAL TEMPERATURE AT CRETE The calculations of isochrones at Crete are based on assumption that re has never been melting at hte bedrock, i.e. bedrock temperature has never exceeded pressure-melting point of -2.4 cc. Paterson and Waddington (1986) have discussed range of geormal heat fluxes that would give non-melting conditions at bedrock. It was concluded that, with a heat flux of less than 48 mw / m2, bedrock temperature is lower than pressure-melting point. If bedrock is Precambrian rock with a heat flux of 38.7 ± 8 mw/ m2 (Lee and Uyeda, 1965), ice has most likely been frozen to bedrock throughout last glacial-interglacial period. The non-steady-state temperature model used to calculate temperature profile at Dye 3 (Dahl-Jensen aned Johnsen, 1986) is applied to Crete. The velocity and stresses used in temperature calculations are those derived from steady-state flow model for Crete. The results agree reasonably well with those of Paterson and Waddington (I986). In order to avoid repeating results presented by Paterson and Waddington, we shall discuss here basal temperature-dependence on glacial accumulation rate. The geormal heat flux is set to mean value for Precambrian bedrock, i.e. 4 mw / m2, and climatic input; in this case, surface temperature and accumulation rate are kept simple as shown in Figure 5. It is known that temperatures and precipitation rate during Wisconsinan stage was not constant but, for this purpose, it is sufficient to use average values. The glacial surface temperature is C, 12 C colder than prese nt surface temperature at Crete. It is seen in Figure 6 that basal temperature changes 8 C when accumulation rate during glaciation is one-third to two-thirds of its present value. When accumulation rate is two-thirds of present, basal temperature is close to steady state at end of glaciation (I5 kyear B.P.). When warm period begins, basal temperature first decreases because Ag1 =.lom/yr u o Ag1 =.15m/yr -7.5 _.- Ag1=.2m/yr '--"" ,# ",.-' /'., j' "'''' ' --'-' -'-, r-r-'--t--r-r---r---r-...,.--r-...,...t'-i o 1 Years B.P. Fig. 6. Calculations of basal temperatures through 115 kyear glacial-interglacial cycle shown at top of Figure 5. In all of three cases presented, mean glacial temperature is Cc.

5 Dahl-Jensen: increased accumulation rate increases advection of cold from near surface ice to bedrock. It takes 15 kyear after termination of warm period for "warm" wave from this warm period to reach bedrock and for basal temperatures to reach maximum values. When glacial accumulation rate is reduced to one-third of its present value, temperature decrease under warm period is more pronounced, while temperature increase after termination of warm period is smood out more and a stable basal temperature at termination of glacial period (12.5 kyear B.P.) is not reached. In general, a decreased glacial accumulation rate results in increased temperatures at base. For each case, basal temperature changes 2 C throughout glacialinterglacial cycle. The basal temperatures depend strongly on unknown geormal heat flux and past precipitation rate at Crete. DISCUSSION order to discusss assumptions for In "non-steady-state" ice-divide model and possibility of observing rises in isochrones, basal temperatures, and surface strain-rates at an ice divide, we must investigate time-scales needed to reach steady-state conditions. The temperature calculations made through a glacial cycle show that basal temperature reaches a constant value after 1 year or more, which must be time-scale needed for temperature to reach static conditions at an ice divide. Outside ice-divide area, steady state of ice thickness, i.e. mass balance of ice sheet, depends on temperature of ice and on ice rheology. Since re are internal layers in ice consisting of intermediate ice with an enhancement factor of I and of glacial ice with an enhancement factor of 3 that moves with ice flow, rheology changes with time. The depth<lge profiles in Figure 5 show that several glacial periods must be expected in ice, all slowly sinking down through ice. The time-scale needed for ice sheet to reach mass balance is refore more than a glacial-cycle length of 115 year. Finally, velocity and stress distributions through ice sheet are believed to adjust mselves within some thousands of years to actual temperature distribution, accumulation, and geometry of ice sheet because y are mostly determined by momentum balance of ice. At ice divide, ice thickness is assumed to be constant, and ten ice thicknesses away from ice divide ice thickness changes less than I% between situations (a)-(g) in Figure 4. The area considered is refore always close to mass balance. The vertical velocities in this area are somewhat independent of temperature distribution and depth variation of enhancement factor, so positions of isochrones found by "non-steady-state" model are believed to be reasonable. The isochrone and temperature results are very dependent on temporal changes in precipitation rate, as well as assumption of constant ice thickness at ice divide. The precipitation rates during last glacial period and earlier are poorly known, so isochrones can only be considered reliable as far back as Wisconsin / Holocene transition at z = 15 m, 1 year B.P. (Reeh, in press). The very simplified climatic input and approach to a non-steady-state model by combining two steady-state models is refore believed to give a view of what can be expected at Crete. The phenomenon of an isochrone bump, which implies that amount of ice-core material older than 115 year from a core drilled at divide is twice as large as in an ice core drilled 15 km away, is interesting. It is deduced from model calculations with any climatic history, as long as no bottom melting has occurred. The presence of an increased amount of old ice at Crete depends on how stable position of ice divide has been. The isochrone bump is very narrow and is only found in a 12 km area around ice divide. Model calculations on glacial ice cover of Greenland with a perfectly plastic ice-sheet model (Reeh, 1982, 1984) estimate Mod elling of flow in central Greenland an ice-divide movement of less than 5 km to west. Earlier positions of ice divide are unknown but even small movement would make isochrone bump and temperature bump less pronounced. The increase in surface strain-rate, however, is expected to adjust within some thousands of years to present position of ice data do suggest surface strain-rates as divide. E.G.I.G. large as 1.3 x 1-3 year- l at ice divide (Hofmann, 1986; personal communication from S.J. Johnsen). The situation of Crete on north-south-trending ice divide (Fig. I) and south-trending surface slope suggests a small south-trending flow along ice divide. A study of how this affects west-east-trending bump cannot be done by applied two-dimensional model but this probably also tends to smooth bumps of temperatures and isochrones. Internal layers seen from radar echo-sounding in this area do not show a rise near Crete. However, quality and cover of area are insufficient to disprove that internal layers have a local bump (Gudmandsen, 1973). New radio echo-soundings made in 1987 could change this picture. CONCLUSIONS The model calculations presented here show that a deep core at an ice divide has advantage of yielding more old ice. It is predicted that re would be up to twice amount of ice-core material older than 1 kyear in an ice core from ice divide than 15 km away. The isochrone rise at ice divide at Crete is probably less pronounced because of slow flow along ice divide from site Summit, 1 km north of Crete, towards Crete, and a possible movement of ice divide with time. The estimates of basal temperatures at Crete show a wide range of varying values, mainly due to unknown geormal heat flux and past accumulation rates. Most of estimates, however, suggest that basal ice has been frozen throughout last glacial period, so old ice is preserved at Crete and along first part of flow line. Since data material from Crete is so sparse, model predictions are bound to be uncertain. Surface velocities and surface strain-rates in area would be very helpful. And eventually, one day, perhaps data from an ice core and a bore hole at Crete will increase our knowledge of past climatic conditions and of ice flow close to an ice divide. ACKNOWLEDGEMENTS This work has been funded by Danish Science Research Council, European Economic Communities, XII Directorate General (contract CLI.67.DK), and Danish Commission for Scientific Research In Greenland. REFERENCES Beer, J., and 6 ors Temporal vanatlons in lobe concentration levels found in Dye 3 ice core, Greenland. Ann. Glaciol., 5, Budd, W.F. and N.W. Young Application of modelling techniques to measured profiles of temperatures and isotopes. In Robin, G. de Q., ed. 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