Journal of Glaciology, Yol. 31, No. lOS, 1985
DETERMINATION OF THE FLOW PROPERTIES AT DYE 3, SOUTH GREENLAND, BY BORE-HOLE-TILTING MEASUREMENTS AND PERTURBATION MODELLING
By DORTHE DAHL-JENSEN (Geophysical Institute, Department of Physical Glaciology, University of Copenhagen, Haraldsgade 6, DK-2200 Copenhagen N, Denmark)
ABSTRACT. The ice-flow properties at great depths que de la faible taille des cristaux de glace au cours du have a strong impact on the depth-age relationship of Pleistocene. La parametre de la loi d'ecoulement ainsi the strata in the Gree::land ice sheet. Previous attempts determine et la topographi~ du lit sont utilises dans un to calculate this relationship have used Glen's flow law modele de perturbat;on du premier ordre a deux with a temperature-depcndent flow-law parameter. The dimensions, multi couches pour calculer les ondulations data from the bore-hole-tilting at Dye 3, south de surface et les taux de deformation. Les resultats sont Greenland, make it possible to calculate the flow-law en bon accord avec les observations. lis sont tres parameter versus depth. The flow-law parameter shows a sensibles aux variations due profil du parametre de la big change at the Holocene/ Pleistocene transition with a loi de deformation, qui peut cependant etre estime dans mean flow enhancement factor of 3 for the ice d'autres zones par le modele de perturbations le long des deposited during the last slaciation. This high lignes d'ecoulement lorsque les topographies en surface et enhancement factor is believed to be due to high sur le lit sont connues. concentrations of dust and other impurities and small ice~rystal size in the Pleistocene ice. The derived ZUSAMMENFASSUNG. Bestimmung der Fliessverhiiltnisse in flow-law parameter and the up~tream basal topography Dye 3, Sudgrollland, mit Hi/fe von Neigungsmessungen in are used in a two-dimensional, multi-layer, first- INTRODUCTION bedrock profiles. Whillans and lohnsen (1983) used a linear viscous flow law to model the Byrd Station, West Several authors have treated the relationship Antarctica, flow line, 10hnsen and others (1979) and between glacier surface and bedrock topographies. Nye Rasmussen (unpublished) introduced 'a multi-layer model (1959) calculated a first approximation using Glen's with Newtonian viscous flow, which was used by flow law in a simple way. As more and better data on Overgaard and Rasmussen (1979) on the EGIG line, surface and base profiles became available, perturbation south Greenland. Reeh and others (in press) modelled the models were developed to explain the small~cale flow-line up-stream from Dye 3, south Greenland, using undulations caused by an irregular base. Budd (1970) Glen's flow law with a flow-law parameter which varies described a perturbation method for Newtonian viscous exponentially with depth to account for changes in me~ia, and Hutter and others (19SI) gave a systematic stress, temperature, ice fabrics, etc. The predicted Dye 3 analysis of the governing equations and boundary surface undulations were 50% smaller than observed. conditions for first- 92 Downloaded from https://www.cambridge.org/core. 05 Oct 2021 at 17:35:54, subject to the Cambridge Core terms of use. Dahl-Jensen: Flow properties at Dye 3. south Greenland measurements of the temporal changes of the hole suggested by Nye (I959). The dashed lines in Figure 2 geometry. It will be used in a first~rder, multi-layer, represent the lower and upper boundaries to the basic two- THE FLOW-LAW PARAMETER In order to simplify the calculation of the f1ow-law 65.1 parameter, the flow is assumed to be two- -'-----....>...,-"'--->--.l...... J'---'--'--"---->--,-'---'-'--'-_-'-_-'-65 .0 deformation is: 44.5 IOWI (Ju t -+-(Jwl (I) Fig. I. Surface elevation map of the Dye 3 region (Jz (Jx (Overgaard and Gundestrup. in press). elevations in t metres above sea-level. The Dye 3 drill hole is located at point lB. Basal and surface elevation profiles are u and w being the horizontal and vertical velocities, B measured along the three parallel lines A. B. and C . The the f1ow-law parameter, ax' az , TXZ the stress filled circles mark the ends of the line intervals used in components, x the axis parallel to the basal trend line, the model. and z the axis perpendicular to the x-axis. Near the bottom of the bore hole the deformation is dominated by shear, which allows the following approximation to Equation (J): others, 1984} and repeated satellite fixed p~i!lts (Zwally and others, 1983) the flow line (dotted line on Fig. I) is au 3 t - = B T • (2) estimated. Between 14B and -7D it is close to the B-line. (Jz xz In Figure 2 the surface and base elevations are shown along the A, B, and C lines. Obviously, the bedrock The f1ow-law parameter B is considered to be a topography has undulations of up to 15% of the ice function of temperature, ice fabric, ice crystal size, and thickness. The dashed lines in Figure 2 are trend lines; impurity concentration in the ice. the base trends are determined as the least-iquare linear The shear stress T xz is calculated as fit to the observed base-eleva tion profiles, the surface trends as a least-iquare fit to an ice-iheet profile TXZ = pg sin 00 (Ho - z), (3) -, .jj====:J ij~~!-I it:3 0.0 10.0 20.0 30.0 10.0 0.0 1 ~ . 0 2~. 0 3~ . 0 1~.O 50. 0 10 .0 20.0 30.0 10.0 ~~- ~ ~~- ~ ~~~ ~ 0,-______..., o ,------r---, g .... 0 b) '"0 o § 0 .... o ...... o 0 0 1:g ...... 0 In 8 ...... '" In o o 0 0+r~~~~_~~~~~~~_~ 0+r~~~_~~~-~~---~-~~ 0 0.0 10 .0 20.0 30.0 10.0 0.0 10 .0 20 .0 30.0 10.0 50 . 0 15 . 0 25 .0 35 . 0 Obta>co ~.-un. (km) OlS1Once ~ e- Fig. 2. Sur face and basal elevation profiles along the A . B. and C lines. The full curves are observed and the dashed curves are trends of the profiles. The X- 93 Downloaded from https://www.cambridge.org/core. 05 Oct 2021 at 17:35:54, subject to the Cambridge Core terms of use. Journal of Glaciology DUDZ, DYE 3 B, DYE 3 E. DY[3 According to Equation (4), E equals unity for ' 10'· 250 m < z , ';00 m, and this is assumed to hold in all 000 O ~ 0.0 M of the Holocene ice (z > 250 m). In the Pleistocene ice (0 < z , 250 m), E is calculated using Equation (5) and ') b ) ~ ~ c) shown on Figure 3c. This part of the B-profile is probably to be explained by varying impurity ;; concentration, crystal size, and ice fabric, ~ The measurements of ice-crystal size (Herron and others, in press), dust concentration, and pH (Hammer and others, in press) along the deep ice core all give b ;; profiles characterized by nearly constant values in the ~ b b Holocene ice, and different and highly variable values . in the Pleistocene ice. The crystal size decreases by a ~ factor of two at the Holocene to Pleistocene transition; !} the dust concentration increases by one to two orders of § 1 ~ ~ I ! magnitude, and the ice becomes alkaline. The shape of I I I the dust concentration profile is very similar to that of I I I the deformation rate au/az profile (Gundestrup and ~ o 0 h rl Hansen, 1984). 0.00 0.• 0.0 '0 2 B ( Po"",, ' ) ' 10'· E The single-maximum c-axis fabric of the ice crystals intensifies with increasing depth (Herron and others, in Fig. 3. Flow-law parameters derived from the Dye 3 press), but it does not change drastically at the said bore-hole-lilting measurements. transition to judge from the few measurements available (a) The full curve shows au/ az for the bottom 700 m, in Pleistocene ice. Hence, the high deformation rates in points are the 50 m averaged au/ az values and the Pleistocene ice cannot be explained by a c-axis dashed curve represents values calculated from the alignment. This is why Gundestrup and Hansen (1984) model. concluded that dust concentration, and possibly other (b) The flow-law parameter estimated by Equation (2) parameters that vary in a similar way, are responsible for the bottom 700 m is shown as the full curve. The for the easy flow in Pleistocene ice. It should be dashed curve is the best fit to the Arrhenius equation mentioned that the enhancement factor defined by (3). The B-profile used in the model is the full curve Equation (5) agrees with the observed variations of dust, for z'250 m and the dashed curve for z > 250 m. etc., over the entire ice-sheet thickness, (c) The enhancement factor E. The enhancement factor is In the perturbation model to be designed below, we the factor of the flow-law parameter B which depends on also need the longitudinal strain rate au/ax(z), which is the impurity concentration. size, etc., in the ice. The given by Glen's flow law (Glen, 1955): enhancement factor does not depend on the ice temperature. (6) p being the ice density, g the gravitational acceleration, 00 the mean surface slope at Dye 3 (averaged over Glen)s flow law (Equation (I) and (6» describes the 10 km (5Ho»' and Ho the mean ice thickness at Dye 3 deformation rates of istotropic ice. As the enhancement (averaged over 10 km (5Ho»' . factor E is assumed to depend on parameters which Perturbation models (Re eh and others, in press) hardly affect the isotropic properties of the ice, Glen's indicate that the simple linear equation (3) gives a flow law remains valid and the flow-law parameter B in satisfactory description of the shear stress at the deep Equation (6) is given by Equation (5). hole, in spite of the mountainous bedrock topography in Observations of the mean-diameter profile in the the area. Dye 3 bore hole (Gundestrup and Hansen, 1984) support The estimate of au(z)/az from the bore-hole tilting these considerations: the slowly increasing diameter with and the calculated shear stress TX Z from Equation (3) depth is caused by overpressure in the hole fluid since allow calculation of B for the bottom 700 m The full the hole was drilled, but the big diameter increase at curve in Figure 3b shows the calculated B-profile. At the Holocene/Pleistocene ice are features similar to the the Holocene to Pleistocene transition (z = 250 m), B observed variations of the au/az and dusH:oncentration increases by a factor of three, in agreement with the profiles. These diameter changes in the Pleistocene ice experiments of Shoji and Langway (in press), and in the must result from higher and variable values of the 25 m of sil ty ice close to the base, B reaches a level flow-law parameter in Equation (6), resulting in a approximately seven times higher than the Holocene B-profile here similar to the B-profile in Equation (5), value, which is 1.0 x 10- 17 Pa- 3 year- 1, in agreement The enhancement factor E is therefore used to with the flow-law parameter suggested by Paterson model ~he softer ice, deposited during the last glaciation, (1981), In the Holocene ice, B decreases slowly upwards, and E IS not used to model anisotropic ice, These trends will be interpreted below. As a final test on the derived flow-law parameter B As to the Holocene ice, the decreasing temperature (Equation (5», the au/az profile has been calculated by upward in the bore hole (Gundestrup and Hansen, 1984), Equation (I). TXZ is calculated by Equation (3), aw/ax is essentially explains the variation of B, as can be neglected and the longitudinal stress deviation (a - a ) verified by applying the Arrhenius equation in the is modelled: x z interval 250 m < z , 700 m: (I) using the horizontal velocity profile u(z) from the ..4(T) - Ao exp [~] , (4) bore-hole-tilting measurements, Q being the activation energy (60 kl/mol) (Paters on, (2) assuming the profile form of u(z) to be slowly 1 1 1981), R the gas constant (8.314 1 mol- K- ), and T the changing in x along the flow line, absolute temperature. The constant Ao is calculated to 6 3 1 be 9.8 x 10- Pa- year- by inserting the measured B (3) using accumulation rates along the flow-line and T profiles through the said interval. (Reeh and others, in press; Ragle and Davis, 1962). In order to explain the Pleistocene B-profile, it is practical to divide B into a temperature- 94 Downloaded from https://www.cambridge.org/core. 05 Oct 2021 at 17:35:54, subject to the Cambridge Core terms of use. Dahl-Jensen : Flow properties at Dye 3. south Greenland Q being the mass flux determined either by the where u, W , ax ' a z ' and T;CZ are the velocities and the stresses caused by the baSIC flow. So(x) is the surface accumulation rates along the flow line Q = fa(X' )dx' trend line, which is the upper boundary for the basic flow, and the second o Ho terms on the right-hand sides are the perturbation terms caused by the basal perturbations or by the bore-hole tilting measurements Q = J u(z)dz. (9) (Hutter and others, 1981). The basic terms in (9) have o an x -variation, so they can describe a "realistic" basic flow between the smoothed (a - a ) is calculated boundaries Ro(x ) and x z by inserting Equation (7) into So(X).In the perturbation Equation (6). model the perturbation terms in Equation (10) will be considered to The calculated {Ju / {Jz profile (Fig be sinusoidal in x . 3a) agrees with due to the harmonic basal that observed within the measuring perturbation given b y accuracy. Equation (9). this requires the In principle, the x-variation of the basic bore-hole-tilting measurements terms to be small compared should reveal a possible to the perturbation terms, deformation effect of the which limits the wavelengths increasing single-maximum in Equation (9) to a length c-axis orientation downward scale not asymptotically large in the Holocene ice compared with the ice . However, this effect must be thickness. The resulting negligible in perturbation model however the Dye 3 area, because the observed yields more "realistic· increase results by allowing the basic flow of {Ju / {Jz and B downwards are fully explained a small b x~ependence instead of using a theoretically y A(D in the Holocene ice. more co rrect basic flow independent of x (Hutter, [c 1983]; Reeh and others, in press). If the representations (10) are substituted into the field-equations (balance laws of mass and momentum, PER TURBA TION MODELLING OF THE ICE and Glen's flow law given by Equations (I) and (6», DEFORMA TION and if the terms of equal powers of E are collected, the first- R(x) = 2Rn cos (~x - 4>n) , (8) F3 = ·rx~ + h- (axo- azo )2 , n=1 where the basic shear stress T is where N in this analysis is 12. X Z O determined by Equation (3) and the longitudinal stress deviation For a first- a a , z x ~dSrdx at z S(x), (12) dS (10) dx U - w = at a being the accumulation rate at Dye 3. The first-{)rder boundary conditions for the field equations (11) are derived by substituting Equations (9) 95 Downloaded from https://www.cambridge.org/core. 05 Oct 2021 at 17:35:54, subject to the Cambridge Core terms of use. Journal of Glaciology and (IO) into the boundary conditions (12) and m m+l approximating them to the conditions at the mean base u 1 (Z m) = u l (0 ), Ro(x) and surface So(x) by Taylor series expansions. Note that the surface perturba tions Sl(x) appearing in m m+l w w (0 ), the approximated boundary conditions are considered as 1 (Zm) = l unknown. The field eql!ations (I I) were also used in the m m+l (14) modelling by Reeh and others (in press). However in a Zl (Zm) = aZl (0 ), that study F 1 was neglected, and F2 was put approximately equal to Fs' the depth variation of which m m+l was approximated by an exponential variation. In this TXZl (Zm) = T XZ1 (O), for m = 1, M - I. study the latter, rather rough approximation is avoided The M systems of field equations, and the and the ice is divided into M layers, in which F 2 , Fs' and B are assumed to be constant (Fig. 4). The terms corresponding boundary conditions, define the first-order containing Flare still neglected as they are an order of boundary-value problem. As the x-vanatIon of the magnitude smaller than the terms containing F 2 and F s first-order solution is entirely caused by the harmonic near the base, where (axo - '20) « TXZO (Dahl-Jensen, form of the basal perturbation ERl(x) = EHO cos WX, the unpu blished). expressions for the resulting perturb at ions will also be sinusoidal in x, with the same frequency. Specifically, the surface perturbation ESl(X) will be sinusoidal, and Ho Z ------Z-M----- with a phase shift ca,(w) relative to the base perturbation: m=M ESl = ESl cos wx + ES2 sin wx = F(w)Ho cos(wx - ca,(w) , o where the functions (15) m=M-1 o and (16) are called the filter function and the phase shift, respectively. When used on the basal Fourier series (9), they produce a Fourier series for the surface undulations. RESULTS Calculations of the filter function and the phase Fig. 4. Definition of the coordinate systems used in the shift for the Dye 3 region are shown in Figure 5. perturbation model. The filter function increases with the wavelength ). between 5 and 25 km (Fig. 5). The phase shifts are close to - ,,/2 for ). > 4 km. With these three assumptions the mth layer has the following field equations: Filterfunction, DYE 3 o 5 10 15 20 25 30 l..----' --- lL.1/") I/") o o m m ci ---- ci .5+ ~]__ (mm] m t [ 3 BFs TXZl' (13) o --- o 8z &x o o ci ci r 5 10 15 20 25 30 m m o ~l +~l 0, Wavelength /Ho 8x &z m m Phaseshift, DYE 3 ~ + ~l = 0, for ZE [0 , Zm1. 0 8z 8x 0 ci These field equations are nearly identical to the field equations for a Newtonian viscous fluid and can ~ ID""! I/") be solved accordingly by the use of the stress function (/)0 4-~==_+------+-----_+------+_----~----~~ci1\ (Hutter and others, 1981). The field equations have an 0 1 I . . m1]J mm .£ analytIcal solutIon because BI' 2 and B Fs are assumed a.... constant in each layer. It is an advattage to have ~ ~ "I 4-rT~~~-r~-.~-r+o~~~~~r+~-.~"1 analytical solutions, still with the use of non-linear 0 5 10 15 20 25 30 rheology. It greatly simplifies the problem, and hence it is more feasible for computer calculations. Wavelength /Ho As boundary conditions between the layers it is Fig. 5. The filter function and the phase shift calculated assumed that by the model for the Dye 3 region. 96 Downloaded from https://www.cambridge.org/core. 05 Oct 2021 at 17:35:54, subject to the Cambridge Core terms of use. Dahl-Jensen: Flow properties at Dye 3, south Greenland SIrdn Rat •• A-un. DYE 3 Strain Rat .. B-IIno DYE 3 SIrdn Rat .. c-un. DYE 3 20 JO .. " 20 '" a) ,. !l .. ,Ii' I ~. r. r" f\ r"1ci. / 1\ 1\ rlf\ jV - [Zj V lot-V'' , V\ ll-\, , [-1 I ..! 1 '>.0 lA h\' ~ '\ --V ~\ .v--: -} ~v: I \-ij d if "'0o ~ '- -- or \If .; - I IV V 1\ I I I 1 1 1 , 20 JO ., ,. 20 30 .. 10 ~ .. 10 Obtance -.g A-IIno (km) Dlstanc. clang" B-IIno (km) 0fs1 Surface Fit. A-iJne DYE .3 Surface m, 8-f." DYE 3 Su1cx:. Fit, C-une OYE .3 JO .. ~ ,. .. b) 11 '" 2 2 " 2 lI+e __ ~ __ ~"~~ __ ~~~ __ ~-f4'~ (\ ~ ~o.l f ~ wrJ~ \., .../ \ '.; \f.; .... 7+e--~~-,.~~---~~--~--t07 Distanc. ciong c-r",. (km) 80se Deviations. A-Line O'1! j 80se Deviations. 8-f1ne DYE .3 Bas. Qov;ations. c-un. DYE 3 ,. ~ 20 '" ,. " §e c) I I ( I I \.. j\ f' I 1\ ::1-0 1-\/\ h '\ r\ l0 1\ V\ -V '-' V ~ I\l \ r 1\ (U '-r... ~\; \J , V: , ,. 30 .. 10 ,. " .. ~ .. Ilbtanc:. -.g ,,__ (km) .. 0Ist0nc. clang B-IIno (km) 0I0t Fig. 6. Comparison of the calculated surface undulations and strain-rates with those observed for the A. B . and C lines. (a) 12-component Fourier representation of the modelled strain-rates (full curves). The dashed step curves are the average strain-rates measured by Whillans and others ( 1984). ( b) J 2-component Fourier representation of the modelled surface undulations (full curves) and the observed undulations (Overgaard and Gundestrup. in press) (dashed curves). (c) Basal deviations from the trend lines in Figure 2b (full curve). A 12-component Fourier representation of the curves (dashed curves) coincides completely with the full curves. Comparison with the observed surface deviations the calculated strain-cate amplitudes seem to be too big. (Fig. 6b) shows good agreement for the A, B, and C This might be expected (in particular for wavelengths lines in Figure 1. The dominating wavelength of the smaller than 3Ho = 6 km) in view of the approximations surface and basal undulations is 8 km (4Ho)' so with in the first- 97 Downloaded from https://www.cambridge.org/core. 05 Oct 2021 at 17:35:54, subject to the Cambridge Core terms of use. Journal 0/ Glaciology required computer time. The number of layers can be Hutter, K. [cI983.] Theoretical glaciology; material science chosen to agree with the amount of information of ice and the mechanics 0/ glaciers and ice sheets. available regarding the flow-law parameter. Dordrecht, etc., D. Reidel Publishing Company; Tokyo, The flow-law parameter derived from the Dye 3 Terra Scientific Publishing Company. deep-bole measurements, and the basal topographic Hutter, K., and others. 1981. First-order stresses and observations are used as inputs in the perturbation deformations in glaciers and ice sheets, by K. Hutter, model to predict the surface profile and the strain-rates. F. Legerer, and U . Spring. Journal of Glaciology, Vo!. The agreement with the obsei"ved surface profiles and 27, No. 96, p. 227-70. strain-£ates shows that the applied flow-law parameter Johnsen, S.1 ., and others. 1979. Theory of deformations profile is realistic. within ice sheets due to bottom undulations, by S.J. Analyses show, that the results from the Johnsen, K. Rasmussen, and N. Reeh. Journal of perturbation model are very sensitive to the flow-law Glaciology, Vo!. 24, No. 90, p. 512. [Abstract.] parameter used as an input. Beyond being used with a Nye, J.F. 1959. The motion of ice sheets and glaciers. basic flow model, to calculate the age~epth relationship Journal 0/ Glaciology, Vo!. 3, No. 26, p. 493-507. and the flow, the perturbation model can be used on Overgaard, S., and Gundestrup, NS. In press. Bedrock other flow lines where base and surface profiles are topography of the Greenland ice sheet in the Dye 3 known, to estimate the flow-law parameter "·:versus area. (In Langway, C.C., jr, and others, ed. The depth. Greenland Ice Sheet Program. Edited by C.C. Langway, jr, H. Oeschger, and W. Dansgaard. Washington, D.C., American Geophysical Union. (Geophysical Monograph Series.» ACKNOWLEDGEMENT Overgaard, S., and Rasmussen, K . 1979. Study of deformations within ice sheets due to bottom The Department of Physical Glaciology, Geophysical undulations by means of radio echo-sounding. Journal Institute, have given valuable help and discussions. Lyle of Glaciology, Vo!. 24, No. 90, p. 513. [Abstract.] Hansen and Niels Gundestrup very kindly provided the Paterson, W.s.B. 1981. The physics of glaciers. Second data from the logging of the Dye 3 deep hole. This edition. Oxford, etc., Pergamon Press. (Pergamon work has been funded by Danish Natural Science International Library.) Research Council and the European Economic Paterson, W.s .B. 1983. Deformation within polar ice Community, XII Directorate General (Contract sheets; an analysis of the Byrd Station and Camp Century borehole4ilting measurements. Cold Regions CLI.D67 DK.l). Science and Technology, Vo!. 8, No. 2, p. 165-79. Ragle, R.H., and Davis, T.C., jr. 1962. South Greenland traverses. Journal of Glaciology, Vo!. 4, No. 31 , p. 129-31 . [Correspondence.] Rasmussen, K. Unpublished. Isbevaegelse over en REFERENCES undulerende bund. [MSc. thesis, University of Copenhagen, 1977.] Budd, WF. 1970. Ice flow over bedrock perturbations. Reeh, N., and others. In press. 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MS . received 7 August 1984 and in revised form 21 September 1984 98 Downloaded from https://www.cambridge.org/core. 05 Oct 2021 at 17:35:54, subject to the Cambridge Core terms of use.