late wisconsin ice-surface profile calculated from esker paths and types, katahdin esker system,...

QUATERNARY RESEARCH 23, 27-37 (1985)

Late Wisconsin Ice-Surface Profile Calculated from Esker Paths and Types, Katahdin Esker System, Maine

RONALD L. SHREVE

Department of Earth and Space Sciences and Institute of Geophysics and Planetary Physics. Unitvrsity of Cal(fornia. Los Angeles. Caljfornia 90024

Received November 21, 1983

Values of the gradient of the former ice surface can be inferred at points along a flow line from deviations of esker paths or transitions in esker type and numerically integrated to give the profile. A profile calculated in this way shows that during formation of the Katahdin esker system about 12.700 yr ago the ice thickness at distances of IO, 20. 50. 100, and 140 km from the terminus, which is about two thirds of the distance to the ice divide. was approximately 200, 300, 600, 750. and 900 m. The terminal reach was computed by assuming an unknown uniform basal drag and matching the profile to its known elevation at the terminus and known gradient IO km upglacier. Correction for isostatic rebound based on the elevations of contemporaneous deltas and of the marine limit proved unnecessary, because the tilt due to the difference in uplift at the two ends of the profile is only 0. I m km-‘. With other plausible assumptions as to sea levels in the past, elevations of the marine limit, or exact location of the terminus the profile could be as much as roughly 100 m higher. It hits Mount Katahdin about 500 m below its summit, which is at 1600 m, in agreement with the geological evidence farther west. The steepening of the upper part of the profile suggests that the mountain dammed and diverted the ice. Basal drag computed from the profile varies from about 20 kPa (0.2 bar) near the terminus to 30 kPa (0.3 bar) at 100 km to 70 kPa (0.7 bar) at 140 km. The relatively low values away from the influence of Mount Katahdin agree with independent evidence from deep-sea cores of substantial late Wisconsin ice-sheet thin- ning without comparable areal reduction. The method has potential for application over wide areas that were occupied by the Laurentide and Scandinavian ice sheets. c‘ 198 Umvercity of Washington

INTRODUCTION

Where large eskers cross transverse hills and valleys having such gentle slopes that basal pressures in the ice deviated only slightly from glaciostatic, the local gradient of the former ice surface can be inferred from oblique reaches of esker paths or cer- tain transitions in esker type provided the inclination of the ground surface at the time of esker formation can be determined (Shreve, 1985). If this can be done at enough places along a single flow line, the profile of the surface can in turn be found by numerical integration of the curve of gradient versus distance from the terminus. Thus, the method potentially can give the profile far from the ice margins without ref- erence to any detailed model of glacier flow, basal sliding, ice temperatures, or mass balance.

The Katahdin esker system of eastern Maine provides an interesting test case, inasmuch as it traverses several suitable sites well spaced along a single flow line 140 km long (Fig. l), which is about two thirds the distance from the glacier margin to the ice divide. The whole esker system formed si- multaneously in a relatively short period during a minor maximum in which the ice terminated in tidewater near the present Pineo Ridge for a few hundred years about 12,700 yr ago, just prior to its final rapid retreat from Maine (Barns, 1973, p. 42, 1978, p. 104; Stuiver and Borns, 1975, p. 100; Borns and Hughes, 1977, p. 205; Thompson, 1980, p. 216; Shreve, 1985; H. W. Borns, Jr., 1983, personal communication). At that time the Saint Lawrence Valley probably was ice free, so that the ice in Maine flowed from an elongate center in the mountains along the Canadian border

27 0033-589465 $3.00 Copyright ,D 1985 by the Umverhity of Wathmgton All rights of reproductmn in any form reerved.

28 RONALD L. SHREVE

(Borns and Hughes, 1977, p. 204). Radio- carbon dates on high-level ice-contact lake deposits indicate that the ice surface lay below the summits of some of the higher peaks in western Maine, whose elevations are 900 to 1200 m, and hence well down the flanks of Mount Katahdin, whose elevation is 1600 m (H. W. Borns, Jr., 1983, personal communication). Thus, a rough independent test of the elevation of the computed profile at approximately 140 km from the terminus is possible.

DETERMINATION OF ICE-SURFACE GRADIENTS

Eight suitable localities were found on the Katahdin system (Table 1 and Fig. 1), of which six lie along a single flow line. Unfortunately, of these the one at Morrison Ponds cannot be used because the reach of broad-crested esker there, and hence the area over which the basal drag was effec- tively zero (Shreve, 1985), has a width and length greater than the probable ice depth, thus making the ice-surface gradient anomalously low, as Table 1 confirms. Thus, five localities spaced 25 to 46 km apart were usable.

The gradients were determined by two somewhat independent methods: the oblique-path method, which utilizes information from esker paths, and the transitiorz method, which utilizes changes in esker type. Both methods derive from the fact that the water pressure in subglacial tunnels beneath the interiors of ice sheets very nearly equals the glaciostatic pressure of the surrounding ice, which in turn is governed by the topography of the glacier bed and the ice surface. Quantitatively, the water pressure at the glacier bed is given to good approximation by

PW = Pig (h, - hG) (1)

(Shreve, 1972, p. 207). where h, and h, are the elevations of the ice surface and the glacier bed, pi is the density of the ice, and g is the acceleration due to gravity.

This equation neglects the small pressure differentials pIO and pLD due to ice inflow or outflow near the tunnel and to local dis- tortions of ice flow over hills and valleys. The order of magnitude of pIo can be estimated from the formula of Nye (1953, p. 482) for ice inflow to an open circular- cylindrical passage in a glacier. Assuming a

TABLE 1. LOCAL ICE-SURFACE GRADIENTS ON THE KATAHDIN ESKER SYSTEM

Distance from Ice-surface gradient” tm km-‘)

terminus Oblique-path Transition Value Locality Symbol (km) method method adopted

Trout Pond TP 142 9.@” - 9.6 Medway MW (110)” 4.0’ 4.6’,’ Not used Ayers Stream AS 108 5.P - 5.2 South Lincoln SL 81 4.Ih 4.1 Cardville cv (57)d - 4.5h Not used Otter Brook OB 56 4.4” 4.0” 4.2 Morrison Ponds MP 36 1 ,9”.” Not used Rocky Pond RP 10 - , 1 ,f.r II.

u Uncorrected for tilting due to isostatic rebound. b Ground surface poorly defined. c Independent measurements agree on two esker segments differing 15” in direction (Table 2). d Not on flow line, hence not used. e Ground surface well defined.

f Transition not verified in field. K Ground surface moderately defined. h Anomalously low, hence not used, because dimensions of broad-crested reach greater than ice depth

(Shreve, 1985).

LATE WISCONSIN ICE-SURFACE PROFILE 29

,

FIG. 1. Katahdin esker system, Maine. Flow line of ice (dotted) is mainly based upon esker direction in level areas. The ice divide (dashed) is from Borns and Hughes (1977, p. 204). Localities along the esker system at which ice-surface gradients were determined are listed in Table I. Other localities of importance are Mount Katahdin (MK). Hunt Mountain (HM), Whet- stone Mountain (WM). Mattamiscontis Mountain (MM), Passadumkeag Mountain (PM), Springy Brook Mountain (SBM). Lead Mountain (LM). Pleasant Mountain (PLM). Pineo Ridge (PR). Grindstone (G), and Bangor (B).

water discharge of 1000 m3 set -’ in the tunnel, which corresponds to an average ablation rate of 7 m yr- 1 over the whole drainage area of the Katahdin system, and using the ice-surface protile calculated farther on gives pto at most about 30% of pw computed from (1) and generally much less. The order of magnitude of pLD can be estimated by an approach analogous to that of Weertman (1964, p. 290) for strain rates near basal irregularities. The vertical normal component of strain rate over the stoss face of a hill will be roughly the upward velocity of the basal ice divided by the length of the face; the upward velocity in turn will be roughy the forward velocity multiplied by the gradient of the face; and the forward velocity can be approximated by means of the flow law of ice (Nye.

1952a, p. 84). Finally, calculating the vertical normal component of deviatoric stress from the strain rate by means of the flow law gives pin at most about 1% of pw. Thus, except within a few kilometers of the terminus, (1) is a reasonable approximation.

In the calculations to follow, however, what counts is not pw itself but its rate of change downglacier. Combining the formulas of Nye (1953, p. 482) and Shreve (1972, p. 209) with those for fully turbulent pipe flow (Schlichting, 1955, p. 422), assuming a relative roughness of 10PZ. and taking the same ablation rate and ice-surface profile as before gives the rate of change of pro less than about 10% of the rate of change of pw calculated by differentiation of (1). Likewise, dividing pLD by the length of the face gives its rate of change as at most about 35% that of pw and generally much less. Thus, again except very near the terminus, differentiation of (1) is justified.

Obliqlle-Path Method

Subglacial tunnels, and hence esker paths, go in the direction of water flow beneath the ice (Shreve, 1972, p. 211). The flow along the glacier bed, as elsewhere, is driven, not by differences in water pressure pw, but by differences in the hydraulic head h,, which can be defined either in the con- ventional way or in the slightly more con- venient unconventional form

Pigh, = Pw + PwghG (2)

(Shreve, 1972, p. 207), where pw is the density of water. Substituting (1) into (2), dividing through by pig, defining Ko, = (p, - pi)/pi. or approximately l/l 1, where the subscript stands for “oblique path,” taking the gradient of both sides (noting that h, and ho, and hence h,, are functions only of geo- graphic position and not elevation), substituting the (horizontal) vectors I, G, and H for -grad h,, -grad h,, and grad h,, and rearranging leads finally to the basic relationship

I + K,,G + H = 0. (3)

30 RONALD L. SHREVE

Vector I points in the direction of ice flow, which is most readily found from the direction of nearly level reaches of the esker, and has magnitude I equal to the tan- gent of the slope angle of the ice surface, that is to the ice-surface gradient. Vector G points in the downslope direction of the glacier bed and has magnitude G equal to the ground-surface gradient. Vector H is more complicated to interpret. The esker path goes perpendicular to the lines of equal 1zu, or equipotentials, on the ground surface. If these lines are projected onto a horizontal datum surface, the result is an equipotential-contour map of the glacier bed, exam- ples of which have been constructed for the Medway area by Shreve (1985). Vector H points upglacier in a direction perpendicular to the equipotential contours on the map. Thus, where the ground surface is horizontal, it points upglacier in the direction of the esker path: but, where the surface is obliquely inclined, it deviates toward the upslope direction. The deviation is negligible, however, for the small ground-surface gradients already assumed. Thus, to good approximation, H is parallel to vector E, which points upglacier it the direction of the esker path and has magnitude E equal to the esker-path gradient.

The significance of (3) is that the three vectors form a triangle when plotted head to tail (Fig. 2). The directions of all three sides and the length of side K,,G are known. Letting (Y and p be the included an- gles opposite sides I and K,,G, and solving for I by means of the sine law gives the desired formula,

I = K,,G sin a/sin p. (4)

Details of the computations made using this formula are given in Table 2.

Transition Method

Esker type is governed primarily by the rate of melting or freezing of the tunnel walls (Shreve, 1985): melting produces sharp-crested and multiple-crested eskers whereas freezing produces broad-crested

FIG. 2. Relationship among the magnitudes of the three vectors I, which points in the direction of ice flow, I&G, which points in the direction of down- ward ground slope, and H, which to good approximation points in the upglacier direction of the esker path.

ones. The heat for melting comes almost entirely from viscous dissipation in the flowing water. Not all of the heat generated is available for melting, however, because some is needed to warm the water to the increasing melting temperatures associated with the decreasing pressures downstream (Rothlisberger, 1972, p. 179; Shreve, 1972, p. 208). Even in level tunnels this uses about 30% of the total: and in ones that ascend downstream the pressure can decrease so fast that, instead of melting, the walls freeze.

For small ground-surface gradients the melting rate (or, if negative, the freezing rate) is given to good approximation by

M = Q[tl - k)PigH + kP&El Pixc (5)

(Shreve, 1972, p. 208), in which H is the magnitude of vector H, Q is the volume discharge of water through the subglacial tunnel, C is the circumference of the tunnel, A is the latent heat of fusion of ice, and the dimensionless coefficient k = pWc,y, or approximately 0.3, where c, is the specific heat capacity of water and y is the rate of decrease of the melting temperature of ice with pressure. At the point of transition from multiple-crested to broad- crested the melting rate is zero and there- fore, setting M = 0. dividing through by (I - k)pig, and letting KT = K,p + (p,ipi)ki (1 -k), or approximately 111.7, where the subscript stands for “transition,”


TABLE 2. COMPUTATION OF UNCORRECTED ICE-SURFACE GRADIENTS

Locality

Ground- Esker- Computed Direction Direction Angle Angle surface path ice-surfdce

Direction of ice of ground 01 P gradient gradient gradient Method” of esker surface surface (deg) (deg) (m km-‘) cm km-‘) (m km-‘)

Trout Pond lb OP N45”E SIYW N08”W 127 30 67 9.7 Trout ?ond 2” OP N30”E S15”W N08”W 142 15 44 - 9.5 Ayers Stream” OP N6Y”W S14”E NOTE 109 55 50 - 5.2 South Lincoln” T N07”E SOO”E 7 7 4.1 Otter Brook 1 t OP N30”W S14”E N34”E 116 16 IS 4.4 Otter Brook 2’ T N30”W S14”E N34”E 116 16 15 - 4.0 Rocky Pond” T N4Y”W S60”E II 18 II.

‘ l OP, oblique-path method: T, transition method. b 750 m N25”E of north end of Trout Pond. ( 500 m N21”E of north end of Trout Pond. Cl 310 m N86”W of summit of pass between Mud Brook and Sam Ayers Stream. r 2350 m S30”E of junction of Dodlin Road with Pine Tree Trail. ’ 1000 m N16”W of where Spring Bridge Road crosses Otter Brook. x 1250 m S33”E of outlet of Rocky Pond.

H = -(KT - K&E. (6)

Inasmuch as the basic relationship (3) applies everywhere along an esker, the triangle diagram (Fig. 2) applies and

H = I cos (3 + KopG cos a. (7)

Noting that -G cos (Y = E, eliminating H between (6) and (7), and solving for I then gives the desired formula in two forms,

I = K,E/cos p = - K,G cos &OS 6. (8)

In some cases one form can be used when the other cannot.

Details of the computations made using these formulas are given in Table 2.

Correction for Rebound

As a result of postglacial isostatic rebound the region of the Katahdin esker system has undergone considerable uplift since the time of esker formation about 12,700 yr ago. Deltas that formed contem- poraneously with the eskers near Pineo Ridge, for example, now have an elevation of 80 m (Thompson, 1980, p. 216). Inas- much as sea level at the time was about 70 m lower than that today (Dillon and Oldale. 1978, p. 59), the subsequent uplift has been

150 m. If uplift elsewhere along the esker system differed significantly from this amount, the ice-surface gradients computed from present ground-surface and esker-path gradients would have to be corrected for tilt. As will be seen, the correction is in fact negligible, but it is not possible to make a convincing case without carrying through the calculation.

What is needed is an estimate of uplift in the last 12,700 yr at a locality toward the headwaters of the esker system. Then the subsequent tilt could be computed as the difference of the two uplifts divided by the distance from Pineo Ridge, and the gradients of the ground surface and ice surface corrected accordingly. Because the area of esker formation was under ice 12,700 yr ago, however, uplift cannot be estimated di- rectly, as at Pineo Ridge, but must be calculated from a model of the rebound history, which necessarily involves the changing thickness of the ice as it retreated. Thus, the ice-surface profile and the rebound correction have to be calculated jointly. The simplest procedure is to calculate the profile using the uncorrected gradients, then to use it to compute the rebound and thence the tilt, and finally to cor- rect the gradients and recompute the

32 RONALD L. SHREVE

profile, repeating the process one or two times until the corrections become negligible.

A suitable locality for the rebound calculation is where the marine limit crosses the East Branch of the Penobscot River near Grindstone at an elevation of 91 m approximately 120 km from Pineo Ridge (Thompson, 1980, p. 213). This locality was at sea level when the ice left it about 300 yr after the time of esker formation (Stuiver and Borns, 1975, p. 100). Thus, inasmuch as sea level 12,400 yr ago was about 68 m lower than that today (Dillon and Oldale, 1978, p. 59), the subsequent uplift has been 159 m. The problem is to calculate the unknown uplift that occurred as the ice retreated during the preceding 300 yr.

A good assumption, borne out by nu- merous observed rebound curves (An- drews, 1968, p. 408), is that the rebound rate is proportional to the isostatic anomaly (Broecker, 1966), that is that

dyidt = K(y, - y - RA), (9)

where y = y(t) is the elevation of the land relative to present-day sea level, yr = y(m). which is unknown, is the elevation it will reach when rebound is complete, A is the varying ice thickness, which by definition has been zero during postglacial time, K is a constant, and R is the ratio of ice density to mantle density, or about 0.28.

A reasonable way of approximating A is to assume that the ice-surface profile simply shifted at constant speed 120 m upglacier in 300 yr without change of shape, as did Broecker (1966) in a similar calculation. Inasmuch as the ice surface actually probably steepened somewhat, because the retreat occurred principally by calving into the ocean, this will lead to overestimation of the uplift, which, as will be seen, is in the direction of conservatism.

The constant K = In 2/t,, where tlh is the half-rebound time for postglacial uplift. Stuiver and Borns (1975, p. loo), in a calculation for the region southwest of the Ka- tahdin system using formulas of Broecker

(1966), adopted t,b = 700 to 1500 yr, based in part on rebound curves for northeast Greenland. Andrews (1968, pp. 418-419), however, found the values for Greenland to be anomalously low when compared to the values found for arctic Canada, Fennos- candia, and highlands Britain. Hence, in the absence of better information the best value for Maine, and the one adopted in this paper, appears to be that found elsewhere for the Laurentide Ice Sheet, namely, 1700 yr (Andrews, 1968, p. 419).

Letting ye = y(t,), which like yr is unknown, ym = y(t,) = -68 m, and A = A(t - t,), where t, = - 12,700 yr is the time of esker formation and t, = - 12,400 yr is the time of retreat to Grindstone, the so- lution of (9) for te c t 8 t, is

Y = Yx - i Yz - y, + KR

I

t r, A(Z) exp KT dT

0 1

eXp -K(t - te), (10)

and for t, < t < x it is

Y = Yr - t.Y, - y,) eXp - K(f - tm). (11)

Substituting y = y. = y(O) = 91 m in (11) and solving for y, gives

Y7_ = (Y. - ym exp K&,ji(l - exp Kt,), (12)

from which y, = 92 m. Finally substituting y = ym in (IO) and solving for ye gives

Ye = Yz - b, - Y,) exp K($,, - ‘J

J-

fm-fe + KR A(T) exp KT dT, (13)

0

from which ye, the elevation of the Grind- stone locality at the time of esker formation, can be computed jointly with A.

INTEGRATION OF ICE-SURFACE GRADIENTS

The simplest and, considering the uncer- tainties in the gradients, probably the most appropriate method of computing the profile is a variation of trapezoidal integration in which the gradients are linearly interpo- lated between the known localities and the


resultant polygonal curve integrated to give a smooth profile consisting of a series of parabolic segments. This method is not suitable for the terminal reach, however, not only because it requires the gradient at the terminus, which is unknown, but also because it assumes a uniform rate of change of gradient between localities, whereas the ice-surface gradient must have decreased upglacier far more rapidly near the terminus than near Rocky Pond.

Terminal Reach

The procedure followed in this paper is to compute an ice-surface profile for the terminal reach that matches the known gradient at Rocky Pond (Table 1) using the commonly employed assumption (Nye, 1952b, p. 529) of uniform basal drag. Ice- surface gradients, provided they are small, are related to the drag T according to

dh 7 TY= pi&T!(h, - 17,)

(14)

(Nye, 1969, p, 207), where x is a horizontal coordinate measured along the flow line upglacier from the terminus. For given 7 and terminal h,, this is a differential equation for the ice-surface profile. It has to be integrated numerically, inasmuch as the ground-surface profile is not something simple, like a horizontal plane. Thus, 7 must be found by trial and error.

Stratified end moraines, ice-contact deltas, and other evidence demonstrate that at Pineo Ridge the glacier terminated against the sea (Borns. 1973; Thompson, 1980, p. 214). The relatively static position of the ice margin probably was stabilized at the former shoreline by the extreme dis- parity between terrestrial and marine ablation of the terminus (Borns, 1973, p. 42). Thus, the terminal ice must have been just grounded, or very nearly so, which means that the ice thickness was approximately pw/pi times the water depth, or about 40 m.

Using this terminal thickness, interpo- lating the ground elevation linearly between measurement points 1 km apart, and inte-

grating (14) gives the known ice-surface gradient of 11 m km- ’ at Rocky Pond (Table 1) when the value of 7 is approximately 18.3 kPa (0.183 bar). The corresponding terminal ice-surface gradient is 50 m km - ’ , which is marginally small enough to assure the validity of (14). At Rocky Pond the calculated ice-surface elevation is 273 m. It serves as the starting point of the trapezoidal integration that gives the remainder of the profile.

Pro$le of Ice Surface

Computing a preliminary profile from the uncorrected gradients and using the re- sulting A in (13) gives ye = -71 m at Grind- stone, from which the uplift since the time of esker formation is 162 m. The tilt. there- fore, was only about 0.1 m km-‘, which is less than the uncertainty in determining the gradients. Thus, the ice-surface gradients can be used uncorrected for tilt. The result is the lower profile in Figure 3.

This conclusion would not change were somewhat different sea-level curves used, such as those of Curray (1965) or Milliman and Emery (1968), which, respectively, lie above and below the curve of Dillon and Oldale (1978) used in this paper. But it would change were new observations to show that the elevation of the marine limit at Grindstone is actually significantly higher than 91 m. If, for example, it were 130 m, as is the case near Bingham, 125 km to the southwest of Grindstone (Thompson, 1980, p. 213), the tilt would be approximately 0.5 m km- ’ and the ice-surface profile at Trout Pond would be roughly 100 m higher.

Another uncertainty, left unmentioned so far, is the exact position of the terminus. The position shown for the lower profile in Figure 3 is the southernmost point at which the esker can be found. On the basis of the Pineo Ridge moraines the actual position is likely to have been approximately 5 km farther southwest, in which case the whole profile would be about 80 m higher (upper profile in Fig. 3).

34 RONALD L. SHREVE

o,, I1 ,,,,, I,,,,,, 150 100 50 0

Distance fkml

F[G. 3. Calculated profile of ice surface approximately 12,700 yr ago. The lower profile assumes terminus at southernmost point esker can be found. The upper profile assumes it at projection of Pineo Ridge ice front. Vertical exaggeration is 50 x The glacier terminated against the sea, whose level relative to the land was 80 m higher than that today. Arrow indicates elevation of shoulders on Lead Mountain. Locality symbols are defined in Table 1 and Fig. I.

Probably the principal difficulty of the method is finding suitable localities. The best sites are not randomly distributed, but tend to be where the ground surface rises downglacier, which introduces a systematic bias that is hard to evaluate, though it doubtless is small where slopes are gentle.

Once a site is found, the main problem is determining the ground-surface gradient and direction. What is required is the ground surface as it existed beneath the glacier, whereas what is available is the surface as it exists today. In eastern Maine the major difficulties typically are due to postglacial alluviation, commonly by debris shed from the eskers themselves. A quali- tative evaluation of this uncertainty at each locality is given in Table 1.

When projected 20 km to the west, the calculated profile hits Mount Katahdin at an elevation of about I100 m, in agreement with the geological evidence still farther west. The greater steepness of the protile at Trout Pond may be due in part to the presence of Mount Katahdin and its foot- hills, which rise well above the general level and must have dammed and diverted the ice flowing from the northwest. Thus, the ice surface would have been higher than the profile on the northwest slopes of the mountain, lower on the southwest ones,

and steeper at the sides. It seems likely that the Ayers Stream branch of the Katahdin esker system and the trunk esker downstream follow the locus of a trough in the ice surface that formed in the lee of Mount Katahdin.

The profile shows that not only Mount Katahdin but also Lear! Mountain, 9 km northeast of the esker and 12 km from the terminus, rose above the ice (Fig. 3). Inter- estingly, the topographic map shows prom- inent shoulders on this mountain, 25 m higher on the north than on the south, within a few meters of the corresponding part of the upper profile. The 25-m drop is about three times steeper than the profile. which seems about right for the ice surface near a nunatak. Similar shoulders about 5 m higher are present on Pleasant Mountain, 8 km farther northeast (pointed out by J. A. Maccini, 1983, personal communication). If these features can be associated with the ice surface at the time of esker formation, like the high-level lake deposits in western Maine, they would be a striking confirma- tion of the method.

Basal Drag With the profile known, the drag (Fig. 4)

can be computed from (14). It increases slowly from about 20 kPa (0.20 bar) near the terminus to about 30 kPa (0.30 bar) 100 km upglacier, then increases much more rapidly, reaching 70 kPa (0.70 bar) at 140 km, perhaps, as already suggested, because of the influence of Mount Katahdin.

For the margin of the late Pleistocene Laurentide Ice Sheet at its maximum extent in the coastal mountains of northern Lab- rador, a terrain somewhat like Maine, Ma- thews (1974, p. 41) deduced a basal drag of 75 kPa (0.75 bar), assuming that the drag is uniform so as to give a square-root ice-surface profile (Nye, 1952b, p. 529). For the Puget lobe of the Cordilleran Ice Sheet in western Washington at its maximum extent Thorson (1980, p. 308) deduced a profile that gives about 45 kPa (0.45 bar) if uniform drag is assumed.


FIG. 4. Basal drag computed from profiles of Fig. 3. Locality symbols are defined in Table 1.

Several computations based on (14) have been made for present-day ice sheets. For Antarctica Budd et al. (1971) found values ranging from 20 to 60 kPa (0.20 to 0.60 bar) over most of the interior, with a sharp rise to more than 100 kPa (1 .OO bar) within about 400 km of the margins. For Green- land, Bull (1957, p. 69) obtained values ranging from 30 to 70 kPa (0.30 to 0.70 bar) along a 750-km profile at about latitude 77”N and Haefeli (1961, p. 1141) calculated values ranging from zero at the ice divide, as expected where the gradient is zero, to 50 kPa (0.50 bar) 75 km from it to 100 kPa (1 .OO bar) 310 km farther away on a line about latitude 70”N. Along the “trilatera- tion net” flow line of the Barnes Ice Cap on Baffin Island R. LeB. Hooke (1983, personal communication) found about 60 kPa (0.60 bar).

Perhaps the present-day glacier most comparable to the ice in Maine at the time of esker formation is the Malaspina, about 70 km northwest of Yakutat in southeastern Alaska. It is a fan-shaped Piedmont glacier roughly 70 km wide that flows more or less radially outward about 45 km from its principal source, the lower Seward Glacier, a valley glacier that arises in the Saint Elias Mountains. According to data given by Allen and Smith (1953) the drag near the center of the Malaspina is about 65 kPa (0.65 bar).

The lack of a clear trend in all these data, except for the low values near ice divides, is not surprising. The drag is governed by position with respect to the ice divide, the temperature of the ice, the ease of basal sliding, and the activity of the glacier, with lower drag being associated with ice di-

vides, warmer temperatures, easier sliding, and lower activity.

Even with allowance for this inherent variability, however, the basal drug computed for the Maine ice away from the possible influence of Mount Katahdin is clea; ly the lowest of the seven cases. This agrees with the conclusion of Ruddiman and McIntyre (198 1, p. 128) from paleontologic. sedimentologic, and isotopic analyses of deep-sea cores from the North Atlantic Ocean that more than 50% of the Wisconsin ice volume had disappeared by 13,000 yr ago even though the glaciers still occupied 75 to 80% of their full-glacial areas of about 5000 yr earlier, which implies a substantial decrease in basal drag from its earlier values.

DISCUSSION

Very few methods exist for determining the surface profiles of former ice sheets.

One of the most widely used is theoret- ical calculation. This ranges from simply assuming uniform basal drag of some “reasonable” magnitude and calculating the corresponding square-root profile from the re- quirement for mechanical equilibrium to assuming a detailed model of ice flow, basal sliding, ice temperatures, and mass balance, and calculating the profile by numerical methods (for an example from the Maine region see Fastook and Hughes (1980)). A major disadvantage of this method is that initial conditions and such important parameters as surface temperatures, energy fluxes, and accumulation and ablation rates are generally almost as poorly known as the profile.

Another widely used method is to map the ice surface from moraines or other evidence of glacier margins preserved on former nunataks and elsewhere. The elevations are projected to specific profiles along curves drawn perpendicular to ice- flow indicators such as drumlins and grooves (for a nice example see Thorson (1980)). This method has the considerable virtue of directness. It works best for de-

36 RONALD L. SHREVE

termining profiles at times of maximum ice extent, inasmuch as the highest ice levels at disjoint localities are presumably syn- chronous, which solves the otherwise dif- ficult problem of correlation. Its major lim- itation is the necessity for relatively thin ice, which tends to restrict it to areas near ice margins.

A third method is to deduce former maximum ice loads, and hence profiles at maximum ice extent, from the preconsolidaion pressures of lake sediments overridden by the ice (for the only example known to me see Harrison (1958)). It is based on the fact that, when freely draining saturated silts are compacted under increasing load, then released, and finally slowly reloaded, they cannot be further compacted significantly until the new load exceeds the maximum value of the old, that is until it reaches the preconsolidation pressure. A major limita- tion of this method is that the sediments cannot have been dried, frozen, or dis- turbed from the time they were deposited until they are tested. Moreover, the required free drainage may be impeded beneath glaciers, giving preconsolidation pressures, and hence ice thicknesses, that are too low.

The present method gives ice-surface profiles from the late stages of final retreat, rather than from times of maximum ice extent. It requires sufficient information to enable correction for tilting due to isostatic rebound as well as independent knowledge of the elevation of some point on the profile, which must be found either by theo- retical calculation using available data (as done in this paper) or from direct evidence such as datable ice-surface deposits on a former nunatak (as potentially may be possible on Mount Katahdin or Lead Moun- tain). It does not require that the profile lie along a single esker system, nor that the eskers be continuous, provided that all segments used were formed at the same time. Its great advantage is that the plethora of large eskers traversing suitable terrain in many parts of the regions covered by the

late Wisconsin Laurentide and Fennoscan- dian ice sheets give it the potential for wide application.

ACKNOWLEDGMENTS 1 thank Professor Harold W. Borns. Jr.. of the Uni-

versity of Maine, Orono, for much vital information on the glacial history of Maine, assistance that greatly expedited the field work, and a variety of other con- tributions to the project; Mr. Wilbur Tidd and Mr. Charles Norberg, geologists with the Maine Depart- ment of Transportation. and Ms. Bea Foster and her colleagues in the Soil Conservation Service, U.S. De- partment of Agriculture, Bangor, Maine, for the use of air photos of the localities studied; Professor Roger LeB. Hooke of the University of Minnesota, Minne- apolis, for his long-standing interest in the project and for copies of figures from the report by W. F. Budd, D. Jenssen, and U. Radok (1971); Professor William M. Bruner of the University of Washington. Seattle, for many fruitful discussions of glacier physics; Dr. John A. Maccini of the National Science Foundation. Washington. D.C., for noting the apparent correspon- dence of the shoulders of Pleasant Mountain to those on Lead Mountain. and Messrs. Derek Booth. Ber- nard Hallet. Roger LeB. Hooke. and an anonymous reviewer for their suggestions for improvement of the paper. This paper is based upon work supported by the National Science Foundation under Grant EARS l- 21051. It is Publication 2483 of the Institute of Geo- physics and Planetary Physics, University of Cali- fornia, Los Angeles.

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late wisconsin ice-surface profile calculated from esker paths and types, katahdin esker system,...

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