influence of glacier hydrology on the dynamics of a large quaternary ice sheet

JOURNAL OF QUATERNARY SCIENCE (1992) 7 (2) 109-1 24 0 1992 by John Wiley & Sons, Ltd.

0267-81 79/92/0201 09-1 631 3.00

Influence of glacier hydrology on the dynamics of a large Quaternary ice sheet NEIL ARNOLD and MARTIN SHARP Department of Geography, University of Cambridge, Downing Place, Cambridge CB2 3EN, England

Arnold, N. and Sharp, M. 1992. influence of glacier hydrology on the dynamics of a large Quaternary ice sheet. journal of Quaternary Science, Vol. 7, pp. 109-124. ISSN 0267-8179

Received 27 September 1991 Accepted 20 March 1992

ABSTRACT: The influence of glacier hydrology on the time-dependent morphology and flow behaviour of the late Weichselian Scandinavian ice sheet is explored using a simple one- dimensional ice sheet model. The model is driven by orbitally induced radiation variations, ice- albedo feedback and eustatic sea-level change. The influence of hydrology i s most marked during deglaciation and on the southern side of the ice sheet, where a marginal zone of rapid sliding, thin ice and low surface slopes develops. Such a zone is absent when hydrology is omitted from the model, and its formation results in earlier and more rapid deglaciation than occurs in the no- hydrology model. The final advance to the glacial maximum position results from an increase in the rate of basal sliding as climate warms after 23000 yr BP. Channelised subglacial drainage develops only episodically, and is associated with relatively low meltwater discharges and high hydraulic gradients. The predominance of iceberg calving as an ablation mechanism on the northern side of the ice sheet restricts the occurrence of surface melting. Lack of meltwater penetration to the glacier bed in this area means that ice flow is predominantly by internal deformation and the ice sheet adopts a classical parabolic surface profile.

J ~ ~ ~ ~ I of Quaternary Science

KEYWORDS: Glacier hydrology, Quaternary, Scandinavia, ice sheet dynamics.

Introduction

It is now widely recognised that fast ice flow, as manifested by ice streams and outlet glaciers, plays an important role in the dynamics of the present-day Antarctic and Greenland ice sheets (e.g. Clarke, 1987; Hughes, 1987), and several authors have suggested that this role might have been equally important in the case of Quaternary mid-latitude ice sheets (Hughes et a/., 1985; Dyke and Morris, 1988). Evidence in support of this includes reconstructed low-gradient ice surface profiles (Mathews, 1974; Beget, 1 986; Clark, 1980, 1 985)’ localised zones of long distance erratic transport (Boothia type erratic trains) (Dyke and Morris, 1988), lobation of ice sheet margins during deglaciation (Boulton and Jones, 1979; Punkari, 1982, 1984) and evidence for rapid, asynchronous fluctuations of ice margins during overall retreat (Clayton et a/., 1985). The causes of fast flow are, however, less well understood. It has been explained in terms of rapid sliding over a rigid, water-lubricated bed (Budd et a/., 1979; Iken, 1981; Kamb, 1987; Fowler, 1987a,b), motion over a deforming sediment substrate (Boulton and Jones, 1979; Boulton and Hindmarsh, 1987), and of enhanced deformation of glacier ice (Echelmeyer and Harrison, 1990). Since it i s clear that both rigid and potentially deformable glacier beds exist, it may be that al l three mechanisms can operate, albeit in different parts of an ice sheet and at different times in its history. It is important to realise, however, that both the

lubricated bed and deforming substrate models of fast glacier flow require high subglacial water pressures, and are therefore critically dependent on the character and behaviour of the subglacial drainage system.

To date, large scale, time-dependent models of Quaternary ice sheets (e.g. Budd and Smith, 1981, 1982; Oerlemans, 1981; Payne et a/., 1989) generally have not incorporated representations of those processes that may permit fast flow to occur. Such models appear to be able to reproduce geologically determined ice-margin positions (and possibly also ice volumes) with a reasonable degree of accuracy, but they are less effective at depicting the distribution of velocity within ice sheets. For instance, Mclnnes and Budd (1984) obtained good agreement between modelled and observed velocities at the grounding line of Ice Stream B, Antarctica, but found that modelled velocities decreased inland much more rapidly than observed velocities. The Antarctic Peninsula Ice Sheet model of Payne et a/. (1 989) also shows discrepancies between modelled and observed ice velocities for the present day, and some discrepancies over the distribution of floating and grounded ice. One possible explanation for these disagreements i s the rapid decrease of velocity away from the grounding line in the model, which allows thick ice to remain in areas which in reality are now occupied by floating, rather than grounded ice. These problems arise primarily because water pressures calculated in these models relate to hydrostatic pressures imparted by standing water at the ice sheet margins, rather than to subglacial drainage of basally and supraglacially

110 JOURNAL OF QUATERNARY SCIENCE

derived meltwater. Oerlemans (1 982) included a relationship between meltwater production at the base of the ice sheet and basal sliding, in which sliding was evaluated such that an ice sheet which experienced basal sliding would, for a given steady state, be half the thickness of one in which no sliding occurred. The aim of Oerlemans' study, however, was to investigate, and to try to reproduce, asymmetric glacial cycles and the controls on the periods of such cycles. As such, it did not include an evaluation of the basal hydrological system, and the effect this may have on sliding velocity. The failure of models that aim to simulate ice sheet dynamics to reproduce the essential features of fast flow in reconstructions of contemporary ice cover suggests that they will also fail to model the evolution of fast flow through time. If this is so, it i s unlikely that they will be able to predict accurately the time-dependent evolution of the morphology and flow dynamics of palaeo-ice sheets. Thus, if the aim of a given study is to assist in the interpretation of the geological record left by those ice sheets, such models will be of limited value.

In the light of the above discussion, it seems desirable that if time-dependent ice sheet models are to be used in such studies they should incorporate some explicit treatment of subglacial hydrology and its influence on ice flow dynamics. As a first step in this direction, we have developed a one- dimensional model of an ice sheet resting on a rigid bed which includes a water-pressure dependent sliding law and evaluates the subglacial water pressure as a function of meltwater discharge and the configuration of the subglacial drainage system (major channels or a distributed, linked- cavity system). In this paper we seek to compare the response to imposed environmental forcing of such a model with a simpler model, which does not explicitly include subglacial hydrology. This latter model is similar to the models of Budd and Smith (1981, 1982) and Payne et a/. (1989), in that it treats the subglacial water pressure as dependent upon hydrostatic pressure arising from marginal water bodies. A schematic view of the full model i s given in Fig. 1 . The models are applied to a transect through the Scandinavian ice sheet from the Lofoten Islands to Estonia, and to the last 40 000 yr.

We stress that the goal of the paper i s to evaluate the importance of the hydrology of an ice sheet in regulating its response to environmental change. We hope that the study will provide insights into ice sheet behaviour which will assist in the interpretation of geological evidence. At this stage,

however, we have not attempted to adjust the parameterisation of the model or the nature of the environmental forcing to which it is subjected to make model output consistent with available geological evidence. Equally, we have not tried to investigate the sensitivity of model response to variations in environmental forcing.

The ice sheet model

The models used in this study are based on a one-dimensional version of the continuity equation for ice thickness

dZldt = A - rn - dldx (Us + u d ) z (1 1

where Z is ice thickness, t is time, A i s net mass balance, m is rate of marine losses, U, is ice sliding velocity, U d i s ice deformation velocity and x is distance. This equation was solved using a forward-time, centred space finite difference scheme with a 20 km grid size, and a 2.5-year time step. Because of inherent instabilities in the finite difference scheme used to solve the continuity equation (which are manifest as high-frequency waves within the solution), it was necessary to smooth the model output periodically. This was done using a five-point weighted average of the form

Z:= 1/16(10Zi+4(Zi+1 +Zi-l)-Zi+2-Zi-z) (2)

where Zi is the value of ice thickness or bed elevation in a given cell. This procedure was carried out once every 10 iterations. Runs were also carried out with smaller time steps, and less frequent smoothing, to ensure that the smoothing was not unduly affecting model results.

Ice flow

Ice flow is assumed to occur by a combination of internal deformation and basal sliding. Vertically integrated deformation velocity i s calculated from basal shear stress, ?br

assuming simple shear, using the equation

Ud = A T ~ Z (3) i s calculated by where A and n are flow parameters, and

Figure 1 Outline of the structure of the full ice sheet model (Mqdel 2).

INFLUENCE OF GLACIER HYDROLOGY ON LARGE QUATERNARY ICE SHEET 111

?b = p,gZ sin a (4)

Sliding velocity i s determined using the relation determined where a is the ice surface slope and p, is ice density.

by Mclnnes and Budd (1984)

U s = kI?d(N + k,N2) (5)

where kl and k, are flow parameters and N i s the effecti.de pressure (ice overburden minus subglacial water pressure). To prevent excessive changes in sliding velocity over short distances, which might arise from approach to the ice sheet grounding line or from variations in N linked to changes in the subglacial drainage configuration, ice velocity is con- strained by imposing a maximum horizontal longitudinal strain rate of 0.004 yr-' (cf. Mclnnes and Budd (1 984)). This can be thought of as simulating the role of longitudinal stresses and the buttressing effect of downstream ice (Mclnnes and Budd, 1984).

If ice is found to be floating (i.e. Z < pw/pi x W, where W i s the depth of water), ice flow in the resulting ice shelf is approximated by using a prescribed constant horizontal strain rate of 0.004 yr-'. This is the observed strain rate at the grounding line of Pine Island Glacier in West Antarctica, a floating ice tongue that is not buttressed by ice rises (Mclnnes and Budd, 1984). This is a situation that seems likely to be comparable to the Atlantic margins of the Weichselian Scandinavian ice sheet. It is thus assumed that, within an ice shelf, ice velocity changes as a result of spreading and thinning of the ice under its own weight, given an initial velocity at the grounding line.

Mass balance

Mass balance of the ice sheet i s determined by the interaction of four separate factors: accumulation due to precipitation on the ice surface, surface melt, basal melt from ice shelves, and calving of icebergs where the ice terminates in standing water.

Accumulation rates, which must be considered largely unknown for the area and time period concerned, were taken to be equal to modern precipitation rates. The latter were taken to be 90% effective (i.e. given the relative densities of ice and water, 1 m of precipitation in a given time period would be converted to 1 m of ice by the end of that period). This approach, which minimises the complication of including precipitation as an interactive variable and saves computing time, was also used by Budd and Smith (1981). Whilst this is undoubtedly a simplification of reality, it i s justified here on the grounds that we are not, at this stage, trying to use the model to precisely simulate reality. The initial accumulation values were modified subsequently by the incorporation into the model of an 'elevation desert effect', which simulates the reduction of precipitation observed at high altitudes on modern ice sheets. Thus precipitation was reduced by a factor of two for every kilometre of ice surface elevation above 2 km (cf. Budd and Smith, 1981).

Ablation (surface melt) was calculated using the regression equation derived by Budd and Smith (1981), which predicts the current distribution of ablation rates as a function of elevation and latitude

logloA = l l rn (E, - €1 (6)

Here A i s the ablation rate in m yr-I, Eo i s the elevation of the 1 m yr-' ablation contour at a given latitude at a given time (as determined from present-day distributions of ablation,

elevation and latitude), E i s the ice-surface elevation, and m = 1200 m. Changing insolation receipts over time were assumed to modify the value of E, (see below).

Marine melting from the bottom of ice shelves is influenced by many factors, such as sea-surface temperatures, salinity and turbulence. Since very few of these could be determined for the case of the late Weichselian Scandinavian Ice Sheet, a constant rate of melt of 0.5 m yr-I was assumed, following Payne et a/ . (1989). This simplification also was thought to be justified given the aims of the paper.

Iceberg calving from tidewater glaciers was shown by Brown et a/. (1982) to be proportional to depth of water at the margin of the glacier. Although their relationship i s not strictly applicable to ice shelves, it i s used here in the absence of any studies of calving rates from ice shelves. The relation takes the form

V, = 28.75 W (7)

where V, is the calving fiux in m yr-' and W the water depth.

Me1 twater discharge

Meltwater discharge, which is required for calculations of effective pressure, was calculated on the assumption that, in the ablation area of the ice sheet, all of the ablated ice would reach the bed as surface-derived meltwater. It was further assumed that in the accumulation area, all of this water would refreeze as it flowed through the ice sheet, and none would reach the bed. This implies that the base of the ice sheet is temperate in the ablation zone only, an assumption supported by time-dependent thermomechanically coupled modelling of ice masses (Hindmarsh et a/., 1989).

This assumption was tested in the current study as follows. Hindmarsh (pers. comm.; 1990) has shown that for velocity due to deformation or sliding, the heat production over a column of ice i s

H = pig dEldx Qi (8)

where H i s the heating and Q, is ice discharge. It can then be assumed that this heating occurs only at the bed of the ice sheet. This can then be compared with the temperature gradient needed to conduct the heat away

H = K dTldx (9)

where K is the thermal conductivity of ice and dTldx is the temperature gradient. If the surface temperature of the ice is known, it i s then possible to calculate the basal temperature as follows

T b = T, + ZHIK (1 0)

where Tb is basal temperature and T, i s surface temperature. Oerlemans (1 982) gives a relationship for the surface temperature of an ice sheet

(1 1)

where 7, is the temperature at the equilibrium line, E, is the equilibrium line elevation and h i s the atmospheric lapse rate, here taken to be 6.5"C km-I. For the present-day Greenland ice sheet, studies suggest that T, i s in the range - 12 to - 15°C (Oerlemans, 1982). The latter value is used in this study. These calculations were included in some runs of Model 2 , and show that during ice sheet growth, areas within 60-100 km of the margin are at the melting point (although if ice in the marginal cell (20 km) is very thin, this one cell is sometimes

112 JOGRNAL OF QUATERNARY SCIENCE

below the melting point), whereas interior areas are below the melting point. This temperate area is slightly larger than the ablation area. During decay, the marginal 200-250 km of the ice sheet is temperate; again, this area i s slightly larger than the ablation area. This seems to suggest that only allowing water to reach the bed of the ice sheet in the ablation area will, i f anything, underestimate the role of basal hydrology on ice sheet dynamics. These calculations do, of course, neglect horizontal advection of colder ice from upstream into the ablation area, but vertical advection in the ablation area hinders cooling (e.g. Paterson, 1981, p. 2041, and should ensure that the base of the ice sheet i s at the melting point.

The assumption that surface-derived meltwaters did reach the glacier bed within the ablation areas of Quaternary mid- latitude ice sheets is supported by geological evidence. Allen (1 971) argued that climbing-ripple sequences in the Uppsala esker result from discharge variations with a time-scale of a few hours, consistent with the diurnal variation observed in meltwater stream discharges. Banerjee and McDonald (1 975) aruged that cyclic sequences of sand and gravel, with thicknesses of one to a few metres, which are frequently found in the core of large eskers, may be due to annual discharge variations. These observations are consistent with surface-derived water comprising the greatest component of discharge, because the discharge of basally derived meltwater would not be expected to vary significantly on annual or diurnal scales.

Meltwater discharge (in m3 s-l) was calculated by multiply- ing the ablation rate by the relative densities of ice and water (0.9), the grid interval and the flowband width, and then integrating downstream from the ice divide. Flowband width was taken to be 30 km, on the basis that the average spacing between eskers in southern Finland is approximately 30 km (Geological Survey of Finland, 1984).

Meltwater drainage and subglacial water pressure

The two ice sheet models used in this study are differentiated in terms of the way in which effective pressure (N) is calculated. In the simplest model (Model 11, N is taken to be the height of ice above buoyancy, following Mclnnes and Budd (1984) and Radok et a/. (1989). In the other model, N i s calculated using equations taken from Fowler (1 987a,b), and varies according to whether tunnels or linked cavities carry the basal water. Model 2 (the full model) allows drainage to occur via either major tunnels or a system of linked cavities (cf. Walder, 1986; Kamb, 1987).

For a system of tunnels

(1 2)

where NR i s effective pressure for a tunnel-based system, pw is water density, g i s the acceleration due to gravity, QR i s the volume flux of meltwater, pi is ice density, A i s the multiplier in Glen’s flow law, L i s latent heat, n i s the exponent in Glen‘s flow law, SR i s the tunnel cross-sectional area, and I$ is the hydraulic gradient, defined as

I$ = PI{^ + l(pw + p i ) /pw lP} (1 3 )

Here p i s the bed slope. The tunnel cross-sectional area, SR, is calculated as

(1 4)

where f i s an empirical constant related to turbulent channel flow. Fowler (198713) uses the bedslope (p) instead of the hydraulic gradient (4) to calculate the effective pressure, arguing that for a glacier the pressure change due to the ice- surface slope is small. For an ice sheet, however, where the depth and length scales are much greater, the influence of surface slope does become important.

For a system of linked cavities, the small linking channels must each carry a lower discharge than the equivalent large tunnel, so water pressure will be higher, and effective pressure therefore lower, for a given discharge. Fowler (1 987a) shows

(1 5)

where NK is effective pressure for a cavity-based system, s is a shadowing function (Lliboutry, 1978), defined as the probability that a randomly selected area of the bed is in contact with the ice, QK = QR the volume flux of meltwater, f l K i s the number of passageways across the width of the glacier and SK is the cross-sectional area of a typical passageway. Variables f l K and SK are both determined empirically-Fowler (1 987a) argues for values of 1 0-2 m2 for SK, and of 10’ cavities per kilometre of glacier width for nK,

given a bedrock wavelength of 1 m. Typically, effective pressure for a cavity system is a factor of at least two lower than for a tunnel system carrying equivalent discharge.

Since Model 2 allows meltwater to drain via either major tunnels or a system of linked cavities, this model includes an evaluation of the likely configuration of the subglacial drainage system. It should be emphasised here that when major tunnels occur it seems probable that they will coexist with a linked cavity system rather than completely replace it (Fowler, 1987a,b). Fowler (1987a,b) suggests that a tunnel system will be stable if the value of a stability criterion, A, calculated from

A = vU,/IAN” (1 6 )

(where v = (a//), with a the typical bedrock bump amplitude and I the typical bump wavelength) i s less than a critical value, equal to

AC = ( ~ ~ S R / A * ) ‘ ~ - * ’ ~ ’ (1 7)

(where A* i s the total cavity cross-section area and i s the power function for self-similar bedrocks (Fowler, 1987a,b)). The critical stability factor, A,, has a typical value of 0.25 (Fowler, 1987a). For n = 3, p has values from 2 to 2.5. Values for a and I are determined empirically; Fowler (1987b) gave values of 1 m and 5 m, respectively, and these values also are used in this study (Table 1).

In Model 2, effective pressure i s calculated from either equation (1 2) or (1 5), depending on the value of the stability criterion (A) relative to A, in the previous time interval. (This assumes that the drainage system configuration can change within one time period. Evidence from Variegated Glacier, Alaska, suggests that 2.5 years is more than adequate (Kamb et a/., 1985)). In Model 2, the value of N calculated above is compared with that calculated from ice thickness above buoyancy, and the lower value is used (i.e. the higher water pressure). This simulates, albeit in a very simple manner, the effect on subglacial water pressure of a head of water at the ice sheet margin. If the calculated effective pressure is greater than the ice overburden pressure (implying a negative water pressure, which i s impossible in reality, but actually indicates that subglacial water i s at atmospheric pressure and tunnels are not full (Fowler, 1987a)), ice overburden pressure is used instead of the calculated effective pressure.

INFLUENCE OF GLACIER HYDROLOGY ON LARGE QUATERNARY ICE SHEET 113 ~

Table 1 Parameter values for the models

Parameter

Ice flow Deformation:

multiplier power

1st multiplier 2nd multiplier

latent heat channel flow number of cavities cavity cross-section shadowing function bedrock amplitude bedrock wavelength ratio all power function

Sliding:

Drainage configuration:

Ice conductivity Ice density Water density Gravity lsostas y Mantle density Mantle diffusivity

Symbol Value Units

A n

k l k2

L f nK SK

a I

CL K PI Pw

g

Prn

5

V

D a

5.3 x 10-15 3.0

6.3 X lo-' 400

3.3 x 105 700 30 000 10-2 0.5 1 5 0.2 2.0 2.1 900 1025 9.81

3300 1.11

s - l kPaP3

m2 s-l kPa-' m

I kg-' g m-8/3 k

m2

m m

J s - l ,-I K-I kg m-3 kg m3 m s - ~

kg m-' m2 s-l

Glacial isostasy

Ice sheet behaviour i s known to be strongly influenced by changes in bed elevation caused by isostatic effects (Budd and Smith, 1981). This i s incorporated into the model using a diffusion equation based on ice load and the deflection of the bed away from an initially relaxed condition (Oerlemans and Van der Veen, 1984). The rate of response is controlled by a diffusivity constant, and the relationship is

dBfdt = D,d2/dX2 (6, - B + L, + L,) (1 8)

where B i s bedrock elevation, B, i s the initially relaxed bedrock elevation, D, i s the diffusivity constant, Li i s the ice load and 1, i s the water load, calculated as

4 = ZPifPm for grounded ice L, = Z*pi/pm for ice with its bed below sea-

level 1, = 0 where ice is floating 1, = pw/pm (B, - B) where current bedrock or initial

bedrock are below sea-level L, = 0 where both are above sea-level

where pm is mantle density, and Z* = z - pw/plW.

Model inputs

The input data for the model comprised the initial bed morphology, initial ice thickness, accumulation-rate data, ablation relationships and sea-level. External forcing comprised changes in solar radiation receipts owing to orbital changes, ice sheet area (affecting global albedo) and changing eustatic sea-level. Solar radiation receipts and albedo feedback

were used to alter the ablation relationships within the model (see below).

Bed elevation was defined at 20 km intervals along a transect from approximately 69"30'N, 12"30'E to 55"N, 26"E (Fig. 21, giving a total of 75 grid cells. Elevations were taken from topographic maps of Scandinavia (Office of Geodesy and Cartography, German Democratic Republic, 1967; Main Administration of Geodesy and Cartography under the Council of Ministers of the USSR, 1972). Initial ice thickness was assumed to be zero. The extent of the Scandinavian ice sheet at about 40OOOyr ago is still very much a matter of debate. There is evidence that there was an interstadial at about this time (the Alesund interstadial, Larsen and Sejrup (1 990)), but the extent of ice retreat during this period is uncertain. Because of this, it was decided to assume no ice cover (following Boulton et a/., 1985). Bedrock elevation was assumed to be in isostatic equilibrium at the start of model runs, although this is a difficult assumption to test.

Accumulation rates were taken to be equal to the average annual precipitation given by the UNESCO climatic atlas of Europe, as read at 20 km intervals along the transect.

The effect on ablation rates of changing radiation receipts linked to variations in the Earth's orbital geometry was incorporated into the model using the method of Budd and Smith (1 981 1. These authors argued that changes in radiation receipts over time at a given latitude, as calculated by Vernekar (1972), would result in changes in the elevation of the 1 m yr-' ablation level (€,-equation (6)). These changes would be equivalent in magnitude to present-day changes in the elevation of Eo that occur between latitudes which show spatial differences in radiation receipts comparable to the difference between present-day receipts and those at the time period in question. Thus, the elevation of E, i s argued to vary by 30 m/(ly/day). The changes in Eo induced by varying radiation receipts at latitude 69"N over the last 40000 yr are shown in Fig. 3a.


a I

b A B

-2000 0 250 500 750 1000 1250 1500

Distance (krn)

Figure 2 (a) Location of the transect through the Scandinavian ice sheet for which calculations were performed. (b) Current bedrock topography along the line of the transect shown in 2a.

These changes in En can be converted to equivalent changes in air temperature by assuming a moist adiabatic lapse rate of 6.5"C km-' (Budd and Smith, 1981). The results suggest that only a part of the temperature changes inferred for the Last Glacial Maximum by climate model studies can be attributed directly to radiative forcing (Budd and Smith, 1981). Budd and Smith attributed the residual temperature change to ice-albedo feedback effects, which were argued to vary linearly in magnitude with ice sheet extent. In reality, these feedback effects also may involve such variables as atmospheric composition and turbidity, a fact acknowledged by Budd and Smith (1981). The values calculated by Budd and Smith (1981) are used in this study. The resulting variations of En through time for the northern end of the transect used in this study are shown in Fig. 3a.

These changes in En were used as the input to the models in the current study. The present-day value of Eo at the northern end of the transect was taken to be 900 m, and a latitudinal gradient of 10 m increase per degree south of this point was used (Budd and Smith, 1981). This gradient was assumed not to vary through time. It i s planned to include the complication of allowing the temporal pattern of change

in the elevation of En to vary with latitude in future studies. At this stage, the models were not run over the whole of the last glacial cycle because of limitations on computing time. Since the climatic curves of Budd and Smith (1981) suggest that the elevation of the 1 m a-I ablation rate cmtour at 37000 yr BP was similar to that of today, and because geological evidence suggests an interstadial at about this time (Larsen and Sejrup, 19901, the models were run from 40000 yr BP to the present day.

Changing eustatic sea-level was the second forcing variable used. Since the volume of the Scandinavian ice sheet at its maximum extent i s thought to have been only about one- quarter of that of the Laurentide ice sheet (Boulton et a/., 19851, it was assumed that eustatic sea-level was external to the model. The eustatic sea-level curve used was taken from Shackleton (1987) (Fig. 3b).

Model results

The results of the model runs are presented in two parts. In the first, the results of the two models are compared. In the


40000 30000 20000 10000 0 Time (yrs BP)

I b

I " " I " " I " " I

40000 30000 20000 10000 0 Time (yrs BP)

Figure 3 Environmental forcing functions used to drive the models: (a) time series of elevation of the 1 m yr-' ablation contour (Eo)--(i) with, and (ii) without ice-albedo feedback; (b) eustatic sea-level curve (after Shackleton, 1987).

second part the sensitivity of the full model to parameter variations is examined. The two initial runs used the parameter values summarised in Table 1, and the forcing functions shown in Fig. 3.

Influence of glacier hydrology on ice sheet dynamics

The response of ice volume to environmental forcing in the two model runs is shown in Fig. 4. The inclusion of hydrology results in an earlier and smaller glacial maximum, and earlier and more rapid deglaciation than occurs in Model 1.

The morphologies of the two model ice sheets for 3000 yr before maximum volume, maximum volume, and 3000 yr after maximum volume are shown in Fig. 5a-c. Both models show very similar, parabolic, profiles at the northern margin of the ice sheet for all three periods. The southern margin, however, behaves differently. During growth (Fig. 5a), Model 2 shows a more restricted extent, and has developed a small, flatter marginal area, with a distinct inflection in the surface profile. By the time the ice maximum is reached (Fig. 5b), this feature has enlarged somewhat, and the areal extent of

the ice sheet is closer to that of Model 1. The flatter marginal area becomes greatly exaggerated during deglaciation (Fig. 5c), when the southern side of the ice sheet shows a virtually straight surface profile. Model 1 retains a parabolic surface profile throughout. By 3000 yr after maximum, the margin of Model 2 has retreated over 200 km further than Model 1.

These morphological patterns can be explained by the processes of ice flow in the two models. Figures 6-8 show the total, deformation and sliding velocities for the two models for the same periods as Fig. 5. The northern margins again show similar behaviour in both models for all three time periods. Internal deformation makes up one-third to over one- half of the total velocity, and generally increases towards a maximum at the margin of the ice sheet.

The two models show quite different behaviour on the southern margin, however. Internal deformation is again an important component of ice flow for Model 1 , contributing at least one-half of the total velocity, and increasing to a maximum at the ice margin. The magnitude of total velocity is highest at the ice sheet maximum. Model 2 shows much higher total velocities for all three time periods, with the magnitude increasing through time. Internal deformation makes up a much smaller proportion of total ice flow near the margins, and this proportion decreases from growth to


'O01

........ . . . . . .

40000 30000 20000 10000 0 Time (yrs BP)

Figure 4 Time series of ice volume for the two model runs.

decay. The actual maximum value for internal deformation, however, increases through time and also occurs an increasing distance from the ice sheet margin. This maximum occurs at the inflection in the surface profile noted above, owing to the steeper slopes and consequent increase in basal shear stress. These changes are due to the much greater importance of sliding in the model that includes basal hydrology, particularly at maximum ice sheet extent and during deglaciation.

The morphological and dynamical contrasts between the northern and southern margins of the ice sheet produced by Model 2 can be explained in terms of the contrasting modes of ablation that occur at the two margins (Fig. 9). In the north, the ice sheet advances across the relatively narrow continental shelf and into deep water at an early stage in its history. As a result, calving of icebergs becomes the dominant mode of ice loss, and it strongly restricts the subsequent growth of the ice sheet. The ice sheet in this area does not, therefore, develop a significant zone of surface melting, and meltwater does not penetrate to large areas of the glacier bed. Therefore, hydrology never exerts a major influence on the flow dynamics or geometry of the ice sheet in this area. Even during deglaciation, surface slopes are steep, so the increase in Eo that occurs has little effect on the size of the ablation area. As a result, the increase in amounts of water reaching the base of the ice sheet is quite slow. The steep slopes also lead to higher effective pressures, and large increases in sliding velocity do not occur. Retreat of the ice sheet margin takes it into shallower water where the calving rate i s reduced, counterbalancing any increase in surface melting.

In the south, however, the ice sheet does not have to advance into deep water, so calving never becomes a significant form of ice loss. The ice sheet is therefore able to expand much further than in the north and develop a relatively large surface ablation zone. Water reaches the glacier bed below this zone and provides the lubrication necessary for rapid sliding. As melting i s the dominant form of ablation at the southern margin, the impact of climatic warming is much greater. This is particularly so because the flat marginal zone

allows a rise in Eo to significantly enlarge the ablation area. This produces a strong positive feedback effect, whereby increased amounts of meltwater reach the glacier bed, increasing both the extent of the area affected by rapid sliding and the rates of sliding within it. This initiates a drawdown effect, in which increasing amounts of ice are transferred into the ablation zone (by the increase in deformation velocity at the equilibrium line) and removed by surface melting, and consequent thinning of the ice sheet interior further expands the ablation area and the region of fast flow.

Thus, penetration of increasing amounts of meltwater to the glacier bed in response to climatic warming initiates marginal regions of fast flow, which expand headwards over time, producing volumetric deglaciation of the ice-sheet interior. This process does not, however, occur in those parts of the ice sheet where mass loss is dominated by calving, nor in Model 7 , which takes no account of the influence of glacier hydrology on ice flow. The inclusion of hydrology into an ice sheet model thus produces a type of dynamic behaviour that i s absent in models that omit it, and which has a profound influence on the evolution of ice sheet morphology over time.

Modelling the thermal evolution of the ice sheet including hydrology, using equations (8141 l ) , supports the idea of rapid headward expansion of fast flowing, wet-based ice. Heat production i s high at the equilibrium line, owing to the high deformation velocities, and the relatively thick ice prevents the easy escape of this heat. This suggests that there wil l be little lag between surface warming and basal warming, and that the area of temperate ice wil l increase in line with increases in the ablation area. Further down-glacier, even though the ice sheet is thinner, the basal ice remains at the melting point owing to the high sliding velocities.

Figure 10 shows the evolution of the subglacial drainage system in Model 2. Tunnel-based drainage systems are most common 8000 to 4000 yr before the ice sheet maximum. This seems to be due to the moderate water discharges during this period, when climate is cold, but the large extent of the ice sheet allows a fairly extensive ablation area. This results in moderate sliding velocities. As discharge increases with


a 30001 J

2000- E - 1000-

0-

-1000-

-2000 ~ ~ ~ ~ ~ ~ ~ " ~ " ~ ' " " ' " " ' " " ~ ~

- .- 5 L

6 --- w - 1 El

0 250 500 750 1000 1250 1500 Distance (km)

2000- T' ; 1000- 9 0- 8 0 .- L -

-" w - 1 El -moo-

-2000 I I I I ~ ~ ~ ' ~ " 1 " " 1 " " 1 " ' ' 1

0 250 500 750 1000 1250 1500 Distance (km)

30001

0 250 500 750 loo0 1250 1500 Distance (km)

Figure 5 (a) Surface morphology of the ice sheets produced by the two models 3000 yr before maximum extent. (b) Surface morphology of the ice sheets produced by the two models at maximum extent. (c) Surface morphology of the ice sheets produced by the two models 3000 yr after maximum extent.

warming climate, the higher water discharges (with consequent higher effective pressures) are offset by the headward migration of the ablation area into thicker, more dynamic ice, which results in an increase in sliding velocity at this time, and the destabilisation of the tunnels. Within the main episode of tunnel-based drainage, however, tunnels are repeatedly destroyed and reformed. This suggests that quite local and short-lived ice and bed conditions combine to destroy tunnel- based drainage systems, given generally favourable longer term conditions. The converse of this also seems true; the occurrence of tunnels outside of this main episode seems to indicate that such short-lived, local effects can also outweigh generally unfavourable conditions, and allow short-lived phases of tunnel-based drainage to occur.

Sensitivity of the model

Since the aim of this paper is to examine the influence of subglacial hydrology on ice sheet dynamics and on the

response of the ice sheet to a given pattern of environmental forcing, sensitivity studies have been limited to those parameters that may alter this. These include the parameters in the deformation and sliding relationships, bed roughness, and tunnel and cavity characteristics. No attempt is made to vary the environmental forcing itself.

Some 30 model runs have been carried out, in which parameter values were varied by a factor of up to two above and below the standard values, as given in Table 1 . Some runs also were carried out in which the time step and frequency of smoothing were varied. The results of all these runs are summarised in Table 2 .

Only the sliding parameter k , produces large changes in the response of the ice sheet, and only then when a critical value of almost twice the standard value is used. All the other parameters affect only the details of the ice sheet response, such as the time of maximum volume (to within 100-200 yr), and the actual maximum volume (to 25000 m total thickness (cf. 80 000 m maximum total thickness, equivalent to volume in a one-dimensional model)). These parameters also influence the frequency of tunnel-based drainage systems,


a 500

I:!i,, , , ,~ 8 200 f -

100

0 . - I '

i " " I " " I " " I 0 250 500 750 1000 1250 1500

Distance (km)

0 250 500 750 1000 1250 1500 Distance (km)

200 301 100 A 0

0 250 500 750 1000 1250 1500 Distance (km)

Figure 6 Distribution of deformation velocity within the ice sheets produced by the two models 3000 yr before maximum extent. (c) Distribution of basal sliding velocity within the ice sheets produced by the two models 3000 yr before maximum extent. Note the different vertical scales.

(a) Distribution of total flow velocity within the ice sheets produced by the two models 3000 yr before maximum extent. (b)

and the periods during a growth-decay cycle when tunnel- based systems are most common. For smoother beds (smaller u) , tunnel-based systems are generally more stable throughout the cycle, because equation ( 1 7) predicts lower values of A. Longer bump wavelengths have a similar effect. The power function, p,, has a more complex effect, however. As p, increases, again representing a smoother bed, low Q, (20-80 m3 s-') and resulting low SR, gives higher values of A,, resulting in more stable tunnel-based systems, but for higher Q, (over 100 m3 s-'), A, decreases, resulting in less stable tunnels. Thus, for larger values of p, tunnels are more common during the growth phase of the cycle, but less so during ice sheet decay.

Freer drainage (smaller f , has a similar effect to a higher p.. Smaller values of SR result in lower values of A,, which result in tunnel-based drainage being less stable during deglaciation. This effect i s complicated in this case, however, by the higher effective pressures that result from freer drainage, which lead to lower sliding velocities, and hence more stable tunnels. Increasing f has the reverse effect. increasing the

area occupied by cavities (possibly owing to a rouzher bed) also reduces tunnel stability.

These results support the contention that the actual occurrence of tunnel-based or cavity-based drainage depends on a complex interplay of local ice and bed parameters, making prediction of tunnel location in the field very difficult, unless ice dynamics and bed parameters are very well known. They also suggest that smoother beds favour tunnel-based drainage systems, particularly during the growth phase of an ice-age cycle.

Discussion and conclusions

Influence of glacier hydrology on ice sheet dynamics

The inclusion of hydrology in the ice sheet model has a significant impact on ice sheet dynamics and morphology,


6 f 300- U

Y A .- 4 200-

400 - 6 f 300- U

Y x .- 4 200- >

Wl 100-

0- " " I " " I 0 250 500 750 1000 1250 1500

Distance (km)

b ""1

0 250 500 750 1000 1250 1500 Distance (km)

400 5001 n

0 250 500 750 1000 1250 1500 Distance (krn)

Figure 7 (a) Distribution of total flow velocity within the ice sheets produced by the two models at maximum extent. (b) Distribution of deformation velocity within the ice sheets produced by the two models at maximum extent. (c) Distribution of basal sliding velocity within the ice sheets produced by the two models at maximum extent. Note the different vertical scales.

but only at certain times and in certain locations. This is particularly true during deglaciation of the southern side of the ice sheet, which develops a lower, flatter profile in its marginal areas, where fast ice flow occurs. Absence of such a profile during most of the growth history of the ice sheet makes i t dangerous to assume symmetry of both form and flow processes during growth and decay (cf. Boulton et a/., 1985).

The more extensive ablation zone that develops during deglaciation allows water to reach the bed more widely than during growth, when continued climatic cooling restricts the size of the ablation area. As a result, enhanced sliding becomes the dominant flow process during deglaciation. This changes the responsiveness of the ice sheet to climatic change and accelerates deglaciation, thus accentuating the asymmetry of glacial cycles. It does this by a process analogous to 'marine downdraw', in which flow rates increase both at the margins of the ice sheet (as a direct result of increased sliding), and in internal areas (owing to the development of inflections

in the surface slope of the ice sheet at the head of the zone of rapid flow). This more rapid flow increases the flux of ice to the margin, and ice i s removed by high rates of ablation over the flattened marginal zone.

Nort h-sou th asymmetry of i ce-s heet be haviou r

The model results highlight an important set of interactions between bedrock topography and mechanisms of mass loss from the ice sheet, which result in a strong north-south asymmetry in flow dynamics and ice sheet morphology. At the northern end of the transect, the ice sheet advances rapidly across a narrow continental shelf until its advance is halted by rapid calving in deep water. Calving maintains steep ice-surface slopes and prevents the development of extensive areas of surface melting, so penetration of meltwater to the glacier bed is severely limited. Internal deformation is

120 IOURNAL OF QUATERNARY SCIENCE

a I

0

150-

f: loo- € h -

3 50- >

U

250 500 750 1000 1250 1500 Distance (krn)

l"-w 1 I

0 250 500 750 1000 1250 1500 Distance (krn)

0 250 500 750 1000 1250 1500 Distance (km)

Figure 8 (a) Distribution of total flow velocity within the ice sheets produced by the two models 3000 yr after maximum extent. (b) Distribution of deformation velocity within the ice sheets produced by the two models 3000 yr after maximum extent. (c) Distribution of basal sliding velocity within the ice sheets produced by the two models 3000 yr after maximum extent. Note the different vertical scales.

an important component of ice flow and the ice sheet displays a classic parabolic surface profile. This situation changes only during the final stages of deglaciation, when the ice margin becomes grounded on land and the warming climate allows an ablation area to develop. Water then reaches the bed, and there is a short episode of rapid sliding.

At the southern end of the transect, the ice sheet does not encounter deep water during its growth phase, so calving is never an important ablation process. Surface melt is also limited by the cooling climate until 23000 yr BP, after which time it does become increasingly important. The ice sheet then develops a significant ablation area, water reaches the bed, and a flat marginal area of rapid sliding develops. This initiates a train of positive feedback effects in which lowering of the surface profile of the ice sheet contributes to the expansion of the ablation area and associated region of rapid sliding. This then propagates headwards into the ice sheet as a result of rapid ice deformation driven by high basal shear stresses, which occur in the region of locally steepened surface slope at the head of the zone of fast flow. Volumetric

deglaciation thus occurs by the formation and headward growth of what are, in effect, ice streams.

The results of our modelling are thus consistent with the suggestion of Alley (1 990) that ice-flow-drainage systems in large ice sheets can exhibit multiple steady states. Where subglacial drainage is absent or driven by high hydraulic gradients, water pressures are low, rapid sliding does not occur, and internal deformation is a significant component of ice flow. Where meltwater drainage occurs and hydraulic gradients are lower, however, drainage is more difficult and water pressures rise. This allows rapid sliding to occur and greatly reduces the importance of internal deformation within the ice. Which of these two states actually occurs can depend upon the dominant ablation mechanism, which in turn depends upon the topographic and climatic setting in which the ice sheet develops.


600 7001 200-

40000 30000 20000 10000 0 Time (yrs BP)

3001 b

50i 0 40000 30000 20000

Time (yrs BP) loo00 0

Figure 9 (a) Time series of meltwater discharge at the northern and southern margins of the ice sheet produced by Model 2. (b) Time series of iceberg calving rates at the northern and southern margins of the ice sheet produced by Model 2.

Fluctuations of the ice sheet margin

Our results suggest that short-lived changes in drainage configuration can lead to increases in ice flux, which in turn lead to significant marginal fluctuations. During growth, a surge-type mechanism seems to exist, in which the gradual growth of the ablation area results in increased amounts of wet-based ice, and hence in increased sliding. The margin therefore advances into warmer areas faster than it would if basal sliding were not present. This results in a temporary imbalance between ice flow to the margin and ablation at the margin, which leads to a temporary halt during ice advance, as seen at 580 km and 750 km in Fig. 10. The change to a warming climate stops this mechanism, and results in a more permanent change in ice-sheet morphology owing to an increase in the rate of basal sliding. Indeed, the advance of the southern margin to its maximum position seems to be the result of an adjustment in ice sheet morphology of this kind. This was triggered by increased meltwater production in response to climatic warming. Thus, the response of the ice margin to climatic fluctuations can result from changes in flow process, as well as from changes in mass balance. This may mean that the direction of marginal fluctuations could be directly opposite to that which would

be expected from simplistic considerations of the effects of climatic change on mass balance (e.g. an advance when climate is warming, or a retreat (or stillstand) as the climate cools). This also i s consistent with Alley (1990), who argues that small climatic changes can lead to large ice-sheet responses as the ice sheet adjusts between steady states.

Configuration of the drainage system beneath a large ice sheet

The model results suggest that the development of channelised drainage beneath the ice sheet is localised in both time and space, and dependent on quite local factors, such as bedrock topography and roughness, ice thickness and surface slope, and the levels of meltwater discharge, as well as longer term factors, such as whether the ice sheet i s growing or decaying. On the southern side of the ice sheet, channels are more common during growth than during deglaciation. This would seem to be because of both the generally lower water discharges and smaller areas influenced by basal sliding during growth, and because during decay, the zone of sliding


0 250 500 750 1000 1250

Distance (km)

Figure 10 Configuration of the subglacial drainage system in time and space in Model 2.

Table 2 Summary of sensitivity tests

Ice sheet morphology Drainage system" -

Variable Gross response Detailed response Increase favours Decrease favours

Ice flow Deformation 'Sliding, k , Sliding, k,

Bed 'Roughness, p. Roughness, v Wavelength, I Channel, f Cavity area Time step Smoothing

No Yesb No

No No No No No No No

Yes Yes Yes

Yes Yes Yes Yes Yes Yes Yes

T T

"T = favours tunnel-based drainage; C = favours cavity-based drainage bAbove a critical value (1.15 x 'Reversed during deglaciation.

m2 s-' kPa-I).

is moving headwards into thicker, more dynamic ice, which leads to generally higher velocities.

The impact of a cooling phase during deglaciation, such as the Younger Dryas, has not been investigated in this study owing to the lack of an apparent cause of the Younger Dryas in the history of global radiation receipts that we use as a forcing variable. Such a cooling would reduce meltwater discharge, which would lower water pressure in a cavity-based drainage system, and might thus lead to the establishment of tunnel drainage. The apparently episodic development of channelised drainage helps to explain why esker systems, deposited in subglacial channels, often appear to comprise discrete sets of landforms associated with what presumably are relatively transient ice sheet geometries.

Geomorphic implications

Consideration of the manner in which flow processes within the ice sheet adjust to changing subglacial hydrological conditions through time may provide insight into the likely nature of the erosional record left by the ice sheet. It has been argued that the rate of subglacial abrasion scales with the square of the sliding velocity (Hallet, 1979, 19811, while the rate of erosion by quarrying scales directly with the sliding velocity (Shoemaker, 1986). Since different parts of the area covered by the ice sheet had different sliding velocity histories, they might also be expected to exhibit erosional landforms that reflect flow at different stages of the ice sheet's history. In the north, for instance, there would be one prolonged


erosional episode, the intensity of which would decrease during deglaciation, perhaps allowing preservation of features produced close to the glacial maximum beneath later ornamentation. In the south, however, except at the maximum position reached by the ice margin, there would be two phases of erosion, separated by a period of limited erosion when the marginal zone of sliding has moved south of the area of interest as the ice sheet grows to its maximum extent. The deglacial phase would be far more intensive than the growth phase, and would therefore tend to dominate the erosional landscape. This landscape would, however, be diachronous, reflecting the northward migration of the zone of fast flow during ice-margin retreat. At the maximum position of the ice sheet margin, there would be one strong episode of erosion at the time of the glacial maximum, and features resulting from this episode probably would be preserved. Thus it would seem erroneous to assume that large-scale erosional landscapes can be related to conditions at the ice sheet maximum (cf. Sugden, 1978).

Limitations of the present analysis

Several points of caution with regard to the current analysis must be emphasised. The purpose of this study was to evaluate the likely impact of hydrological changes on the dynamics of a large Quaternary ice sheet. The results should provide insights that wi l l guide the interpretation of field evidence, and help to identify appropriate questions to be asked of such evidence, but they do not yet reproduce reality as expressed in the geological record of specific locations. The results obviously are quite deficient in this regard, because the maximum ice extent predicted is not as great as that inferred from geological evidence for the late Weichselian, and the ice sheet disappears 2000-3000 yr too early.

These deficiencies arise from the relatively simplistic environmental forcing which we have used to drive the model, from the nature of the model itself, and from the way in which it is parameterised. It clearly is unrealistic to apply a uniform environmental forcing to a continental-scale ice sheet, and to assume that precipitation does not vary over time. Equally, it is impossible to reproduce such an event as the Younger Dryas when there i s nothing in the environmental forcing to explain it. The one-dimensional nature of the model does not permit the movement of ice across flowbands, a process that is likely to be extremely important as ice sheet morphology evolves in response to the initiation of fast flow during deglaciation. Neither does it allow the movement of water across flowbands, which would allow fast flow to occur in more localised areas (ice streams), rather than around the entire margin of the ice sheet.

The treatment of the temperature regime i s also simplistic, but given the evidence discussed earlier, and the fact that surface temperatures of ice sheets during the Quaternary must be considered largely unknown, this seems justified given the aims of the study. Full thermomechanical coupling, in particular, seems unnecessary at this stage.

In future work, we intend to develop more sophisticated climatic forcing curves, which allow for changes in atmospheric composition and spatially and seasonally varying radiation receipts. We also intend to develop a two-dimensional version of the full model and conduct more extensive sensitivity tests. The results presented here, however, clearly emphasise the importance of incorporating a treatment of subglacial hydrology in more sophisticated ice sheet models and of reassessing geomorphological evidence to gain an understand-

ing of temporal changes in drainage conditions and flow processes within large Quaternary ice sheets.

Acknowledgements NSA acknowledges receipt of a NERC stud- entship. The manuscript was completed while MIS was on sabattical leave at the University of Alberta, Edmonton, Canada. MIS wishes to thank Professors John England and John Shaw for their hospitality and for comments on an earlier draft of the manuscript, and acknowledges the support of a Royal Society/NSERC scientific exchange award. Richard Hindmarsh helped with ideas and maths for the treatment of temperature. The two referees also contributed valuable ideas.

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OFFICE OF GEODESY AND CARTOGRAPHY, GERMAN DEMO-

influence of glacier hydrology on the dynamics of a large quaternary ice sheet

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