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Diurnal flow variations in a temperate valley glacier67.5.2

Basal motion

Field research indicates that basal motion is essentially governed bya small number of factors, including basal shear traction, bed temperature and the presence and pressure of water at the basal interface (whether hard or soft). Basal water pressure is the least-wellconstrained of these controls, being governed by a balance between
water delivery to the glacier base and water removal from it.
Although our understanding of spatial and temporal variations inbasal water pressure at temperate valley glaciers is now fairly welladvanced, little is known about these variables beneath larger
and/or thermally complex ice masses. There is therefore a pressing
need for field programmes that simultaneously record motion components and their principal controls at these ice masses. Indeed,some work has already begun in this arena. Bingham et al. (2003),
for example, report systematic annual and seasonal motion variations at polythermal John Evans Glacier in the Canadian Arctic.

345This lumped basal motion approach should develop in tandemwith further detailed investigations of specific processes of basalmotion. For example, the effects of variations in bedrock roughness and the integrity of the basal drainage system are currently
neglected in sliding models. The balance between sliding by slip,
enhanced deformation and regelation also needs to be formalized,
perhaps as functions of stress regime, location and basal temperature. Similarly, controls over processes and rates of subglacial
sediment deformation need further investigation. Ultimately, a
constitutive relation for subglacial sediments is required. This
relation should include terms for processes such as ploughing andperhaps even terms for grain-size evolution and the developmentof clast fabric that models could calculate in a time-evolving
manner. Finally, basal motion at subfreezing temperatures also
needs to be investigated more thoroughly and at the glacier-wide
scale, both at hard- and soft-bedded glaciers.

SIXTY-EIGHT

Measurements andmodelling of diurnal
flow variations
in a temperate
valley glacierShin Sugiyama
Section of Glaciology, Versuchsanstalt für Wasserbau Hydrologie
und Glaziologie, ETH, Gloriastrasse 37/39, CH-8092 Zürich,
Switzerland

68.1

Introduction

During the ablation season, surface melting causes diurnal flow
variations in temperate glaciers. Because basal motion plays a keyrole in short-term velocity fluctuations, measurements of diurnal
flow variations provide insight into subglacial conditions andtheir influence on the glacier flow field.
This contribution reports high-frequency measurements ofdiurnal flow speed and strain rate variations in a temperate valley
glacier in the Swiss Alps. Numerical simulations of the observed
flow field using a two-dimensional finite-element flow modelindicated that basal motion was non-uniformly distributed along
the glacier and its pattern varied in a diurnal manner.

68.2

Field measurements

Measurements were carried out in the ablation area of Lauteraargletscher, Switzerland (Fig. 68.1) from 22 to 27 August 2001.
A 6.5-m-long aluminium pole was installed on the surface andcontinuously surveyed with GPS (Global Positioning System) toobtain hourly records of horizontal flow speed and vertical displacement. Three other poles were also surveyed twice a day at0600 hours and 1800 hours to compute the horizontal surfacestrain rate. The precise distance (±5 mm) from the base of a ca.
173-m-long borehole to a surface reference was also measured inorder to calculate the vertical strain of ice over the depth of theborehole (Gudmundsson, 2002; Sugiyama & Gudmundsson,6/7/06

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S. Sugiyama

160

vertical displacement (m)

w
flo00

resu
es
prthng
le25

dee
sp

2500
00
24

656
easting (km)

657

0.10
0.05

0.02
0.00
−0.02

173.54173.52173.50Figure 68.1 Map of the study site on Lauteraargletscher,showing the locations of the surface and borehole measurement sites and the strain grid. The two ellipses are strain rate
ellipses during the indicated time periods of the days from 22
to 27 August 2001. These ellipses represent the deformation of
the circle (broken line) subjected to the computed surface
strain rate in units of 10−4 day−1.

2003). A water-pressure transducer was also installed in a nearby
borehole to measure the water level every 10 min.
Results of the measurements are shown together with air temperature recorded at the glacier flank, in Fig. 68.2. During thestudy period, 40–60 mm w.e. of daily ablation took place, and theborehole water level oscillated more than 100 m diurnally. The
surface flow-speed showed clear diurnal variations, reaching amaximum in the afternoon and a minimum in the morning.
Diurnal variations are also observed in the vertical displacement
and the borehole length. Peaks in flow speed coincide with waterlevel maxima, suggesting that basal motion was enhanced by highsubglacial water pressure. These peaks also coincide with themaximum rate of upward vertical displacement. Surface upliftduring fast-flow events has been considered to result from the formation of water-filled basal cavities (Iken et al., 1983). Nevertheless, the observed uplift cannot be attributed totally to basal cavity
formation as the borehole length measurements showed verticaltensile strains at times when the surface moved upward. Thesevertical strain rates reached 5 × 10−4 day−1. The horizontal surface
strain data (Fig. 68.1) indicate compression along the flow direction during the daytime (0600–1800 hours), which is consistentwith the vertical extension. These changes in the strain regime
indicate consistent systematic diurnal variations in the spatial
pattern of flow speed along the glacier.
During the same period, we operated two further GPS receivers
located 1.5 km up-glacier and down-glacier from the main study
site. Although diurnal velocity variations were recorded at all
three sites, those up-glacier of the main study site were greatest.
This suggests that the measured vertical extension during thedaytime was probably caused by a compressive flow establishedat the study site. Because the seasonal snow line was about 1 kmup-glacier, it is plausible that the highest peak diurnal pressure

0.15
100temperature (oC)

158655200

flow speed (m day−1)

0600−1800 h

300

173.4820borehole length (m)lehorebo159

water level (m)1800−0600 hnorthing (km)KGS_Part4

102325
August 2001

27

Figure 68.2 Records of water level in a borehole, horizontal flow speed, vertical displacement of the surface relative tothe mean elevation, borehole length and air temperature. The
dash–dot line at the top indicates the water level corresponding to the ice overburden.

was greater in the upper reach owing to less developed subglacial
drainage conditions (Nienow et al., 1998). The compressive flow
field dissipated overnight as subglacial water drained downglacier and flow speed slowed down.

68.3

Numerical modelling

To test the hypothesis proposed above, the englacial flow field wasinvestigated with a finite-element glacier flow model. The aim of
the modelling was to prescribe basal conditions to reproduce themeasured diurnal variations in flow speed and vertical strain rate.

68.3.1 Model configuration
The model computes flow-velocity fields in a longitudinal crosssection of a slab, based on Stokes’ equations and Glen’s flow law
with n = 3 (Sugiyama et al., 2003). A finite-element scheme wasused to solve the differential equations for two components of theflow velocity, using quadratic triangle elements with horizontal
and vertical grid sizes of 500 m and 50–100 m, respectively. Basal
motion was introduced in the form:
ub(x) = c(x)t bn′
where ub(x) is the basal sliding speed at the longitudinal coordinate x, tb is the basal shear stress, and n′ and c(x) are parameters

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Diurnal flow variations in a temperate valley glacier

z (m)

z (m)

z (m)

z (m)

z (m)

z (m)

(b)

20

400
0800
0.08
200
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1000
– 0.5
0.12
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0.
0.04
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200
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400
1500 –1 –0.5
6–1
0.1 0.12
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0.
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1800
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0.080.04
0–3000 –2000 –1000
01000
2000
3000
–2000
x (m)

0800

1000

0.5
1

1200

0.5

0.

z (m)

z (m)

z (m)

(a)

347

1

1500

0.5
1

1.5

1800

0.5
1

2100
0.5

0000
0.5

0300
–0.5

0600
–0.5

–1000

0x (m)

1000

2000

3000

Figure 68.3 Englacial flow fields in a longitudinal cross-section computed for the times indicated in the figures. (a) Horizontal flow speed (m day−1). (b) Vertical strain rate (10−4 day−1). Positive value indicates tensile strain rate.
of the sliding law. We assume n′ = 1, and prescribe c(x) in orderto reproduce the observed diurnal flow variations.

68.3.2Model output

Horizontal flow speed and vertical strain rate were computed fora region between 3 km up-glacier and down-glacier of the studysite (Fig. 68.3). Ice thickness at the study site was modelled to be
400 m. No sliding c(−3000 < x < 3000) = 0 was assumed at 0800
hours when the surface speed was at a minimum, and then c(x)was determined for each time of day so that the computed
surface-flow speeds at x = 0 agree to within 10% of the measurements on 25 August (Fig 68.3a). This 10% tolerance was within
GPS measurement errors. Based on the hypothesis that the basal
motion was higher in the upper reach, a sliding zone with greater
value of c was allocated up-glacier of the study site at 1000 hours
and its magnitude increased during the daytime. The position of
the sliding zone was then gradually moved down-glacier at night,
reducing the magnitude of c toward next morning.
Output from the model recreates the principal features of thefield measurements:
1

vertical strain rate is tensile during the day from 1000 to 1800
hours;
2 magnitude of the strain rate reaches 10−4 day−1 at 1500 hours
at x = 0;
3 diurnal velocity variation is greater up-glacier.

Although the computed strain rate is still smaller than that measured in the borehole, the basal sliding zone established in theupper reach and its temporal evolution was able to explain thediurnal flow variations in Lauteraargletscher.

68.4

Discussion

High temporal resolution measurements of surface velocity andvertical strain rate indicated systematic diurnal flow variations ina temperate valley glacier. These data are consistent with a causerelated to spatially and temporally non-uniform basal motion.
These surface flow fluctuations are not simply the results ofvariations in local basal motion, but also of variations in neighbouring basal and englacial motion, as shown in Fig. 68.3. Thus,
the spatial distribution of basal motion and its transfer to theenglacial flow field control short-term flow variations in temperate glaciers. High-frequency measurements of the englacial flow
field are crucial to improve our understanding in this research
field and accurate computation of the glacier flow field is important to infer bed conditions and basal-flow processes from surface
observations.

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SIXTY-NINE

Using field data toconstrain ice-flow
models: a study of asmall alpine glacier
Alun Hubbard
Department of Geography, University of Edinburgh, Edinburgh
EH8 9XP, UK

This case study is concerned with the application of a threedimensional ice flow model to Haut Glacier d’Arolla, a small temperate glacier in the Swiss Alps. The study illustrates both thepotential and certain limitations of using high-resolution, threedimensional flow modelling constrained by observations to aid
our understanding of alpine glacier systems. In particular, thecombination of flow modelling with detailed field measurements
at Haut Glacier d’Arolla provides a powerful tool to investigate its
rheological properties, the interaction between basal hydrology
and ice dynamics, the occurrence and formation of surface structures, its past and present response and, ultimately, its future trajectory to climate change.
The model is based on Blatter’s (1995) solution of the mass andforce balance equations using a non-linear rheology (with theflow law exponent: n = 3) applied to the surface and bed topography of Haut Glacier d’Arolla at 70 m horizontal resolution with
40 vertical layers scaled to thickness. The model is steady-state; itcomputes the instantaneous stress and strain distribution basedon Glen’s (1958a) flow law:.
eij = Aten−1tij.
where eij is the strain rate tensor, A is the rate factor reflecting ice
hardness, tij is the imposed stress tensor and te is the effective
stress given by
222+ t yz2 ) + t ′xx+ t ′yy+ t′zz22t e2 = 2(t xz2 + t xy

The model computes first-order terms, that is longitudinal andtransverse deviatoric stresses (t ′xx and t ′yy) which result in compression and tension throughout the ice mass and which, inextreme circumstances, lead to observed structural failure such as
overthrusting, shearing and crevassing. The resulting solution ishighly dependent on the basal boundary condition, which can be
specified as either a velocity or drag distribution or a combination of both, which enables the spatial interaction of slip/stick
patchiness resulting from heterogeneity in the bulk subglacialproperties, such as roughness, hydrology, effective pressure andsediment strength, to be modelled. This is achieved through theprescription of zero or reduced basal shear traction to replicate
decoupled, low drag zones, whereas zero velocity or increased
shear traction can be prescribed to simulate ‘sticky’ zones. For thepurposes of initial model optimization though, the model is first
specified with zero sliding across the whole of the bed.
By holding the flow-law exponent (n) constant then the only
parameter that requires ‘tuning’ is the rate factor (A) related to iceviscosity. Assuming Haut Glacier d’Arolla to be temperate withnegligible basal motion during winter, then comparison of modelled with observed winter surface velocities provides an effectivemeans for calibrating A. Observations that winter surface velocities are consistently low and, furthermore, that there is virtuallyno supraglacial melt-water available to drive ice–bed separation,
lend tentative support for this assumption. Tuning of the model
against surface velocities measured over 10 days in January 1995
yields an optimum value of A = 0.063 yr−1 bar−3, corresponding toan R2 = 0.74 on a bivariate plot of modelled against measured
surface velocities (Fig. 69.1). Although this value is half that
expected (Paterson, 1994) it does lie within a narrow range of 0.07
± 0.01 yr−1 bar−3 reported for other temperate glaciers modelledusing higher order solutions (e.g. Gudmundsson, 1999). Suchconsistency lends confidence in the predictive quality of the internal strain component of these models but also suggests that they
may be applied to temperate ice masses without significant tinkering with the rate factor.
Computation of the full stress and strain field enables modelcomparison with additional observables such as the occurrence
and orientation of crevassing and measured principal strains.
Comparison of zones of maximum computed surface tensile
strain and their direction with the actual distribution and orientation of crevassing reveals a good general correspondence (Fig.
69.2a & b). Furthermore, the orientation of modelled principal
strain faithfully reproduces the pattern measured from repeat
survey of a dense network of strain diamonds from 1994 to 1995
(Fig. 69.3a & b). However, it is apparent that even though their