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Conduit roughness and dye-trace breakthrough curves:
why slow velocity and high dispersivity may not reflect flow in
distributed systems
J.D. GULLEY,
1,2
P. WALTHARD,
2,3
J. MARTIN,
4
A.F. BANWELL,
5,2
D.I. BENN,
2,6
G. CATANIA
1,7
1
Institute for Geophysics, University of Texas at Austin, Austin, TX, USA
E-mail: gulley.jason@gmail.com
2
Department of Arctic Geology, The University Centre in Svalbard, Longyearbyen, Norway
3
Department of Environmental Sciences, University of Basel, Basel, Switzerland
4
Department of Geological Sciences, University of Florida, Gainesville, FL, USA
5
Scott Polar Research Institute, University of Cambridge, Cambridge, UK
6
School of Geography and Geosciences, University of St Andrews, St Andrews, Fife, UK
7
Department of Geological Sciences, University of Texas at Austin, Austin, TX, USA
ABSTRACT. Dye-trace breakthrough curves (BTCs) that increase in velocity and decrease in dispersivity
through a melt season have been interpreted as indicating a switch from a distributed to a conduit
subglacial drainage system, but this interpretation has not been validated in glaciers where the drainage
system configuration was independently known. To test if processes other than a change in the
configuration of the subglacial drainage system could produce similar BTCs, we measured BTCs from a
persistent, mapped subglacial conduit beneath Rieperbreen, Svalbard, which lacks a distributed system
because it is frozen to its bed. This conduit produced slow and highly dispersed BTCs early in the melt
season when meltwater delivery rates were low, and fast and sharply peaked BTCs after the snowpack
had retreated past the injection moulin. At Rieperbreen, the seasonal evolution of BTCs was controlled
by decreases in conduit roughness as increased rates of meltwater delivery increased the relative
submergence depths of rocks on the conduit floor. Because seasonal changes in roughness can produce
slow and highly dispersed BTCs, dye-tracing studies may not be capable of uniquely identifying
subglacial drainage system configurations. As a result, conduits may form earlier in melt seasons than
previously recognized.
1. INTRODUCTION
The configuration of the subglacial drainage system that
dominates a glacier bed is thought to be an important
control on basal water pressure and ice sliding speeds
(Fountain and Walder, 1998; Benn and Evans, 2010).
Distributed systems have low hydraulic capacity, so modest
increases in the rate of meltwater delivery can generate large
increases in basal water pressure (Kamb, 1987; Fountain and
Walder, 1998). Increased basal pressure and meltwater flow
are thought to destabilize distributed systems, forcing them
to collapse into integrated networks of conduits (Kamb,
1987). In contrast, conduits have greater hydraulic capacity,
which allows them to better drain surface meltwater inputs
as well as the adjacent distributed system, ultimately
decreasing subglacial water pressure and sliding speeds
(Mair and others, 2002; Anderson and others, 2004).
Shifts between distributed and conduit subglacial drain-
age systems have widely been inferred from temporal
changes in breakthrough curves (BTCs) obtained from dye-
tracing studies (Seaberg and others, 1988; Willis and others,
1990; Fountain, 1993; Hock and Hooke, 1993; Nienow and
others, 1998). In these studies, dye is injected into moulins
and BTCs are recorded at discharge points or proglacial
streams near glacier termini. These data are used to
calculate the velocity and dispersivity of tracer flowing
through subglacial hydrologic systems so that traces
conducted at different moulins or from the same moulin at
different times can be compared (Hubbard and Glasser,
2005). Transit velocity, v, is typically calculated from the
straight-line distance between the injection point and
the discharge point divided by the time from injection until
the time of peak concentration. Dispersivity, d(m), is
commonly used to define the rate of spreading of dye from
BTCs. It is the ratio of the degree of spreading of dye
(dispersion, D(m
2
s
–1
)) to transit velocity (m s
–1
).
Changes in velocity and dispersivity have been widely
used to infer shifts between conduit and distributed sub-
glacial drainage systems at individual moulins and across
glaciers (Seaberg and others, 1988; Willis and others, 1990;
Fountain, 1993; Hock and Hooke, 1993; Nienow and
others, 1998; Bingham and others, 2005). One widely cited
study was conducted on Haut Glacier d’Arolla, Switzerland,
where seasonally increasing velocities and declining dis-
persivities were interpreted as indicating a switch from a
distributed to a channelized subglacial drainage system
(Nienow and others, 1998). The increased velocity and
decreased dispersivity generally coincided with the passage
of the retreating snowline up-glacier, suggesting the timing
of this switch was controlled by the increased volume and
diurnal variability of meltwater inputs associated with the
thinning and depletion of the snowpack (Nienow and others,
1998; Willis and others, 2002). In particular, the large
diurnal variations in supraglacial recharge that occur
following the removal of the snowpack have been suggested
to be important in driving the pressure perturbations
necessary to destabilize linked cavities within the distributed
Journal of Glaciology, Vol. 58, No. 211, 2012 doi: 10.3189/2012JoG11J115 915
system and ultimately form conduits (Nienow and others,
1998; Willis and others, 2002).
Other studies show that multiple processes other than
changes in the configuration of subglacial drainage systems
affect BTCs (Nienow and others, 1996; Schuler and others,
2004). Repeat tracer tests over diurnal melt cycles from
single moulins produced BTCs with diurnal variations in
velocity and dispersivity that were of similar magnitude to
seasonal changes. Generally, these diurnal traces generated
higher velocities and less dispersed BTCs when surface melt
rates were higher during the day than at night when surface
melt rates were low, indicating that the rate of meltwater
delivery may also affect BTCs. Because these diurnal
investigations were conducted late in the melt seasons after
conduit flow had become well established, these changes in
BTC could be driven by changes in the rate of meltwater
delivery to subglacial conduits in lieu of changes in the
configuration of the subglacial drainage system.
The rate of meltwater delivery to an existing conduit may
alter BTCs in multiple ways, including changes in conduit
diameter, variations in hydraulic gradients, and changes in
tracer storage time within moulins. Sharply peaked BTCs
with high velocities could result if conduits enlarge during
the day as rapid delivery of water maintains high conduit
water pressure, offsets creep closure, and allows melting of
the walls to increase conduit diameter and hydraulic
capacity. Conversely, low delivery of water at night will
allow creep closure, make conduits less efficient, and
generate more dispersed BTCs with lower velocities (Schuler
and Fischer, 2003, 2009). Although creep closure has
negligible control on conduit hydrology over diurnal time-
scales where glaciers are only a few hundred meters thick
(Werder and others, 2010), it becomes increasingly import-
ant as ice thicknesses approach 1000 m (Covington and
others, 2012). Hydraulic gradient and velocity are positively
correlated, resulting in elevated velocities during the day
when melt rates increase, and lower velocities at night when
melt rates decrease (Nienow and others, 1996). This effect is
offset when tracer is stored in moulins, thereby delaying the
BTC (Schuler and Fischer, 2009; Werder and others, 2010).
Changes in roughness have also been proposed to exert
an important, though largely unconstrained, control on BTCs
(Nienow and others, 1996; Schuler and Fischer, 2003).
Loosely defined, roughness is a resistance force caused by
bedforms or projections from the channel (i.e. rocks). This
force is empirically parameterized relative to an idealized
frictionless circular conduit to account for reduction in flow
velocity. In glaciological studies, roughness is commonly
parameterized as Manning roughness, n, though other types
of hydraulic roughness parameters exist. Holding hydraulic
gradient, flow path length and conduit diameter constant,
conduits with higher roughness should have lower velocities
and more highly dispersed BTCs than conduits with lower
roughness (Schuler and Fischer, 2003). Effects of roughness
on BTCs are poorly known in subglacial flow because
hydraulic diameters of subglacial conduits, which are
needed to solve for roughness, are largely inaccessible. As
a result, how variations in roughness affect BTCs remains
largely unknown.
This prior work indicates that BTCs have similar velocity
and dispersivity responses to multiple forcings, suggesting
unique interpretation of processes controlling their shapes
may not be possible (Nienow and others, 1996; Schuler and
Fischer, 2003, 2009; Schuler and others, 2004; Werder and
others, 2010). Here we contribute to understanding of
controls on BTCs by using field data to evaluate how
seasonal changes in roughness in conduits affect transit
velocities and dispersivities derived from BTCs. We report on
nine traces conducted in a subglacial conduit beneath the
thin, cold-based Rieperbreen, Svalbard. This subglacial
conduit has a known and fixed geometry and no subglacial
tributaries downstream of the injection point, fixing many of
the variables that are unconstrained in other locations and
allowing us to assess the effects of meltwater delivery and
roughness on BTCs. Our results suggest that changes in
roughness may produce BTCs that mimic the shape of BTCs
that have been interpreted to reflect the shift between conduit
and distributed systems.
1.1. Rieperbreen
Rieperbreen is an outlet glacier of Foxfonna ice cap (Fig. 1)
and is located near the main settlement of Longyearbyen in
Svalbard (Lysa
˚and Lønne, 2001). The glacier and the
subglacial conduit beneath it have many attributes that make
it ideal for investigating recharge effects on dye BTCs.
Rieperbreen is frozen to its bed (Lysa
˚and Lønne, 2001) and
thus lacks a distributed subglacial drainage system. A
conduit exists at the bed, however, which evolved from a
supraglacial stream by the process of incision and roof
closure (termed ‘cut and closure’ by Gulley and others,
2009). Conduit flow processes alone thus control BTCs
because exchange with porous subglacial till or linked
cavities is not possible. The conduit persists between years
and, based on repeat visits between September 2006 and
March 2010, its morphology varies little from one year to the
next. The ice overlying the conduit is <30m thick, making
creep closure rates negligible. The conduit is incised in
frozen till, limiting its enlargement over seasonal timescales.
It has large cross sections (Fig. 2) and, since it is incised in
frozen till, is likely to remain at atmospheric pressure for
most of the melt season. If the conduit experienced
pressurized flow conditions for extended periods of time,
Fig. 1. Rieperbreen location map. Dye was injected in the moulin
at point A. Moulin B was used as the access point to the
englacial and subglacial drainage system in all years. A small hole
in the roof of the subglacial conduit at point C provided access to
the most downstream portions of the drainage system in spring and
summer 2010. Water discharged from the glacier at point D, and
the field fluorometer was installed at the location indicated by the
black square.
Gulley and others: Conduit roughness and dye-trace breakthrough curves916
the overlying ice would have melted to enlarge the conduit
rather than incising into till. Seasonal changes in hydraulic
gradient are thus small compared to glaciers where
hydraulic gradients increase as water backs up in moulins
(Schuler and Fischer, 2009; Werder and others, 2010).
These conduit characteristics make Rieperbreen ideal to
assess the effects of changing meltwater delivery rates and
the associated changes in roughness on BTCs. Although a
similar study could have been carried out on a proglacial
stream, its flow would be modulated by the subglacial
drainage system, which would be less constrained than at
Rieperbreen (Jobard and Dzikowski, 2006; Flowers, 2008).
In addition, snow and ice deposits can dam proglacial
streams and alter the BTCs. Snow and ice blockages cannot
occur within the Rieperbreen conduit because it is covered
by the glacier. While the seasonal changes in roughness that
occur in the Rieperbreen subglacial conduit result from
changes in flow depth in an open channel, we can relate
these changes in roughness that would occur during
subglacial conduit enlargement using the Manning equation.
The Manning equation calculates flow in non-circular pipes
and open channels by converting irregular cross sections to
circular pipes of equivalent diameter. The Manning equation
is discussed in greater detail in Section 2.4.
1.2. Conduit description
Three major supraglacial streams sink into moulins on
Rieperbreen (Fig. 1). These moulins are the upstream ends
of cut-and-closure conduits (Gulley and others, 2009) and
were not formed by water sinking into a crevasse (cf. Sten-
borg, 1969). Field observations and maps from 1996 indicate
there was no subglacial conduit at that time and the only
drainage beneath the surface of the glacier was along a lateral
meltwater channel a few tens of meters long (Lysa
˚and Lønne,
2001). Although conduits in cold ice in Svalbard can persist
for decades (Gulley and others, 2009), the lack of subglacial
drainage features in 1996 (Lysa
˚and Lønne, 2001) suggests the
conduit below Rieperbreen may have formed recently.
The entire section of englacial and subglacial conduit was
humanly traversable from the lowest moulin (point B in
Fig. 1) to the discharge point at the glacier snout (point D in
Fig. 1) in September 2006 when we mapped a combined
total of 671 m of englacial and subglacial passage (Fig. 2).
The moulin entrance was 44m above the subglacial
meltwater resurgence at the glacier snout. The floor of the
subglacial conduit consisted of large boulders (Fig. 3), and
gaps between individual boulders and the ice ceiling were
only a few decimeters in some conduit sections (e.g. A28).
The sorted floor contrasted strongly with the frozen, poorly
sorted matrix-supported diamict in the walls of the Nye
channel (Fig. 3). These passage morphologies are consistent
with incision of the floor by winnowing of fine-grained
matrix and concentration of boulders.
The cave was remapped in October 2009 and revisited in
March 2010 to determine the rate of alteration by collapse or
creep closure. In both October and March, the short section
of conduit between cross sections A11 and A13 remained
wide, but was too low to permit passage due to build-up of
aufeis on the conduit floor. Sections of the conduit down-
stream of A11 were accessible via a small hole in the ceiling
near station A8 (point C, Fig. 1) that appeared in summer
2010. Finally, collapse of a 30 m long block of thin ice near
the discharge point in 2009 made passage through the lower
exit impossible. Although these minor changes occurred
between the two mapped times, the conduit changed little,
reflecting its stability over interannual timescales.
In September 2006, the lowest of the three main moulins
on the glacier was at the position marked ‘Moulin entrance’
in Figure 2 (point A, Fig. 1). By October 2009, the
supraglacial stream had incised its way into another, pre-
existing portion of the subglacial conduit and sank at the
point marked with a black star (point B in Fig. 1). The new
moulin bypassed most of the englacial cut-and-closure
conduit and delivered meltwater to the bed directly and
water no longer flowed into the original moulin B (Fig. 1).
Dye traces were injected into the new moulin (indicated
with the black star in Fig. 1) starting on 14 June 2010 when
meltwater first started emerging from the front of the glacier,
and continued nearly weekly through 4 August 2010.
2. METHODS
2.1. Dye tracing
We measured 10–25 mL of Rhodamine WT (20% active
ingredient) in the field using a graduated cylinder calibrated
at 1 mL increments, and added it to a 1 L opaque nalgene
Fig. 2. Map of the subglacial conduit beneath Rieperbreen. Dotted
lines indicate passage continues but was not surveyed. Black star
represents supraglacial recharge point in October 2009 and dye
injection point. Note the difference in north arrow direction between
Figures 1 and 2. Conduit cross sections drawn 4plan scale.
Gulley and others: Conduit roughness and dye-trace breakthrough curves 917
bottle. Complete transfer of dye was accomplished by
adding small volumes of stream water to the cylinder and
decanting into the 1 L bottle until the graduated cylinder was
completely clear. The 1 L bottle was topped up with water
from the proglacial stream and mixed thoroughly. Snow was
excavated from the moulin, and the dye solution was
injected directly into water cascading into the moulin. All
injections reported here were made from the same location
(point A, Fig. 1) beginning on 14 June 2010, less than 1 week
after meltwater began emerging from the front of the glacier.
Dye traces were conducted through 500 m of mapped
subglacial conduit. Dye concentrations were measured and
logged at 30 s intervals with a Turner Model 10 AU field
fluorometer 100 m downstream of the subglacial melt-
water resurgence. The total distance of the trace was
600 m. Dye injections continued through the summer
and were conducted near the same time each day, within
2 hours of 13:00. Dispersivity was calculated using the
techniques of Seaberg and others (1988).
2.2. Discharge
We were unable to gauge rates of meltwater delivery to the
moulin during the early season due to ponding in the
snowpack. A stage–discharge relationship could not be
developed for the proglacial stream due to channel
migration. Continuous discharge data are therefore not
available. Discharge was calculated from the dye-trace BTCs
using the dilution method (Hubbard and Glasser, 2005)
assuming a 100% dye recovery. We consider complete
recovery to be reasonable because the subglacial conduit
consisted only of a short, unbranching passage and
exchange with the frozen till was unlikely. Because the
recession curve of the 14 June trace was incomplete, we
measured discharge for this trace by timing how long it took
for a float to travel 2m of the proglacial stream and
multiplied the velocity by the channel area.
2.3. Channel flow conditions
Plots of the log of the discharge versus the log of the velocity
can be used to distinguish, roughly, between flow in open
channels and pipe-full conditions at glacier beds (Nienow
and others, 1996; Palmer, 2007) according to
v¼kQmð1Þ
where kand mare empirically derived constants; mshould
equal 1 if conduits are pipe-full since any change in
discharge, Q, must be caused by an increase in velocity, v.
In conduits with free-surface streams, some of the increase
in Qcan be accommodated by an increase in cross-
sectional area of flow within the open conduit without an
increase in vand, as a result, mwill be <1. Equation (1)
reflects flow conditions at Rieperbreen because it contains a
single conduit and because it does not exchange water with
the surrounding bed. Networks of conduits allow water to
back up into tributary conduits, invalidating assumptions
used in Eqn (1) (cf. Nienow and others, 1996; Schuler and
Fischer, 2009). Additionally, velocity–discharge relation-
ships will be affected where conduits can exchange signifi-
cant amounts of water with their beds or where glaciers are
hydraulically lifted from their beds.
Since conduit geometry is known at our site, we can also
calculate how flow depth changes with discharge at each
Fig. 3. Large boulders on passage floors contrast with poorly sorted, fine-grained subglacial till in passage walls (at left). Photo taken at
survey station A19 (Fig. 1).
Gulley and others: Conduit roughness and dye-trace breakthrough curves918
conduit cross section. First we calculate the cross-sectional
area, A, of flow,
A¼Q
vð2Þ
and solve for the conduit flow depth, d
flow
, at each station by
dividing Aby the width, w, of the conduit:
dflow ¼A
wð3Þ
If d
flow
is lower than the measured height of the ceiling, flow
is in an open channel.
2.4. Manning roughness
Manning roughness coefficient, n, has units of s m
–1/3
and
can be calculated by solving the Gauckler–Strickler–
Manning equation for n:
n¼R2=3
hS1=2
0
vð4Þ
where R
h
is the hydraulic radius (defined as the cross-
sectional area divided by wetted perimeter), S
0
is the
hydraulic gradient and vis the average cross-sectional flow
velocity. Calculation of the hydraulic radius converts flow in
open channels or non-circular cross sections to a circular
pipe with equivalent hydraulic properties. We use data from
each of our dye traces to determine how nvaries through the
melt season. We calculate R
h
using an average stream width
of 5 m in the subglacial conduit that was determined from
conduit surveys and the flow depth that was calculated using
Eqns (2) and (3). S
0
is taken to be the surveyed hydraulic
gradient of the subglacial portions of the conduit (0.043).
3. RESULTS
3.1. Snow depth
On 14 June, snow depth in a pit excavated by the moulin
was 120 cm, and depth of the water layer at the base of the
snowpack was 28 cm (Fig. 4). While water level was not
recorded on 17 June, the snowpack had decreased an
additional 10 cm. By 28 June, the water level was nearly the
same height as the snowpack, and the snowline had
retreated past the moulin by 4 July.
3.2. Breakthrough curves
The shape of BTCs evolved systematically over the period
that the snowpack was melting out. They became more
peaked and had fewer tailing effects as the snowpack
thinned (Fig. 5a). The 14 June trace had a slow rise to peak
concentration over 45min and then concentrations con-
tinued to decrease for another 2.5 hours. Unfortunately, dye
concentration measurements had to be terminated before
background concentrations were reached, so the full length
of the tail is unknown. The 17 June trace had a much faster
rise to peak concentration (25 min) than the 14 June trace,
and tailing effects, while still present, were less prominent.
By 4 July, BTCs had greater symmetry, and most traces for
the remainder of the study rose to peak concentrations in
5 min (Fig. 5a and b).
3.3. Proglacial discharge
On average, discharge increased nearly linearly from the
beginning of the melt season from a low of 0.04 m
3
s
–1
on
14 June to 0.77 m
3
s
–1
on 4 July. Discharge dropped back to
0.44 m
3
s
–1
on 23 July and then rebounded to a maximum of
1.07 m
3
s
–1
on 4 August (Fig. 6a). We calculated a discharge
of >3 m
3
s
–1
on 11 July, but, based on flow observations in
the field, this value is unreasonable and is not plotted.
3.4. Tracer transit velocity
Velocities increased through the early part of the melt season
and peaked along with the estimated peak discharge in early
August (Fig. 6b). The lowest velocity of 0.07 m s
–1
was
recorded on 14 June. Velocities increased systematically
Fig. 4. Snow depth and water depth within the snowpack.
Fig. 5. (a) Dye-trace BTCs during the retreat of the snowpack past the injection moulin. The snowline migrated past the moulin on 4 July.
(b) Dye-trace BTCs from traces conducted from 11 July to 5 August. Note the change in scale on the xaxis between (a) and (b).
Gulley and others: Conduit roughness and dye-trace breakthrough curves 919
to 0.77 m s
–1
through 11 July before dropping back to
0.5 m s
–1
on 23 and 27 July. The highest velocity of
0.88 m s
–1
was recorded on 4 August.
3.5. Dispersivity
Early-season BTCs show lower velocities and higher
dispersivities than BTCs recorded later in the season
(Figs 5a and 6c; Table 1). For example, the 14 June trace
had a dispersivity of 19.9 m (Fig. 6c). Dye discharged from
the glacier for 4 hours, and the difference between peak and
background concentration was small (Fig. 5a). Dispersivity
dropped sharply to 4.3 m on 17 June (Fig. 6c) and continued
declining to 1.2 m on 4 July when the snowpack retreated
past the injection moulin. Decreases in dispersivity also
coincided with decreased tailing in BTCs. For instance, the
14 June trace rose from background to peak concentration in
45 min but the decline to background continued for
>3 hours before the trace had to be terminated. Dispersivity
remained below 2 m for all the traces conducted between
28 June and 27 July and increased slightly, to 2.7 m, on
4 August. Visually, these decreases in dispersivity are seen as
BTCs that become progressively more peaked, and exhibit
progressively less tailing, during the removal of the snow-
pack (Fig. 5a and b). Once the snowpack has retreated past
the moulin, BTCs have similar shapes (Fig. 5) and
dispersivities (Fig. 6c).
3.6. Manning roughness
Manning roughness, n, was highest at the beginning of the
season (0.67 s m
–1/3
) and declined sharply through the first
Fig. 6. (a) Discharge, Q, at the glacier snout, (b) tracer velocity, v, (c) tracer dispersivity, d, and (d) Manning roughness, n, through time.
Table 1. Dye-tracing data and Manning roughness
Date (2010) Qv dCSA Avg. depth PR
h
n
m
3
s
–1
ms
–1
mm
2
mmmsm
–1/3
14 Jun 0.04 0.07 19.89 0.57 0.114 5.23 0.11 0.68
17 Jun 0.06 0.09 4.34 0.67 0.134 5.27 0.13 0.58
24 Jun 0.12 0.21 2.3 0.57 0.114 5.23 0.11 0.23
28 Jun 0.37 0.36 1.92 1.03 0.206 5.41 0.19 0.19
4 Jul 0.59 0.5 1.24 1.18 0.236 5.47 0.22 0.15
11 Jul 0.77 1.75
23 Jul 0.53 0.44 1.5 1.2 0.24 5.48 0.22 0.17
27 Jul 0.94 0.5 1.36 1.88 0.376 5.75 0.33 0.20
4 Aug 1.07 0.88 2.65 1.22 0.244 5.49 0.22 0.09
Notes:Q: proglacial discharge; v: tracer velocity; d: dispersivity; CSA: cross-sectional area; Avg. depth: flow depth calculated assuming average conduit width
of 5 m; P: wetted perimeter; R
h
: hydraulic radius; n: Manning roughness.
Gulley and others: Conduit roughness and dye-trace breakthrough curves920
four traces, dropping below 0.2 s m
–1/3
on 28 June (Fig. 6d).
Values of nremained below 0.22 s m
–1/3
for the remainder of
the season. Manning roughness values decreased rapidly
with hydraulic radius, R
h
, at the beginning of the melt season
(Fig. 7a), and relationships between R
h
and ncould be fit
with a power law if data from the 24 June trace were
excluded. Once the hydraulic radius decreased below
0.2 m, values of nwere relatively constant between 0.1
and 0.2 s m
–1/3
. Manning roughness also exhibited nonlinear
relationships with tracer velocity and dispersivity that could
be fit with a power law (Fig. 7b and c respectively), with
velocity increasing and dispersivity decreasing as nde-
creased. Dispersivity values became relatively constant at
1–2 m once ndecreased below 0.4 s m
–1/3
, whereas velocity
continued to increase with decreases in n.
3.7. Channel flow conditions
Equation (1) was fit to the log of the discharge and the log of
the velocity, resulting in an r
2
of 0.87, and the parameters k
and mwere 0.70 and 0.69 respectively, indicating flow in
open channels (Fig. 8). We also calculated changes in water
depth at the subglacial conduit cross section at A16. At 14 m
wide and with a maximum vertical extent of 1 m between
the highest point on the ceiling and the lowest point on the
floor (Fig. 2), cross section A16 is the lowest conduit cross
section in the subglacial conduit. At A16, flow depths varied
from 0.04 m at low flow to 0.09–0.13 m at high flow, leaving
a minimum (though irregularly shaped) air space of 0.87 m.
4. DISCUSSION
4.1. Comparison with other dye-trace breakthrough
curves
Velocities and dispersivities from Rieperbreen are generally
comparable to other glaciers (summarized in Willis and
others, 2009) despite all our traces having occurred under
atmospheric pressure. Our lowest velocity of 0.09 m s
–1
was
recorded for the first dye trace of the melt season and is
slightly higher than the lowest velocities of 0.01 m s
–1
reported from both Midtdalsbreen, Norway (Willis and
others, 1990), and South Cascade Glacier, USA (Fountain,
1993). Our highest velocity of 0.88 m s
–1
was on 5 August
and is of a similar magnitude to the maximum reported
velocity of 0.99 m s
–1
recorded at Haut Glacier d’Arolla
(Nienow, 1993). Our dispersivity values ranged from a
minimum of 1.3 m on 14 July to a maximum of 19.9 m on
14 June and are within ranges reported from other glaciers.
For example, dispersivity ranged from 0.7 to 71 m on
Midtdalsbreen (Willis and others, 1990) and from 0.9 to
53 m on Haut Glacier d’Arolla (Nienow, 1993).
The lower dispersivities at Rieperbreen than at other
glaciers may reflect differences in flow path length, since
dispersivity increases with flow path length in natural
systems (Moltz and others, 1983; Gelhar and others, 1992;
Glimm and others, 1993; Hauns and others, 2001). Flow
path length increases dispersivity through multiple pro-
cesses, including changes in the velocity structure of water
and directions along flow paths (Levy and Berkowitz, 2003),
dilution by additional inputs of water downstream of the
injection site (Li and Loper, 2011), and temporary storage of
water in eddy currents or pools (Hauns and others, 2001).
These processes have been observed as temporary storage of
tracer in water-filled moulins (Werder and others, 2010),
hydraulic damming of up-glacier moulins caused by water
backing up into tributary conduits (Nienow and others,
1996; Schuler and Fischer, 2009), dilution by multiple
downstream inputs of water in conduit networks, and
temporary storage of water and dye in plunge pools in
englacial passages (Fountain, 1993). The lower dispersivity
values observed at Rieperbreen may therefore reflect the
Fig. 7. (a) Manning roughness coefficient, n, plotted as a function of hydraulic radius, R
h
. (b) Dispersivity, d, plotted as a function of Manning
roughness, n. (c) Tracer velocity, v, plotted as a function of Manning roughness, n. Note the goodness of fit for the power-law relationship
between nand R
h
was improved by omitting data from 24 June outlier (shown as black square in (a)).
Fig. 8. The log of the discharge, Q, plotted against the log of the
velocity, v. If flow occurred in a full pipe, the slope of this line
would be 1 (shown here as a dashed line); however, our data have a
slope of 0.69, indicating flow in an open channel.
Gulley and others: Conduit roughness and dye-trace breakthrough curves 921
fact that the conduit at Rieperbreen is shorter (600 m) than
other sites, which are typically >1 km. Other potential
causes for lower dispersivity at Rieperbreen relative to other
glaciers include the lack of additional meltwater inputs
downstream of the injection point at Rieperbreen, which
dilute tracers, as well as the lack of hydraulic damming,
which increases tracer travel time. Both dilution and
damming are common to most other studied glaciers and
are known to increase dispersivity values.
Changes through time of BTCs measured at Haut Glacier
d’Arolla are interpreted as examples of a distributed system
evolving into a channelized system (Hubbard and Glasser,
2005; Cuffey and Patterson, 2010). The timing of the shift
from distributed to channelized systems has been estimated
primarily from changes in velocity, with velocities lower
than 0.4–0.5 m s
–1
indicating flow in distributed systems,
and velocities higher than this indicating flow in conduits
(Nienow and others, 1998; Mair and others, 2002; Hubbard
and Glasser, 2005). Dispersivity values greater than 10 m are
also considered to indicate flow in distributed systems, but
this variable is less commonly used to identify types of flow
systems. The progression of BTCs used to infer a switch from
distributed to conduit systems at Haut Glacier d’Arolla
(Fig. 9b) corresponded to the arrival of the snowline at the
moulin (Nienow and others, 1998).
The seasonal evolution of BTCs, in both velocity and
dispersivity (Nienow and others, 1998), recorded at Haut
Glacier d’Arolla has been proposed to be caused by a shift
from a distributed to a conduit subglacial drainage system.
We obtained a similar progression of BTCs from traces
conducted in the subglacial conduit at Rieperbreen, although
our mapping of the Rieperbreen conduit clearly shows the
lack of a distributed system (Fig. 9). Similar to Haut Glacier
d’Arolla, tracer velocities at Rieperbreen are low prior to the
retreat of the snowpack, with velocities increasing system-
atically from 0.07 m s
–1
on 14 June to 0.36 m s
–1
on 28 June.
These velocities are less than the criterion of <0.5 m s
–1
assumed to indicate distributed systems (Nienow and others,
1998; Mair and others, 2002) but were generated in a
conduit at Rieperbreen. Similar results of flow less than
0.5 m s
–1
have been found in drainage systems that were
presumed to have been conduit-dominated (Nienow and
others, 1996; Schuler and others, 2004), but the decrease in
velocities could not be unambiguously established since the
conduit size and morphology were unknown. Possible
causes of changes in BTCs in these studies are changes in
hydraulic gradient, changes in roughness, or conduit
enlargement and creep.
4.2. Roughness and hydraulic radius control of
breakthrough curves
Recent modeling studies have advanced understanding of the
controls on BTC evolution (Schuler and Fischer, 2009;
Werder and Funk, 2009; Werder and others, 2010). Model
solutions are likely non-unique, however, because changes
in hydraulic gradient, roughness or hydraulic radius can
generate similar changes in BTCs. Additionally, these models
rely on tuning parameters to fit data from complex conduit
networks consisting of multiple moulins that receive melt-
water inputs at different times to the outputs of models that
treat these complex networks as one or two pipes. Our
observational results are, however, unambiguous. Log–log
plots of velocity and discharge (Fig. 8) and measured
discharge values compared with channel cross-sectional
area show that flow in the single conduit beneath Rieper-
breen was at atmospheric pressure. As a result, because
changes in hydraulic gradient are fixed by the slope of the
bed, which is known, and because changes in hydraulic
radius can be calculated from changes in flow depth, we can
use the Manning equation to solve for changes in roughness.
Our study therefore allows us to more directly assess the
influence of roughness on tracer dispersivity and velocity
than previous studies, where drainage system configurations
were not known and where tracer velocities could have been
influenced by conduit network effects.
The floor of the conduit beneath Rieperbreen consists of
roughness features, such as large boulders and cobbles
(Fig. 3), that affect BTCs in several complementary ways.
Wetted surface area decreases relative to cross-sectional area
as flow depth increases, thereby increasing velocity and
decreasing dispersivity. At high discharges, boulders and
cobbles on the floor of the conduit are completely
submerged, reducing tortuosity of flow around boulders.
Flow above boulders will reduce the amount of back-eddy
currents and immobile regions that can delay and tempor-
arily store small quantities of tracer (Hauns and others, 2001).
Limited ponding and back-eddy currents will increase
velocity and decrease dispersivity. In contrast, during low
discharges early in the melt season, emergent boulders will
increase ponding and back eddies and could cause the
tailing, slow velocity and elevated dispersivities observed
prior to the 24 June trace (Fig. 5). Changes in roughness in
open channels, such as at Rieperbreen, may also explain
diurnal variability in velocity and dispersivity found late in
melt seasons after conduit flow had had time to become
established (cf. Nienow and others, 1996; Schuler and others,
2004). Conduit diameters will adjust to accommodate peak
meltwater inflows that occur during the day, but during the
night, when surface melt rates decrease, lower inflow could
result in open-channel flow and high conduit roughness.
Dispersivity values at our site also declined more rapidly
than some of the BTCs recorded at other glaciers. Because it
seems likely that changes in dispersivity at Rieperbreen are
caused primarily by changes in roughness, the more rapid
decrease in dispersivity at Rieperbreen may reflect the fact
that increased flow depth can increase hydraulic radii in
open channels, and thus decrease roughness, much faster
than pipe-full subglacial conduits can increase their hy-
draulic radius through melt to decrease roughness. Conduit
diameters can increase with greater influx of meltwater early
in the season and then decrease through creep late in the
melt season. Such variation in conduit size will change
roughness because the size of boulders on the conduit floors
is fixed (Fig. 10). At Rieperbreen, the largest roughness
changes occurred when conduit hydraulic radii were
<0.2 m. Small conduit hydraulic radii, coupled with large
boulders or other roughness elements at subglacial beds,
will result in high Manning’s roughness, providing a mech-
anism to generate the large values of n, which have
previously been disregarded as physically implausible (cf.
Nienow and others, 1996). Large magnitudes of roughness,
coupled with rapid, nonlinear changes in roughness as
conduits enlarge, may explain why pipe models adapted to
glacier hydrological systems are more successful in reprodu-
cing proglacial hydrographs in the middle of the melt season
when large conduits have formed than early in the melt
season, when drainage systems are more constricted (cf.
Arnold and others, 1998; Werder and Funk, 2009).
Gulley and others: Conduit roughness and dye-trace breakthrough curves922
4.3. Implications of the configuration and evolution of
subglacial drainage systems
While temporal increases in velocity and decreases in tracer
dispersivity may reflect a switch from a distributed to a
conduit subglacial drainage system (Nienow and others,
1998), similar velocity and dispersivity values derived from
BTCs at Rieperbreen simply result from changing relative
roughness in a fully developed conduit system. Diurnal
changes in tracer velocity may be caused by changes in flow
depth in open channels, and seasonal changes may be
caused by the enlargement of constricted conduits or other
preferential, channelized flow paths. Slow and highly
dispersed BTCs can also result from changes in hydraulic
gradient or changes in moulin residence time (Schuler and
Fischer, 2009; Werder and others, 2010). Consequently,
changes in velocity and dispersivity alone may provide
insufficient evidence to independently distinguish between
a conduit and a distributed system.
Several studies have demonstrated that flow in conduits
connected to moulins can produce slow and highly
dispersed BTCs, but few studies have examined, either
directly or through models, what types of BTCs a distributed
system will generate. A few studies, however, have been
conducted where dye was injected into boreholes. Conduits
represent a small fraction of the glacier footprint, so
boreholes are unlikely to be installed close to conduits
(Gordon and others, 1998). Consequently, the few dye traces
conducted from boreholes are more likely to represent
distributed systems. In borehole injections, tracer travel
times are measured in days or weeks instead of hours, and
dye continues to discharge for weeks, resulting in BTCs that
are too highly dispersed for dispersivity values to be
calculated (Hooke and Pohjala, 1994; Hock and others,
1999). Transit times from moulin injections are orders of
magnitude faster than injections in boreholes, indicating
moulins are connected to the fastest portions of the
subglacial drainage system and that subglacial drainage is
already channelized early in the melt season. If flow was not
channelized, BTCs from injections made in moulins at the
beginning of the melt season should be similar to BTCs
collected from traces injected in boreholes.
Channelization of the drainage system during the early
melt season could be caused by several processes. Past
workers have suggested that creep closure of ice should
cause subglacial conduits to collapse during the winter and
re-form from a distributed system each melt season
(Fountain and Walder, 1998; Nienow and others, 1998).
Recent work suggests that constricted conduit systems may
be kept open during winter by meltwater backing up into
the coupled englacial system (Benn and others, 2009;
Catania and Neumann, 2010). In addition, moulins are
frequently (but not always) recharge points for subglacial
conduits (Gulley and others, 2009) and form in approxi-
mately the same location year after year (Stenborg, 1969).
The location of moulins may thus allow subglacial conduits
to form along similar flow paths each year. Subglacial
conduits have been mapped in the same location with
similar morphologies in multiple years (Gulley and others,
in press). Sediments on the floors of subglacial conduits that
form in the same locations during multiple years can be
winnowed of fine-grained materials (Harbor and others,
1997; Gulley and others, in press), generating high-permea-
bility zones that could channelize flow and rapidly enlarge
each melt season (Figs 3 and 12). In this scenario, the
seasonal evolution of BTCs could be explained by the
continued enlargement, and decreased roughness, of this
channelized zone. While it could be argued that flow
bifurcation by rocks at low flow in our conduit at
Rieperbreen could be seen as analogous to flow in a
distributed system, it is important to note that our data
indicate this ‘distributed’ system need only be a few meters
wide to generate highly dispersed BTCs and that flow in this
system can be described using conduit flow equations.
While we agree that subglacial drainage systems enlarge
and become more efficient through melt seasons, and that
past studies have recorded increases in the hydraulic
Fig. 9. BTCs collected from a persistent subglacial conduit during the retreat of the snowpack in our study (a) are qualitatively similar to BTCs
from a 1990 study at Haut Glacier d’Arolla (b) (reproduced from Nienow and others, 1998). While most studies have interpreted a transition
from highly dispersed BTCs to more-peaked BTCs as indicating a transition from a distributed to a channelized subglacial drainage system,
all the BTCs in Figure 9a were obtained in a subglacial conduit with no possibility of influence from a distributed system.
Fig. 10. Early in the melt season (T
1
), constricted conduits have
small hydraulic radii relative to the height of bed roughness
features, such as rocks, resulting in high roughness. Conduit
enlargement by melt increases in the hydraulic radii of conduits
(T
2
), reducing roughness.
Gulley and others: Conduit roughness and dye-trace breakthrough curves 923
efficiency of subglacial drainage systems by analyzing
systematic variations in dye BTCs, we do not believe these
observations uniquely identify a switch from a distributed
system to a channelized system. Our results, similar to other
studies (cf. Schuler and others, 2004), suggest that BTCs that
are frequently interpreted as indicating flow in a distributed
system can be generated in conduits. As a result, subglacial
conduits may form earlier in the melt season than previously
thought, such as after snowpack has begun melting but
before it has retreated past moulins, or conduits could be
reused between years (cf. Benn and others, 2009; Catania
and Neumann, 2010) but their presence might not be
indicated with current interpretations of BTCs (cf. Nienow
and others, 1998).
The results of this study are qualitatively similar to a recent
investigation of the controls on proglacial glacier discharge
hydrographs (Covington and others, 2012). Although such
hydrographs were once thought to contain unique informa-
tion about the configuration of subglacial drainage systems,
Covington and others (2012) showed that proglacial dis-
charge hydrographs primarily respond to changes in the rate
of meltwater delivery to the subglacial drainage system and
that hydrographs do not carry unique information about the
configuration of subglacial drainage systems. The Covington
and others (2012) study, combined with our present investi-
gation, indicate that two of the main sources of field data that
have been used to infer a switch in the configuration of the
subglacial drainage system are not as conclusive as once
thought. Consequently, seasonal changes in subglacial
hydrological systems could be strongly influenced by
seasonal changes in the hydraulic capacity of channelized
zones rather than simply by changes in the configuration of
the subglacial drainage system.
5. CONCLUSIONS
Seasonal changes in BTCs in a subglacial conduit beneath
Rieperbreen were similar to BTCs that have been used to
infer switches from distributed to conduit drainage systems.
Beneath Rieperbreen, however, drainage was through a
known conduit with a fixed geometry and changes in
drainage system configuration can be ruled out. Instead,
systematic changes in BTCs were caused by changes in
conduit roughness. At Rieperbreen, seasonal decreases
in roughness were caused by seasonal increases in water
flow depth that occurred as the retreat of the snowpack
increased the rate of meltwater delivery to the conduit.
Increases in flow depth resulted in increased relative
submergence of roughness elements at the bed, such as
boulders and rocks. Tracer velocity increased and disper-
sivity decreased because water had less interaction and
contact with rocks on the conduit floor. These changes in
roughness at Rieperbreen are directly analogous to how
roughness would change in pipe-full subglacial conduits
beneath thicker glaciers where conduit enlargement reduces
the interaction of subglacial water with rocks at the bed.
Dispersivity values recorded in BTCs may therefore be
higher for longer periods of time on glaciers undergoing
conduit enlargement than in glaciers that experience open-
channel flow because conduit enlargement requires a much
longer timescale than increasing flow depth in open
channels and because the wetted perimeter of closed
conduits is greater than the wetted perimeter in open
channels. Because changes in roughness can generate the
slow and highly dispersed BTCs which have been inter-
preted to indicate flow in distributed drainage systems, it
may not be possible to infer system morphology or
configuration using BTCs. Further, because dispersivity
increases with the length scale of the studied flow path,
comparisons of dispersivities between different glaciers and
traces conducted in different moulins on the same glacier
may not be comparing like with like.
Our results do not, of course, disprove that seasonal
variations in BTCs might be influenced by changes in
drainage system configuration, but they do complement a
growing number of studies that indicate slow and/or highly
dispersed BTCs can be generated in conduits. Using changes
in BTCs to infer changes in the configuration of subglacial
drainage systems will remain problematic until the full range
of BTCs that can be produced in conduit systems under
different recharge conditions is known. While changes in
BTCs may be insufficient evidence for a change in the
configuration of subglacial drainage systems, dye tracing
remains an effective method of delineating subglacial
drainage basins (Fountain, 1993).
ACKNOWLEDGEMENTS
J. Gulley acknowledges funding from the American Philoso-
phical Society, the Lewis and Clark Fund for Exploration and
Field Research, the University of Florida and the US National
Science Foundation (NSF) in the form of a Graduate Research
Fellowship and a NSF Division of Earth Sciences (EAR)
Postdoctoral Fellowship (No. 0946767). Gulley and Benn
also acknowledge funding and logistical support from the
University Centre in Svalbard. We thank A. Bergstrom,
M. Temminghoff, Z. Luthi, M. Covington and I. Willis for field
assistance. We thank I. Willis for conversations, and
M.A. Werder and an anonymous reviewer for comments
that greatly improved the manuscript.
REFERENCES
Anderson RS and 6 others (2004) Strong feedbacks between
hydrology and sliding of a small alpine glacier. J. Geophys.
Res., 109(F3), F03005 (doi: 10.1029/2004JF000120)
Arnold N, Richards K, Willis I and Sharp M (1998) Initial results
from a distributed, physically-based model of glacier hydrology.
Hydrol. Process., 12(2), 191–219 (doi: 10.1002/(SICI)1099-
1085(199802)12:2<191::AID-HYP571>3.0.CO;2-C)
Benn DI and Evans DJA (2010) Glaciers and glaciation, 2nd edn.
Hodder Education, London
Benn D, Gulley J, Luckman A, Adamek A and Glowacki PS (2009)
Englacial drainage systems formed by hydrologically driven
crevasse propagation. J. Glaciol., 55(191), 513–523 (doi:
10.3189/002214309788816669)
Bingham RG, Nienow PW, Sharp MJ and Boon S (2005)
Subglacial drainage processes at a High Arctic polythermal
valley glacier. J. Glaciol., 51(172), 15–24 (doi: 10.3189/
172756505781829520)
Catania GA and Neumann TA (2010) Persistent englacial drainage
features in the Greenland Ice Sheet. Geophys. Res. Lett., 37(2),
L02501 (doi: 10.1029/2009GL041108)
Covington MD, Banwell AF, Gulley J, Saar MO, Willis I and Wicks
CM (2012) Quantifying the effects of glacier conduit geometry
and recharge on proglacial hydrograph form. J. Hydrol.,
414(415) (doi: 10.1016/j.jhydrol.2011.10.027)
Cuffey KM and Paterson WSB (2010) The physics of glaciers, 4th
edn. Butterworth-Heinemann, Oxford
Gulley and others: Conduit roughness and dye-trace breakthrough curves924
Flowers GE (2008) Subglacial modulation of the hydrograph from
glacierized basins. Hydrol. Process., 22(19), 3903–3918 (doi:
10.1002/hyp.7095)
Fountain AG (1993) Geometry and flow conditions of subglacial
water at South Cascade Glacier, Washington State, U.S.A.; an
analysis of tracer injections. J. Glaciol., 39(131), 143–156
Fountain AG and Walder JS (1998) Water flow through temperate
glaciers. Rev. Geophys., 36(3), 299–328 (doi: 10.1029/
97RG03579)
Gelhar LW, Welty C and Rehfeldt KR (1992) A critical review of
data on field-scale dispersion in aquifers. Water Resour. Res.,
28(7), 1955–1974 (doi: 10.1029/92WR00607)
Glimm J, Lindquist WB, Pereira F and Zhang Q (1993) A theory of
macrodispersion for the scale-up problem. Transport Porous
Med., 13(1), 97–122
Gordon S, Sharp M, Hubbard B, Smart C, Ketterling B and Willis I
(1998) Seasonal reorganization of subglacial drainage inferred
from measurements in boreholes. Hydrol. Process., 12(1),
105–133
Gulley JD, Benn DI, Screaton E and Martin J (2009) Mechanisms of
englacial conduit formation and their implications for subglacial
recharge. Quat. Sci. Rev., 28(1920), 1984–1999 (doi: 10.1016/
j.quascirev.2009.04.002)
Gulley JD, Grabiec M, Martin JB, Jania J, Catania G and Glowacki P
(in press) The effect of discrete recharge by moulins and
heterogeneity in flow path efficiency at glacier beds on
subglacial hydrology. J. Glaciol.,58(211), 926–940
Harbor J, Sharp M, Copland L, Hubbard B, Nienow P and Mair D
(1997) The influence of subglacial drainage conditions on the
velocity distribution within a glacier cross section. Geology,
25(8), 739–742
Hauns M, Jeannin PY and Atteia O (2001) Dispersion, retardation
and scale effect in tracer breakthrough curves in karst conduits.
J. Hydrol., 241(3), 177–193
Hock R and Hooke RLeB (1993) Evolution of the internal drainage
system in the lower part of the ablation area of Storglacia
¨ren,
Sweden. Geol. Soc. Am. Bull., 105(4), 537–546
Hock R, Iken A and Wangler A (1999) Tracer experiments and
borehole observations in the overdeepening of Aletschgletscher,
Switzerland. Ann. Glaciol.,28, 253–260 (doi: 10.3189/
172756499781821742)
Hooke RLeB and Pohjola VA (1994) Hydrology of a segment of a
glacier situated in an overdeepening, Storglacia
¨ren, Sweden.
J. Glaciol., 40(134), 140–148
Hubbard B and Glasser N (2005) Field techniques in glaciology and
glacial geomorphology. Wiley, New York
Jobard S and Dzikowski M (2006) Evolution of glacial flow and
drainage during the ablation season. J. Hydrol., 330 (3–4), 663–
671 (doi: 10.1016/j.jhydrol.2006.04.031)
Kamb B (1987) Glacier surge mechanism based on linked cavity
configuration of the basal water conduit system. J. Geophys.
Res., 92(B9), 9083–9100 (doi: 10.1029/JB092iB09p09083)
Levy M and Berkowitz B (2003) Measurement and analysis of non-
Fickian dispersion in heterogeneous porous media. J. Contam.
Hydrol., 64(3–4), 203–226 (doi: 10.1016/S0169-
7722(02)00204-8)
Li G and Loper DE (2011) Transport, dilution, and dispersion of
contaminant in a leaky karst conduit. Transport Porous Media,
88(1), 31–43 (doi: 10.1007/s11242-011-9721-1)
Lysa
˚A and Lønne I (2001) Moraine development at a small High-
Arctic valley glacier: Rieperbreen, Svalbard. J. Quat. Sci., 16(6),
519–529
Mair D, Nienow P, Sharp M, Wohlleben T and Willis I (2002)
Influence of subglacial drainage system evolution on glacier
surface motion: Haut Glacier d’Arolla, Switzerland. J. Geophys.
Res., 107(B8), 2175 (doi: 10.1029/2001JB000514)
Molz FJ, Gu
¨ven O and Melville JG (1983) An examination of
scale-dependent dispersion coefficients. Ground Water, 21(6),
715–725 (doi: 10.1111/j.1745-6584.1983.tb01942.x)
Nienow P (1993) Dye tracer investigations of glacier hydrological
systems. (PhD thesis, University of Cambridge)
Nienow PW, Sharp MJ and Willis IC (1996) Velocity–discharge
relationships derived from dye tracer experiments in glacial
meltwaters: implications for subglacial flow conditions. Hydrol.
Process., 10(10), 1411–1426
Nienow P, Sharp M and Willis I (1998) Seasonal changes in the
morphology of the subglacial drainage system, Haut Glacier
d’Arolla, Switzerland. Earth Surf. Process. Landf., 23(9),
825–843
O
¨zger M and Yildirim G (2009) Determining turbulent flow friction
coefficient using adaptive neuro-fuzzy computing technique.
Adv. Eng. Softw., 40(4), 281–287 (doi: 10.1016/j.advengsoft.
2008.04.006)
Palmer AN (2007) Cave geology. Cave Books, Dayton, OH
Schuler T and Fischer UH (2003) Elucidating changes in the degree
of tracer dispersion in a subglacial channel. Ann. Glaciol.,37,
275–280 (doi: 10.3189/172756403781815915)
Schuler TV and Fischer UH (2009) Modeling the diurnal variation of
tracer transit velocity through a subglacial channel. J. Geophys.
Res., 114(F4), F04017 (doi: 10.1029/2008JF001238)
Schuler T, Fischer UH and Gudmundsson GH (2004) Diurnal
variability of subglacial drainage conditions as revealed by
tracer experiments. J. Geophys. Res., 109(F2), F02008 (doi:
10.1029/2003JF000082)
Seaberg SZ, Seaberg JZ, Hooke RLeB and Wiberg DW (1988)
Character of the englacial and subglacial drainage system in the
lower part of the ablation area of Storglacia
¨ren, Sweden, as
revealed by dye-trace studies. J. Glaciol., 34(117), 217–227
Stenborg T (1969) Studies of the internal drainage of glaciers.
Geogr. Ann., 51A(1–2), 13–41
Werder MA and Funk M (2009) Dye tracing a jo
¨kulhlaup: II. Testing
ajo
¨kulhlaup model against flow speeds inferred from measure-
ments. J. Glaciol., 55(193), 899–908 (doi: 10.3189/
002214309790152375)
Werder MA, Schuler TV and Funk M (2010) Short term variations of
tracer transit speed on alpine glaciers. Cryosphere, 4(3),
381–396 (doi: 10.5194/tc-4-381-2010)
Willis IC, Sharp MJ and Richards KS (1990) Configuration of the
drainage system of Midtdalsbreen, Norway, as indicated by dye-
tracing experiments. J. Glaciol., 36(122), 89–101
Willis IC, Arnold NS and Brock BW (2002) Effect of snowpack
removal on energy balance, melt and runoff in a small
supraglacial catchment. Hydrol. Process., 16(14), 2721–2749
(doi: 10.1002/hyp.1067)
Willis I, Lawson W, Owens I, Jacobel R and Autridge J (2009)
Subglacial drainage system structure and morphology of
Brewster Glacier, New Zealand. Hydrol. Process., 23(3),
384–396 (doi: 10.1002/hyp.7146)
MS received 7 June 2011 and accepted in revised form 19 May 2012
Gulley and others: Conduit roughness and dye-trace breakthrough curves 925
