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Large values of hydraulic roughness in subglacial
conduits during conduit enlargement: implications
for modeling conduit evolution
J. D. Gulley,
1
*P. D. Spellman,
2
M. D. Covington,
3
J. B. Martin,
4
D. I. Benn
5†
and G. Catania
6
1
Michigan Technological University, Department of Geological and Mining Engineering and Sciences
2
Michigan Technological University, Department of Civil and Environmental Engineering
3
University of Arkansas, Department of Geosciences
4
University of Florida, Department of Geological Sciences
5
University Centre in Svalbard (UNIS), Department of Arctic Geology
6
University of Texas, Institute for Geophysics
Received 16 October 2012; Revised 20 May 2013; Accepted 28 May 2013
*Correspondence to: J. D Gulley, Department of Geological and Mining Engineering and Sciences, Michigan Technological University, MI, USA. E-mail: gulley.
jason@gmail.com
†
Current Address: University of St Andrews, Department of Geography and Sustainable Development.
ABSTRACT: Hydraulic roughness accounts for energy dissipated as heat and should exert an important control on rates of
subglacial conduit enlargement by melting. Few studies, however, have quantified how subglacial conduit roughness evolves
over time or how that evolution affects models of conduit enlargement. To address this knowledge gap, we calculated values
for two roughness parameters, the Darcy–Weisbach friction factor (f) and the Manning roughness coefficient (n), using dye
tracing data from a mapped subglacial conduit at Rieperbreen, Svalbard. Values of fand ncalculated from dye traces were
compared with values of fand ncalculated from commonly used relationships between surface roughness heights and conduit
hydraulic diameters. Roughness values calculated from dye tracing ranged from 75–0.97 for fand from 0.68–0.09 s m
-1/3
for
n. Equations that calculate roughness parameters from surface roughness heights underpredicted values of fby as much as a
factor of 326 and values of nby a factor of 17 relative to values obtained from the dye tracing study. We argue these large
underpredictions occur because relative roughness in subglacial conduits during the early stages of conduit enlargement
exceeds the 5% range of relative roughness that can be used to directly relate values of fand nto flow depth and surface
roughness heights. Simple conduit hydrological models presented here show how parameterization of roughness impacts
models of conduit discharge and enlargement rate. We used relationships between conduit relative roughness and values of
fand ncalculated from our dye tracing study to parameterize a model of conduit enlargement. Assuming a fixed hydraulic
gradient of 0.01 and ignoring creep closure, it took conduits 9.25 days to enlarge from a diameter of 0.44 m to 3 m, which
was 6–7-fold longer than using common roughness parameterizations. Copyright © 2013 John Wiley & Sons, Ltd.
KEYWORDS: glacier hydrology; roughness; friction factor; englacial; subglacial
Introduction
Much previous work on links between glacier hydrology, ice
motion and hydraulic capacity has emphasized the importance
of changes in the physical configuration of subglacial drainage
systems in controlling changes in their hydraulic capacities.
Distributed drainage systems, often conceptualized as net-
works of cavities kept open by ice sliding over bumps and
linked to one another by narrow orifices, have been proposed
as a hydraulically inefficient drainage system configuration that
would offer high resistance to flow (low hydraulic capacity)
(Kamb, 1987). At the beginning of melt seasons, delivery of
surface melt to linked cavities would exceed the hydraulic
capacity of this inefficient drainage system, increasing subglacial
water pressure and ice sliding speeds. High water pressure and
sliding would persist until this linked cavity system evolved into
efficient, high hydraulic capacity conduits, which would draw
down subglacial water pressure and decrease ice sliding speeds
(Kamb, 1987; Schoof, 2010; Bartholomaus et al., 2011).
The necessity for different configurations of subglacial drain-
age systems to result in different relationships between subgla-
cial water pressure and sliding speeds can be traced to early
conceptual models of conduit flow. These widely-cited models
assumed that conduits received meltwater inflow from diffuse
englacial or subglacial sources of water and existed in steady
state (Rothlisberger, 1972; Shreve, 1972). As a result, conduits
were thought to always function as low pressure, hydraulically
efficient drains (high hydraulic capacity) of higher pressure
distributed systems. Under this model, conduits always
decrease subglacial water pressure and ice sliding speeds and
EARTH SURFACE PROCESSES AND LANDFORMS
Earth Surf. Process. Landforms 39, 296–310 (2014)
Copyright © 2013 John Wiley & Sons, Ltd.
Published online 10 July 2013 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/esp.3447
a configuration of subglacial drainage system that was
physically different and less hydraulically efficient than conduits
was required to increase subglacial water pressure and glacier
sliding speeds (Gulley et al., 2012a).
A growing number of studies, however, suggest that changes
in the configuration of subglacial drainage systems are not
necessary to promote high subglacial water pressure and ice
sliding speeds (Bartholomaus et al., 2008). High glacier sliding
speeds occur when the rate and volume of meltwater delivery
exceeds the hydraulic capacity of the subglacial drainage
system (Hubbard et al., 1995; Anderson et al., 2004; Kessler
and Anderson, 2004; Walder et al., 2006; Bartholomaus et al.,
2008, 2011); the configuration of the subglacial drainage
system (i.e. conduit or linked cavity) is largely irrelevant
(Bartholomaus et al., 2008; Gulley et al., 2012a).
Because of interrelationships between meltwater delivery, sub-
glacial hydraulic capacity and subglacial water pressure, the
effect of meltwater delivery to glacier beds on ice sliding speeds
is critically dependent on the rate of meltwater delivery relative
to antecedent hydraulic capacity of the subglacial drainage
system, which is determined by conduit diameter and hydraulic
roughness (Munson, 2005). Little is known, however, about
magnitudes or seasonal evolution of hydraulic roughness in
subglacial conduits. As a result, the ability to model accurately
the influence of changes in the hydraulic capacity of subglacial
drainage systems on ice sliding speeds remains limited.
In this paper, we use a combination of previously published
data as well as new observations and field experiments in
subglacial conduits to provide an integrated overview of how
changes in conduit diameters affect the hydraulic capacity of
subglacial conduits through their effects on roughness. We
begin with a review of fundamental hydraulic processes and
relate them to hydraulic capacity and roughness in rigid pipes.
This section builds off similar roughness research in rivers and
streams (Wohl and Thompson, 2000; Lane, 2005; Ferguson,
2007, 2010; Morvan et al., 2008) and is particularly relevant
because no papers have specifically addressed how hydraulic
roughness equations that were developed for open channels
or rigid pipes relate to subglacial conduits. We then adapt these
concepts of roughness to subglacial conduits, where large
changes in hydraulic capacity, flow depth and hydraulic
gradients occur in response to conduit enlargement by melting
and closure by creep. Direct observations or measurements in
subglacial conduits that could be used to constrain hydraulic
capacity or roughness have not been published. We address
this gap in knowledge with maps of conduits that were made
using caving techniques (Gulley, 2009) and by interpreting
dye trace data collected from subglacial conduits that were
mapped from input to output (data from Gulley et al., 2012b).
Finally, we discuss the implications of observational and dye
trace data for models of subglacial hydrology and suggest
methods for model improvement.
Hydraulics of Friction within Conduits
Water flow
Energy conservation is related to fluid flow by Bernoulli’s
equation (Figure 1):
pw
γþV2
2g þz¼C(1)
which states that the sum of the pressure head (p
w
/γ; where p
w
is the pressure of water, and γis the specific weight of water),
velocity head (V
2
/2g; where Vis velocity and gis the
acceleration of gravity) and the elevation head (z) remain
constant (C) along a streamline. In an idealized system, where
a fluid is both irrotational and inviscid, the sum of the pressure,
velocity and elevation heads equals the total head in a system,
shown as the Energy Line in Figure 2. As water flows from point
1 to point 2, velocity increases and therefore pressure must
decrease. As a result, the sum of the elevation and the pressure
heads, which is the elevation to which water would rise if a
vertical pipe were inserted into the flow, also decreases. The
difference between this elevation, also known as the Hydraulic
Grade Line, and the Energy Line, is accounted for by the
velocity head component.
While the Bernoulli equation is useful for understanding the
first-order controls on fluid flow, in real fluid systems, energy
is also dissipated due to friction and turbulence, resulting in a
total head that is lower than what is predicted by Equation 1
(Figure 2). This departure is termed head loss. Energy is still
conserved in systems subject to head loss, so that:
pw
γþV2
2g þzþhL¼C(2)
because head losses (h
L
) reflect energy that has been dissipated
as heat. Heat generated by viscous effects (head loss) is
ultimately what supplies the thermal energy used to enlarge
conduits in glacier by melt, a topic that will be revisited later.
Calculating head losses
The two most widely used equations to predict head loss
are the Darcy–Weisbach and Gauckler–Strickler–Manning
(referred to hereafter as simply the Manning equation) equa-
tions. The Manning equation is based on empirically derived
relationships between fluid velocity, head loss and the physical
properties of open channels. The Darcy–Weisbach equation is
phenomenological and can be derived using simplifications
of the Euler equation (Covington et al., 2009) or via dimen-
sional analysis. However, applications of the Darcy–Weisbach
equation require empirically-derived relationships between the
Darcy friction factor (f) and other parameters. While both
equations have been applied to open channel and pipe full
flow, the experiments used to parameterize the Darcy–
Weisbach friction factor were conducted in full pipes and the
experiments used to parameterize the Manning equation were
conducted in open channels (Morvan et al., 2008). These em-
pirically-derived relations cannot be meaningfully extrapolated
beyond the range of data from which they have been derived.
The Darcy–Weisbach equation (Munson, 2005), relates
changes in velocity (v) to hydraulic diameter (D
H
), acceleration
of gravity (g) and uses a dimensionless friction factor (f)to
account for changes the hydraulic gradient (h
L
/L; head loss
per unit length) caused by head loss:
v¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2hLDHg
fL
r(3)
While the goal of the Manning equation is the same as the
Darcy–Weisbach equation, to predict velocity, the Manning
equation uses a different parameterization scheme and lacks
the theoretical underpinning of the Darcy–Weisbach equation.
While the Manning equation is seldom a good choice for
calculation of flow velocity or discharge because hydraulic
roughness values are usually stage dependent and the
equation tends to underestimate flow resistance at high
flows (Ferguson, 2010), the Manning equation has been
297HYDRAULIC ROUGHNESS IN SUBGLACIAL CONDUITS
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
widely used in glaciology studies and it is included in this work
for completeness.
The Manning equation relates velocity to the hydraulic
radius of a pipe or channel (R
h
) surface slope of the water (S
0
;
which can also expressed as h
L
/L) and a Manning roughness
coefficient (n):
v¼R2=3
h
hL
L
1=2
n(4)
While the Manning equation is widely used, conventions for
the units of nare inconsistent, which can make it difficult to
relate nto physical features of a channel using dimensional
analysis. Ideally, resistance terms should be non-dimensional,
as with the Darcy–Weisbach equation’s friction factor. According
to Equation (4), however, nrequires units of s m
-1/3
. Because
dimensions of T L
-1/3
cannot be obtained from the physical
channel parameters that should determine roughness, nis
commonly treated as being non-dimensional, and units of T L
-1/3
are ascribed to a unit conversion factor (which has a value of
1 if SI units are used) that is often not explicitly represented
in the Manning equation. While this approach provides a
workaround for implementing the Manning equation when
values of nare selected from a table of published roughness
coefficients, it creates additional dimensional problems when
nis derived from channel properties, as will be seen in
Equation (10). Despite issues arising from dimensional
inconsistency, the equation persists partly due to tradition
and partly because it provides a computationally simple (if
dimensionally perplexing) method of calculating flow velocity.
Resistance terms in fluid flow
Both the Manning and the Darcy–Weisbach equation use resis-
tance terms nand frespectively, to account for frictional
headloss. Both resistance terms have been related through
other equations to the relative submergence of projections from
a stream bed or pipe walls, termed relative roughness.
Following a short discussion about relative roughness, we will
show how relative roughness is used to calculate nor f.Itis
important to note that while the concept of roughness is
critically important to many fluid modeling schemes, physically
speaking, roughness is merely a model parameterization that is
used to account for momentum and energy dissipation (i.e. the
head loss) that are not explicitly accounted for in simple
models, such as the Darcy–Weisbach or Manning equation
(cf. Lane, 2005).
Relative roughness relates flow depth and cross-sectional
area to the elevation that rocks or other surface roughness
features extend from the walls of a pipe or from the bed of a
stream (defined as the surface roughness height) in order to
quantify the effects of these features on flow velocity. In open
channels, surface roughness height is frequently taken as the
diameter of a specific size fraction of bed material, commonly
the diameter of rocks in the 84th percentile (Gordon, 2004).
Relative roughness is measured in pipes, open channels or full
conduits with non-circular cross-sections using the equation:
Relative roughness ¼ks
DH
(5)
where k
s
is the surface roughness height in an open channel or
pipe and D
H
is the hydraulic diameter of a pipe or conduit or open
channel with non-circular cross-sections (Munson et al., 2005).
g
V
2
2
2
g
V
2
2
3
γ
ρ
2
1
zH =
3
z
2
z
0
11
== pV
0
3
=p
(3)
(2)
(1)
HGL
EL (Total Head)
Figure 1. The Bernoulli equation states that for an inviscid fluid energy and mass are conserved along a flow line, or that the sum of the elevation
head (z), the pressure head (ρ/γ) and velocity head (V
2
/2g) are constant along a flow line. The difference between the head measured at points 2 and 3
(the HGL, Hydraulic Grade Line) and the Total Head in the system (Energy Line), is accounted for by the velocity head. Real fluids depart from this
ideal due to turbulence and viscous effects (Figure 2). Adapted from Munson et al., (2005)
dh
dl
dh
dl = 0
Inviscid
viscous, turbulent
Ideal Energy Line
actual head
A
B
Figure 2. (A) In inviscid fluid flows, all energy is conserved along a
flow line. (B) Real fluids depart from this ideal by dissipating energy
due to viscous effects or turbulence. The difference between the actual
total head (Ideal Energy Line) and the total head in an inviscid flow is
referred to as head loss. Heat dissipated during head loss supplies the
thermal energy needed to melt conduits in glaciers.
298 J. D. GULLEY ET AL.
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
Hydraulic diameter allows flow in open channels or non-
circular cross-sections to be approximated as pipes by relating
the cross-sectional area (A) to the wetted perimeter (P):
DH¼4A
P(6)
In physical terms, the relative roughness equation con-
verts non-circular cross-sections to a circular pipe that
would pass the volume of water per unit time (Figure 3).
Assuming a fixed conduit width, relative roughness de-
creases with flow depth in open channels in a manner that
is directly analogous to how relative roughness would
decrease as melting processes increased the diameter of a
subglacial conduit that remained pipe full. In both
instances, surface roughness heights (the size of rocks on
the conduit floor) are fixed, but the hydraulic diameter is
increasing (Figure 4). While changes in discharge in open
channels are coincident with changes in flow depth, and
hence changes in relative roughness, changes in discharge
in rigid, pipe full conduits do not coincide with a change
in relative roughness because flow depth does not change.
Resistance in Darcy–Weisbach equation
Relative roughness can be used to calculate Darcy–
Weisbach friction factor using the Colebrook–White
equation (Munson, 2005), which also considers viscous
effects that may be present in flow systems. The Colebrook–
White equation relates the dimensionless friction factor (f)tothe
relative roughness (k
s
/D
H
) and to the Reynolds number (Re, a
ratio of inertial to viscous forces) where:
1
ffiffiffi
f
p¼2log ks=DH
3:7þ2:51
Re ffiffiffi
f
p
(7)
with the Reynolds number being calculated according to the
following equation:
Re ¼ρwVDH
μ
(8)
where μis the dynamic viscosity of water.
In general, ftherefore depends on both the Reynolds number
and the relative roughness. However, in the fully rough turbu-
lent flow regime, when either relative roughness values are
sufficiently large or flow velocities (and hence Re) are suffi-
ciently high, surface roughness effects are dominant and the
second term in Equation (7) can be neglected (Munson, 2005).
The value of Re that defines fully rough turbulent flow is inversely
proportional to relative roughness, with fully rough turbulent flow
occurring at a Re of ~6.5 × 10
3
in flow systems with relative
roughness of 5%, compared with an Re of ~10
7
for a relative
roughness of 0.02%. Fully rough turbulent conditions will
frequently hold in subglacial conduit systems, where high
velocity flows occur in conduits with high relative roughness.
As the Colebrook–White equation is valid only for flows where
k
s
/D
H
<0.05 (Özger and Yildirim, 2009), other equations have
been developed for flows where surface roughness heights
approach or exceed flow depths, such as in alpine streams. For
instance, Bathurst (1985) was able to fit relationships between
relative roughness and the Darcy–Weisbach friction factor of
alpine streams with the following equation:
ffiffiffi
1
f
r¼1:987 log ks
5:15Rh
(9)
Bathurst’s equation was empirically derived from field studies
of open channel streams with gradients that were less than 5%.
k @ T1
sk @ T1
s
k @ T2?
s
k @ T1
s
k @ T3?
s
k @ T2?
s
A. B. C.
Figure 3. Illustration of how hydraulic diameter and relative roughness can change with flow depth in subglacial conduits. Wetted perimeter is
shown as a dotted line. The flow cross-sectional area is shaded dark grey. (A) relative roughness is determined primarily by the relative submergence
depth of rocks on the channel floor (k
s
= surface roughness height); (B) irregular channel cross-sections result in more complex relationships between
roughness and flow depth, as some rocks are submerged to greater depths than others; (C) pipe full flow in a subglacial conduit with an irregularcross-
section results in the additional complexity of having to consider the additional effect of glacier ice, which may or may not have a different surface
roughness height than the bed, on conduit roughness.
A. B. C.
Figure 4. Relative roughness changes as a result of conduit enlargement under pipe full conditions. (A) During the early stages of conduit
enlargement, surface roughness features can nearly block conduit flow paths. (B) Increases in conduit diameter decrease relative roughness. (C) Even
large conduits can have large relative roughnesses due to large boulders on conduit floors.
299HYDRAULIC ROUGHNESS IN SUBGLACIAL CONDUITS
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
The equation is subject to uncertainties >30% and its applicabil-
ity to pipe flows or steeper hydraulic gradients has not been
established (Bathurst, 1985).
Resistance in Manning equation
Manning’s roughness coefficient ncan be related to the ratio of
the conduit hydraulic radius R
h
and the surface roughness
height k
s
in meters (typically taken to be D
84
of rocks on a
stream bed) by Morvan et al. (2008):
n¼ks1=6Rh=ks
ðÞ
1=6
18 log 11Rh=ks
ðÞ
!
(10)
Relationships between n,k
s
and R
h
are empirically derived
from experiments in open channels and are only validated for
relative roughness values of 10 <R
h
/k
s
<100. Unfortunately,
this range restricts the utility of Equation (10) to a range of
relative roughness where calculation of nis mostly insensitive
to variations in relative roughness. It is important to note, how-
ever, that the converse is not true, and calculations of relative
roughness from Equation (10) are very sensitive to values of n.
Individual values of Manning’s roughness can be converted
to Darcy–Weisbach friction factors by the relation:
f¼8gn2
R1=3
h
(11)
While Equation (11) is widely used to convert between values
of fand n, the conversion is derived from data used in the
empirical derivation of Equation (10) (Morvan et al., 2008), and
consequently, roughness values derived from this conversion
can be related to surface roughness heights onlyfor the following
range of relative roughness values: 10 <R
h
/k
s
<100 (Morvan
et al., 2008).
Hydraulic capacity
The instantaneous hydraulic capacity of a hydraulic system can
be defined as the maximum volume of a fluid that the hydraulic
system can discharge per unit time for the head gradient
currently available to the system. Hydraulic capacity can be
calculated using either the Darcy–Weisbach or Manning
equation. In steady-state, the hydraulic head gradient has
adjusted such that the rate of recharge is equal to the rate of
discharge and there will be no change in the volume of water
within the system over time. Once more water begins flowing
into a system than flows out of it, water must either be stored
in reservoirs within the hydraulic system or overflow onto the
surface. In many hydraulic systems, such as storm sewers or
caves developed in low-permeability limestones, the only
available storage reservoirs (portions of the hydraulic system
that are not already water filled) are air-filled conduits that exist
at higher elevations than the discharge point. Increases in
storage in these reservoirs result in an overall increase in the
hydraulic head within pressurized portions of the hydraulic
system (Halihan and Wicks, 1998; Covington et al., 2009,
2012). Conversely, if the rate of water delivery decreases below
the hydraulic capacity of the drainage system and more water is
leaving the system than is entering it, hydraulic head will
decrease as water is removed from storage.
In addition to the changes in hydraulic capacity due to
changes in head gradient described above, changes in the
hydraulic capacity of conduits in glacier ice are driven by the
competing effects of enlargement by melt and closure by creep
(Rothlisberger, 1972). Enlargement and creep affect hydraulic
capacity through two related processes. First, enlargement
and creep change the hydraulic diameter of a conduit, thereby
affecting the cross-sectional area available for flow. Second,
and relatedly, as the hydraulic diameter changes, relative
roughness also changes, which ultimately affects values of n
or f. These changes occur because the size of many surface
roughness features in subglacial conduits, such as boulders or
rock projections, are essentially fixed but the hydraulic
diameter changes in response to melt and creep (Figure 4).
Incremental changes in relative roughness (and correspond-
ing changes in nand f) are small when hydraulic diameters
are large, and models are generally insensitive to plausible
changes in relative roughness within this region (Chow,
1959). Large incremental increases in relative roughness occur,
however, when surface roughness heights exceed 5–10% of the
hydraulic diameter (Figure 5). Intuitively, these large increases
in relative roughness would be expected to generate incremen-
tally larger values of fand n, however, empirically derived
relationships between relative roughness are valid only for
relative roughness values that are less than 5%. As a result,
physically meaningful parameterization schemes for roughness
values of conduits that have high relative roughness, such as
subglacial conduits in the early stages of enlargement, are
lacking. Although the problem of roughness parameterization
in shallow streams and rivers has been recognized and is an
area of intense investigation by hydrologists (Wohl, 1998; Wohl
and Thompson, 2000; Lee and Ferguson, 2002; MacFarlane
and Wohl, 2003; Ferguson, 2007), this critical gap in
understanding has not been widely acknowledged within the
glacier hydrological modeling community.
Form roughness
If channels or conduits were perfectly straight and had regular
cross-sectional areas and permanently fixed roughness fea-
tures, roughness would not change with discharge. Few natural
or artificial channels meet these criteria, however, and mobile
roughness features contribute to head losses along flow paths,
causing hydraulic roughness to evolve with changing flow
conditions (Ferguson, 2010). Migratory channel forms, such
as ripples and dunes in sediments (Knight, 1981) and scallops
0 12345
0.0
0.2
0.4
0.6
0.8
1.0
radius (m)
relative roughness
0.35 m
(surface roughness)
0.25 m
0.15 m
0.05 m
ABC D
Figure 5. Plots of relative roughness as a function of the actual radius
of a semicircular conduit with surface roughness heights that vary from
0.05 m to 0.35 m. A horizontal line extends across the graph field at a
relative roughness of 0.05, which is the upper limit of relative rough-
ness for which the Colebrook–White formula remains valid. Values
above this line are too large to calculate the friction factor. Points
A–D on the x axis indicate the radius necessary for relative
roughness to decrease below 0.05 for surface roughness heights
0.05–0.35, respectively.
300 J. D. GULLEY ET AL.
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
and flutes in bedrock (Blumberg and Curl, 1974), are two
mobile roughness features that contribute to evolving hydraulic
roughness. The effect of migratory channel forms on hydraulic
roughness in transient flows can be considerable. For instance,
changes in relative roughness resulting from shifting sediment
dune sequences in tidal channels have resulted in variations
in Manning’snfrom ~0.02 to 0.11 s m
1/3
over a single tidal
cycle (Knight, 1981).
Similar dynamic changes in glacier beds as well as the
overlying ice might be expected to affect subglacial conduit
roughness. First, conduit flow can winnow fine-grained
material from subglacial till, causing boulders to progressively
accumulate on conduit floors (Figure 6). As a result, the
evolution of roughness would depend critically on the relative
size of emergent boulders relative to the increase in hydraulic
diameter. In addition, scallops can form in overlying glacier
ice from turbulent subglacial conduit flow (Figure 7). The size
of scallops are determined by the velocity of water at peak
discharges and create a dynamic roughness feature that can
be overprinted, with the size of scallops, and hence magnitudes
of roughness changing with water velocities in conduits
(Blumberg and Curl, 1974). Scalloping of ice in ice covered
streams results in seasonal increases in roughness coefficients
that are equivalent to roughness coefficients obtained from
flows in equivalent sized wavy boundaries (Ashton, 1978). This
finding indicates that glacier ice, often presumed to be
hydraulically smooth, can contribute significantly to conduit
roughness if the conduit is highly scalloped (cf. Figure 7).
Quantification of the effects of scallop development on
roughness is beyond the scope of this paper and will be
considered in a separate work because feedback between
flow velocity and scallop size (i.e. scallop size is controlled
by velocity, which is controlled by roughness, which affects
velocity), results in complicated relationships.
An additional complicating factor in determining roughness
is that roughness parameters in natural channels are scale-
dependent (Lane, 2005). Channel sinuosity and longitudinal
variability in bedforms both contribute to flow resistance and
the cumulative effects of these features cause roughness to
increase with flow length (Clifford et al., 1992; Lane, 2005).
Roughness, viscous dissipation and subglacial
conduit enlargement
Accurate representations of the factors affecting discharge and
head loss in subglacial conduits (relative roughness and
conduit cross-sectional area) are critically important in
subglacial hydrological models because rates of conduit
enlargement are direct functions of discharge and head loss.
Assuming a semicircular conduit with a radius (R), sediment
or bedrock floor and that all heat generated by viscous dissipa-
tion is locally converted into wall melting (Nye, 1976; Fountain
and Walder, 1998), the rate of conduit enlargement (m)is
directly proportional to conduit discharge (Q) and the rate of
head loss (h
L
/L):
m¼
QρwghL
L
πρihiwR(12)
where ρ
w
is density of water (1000 kg m
-3
), ρ
i
is the density of
ice (917 kg m
-3
), gis the acceleration of gravity (9.8 m s
2
) and
h
iw
is the latent heat of melting (Fountain and Walder, 1998).
A. B. C.
D.
Figure 6. Surface roughness heights can increase due to winnowing fine-grained sediments from conduits floor. (A) Initial surface roughness heights
can be low during the early stages of conduit enlargement due to a high percentage of fine grained sediments. (B) Removal of fine grained sediments
causes large boulders to accumulate on conduit floors. (C) Continued incision increases roughness heights further. (D) Subglacial conduit beneath
Rieperbreen, a cold-based glacier in Svalbard, Norway that enlarged by incision of underlying till and winnowing of fine grained materials from larger
boulders. This figure is available in colour online at wileyonlinelibrary.com/journal/espl
301HYDRAULIC ROUGHNESS IN SUBGLACIAL CONDUITS
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
Equation (12) represents a theoretical maximum conduit
enlargement rate because it assumes that all heat is instanta-
neously and locally transferred to conduit walls and translated
into melt (Clarke, 2003).
To examine the influence that roughness parameterization
can have on calculated melt rates on pipe-full subglacial
conduits, it is useful to consider two limiting cases for
hydrologic boundary conditions. During some time periods,
discharge through a pipe-full subglacial conduit extending
from the base of a moulin will be limited by the rate at
which melt water is delivered to the moulin. This is likely
to occur late in the melt season, when large subglacial
conduits have formed that can easily accommodate the
available melt water, or during low discharge periods
between diurnal melt pulses when water levels are lower
than the elevation of the ice surface at the top of a moulin.
In the second example case, conduit discharges are limited
by the available head gradient. This would be likely to
occur during periods of intense melt early in the season,
when more melt water is available than can be accommo-
dated within the subglacial hydrologic system. Examples of
hydraulic gradient limitation include the early stages of
supraglacial lake drainage by hydrofracturing (Das et al.,
2008, Benn et al., 2009) and where moulins and crevasses
fill and overflow onto the glacier surface.
For the first limit, where available melt water controls
discharge (water level is below the elevation of the ice surface
at the top of a moulin), we can examine how melt rates in the
subglacial conduit leading away from the base of the moulin
vary with assumed friction factor by using the Darcy–Weisbach
equation (Equation (3)) to eliminate head gradient as a variable
in Equation (12). For a given conduit cross-sectional geometry
and a given discharge, melt rates scale such that:
m∝f(13)
Therefore, melt rates in the subglacial conduit extending
from the moulin depend linearly on the friction factor, such that
any error in friction factor results in a proportional error in melt
rate. Alternatively, for the limit where discharges are controlled
by the maximum available gradient (water in moulins is
overflowing on the surface or during a supraglacial lake
drainage event), one can use the Darcy–Weisbach equation (3)
to eliminate discharge as a variable. Then holding conduit
cross-sectional geometry and head gradient fixed, melt rates
scale such that:
m∝f1=2ðÞ (14)
In this case, melt rates are most sensitive to errors in friction
factor for f<1, where small changes in fresult in large changes
in melt rate. These scaling relations are primarily useful in
understanding non-equilibrium conduit systems that are
adjusting to changes in available melt water, demonstrating
how friction factor may influence the rate of evolution via melting
in a subglacial conduit during these adjustment periods.
However, it is specifically these cases that are of most interest
in studies of conduit network evolution. Under an assumption
of equilibrium, the relationships between head gradient,
discharge, and friction factor are further constrained.
A. B. C.
D.
Figure 7. Roughness features can also form in ice as a result of turbulent flow. (A) Initial water flow in small conduits with smooth sediment floors.
(B) Winnowing of fine-grained sediments during conduit enlargement can excavate larger boulders. (C) Turbulent flow patterns around boulders on
conduit floors can generate large scallop features in the ice. (D) Large scale scallops on the walls and ceiling of a subglacial conduit beneath
Hansbreen, a polythermal glacier in Svalbard, Norway. Smaller scale scallops than shown above can also form as a result of turbulent flow in smooth
conduits and can also affect hydraulic roughness. This figure is available in colour online at wileyonlinelibrary.com/journal/espl
302 J. D. GULLEY ET AL.
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
The scaling relations can be qualitatively interpreted by
considering that the melt rate is controlled by the rate of
dissipative energy loss within the conduit, which, in the
context of subglacial hydrology typically restricts application
of Equations (13) and (14) to water-filled portions of the
conduit system. This dissipation rate within the water-filled
portion of the system is primarily a function of the total hydrau-
lic head loss in water filled portions of the conduit (i.e. the
change in elevation between moulin water level and the
glacier snout), and the rate at which that water is flowing
through the system and losing its energy. For cases where dis-
charge is fixed, the second factor is constant, and the energy
loss is determined by head loss along the flow path. Higher f
values will require larger head gradients to drive an equivalent
discharge, so water backs up to higher elevations in moulins
and total head loss along water-filled portions of the conduit
increases with f, as does melt rate (Equation (13)). If head
gradient is fixed, such as when moulins are completely water
filled and excess water overflows on the surface (or at the
beginning of a supraglacial lake drainage event), then the
amount of head loss along water-filled portions of the conduit
is fixed. In this case the rate of dissipative energy loss will be
controlled by the flow rate, and higher fvalues lead to lower
discharge, and therefore lower melt rates (Equation (14)).
Both of these relations show that errors in fcalculated from
relative roughness can result in substantial over or underestima-
tion of the melt rates in a conduit under the same hydrologic
boundary conditions. Consequently, uncertainty in roughness
values which control discharge and head loss in conduits
(Equation (3)), which in turn control melt rate (Equation (12)),
may lead to significant uncertainty in the evolution time scales
of subglacial conduits. Similar issues of sensitivity to roughness
parameterization have been acknowledged in models of river
bed and landscape evolution, where accurate calculations of
stream velocity are necessary to predict sediment transport
(Ferguson, 2010).
Hydraulic roughness in glacier hydrological studies
Nearly all characterizations of subglacial hydrological systems
involve some parameterization of hydraulic roughness, but the
physical processes reflected in these parameterizations are
surprisingly rarely discussed. Nor is there attention to whether
the roughness values being used fall within the calibrated limits
of the empirically-derived data sets used to determine rough-
ness parameters. To a large degree, this oversight is because
roughness parameters used in numerical models have seldom
been attributed to specific physical features, such as conduit
diameters relative to surface roughness heights.
One consequence of not relating roughness in hydrological
models to physical features of hydrological systems has been
the lack of a systematic approach to dealing with roughness
in hydrological models. Table I lists a selection of field and
modeling glacier hydrological studies that used or calculated
roughness parameters, primarily Darcy–Weisbach friction
factors or Manning roughness values. Roughness parameters
used in model studies are generally much lower than roughness
values calculated from field studies. In addition, while several
field studies have indicated that values of nand fcan vary
significantly over seasonal and diurnal timescales, many
hydrological models that simulate conduit enlargement and
creep rely on a fixed roughness parameter (Clarke and
Waldron, 1984). The assumption of fixed roughness parameters
is commonly employed in hydrological models of rivers,
however, field studies indicate roughness parameters vary widely
as a function of river stage (Lane, 2005; Ferguson, 2010).
An additional problem with studies that do allow roughness
parameters to fluctuate with conduit diameter is that variations
Table I. Comparison of select roughness parameters from models and field studies of subglacial conduits
Darcy–Weisbach friction factors
Models Field studies
f source f source
0.01-0.02 Colgan et al., 2011 0.97-75 Gulley et al., 2012a, 2012b
0.05 Fowler 2009
0.25 Spring and Hutter, 1981
0.01-0.5 Covington et al., 2012
0.5 Melvold et al., 2003
0.008-0.6 Boulton et al., 2007
Manning roughness coefficient
Models Field Studies
n(sm
-1/3
)Source n (sm
-1/3
)source
0.08 Björnsson 1991 0.1-6 Willis et al., 2012
0.12 Clarke and Mathews, 1981 0.105-0.23 Clarke 1982
0.09 Nye 1976 0.2 Seaberg et al., 1988
0.09 fowler 2009 0.1-0.74 Nienow et al., 1996
0.05-0.2 Hooke and Pohjola 1994 0.09-0.68 Gulley et al., 2012b
0.105-0.49 Werder and Funk 2009
0.2 Cutler 1998
0.1 Rothlisberger, 1972
0.01-0.1 Boulton et al., 2007
0.033 Werder and Funk, 2009
0.05-0.25 Arnold et al., 1998
0.12-0.20 Clarke 2003
0.032 Pimentel and Flowers 2011
0.02-0.05 Flowers 2008
f–Darcy–Weisbach friction factor (dimensionless); source –citation for roughness parameter value; n –Manning roughness coefficient
303HYDRAULIC ROUGHNESS IN SUBGLACIAL CONDUITS
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
in roughness are not applied in a consistent manner. While
Boulton et al. (2007) used relative roughness to evolve
roughness parameters in concert with conduit hydraulic
diameters in their models, other workers have used different
approaches. Arnold et al. (1998) varied hydraulic roughness
linearly between a high and a low endmember, although
processes that change roughness should not vary linearly
(Figures 3, 6, and 7). Other modelers have selected a Manning
roughness coefficient that was then converted to a Darcy–
Weisbach friction factor, which was then varied in response
to changes in hydraulic diameter (Clarke, 2003; Flowers
et al., 2004; Flowers, 2008; Pimental and Flowers, 2011).
Strategies for dealing with hydraulic roughness in subglacial
conduits are also complicated by the fact that conduit cross-
sections consist of a ‘floor’of rough glacier bed material (i.e.
rocks, sediment) but are covered by a ceiling of glacier ice.
Modelers have generally regarded glacier ice to be hydrauli-
cally smooth (Clarke, 2003; Flowers et al., 2004; Fowler,
2009), although other workers have treated glacier ice as
having similar roughness to conduit floors (Boulton et al.,
2007). Scalloping of ice by turbulent water flow over glacier
ice (Figure 7) generates surface roughness that is inversely
proportional to flow velocity (Blumberg and Curl, 1974),
indicating glacier ice in low flow conduits may have larger
surface roughness features than high flow conduits.
Comparing Field Measurements of Roughness
Calculated from Velocity versus from
Conduit Properties
In the following sections we use dye tracing data from a
671-m-long, single, unbranching subglacial conduit beneath
Rieperbreen, a cold-based glacier in Svalbard, Norway (Gulley
et al., 2012b; Figure 8), to constrain how the roughness
parameters might evolve in response to conduit enlargement.
The ice thickness above the conduit was <30 m, which when
combined with the cold-based nature of the glacier resulted
in negligible creep closure rates, and the conduit was incised
in frozen till (Figure 6(D)), which resulted in negligible conduit
enlargement and precluded the formation of scallops on glacier
ice. Generally unchanging conduit morphologies and lack of
contact of water with glacier ice were confirmed by using
speleological techniques (Gulley, 2009) to map the entire
conduit immediately before and after the melt season.
Comparison of flow depths that were calculated from discharge
data and conduit cross-sectional areas indicated that all traces
occurred under atmospheric conditions (all flow occurred as
open channel; pipe full conditions did not occur). As a result,
all changes in dye trace data and tracer velocities resulted from
changes in the rate of meltwater delivery to the conduit that
increased or decreased water flow depth relative to boulders on
the conduit floor (i.e. relative roughness; Figure 6). Changes in
relative roughness could therefore be calculated directly from
discharge data and conduit cross-sectional areas, which would
not have been possible if the conduit were pipe full and relative
roughness was also changing due to conduit enlargement.
Discharge in the subglacial drainage system varied from
0.04 m
3
s
-1
to 1.07 m
3
s
-1
, corresponding to an increase in
average flow cross-sectional area from 0.57 m
2
to 1.88 m
2
.
We can use data from Gulley et al. (2012b) to predict rough-
ness and water velocity using values of relative roughness and
compare these results with roughness and water velocity
calculated directly from the dye tracing data (referred to as
‘field-derived’). Comparison of the predicted versus field-
derived data allows assessment of the accuracy of roughness
parameterizations that are commonly used in dynamic models
of subglacial conduit enlargement (Spring and Hutter, 1981;
Arnold et al., 1998). Assuming a fixed conduit width, relative
roughness decreases with flow depth in open channels in a
manner that is directly analogous to how relative roughness
would decrease as melting processes increased the diameter
of a subglacial conduit that remained pipe full. In both
instances, surface roughness heights (the size of rocks on the
conduit floor) are fixed, but the hydraulic diameter is increasing.
Predicting n,fand vfrom conduit
physical properties
We use an average conduit width of 5 m to calculate an
average flow depth, wetted perimeter, hydraulic radius and
hydraulic diameter (Equation (6)) from discharge data (Table II).
Taking 15 cm to be a representative surface roughness height
(15 cm was found to correspond to the b-axis of d
84
of rocks
at the location of the photograph in Figure 6) and using the
Figure 8. Subglacial conduit beneath Rieperbreen, a cold based
glacier in Svalbard, Norway (from Gulley et al., 2012b). The conduit
was 671 m in length and ice overlying the conduit was a maximum
of 30 m thick, making creep closure negligible. Note the low, flattish
roofs in conduit cross-sections. Additional information about the
conduit and dye tracing study can be found in Gulley et al. (2012b).
304 J. D. GULLEY ET AL.
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
hydraulic diameter calculated using Equation (6), we can
calculate an average relative roughness for the conduit for each
of the dye trace experiments presented in Table II. We then use
these relative roughness values to calculate a Darcy–Weisbach
friction factor fusing the Colebrook–White (Equation (7)) and
the Bathurst (1985); Equation (9) equations and predict tracer
velocity using the Darcy–Weisbach equation (Equation (3))
and the surveyed conduit slope of 0.04. Similarly, we use
relative roughness data to calculate Manning roughness
coefficient nusing (Equation (10)) and predict velocity using
the Manning equation (Equation (4)). We then compare these
predicted values with velocities measured with dye traces and
values of nand fcalculated from dye tracing velocities and
from solving the Manning and Darcy–Weisbach equations for
their respective roughness parameters.
Roughness
Estimates of Darcy–Weisbach friction factor and Manning rough-
ness predicted on the basis of relative roughness underpredicted
values of fand nthat were field-derived from every dye trace
(Figure 9; Tables III–V). Predicted values showed the greatest
divergence with field-derived values of fand nearly in the melt
season, when relative roughness was high. Later in the melt
season, as increased rates of meltwater delivery increased flow
depths in the channel and decreased relative roughness,
predicted and field-derived values agreed more closely.
The Bathurst (1985) equation (Table IV) generally predicted
field-derived values of fbetter than the Colebrook–White
equation (Table III), although neither equation was particularly
accurate. Field-derived values of fdecreased from a high of 75,
corresponding to relative roughness of 0.34, on 14 June, to a
low of 0.97, corresponding to a relative roughness of 0.17, on
4 August. Excluding a 24 June outlier, relationships between
decreasing values of fand decreasing values of relative
roughness could be fit with a power law (Figure 9(A)). In
contrast, values of fpredicted from the Colebrook–White
equation ranged from a high of 0.23 to a low of 0.11, and values
of fpredicted from Bathurst (1985) equation ranged from a high
of 0.77 to a low of 0.23 over the same time period. Values of fpre-
dicted by the Colebrook–White and Bathurst (1985) equations
were 326 and 97 times greater, respectively, than field-derived
values of fwhen relative roughness was high (0.34). Discrepancies
between field-derived and predicted values decreased at lower
relative roughness values, with field-derived values of fdecreasing
to 6.9 (Colebrook–White) and 1.7 (Bathurst, 1985) times the
predicted value once relative roughness had decreased to 0.17.
Predicting Manning roughness coefficients from relative
roughness data (Equation (10)) was similarly unsuccessful.
Field-derived values of ndecreased from a high of 0.68 s m
-1/3
on 14 June to a low of 0.09 s m
-1/3
on 4 August (Figure 9(B);
Table V). In contrast, values of npredicted from relative rough-
ness data and Equation 9 showed little variability, fluctuating
between 0.04 and 0.03 s m
-1/3
over the same time scale. Differ-
ences between predicted values of nand field-derived n
decreased from a high of 17 times the predicted value on 14 June
(relative roughness 0.34) to2.3 times the predicted value on the 4
August (relative roughness 0.17). Relationships between decreas-
ing values of field-derived nand decreasing values of relative
roughness could be fitted with a power law (Figure 9(B)).
Table II. Conduit parameters calculated from dye tracing data
Date Q (m
3
s
-1
) v (m s
-1
) CSA (m
2
) avg.depth (m) P (m) R
h
(m) D
H
(m) Re (x 10
4
) n (s m
-1/3
)f
6/14/2010 0.04 0.07 0.57 0.11 5.23 0.11 0.44 1.10 0.68 75.01
6/17/2010 0.06 0.09 0.67 0.13 5.27 0.13 0.51 1.64 0.58 52.93
6/24/2010 0.12 0.21 0.57 0.11 5.23 0.11 0.44 3.30 0.23 8.33
6/28/2010 0.37 0.36 1.03 0.21 5.41 0.19 0.76 10.02 0.19 4.95
7/4/2010 0.59 0.50 1.18 0.24 5.47 0.22 0.86 15.92 0.15 2.91
7/23/2010 0.53 0.44 1.20 0.24 5.48 0.22 0.88 14.29 0.17 3.81
7/27/2010 0.94 0.50 1.88 0.38 5.75 0.33 1.31 24.84 0.20 4.41
8/4/2010 1.07 0.88 1.22 0.24 5.49 0.22 0.89 28.84 0.09 0.97
Q–discharge; v –velocity; CSA –conduit cross-sectional area; avg. depth –average depth of water in conduit;
P–wetted perimeter; R
h
–hydraulic radius; D
H
–hydraulic diameter; Re –Reynolds number;
n–Manning roughness coefficient; f –Darcy–Weisbach friction factor. Dye tracing data from Gulley et al. (2012b).
k /DHs k /DHs
0.1 0.2 0.3 0.4
100
80
60
40
20
0
f
0.0
0.8
0.6
0.4
0.2
0
n (s m )
-1/3
A. B.
0.1 0.2 0.3 0.4
0.0
f = 4319 3.75
r = 0.97
n = 6.36 2.06
K /DH
s
()
K /DH
s
()
r = 0.91
2
K /DHs = 5%
K /DH
s= 5%
factual
CW
Bathurst
f
f
n
nactual
calc
Darcy-Weisbach Manning
Figure 9. Comparison of roughness parameters calculated on the basis of surface roughness heights with roughness parameters calculated from dye
tracing data. (A) Darcy–Weisbach friction factor (f). Values of fcalculated from dye tracing data are plotted as f
actual
. Plots of fderived using the
Colebrook–White equation (f
CW
) and Bathurst (1985; f
Bathurst
) plot directly on top of each other at this scale. The Colebrook–White equation is valid
only for relative roughness values that are less than 0.05, shown as a dashed line at left. (B) Manning roughness coefficient (n). Values of n
actual
were
derived from dye tracing data (n
actual
) and values of n
calc
were derived from Equation (10). Note: k
s
–surface roughness height (0.15 m); D
H
–hydrau-
lic diameter; R
h
–hydraulic radius.
305HYDRAULIC ROUGHNESS IN SUBGLACIAL CONDUITS
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
Velocity
As expected, underestimates of roughness parameters affected
predictions of tracer velocities. Predicted velocities were much
higher than field-derived velocities, and the difference between
predicted and field-derived values decreased with decreasing
relative roughness. The Bathurst (1985) equation came closest
to predicting tracer velocities, and velocities predicted using
the Manning equation and Colebrook–White equations
showed increasing divergence from field-derived values.
Using values of fpredicted in the previous section from the
Bathurst equation (1985; Table IV), velocities calculated from
the Darcy–Weisbach equation were overpredicted by a factor
of 9.9 on 14 June when relative roughness was 0.34 but this
overprediction decreased to a factor of 1.7 by 4 August, when
relative roughness had decreased to 0.17 (Figure 10). Similarly,
using values of fpredicted by the Colebrook–White equation
(Table III), velocities calculated from the Darcy–Weisbach
equation were overpredicted by a factor of 18 on 14 June
(relative roughness 0.34) but this overprediction decreased to
a factor of 2.6 by 4 August (relative roughness 0.17). Velocity
overpredictions by the Manning equation ranged from a factor
of 16 to a factor of 2.4 over the same time period.
Discussion
Similar to work in rivers (Wohl and Thompson, 2000; Lane,
2005; Ferguson, 2007), our results suggest that even if conduit
hydraulic gradient, hydraulic diameter and surface roughness
heights are known to good approximations, values of for n
calculated solely from surface roughness heights are likely to
be lower than actual values, resulting in modeled velocities
that are faster than true velocities. Also similar to rivers
Table III. Roughness and velocity data calculated from Colebrook–White equation
Date D
H
(m) k
s
/Dh f
CW
f
actual
f
actual
/f
CW
v
CW
(m s
-1
)v
actual
(m s
-1
)v
CW
/v
actual
6/14/2010 0.44 0.34 0.23 75.0 326.1 1.25 0.07 17.9
6/17/2010 0.51 0.29 0.21 52.9 252.0 1.44 0.09 16
6/24/2010 0.44 0.34 0.23 8.3 36.2 1.25 0.21 6.0
6/28/2010 0.76 0.20 0.15 5.0 33.0 2.04 0.36 5.7
7/4/2010 0.86 0.17 0.14 2.9 20.8 2.26 0.50 4.5
7/23/2010 0.88 0.17 0.14 3.8 27.2 2.29 0.44 5.2
7/27/2010 1.31 0.11 0.11 4.4 40.1 3.17 0.50 6.3
8/4/2010 0.89 0.17 0.14 1.0 6.9 2.32 0.88 2.6
D
H
–hydraulic diameter; k
s
–surface roughness height (taken to be D
84
or 0.15 m); f
CW
–Darcy–Weisbach friction factor calculated from k
s
/D
H
using
the Colebrook–White equation; f
actual
–f calculated from dye trace data (Table I); v
CW
–velocity calculated using the Darcy–Weisbach equation, f
CW
and a slope of 0.043; v
actual
–velocity calculated from dye trace data.
Table IV. Roughness and velocity data calculated from Bathurst (1985)
Date R
h
(m) k
s
/R
h
f
Bathurst
f
actual
f
actual
/f
Bathurst
V
Bathurst
(m s
-1
)v
actual
(m s
-1
)v
Bathurst
/v
actual
6/14/2010 0.11 1.38 0.77 75.01 97.4 0.69 0.07 9.9
6/17/2010 0.13 1.18 0.62 52.93 85.4 0.83 0.09 9.2
6/24/2010 0.11 1.38 0.77 8.33 10.8 0.69 0.21 3.3
6/28/2010 0.19 0.79 0.38 4.95 13.0 1.30 0.36 3.6
7/4/2010 0.22 0.70 0.34 2.91 8.6 1.47 0.50 2.9
7/23/2010 0.22 0.69 0.33 3.81 11.6 1.50 0.44 3.4
7/27/2010 0.33 0.46 0.23 4.41 19.2 2.19 0.50 4.4
8/4/2010 0.22 0.67 0.33 0.97 2.9 1.52 0.88 1.7
D
H
–hydraulic diameter; k
s
–surface roughness height (taken to be D
84
or 0.15 m); f
CW
–Darcy–Weisbach friction factor calculated from k
s
/D
H
using
the Bathurst (1985) equation; f
actual
–f calculated from dye trace data (Table I); v
Bathurst
–velocity calculated using the Bathurst (1985) equation, f
Bathurst
and a slope of 0.043; v
actual
–velocity calculated from dye trace data.
Table V. Roughness and velocity data calculated with the Manning equation
Date R
h
(m) R
h
/k
s
n
calc.
(s m
-1/3
)n
actual
(s m
-1/3
)n
actual
/n
calc.
v
calc
(m s
-1
)v
actual
(m s
-1
)n
Calc
/n
actual
6/14/2010 0.11 0.73 0.04 0.68 17 1.11 0.07 15.9
6/17/2010 0.13 0.85 0.04 0.58 14.5 1.29 0.09 14.3
6/24/2010 0.11 0.73 0.04 0.23 5.8 1.11 0.21 5.3
6/28/2010 0.19 1.27 0.04 0.19 4.8 1.86 0.36 5.2
7/4/2010 0.22 1.44 0.04 0.15 3.8 2.08 0.50 4.2
7/23/2010 0.22 1.46 0.04 0.17 4.3 2.11 0.44 4.8
7/27/2010 0.33 2.18 0.03 0.20 6.7 2.94 0.50 5.9
8/4/2010 0.22 1.48 0.04 0.09 2.3 2.13 0.88 2.4
R
h
–hydraulic radius; k
s
–surface roughness height (taken to be D
84
or 0.15 m); n
calc
is the value of n calculated from k
s
and R
h
using Equation (9);
n
actual
is the value of n calculated from dye tracing data; v
calc
is the velocity calculated using n
calc
and a slope of 0.043.
306 J. D. GULLEY ET AL.
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
(Ferguson, 2010) these discrepancies will be largest when
relative roughness is high, such as in the early stages of conduit
enlargement, and decrease as relative roughness decreases, such
as during enlargement of a pipe-full subglacial conduit (Figures 4,
9). In spite of these limitations, we found that predicted values of f
and napproached field-derived values as relative roughness
decreased below 0.17. These findings suggest that once conduit
relative roughness is <0.10, calculating nor ffrom relative
roughness data may become reasonably accurate.
Differences between field-derived and predicted roughness
parameters likely stem from two main sources. First, equations
used to calculate roughness are empirically derived and our
entire dataset falls outside the calibration range. In order for
the Colebrook–White equation to be valid (relative roughness
<5%), open channel flow depth in the conduit beneath
Rieperbreen would have had to have been ~1 m. Likewise,
the equation relating surface roughness height and Manning
roughness coefficient is valid only where 10 <R
h
/k
s
<100,
requiring an open channel flow depth of ~3.8 m. The Bathurst
(1985) equation was parameterized using data from streams
where surface roughness heights exceeded the hydraulic
diameter, however, there was considerable uncertainty in
relationships between R
h
,k
s
and f. Despite these limitations,
the Bathurst (1985) equation reasonably predicted velocities
by the end of the melt season (Figure 10). The second reason
that roughness parameters are underpredicted is that the scale
dependence of roughness is not taken into account when
calculating for nsolely from relative roughness (Clifford
et al., 1992; Lane, 2005). Channel sinuosity, variability in the
length of step-pool segments and changes in bed geometry all
contribute to increased roughness values and are not captured
by relative roughness (Lane, 2005).
Errors in roughness parameter calculation also affected
velocity predictions. Velocities were overpredicted by a
minimum of a factor of 10 at high relative roughness but were
roughly within an order of magnitude of field-derived velocities
at lower relative roughness. Velocities predicted using values of
fcalculated from the Colebrook–White equation and values of
nfrom the Manning equation were within a factor of 2.6 and
1.7, respectively, of measured velocities once relative
roughness decreased to ~0.17. Velocities predicted using
values of ffrom the Bathurst (1985) equation were within a
factor of 1.7 of field-derived velocities, suggesting the equation
may do a reasonable job of predicting velocities at moderately
high relative roughness values (<0.15). The applicability of the
Bathurst (1985) equation to pipe flow conditions, however,
remains to be validated.
Roughness effects on conduit enlargement rates
In the following section, we use Equation (12) to conduct a
simple modeling exercise to quantify the degree to which dif-
ferent parameterizations of roughness affect modeled glacier
conduit enlargement rates. Specifically, we explore the effects
of high roughness that occur in early conduit development.
For the sake of comparison with existing models, we constrain
our simulations to circular conduits. In four separate simula-
tions, we model roughness evolution differently as conduit
diameters increase. We parameterize one simulation by vary-
ing Manning roughness linearly between 0.25 s m
-1/3
and
0.05 s m
-1/3
during conduit enlargement (Arnold et al., 1998).
In a second simulation, we parameterize roughness by
calculating the Darcy–Weisbach friction factor using the
Colebrook–White equation (Equation (7)), a fixed surface
roughness height of 15 cm and an evolving hydraulic diameter.
For the third simulation, we hold fconstant at 0.08. In a fourth
simulation, we parameterize roughness using the empirically
derived relationship between fand relative roughness
identified from Gulley et al.’s (2012b) dye tracing study
(Figure 9(A)). Because we were only interested in comparing
how different roughness parameterizations affect conduit
enlargement rates, we assume a constant head boundary
condition, with the hydraulic gradient fixed at 0.01 and do
not consider creep closure. Creep closure would vary between
simulations in addition to variations in roughness, confounding
our ability to independently assess the effects of roughness
parameterization on enlargement rates. Conduit growth was
simulated from 0.44 m, the smallest hydraulic diameter
obtained from our dye tracing studies, through a hydraulic
diameter of 3 m, corresponding to a relative roughness of 5%,
or the highest relative roughness value for which the
Colebrook–White equation has been validated.
Conduit hydraulic diameters were integrated numerically in
MATLAB using a variable time step set equal to the time
required to produce 0.001 m of wall melt. During each time
step, the corresponding roughness parameter was determined
using one of our four roughness parameterization schemes
and used to calculate discharge.
Timescales of conduit enlargement were strongly affected by
roughness parameterization (Figure 11(A)). Parameterizing
roughness with a fixed value of fresulted in the shortest conduit
enlargement timescale. Conduit diameters reached 3 m in
~0.9 days (Figure 11(A)). The Colebrook–White formula
resulted in the second fastest conduit enlargement rates. The
conduit diameter reached 3 m in slightly more than 1 day.
When Manning roughness was linearly varied between 0.25
and 0.05 s m
-1/3
, conduits required ~2.5 days to reach 3 m in
diameter. Using the empirically derived relationship between
relative roughness and f(from Gulley et al., 2012b) required
9.25 days for conduits to reach 3 m in diameter. In all cases,
rapid conduit enlargement occurs due to nearly exponentially
increasing discharge (Figure 11B) and the model assumption
that all head loss is instantaneously translated into melt. In
actual glaciers, growth rates are unlikely to be as fast as
simulated by our simple model because there is not an infinite
source of water available to maintain fixed head gradients and
CW
Bathurst
Manning
f
n
f
calculated v / actual v
20
16
12
8
4
0
0.0 0.1 0.2 0.3 0.4
k /DHs
K /DH
s= 5%
1:1 line
Figure 10. Comparison of the flow velocity calculated using rough-
ness parameters (for n) based on surface roughness heights with the
field-derived velocity from dye tracing data. Note: Calculated / actual
velocity –is the ratio of the flow velocity calculated using roughness
parameters derived from surface roughness heights and the field-
derived velocity from dye tracing data (Table II); k
s
/D
H
–relative
roughness (k
s
- surface roughness height/D
H
–hydraulic diameter); f
CW
–
velocity calculated using fderived from k
s
/D
H
and the Colebrook–
White equation.
307HYDRAULIC ROUGHNESS IN SUBGLACIAL CONDUITS
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
heat must be transferred from the location of creation within the
water column to the conduit walls (Clarke, 2003).
Our results suggest that high hydraulic roughness during
early conduit enlargement should result in conduits with
lower hydraulic capacity than previously anticipated,
leading to lower discharges, slower conduit enlargement
rates and higher subglacial water pressures. This low
hydraulic capacity could allow subglacial conduits to mimic
many aspects of linked-cavity drainage systems (Kamb,
1987). Water flowing through conduits with low hydraulic
capacity will have long residence times. Dye traces
conducted in these conduits should have slower velocities
and higher dispersivities than in conduits that have higher
hydraulic capacity. The Gulley et al. (2012b) dye tracing
study at Rieperbreen showed that dye trace breakthrough
curves in conduits can have patterns of velocity and
dispersivity that are very similar to patterns that have been
interpreted to indicate flow in linked cavities. Additionally,
high relative roughness can cause flow in conduits to be
inefficient, generating high subglacial water pressure in
conduits when meltwater delivery by moulins exceeds
hydraulic capacity (Gulley et al., 2012a).
Ourdataraisethepossibility that changes in the hydrau-
lic capacity of subglacial conduits could account for many
of the changes in subglacial water pressure that are
observed at glacier beds; the assumption of linked cavity
systems may be unnecessary. This finding could explain
why ‘linked-cavity behavior’, such as glacier surging
(Harrison and Post, 2003) or high subglacial water pressures
(Humphrey and Raymond, 1994), has been reported from
glaciers that have soft sediment beds and are therefore
unlikely to have cavities due to the absence of stepped
bedrock topography.
Needs for, and challenges to,
understanding roughness
Our findings indicate that more research on the magnitudes
and evolution of hydraulic roughness in subglacial conduits is
necessary in order to improve understanding of conduit
hydraulic processes, as well as simulate more accurately
conduit hydrology. Direct application of the roughness
parameterization from our study to other studies is limited
because the size of boulders on the floors of conduits beneath
other glaciers is unlikely to be similar to the conduit beneath
Rieperbreen and because the roughness evolution of pipe-full
subglacial conduits will also be affected by feedbacks between
flow velocity and scallop size. Nonetheless, roughness values
obtained from our study at Rieperbreen are useful for introduc-
ing a research problem that has largely gone unrecognized and
highlights the importance of filling this knowledge gap.
Realistic values of surface roughness heights in subglacial
conduits remain almost entirely unconstrained by direct
observations. In addition to the need for more surface
roughness height information, roughness evolution due to con-
duit enlargement under high relative roughness scenarios
needs to be quantified. Some work in this area has already been
conducted in rivers and streams (Bathurst, 1985; Ferguson,
2007) but the applicability of the work to pipe-full conduits
has yet to be demonstrated. Understanding and quantifying
feedbacks between decreased relative roughness, increased
discharge and conduit enlargement needs to be addressed
before models of glacier hydrological systems can begin to
model conduit systems with relative roughness >5%.
Despite the need for better means of quantifying roughness
in glacier hydrological systems, studies in other conduit-
dominated flow systems suggest that finding universal relation-
ships between roughness and conduit parameters will be
difficult. Values of fhave been investigated in a large number
of conduits in karst aquifers, where research frequently has
the benefit of direct access to the hydrological systems during
low flow conditions. Values of fcalculated from these individ-
ual studies have been found to range from 0.039 to 340,
demonstrating considerable variability between different
conduits in different limestone karst aquifers (Ford and
Williams, 2007). Because conduit hydraulic diameters are
typically unknown in glaciers, and D
H
and fwill co-vary, the
range of values of fhave been poorly constrained in glacier
hydrological studies. Due to morphological similarities
between conduits in glaciers and karst aquifers, however, the
range of values of fis likely to be similar in the two systems.
Indeed, we found that values of fat Rieperbreen varied
between 0.97 and 75. Larger values would be expected in
subglacial conduits that had similar cross-sectional areas but
were pipe full. This increase occurs because the wetted
perimeter of a pipe full conduit is larger, resulting in a higher
relative roughness, than in an open channel of equivalent flow
cross-sectional area.
Summary and Conclusions
The hydraulic roughness of subglacial conduits can be very
high during early conduit enlargement because the distance
boulders project into water flow is large relative to conduit
hydraulic diameters. High relative roughness results in conduits
that have low hydraulic capacity and may allow conduits to
mimic processes that are normally attributed to linked cavities.
Roughness decreases during conduit enlargement, and
relationships between relative roughness and values of fand
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
Days
0 1 2 3 4 5 6 7 8 9 10
0
5
10
15
20
25
30
35
Days
A.
B.
Colebrook−White
Empirically Derived
Manning roughness
Constant at 0.08
DH (m)
Q (m3 s-1)
Figure 11. (A) Evolution of the diameter of circular, ice walled
conduits calculated using a simple melt model using four different
roughness parameterization schemes. (B) Discharge history for the four
conduit enlargement models shown in panel A.
308 J. D. GULLEY ET AL.
Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)
nwere well-represented by power laws in our study. Existing
equations that relate changes in relative roughness to rough-
ness parameters fand n, such as the Colebrook–White formula,
are inadequate for describing glacier hydrological systems
during the early conduit enlargement because these equations
are valid only for relative roughness <5%. Because high initial
values of fand nthat occur early in melt seasons delay conduits
from enlarging to a relative roughness of 5% by 7–10-fold,
existing models of conduit hydrology are likely to overpredict
rates of conduit enlargement until new roughness parameteri-
zation schemes are developed. These findings have important
implications for models of glacier hydrology, because many
of the important links between glacier motion and glacier
hydrology occur early in the melt seasons, when conduit
diameters are likely to be small and relative roughness is>>5%.
Acknowledgements—J. Gulley acknowledges funding from the
American Philosophical Society, Lewis and Clark Fund for Exploration
and Field Research, University of Florida and NSF in the form of a
Graduate Research Fellowship and an EAR Postdoctoral Fellowship
(#0946767). Gulley and Benn also acknowledge funding and logistical
support from the University Centre in Svalbard. We acknowledge
helpful comments from two anonymous reviewers. Thanks to A.
Banwell, A. Bergstrom, Z. Luthi, M. Temminghoff, P. Walthard and I.
Willis for field assistance.
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