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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.

References

Anderson RS, Anderson SP, MacGregor KR, Waddington ED, O’Neel S,

Riihimaki CA, Loso MG. 2004. Strong feedbacks between hydrology

and sliding of a small alpine glacier. Journal of Geophysical Research

109: 17 PP. DOI: 200410.1029/2004JF000120

Arnold N, Richards K, Willis I, Sharp M. 1998. Initial results from a

distributed, physically based model of glacier hydrology. Hydrological

Processes 12:191–219.

Ashton GD. 1978. River ice. Annual Review of Fluid Mechanics 10:

369–392.

Bartholomaus TC, Anderson RS, Anderson SP. 2008. Response of

glacier basal motion to transient water storage. Nature Geoscience

1:33–37.

Bartholomaus TC, Anderson RS, Anderson SP. 2011. Growth and

collapse of the distributed subglacial hydrologic system of Kennicott

Glacier, Alaska, USA, and its effects on basal motion. Journal of

Glaciology 57: 985–1002.

Bathurst JC. 1985. Flow resistance estimation in mountain rivers.

Journal of Hydraulic Engineering 111: 625–643.

Benn D, Gulley JD, Luckman A, Adamek A, Glowacki P. 2009.

Englacial drainage systems formed by hydrologically driven crevasse

propagation. Journal of Glaciology 55: 513–523.

Björnsson H. 1991. Jökulhlaups in Iceland: prediction, characteristics

and simulation. Annals of Glaciology 16:95–106.

Blumberg PN, Curl RL. 1974. Experimental and theoretical studies of

dissolution roughness. Journal of Fluid Mechanics 65: 735–751.

Boulton GS, Lunn R, Vidstrand P, Zatsepin S. 2007. Subglacial drainage

by groundwater–channel coupling, and the origin of esker systems:

part II—theory and simulation of a modern system. Quaternary

Science Reviews 26: 1091–1105.

Chow V. 1959. Open-channel Hydraulics. McGraw-Hill: New York.

Clarke GKC. 1982. Glacier outburst floods from “Hazard Lake”, Yukon

Territory, and the problem of flood magnitude prediction. Journal of

Glaciology 28:3–21.

Clarke GKC. 2003. Hydraulics of subglacial outburst floods: new

insights from the Spring-Hutter formulation. Journal of Glaciology

49: 299–313.

Clarke GKC, Mathews WH. 1981. Estimates of the magnitude of glacier

outburst floods from Lake Donjek, Yukon Territory, Canada.

Canadian Journal of Earth Sciences 18: 1452–1463.

Clarke GKC, Waldron DA. 1984. Simulation of the August 1979 sudden

discharge of glacier-dammed Flood Lake, British Columbia.

Canadian Journal of Earth Sciences 21: 502–504.

CliffordNJ,RobertA,RichardsKS.1992.Estimationofflow

resistance in gravel-bedded rivers: a physical explanation of the

multiplier of roughness length. Earth Surface Processes and

Landforms 17:111–126.

Colgan W, Rajaram H, Anderson R, Steffen K, Phillips T, Joughin I,

Zwally HJ, Abdalati W. 2011. The annual glaciohydrology cycle in

the ablation zone of the Greenland ice sheet: Part 1. Hydrology

model. Journal of Glaciology 57: 697–709.

Covington MD, Wicks CM, Saar MO. 2009. A dimensionless number

describing the effects of recharge and geometry on discharge from

simple karstic aquifers. Water Resources Research 45: W11410.

Covington M, Banwell A, Gulley J, Saar M, Wicks C, Willis I, Arnold N.

2012. Quantifying the effects of recharge and system geometry on

proglacial hydrograph form. Journal of Hydrology 414–415:59–71.

Cutler PM. 1998. Modeling the evolution of subglacial tunnels due to

varying water input. Journal of Glaciology 44: 485–497.

Das SB, Joughin I, Behn MD, Howat IM, King MA, Lizarralde D, Bhatia

MP. 2008. Fracture propagation to the base of the Greenland Ice

Sheet during supraglacial lake drainage. Science 320: 778–781.

Ferguson R. 2007. Flow resistance equations for gravel-and boulder-

bed streams. Water Resources Research 43: W05427.

Ferguson R. 2010. Time to abandon the Manning equation? Earth

Surface Processes and Landforms 35: 1873–1876.

Flowers GE. 2008. Subglacial modulation of the hydrograph from

glacierized basins. Hydrological Processes 22: 3903–3918. DOI:

10.1002/hyp.7095

Flowers GE, Björnsson H, Pálsson F, Clarke GKC. 2004. A coupled

sheet-conduit mechanism for jökulhlaup propagation. Geophysical

Research Letters 31: 05401. DOI: DOI: 10.1029/2003GL019088

Ford DC, Williams P. 2007. Karst Hydrogeology and Geomorphology,

Revised. Wiley: Chichester.

Fountain AG, Walder JS. 1998. Water flow through temperate glaciers.

Reviews of Geophysics 36: 299–328.

Fowler AC. 2009. Dynamics of subglacial floods. Proceedings of the

royal society a: mathematical. Physical and Engineering Science

465: 1809–1828.

Gordon ND. 2004. Stream Hydrology: an Introduction for Ecologists.

John Wiley and Sons.

Gulley J. 2009. Structural control of englacial conduits in the temperate

Matanuska Glacier, Alaska, USA. Journal of Glaciology 55: 681–690.

Gulley J, Grabiec M, Martin J, Jania J, Catania G, Glowacki P. 2012a.

The effect of discrete recharge by moulins and heterogeneity in flow

path efficiency at glacier beds on subglacial hydrology. Journal of

Glaciology 58: 926–940.

Gulley J, Walthard P, Martin J, Banwell A, Benn D, Catania G. 2012b.

Conduit roughness and dye-trace breakthrough curves: why slow

velocity and high dispersivity may not reflect flow in distributed

systems. Journal of Glaciology 58: 915–925.

Halihan T, Wicks CM. 1998. Modeling of storm responses in conduit

flow aquifers with reservoirs. Journal of Hydrology 208:82–91.

Harrison WD, Post AS. 2003. How much do we really know about

glacier surging? Annals of Glaciology 36:1–6.

Hooke R, Pohjola V. 1994. Hydrology of a segment of a glacier situated

in an overdeepening, Storglaciaren, Sweden. Journal of Glaciology

40: 140–148.

Hubbard BP, Sharp MJ, Willis IC, Nielsen MK, Smart CC. 1995.

Borehole water-level variations and the structure of the subglacial

hydrological system of Haut Glacier d’Arolla, Valais, Switzerland.

Journal of Glaciology 41: 572–583.

Humphrey NF, Raymond CF. 1994. Hydrology, erosion and sediment

production in a surging glacier: Variegated glacier, Alaska, 1982–83.

Journal of Glaciology 40:539–552.

Kamb B 1987. Glacier Surge Mechanism Based on Linked Cavity

Configuration of the Basal Water Conduit System. Journal of

Geophysical Research 92: 9083–9100.

Kessler MA, Anderson RS. 2004. Testing a numerical glacial hydrolog-

ical model using spring speed-up events and outburst floods. Geo-

physical Research Letters 31: L18503.

Knight DW. 1981. Some field measurements concerned with the be-

haviour of resistance coefficients in a tidal channel. Estuarine,

Coastal and Shelf Science 12: 303–322.

Lane SN. 2005. Roughness–time for a re-evaluation? Earth Surface Pro-

cesses and Landforms 30: 251–253.

309HYDRAULIC ROUGHNESS IN SUBGLACIAL CONDUITS

Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)

Lee AJ, Ferguson RI. 2002. Velocity and flow resistance in step-pool

streams. Geomorphology 46:59–71.

MacFarlane WA, Wohl E. 2003. Influence of step composition on step

geometry and flow resistance in step-pool streams of the Washington

Cascades. Water Resources Research 39: 1037.

Melvold K, Schuler T, Lappegard G. 2003. Ground-water intrusions in a

mine beneath Hoganesbreen, Svalbard: assessing the possibility of

evacuating water subglacially. Annals of Glaciology 37: 269–274.

Morvan H, Knight D, Wright N, Tang X, Crossley A. 2008. The concept of

roughness in fluvial hydraulics and its formulation in 1D, 2D and 3D nu-

merical simulation models. Journal of Hydraulic Research 46: 191–208.

Munson BR, Young DF, Okiishi TH. 2005. Fundamentals of Fluid Me-

chanics. 5th edn. John Wiley & Sons.

Nienow PW, Sharp M, Willis IC. 1996. Velocity-discharge relationships

derived from dye tracer experiments in glacial meltwaters.

Hydrological Processes 10: 1411–1426.

Nye JF. 1976. Water flow in glaciers: jökulhlaups, tunnels and veins.

Journal of Glaciology 17: 181–207.

Özger M, Yildirim G. 2009. Determining turbulent flow friction

coefficient using adaptive neuro-fuzzy computing technique.

Advances in Engineering Software 40: 281–287.

Pimentel S, Flowers GE. 2011. A numerical study of hydrologically

driven glacier dynamics and subglacial flooding. Proceedings of the

Royal Society A: Mathematical, Physical and Engineering Science

467: 537–558.

Rothlisberger H 1972. Water pressure in intra and subglacial channels.

Journal of Glaciology 11: 177–203.

Schoof C 2010. Ice-sheet acceleration driven by melt supply variability.

Nature 468 : 803–806.

Seaberg S, Seaberg J, Hooke R, Wiberg D. 1988. Character of the

englacial and subglacial drainage system in the lower part of the

ablation area of Storglaciaren, Sweden, as revealed by dye-trace

studies. Journal of Glaciology 34: 217–227.

Shreve RL. 1972. Movement of water in glaciers. Journal of Glaciology

11: 205–214.

Spring U, Hutter K. 1981. Numerical studies of jökulhlaups. Cold

Regions Science and Technology 4: 227–244.

Walder JS, Trabant DC, Cunico M, Fountain AG, Anderson SP,

Anderson RS, Malm A. 2006. Local response of a glacier to annual

filling and drainage of an ice-marginal lake. Journal of Glaciology

52: 440–450.

Werder MA, Funk M. 2009. Dye tracing a jökulhlaup: II Testing a

jökulhlaup model against flow speeds inferred from measurements.

Journal of Glaciology 55: 899–908.

Willis IC, Fitzsimmons CD, Melvold K, Andreassen LM, Giesen RH.

2012. Structure, morphology and water flux of a subglacial

drainage system, Midtdalsbreen, Norway. Hydrological Processes

26:3810–3829.

Wohl EE. 1998. Uncertainty in flood estimates associated with

roughness coefficient. Journal of Hydraulic Engineering 124:

219–223.

Wohl EE, Thompson DM. 2000. Velocity characteristics along a

small step–pool channel. Earth Surface Processes and Landforms

25: 353–367.

310 J. D. GULLEY ET AL.

Copyright © 2013 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 39, 296–310 (2014)