Introduction
Textbook descriptions of glacial erosion detail mechanisms of abrasion and
quarrying, but mention erosion by subglacial meltwater as a potential,
unquantified, additional incision mechanism
e.g.. This imbalance
reflects the deficiency in our understanding of the latter. In fact,
subglacial meltwater loaded with sediment has been inferred to carve
metre-scale channels in bedrock e.g., often
called Nye channels (N-channels; ,
) and kilometre-scale tunnel valleys
e.g.,
and more recently its role has been invoked as a necessary mechanism in the
carving and deepening of inner gorges
.
Although the ability of subglacial water flow to flush subglacial sediment is
well established
e.g.,
there has been little work quantifying subglacial sediment transport
, and no work on bedrock erosion by subglacial meltwater.
Most studies of glacial erosion that use measurements of proglacial sediment
yield rely on the hypothesis that subglacial meltwater flow is the most
important process removing sediment from the glacier bed
e.g..
In numerical models of glacial erosion, sediment transport by subglacial
water flow is usually neglected
e.g.,
often under the assumption that sediment is removed instantaneously. Bedrock
erosion resulting from subglacial water is likewise neglected. A better
understanding of the processes that lead to the formation of tunnel valleys
or inner gorges is also important for the evaluation of deep geological
repositories for nuclear waste in regions facing a potential future
glaciation e.g..
Subglacial water flows through two main types of drainage systems:
distributed and channelized. In numerical models (see
, for a review), a distributed drainage system is
typically represented by a network of connected cavities
e.g.,
a macroporous sheet of sediment
e.g., or a water film
e.g.. These
representations reflect field observations of an increase in water pressure
with discharge e.g.. A
channelized drainage system is most often described by a single
Röthlisberger channel or a network thereof
e.g..
Water velocities are relatively high in the channelized system and, under
steady-state conditions, water pressure decreases with increasing discharge
. Conduits carved both in sediment and ice,
so-called canals
e.g., have
properties closer to those of a distributed system as pressure often
increases with discharge . In winter, the
distributed drainage system evacuates most basal water, and water pressures
tend to be relatively high. As surface melt becomes significant, the water
input becomes too large for the distributed drainage system alone, water
pressure increases, and R-channels start to form. Once an efficient drainage
system is established, meltwater is routed relatively quickly downstream and
baseline water pressures are generally lower than in winter, with large daily
fluctuations. As surface melt decreases, channels are reduced in size by ice
creep and may eventually close. We hypothesize that this cycle may have a
significant effect on bedrock erosion by subglacial meltwater flow
e.g..
Three main processes produce bedrock erosion in rivers: abrasion,
macro-abrasion, and quarrying
e.g.. Abrasion is the result of
particles entrained by the flow (saltating or in suspension) colliding with
the bedrock and is governed by the tools and cover effect, whereby particles
(i.e. tools) entrained by the flow impact exposed bedrock but can also
shield it if they are immobile i.e. cover;
e.g..
Macro-abrasion and quarrying both result from dislodgement of blocks and
require a relatively high joint density in the bedrock. Macro-abrasion occurs
when blocks are dislodged as a result of the impact of moving particles,
while quarrying is the result of dislodgement by pressure gradients caused by
water flow . Over highly jointed
bedrock, quarrying and macro-abrasion can produce large canyons under extreme
flow conditions e.g.. In
this study, we limit our analysis to abrasion
and use
well-established models to estimate the erosion due to total sediment load
and saltating load only .
suggest that the sediment transport capacity of an
R-channel is most affected by changes in water discharge and hydraulic
potential gradient and further posit that, in most cases, the hydraulic
potential gradient increases downstream (due to steep ice-surface slopes
close to the terminus), so that the transport capacity should also increase.
calculate an erosional potential of
subglacial water based on the hydraulic potential gradient under a valley
glacier and find that the erosional potential increases toward the terminus
and could explain the deepening of inner gorges during a glaciation
e.g.. Both studies are, however, quite
speculative regarding the processes behind subglacial meltwater erosion.
were the first to couple subglacial water flow in a
distributed drainage system, basal refreezing, and sediment transport in a
numerical model to explore the evolution of bed slopes adverse to ice flow
close to the terminus. They demonstrate strong feedbacks between sediment
deposition/entrainment and hydraulic conditions as well as the importance
of daily fluctuations in water input on the sediment flux.
In proglacial studies of seasonal sediment yield, hysteresis between sediment
and water discharge is often observed
e.g.. It is usually
attributed to changes in sediment availability, tapping of new areas of the
bed by the developing drainage system, or increased mobilization caused by
sudden changes in the subglacial hydraulic system
e.g.. An event during which sediment transport
peaks before discharge is usually defined as clockwise hysteresis and is
interpreted as the manifestation of an unlimited sediment source
e.g.. The opposite is true for anticlockwise
hysteresis. In a study of bedload transport by a proglacial stream in the
Italian Alps, identify a transition throughout the
melt season from hysteresis dominated by clockwise events to hysteresis
dominated by anticlockwise events. The authors infer that this transition
is due to the activation of different sediment sources across the drainage
basin.
Evidence of the erosional action of subglacial meltwater flow is widespread
in formerly glaciated regions and appears in the form of N-channels, tunnel
valleys, and inner gorges. Tunnel valleys are large (a few hundred metres to
kilometres wide and up to tens of kilometres long) channel-like features
found within the limits of former continental ice sheets
e.g.
or in Antarctica e.g. in
substrata varying from loosely consolidated sediment to bedrock. Their
formation is attributed to the action of pressurized subglacial meltwater,
and three particular mechanisms have been proposed
e.g.: (1) sediment creep toward preferential groundwater flow paths, (2) carving by
sediment-loaded subglacial water flow, and (3) erosion caused by large
subglacial water floods.
Inner gorges are narrow canyons incised at the bottom of an otherwise
U-shaped valley e.g., and are found
extensively in formerly glaciated mountain ranges like the Alps
e.g.. The
origin of inner gorges was originally entirely attributed to postglacial
fluvial erosion, although conclude that
such features persist through repeated glaciations instead of being reset by
glacial erosion. For example, find that fluvial
incision rates on the order of a centimetre per year occurred during at least
the past 4000 years in a gorge in the French Western Alps. Recently
showed that pressurized water flow is
necessary to explain the longitudinal profile of an inner gorge in the
foothills of the Alps, and infer from cosmogenic
nuclide exposure that the timing of carving of seven inner gorges in the
Baltic Shield matches the timing of glacial cover. Inner gorges are therefore
most likely the combined product of fluvial erosion during interglacial
periods and subglacial meltwater erosion during glacial periods, although the
importance of fluvial vs. glacial conditions is probably dependent on
surrounding topography. Nye channels, tunnel valleys, and inner gorges share
some characteristics suggesting a common genetic origin, although the
specific combination of processes responsible for their evolution may differ
slightly.
We implement a one-dimensional (1-D) model of subglacial water flow in which
a network of cavities is dynamically coupled to one or a few channels. We
then compute the shear stress exerted on the bed and use it to compute
transport stages and bedrock erosion rates by abrasion caused by particle
impacts. We compare the results of models that treat erosion by saltating
particles only and erosion by both the saltating
and suspended particles . We use the word “erosion”
only to describe the carving of bedrock, while “transport”,
“mobilization”,
or “entrainment” refer to the movement of unconsolidated sediments.
We first perform steady-state simulations with the channelized drainage
system alone to demonstrate basic model behaviour. We investigate the role of
ice geometry, surface melt, and sediment supply. We then introduce the coupled
(distributed and channelized) hydraulic system and water forcing with
sub-seasonal fluctuations to test the importance of transients in the
subglacial drainage system. Finally, we discuss the implications of our
results for the formation of N-channels and consequently of tunnel valleys
over bedrock and the persistence of inner gorges through repeated
glaciations. Our specific research questions are as follows. (1) What are the major
controls on subglacial meltwater erosion? (2) How important is subglacial
meltwater erosion compared to overall glacial erosion? (3) Can ordinary
seasonal melt processes lead to subglacial bedrock channel incision (and
potentially the formation of an incipient tunnel valley or persistence of an
inner gorge)? (4) What are the implications of the water flow regime in
channels for hysteresis and sediment transport?
Modelling strategy and rationale
The model outlined above involves numerous variables and parameters leading
to feedbacks. Applications of the model of erosion by abrasion in rivers
e.g.
show that the primary dependencies are the transport stage, the relative
sediment supply, and the hydraulic potential gradient. Numerical modelling
studies of subglacial water flow emphasize the importance of the frequency
and amplitude of the water input forcing and of the ice and bed geometry
e.g.,
although few studies have discussed the shear stress on channel walls
e.g.. In an effort to identify the key variables,
parameters, and feedbacks, we start with simple experiments and build up the
complexity. The simulations are separated in two subsections: (1) steady-state decoupled simulations of a channelized drainage system only
(Table ) and (2) transient simulations of a coupled
channelized and distributed drainage system (Table ).
In the steady-state simulations (Table ), we first
assess the erosion pattern resulting from a constant discharge along the
glacier bed. Then we introduce a water forcing that increases with decreasing
ice-surface elevation and test the effect of ice geometry and sediment
supply. Additional experiments assessing the effect of water flow through a
network of cavities, water input, sediment size, and channel wall and bed
roughness are shown in the Supplement. In the transient simulations
(Table ) we first analyse the role of a synthetic
melt season, then use water forcing following realistic melt seasons, and test
the role of ice geometry, channel density, and sediment supply.
As per the theory on which the SEM and TLEM
are based, sediment supply, through the tools and cover
effects, exerts a major control on erosion rates and patterns by water flow
beneath glaciers. Although it is possible to estimate sediment supply to
rivers
e.g.,
no method exists to quantify subglacial sediment supply or transport. In
glacierized catchments, measurements are usually made in the proglacial
stream, relatively close to the terminus
e.g.,
where water flow is subaerial and potentially influenced by channel dynamics
between the glacier terminus and the measurement station
. In the
steady-state simulations we impose a sediment supply that leads to an
interesting and diverse range of simulations illustrating most processes and
their feedbacks, whereas in the transient simulations we choose a sediment
supply that leads to a sediment yield at the last node equivalent to a few
millimetres of glacial erosion per year.
We use a flat bed and a parabolic surface for all but two geometries: STP and
WDG (Fig. ). The STP geometry aims at reproducing the steep
front of an advancing ice sheet, while the WDG geometry has a wedge shape
that resembles the profile of a thinning and retreating ice sheet margin. We
assume that water input is distributed uniformly (no moulins) along the bed
except for the simulation S_MOULIN. For almost all simulations we fix the
sediment supply per unit width; the total sediment supply therefore increases
with channel size. This is similar to the assumption of a uniform till
distribution across the width of the glacier in which the channelized water
flow sources its sediment.
(a) Different ice geometries considered (see
Tables and ) and (b) corresponding ice-surface slopes. The geometries REF, IS1300, and IS700 are
parabolas such that the surface elevation is given by zs(x)=zs,maxx, where zs,max is the maximum
ice-surface elevation and x∈[1,0]. In order to obtain the steeper front
in STP we use a cubic root instead of the square root: zs(x)=zs,maxx1/3. The wedge-like geometry WDG is defined by a
straight line with the same elevation change as REF.
Summary of steady-state simulations. For simulations in which
meltwater input is a function of ice-surface elevation zs, we
compute f(zs(x))=b˙ssmax×1-(zs(x)-zs,min)/zs,max, where b˙ssmax=8.5×10-7ms-1 is the
maximum meltwater input rate to the channelized drainage system, and
zs,min and zs,max are respectively the minimum
and maximum ice-surface elevations. Note that b˙ssmax=8.5×10-7ms-1 corresponds to 7.6 cm of ice melt
per day assuming ρi=910kgm-3. The reference
sediment supply used for the steady-state simulations is qs,ref=3.6×10-3m2s-1 (Table ).
Simulation
Purpose
Forcing
Difference from reference run
Section
S_MOULIN
R-channel only
Qch(x=0,t)=18m3s-1
Localized input upstream boundary
S_REF
Reference
b˙ch,ref(x,t)=f(zs(x))
S_WDG
Ice geometry
b˙ch(x,t)=b˙ch,ref
Constant ice-surface slope dϕ0/dx=cst
S_STP
Ice geometry
b˙ch(x,t)=b˙ch,ref
Steeper and thicker terminus
S_1300
Ice geometry
b˙ch(x,t)=b˙ch,ref
zs,max=1300 m
S_700
Ice geometry
b˙ch(x,t)=b˙ch,ref
zs,max=700 m
S_SSP
Sediment supply
b˙ch(x,t)=b˙ch,ref
qs=qs,ref/20-qs,ref×25
Summary of transient simulations. In this series of simulations the
basal sliding speed is ub=5ma-1 and the
reference sediment supply is qs,ref=9.1×10-3m2s-1 (Table ). Water is fed to the
network of cavities rather than the channel, hence b˙ch=0ms-1, and the function of surface elevation is that of the
steady-state simulations (Table ).
Simulation
Purpose
Description
Section
T_REF
Reference transient
Reference geometry and synthetic water input
T_2007
Realistic forcing
2007 surface-melt time series
T_2008
Realistic forcing
2008 surface-melt time series
T_1300
Ice geometry
zs,max=1300 m
T_700
Ice geometry
zs,max=700 m
T_W500
Drainage catchment width
W=500 m, i.e. 2 R-channels
T_W333
Drainage catchment width
W=1000/3 m, i.e. 3 R-channels
T_W250
Drainage catchment width
W=250 m, i.e. 4 R-channels
T_SSP/4
Sediment supply
qs=qs,ref/4
T_SSP/2
Sediment supply
qs=qs,ref/2
T_SSPOPT
Sediment supply
Optimized erosion; qs=0.6×qtc
Results
All quantities related to the erosion model are given per unit width of flow.
The cross-sectional area of a subglacial channel (S), and thus its width
(Wch=22S/π), changes more rapidly with distance along
the flow path than a typical subaerial river. We need to account for these
changes when displaying the results, and therefore introduce three
quantities: the total transport capacity Qtc=qtcWch(m3s-1), the total erosion computed with the
TLEM Etot=e˙totWch(m2s-1), and the total erosion computed with the SEM
Esalt=e˙saltWch(m2s-1).
Steady-state decoupled simulations
We examine basic steady-state behaviour of key model variables such as
R-channel cross-sectional area (S), transport capacity (qtc),
impact velocity (wsi and wi,eff), transport stage
(τ∗/τc∗), and relative sediment supply (qs/qtc). In the steady-state simulations, the drainage system is
composed of a single R-channel. Simulations are terminated once dependent
variables (S and ϕch) reach steady state. In this series of
simulations, unless stated otherwise, we impose the sediment supply
qs=3.6×10-3m2s-1 to produce an
interesting and diverse range of model behaviour.
R-channel with constant discharge
In the S_MOULIN simulation we use the reference glacier geometry
(Fig. , REF) and the water input is imposed at the uppermost
node only, i.e. as if a moulin were feeding the system. This permits us to
drive the system with a constant water discharge and to analyse the resulting
relation between discharge, channel cross-sectional area, velocity,
instantaneous erosion rates, and transport capacity patterns.
Relationship between the hydraulic and erosion variables in a
steady-state R-channel with constant discharge (Table ,
S_MOULIN; Sect. ). Components of TLEM and SEM
are also compared. (a) R-channel discharge, Qch, and
cross-sectional area S; (b) velocity, u, and gradient in hydraulic
potential, ∇ϕch; (c) sediment transport capacity,
qtc, prescribed sediment supply rate, qs, and
fraction of bed exposed, Fe; (d) near-bed sediment
concentration, cb (TLEM), and impact rate, Ir (SEM);
(e) impact velocity in the TLEM, wi,eff, and the SEM, wsi; and (f) erosion rate calculated with the TLEM, e˙tot, and
the SEM, e˙salt.
Although discharge is constant (Fig. a), the cross-sectional
area of the R-channel changes along the profile (Fig. a) due
to the ice geometry (Fig. ). Over the first 46 km, the
cross-sectional area decreases in response to the steepening hydraulic
potential gradient (Fig. b), the latter being a function of
ice-surface slope (Fig. ). Close to the terminus (last 4 km
of the profile), the ice thins significantly, the hydraulic potential
gradient shallows (by a factor of 3), and the cross-sectional area increases. The
average water velocity assumes the opposite pattern
(Fig. b). Because the grain size is kept constant,
qtc∝(τ∗-τc∗)3/2, the rate of
sediment transport (Fig. c) is amplified relative to the
velocity (qtc drops by a factor of 6, while u is reduced by about
50 %); both have maxima at 46 km and decrease sharply near the terminus. In
this simulation the sediment transport capacity is always larger than the
supply rate (Fig. c), exposing most of the bed (Fe>0.5).
In both the TLEM and SEM, near-bed sediment concentration and impact rate are
described as a function of the sediment supply and the hop trajectory of a
particle, so they are similar (Fig. d). As the velocity
increases (km 0–46) the sediment is transported faster and further from the
bed and the near-bed sediment concentration and impact rate decrease.
Impact velocities (Fig. e) vary depending on the model. In
the SEM the impact velocity tends toward zero as the hop length increases and
the particles approach transport in suspension, which leads to a local
minimum around km 46. The TLEM accounts for the effect of turbulent eddies on
the trajectory of particles close to the bed, and thus the effective impact
velocity is commensurate with the velocity and peaks around km 46.
Erosion rates (Fig. f) in both the SEM and TLEM show a
minimum at 46 km and a maximum close to the terminus. The minimum and
maximum correspond respectively to the minimum and sharp rise in near-bed
sediment concentration or impact rate. Even under a constant discharge, ice-surface slope and channel size produce a peak in velocity just upstream of
the terminus. At this peak, the flow has the power to lift particles far
enough from the bed to reduce erosion in both the SEM and TLEM. Erosion rates
are higher with the TLEM than the SEM due to the difference in impact
velocities when the sediment transport regime approaches suspension. For the
transport stages we obtain, impact velocities with the TLEM are consistently
higher than with the SEM. Moreover, for relatively large transport stages,
more of the bed is exposed in the TLEM: when some fraction of the total load
travels in suspension, qs>qb.
Ice geometry
The surface slope of a glacier is a first-order control on subglacial water
flow (Eqs. –). We experiment with
thicker ice (IS1300, Fig. ), thinner ice (IS700,
Fig. ), and steeper ice-surface slopes close to the terminus
(STP, Fig. ), all compared to the reference ice geometry (REF,
Fig. ). Shallower surface slopes close to the terminus are
tested with a wedge-like geometry of constant surface slope (WDG,
Fig. ). All these simulations (Fig. ; see
Table ) employ the surface-melt profile of S_REF and
therefore yield nearly identical discharge profiles (Fig. a).
The water input is a function of the ice-surface elevation
(Table ); hence, the discharge is largest close to the
terminus.
Comparison of steady-state simulations with varying ice geometries,
reference water input and drainage through a single R-channel
(Table ). (a) Discharge in the channel,
Qch,
(all curves overlap); (b) channel cross-sectional area, S; (c) transport
stage, τ∗/τc∗; and (d) total erosion computed with the
TLEM (Etot).
The thinner the ice close to the terminus the larger the channel
(Fig. b) and the lower the transport stage
(Fig. c). In the case of S_STP, the combination of a
particularly steep hydraulic potential gradient and relatively thick ice near
the terminus inhibits channel enlargement over the last 5–10 km as observed
in other simulations (Fig. b). Since the discharge profile is
the same for the simulations presented, the smaller the channel, the faster
the flow and hence the larger the transport stage. The maximum transport
stage for the simulation with the steepest terminus (S_STP,
τ∗/τc∗≈18.5, Fig. c) is almost
4 times larger than that for the simulation with the shallowest terminus
(S_WDG, τ∗/τc∗≈5.5, Fig. c; see
Fig. and Table ). Over the first 25 km
of the profile the simulation with a constant ice-surface slope (S_WDG)
shows the steepest hydraulic potential gradient, creating comparatively
higher transport stages than other models.
Erosion begins around km 4 for S_WDG (Fig. d), and around
km 19–20 for S_STP and S_700 (Fig. d), because for the
prescribed sediment supply, the bed becomes exposed for transport stages
τ∗/τc∗≳2.5. All simulations but S_WDG have a
local maximum in total erosion (Etot, Fig. d)
between km 30 and 37 and a local minimum between km 45 and 49. The sediment
supply per unit width is constant; therefore, the relative sediment supply
decreases as the transport stage increases and thus the erosion rate per unit
width decreases when qs/qtc<0.5 because the number of
tools decreases. For simulation S_WDG the increase in channel size
compensates for the small decrease in erosion rate per unit width (not shown)
because the transport stage remains relatively low (τ∗/τc∗<6). Over the last few kilometres of the profile, total erosion
increases again for the simulations in which transport stage drops to
moderate values (S_REF and S_1300). Total erosion drops sharply at the
terminus for S_700 as the transport stage drops below 2.5. When the
transport stage remains relatively high (τ∗/τc∗>15)
the number of tools remains low, as does total erosion (S_STP,
Fig. c and d). Ice geometry exerts a primary
influence on transport stage and erosion patterns via its control on
hydraulic potential gradients and channel size. Simulation S_WDG yields the
most erosion, despite relatively low transport stages, illustrating the
importance of the tools effect e.g..
Sediment supply
Tools and cover compete so that both a lack and an overabundance of tools
hinder erosion. and have
shown that erosion peaks for a given flow regime at an optimum relative
sediment supply. We thus investigate how varying sediment supply (qs=1.8×10-4 to 8.9×10-2 m2s-1)
affects the rates and patterns of subglacial meltwater erosion, while the
subglacial hydraulic regime remains that of S_REF (Fig. ;
Table ).
Influence of sediment supply rate qs (S_SSP,
Table ) on total erosion with the TLEM and SEM for the
same hydrology as the reference simulation (S_REF,
Table ). The legend applies to all panels. (a) Relative
sediment supply (qs/qtc), (b) total erosion rate
computed with the TLEM (Etot), and (c) total erosion rate computed
with the SEM (Esalt). Note the logarithmic scale on the y axis
of (b) and (c).
For sediment supply rates qs≤qs,ref×2
(Fig. a) the patterns of total erosion with the TLEM and
SEM (Fig. b and c) show the same features
as in Fig. : no erosion in the uppermost part of the profile
followed by a local maximum around mid-profile, a local minimum around
km 46–47 and a sharp rise in the last 3 km. For larger sediment supply
qs≥qs,ref×5, the patterns change and have
a single maximum only. If the relative sediment supply satisfies qs/qtc>0.3 (Fig. a) the number of tools
remains high enough for total erosion to increase with transport stage
(τ∗/τc∗) as the increase in channel size compensates
for the small drop in erosion per unit width e˙tot when
qs/qtc<0.5.
Erosion occurs over 48 km of the bed (S_SSP, qs,ref/20) in both
the SEM and TLEM for the lowest sediment supply, whereas the bed is almost
completely shielded for the second largest sediment supply (S_SSP,
qs,ref×20) and erosion occurs only between km 46 and 48 in
the TLEM (Fig. ). The SEM treats the whole sediment supply
as participating to the cover effect, and the bed is shielded as soon as
qs/qtc≤1. The TLEM discriminates between
transport as bedload and in suspension; therefore, if qb<qs the bed can remain partially exposed even for a relative
sediment supply qs/qtc≥1. The erosion window is
therefore larger for the TLEM Fig. 6 in. The fact
that erosion rates computed with the TLEM are higher than those computed with
the SEM is inherent to the model formulation .
An interesting conclusion arising from Fig. is that total
erosion is significant over most of the bed (Fig. b and
c) when the sediment supply rate is relatively low
(qs≤qs,ref×2; Fig. a).
Total erosion becomes more localized at higher relative sediment supply
rates. Changing the particle diameter (D), instead of sediment supply
(qs), leads to similar changes in relative sediment supply and
therefore in erosion patterns (see Supplement). The transport stages
calculated are large enough to produce significant differences between the
SEM and TLEM for large relative sediment supplies. Hereafter, we focus on the
TLEM only in the results as it is more appropriate for the flow conditions
encountered beneath glaciers.
Transient coupled simulations
Arguably, most erosion and sediment transport in fluvial systems occur during
flood events
e.g..
In glacial environments large daily variations in meltwater input to the
glacier bed can be likened to periodic flooding
e.g.. We perform a series of transient simulations
with a cavity network coupled to an R-channel
(Eqs. , , and ) to explore how the transience in subglacial water flow is
affected by changes in ice thickness, surface melt, and sediment supply, and
how this transience impacts instantaneous and annually integrated erosion
rates. In this series of simulations, we choose the sediment supply
(qs,ref=9.1×10-3m2s-1) such that the
modelled sediment yield (∫min(qs(XL)Wch(XL),Qtc(XL))dt)
corresponds to an inferred basin-wide erosion rate of a few millimetres of
erosion per year
e.g..
Reference model
The reference model uses a synthetic forcing in the form of water supply to
the distributed system (b˙ca, Eq. ();
sinusoid with a period of 120 days on which we superimpose daily
fluctuations) and the reference glacier geometry (Fig. , REF;
Table , T_REF). The sediment supply rate per unit
width is assumed to be uniform along the bed. With this simple test we
explore how the transience in water input and response of the subglacial
drainage system affect transport stage and erosion rate. Modelling subglacial
water flow through coupled distributed and channelized drainage systems has
been described extensively in recent literature (see Table 3 in
), so we omit a
discussion of the drainage system itself and focus instead on how subglacial
hydrology affects transport stage and erosion.
Time series at six different distances from the divide (km 50, 45,
40, 35, 30 and 25) for simulation T_REF (Table ).
(a) Transport stage (τ∗/τc∗) and normalized meltwater
input (b˙ca/b˙ssmax) at the terminus
(light grey). The dashed line represents the threshold for sediment motion:
τ∗/τc∗=1. (b) Relative sediment supply qs/qtc. (c) Erosion per unit width e˙tot
(ma-1, thin lines) and total erosion Etot
(m2a-1, thick lines) computed with the TLEM.
Since the sediment supply is fixed, transport stage
(Fig. a) and relative sediment supply
(Fig. b) are anti-correlated. The time transgression
in transport stage and relative sediment supply is a result of the up-glacier
incision of R-channels. Once a channel is well developed, water pressure
decreases in the channel, as does water velocity and transport stage
(Fig. a). Daily fluctuations are only detectable
close to the terminus, where a channel is relatively well developed. Similar
to what we find in the steady-state simulations, the largest transport stages
are found a few kilometres up-glacier from the terminus. The bed remains
shielded over the first 25 km from the ice divide. The time window during
which erosion occurs (qs/qtc≲1-1.5) is
longest at km 45 and decreases up- and down-glacier. The size of the
R-channel at the terminus (km 50) after day 80 becomes large enough that the
transport stage (Fig. a) is insufficient to maintain
an exposed bed. Further upstream from km 45, the hydraulic potential gradient
shallows and the discharge decreases so that the bed is exposed for a shorter
time.
Erosion per unit width (e˙tot,
Fig. c) peaks when the relative sediment supply
satisfies 0.25≤qs/qtc≤0.4, i.e. at the
onset of R-channel formation, even before the peak in transport stage for the
lowermost 15 km (km 35–50 in Fig. ). The peak in
erosion is followed by relatively constant values and eventually an abrupt
drop. At the terminus, erosion ceases due to low shear stress in the
relatively large channel, while upstream erosion ceases due to declining
water supply. Note that erosion can occur after the melt season ends (day 120) at km 45 because water remains stored englacially and subglacially
(Eq. ). Further up-glacier (km 25–30) the maximum
erosion per unit width coincides with the minimum in relative sediment
supply, as the latter remains larger than 0.5 and enough tools are available
close to the bed.
Patterns of total erosion (Etot,
Fig. c, thick lines) all peak between day 65 and 75,
except at the terminus (km 50). The initial peak in erosion per unit width
occurs while the channel is small, and thus total erosion is largely controlled
by channel size over most of the record. Simulations with transient meltwater
input highlight the role of channel size in controlling transport stages and
erosion close to the terminus.
Surface melt
We vary the amount of water reaching the bed to explore the differences
between using synthetic and realistic melt records. The realistic melt
records come from the ablation area of an unnamed glacier in the Saint Elias
Mountains, Yukon, Canada, in 2007 and 2008 . We
scale the 2007 melt record (T_2007, Table ) so that
the total volume of water is identical to the synthetic input. The melt
time series from 2008 (T_2008, Table ) is then scaled
such that the ratio of 2007 to 2008 melt volumes is preserved. This test is
intended to highlight the importance of total melt volume and the temporal
structure of meltwater input.
Time series of transport stage and erosion rates for two realistic
water input time series (Table , T_2007 and T_2008)
at five different distances from the divide (km 20, 27.5, 35, 42.5, and 50).
(a) Transport stage (τ∗/τc∗) and normalized meltwater
input (b˙ca/b˙ssmax) at the terminus
(light grey) for simulation T_2007. (b) Erosion rate per unit width
e˙tot (thin lines) and total erosion Etot
(thick lines) for simulation T_2007. (c) Transport stage
(τ∗/τc∗) and normalized meltwater input
(b˙ca/b˙ssmax) at the terminus (light
grey) for simulation T_2008. (d) Erosion rate per unit width
e˙tot (thin lines) and total erosion Etot
(thick lines) for simulation T_2008. The dashed lines in (a) and (c)
represent the threshold for sediment motion: τ∗/τc∗=1.
When we apply the realistic forcing from 2007 (T_2007), the transport stage
exhibits four to five peaks (Fig. a) at km 35, 42.5, and
50. At these three locations, the first peak occurs once enough water is
supplied to the bed to form a channel (Fig. a; after 45 days at 42.5 and 50 km and after 75 days at 35 km). The subsequent
peaks in transport stage (after day 60 at 42.5 and 50 km and after day
80 at 35 km) follow periods of high melt. After day 45, the transport stage
remains highest at 42.5 km because creep closure prevents R-channels from
becoming too large and water discharge is high enough to maintain high
velocities. At km 20 and 27.5 it takes about 60 days for the first peak in
transport stage to occur. The subsequent peaks at these two locations (around
day 90, 105, and 129) lag high melt periods even further.
Given the prescribed sediment supply rate, the bed is only exposed at
transport stages larger than 3.5. Erosion (Fig. b;
e˙tot and Etot) at km 27.5 and 50 therefore
only occurs during peaks in transport stage; at km 20 the bed is always
covered. Erosion rate per unit width (e˙tot,
Fig. b) plateaus at moderate transport stages, thus it
remains relatively constant once the bed is partially exposed. On the other
hand, total erosion (Etot, Fig. b) peaks
with transport stage (Fig. a). Similar results are
obtained for T_2008 (Fig. c and d),
where a different melt time series is employed. While the amplitudes of the
fluctuations of meltwater input are a few times larger in T_2008 than
T_2007 (Fig. a and c), the total
melt in T_2008 is about 80 % that of T_2007. These realistic meltwater
forcings produce episodic variations in transport stage and erosion rate that
suggest multi-day fluctuations in meltwater input are important.
Comparison of time-integrated erosion per unit width (∫e˙totdt) and total erosion (∫Etotdt) for different water inputs. The first 20 km of the profile are
not shown because the bed is alluviated and erosion rates are negligible.
These multi-day variations in water input also lead to a succession of
channel enlargement events (Fig. a and
c), represented by multiple peaks in transport stage
(τ∗/τc∗). On the timescale of several days, creep
closure near the terminus is low enough that a channel is sustained between
the melt events, leading to an up-glacier migration of relatively large
transport stages and integrated erosion. Thus, the pressure in the channel
close to the terminus, and hence transport stage, is low. This results in the
integrated total erosion (∫Etotdt) being about
3 times lower for realistic inputs (Fig. , T_2007 and
T_2008) than for the synthetic one (Fig. , T_REF) at the
terminus. For the same total water input (T_REF and T_2007), the realistic
melt season produces more erosion averaged over the glacier bed than the
synthetic input (8×10-2mm for T_2007 vs. ∼7×10-2mm for T_REF).
Comparison of time-integrated erosion per unit width (∫e˙totdt) and total erosion (∫Etotdt) for different ice geometries. The first 20 km of the profile
are not shown because the bed is alluviated and erosion rates are
negligible.
Ice geometry
Studies of sediment yield from glacierized catchments
e.g.
conclude that glaciers are more erosive during retreat than during advance
due to the amount of meltwater production. Glacier thinning (or thickening)
during a phase of retreat (or advance) will also impact the development of
the subglacial drainage system and hence its ability to flush sediments and
erode the bed. In this series of model tests we hold the sediment supply
fixed and vary the glacier geometry by changing the maximum ice thickness
(Fig. , T_REF, T_1300, T_700,
Table ), while the water input remains the same. As
we have already described the principal mechanisms responsible for
fluctuations in erosion rates in previous sections, we now focus on annually
integrated erosion.
For all ice geometries tested in Fig. (T_1300, T_700 and
T_REF, Table ), significant erosion only occurs
down-glacier of km 20. The thicker the ice, the further up-glacier significant
erosion (both ∫e˙totdt and ∫Etotdt) occurs (up to km 21 for T_1300 and km 29 for T_700). In
these tests, thicker ice also means steeper surface slopes
(Fig. ). Since water input is identical for these simulations,
steeper surface slopes lead to faster water flow and the possibility of
initiating sediment motion further up glacier. At the terminus almost 4
times as much erosion occurs for T_1300 than T_700 because thick ice
prevents the growth of a large channel.
Subglacial drainage catchment width of a channel
Hydraulic properties of the distributed drainage system determine the density
of channels that form e.g.. The smaller the channel
spacing, the lower the discharge in a single channel. A smaller channel, at
equilibrium, yields larger water pressures, so we expect that more water
would be evacuated through the cavity network. In this test we fix the total
glacier width at 1000 m and allow two, three, or four channels to form such
that channel catchment widths (W) are, respectively, 500, 333, and 250 m
(T_W500, T_W333, T_W250; Table ).
Influence of drainage catchment width (W) on time-integrated
erosion. (a) Time-integrated erosion per unit width (∫e˙totdt) and total erosion (∫Etotdt)
for an individual R-channel. (b) Time-integrated total erosion (∫Etotdt) summed over all R-channels in each simulation.
The first 20 km of the profile are not shown because the bed is alluviated
and erosion rates are negligible.
If we consider the erosion in a single R-channel per simulation
(Fig. a), the smaller the drainage catchment width, the smaller
the discharge, and the smaller the time-integrated erosion
(Fig. a; ∫e˙totdt and ∫Etotdt). The feedback causing erosion rate per unit
width to decrease at large transport stages (see Fig. ) is
such that even for the simulations where the drainage catchment width is
relatively small, erosion rates are comparable (∫e˙totdt, Fig. a) despite the lower transport stages. The
differences are, however, relatively large for the annually integrated total
erosion (maximum ∫Etotdt for T_REF is more than
twice that of T_W250, Fig. a) because of the effect of channel
size.
The hierarchy in total integrated erosion is inverted when all R-channels
within a fixed glacier width are accounted for (Fig. b, between
km ∼37 and ∼48). Once the number of R-channels present is taken
into account, integrated total erosion is largest for the smallest channel
catchment (almost twice as large for T_W250 than T_REF,
Fig. b). In this case, numerous small channels therefore
produce more erosion than few large ones.
At the terminus (km 50), the simulations with a catchment width per channel
smaller than that in T_REF show values of annually integrated erosion (∫Etotdt) of about half that of the reference simulation
(Fig. b). The relatively smaller R-channels in these
simulations remain more pressurized and thus drain less water from the cavity
network (Eq. ); the relative discharge in the
cavity network near the terminus is therefore larger than in the reference
simulation (T_REF), further diminishing transport stage near the terminus
(see Figs. –).
Sediment supply
In the present model the values and patterns of sediment supply are amongst
the key unknowns. Most till is produced subglacially
e.g. and the amount and size distribution of
till depends on the history and patterns of production (quarrying) and
comminution (abrasion). We test the sensitivity of erosion rates and patterns
to different values of input sediment supply. In two simulations (T_SSP/2
and T_SSP/4; Table ) the sediment supply rate per
unit width is constant in space and time and is taken as a fraction of the
reference supply rate in T_REF (Table ). The largest
erosion rate in the SEM occurs when the relative sediment supply is
qs/qtc=0.5. For the TLEM and transport stages
τ∗/τc∗<100, the maximum erosion rate is obtained for a
relative sediment supply of 0.5≤qs/qtc<0.8
. We determine a ratio qs/qtc
close to optimum and examine the resulting erosion rates and patterns
(T_SSPOPT; Table ). This provides us an upper bound
on subglacial meltwater erosion rates. The hydraulic conditions in this suite
of simulations are that of T_REF (Fig. ).
Time-integrated erosion per unit width (∫e˙totdt) and time-integrated total erosion (∫Etotdt) as a function of sediment supply rate (qs)
(Table ). The first 20 km of the profile are not
shown because the bed is alluviated and erosion rates are negligible.
Decreasing the sediment supply leads to a decrease in the maximum integrated
erosion (Fig. , T_SSP/2 and T_SSP/4; ∫e˙totdt and ∫Etotdt)
and to the bed being eroded further up-glacier. For a relatively low sediment
supply (T_SSP/4), the peak in annually integrated erosion per unit width
(∫e˙totdt, Fig. ) is
hardly discernible and the peak in annually integrated total erosion (∫Etotdt, Fig. ) is controlled by
channel size. In order to estimate the maximum erosion that can occur under
the given hydraulic conditions and sediment size, we optimize the sediment
supply rate by expressing it as a function of the transport capacity
(qs/qtc≈0.6). The resulting patterns of
annually integrated erosion (∫e˙totdt and
∫Etotdt, Fig. , T_SSPOPT)
mimic the transport stage patterns (see τ∗/τc∗,
Fig, a), and peak at nearly twice the values of
T_REF.
Discussion
Significance of model simplifications
We have detailed the simplifications and underlying assumptions of the model
while describing the model and the strategy; we therefore focus on the
potential implications of the most important simplifications. At the onset of
the melt season, sliding is expected to accelerate as a response to increased
water supply to a distributed drainage system
e.g.
which would promote cavity enlargement and water flow through the distributed
rather than the incipient channelized drainage system. This sliding feedback
alone could produce a small decrease in water pressure
e.g. and hence a decrease in transport stage.
In this study, we treat only the case of bedrock erosion by abrasion and we
neglect the effect of quarrying. Although the latter can lead to erosion
rates up to an order of magnitude larger than abrasion, it requires that the
bedrock be highly jointed . Quarrying
is a two-step process: (1) loosening of blocks around pre-existing cracks (or
possibly opening of new cracks) and (2) mobilization and transport of loose
blocks
.
The depth of loose cracks could be related to sediment availability
and mobilization and transport of
quarried blocks scale with the transport stage
. Therefore we expect that the
patterns of quarrying would be similar to the transport stage, yet limited by
the thickness of the loosened layer.
We compute erosion with only a single particle size that is assumed to be the
median of size of the sediment mixture
e.g..
The SEM was generalized for a grain size
distribution by , a study in which they, however,
omit a discussion of the implications of the generalization of the SEM. As
for the TLEM, suggest that a generalization to grain
size distribution would require re-evaluation of some of the equations to
account for the interactions between particles of different sizes within the
bedload layer. A decrease in median sediment size would probably result in an
erosion profile more spread out along the bed and an increase in median
sediment size would result in a localization of erosion (see Supplement). The
changes in erosion would be quantitatively similar to a decrease in sediment
supply (qs) and thus a decrease in relative sediment supply
(qs/qtc), which strongly controls erosion patterns.
We make the assumption of a supply-limited glacier bed and hence neglect the
effect of sediment transport and the interactions between sediment thickness
and water flow. The mobilization and particularly deposition of sediment
affect the flow regime by enlarging or reducing the cross section of flow
. On a timescale of days, when sediment is mobilized, the
cross section of flow is enlarged and could result in a drop in channel water
pressure and a corresponding loss of flow strength. The opposite effect,
leading to flow strengthening, could occur when sediment is deposited. We do
not treat the case of transport-limited conditions, where the channelized
drainage system would more closely resemble canals
. To implement
sediment transport adequately, it is necessary to improve existing models for
subglacial water flow through canals
with
time-evolving effective pressure. , however, argue
that in the case of subglacial water flow through canals, the water pressure
remains relatively high and very little water would be drained from the
distributed system, limiting the capacity of canals to transport sediment. In
contrast, the steady-state water pressure in an R-channel decreases with
increasing discharge, favouring water flow in the channelized system and
enhancing transport and erosion.
Accounting for the production of sediment and the evolution of particle
diameter at the glacier bed would also largely influence sediment supply
patterns. For example, we can speculate that if the sediment sources are
localized in areas of more easily eroded bedrock
e.g., tools would only be present downstream
from these areas. If, instead of fixing the sediment supply per unit width, we
fix the total sediment supply (simulation not shown), tools are less
available at peak flows, reducing erosion, whereas the cover effect is
enhanced for a relatively small channel. We also tested a simple power-law
downstream fining function (e.g. ; simulation not shown). The results were very
similar to those obtained with a decrease in relative sediment supply,
because particles were smaller than the reference size of 60 mm in the
region of the bed where channels form. Another means of obtaining insight
into sediment supply rates and patterns would be through the use of a
comprehensive model of glacial erosion, i.e. a model encompassing transient
subglacial hydrology (between distributed and channelized systems), ice
dynamics, glacial abrasion, and quarrying. Such a model is, however, yet to be
developed as patterns of glacial erosion remain poorly understood
see. Finally, subglacial water flow evacuates a
significant volume of sediment despite the small area over which R-channels
operate and the tendency of these channels to remain stably positioned in
association with moulins e.g.. The mechanism by
which large volumes of sediment are delivered to the channels remains elusive.
More work is therefore required to quantify subglacial sediment production
patterns.
What are the major controls on subglacial meltwater erosion?
Synthesis of transient simulations (Table )
through comparison of the following quantities calculated for one model year.
(a) Erosion rate averaged over the whole glacier bed (∫∫Etotdtdx). (b) Apparent erosion rate calculated as the
volume of sediment that is transported across the last grid node, i.e.
terminus, (∫min(qsWch,Qtc)dt) averaged over the glacier area. This quantity corresponds to
what one would measure as the sediment flux in a proglacial stream.
(c) Maximum incision depth (max∫e˙totdt). Simulations are ranked by averaged erosion rate and
the colours represent different simulation suites: black for “reference”,
blue for “water input”, purple for “ice geometry”, red for “drainage
width”, and orange for “sediment supply”
(Table ).
We rank the transient simulations by glacier-area-averaged erosion rate in
Fig. a. Because we prescribe water input rates sufficient
to form a channelized drainage system, it stands out from the model
formulation that sediment supply is the most important parameter. A lack or
overabundance of tools inhibits erosion. In our results this is shown by the
fact that T_SSPOPT (sediment supply optimized for erosion;
Table ) produces the most erosion and T_SSP/4
(smallest sediment supply; Table ) the least.
Changing the ice geometry also leads to a relatively large range of averaged
erosion rates as T_1300 (thick ice; Table ) yields
twice as much erosion as T_700 (thin ice; Table ;
Fig. a). Larger hydraulic potential gradients in T_1300
cause the shear stress to be large enough over larger portions of the bed to
create erosion (Fig. ). Subglacial drainage catchment width,
within the range tested, plays a lesser role than sediment supply or ice
geometry although the averaged erosion rate in T_W250 (four channels;
Table ) is ∼30 % more than that of T_REF. The
fact that T_2007 (realistic melt season from 2007 record;
Table ) produces more averaged erosion than T_REF
suggests that an increase in the multi-day variability of the water input
enhances erosion.
The relations are different for apparent erosion rate
(Fig. b), here defined as the equivalent thickness of bed
material evacuated by the integrated sediment flux (∫min(qsWch,Qtc)dt) at the
terminus (km 50). The apparent erosion rate corresponds to the quantity
estimated by studies of sediment yield in proglacial channels, lake, or
fjords. Relatively large hydraulic potential gradients and relatively thick
ice close to the terminus, both of which inhibit R-channel growth, compete
against the loss of transport capacity. Therefore the largest apparent
erosion occurs for the thickest ice (T_1300, Fig. ).
Interestingly, the lowest drainage density (T_REF) yields more apparent
erosion than the highest (T_W250). Discharge through the cavity network
close to the terminus increases with R-channel density; the smaller the
channel, the larger the water pressure and the lower the pressure gradient
between the two systems. This feedback, in addition to the discharge in the
R-channel being smaller due to the R-channel drainage catchment size, reduces
the transport stage close to the terminus.
The results in Fig. b suggest that, despite the increase in
apparent erosion that accompanies an increase in meltwater input (e.g. the
total melt in T_2007 is about 1.25 times that of T_2008), the thinning
associated with the retreat of an ice mass would have a competing effect by
decreasing the hydraulic potential gradient (see Fig. ). The
flushing power of subglacial water flow is conducive to the removal of
subglacial sediment enabling glacial abrasion and quarrying to be efficient.
Our results suggest that the subglacial drainage conditions most favourable for
glacial erosion occur where significant surface melt and relatively steep
surface slopes occur simultaneously, i.e. during an ice sheet maximum advance
or during early phases of retreat. This corroborates the hypothesis of
that some Danish tunnel valleys were excavated
during the stagnation of the Scandinavian ice sheet. However, these findings
challenge the hypothesis that glaciers deliver more sediment to proglacial
areas during retreat than during advance
e.g.,
yet more work is required to explore this hypothesis. The lack of flow
strength in the upper reaches of the glacier (upstream from km 20 for most
simulations) suggests that subglacial sediment in the accumulation area is
transported almost solely by entrainment due to sliding at the ice–bed
interface.
In the steady-state simulations we find that significant erosion can occur in
a network of cavities (see Supplement). In the transient simulations,
however, the coupling with R-channels prevents large shear stresses from
developing in the distributed drainage system, and the threshold for sediment
motion is not even reached for particles of 1 mm diameter. We thus argue
that bedrock erosion in the distributed drainage system is limited unless
specific conditions are satisfied, for example a subglacial flood or a surge.
How important is subglacial meltwater erosion compared to overall glacial erosion?
In most literature on modelling landscape evolution by glacial erosion it is
assumed that subglacial meltwater efficiently removes sediment from the
glacier bed, while its effect on bedrock erosion is neglected
e.g..
On the other hand, in formerly glaciated landscapes, erosional features like
tunnel valleys
e.g.
indicate that subglacial water can produce significant bedrock erosion. The
results we obtain with our simple ice geometries and water input forcings
indicate that the areally averaged bedrock erosion produced by subglacial
water flow is on the order of 10-1-10-2mma-1
(Fig. ), while glacial erosion rates are most often on the
order of 1-10mma-1
e.g..
Bedrock erosion by abrasion from sediment-bearing subglacial water appears
negligible compared to reported erosion rates in proglacial areas. Our
results corroborate the assumption that subglacial meltwater efficiently
removes sediment from the bed and we postulate that this flushing action is
necessary for glacial abrasion and quarrying to access an exposed bed and
remain efficient.
Can ordinary seasonal melt processes lead to subglacial bedrock channel incision?
We find maximum modelled vertical bedrock incision ranging from ∼50 to
∼200 mma-1 (Fig. c). Assuming that
over a period of 20 years climate is relatively steady and the bedrock does
not change significantly, the location of moulins would remain relatively
fixed laterally and so would the channel paths .
Using the lowest incision rate (T_SSP/4), an N-channel almost a metre deep
and a few metres wide could be carved near an ice sheet margin in 20 years. A
similar N-channel would be carved in only five years assuming the largest
incision rate (T_SSPOPT).
Landforms created by former continental ice sheets indicate that subglacial
waterways can occupy persistent paths throughout a deglaciation. Eskers
deposited by the retreating Laurentide ice sheet can be traced for up to
several hundred kilometres and show a dendritic pattern almost as far
upstream as the former divide e.g.. Some
tunnel valleys show several cut-and-fill structures suggesting different
excavation events; moreover, tunnel valleys carved during different
glaciations tend to follow the same paths
e.g.. Eskers also commonly lie inside tunnel
valleys e.g.. Assuming an
incision rate of 100 mma-1 (e.g. Fig. ),
a simple volume calculation suggests that it would take about 15 000 years to
carve a 30 m deep and 100 m wide V-shaped tunnel valley, similar to the
dimensions of tunnel valleys observed in Ireland .
In the context of an alpine glacier, valley geometry tends to focus
subglacial water flow paths toward the thalweg. Assuming that a glacier
occupies topography strongly imprinted by fluvial processes, erosion by
subglacial water flow may tend to preserve if not enhance the pre-existing
fluvial features along the valley centreline. For an alpine glacier eroding
its bed at a pace of 2 mma-1
e.g., in
the case of simulation T_2008 (Table , apparent
erosion of ∼2 mm a-1; ice geometry comparable to that of a large valley
glacier), the maximum incision depth in one year is ∼125 mm
(Fig. c). The relief of a canyon with the maximum width of
the N-channel (∼4.5 m, for T_2008) would increase by more than ∼120 mma-1 (rate of vertical bedrock incision minus rate of
surrounding glacial erosion). If the canyon were 5 times as wide (∼22.5 m), the maximum rate of relief increase would still be ∼24 mm a-1, about twice the measured incision rates in a metres-wide gorge in
the French Western Alps e.g., highlighting the
erosional power of localized subglacial meltwater action.
What are the implications of the water flow regime in channels for hysteresis and sediment transport?
Hysteresis between water discharge and transport stage for
simulation T_2008 (see Table and
Fig. ). Time series at four locations (km 40, 43, 46, and
50) distributed over the last 10 km of the glacier profile of (a) transport
stage (τ∗/τc∗) and normalized water input
(b˙ca/b˙ssmax) at the terminus (light
grey); (b) discharge in the channel, Qch; and (c) calculated
direction of the daily hysteresis when transport stage is plotted against
water discharge. The hysteresis is clockwise when transport stage peaks
before discharge over a daily cycle. Undefined events represent days where
the fluctuations in transport stage or discharge are either simultaneous or
not strong enough to produce hysteresis.
We calculate the direction of daily hysteresis between modelled transport
stage (Fig. a) and water discharge (Fig. b)
at four locations within the last 10 km of the glacier profile
(Fig. c) for simulation T_2008
(Table ; Fig ). Overall,
hysteresis is dominated by clockwise events, with anticlockwise events
only occurring during the second half of the melt season. Clockwise
hysteresis correlates well with the rising limb of multi-day water discharge
and transport stage peaks, while anticlockwise hysteresis correlates with
the falling limb, particularly at km 46 (Fig. ). During the
rising limb of a multi-day melt event, changes in channel size are dominated
by enlargement; the pressure in the channel therefore peaks before the
discharge, as do the averaged water flow velocity and transport stage. During
the falling limb of the melt event, if the channelized drainage is relatively
well established, changes in channel size are dominated by closure, and the
pressure peak can occur after the peak in discharge. In the case of a
proglacial stream carrying a sediment load smaller than its transport
capacity, peaks in transport stage would act as mobilizing events propagating
sediment pulses downstream. We therefore surmise that the direction of
hysteresis in sediment transport and discharge is not necessarily linked to
changes in sediment supply conditions or the tapping of new sediment sources,
but may be the result of changes in subglacial sediment mobilization in the
vicinity of the glacier terminus.
Maximum particle diameter for which movement would be initiated in
simulation T_2008 assuming τc∗=0.03 (see
Table and Fig. ).
In coarse-bedded streams, grain hiding has a significant effect on sediment
transport e.g., as
mobile grains can be trapped behind larger immobile particles. We calculate
the maximum particle diameter for which movement would be initiated in
simulation T_2008 (Fig. ) and find that boulders of up
to 70 cm in diameter can be transported within the last 10 km of the profile
and would correspond to flood-like conditions in rivers. The analogy to river
systems might have influenced the interpretation of glacial deposits such as
eskers, where the presence of boulders or lack of fines is often used to
infer emplacement during flood events
e.g..
For simulation T_2008, we find that transport stage exhibits a sharp
decrease close to the terminus (Figs. a,
a and c) which leads to a
correspondingly sharp decrease in the size of particles transported
(Fig. ) and could lead to a bottleneck in sediment
transport. This bottleneck effect could lead to the deposition of sediment,
filling the channel toward the end of the melt season. A similar process, but
operational over a longer timescale, would be consistent with
time-transgressive deposition of eskers near the mouths of R-channels beneath
retreating ice margins
e.g..
Conclusions
This study is the first attempt to quantify bedrock erosion rates by
transient subglacial water flow with a numerical model. We implement a 1-D
model of subglacial drainage in which a network of cavities and R-channels
interact. We compute the shear stress exerted on the bed and the resulting
bedrock erosion by abrasion (saltation erosion, after
, and total load erosion after
). Because of the large calculated transport stage we
argue that, in the case of subglacial meltwater erosion, it is probably more
appropriate to use the TLEM than the SEM. Assuming that a significant amount
of meltwater is produced and reaches the bed, the main drivers of subglacial
water erosion that we isolate are the rate of sediment supply, particularly
the relative sediment supply, and ice geometry.
From this exercise, we conclude the following:
Bedrock erosion and transport stage in the subglacial drainage system do not
scale directly with water discharge. Instead, transport stage and erosion are related to the hydraulic potential
gradient and hence a combination of water discharge, ice-surface slope, and channel (or cavity) cross-sectional
area. In our simulations, this combination of discharge, slope, and channel cross-sectional area leads to a
drop in transport stage close to the terminus as water pressure approaches atmospheric.
Erosion rates due to the action of subglacial water flow averaged over the whole glacier bed are
negligible compared to the rates of glacial erosion necessary to produce the sediment supply rates we impose.
In our transient simulations, a bedrock channel a few to several decimetres in depth could be carved
over a single melt season as erosion is concentrated at the base of R-channels.
The vertical incision rates we calculate are a few to several times larger than published rates of
fluvial incision in gorges. Therefore, this mechanism may explain the gradual excavation of tunnel valleys
in bedrock and the preservation or even initiation of inner gorges.
Though we have demonstrated the potential for subglacial water flow to incise
bedrock on seasonal timescales, site-specific and quantitative assessments of
its importance will require more realistic 2-D hydrology models
e.g. and simulations over
timescales of glacial advance and retreat.