Introduction
Since the initial observation of “large, flat, circular basins”
in the ice surface of Antarctica by Russian pilots during the International
Geophysical Year , there has been significant interest about the role of
lakes within the larger ice-sheet system. Beginning in 1972, radio-echo
sounding (RES) began confirming that these surface features reflect storage
of free water at the ice-sheet base and subsequently
continued to be a primary technique to identify subglacial lakes, including
Lake Vostok, one of the largest freshwater lakes in the world
. With increased availability of RES data, and
as our ability to observe the ice surface precisely has improved, the number
of known subglacial lakes in Antarctica has increased. At the time of
writing, over 300 subglacial lakes have been discovered throughout the
continent using a variety of geophysical methods . Until
the mid-2000s RES was the primary technique for identifying subglacial lakes
e.g.,. Most of the lakes found in RES
surveys tended to be located beneath the slow-moving ice near the divides
(Fig. 1a), driving initial research questions on whether lakes were open or
closed systems e.g.,, with considerable
speculation about their impact on local ice dynamics
e.g.,.
Since 2005, a variety of repeat observations of the ice surface from
satellite missions have revealed patterns of surface uplift and subsidence
consistent with the filling and draining of subglacial water bodies
e.g.,. In contrast to “RES
lakes”, these “active lakes” have been found beneath fast-flowing ice
streams and outlet glaciers (Fig. 1a, b). Many of the active lakes that
have been surveyed with RES e.g., lacked the characteristic basal reflections traditionally used to
identify the presence of subglacial lakes (hydraulic flatness, specularity,
and brightness relative to surroundings; see ). Multiple
hypotheses have been proposed to explain this discrepancy, mostly relating to
unconstrained hydrodynamics of the system or data processing artifacts
.
Active lakes are of particular glaciological interest due to their potential
impact on the dynamics of fast-flowing ice streams. Surface altimetry
observations suggest that these lakes can hold back and then episodically
release large volumes of water into the subglacial environment
e.g.,, which could alter
ice-stream dynamics for hundreds of kilometers downstream. Repeat measurements of ice
velocity coincident with the existing record of lake dynamics are sparse,
with only three published instances: Byrd Glacier, East Antarctica
; Crane Glacier, Antarctic Peninsula ;
and Whillans Ice Stream, West Antarctica . In all three
cases, subglacial lake activity inferred from surface height anomalies
correlated to episodes of temporary ice acceleration, but understanding of
the longer-term impact of lake drainage on ice dynamics is significantly
limited by a short (maximum 12-year) observational window
.
Critical to resolving the link between ice dynamics and lake activity is
determination of the mechanism by which lake drainage occurs. While some
ice-sheet models have started to incorporate primitive elements of subglacial
lake dynamics e.g.,, they still do not
have a realistic treatment of observed lake-drainage processes. Most
process-based treatments of Antarctic subglacial lakes
e.g., have
hypothesized that active lakes drain via a mechanism similar to that of
ice-dammed lakes in temperate glacial environments, in which narrow,
semi-circular conduits are melted into the basal ice (“R-channels”). More
recently, however, several complementary lines of evidence have called into
question the ability of an R-channel to form and close
in subglacial conditions typical of Antarctica.
suggested that channels incised into the underlying
sediment may be the preferred mechanism.
In this paper we describe the development of a new model for the filling and
drainage of Antarctic subglacial lakes on ideal and real domains based upon
several well-studied lakes in the hydrological system of Mercer and Whillans
ice streams, West Antarctica . Lake drainage occurs in our model through
channels in the underlying sediment termed “canals” by
and we compare the output lake volumes with those from an existing R-channel
model following. We aim to (i) develop a
lake-drainage model that reproduces the recurrence interval and inferred
volume ranges for lake-drainage events on the Siple Coast, (ii) provide
better context for ongoing observations of lake-volume change and lake
distribution with respect to the subglacial hydrology, and (iii) move towards
a consistent parameterization of subglacial lake activity in ice-sheet
models.
This paper begins with background on existing subglacial water models, both
for Antarctica and for ice-dammed lakes in temperate environments (Sect. 2).
Next, we describe the theory of the models used in this paper, including
background hydropotential theory, our implementations of R-channel and canal
theory, and our coupling between a background distributed water system and a
channelized system. We also detail the domains over which we apply the models
(Sect. 3). We then highlight the results from our experiments using the
R-channel and canal models, compare these results to published observations
of Antarctic subglacial drainage events, and explore the sensitivity of our
model to various physical parameters required as model inputs (Sect. 4). We
discuss the implications of these model sensitivities, suggest
reinterpretations for existing observations of subglacial hydrology in
Antarctica based on the viability of a canal mechanism for lake drainage, and
consider issues related to the inclusion of realistic models of subglacial
lake drainage into large-scale ice-sheet models (Sect. 5). Finally, we
summarize our findings (Sect. 6).
Basal water models and subglacial lake drainage
Antarctic basal water models
Models for subglacial water transport and distribution include at least one
of three processes: distributed sheet flow, groundwater, and channelized
flow. The simplest and most common models for ice-sheet basal water flow
invoke some form of distributed system that spreads water laterally
e.g.,. In such systems, water pressure increases with
water flux, while basal traction decreases. More sophisticated models,
however, prefer to accommodate sliding by deformation of the subglacial
sediment, making basal traction decrease through increasing sediment porosity
e.g.,. Given that subglacial sediment is widely
understood to lack the transmissivity necessary to accommodate the water
fluxes at the base of the Antarctic Ice Sheet , changes in
sediment water storage have been regulated by exchange with a distributed
system e.g.,. These more
sophisticated distributed–groundwater exchange models show the most
consistency with borehole
and seismic observations of the basal environment.
Channelized systems are those in which water flux is concentrated in one or
more discrete conduits. The type of conduit that is most commonly modeled is
referred to as an “R-channel”, which is eroded thermally into the ice by
turbulent heat generated by water moving down a hydraulic gradient
. As the relative area of the ice–bed
interface occupied by these systems is small, they can support lower water
pressures. R-channels have been well studied in Greenland, where they have
been directly observed at the ice-sheet margin, and are understood to be
supplied primarily by surface water, much of which enters at discrete
recharge points known as moulins e.g.,. Basal meltwater
systems in Antarctica differ in two significant ways from those in Greenland:
they are isolated from the atmosphere and so do not receive surface
meltwater; and they are located in regions of low hydraulic slopes, such that
the heat generated by water moving down gradient is likely not sufficient to
erode an R-channel . Therefore, channelization has
typically not been considered in large-scale basal water models for
Antarctica e.g.,. In the last decade,
however, increased consideration has been given to the role of channelization
in the drainage of subglacial lakes e.g.,
and to the possibility of subglacial channels incised in the sediments
instead of the overlying ice e.g.,.
Schematic diagram showing the various stages of lake activity (right
panels) and the corresponding evolution of the channel (left panels). Red
arrow denotes net erosion rate, white arrow denotes closure rate, brown arrow
denotes canal effective pressure, and blue bar indicates hydropotential in
the distributed system: (a) initially the lake is filling and no outflow
takes place; (b) as lake level increases, effective pressure differences
between lake and seal allow for some water to escape via a distributed system
(a.k.a. “water sheet”); (c) as the distributed system grows, a low
pressure channel is eroded that begins to draw water from the surroundings;
(d) while the low pressure channel allows for the lake to drain below the
level at which outflow was initiated, as pressure lowers, sediment creeps
inward; and (e) the channel contracts until its flow is negligible and the
lake refills.
Lake-drainage theory
Most of our understanding of the drainage of subglacial lakes derives from
the model for the drainage of ice-dammed lakes on temperate
glaciers in alpine environments e.g.,, where
floods descend hundreds of meters over 1's to tens of kilometers on timescales of days, and
channelization is well documented e.g.,.
explained the repeated drainage in these systems with the
following model (see also Fig. 2): (i) when the lake is at low stand, water
is trapped behind a local maximum in hydropotential, known as “the seal”;
(ii) as lake water levels rise, a hydraulic connection forms over the seal
initiating thermal erosion by outflow from the lake; (iii) during the early
stages of lake drainage, the potential gradient is relatively steep and
effective pressure is low causing melt to exceed creep closure; (iv) with
ongoing drainage, the level of the lake lowers causing a decrease in the
hydraulic gradient, while the effective pressure at the seal increases,
allowing the channel to continue to siphon outflow despite the lower
lake level. The effective pressure change also increases the rate of creep
closure of the channel by the overlying ice, while the energy available for
thermal erosion decreases simultaneously. (v) As the channel closure rate
increases, a reduction in effective pressure ultimately reforms a hydraulic
seal between the lake and points downstream.
Antarctic subglacial floods occur on larger spatial and longer temporal
scales than alpine subglacial floods: water typically descends tens of meters
over hundreds of kilometers and drainage sometimes persists for multiple years
. Limited evidence from borehole observations
of the basal environment along major flow paths connecting lakes indicates
that water travels via distributed systems e.g.,.
Although the patterns of lake volume and outflow over time are qualitatively
similar to those observed during the drainage of alpine ice-dammed lakes,
both historical and more recent work has shown that R-channels are unlikely
to play a major role in the Antarctic subglacial water system due to several
issues:
The inability of water to melt a channel on an adverse bed slope: a substantial
number of flow paths that drain known subglacial lakes appear to exist on an adverse
bed slope from thicker ice into thinner ice, where the pressure melting point is higher.
showed that once the basal slope exceeds 1.2–1.7 times the surface
slope, the heat generated by turbulent dissipation is insufficient to maintain
the water at the pressure melting point and therefore cannot melt surrounding ice to form an
R-channel.
Thermal issues related to polar ice: most models for R-channel formation
imply that the surrounding ice is temperate and isothermal. In polar ice, where a
significant temperature gradient is likely to be present in the ice immediately above
the bed, more turbulent heat will be required to melt a given amount of ice than
would be needed for temperate ice .
The slow rate of closure predicted for R-channels at pressures observed
under Antarctica: creep closure of R-channels requires a drop in water pressure
on the order of several tens of meters water equivalent, 5–10 times higher than the
surface drawdown observed during the drainage of most Antarctic subglacial lakes
. Field reports suggest that subglacial Lake Whillans (SLW), for example,
contained water even at low stand ,
suggesting that the channel may have closed before the lake was completely drained.
In our work we directly compared output from a model for lake drainage via an
R-channel with a model in which channels are formed via
mechanical erosion of underlying sediment (canals) on a domain with geometry
similar to that found for flow paths draining Antarctic subglacial lakes. By
replacing a channel incised into the overlying ice with one incised into the
sediment we address issues (1) and (2), as the erosion of sediment is less
sensitive to temperature than the erosion of ice. This change also addresses
(3) because the deformability of sediment is more sensitive to small changes
in water pressure than deformability of ice . We tested the
output from these models against observations to determine which model was
better able to reproduce estimates of the inferred magnitude and recurrence
interval for known subglacial lake-drainage events.
Model description
Our model is a prototype for lake drainage via a single channel incised into
a linearly viscous subglacial sediment. The overall principle of lake
drainage via a channel where water pressures are significantly below
overburden pressure (in our case 2–8 m water equivalent, or w.e.) is well established in the
literature e.g.,. Only recently has
it been suggested that such drainage may occur via channels forming in
sediment rather than the overlying ice e.g.,. In order
to accommodate a channel incised into sediment rather than in ice, we adopted
principles of sediment mechanics from theoretical work presented in
and .
Given that borehole observations on the Siple Coast region have indicated a
distributed system at the ice–bed interface e.g.,, we
needed a simple model for the coupling of the channel to a distributed
system. Therefore, we adapted a formulation presented in
which, although designed for a rocky-bottomed mountain
glacier lake drainage, contained the four most essential elements we required
for our ice-stream experiment: a lake, a distributed-flow system, a
channelized system, and a means for communication between all parts of the
system. We developed two models based on the
foundation: one model with channelization through conduits melted into the
basal ice based on R-channel theory (henceforth called the R-channel model)
and one model with channelization through canals eroded into the basal
sediment (henceforth called the canal model). Although there may be more
appropriate models for each of the individual components in the Antarctic
subglacial environment (see for a review of existing
water models), we found the ease of coupling between the lake, channelized
system, and the distributed system provided by the
formulation to be the most effective for proof of concept. Input data for the
model come from several sources: previous water-budget studies for
lake-inflow estimates, RES for ice thickness, and satellite and airborne
radar altimetry for surface height changes. By comparing the lake-volume
change inferred from satellite observations with lake-volume change as
simulated with R-channel and canal models, we can (i) perform a diagnostic
test for the hypothesis by that drainage could occur for
sediments that behave like erodible-deformable ice and (ii) develop a
potential prototype for simulating Antarctic subglacial lake drainage.
Our model consists of a system of equations on a 1-D finite
difference grid, with scalar values, such as ice base elevation, ice
thickness, water pressure, and channel width calculated at the center of the
grid cells, while fluxes of sediment and water are calculated at intermediate
points. The systems of equations for the model are described in Sect. 3.1–3.3.
Methods of solution and time integration are described in Sect. 3.4.
We describe our method of obtaining input data and the size of the domain in
Sect. 3.5.
Theory for subglacial water flow
Subglacial water flows from areas of high hydropotential to areas of lower
hydropotential. Hydropotential at the base of the ice, θ0 (calculated
in m w.e.), is the sum of the water-system elevation,
zb, and water pressure, pw, normalized by water density, ρw, and
gravity, g :
θ0=zb+pwρwg.
Many models e.g., calculate θ0
assuming water pressure at the ice base is equal to the overburden pressure,
po, because the quantities used to calculate po (ice surface elevation
zs, zb, and ice density ρi) are easily measured:
pw=po=(zs-zb)ρig.
In reality, pw is the difference between the overburden pressure and
effective pressure, N, which can only be calculated with seismic data
e.g., or borehole data
e.g.,. Although N does not typically affect
regional water routing, modeling work has shown that temporal change in N
is a critical part of the lake-drainage process e.g.,. As N decreases, it enables the formation of a
temporary hydraulic divide between the lake and points downstream. As N
increases, water can then overcome the divide leading to the onset (and
on-going) drainage of that lake. Most of the theoretical work surrounding
subglacial floods expresses hydropotential, θ, in terms of pressure
units (Pa), which in this work is calculated as follows:
θ=zbρwg+(zs-zb)ρig-N.
While we also use θ (in Pa), we mostly express hydropotential as
θ0 (in m w.e.) to allow more direct comparison between it and the
measurements used to calculate it.
Given that θ and N, as well as channel cross-sectional area (S)
and water flux (Q), are each defined differently for the R-channel, canal,
and distributed-flow systems, we will use the subscripts RC, CC,
and S, respectively, to avoid confusion in the following sections.
Explanations for all symbols not defined explicitly in the text can be found
in Tables 1 and 2.
List of all symbols used in this paper with definitions, dimensions, and value/method used to estimate them.
Symbol
Meaning
Dimensions
Value/method
ACC
Flow law constant for sediment
M0.47 L-0.47 T-1.94
3 × 10-5 Pab-a s-1
AL
Lake area
L2
Measured
a
Constant of sediment deformation
–
1.33
b
Constant of sediment deformation
–
1.8
CVRC
Rate of R-channel deformational closure
L2 T-1
Output by model
CVCC
Rate of viscous sediment closure
L2 T-1
Output by model
c¯
Sediment concentration in water column
–
Output by model
cS
Constant for roughness
M-3 L4 T5
2 × 10-20 m s-1 Pa-3
DCC
Deposition rate
LT-1
Output by model
d15
Characteristic grain size
L
Input (see Table 2)
d50
Median grain size
L
Calculated from input
ECC
Erosion rate
LT-1
Output by model
fr
Hydraulic roughness
L-2/3 T2
0.07 m-2/3 s-2
fCC
Canal roughness parameter
L-2/3 T2
0.07 m-2/3 s-2
FL
Coefficient of lake flexure
–
Input between 1 and 2 based on observations reported in
g
Gravitational acceleration
LT-2
9.81 m s-2
hS
Water layer thickness for sheet-flow system
L
Measured
hi
Ice thickness
L
Measured
KRC
Glen's flow law parameter for ice
L3 T5 M-3
10-24 Pa-3 s-1
KS1
Constant
–
1.1
KT1
Erosional constant
–
0.1
KT2
Constant of deposition
–
6
k
Constant of sheet–conduit transfer
M-1 L3 T
10-9 m2 s-1 Pa-1
kh
Coefficient for partitioning of turbulent energybetween heating water and melting surrounding ice
–
0.309
Lh
Latent heat of fusion
L2 T-2
333 500 J kg-1
MS
Flux into system from melt/inflow
L2 T-1
mRC
Röthlisberger channel melt rate
ML-1 T-1
Output by model
mCC
Canal net erosion rate
ML-1 T-1
Output by model
N
Effective pressure
ML-1 T-2
Output by model
NS
Effective pressure in distributed system
ML-1 T-2
Output by model
NRC
Effective pressure in R-channel
ML-1 T-2
Output by model
NCC
Effective pressure in canal
ML-1 T-2
Output by model
N∞
Effective pressure of sediments
ML-1 T-2
Input (see Table 2)
n
Glen's flow law exponent
–
3
p
Constant for power law sliding
–
4
pw
Water pressure
ML-1 T-2
Calculated
Qb
Initial flow in distributed system
L3 T-1
Qin
Inflow to lake
L3 T-1
Qout
Outflow to lake
L3 T-1
Output by model
QS
Outflow via distributed system
L3 T-1
Output by model
QCC
Outflow via canals
L3 T-1
Output by model
QRC
Outflow via R-channels
L3 T-1
Output by model
Qonset
Outflow necessary for channel initiation
L3 T-1
Input (see Table 2)
Qshutdown
Outflow below which channelization ceases
L3 T-1
Input (see Table 2)
q
Constant for power law sliding
–
1
R1
Characteristic obstacle height
L
Input (see Table 2)
RkRC
Coefficient for transmission efficiency betweenR-channels and distributed-flow systems
–
Input (0.05)
Continued.
Symbol
Meaning
Dimensions
Value/method
RkCC
Coefficient for transmission efficiencybetween canals and distributed-flow systems
–
Input (0.05)
SS
Cross-sectional area of distributed system
L2
Output by model
SRC
Cross-sectional area of R-channel
L2
Output by model
SCC
Cross-sectional area of canal
L2
Output by model
TRC
Flux between distributed and R-channel
L2 T-1
Output by model
systems per unit length
TCC
Flux between distributed and canal systems
L2 T-1
Output by model
per unit length
t
Time
T
Measured
uCC
Mean water velocity downstream
LT-1
output by model
vs
Mean settling velocity
LT-1
Calculated
VL
Subglacial lake volume
L3
Output by model
x
Along flow distance
L
Measured
yS
Cross flow distance
L
Measured
yCC
Canal width
L
Calculated
zb
Ice base elevation
L
Measured initially, but changeswith model output
zs
Ice surface elevation
L
Measured initially, but changeswith model output
zsL
Ice surface elevation over the lake
L
Measured initially, but changeswith model output
zbL
Ice base elevation over the lake
L
Measured initially, but changeswith model output
αCC
Geometry correction
–
Input (see Table 2)
θ
Hydropotential
ML-1 T-2
Measured/calculated
θ0
Base hydropotential
L
Measured/calculated
θS
Hydropotential in distributed system
ML-1 T-2
Measured/calculated
θRC
Hydropotential in R-channel
ML-1 T-2
Measured/calculated
θCC
Hydropotential in canal
ML-1 T-2
Measured/calculated
θL
Hydropotential in lake
M
Measured/calculated
μw
Viscosity of water
ML-1 T-1
1.787 × 10-3 Pa s-1
ρw
Density of water
ML-3
1000 kg m-3
ρi
Density of ice
ML-3
917 kg m-3
ρCC
Density of sediment
ML-3
2700 kg m-3
τb
Basal driving stress
ML-1 T-2
Calculated from initialmeasurements
τk
Critical hydraulic shear stress necessary toinitiate erosion
ML-1 T-2
Calculated
τCC
Hydraulic shear stress
ML-1 T-2
Output by model
List of parameters used for each of the experiments. “n/a” stands for not applicable.
Idealized/control
R-channel
SLW
SLE
SLC
SLM
αT
6.0 × 103
n/a
1.9 × 104
2.3 × 105
1.3 × 104
6.0 × 103
d15 (mm)
0.12
n/a
0.24
1.5
0.25
0.12
Qin (m3 s-1)
5
5
4
16
12
22.5
Qonset (m3 s-1)
0.75
0.75
1.75
1.25
3.5
0.75
Qshutdown (m3 s-1)
0.25
0.25
0.25
0.25
0.25
0.25
R1 (mm)
6.0
6.0
6.0
24
1.5
6.0
AL (km2)
100
100
58
257
247
132
N/ρwg (m w.e.)
2.5
n/a
3.25
9
1.75
2.5
HL
2
2
2
1
1
2
MC (m3 s-1)
0.001
0.001
0.001
0.013
0.025
0.002
Subglacial channel formation
The principal governing equations for the two channelization systems in our
model all come from previous work: (i) the evolution of the R-channel from
and and (ii) the evolution of the
canal model from and . The method for
solving both models has been largely adopted from recent work by
, which specifically described the drainage of an
ice-dammed lake via an R-channel coupled with a distributed system and thus
was already well posed for subglacial lake drainage. Exchange between
channelized and distributed water systems in both models also follows
and is mediated by a term describing water transfer
between the systems (TRC for the R-channel model; TCC for the canal
model), which is a function of the pressure difference between the
channelized and distributed systems. To adapt the model
fully to a domain containing Antarctic subglacial lakes, however, we made
additional modifications, including defining the hydropotential gradient with
Eq. (1), so that it is a function of the slope of the ice base and the
effective pressure as well as the slope of the ice surface as defined in
. We also allow for the change in pressure melting point
of water with change in pressure e.g.,, as
this effect has been reported to be significant to hydrological evolution
over other flow paths following adverse bed slopes in Antarctica
e.g.,.
Channels incised into the ice (R-channels)
In the classic R-channel model, transmissivity through a channel is
controlled by the channel's cross-sectional area (SRC), which is a
balance of channel-wall melt rate (mRC) and viscous-ice deformation rate
(CVRC) such that
∂SRC∂t=mRCρi-CVRC.
This formulation, originally presented in , has
since been adapted for a variety of situations. The melt rate is related to
the flux of water through the channel (QRC) by
mRC=QRC(1-kh)∂θRC∂x+khρwg∂zb∂x+kh∂NRC∂xLh,
which describes the conversion of turbulent heat dissipation into melting
with the term kh from to account for the change in
melting point with pressure. The rate of viscous-ice deformation into the
channel is defined by
CVRC=KRCSRCNRCn,
which describes creep closure as a function of effective pressure (NRC)
and aperture size (SRC), where KRC and n are constants of Glen's
flow law. Conservation of mass governs the change in flux along the flow path
by
∂QRC∂x=mRC1ρw-1ρi+CVRC+TRC,
while transfer between the R-channel and the distributed system (see Sect. 3.2.3) is governed by the transfer term TRC using
the equation
TRC=RkRCk(NRC-NS),
where RkRC is the transfer efficiency, k is a constant for
connectivity from , and NRC and NS are the
effective pressures of the R-channel and distributed system, respectively. It
should be noted that our value for RkRC is near the lowest end of values
explored by , primarily due to model stability issues.
Pressure along the channel co-evolves with water flux through an adaptation
of the Manning friction formula, following :
∂NRC∂x=ρwgfrQRC|QRC|SRC8/3-ρwg∂θ0∂x,
where fr is the hydraulic roughness.
Channels incised into the sediment (canals)
Our model for channelization by mechanical erosion into the sediments is
adapted extensively from concepts in existing models of lake drainage via
R-channels eroded into the ice e.g.,. It includes the basic principle of
semi-circular channels capable of sustaining water pressures several meters w.e.
below flotation , as well as more recent
concepts regarding exchange of water between the channelized and distributed
systems and evolution of pressure . In contrast to
R-channel models, our model replaces melting of ice with sediment erosion and
ice closure with deformation of sediment with linear viscous rheology,
borrowing concepts and formulations from and
. In describing channels eroded into the sediment, our work
follows other efforts to model canals incised into the sediment behaving as
conduits e.g., but, in contrast to
these models, focuses specifically on subglacial lake drainage. Our
descriptions of sediment erosion and deformation and of channel geometry are
all extensively simplified; we regard the model developed below as a proof of
concept that can be further refined if it is able to reproduce lake-volume
change as inferred from satellite and GPS observations
e.g., with realistic
parameter choices.
As with the R-channel model, transmissivity for a channelized system with
canals is governed by the canal cross-sectional area (SCC). In our canal
model, we ignore melting (mRC) and deformation (CVRC) of the ice
above and assume all change to SCC is due to erosion and deformation of
the sediment. Change in aperture in this model therefore is a balance of net
erosion (erosion, ECC, minus deposition, DCC) and sediment
deformation (CVCC, for which we assume a linear viscous rheology):
∂SCC∂t=(ECC-DCC)yCC-CVCC.
Adopted from , ECC is a function of the mean sediment
settling velocity (vs), the channel geometry (αCC), the stress
exerted on the bed by the flowing water (τCC), and the particle size
(d15):
ECC=KT1vsαCCmax(τCC-τk,0)gd15(ρCC-ρw)3/2,
where KT1 is a sediment erosion constant and “max” refers to the
maximum of τCC-τk and zero. αCC is a dimensionless
correction factor between 5500 and 233 000, which accounts for the geometric
differences between the semi-circular channel geometry implied by the
formulation and the actual geometry which is likely more elliptical in nature
e.g.,.
DCC, which also follows from , is defined similarly:
DCC=KT2vsαCCc¯gd15(ρCC-ρw)τCC.
where KT2 is a sediment deposition constant and c¯ is the
concentration of sediment in the water column.
Equations (8a) and (8b) imply that erosion and deposition are both a function
of the hydraulic shear stress (τCC), which is a function of water
velocity. Following , we use the mean water velocity
(uCC) such that
τCC=18fCCρwuCC2,
where fCC is a constant describing roughness. In our model, erosion
begins once τCC exceeds a critical value, τk, which is a
function of median grain size as described by
τk=0.025d15g(ρCC-ρw).
In this description, erosion of the sediments occurs only when water velocity
exceeds a certain value since τCC is only dependent on uCC.
Sediment settling velocity (vs), important for both Eq. (8a) and (8b),
is treated as a function of sediment particle size (d15), the density
contrast between sediment and water (ρCC-ρw), and water
viscosity (μw), by the classic derivation from Stokes' law
:
vs=d1522(ρCC-ρw)g9μw.
Given that the net deposition rate as defined in Eq. (8b) is sensitive to the
concentration of sediment already present, a mechanism is required for
keeping track of sediment concentration. To achieve this, we assume that the
transport rate of sediment downstream comes into equilibrium instantaneously.
Sediment concentration (c¯) at the most-upstream cell where erosion
takes place is given by
c¯=yCCΔxρCCρwECC-DCCQCC.
For each cell downstream, c¯ is calculated as the sum of sediment
eroded within the cell locally and the concentration of sediment in the water
within the next cell upstream (c¯up):
c¯=c¯up+yCCΔxρCCρwECC-DCCQCC.
Closure occurs via viscous sediment creep, CVCC, though this may in
reality be a convenient continuum representation of discrete
sediment-collapse events on the sides of the channel:
CVCC=|NCC|NCCACCSCC|NCC|aa2N∞b.
Here N∞ is the regional sediment effective pressure (which is closely
correlated with sediment strength), NCC is the effective pressure of the
water within the canal, and ACC, a, and b are flow-law constants from
. ACC is similar to KRC in Eq. (4). Although our
formulation for Eq. (8a–h) is largely adapted from work by
, we follow in decoupling NCC from
N∞. In this work, however, we depart from and treat
N∞ as a constant and explore this simplifying assumption in more
detail in Sect. 4.3.
In this system of equations, channel growth via erosion is a function of
water velocity (through the relationship between erosion and shear stress)
and is sensitive to the sediment size and the channel geometry. At lower
water velocity, channelization in the sediment will not initiate and sheet
flow will dominate. Channel closure via deformation is a function of the
effective pressure (NCC) and sensitive to the chosen value for sediment
effective pressure (N∞). Although our formulation of Eq. (8h)
theoretically allows for viscous channel growth if water pressure exceeds
overburden pressure, such pressures are not expected (see also Sect. 4.1.3).
Conservation of water mass is accomplished by
∂QCC∂x=-CVCC+TCC.
TCC is determined by the effective pressure difference between the canal
and the distributed sheet via
TCC=RKCCk(NCC-NS),
with RKCC behaving analogously to RKRC (Eq. 6). Finally we define
the propagation of effective pressure (NCC) along flow using a version
of the Manning friction formula adapted from :
∂NCC∂x=ρwgfCCQCC|QCC|SCC8/3-∂θCC∂x,
where we have introduced the roughness parameter fCC=fr=0.07 m-2/3 s2 based on the assumption of a semicircular channel geometry
following .
Distributed sheet flow
The distributed sheet-flow system, which is a component of both the R-channel
and canal model, is governed by three primary equations all modified from
: two concern the conservation of mass, and a third governs
the evolution of NS. Water flux through the sheet (QS) is function of
the hydraulic gradient (∂θS/∂x) and cross-sectional
area (SS):
QS=SSπR14KS12/36.6ρwgfr1/2∂θS∂x1/2,
where
SS=yShS.
Equations (12) and (12a) assume flow-path width, yS, is constant with
depth, such that water flux increases linearly with water layer thickness,
hS. Evolution of SS is governed by the conservation of mass,
∂SS∂t=MS-∂QS∂x-T,
where MS is a source term for water flowing into the system from the sides
and T is the flux between the sheet and channelized system, equal either to
TCC or TRC depending on whether the distributed-flow system is
coupled to the canal or R-channel model.
Equation (12) implies that transmissivity increases linearly with water
storage, while Eq. (13) implies that local water storage will increase if
outflow from a cell is less than inflow. If water pressure is equal to
overburden pressure then the only way for a water layer to exit an enclosed
basin in the hydropotential is to thicken until it overtops the lowest
saddle. With the inclusion of the effective pressure term (NS) in the
equation for hydropotential (Eq. 1b), distributed water layers have another
way over small obstacles. In a distributed system, NS tends to increase as
water thickness decreases, by the following relation:
NS=πR1csnn4KRCτbpSS1n+q.
Our parameterization for NS is adapted from the
description of a linked-cavity system, but substitutes till cavities as
observed by for hard bed cavities. This formulation
assumes NS reaches steady state instantaneously. Thinner water layers (and
therefore higher values of NS) are maintained over hydropotential maxima,
while thicker water layers (and therefore lower values of NS) are
maintained over hydropotential minima. This coupling allows for a
monotonically decreasing pathway in θS in spite of undulations in
overburden pressure, which would inhibit the flow of water downstream without
taking variable NS into account.
Along-flow hydropotential (left axis) in the distributed (solid) and
channelized (dotted) drainage systems at four important time steps during a
fill–drain cycle (initial state, highstand, peak outflow, low stand) for the
four main lakes in the Whillans–Mercer subglacial system: (a) SLW,
(b) SLE,
(c) SLC, and (d) SLM. Thick black line (right axis) indicates overburden
pressure calculated using Eq. (1a).
Modeling the lake
For a lake, we assume that the hydropotential is the sum of the ice base
elevation (zb) and overburden pressure (derived from hi) at the center
of the lake (Fig. 3). While hydropotential over the lake can then rise and
fall in response to filling and drainage, we assume that the variation of
hydropotential with time is spatially uniform across the lake. With the
subscript L referring to conditions (hydropotential, elevation change,
etc.) across the lake, the change in lake level with time is defined as
dzbLdt=dzsLdt.
Change in lake volume is calculated as
∂VL∂t=Qin-Qout
and
∂θL∂t=FLQin-QoutAL,
where Qin is inflow, Qout is outflow, AL is lake area
(which is held constant in both the R-channel and canal models), and FL is a
dimensionless number that varies between 1 and 2 depending on the degree to
which the overlying ice is supported by flexural bridging stresses
(effectively a simplification of the parameterization of surface deformation
in Evatt and Fowler, 2007). For large lakes that have a large section of
freely floating ice, where the zone of flexural deformation is less than
20 % of the total lake area, FL = 1; subglacial Lake Engelhardt (SLE) and
subglacial Lake Conway (SLC) fall into this category (for lake locations, see
Fig. 1b). For smaller lakes under thicker ice where the flexure zone
comprises the most or all of the lake area and surface deformation can be
approximated by a parabola, FL = 2; subglacial Lake Mercer (SLM) and
SLW fall into this category. We assume that there
is no change in area as the lake state changes, and the implications of this
assumption are explored in Sect. 4.2.2. We also assume that all surface
displacement over the lake results directly from changes in lake volume and
neglect any temporal changes in ice thickness.
We also assume that there is only a single channel. Based on earlier runs of
the R-channel and canal models as well as assertions in
, we
assume that the destination lake is the nearest major low in the hydraulic
potential, empirically defined to be an enclosed basin 3 m w.e. deeper than
its surrounding. This assumption generally agrees with observations
e.g.,. We neglect changes in the destination lake's
level and assume that N=0 at its center. For the canal model, we also
assume (i) no change in sediment water
content or rheology over time; (ii) negligible bridging stresses; (iii) near-instantaneous advection of sediment downstream.
These systems of Eqs. (2–17) are solved on a 1-D finite
difference domain, consisting of a source lake, intermediate points, and a
destination lake. The point at which θ0 reaches a local maximum
downstream of the lake is termed “the seal”. This point is controlled by
ice-sheet geometry and is treated as immobile, in contrast to its definition
in some higher-order models which define the seal as the location where
transmissivity is lowest at any given time e.g.,.
Model implementation
Model spinup and initialization
Given our initial uncertainties about the onset of channelization, it is
simpler to run our model when the source lake is filling, such that outflow
from the lake is minimized, and the lake level is well below discharge level.
We initialize the model assuming that water pressure equals overburden
pressure and that there is a constant supply of meltwater, MC, along the
flow path downstream of the lake, such that initial water flux in the
distributed system (Qb) increases with distance downstream of the lake.
With these initial values, we calculate an initial water layer thickness,
assuming NS=0, reworking Eq. (12) such that
SS=QbπR14KS1-2/36.6ρwgfr-1/2∂θS∂x-1/2.
We then use Eq. (14) to calculate a new value for NS, based on SS and
Eq. (13). For each subsequent time step we calculate QS from Eq. (12),
∂SS/∂t from Eq. (13), as well as change in lake properties
from Eqs. (15–17). With the resulting new water layer thickness, we
finally recalculate NS and SS.
Criteria for onset and shutdown of channelization
Initially water will flow from the seal towards the lake; as the lake level
and lake hydropotential increase, however, water will begin flow out of the
lake over the seal. Due to the inverse relationship between NS and QS,
Qout will initially be much lower than Qin. We assume that the
initial outflow does not erode a channel at the seal but remains as sheet
flow. suggested that water velocity in a distributed
system would tend to be higher in places where irregularities in the
overlying ice base and underlying bed lead to a thicker water layer. When the
bed consists of sediments, τCC would exceed a critical threshold
(τk) in these locations before doing so elsewhere. As the water layer
thickness increases, erosion of the bed and/or melting of the ice above would
be concentrated in these locations, leading to a positive feedback between
local water layer thickness and water flow until a channel develops either in
the bed below (a canal) or ice above (an R-channel).
Ice surface elevation, hydropotential, and ice–bed elevation along
the following flow paths (see Fig. 1b for location of flow paths): (a) an
idealized lake flow path based on SLW, (b) the flow path draining SLW
from, (c) the flow path draining SLC and SLM, and (d) the flow path draining SLE.
Our model tries to simplify these complex dynamics by using a threshold
criterion for channel initiation in which a channel carrying 0.5 m3 s-1 appears spontaneously once QS exceeds a critical threshold
value (Qonset). Similar threshold methods have been employed in
previous works, such as , ,
, and . The value for flux through a
newly formed channel is based empirically on work by ,
which indicated that higher initial values would lead to an unrealistically
rapid growth of outflow. As this value is also well below Qonset, it
allows for a gradual transition between a distributed and channelized system.
Effective pressure is initially set equal to the effective pressure in the
distributed system so that NRC=NS for the R-channel model (or NCC=NS for the canal model). These initial values for water pressure then
allow us to calculate an initial SRC (or SCC) through Eq. (7) (or
Eq. 11). Cessation of channelization occurs once QRC (or QCC)
falls below a threshold value, Qshutdown, at which point it becomes
computationally simpler to eliminate the channelization and revert back to
sheet flow. Although it is possible that small incipient channels are always
present, such as is parameterized by , ignoring them
improves computational speed. In Sect. 4.2, we explore in more detail how
variations in Qonset and Qshutdown affect model
performance.
Evolution of channelized flow
Once the channel is initiated we calculate the geometry of a proto-channel
(Eqs. 7, 11), assuming QRC=Qonset (or QCC=Qonset) everywhere between the source and destination lakes. After
each successive iteration of Eqs. (2–4) (or Eqs. 8–10), we recalculate QRC and NRC (or QCC and NCC) along
the flow path using a shooting method: beginning with an initial guess of
QRC (or QCC) at the outflow point, we use Eq. (7) (or Eq. 11) to
calculate dNRC/dx (or dNCC/dx) locally on the staggered grid and
the corresponding values for NRC and TRC (or NCC and TCC)
at the next point downstream on the regular grid. Using these newly
calculated values for NRC and TRC (or NCC and TCC), we
calculate dQRC/dx (or dQCC/dx) on the regular grid and then
QRC (or QCC) at the next point downstream on the staggered grid.
This process is repeated downstream until we calculate NRC (or NCC)
using Eq. (5) (or Eq. 9) at the destination lake, at which point it is a
known quantity. We compare our calculated NRC (or NCC) with the
“known” NRC (or NCC) to obtain a misfit. We then use a Newton's
method iteration on NRC (or NCC) at the downstream lake (treated as
a function of QRC (or QCC) at the source lake outlet). After a
maximum of 12 iterations we have arrived at a value for QRC (or
QCC) at the source lake that results in a value for NRC (or
NCC) at the destination lake that is within 0.01 m w.e. of the known
NRC (or NCC). Once this value is obtained we then iterate Eqs. (2),
(6), and (7) (or Eqs. 8, 10, and 11) and the corresponding sheet-flow
evolution (Eqs. 12–14). To improve model stability, we apply a
modification of the Courant–Friedrichs–Lewy (CFL)
criterion for the length of the time step. After calculating dSS/dt and
dSRC/dt (or dSCC/dt) we then define the time step Δt by
the following
relation:
Δt=minSSdSSdt0.05,SRCdSRCdt0.05,105,
for the R-channel model, or
Δt=minSSdSSdt0.05,SCCdSCCdt0.05,105,
for the canal model, where “min” refers to the minimum of all expressions
within the parentheses.
Real and idealized model domains
Our idealized model domain (Fig. 4a) was based on a simplified version of
the flow path connecting the well-studied e.g., SLW to the Ross Sea from
(see Fig. 1b for location; Fig. 4b for comparison to
the real domain). In this domain, the segment between the source lake and the
seal, and the seal and the destination lake, is approximated with straight
lines. We tested both the R-channel and canal models on this domain and
compared the model outputs. yS and hi were held constant at 2500 and
500 m, respectively. AL was set at 100 km2. Based on water budgeting
work by Carter et al. (2013), Qb was between 0.24 and 0.35 m3 s-1.
Values for other constants in this model run can be found in Table 2.
To demonstrate the ability of the canal model to reproduce the timing and
magnitude of actual observed lake-drainage events, we also applied it to
several “real” domains for flow paths draining the lakes in lower Whillans
and Mercer ice streams: see Fig. 4b (for SLW), c (for SLC and SLM), and
d (for SLE). For the real-domain values, we obtained values for ice
thickness, hi, ice base elevation, zb, and pathway width, yS, from
RES-derived measurements of ice thickness and surface elevation made between
1971 and 1999 over multiple campaigns (see
for a description of the interpolation strategy and for
descriptions of data used). For the idealized domain we used measured values of
θ0 and θS at the source lake, seal, and destination lake
and then fit a straight line between them, holding ice thickness, and flow
path width constant.
Summary of experiments
Our experiments began with several tests comparing the simulated lake-volume
time series output by the R-channel model against the volume change time
series inferred from observations for SLW .
After experimenting with the unaltered R-channel model we then experimented
with perturbations to parameters controlling the rate of channel grown
(mRC) and closure (CVRC) following hypotheses put forth in
and (Sect. 4.1.1).
All experiments involving lakes on a real domain were run until the lake
had undergone at least one complete fill–drain cycle and at least 10 years
had elapsed. As the sensitivity studies typically involved a small lake,
they were run for only 8 years. In cases where the channel failed to grow the
model was allowed to run for 10 years. In cases where runaway growth in
outflow rate led to unrealistically large ice surface drawdown, the model was
stopped once the lake level had decreased by over 30 m. In order to
facilitate easier comparison of modeled lake-volume change with that
inferred from Ice, Cloud and land Elevation Satellite (ICESat) observations (which spanned 2003–2009) we referred to
the time of observed volume change in years since 2000 and adjusted the
timing of the first model year to coincide with the first observed filling
cycle on the lake being simulated.
Following the suggestion that the channel closure rate inferred from
lake-volume change observations would be more consistent with the rheology of
soft sediment, we focused subsequent experimentation on the canal model.
Lake-volume change as simulated by the canal model on a realistic domain was
compared against lake-volume change as observed for lakes SLW, SLE, SLC, and
SLM (Sect. 4.1.2). In order to explore the possibility of implementing the
lake-drainage model in areas where ice thickness measurements are more
sparse, we compared the model's output on idealized (straight lines) and
realistic (interpolated at 1 km intervals from RES and satellite altimetry)
flow-path geometries (Sect. 4.1.3). The simplifications inherent in the
idealized domain also proved useful for quantifying better how variations in
the flow-path geometries between the source and destination lakes might
affect the magnitude and recurrence interval of lake drainage (Sect. 4.2.1).
This idealized domain was also employed to explore how the time series output
by the canal model varies in response to perturbations both to the input
data, including inflow and lake area (Sect. 4.2.2), as well as changes to
parameters that are more difficult to measure directly, such as αT,
d15, N∞, R1, and Qonset (Sect. 4.2.3 and 4.2.4).
Discussion
Although our models suggest that an R-channel can theoretically grow and
contract in the Antarctic subglacial environment, supporting previous work by
and , their rates of growth and shutdown
do not match those inferred for most active lakes unless the modeled melt
and closure rates are adjusted significantly. If drainage through a canal
system is the dominant mechanism for “active” lakes, then it would suggest
that such lakes are more likely to be found in areas where soft sediment is
widespread. In such an environment many of the criteria used to locate lakes
with airborne RES might fail. Here we explore in more detail (i) the
validity and implications of the assumptions used in achieving our results;
(ii) what might be inferred about Antarctic subglacial hydrology based on our
results in the context of previous work; and (iii) the prospects for
modeling subglacial lake drainage in ice-sheet models.
Reassessment of model simplifications and assumptions
We did not include a model in which erosion and deformation of both the ice
and bed occurred simultaneously. Given the difference in recurrence intervals
predicted for Siple Coast style lake-drainage events via the canal model
vs. an R-channel, we expect that the evolution of the canal will dominate
short-term flow variability. However, as the canal appeared to remain open
throughout the fill–drain cycle, it is possible that ice deformation,
melting, and freezing would all play a role in longer-term water-flow
evolution. In particular if the basal ice is softer than would be implied by
our assumed KRC of 10-24 Pa-3 s-1 and/or
the flow path follows a substantial adverse bedrock slope and is subject to
high rates of basal freeze-on such as observed in Recovery Glacier
, then deformation of the ice into a canal could
significantly affect water flow and drainage system geometry. This process,
which couples canals to ice deformation, should be considered in future
iterations of our model. Additionally, we have not explored the sensitivity
of the model to the initial values for QRC and QCC. Given that in
most model runs the channel does not ever fall below Qshutdown, it
seems that alterations to this initial value for QRC and QCC would
affect only the timing of the first filling and drainage cycle and thus any
issues can be resolved with careful model spinup.
One of the major challenges to our present model is the difficulty in
measuring many of the properties to which our model is most sensitive. We
have limited observations of grain size and sediment effective pressure
e.g.,, limited channel geometry
estimates from RES e.g.,, and limited sediment
effective pressure estimates from seismic surveys
e.g.,. Even where present, estimates
for these parameters are still not sufficiently accurate to constrain our
model effectively. As a consequence, most of our simplifying assumptions were
made due to data constraints, including those regarding the AL remaining
constant over the filling–draining cycle, the till strength as a function of
N∞ remaining constant over time, and our assumption of a
semi-circular channel geometry for the canal. For example, AL likely
correlates positively with lake volume. This effect alone would only
moderately affect the rate of draw down leading to an increase in surface
lowering relative to what would be expected in a constant AL case. Lower
lake levels, however, would lead to faster rates of channel closure,
partially counteracting this effect; these issues remain to be explored.
If N∞ is also increasing as the lake drains and shrinks, then we may
expect the system eventually to come into equilibrium at low stand until flow
from upstream increases. Other results exploring temporal changes in till
strength (which indirectly relates to N∞), however, suggest that
N∞ is likely to be changing significantly over the duration of our
model run as water is exchanged between the sediment pores and ice–bed
interface . When we consider our assumption that
N∞ remains constant over time in light of these results, we can
speculate on how variations in N∞ over time might affect a subglacial
lake's fill–drain cycle. Based on research investigating the exchange of
water between the interfacial flow system and sediment
e.g., we would expect N∞
to co-vary with NS and NCC. In this situation, sediment strength
would increase as lake level declined, NCC and NS increased, and
water was removed from the surrounding sediment. As a result the channel
might remain open longer than predicted by our model and low stands would
last longer. A more detailed exploration of this process including in situ
monitoring of sediment pressures may be key to improving the performance of
our model.
The sensitivity of our model output to small variations in these parameters
(as well as αT) calls into question our assumption that they remain
constant with time. Ideally our model would fully incorporate a more complex
array of glaciofluvial processes with spatial variation in sediment effective
pressure and cohesion, all of which are related to till water content.
Erosion, rather than being uniform along the channel, would then be
concentrated in specific areas. Additionally, our experiments on R-channels
suggest that melting, refreezing, and deformation of the overlying ice may
change the channel significantly, especially in light of our results where
the channels rarely cease once initiated. For this work, however, the
demonstration that reasonable rates for erosion and sediment deformation can
produce changes in channel geometry sufficient to simulate volume change
inferred from observations acts as a first step.
Comparison against other observations of lakes in Antarctica
Observations show that the highstand of active lakes is lower than the
flotation heights of their dams . In all
of our model runs, lake drainage initiated when θ0 was still well
below θ0 at the seal (Fig. 3). Our model is the first to capture
this important dynamic for Antarctic subglacial lakes. We note that onset of
outflow before lake levels reach the flotation height has been previously
observed and modeled for ice-marginal lakes in alpine glaciers (e.g., Fowler,
1999) and the Grímsvötn caldera lake beneath the Vatnajökull ice cap in
Iceland (Björnsson, 2003; Evatt and Fowler, 2007).
Our results are also consistent with a hypothesis put forth in
that the change in lake level (and thus pressure) may be
better explained if channels were incised into deformable sediments rather
than ice. This mode of channelization has two major implications: (i) the
mechanism that allows active lakes to be inferred from satellite observations
might make them difficult to detect with more traditional RES technology;
conversely (ii) lakes detected by RES might drain via a different mechanism.
If the rates of volume change inferred from satellite observations are better
accommodated by channels incised in sediment, then it is likely that, in
addition to the outlet of such a lake existing in soft sediments, the lake
itself exists in soft sediments. The type of sediments that would both erode
easily and quickly deform closed would also tend to have a low angle of
repose such that any lake formed within them would be shallow. This physical
setting may explain why these lakes are not detectable using RES
, as a shallow lake in the presence of saturated sediment
would fail the specularity test . Additionally, a lake
surrounded by saturated sediments might not be significantly brighter than
its surroundings as the reflection coefficient between ice and saturated
sediments is very close to that between ice and lake water
.
Our model for the R-channel suggests that such a mechanism dominates
Antarctica subglacial lake drainage only in very specific circumstances.
While explored the possibility of Antarctic subglacial
lake drainage via such a mechanism, the recurrence interval they found
(thousands
of years between events vs. a few decades of observations) would make it
possible that we have yet to observe a lake draining via an R-channel.
However, the drainage of Lake Cook E2 , which resulted in a surface drawdown of over 50 m (similar in
vertical scale to observed subglacial lake drainages in Iceland), would be a
promising candidate for drainage via an R-channel. Further study of
lake-drainage events outside of the Siple Coast is necessary to understand
fully the respective roles of channels carved into ice and sediment.
Lakes in an ice-sheet model
A number of recent models for ice flow have started to predict the formation
of subglacial lakes in local hydropotential minima
e.g.,. These
models all assume that these lakes simply fill until the water level reaches
the flotation height of the “static seal”, at which point they drain
steadily through a distributed network at the ice–bed interface. The results
of both our R-channel and canal models indicate that channelized drainage is
likely important for many of these lake systems. In particular, the
fluctuations in volumes inferred for lakes in fast-flowing regions of the ice
sheet are likely the result of channelization. As a consequence, ice flow in
areas where subglacial lakes are common may have very different flow
properties relative to a region where subglacial water flows in steady state
.
Regions of the ice sheet underlain by active subglacial lakes will likely
exhibit more variability in ice flow rate with peak ice speed coinciding with
peak distributed flow and peak lake volume (Fig. 6). In chains of lakes,
the lubrication may also be related to the evolution of the region
hydrological system rather than an individual lake
e.g.,. The exact degree to which lake drainage
accelerates ice flow is, however, highly dependent on longitudinal stresses,
which is outside the scope of this paper.
Spatial variations in basal traction in areas of fast flow have previously
been proposed as a mechanism for forming lakes , with
lakes forming in the lee of local traction highs. More recently inversions of
basal traction have inferred bands of stiff sediment impounding water in
areas of moderate to fast flow, but usually outside the regions where active
lakes are found . It may be that these
categories represent two different mechanisms of water storage, one stable
and the other unstable, resulting from subtle changes in sediment properties.
Along with findings of multiple other mechanisms of water storage beneath the
Antarctic Ice Sheet , these conclusions coupled with our
results indicate that simply modeling the filling of an enclosed
hydropotential depression, while an important first step, is not sufficient
to simulate the full nature and impact of lake dynamics in an ice-sheet
model.