Introduction
Nearly 400 subglacial lakes have been identified in Antarctica based on radar
data and satellite measurements , as well as at least two
subglacial lakes in Greenland, all of which satisfy established criteria for
identification of subglacial lakes . Subglacial lakes are
typically located either in the ice-sheet interior, close to ice divides
where surface slopes are low and ice is thick, or in areas of higher ice
velocity where internal ice deformation and sliding add to geothermal energy
to produce melting at the base .
Subglacial lakes have been hypothesized to play a role in the initiation of
fast flow by (1) providing a steady stream of water downstream of their
location, lubricating the base ;
(2) substantially influencing the thermal regime of the ice, gradually warming
the basal ice from below; or (3) providing ice with enough thermal momentum
to resist freezing on downstream of the lake .
The reason why so few lakes have been discovered in Greenland, despite
considerable efforts and relatively dense surveying compared to Antarctica,
is believed to be because of the generally warmer ice and higher surface
slopes in Greenland, which favor rapid and more efficient drainage as well
as increased vulnerability to drainage instabilities . In
addition, the large supply of surface meltwater to the base of the Greenland
Ice Sheet through hydrofracturing means that the drainage networks in place
are probably already highly efficient and, thus, capable of draining
subglacial water effectively and preventing or limiting subglacial lake
development .
Apart from Lake Vostok, the largest subglacial lake in the world, as well as
a few other ones, the typical subglacial lake is around 10 km in
diameter . Recent studies have presented compelling
evidence of rapid transport of water stored in subglacial lakes, indicating
that lakes can either drain episodically or transiently on relatively short
timescales . Lakes appear to form
a part of a connected hydrological network, with upstream lakes draining into
downstream ones as subglacial water moves down the hydrological potential
. Both ice and water transport a substantial amount of
sediment over time which is deposited in the lakes, thus reducing their size
over long timescales. Sedimentation rates can vary from close to 0 to
several millimeters per year, with sediment layers estimated to be up to
several hundreds of meters thick in some lakes
. The dominant mechanism in
transporting sediment to subglacial lakes is thought to be influx of
sediment-laden water for open-system, or active, lakes, and melt-out from the
overlying ice for closed-system lakes, where water exchange happens purely
through melting and freezing . Subglacial lakes thus
change size on a variety of timescales with different mechanisms, from fast
drainage to slow sedimentation.
The presence of a subglacial lake is often signalled by a flattening of the
ice surface, given that the lake is in hydrostatic equilibrium, and as
a local speed-up of ice velocities. This draws down cold ice and deflects
isochrone layers within the ice .
Numerous studies have investigated the effect of spatially varying basal (or
surface) conditions on ice dynamics and the internal layering of ice, such as
melting or basal resistance. Previously, studied the
effect of areas with basal sliding or melting on internal layer architecture,
but not explicitly a subglacial lake. The effect of a subglacial lake on ice
dynamics was investigated by , who used a 2-D
vertically integrated flow equation to study ice-sheet response to transient
changes in lake geometry and basal resistance, and by who
investigated the stability of a subglacial lake to drainage events. The
interaction between lake circulation and ice dynamics and its effect on the
basal mass balance have also been studied by .
The aim of this study is to investigate the influence of a subglacial lake on
ice dynamics and the effect it has on the isochrone structure within the ice.
As all stress components are important for such an interaction, we use
a fully 3-D thermomechanically coupled full Stokes ice-sheet model,
implemented in the commercial finite element software COMSOL. We employ an
enthalpy-gradient method to account for the softening effect of ice
temperature and water content on ice viscosity and we
show how temporally varying basal conditions can lead to the appearance of
flow bands, or arches and troughs, within the internal layering downstream of
the original flow disturbance.
Model description
Ice flow
Ice is treated as an incompressible fluid with constant density, obeying
conservation laws for mass and momentum:
∇⋅u=0
and
∇⋅σ=-ρg,
where u is the velocity vector, g the gravitational
acceleration, ρ the ice density, and σ the Cauchy stress
tensor. The Cauchy stress tensor is given by
σ=τ-pI,
where τ and pI are the deviatoric and the isotropic
parts, p is pressure, and I is the identity matrix. Inertial
forces are assumed negligible and only body forces arising from gravity are
taken into account.
Ice is assumed to follow Glen's flow law , in
which deviatoric stresses are related to strain rates (ε˙) by
τ=2ηε˙
and
ηT′,W,ε˙e=12At(T′,W)-1nε˙e1-nn,
where η is the effective viscosity, ε˙e is
the effective strain rate, (At) is a rate factor that depends on
the homologous temperature (T′) and the water content of the ice (W),
and n is the power-law exponent in Glen's flow law. The homologous
temperature T′ = T + γp corrects for the dependence of the
pressure-melting point (Tm) on pressure, where γ is the
Clausius–Clapeyron constant. The formulation of the rate factor follows
At(T,p,W)=A(T,p)×(1+1.8125W[%]),
where
A(T,p)=A(T′)=A0e-Q/RT′,
and A0 is a pre-exponential constant, Q is the activation energy, and
R is the universal gas constant, with numerical values listed in
Table .
Enthalpy balance
An enthalpy-gradient method is employed, as opposed to
the typically used cold-ice formulation, which is incapable of reproducing
correctly the rheology of temperate layers within ice sheets. The enthalpy
formulation allows for the possibility of including liquid water content
within temperate ice, based on mixture theory, without explicitly tracking
the cold/temperate transition surface . In
the enthalpy-gradient method, enthalpy replaces temperature as the
thermodynamical state variable, such that
ρ∂H∂t+u⋅∇H=∇⋅Ki(H)∇HifH<HTm(p)k(H,p)∇Tm(p)+K0∇HifH≥HTm(p)+Q,
where H = H(T, W, p) is the temperature and water-content-dependent specific
enthalpy and Q = 4ηε˙e2 is the heat
dissipation due to internal deformation. The conduction term in
Eq. () depends on whether the ice is cold
(H < HTm(p)) or temperate (H ≥ HTm(p)). The
conduction coefficient for cold is ice defined as Ki = k/c, where
k is the thermal conductivity and c is the heat capacity, both assumed to
be constant (Table ). HTm(p) is the
specific enthalpy of the pressure-dependent melting point of ice. The
diffusivity for temperate ice is poorly constrained as little is known about
the transport of microscopic water within temperate ice
. In practice, we use the value
K0 = 10-3 Ki, shown by to be sufficiently
low to suppress transport of water by diffusion through the ice matrix, while
still numerically stable.
Age equation
In addition to the balance equations above, we solve a separate equation for
the ice age (χ) to determine the influence of the lake on the isochrone
structure :
∂χ∂t+u⋅∇χ=1+dχ∇2χ,
where χ is the age of ice and the second term on the right represents
a diffusivity term needed for numerical stability, in which dχ is the
numerical diffusivity.
Values for constants used in the study.
Constants
Values
α
bed inclination
0.3, 0.1∘
ρ
density of ice
910 kg m-3
g
gravitational acceleration
9.81 m s-2
n
flow law exponent
3, 1
γ
Clausius–Clapeyron constant
9.8 × 10-8 K Pa-1
Tm0
melting point at atm. pressure
273.15 K
A0
pre-exp. constant (T ≤ 263.15 K)
3.985 × 10-13 s-1 Pa-3
– (T > 263.15 K)
1.916 × 10-3 s-1 Pa-3
Q
activation energy (T ≤ 263.15 K)
60 kJ mol-1
– (T > 263.15 K)
139 kJ mol-1
R
universal gas constant
8.3145 J (mol K)-1
k
thermal conductivity
2.1 W (m K)-1
c
heat capacity
2009 J (kg K)-1
L
latent heat of fusion
3.35 × 105 J kg-1
dχ
diffusion coefficient
10-13 m2 s-1
C
friction coefficient
1013, 1010 (s m2) kg-1
h
ice thickness
1500, 3000m
as
surface accumulation
0 m s-1
Ts
surface temperature
-30 ∘C
qgeo
geothermal flux
55 mW m-2
ηconst
ice viscosity (constant)
1014 Pa s
Boundary conditions (BCs)
At the surface, stresses arising from atmospheric pressure and wind can be
neglected as they are very small compared to the typical stresses in the ice
sheet , resulting in a traction-free BC:
σ⋅n=0,
where n is the normal vector pointing away from the ice. Accumulation
and ablation (as) at the surface are assumed to be 0, giving
the kinematic surface BC as
∂zs∂t+u∂zs∂x+v∂zs∂y-w=as=0,
where zs is the surface elevation. We employ an inverse
Weertman-type sliding law (Eq. ), where the basal drag (τb)
is expressed as a function of the velocity of the
ice (ub) immediately above the ice/base interface, except
over the lake surface where basal traction is set to 0 (full slip). With
basal sliding exponents (p, q) = (1, 0) appropriate for ice-streaming
conditions, the sliding relationship simplifies to a linear relationship
between basal sliding and basal traction. The ice is assumed to be in
hydrostatic equilibrium everywhere and the basal normal pressure (τn)
taken as the ice overburden pressure. Ice accretion and melt at the base are
assumed to be 0 and along with the stress BC at the surface,
a no-penetration condition is used to close the system:
ub=-C-1|τb|p-1τbτnq=-C-1τb⇒τb=Cub,
and
u⋅n=0,
where C is the sliding coefficient (Table ). Periodic
BCs for the inlet/outlet are used, such that the velocity, pressure, and
specific enthalpy are the same at the upstream and downstream extremes. On
the side boundaries of the domain, symmetry for velocity and thermal
insulation is imposed:
uin=uoutpin=poutHin=Hout,
K=μ∇u+(∇u)Tn,K-(K⋅n)n=0,u⋅n=0
-n⋅∇H=0.
At the surface, a value is set for specific enthalpy (Hs) corresponding
to a surface temperature of Ts = -30 ∘C (Ws = 0,
ps = 0) such that
Hs=Hi+cTs-Tm.
At the base, the geothermal flux (qgeo) is used for cold ice and
a zero flux for temperate ice:
n⋅-Ki∇H-K0∇H=qgeo,ifT<Tm0,ifT=Tm,
where Hi is the specific enthalpy of pure ice at the melting
temperature. To determine correctly the basal BC for the
enthalpy field equation, switching between a Dirichlet and a Neumann
condition is necessary , depending on
basal temperature, water availability at the base, and whether a temperate
layer exists immediately above it. Here we opt for a simpler, more
computationally efficient BC where the geothermal flux is
gradually decreased in the specific enthalpy range corresponding to
[(Tm - 0.2 ∘C) (Tm)], with a smoothed
Heaviside function with continuous derivatives.
For the age–depth relation (Eq. ), periodic BCs
are used for the inlet and outlet. At the surface, the age is set to
χs = 0. Zero normal fluxes are used for the side boundaries and
the lower boundary.
-n⋅(-∇χ)=0
Computational domain
We define the computational domain as a rectangular box 350 km long
and 100 km wide with a fixed bed slope (α). Three different
lakes sizes are used, based on typical sizes of subglacial lakes found in
Antarctica . The lakes are all elliptical in
shape, with major and minor axes defined in Table . The major
axis is aligned with the direction of flow. The model experiments are all
either steady-state solutions or start from steady-state solutions.
Values for lake sizes used in the paper.
Lake size
Major axis
Minor axis
LS
10 km
5 km
LM
20 km
10 km
LL
30 km
15 km
Throughout the domain, extruded triangular (prismatic) elements are used with
a horizontal resolution down to ∼ 500 m at the lake edges. This
relatively high resolution is needed in order to capture the effect of strain
softening of ice around the lake edges, where velocity gradients are large,
and to resolve properly the upper surface. The model uses 15 vertical layers,
which become thinner towards the base, where the thinnest layer has
a thickness of ∼ 35 m.
Model experiments
The aim of the study is to show how the presence of a subglacial lake affects
ice dynamics and thermal resolution and to follow its effects on internal
layering through simple temporally dependent and steady-state experiments.
All transient simulations are started from an initially steady-state
configuration, where equations for mass, momentum, and enthalpy are solved
jointly with a direct solver along with equations for surface or grid
evolution. All presented results are based on simulations with a fully
nonlinear ice viscosity (n = 3) unless otherwise stated explicitly, and all
numerical values for model constants are defined in Table .
The lake itself is modeled as a “slippery spot” .
The lake surface is assumed to be fixed to the bed plane. In
reality, any changes to lake size or volume would lead to vertical movement
of the lake surface, which in itself would induce an expression at the
surface of the ice. found that vertical movement of lake
surfaces due to drainage resulted only in small changes to ice velocity and
that the ice surface reached its initial stage after just a few years,
indicating that it would not lead to a significant disturbance in isochrone
structure compared to changes in the horizontal lake extent. Here we consider
only planar changes in lake geometry or changes in lake size fixed to the bed plane.
Two-dimensional surface plots of horizontal velocity (a) in
m a-1 and relative surface elevation (b) in meters. The
outline of the lake (LM lake size, n = 3) and streamlines are shown
in black and flow is from left to right.
Water draining from subglacial lakes is likely to have an impact on ice
dynamics and isochrone structures downstream of the lake, either through
melting of ice or changes in basal water pressure and sliding, although no
representation of subglacial hydrology is included here. The effect the
draining water will have on isochrone layers will depend on the state of the
drainage system and how the water is transported downstream
but will likely be limited by the swiftness of the ice
response compared to the time needed to perturb isochrone layers considerably.
The lack of basal friction over the lake results in increased velocities, not
just over the lake but also a considerable distance both upstream and
downstream of the lake that depends on ice thickness, basal traction
, and the size of the lake. This distance is
around 15 km for the LS lake size, 50 km for LM, and
100 km for the LL lake size, defined as the distance it takes for
the horizontal velocity to drop below 1.05 times the background velocity.
Figure a shows the surface velocity with black lines
indicating streamlines and Fig. b shows the relative change
in surface elevation caused by the presence of the lake, where the general
slope has been subtracted from the surface. The surface responds to the
change in basal conditions by becoming flatter and with more than a doubling
of horizontal velocities over the lake. Streamlines contract and ice is
brought in towards the lake from the sides.
Figure shows different model output in vertical cross
sections, through the center of the lake, in the direction of flow for the
LM lake size. Over the lake, the ice basically moves as an ice
shelf with more or less uniform horizontal velocity throughout the ice column
(Fig. a). At the upstream end of the lake, a strong
downward movement of ice causes cold ice from the upper layers to be drawn
down towards the bottom, steepening the temperature gradient close to the
base. Conversely, at the downstream end a strong upward flow restores
internal layers to their prior depths (Fig. b).
Along flow profiles, through the center of the lake (LM lake
size, n = 3). (a) shows horizontal velocity in m a-1.
(b) shows the homologous temperature in ∘C and water content
in percentage. The black lines represent streamlines. The inset figure shows
a close-up of the temperate layers at the lake edges (same color scales). In
(c) we see the logarithm of the internal deformation energy
W m-3 while (d) shows the logarithm of the viscosity [Pa s].
The temperature and microscopic water content within the ice are shown in
Fig. b. Black lines represent streamlines which
coincide, or line up, with isochrone layers for steady-state simulations
.
The intense internal deformation close to the borders of the lake, where ice
velocities change significantly over short distances, gives rise to a thin
temperate layer of ice both at the upstream and downstream ends of the lake
(Fig. b). The vertical resolution is somewhat limited in
the model but simulations with higher resolution do indicate that a temperate
layer should form for the given velocity field, although its thickness might
be slightly overestimated. This is strongly affected by the chosen sliding
law and sliding coefficients and should not be considered representative of
subglacial lakes in general.
Figure c shows the internal deformation energy, with
high values near the base where velocity gradients are high. Maximum values
are reached near the edges of the lake, in the transition zone between low
and full sliding. Increases in temperature, pressure, and water content, and
larger effective stress, all have the effect of decreasing the viscosity,
which is why the lowest viscosity values are obtained at the base of the ice
sheet, in particular around the edges of the lake
(Fig. d). Generally low viscosity is furthermore
obtained throughout the ice column over the lake boundary, extending all the
way up to the surface, effectively creating a vertical zone of softer ice.
Figure a shows profiles of horizontal surface velocity
for the three different lake sizes considered. As expected, velocity peaks
over the lake surfaces and two fringe peaks at the lake edges are discernible
for the smallest lake size as well.
Surface velocity (a) and elevation (b) along flow
(n = 3), through the center of the lake for various lake sizes. Velocity and
surface profiles for the different lake sizes (L, M, S) are marked with
different colors and the horizontal extent of each lake is marked with
a horizontal bar at the bottom. Note the different horizontal scale for the
two figures.
A characteristic surface dip feature is observed at the upstream end of many
lakes in Antarctica, for instance the Recovery Lakes
and Lake Vostok as well as
a small hump at the downstream end. Figure b shows
surface profiles for the three different lake sizes, each with the
characteristic flattening of the surface as well as the dip and ridge
features on each side of the lake.
Figure a shows a comparison of surface profiles for the
LM lake size, made with a nonlinear viscosity (n = 3 in Glen's
flow law as used in our other experiments) versus a constant viscosity
(ηconst = 1014 Pa s) and a pressure- and
temperature-dependent viscosity (n = 1) but otherwise using the same setup.
The hummocky feature is noticeably absent from simulations with a constant
viscosity. Figure b and c show profiles of horizontal
velocity and viscosity over the center of the lake for the three different
viscosity cases. The horizontal velocity has been scaled with the horizontal
velocity at the surface (the uppermost point of the vertical profile). For
both the n = 1 and n = 3 cases, the horizontal velocity at depth is larger
than at the surface.
A weaker contrast in basal traction inside and outside the lake, or a less
pronounced switch in flow mode such as one might expect beneath an ice stream
, decreases the amplitude of the dip and
ridge features as can be witnessed in Fig. a, where
surface profiles for simulations with different sliding coefficients and ice
thickness are compared (n = 3). Isochrones relative to the ice base are shown
in Fig. b for two of the simulations presented in
Fig. – n = 3 (blue line) and η = 1014 (red
line) – in addition to isochrones for a simulation with a lower contrast in basal
traction and a lower bed inclination than the other two (cyan line). The
structure of internal layers depends heavily both on the contrast in basal
traction and the viscosity formulation used. Low contrast results in small
deflections of internal layers whereas a fixed viscosity (compared to flow
with nonlinear viscosity and a similar surface velocity) results in larger
deflections of internal layers due to the generally smaller ice flux at depth
outside the lake compared to over it.
(a) Profiles of surface elevation in meters for the
LM lake size and three different viscosity cases; fixed viscosity
(ηconst = 1014 Pa s, red), with flow exponent n = 1
(green), and n = 3 (blue) in Glen's flow law (Eq. ). Vertical
profiles of (b) the scaled horizontal velocity and (c) the
logarithm of ice viscosity, over the center of the lake. The horizontal
velocity has been scaled with the velocity magnitude at the surface.
(a) Profiles of relative surface elevation in meters for the
LM lake size for different values of the sliding coefficient and
ice thickness. The blue line is the same surface profile as the blue line in
Fig. a only displaced vertically for comparison. Note the
different vertical scale compared to the previous figure. (b) Isochrone
layers for the n = 3 viscosity case with different values of the
sliding coefficient (cyan and blue lines) and for the fixed viscosity case
(red line). The lake size is LM and the isochrones are presented
relative to the base.
Lakes drain and fill on different timescales. Several studies have
documented relatively rapid drainage events for subglacial lakes in
Antarctica . Typically, drainage
occurs over the course of several years but refilling takes much longer. To
simulate such an event and what effect it could have on the internal
structure of the ice, we set up a model run where the lake diameter shrinks
during a 10-year period from the maximum (LL) to the
smallest lake size used in the paper (LS).
Figure shows four different time slices of horizontal
velocity, with black lines indicating isochrone layers. As the velocity field
adjusts in time to the new BCs, a traveling wave is created
at depth within the isochrone structure that transfers downstream with the
flow of ice.
Four different time slices of the velocity and isochrone adjustment
to a drainage event. The color scale represents horizontal velocity
m a-1 and gray to black lines indicate isochrones. (a) represents
the initial stage (t = 0), (b) 100 years later
(t = 100 a), (c) t = 500 a, and (d) t = 2000 a. The white
lines in the last time slice are the initial isochrone layers from t = 0 and
the red arrow points to the isochrone disturbance that advects downstream
with the flow of ice.
Discussion
The frictionless boundary levels the surface over the lake, changing surface
gradients and causing the ice to speed up in the vicinity of the lake. The
increase in velocity is further amplified by the effect of velocity gradients
on ice viscosity. Outside the lake, the velocity field is affected primarily
by changes in surface gradients and longitudinal stresses and, to a lesser
degree, by changes in ice viscosity and basal velocities. Over the lake, the
ice moves more like an ice shelf, with almost uniform horizontal velocity
throughout the ice column. The increase in velocity
(Fig. a) is due to the lack of basal friction over the
lake, causing the highest velocities to appear there, but secondary velocity
peaks (Fig. ) can also be discerned at the lake edges,
which result from the interaction of surface evolution and ice dynamics. The
velocity increase for the two fringe peaks propagates from the surface and
downward whereas the velocity peak over the lake is mostly caused by
acceleration in the basal layers of the ice sheet.
Thermal regime
For the given model setup, the geothermal flux is sufficient to ensure that
the basal temperature reaches the pressure-melting point everywhere in the
model. Internal deformation then adds to the available thermal energy with
the potential to form a temperate layer with non-zero microscopic water
content in the vicinity of the lake as seen in Fig. b,
where a thin temperate layer has formed (∼ 70 m, 2 vertical
cells). Lakes in ice-streaming areas, where a considerable portion of the
surface velocity arises due to sliding at the base, can be expected to have a
much gentler transition in flow mode at the lake edges and much less
deformational energy available. No temperate layer forms for simulations
where the transition in sliding is considerably less sharp and more
appropriate for streaming areas as in the cases with C = 1010
(Fig. ).
The steeper temperature gradient over the lake efficiently leads away excess
heat created by internal deformation at the upstream end, refreezing whatever
microscopic water that was created upstream. Large quantities of microscopic
water, in reality, would drain to the base (≳ 3 %), although
drainage is not included here as water content is relatively low. Two factors
would contribute to facilitating freeze-on at the interface between ice and
lake water, if it were included in the model. Firstly, draw-down of cold ice
from above increases the temperature contrast in the lower part of the ice
sheet; secondly, heat from internal deformation ceases with the removal of
basal traction over the lake surface. Both result in a steepening of the
temperature gradient close to the base, which removes heat more efficiently
from the base and shifts the balance in favor of freeze-on.
As lake size increases so too do horizontal velocities over the lake
(Fig. ), as well as the effective strain rates and the
available deformational energy for internal heating. Accretion rates at the
ice/lake interface would be limited by the amount of latent thermal energy
that the ice above it can lead away efficiently, and any temperate layer
formed by internal deformation at the lake boundary would, during its
existence, block heat flow and thus accretion, until it completely refreezes.
Transition zone
The softening effect of local increases in effective stress at the lake edges
effectively creates a vertical layer of soft ice in between higher viscosity
ice (Fig. c). Strong vertical flow at the edges of the
lake results from the localized lack of basal traction. A clear difference
can be seen between simulations with constant viscosity and viscosity that
depends on pressure and temperature (Fig. ). The
softening effect of increasing temperature and pressure with depth not only
causes velocity changes in the vertical to be concentrated in the lower
layers of an ice sheet but also means that for areas with varying basal
traction, such as subglacial lakes, the ice at depth will support less
lateral shear and lower longitudinal stresses compared to the upper layers
where viscosity is higher and the ice stiffer. This in return means that as
the ice encounters a slippery spot or a spot with a sharp decrease in basal
traction, such as a subglacial lake, that the force balance will be different
than in the isotropic case, where viscosity is constant. The imbalance in
mass flux at depth must be compensated by a more localized increase in
vertical flow and a subsequent drop in surface elevation at the upstream side
and an upwelling at the downstream side of the lake due to the limited
vertical extent of a typical ice sheet.
For large subglacial lakes, where the basal traction is 0, the horizontal
velocity at depth is predicted to be slightly larger (here about 0.1 %
for n = 3, Fig. b) than at the surface. Lakes, such as
Subglacial Lake Vostok, should therefore experience extrusion flow at the
base, where the basal horizontal velocity exceeds that at the surface.
Extrusion flow is not, however, a requirement for the formation of these dip
and ridge features. To form them it is sufficient to have a sharp transition
in sliding along with a negative downward gradient in ice viscosity. Our
model further predicts that dips and ridges can form in situations where
there is merely a strong decrease, not necessarily a complete disappearance,
in basal traction.
Drainage experiment
Subglacial-lake drainage cycles (including both draining and filling) can
have frequencies on decadal to centennial timescales or potentially even
larger . Although the drainage occurs over
10 years here, it can be seen as instantaneous given the generally
slow flow of ice. The velocity field adjusts rather rapidly (∼ 100 a)
to the new basal BCs relative to the time it takes for the
isochrones to respond, as the flow of ice is relatively slow. For episodic
drainage cycles, the strength of the response will depend partly on how long
it takes for the lake to refill. After roughly 2000 years, the wave
has moved far enough downstream to be more or less separated from its initial
location (Fig. d). As both the upstream and downstream
lake boundaries move during the drainage event, a two-troughed wave is
created. In draining lakes where one end is much deeper than the other
a single wave would be expected, as only the shallower end is likely to move
significantly. The velocity of the moving boundary also affects the amplitude
of the resulting isochrone disturbance , where a slip
boundary, moving with the ice, is capable of distorting isochrone layers to
a much greater extent than stationary slip boundaries. For maximum effect,
the boundary should be moving at a velocity comparable to the averaged ice
column velocity. Only slip boundaries moving with the ice are capable of
distorting layers to a greater extent than stationary boundaries. Boundaries
moving in opposite directions to ice flow, like our downstream lake boundary,
will have a smaller impact on internal layers than a stationary boundary as
the relative horizontal velocity increases.
Water in very active subglacial systems, such as recently discovered in West
Antarctica , has relatively short residence times
and fast circulation, contrary to previous beliefs. The impact of such short
drainage cycles, where both drainage and refilling happen on decadal timescales, is unlikely to have a strong effect on the internal isochrone layers
as it takes a long time for the ice to respond. Regular drainage events in
subglacial lakes that have a much shorter cycle than the time it takes for
a particle of ice to be brought up by vertical flow at the edge of the lake
will therefore probably not be easily detectable in the isochrone structure.
If the frequency of drainage cycles is high, the ice will have little time to
respond and the amplitude of the resulting wave will be small. In contrast, drainage of large lakes in areas with low basal melt rates and
consequently long filling times could be expected to generate traveling
waves, downstream of the lake, with a sufficiently large amplitude to be
detectable within downstream isochrones. The amplitude of the traveling wave
would be expected to be similar in magnitude as the steady-state isochrone
disturbance over the lake itself (Fig. ). In our
simulated scenario of Fig. the traveling-wave
amplitudes of ∼ 100 m would be well within the bounds of
detectability with modern radar systems. Drainage of subglacial lakes where
the transition in flow mode is less abrupt would result in smaller amplitudes
(Fig. b). As layer stratigraphy is often quite complex,
a numerical model of ice age and velocities would be needed to separate the
effect of temporally changing lake size, or basal conditions, from layer
deflections caused by varying basal topography or rheology.
For our particular setup, the isochrone disturbance, or the traveling wave,
should eventually overturn and create a fold as the stress situation
downstream of the lake is essentially one of simple shear without any
longitudinal extension . Resolving this in
the model would require an equally fine resolution downstream of the lake, as
over the lake itself, which is not done here. For a subglacial lake situated
at the onset of streaming flow, a fold might not be expected though, as
overturning would be counteracted by longitudinal extension and vertical
compression, which would tend to flatten all layer disturbances. In general,
both horizontal shear and longitudinal extension can be assumed to be present
and, thus, whether a layer disturbance develops into a fold or flattens out,
eventually to disappear, will be decided by the balance between the two .