TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-10-1883-2016Persistence and variability of ice-stream grounding lines on retrograde bed slopesRobelAlexander A.robel@caltech.eduhttps://orcid.org/0000-0003-4520-0105SchoofChristianTzipermanEliDepartment of Earth and Planetary Sciences,
Harvard University, Cambridge, Massachusetts, USADepartment of Earth and Ocean Sciences, University
of British Columbia, Vancouver, British Columbia, CanadaDivision of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USADepartment of the Geophysical Sciences, University of Chicago, Chicago, Illinois, USAAlexander A. Robel (robel@caltech.edu)25August20161041883189622January201614March20168August20169August2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/10/1883/2016/tc-10-1883-2016.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/10/1883/2016/tc-10-1883-2016.pdf
In many ice streams, basal resistance varies in space and time due to the
dynamically evolving properties of subglacial till. These variations can
cause internally generated oscillations in ice-stream flow. However, the
potential for such variations in basal properties is not considered by
conventional theories of grounding-line stability on retrograde bed slopes,
which assume that bed properties are static in time. Using a flow-line model,
we show how internally generated, transient variations in ice-stream state
interact with retrograde bed slopes. In contrast to predictions from the
theory of the marine ice-sheet instability, our simulated grounding line is
able to persist and reverse direction of migration on a retrograde bed when
undergoing oscillations in the grounding-line position. In turn, the presence
of a retrograde bed may also suppress or reduce the amplitude of internal
oscillations in ice-stream state. We explore the physical mechanisms
responsible for these behaviors and discuss the implications for observed
grounding-line migration in West Antarctica.
Introduction
Theoretical and numerical studies have shown that under certain conditions,
the flux of ice through the grounding line (where ice transitions from
grounded to floating) increases sensitively with bedrock depth
.
Thus, when the grounding line rests on a retrograde
slope (sloping upwards in the direction of flow), any retreat in
grounding-line position leads to greater flux and consequently additional
retreat, and any advance in grounding-line position leads to decreased flux
and additional advance. By preventing a grounding line on a retrograde or
very shallow prograde bedrock slope from achieving a stable steady state
position, this positive flux feedback causes the “marine ice-sheet
instability” .
Ice streams are regions of fast-flowing ice that account for over 90 % of
mass transport from the interior of the Antarctic Ice Sheet to the ocean
. Observations indicate that the flow of some ice
streams exhibits unforced variability on centennial to millennial timescales
that has a significant impact on the mass balance of the Antarctic Ice Sheet
. Other observed changes in the flow
velocity of different West Antarctic ice streams have been attributed to
ocean- and atmosphere-forced melting of ice shelves
e.g.,, which buttress
ice sheets through lateral contact with bedrock and localized basal contact
with bathymetric high points
.
Such changes have raised the possibility that forced ice-stream variability
near topographic transitions to overdeepenings with sections of retrograde
slope may lead to rapid retreat
e.g.,. Studies
have also begun considering additional or alternate physical processes that
play a role in determining the location and stability of grounding lines in
marine ice sheets, including gravitationally driven changes in local sea
level , variations in
ice-stream trough width
,
tidal compaction of till and
sedimentation . Some of these processes have
been shown to potentially stabilize grounding lines on retrograde slopes.
When interpreting observations of grounding-line migration, a critical
question is whether retreat on a retrograde slope necessarily implies
continued irreversible retreat in the future due to the marine ice-sheet
instability. In this study, we explore how unforced, internal ice-stream
variability interacts with forced variability and the consequent limitations
of interpreting observed grounding-line retreat. Previous work has studied
how stable and unstable steady state configurations for marine ice sheets
with constant bed properties depend on forcing parameters such as
accumulation rates , showing that
irreversible transitions across overdeepenings happen at critical values of
these forcing parameters. Similarly, for ice streams on purely
downward-sloping beds but with thermomechanically evolving bed properties,
other work has shown that steady ice streams can undergo bifurcations into
oscillatory behavior when the same forcing parameters are changed
e.g.,. Here we combine
the two approaches and ask how changes in accumulation rate and other
parameters change both the stable steady states and the stably oscillating
configurations of an ice-stream model with evolving bed properties and an
overdeepening. A much richer set of transitions (or bifurcations) is
possible: instead of simple jumping between steady states on either side of the
overdeepening, these steady states can also transition into oscillatory
states, and these oscillatory states can potentially encroach into the
overdeepening or transition to steady states on the other side of the
overdeepening.
This study uses a flow-line model that incorporates dynamically varying till
properties, including temperature and water content. The ice stream has
constant width, geothermal heat flux and surface temperature in both space
and time. We do not include buttressing in our simulations in order to focus
our analysis on time-dependent variations in bed properties and exclude the
potential influence of an ice shelf on grounding-line stability as
discussed extensively in previous studies, such
as,
though we do discuss the importance of buttressing further in Sect. 5. We
examine the processes which play a role in unforced, coupled oscillations in
bed properties, ice-stream flow and grounding-line position on and near
retrograde slopes. We come to two main conclusions. First, allowing bed
properties to dynamically evolve radically changes model predictions of
grounding-line position and variability. Second, ice streams, like those in
the Siple Coast region of West Antarctica, can exhibit behavior that is
unexplained by existing theories of the marine ice-sheet instability and
ice-stream variability. This includes persistence of the grounding line on a
retrograde slope for centuries in the absence of buttressing and a reversal
of migration direction on the retrograde slope. Other notable examples of the
effect of retrograde slopes include the reduction or suppression of unforced
oscillations in the state of an unbuttressed ice stream in ways that are not
predicted by theories of ice-stream variability, which do not consider complex
topography. We conclude that it is important to consider these diverse
dynamical responses to forcing and bed topography when evaluating whether
observed grounding-line retreat onto a retrograde slope is irreversible
and in predicting
future ice-sheet change.
In Sect. 2, we describe the ice-stream flow-line model, idealized bed
configuration and underlying mechanism of unforced ice-stream variability. In
Sect. 3, we analyze the dependence of ice-stream grounding-line variability
on accumulation rate and initial position for a set bed topography. We also
compare the behavior of the ice stream when bed properties are not permitted
to vary and when they vary freely. In Sect. 4, we systematically vary the
idealized topographic configuration and discuss the physical mechanisms which
cause modification of ice-stream variability without a retrograde section.
Specifically, we analyze how the position, length and slope of the retrograde
bed section lead to different types of ice-stream behavior. In Sect. 5, we
discuss the relevance and implications of these findings for more complex
models and ice-stream observations.
Model preliminaries
Active ice streams are strongly resisted by lateral shear stresses resulting
from contact with topography or slow-flowing ice ridges
. Additionally, many ice streams are underlain by
till, which behaves like a Coulomb plastic material with yield stress
depending on water content . In this study, we
employ a shallow ice-stream flow-line model which includes lateral shear
stresses and dynamically evolving subglacial till properties. This model
solves for ice thickness and till water content along a central flow line
(x), with horizontal velocity, vertical velocity and ice temperature also
resolved in the vertical (z). The horizontal force balance is
∂∂x2hA‾-1n∂ub∂x1n-1∂ub∂x=ρigh∂h∂x+τb+Gshub1n-1ub,
incorporating integrated lateral shear stress and basal shear stress from an
undrained Coulomb plastic bed that evolves as meltwater is produced and
refreezes at the ice–till interface. A‾ is the depth-averaged
Nye–Glen Law coefficient which is a function of ice temperature, n is the
Nye–Glen Law exponent, ρi is the density of glacier ice, g is
the acceleration due to gravity and Gs∝W-1 is a
parameter capturing the importance of lateral shear stress where W is the
ice-stream half-width. At the grounding line, longitudinal stress is balanced
by water pressure :
2A‾-1nh∂ub∂x1n-1∂ub∂xx=xg=12ρig1-ρiρwh(xg)2.
The upstream boundary is defined to be the ice divide and, correspondingly,
velocity is set to zero there: ub(x=0)=0.
Vertical shear of horizontal velocity is a function of driving stress
u(z)=ub+2A‾n+1τdnh1-1-z-bhn+1,
and it is added to the basal velocity calculated from
Eq. () to form the full velocity field u(x,z).
Vertical velocity is determined by integrating the mass continuity equation
upwards from the bed at constant x=x0:
w(x,z,t)=-∫bb+h∂u∂xx=x0dz,
subject to the condition w+u∂b∂x=0 at z=b,
where basal melt rate is neglected as it is typically less than an order of
magnitude smaller than vertical velocity in model simulations.
The evolution of ice thickness follows a mass conservation equation with a
source term due to accumulation:
∂h∂t+∂∂xu‾h=ac,
where u‾ is the depth-averaged horizontal velocity and
ac is the accumulation rate. At the grounding line, ice is at the
flotation thickness:
ρih(xg)=ρwb(xg),
where ρw is
the density of seawater. Ice flux through the grounding line (qg=ub(xg)h(xg)) is removed from the grounded
ice-stream system. When the bed slope is locally nonzero at the grounding
line, accumulation and local imbalances of advective ice flux cause changes
in ice thickness which require migration of the grounding line to maintain
this flotation condition.
Basal heat budget determines the basal melt rate:
m=1ρiLfG+τbub+ki∂T∂zz=b,
where, on the right-hand side, the first term is the geothermal heat flux,
the second term is the frictional heat flux and the third term is the
vertical conductive heat flux at the bed. ki is the thermal
conductivity of glacial ice and Lf is the latent heat of fusion.
Till water content (w=eZs, where e is the till void ratio and
Zs is the thickness of unfrozen subglacial till) changes as basal
heating produces or freezes meltwater:
∂eZs∂t=m.
Till behaves as a Coulomb plastic material, and so basal shear stress is
calculated from the basal velocity and void ratio:
τb=τcubub2+ϵu2,
where ϵu is the velocity scale over which till transitions from a
quasi-linear to Coulomb friction law. The critical failure strength of the
till follows the empirical form of :
τc=τ0exp[-b(e-ec)],
where τ0 and b are empirical parameters.
Ice temperature is modeled by an advection–diffusion equation in the x–z
plane
∂T∂t+∂∂x(uT)+∂∂z(wT)=κ∂T∂x2+∂T∂z2,
where κ is the thermal diffusivity of glacial ice. We neglect the strain
heating within the ice that is negligible at the central ice-stream
flow line, though it may be important under specific circumstances within
shear margins (see discussion in Sect. 5). At the ice surface, the
temperature is equal to a prescribed atmospheric temperature: T(z=b+h)=Ts.
The model accurately simulates transient migrations in grounding-line
position and activation waves, propagating fronts associated with the
transition from a strong, non-deforming bed to a weak, deforming bed during
ice-stream activation . The
simulations in this study are run at very high horizontal spatial resolution
in the grounding zone (∼ 100 m) and high resolution elsewhere
(∼ 1 km). At these horizontal resolutions, the range of simulated
grounding-line migration is converged and so small changes in the mesh size
do not significantly change the results discussed. The vertical resolution is
sufficient (∼ 10 m) to resolve the vertical temperature gradient of
basal ice. Additional details of the flow-line model and the numerical
approach used in the following simulations can be found in
.
In certain parameter regimes, the ice-stream grounding line migrates as a
part of internal thermal oscillations in the absence of a retrograde section
on the bed
.
These thermal oscillations (also known as “binge–purge” oscillations) are
due to a similar physical mechanism as thermal surging in mountain glaciers,
described elsewhere
e.g.,.
A typical oscillation proceeds as follows: when the ice stream activates,
till is weak and ice-stream horizontal velocity is high. As ice is advected
from upstream, the grounding line thickens and rapidly advances by
100–150 km to its most seaward position. Ice in the active ice-stream trunk
thins due to an “overshoot” of high advective ice flux which exceeds
accumulation, leading to increased vertical heat conduction, freezing basal
meltwater and strengthening the bed. During the second part of the active
phase, ice-stream velocity begins to decrease and the grounding line
retreats. Eventually, till becomes sufficiently strong that the combined
basal shear strength and lateral shear stress exceeds the driving stress and
the ice stream stagnates. During the stagnant phase, the grounding line
continues to retreat slowly to its minimum grounding-line position. The ice
stream slowly thickens upstream, which reduces the advection of cold ice to
the bed and warms basal ice through conduction. Basal meltwater is produced
and eventually weakens the bed sufficiently that activation occurs. The
dynamics of this process are discussed in further detail in
.
Schematic of bed configuration for model experiments. Solid black
line indicates ice-sheet profile. Brown shaded region is bedrock. Blue shaded
region is seawater. x0 is the horizontal position where the retrograde
section begins. L is the length of the retrograde section. Δxg indicates the range over which the grounding line migrates during
thermal oscillations.
To explore how internal ice-stream variability interacts with retrograde bed
topography typically associated with overdeepenings and sediment wedges, we
add a section of retrograde slope to prograde topography (schematically
illustrated in Fig. ). The resulting bed topography is
given by
b(x)=b0-bxx+br(x),
where b0 is the bed elevation at the ice divide and bx=5×10-4 is the background prograde slope. The added topography,
br(x), has a section of linear retrograde slope, which then
exponentially relaxes back to the prograde profile:
br(x)=0ifx<x0bxr(x-x0)ifx0≤x≤x0+LLbxrexp-2(x-x0-L)L2x>x0+L,
where x0 is the position of the beginning of the retrograde section, L
is the length of the retrograde section and bxr is the slope of
the retrograde section. We choose this simplified bed topography so that we
can systematically vary x0, L and bxr over a wide range of
reasonable bed configurations (as indicated in Table 2).
Parameters used in this study (unless varied as indicated in
Table 2). Other parameters used in the model, but not discussed in the text,
are listed in .
ParameterDescriptionValueb0Ice divide bed height (m)100bxPrograde bed slope5×10-4gAcceleration due to gravity9.81(m s-2)GGeothermal heat flux (W m-2)0.07GsLateral shear stress parameter400(kg s-4/3 m-7/3)nNye–Glen law exponent3TsIce surface temperature (∘C)-15Z0Maximum available till4thickness (m)ρiIce density (kg m-3)917ρwSeawater density (kg m-3)1028Spatiotemporal variations in bed properties change grounding-line behavior
The thermally induced oscillations described in Sect. rely
on till water content and yield strength evolving in response to net basal
heat flux. By contrast, the marine ice-sheet instability paradigm of steady
state grounding lines jumping across overdeepenings
is based on bed properties
(here, the basal yield stress) that remain constant in time, as is frequently
assumed in ice-stream simulations. With the bed configuration defined in
Eqs. () and (), we can reproduce the same
behavior if we set the basal yield stress to a constant value, chosen to be
zero for simplicity. We systematically vary the accumulation rate
ac as a forcing parameter and run the model to steady state for
each value, with the fixed basal yield stress ensuring that no oscillations
occur. Figure 2a shows the resulting steady state grounding-line position on
the horizontal axis as a function of accumulation rate plotted on the
vertical axis. The bed geometry parameters used here are x0=800 km and L=80 km, with bxr=2×10-4. By starting the
simulations with initial grounding-line positions on either side of the
retrograde section, we are able to reproduce the expected hysteresis,
analogous to, for example, Fig. 9a of .
For a range of accumulation rates 0.35≲ac≲ 0.6 m yr-1, multiple stable steady state solutions are possible and
critical values of ac exist at which the large and small ice-sheet
solution branches disappear when the grounding line moves onto the retrograde
slope. In this region, grounding-line advance leads to further advance due to
increased accumulation and decreasing mass loss on the shallowing bed, while
grounding-line retreat leads to further retreat due to decreased accumulation
and increased mass loss on the deepening bed.
Prescribed parameters and their variation in different simulations
listed by figure. Dashed entries indicates no retrograde section. A “yes”
in the vary bed column indicates simulations in which bed properties are
permitted to vary as a function of till water content.
Stable equilibrium grounding-line positions and limit cycle
grounding-line oscillation amplitudes as function of accumulation rate
ac. Solid lines indicate stable equilibrium grounding-line
positions. Shading bounded by dashed lines indicates range over which
grounding line oscillates. (a) Bed properties are static with
τc=0 kPa everywhere (bxr=2×10-4). (b) Bed properties dynamically evolve (bxr=2×10-4). (c) Bed properties dynamically evolve
(bxr=10-3). (d) Bed properties dynamically evolve
with no retrograde section. In panels (a)–(c)x0=800 km and L=80 km and location of retrograde section is indicated by vertical dashed lines. “x” marks indicate location of steady states for simulations
plotted in Fig. .
There are some subtle differences between our model and that used in
or the MISMIP hysteresis experiments in
; instead of a Weertman-type basal friction law
τb=C|u|m-1u, the solutions in Fig. a have
τb=0 and the dominant drag term is the lateral drag
Gsh|ub|1n-1ubsee
also. However, as
demonstrates, the relationship between flux and ice thickness at the
grounding line that underpins the hysteresis in
and carries over to
the case of lateral drag dominated flow with a simple change in coefficients
and exponents.
In Fig. b–d, till water content and basal yield stress are
allowed to dynamically evolve (with other prescribed constants as in Tables 1
and 2). Again, the model is run forward for many different values of
accumulation rate ac until, for each ac, the solution
settles into either a limit cycle or steady state. Note that aperiodic
oscillations of the type identified in
do not appear in our flow-line solutions, likely due to the lack of
interactions with other ice streams. For many values of the accumulation
rate, we see limit cycle solutions, and the range of grounding-line migration
for each of these limit cycles is indicated by the shaded regions in
Fig. b–d. Stable steady states are indicated by solid
lines.
The abrupt transitions from steady states on one side of the retrograde
section to the other (seen in Fig. a) no longer appear when
bed properties are allowed to evolve for the same bed topography
(Fig. b). Instead, we observe that, for small accumulation
rates ac, we have only limit cycles whose amplitude increases with
ac, but no steady states. For small enough ac, the
grounding line remains on the landward side of the retrograde section. For
larger accumulation rates we see the limit cycles eventually encroach on the
retrograde section, and eventually the grounding line migrates all the way
across the retrograde section during a single limit cycle. At a critical
value of ac≈0.37 m yr-1, the limit cycle abruptly
disappears and is replaced by a stable steady grounding line on the far side
of the retrograde section. A further increase in accumulation rate then
destabilizes the steady state and leads to renewed oscillations that see the
grounding line move between the retrograde section and the seaward side, with
the range of oscillation moving gradually away from the retrograde section
for even larger ac. Note that there are two instances of hysteresis
here, too. In this case, however, the hysteresis reflects the possibility of
the ice sheet settling either into a steady state or into a limit cycle for a
small range of values of ac between 0.317 and
0.323 m yr-1 and 0.37 and 0.42 m yr-1. That is
fundamentally different from the hysteresis in Fig. a, with
the possibility of the ice sheet settling into two or more different steady
states.
Using our simple approach, we cannot properly characterize the bifurcations
that mark the appearance or disappearance of a steady state, and our method
may also have difficulty capturing solutions close to the bifurcations.
For the smoother bed and slightly different physics in
, the bifurcations that mark the appearance or
disappearance of the two steady state solution branches in Fig. 2a can be
identified as saddle-node bifurcations. The much simpler box model for
thermally induced oscillations in would
identify the hysteresis associated with the disappearance of the steady state
in favor of a limit cycle as a combination of a subcritical Hopf bifurcation
and a saddle-node bifurcation of limit cycles. Whether those bifurcations
remain relevant to the present spatially extended (that is, high-dimensional)
dynamical system is unclear.
In Fig. c, we recompute steady states and limit cycle
solutions for the same parameter values as in Fig. b but
with the retrograde section slope steepened to bxr=10-3.
Here we observe a much clearer separation between steady states and limit
cycles that remain on one side or the other of the retrograde section. For
small ac, we have only oscillatory solutions in which the grounding
line remains on the landward side of the retrograde section, and for large
ac we have oscillatory solutions where the grounding line remains
on the seaward side. The solution structure is now considerably more
complicated, however, with stable equilibria existing on the seaward side of
the retrograde section for a larger range of ac and yet more
hysteresis permitting the coexistence of that steady state with either one or
two limit cycle solutions. Notably, the limit cycle on the landward side for
small ac never encroaches on the retrograde section, but there is a
limit cycle centered on the seaward side for 0.12≲ac≲ 0.2 m yr-1, during which the grounding line periodically
retreats into the retrograde section, only to readvance out of it again.
Figure b–c should also be contrasted with Fig. d. Here we
recompute solutions for the same parameter values but with no retrograde
section (L=0 km), demonstrating that the more complex dynamics and
hysteresis apparent in Fig. b–c are intrinsically related
to the presence of a retrograde section, without which the same parameter
values invariably lead to oscillations.
Some solution branches found through numerical integration of the model in
this study occupy a narrow region of parameter space. It is unclear whether
real ice streams would ever occupy this part of parameter space given that
natural variability in accumulation rate and other
environmental variables are typically larger than the range associated with
these solutions. Notwithstanding these narrow solution branches, the large
bifurcations that separate most of the different grounding-line solutions
indicate that there are robust differences in ice-stream behavior over
realistic parameter ranges that are readily observable in real ice streams.
The presence of narrow regimes of ice-stream behavior emphasizes the need for
perturbed-physics ensemble studies with more
complex ice-sheet models to ensure that simulated ice-sheet behaviors are
broadly representative of the full range of possibilities.
Comparing the static and dynamic bed cases in Fig. , it is
clear that allowing spatiotemporal variations in bed properties changes not
just the grounding-line position but also the behavior of the grounding line,
over a wide range of possible values of accumulation rate (ac). In
some parameter regimes, thermal oscillations cause the grounding line to
migrate onto the retrograde section (during ice-stream stagnation or
activation) before reversing direction. Such behavior cannot be explained by
appealing to the flux feedback which causes the marine ice-sheet instability
for a static bed, as in Fig. a. That flux feedback relies on
a relationship between grounding-line flux and ice thickness. For sliding at
the ice-sheet bed, relationships of this kind can be justified by boundary
layer theories
and involve prescribed physical properties of the bed such as a friction
coefficient or yield stress. Unstoppable retreat or advance on retrograde
slopes then occurs if these properties do not evolve as the ice-sheet
geometry does. If, for instance, the grounding line retreats onto the
retrograde slope but the basal yield stress increases after it does so, there
is no reason why the grounding line could not readvance subsequently. We
consider the relevant physical processes that permit such behavior in more
detail in the next section.
Ice-stream behavior not explained by simple flux–thickness feedbacks
The goal of this section is to explore the range of ice-stream behaviors that
are caused by the interaction of thermomechanical feedbacks with geometrical
effects for different retrograde bed topographies and explain the physical
mechanisms which cause some of these behaviors to diverge from the
predictions of earlier theories. To do so, the accumulation rate is
initialized at ac=0.1 m yr-1 (with grounding-line position
at approximately 640 km) and then slowly increased to ac=0.3 m yr-1. As discussed in Sect. , buttressing
stress at the grounding line is not included in this study. In a baseline
simulation, there is no retrograde bed topography, and the dynamically
evolving bed causes the grounding-line position to oscillate (as described in
Sect. ) between 690 and 815 km. The simplified nature of the
bed topography (Eqs. –) then permits the
addition of a section of retrograde bed that may modify the baseline
oscillatory behavior. In each experiment, the position (x0), length (L)
or slope (bxr) of the retrograde section can be changed
(Table 2), while all other parameters are held constant (Table 1). Changing
these geometrical parameters controlling bed topography explores a section of
parameter space that is orthogonal to that explored in Sect.
(where accumulation was varied). By comparing the baseline simulation to
ice-stream behavior with an added section of retrograde slope, we can then
explain how natural modes of ice-stream variability interact with bed
topography. In an exploration of the parameter space of potential retrograde
bed configurations, we find four types of ice-stream behavior.
Figure shows (after a period of transient
initialization) four representative simulations of grounding-line migration
(solid lines) with the extent of the retrograde section in these simulations
shaded in grey. For comparison, the baseline simulation is plotted as a
dashed line in all panels. This includes ice-stream grounding lines which
exhibit: (a) persistence on the retrograde section for centuries during
ice-stream stagnation before reversing direction of migration; (b) amplified
variability and reversal of direction of migration while active on the
retrograde section; (c) complete suppression of variability; (d) reduced
amplitude of variability.
Four representative examples of interaction between retrograde
section and ice-stream thermal oscillations. In all panels, the solid line is
simulated grounding-line migration after transient initialization period
(t<20 kyr), the dashed line is grounding-line migration in baseline simulation
run without any retrograde section (L=0 km) and the grey shaded area is the extent
of retrograde section. (a) Minimally modified thermal oscillations
with ice-stream persistence on the retrograde section during stagnation
(bxr=2×10-4). (b) Amplified thermal
oscillations with part of active phase on retrograde section (bxr=2×10-4). (c) Suppressed thermal oscillations
(bxr=6×10-4). (d) Thermal oscillations with
reduced amplitude (bxr=8×10-4). All examples are
initialized with xg=640 km.
Persistence of grounding line on retrograde slope during ice-stream stagnation
When the retrograde section is either short or located far upstream of the
grounding line, oscillations in grounding-line position
(Fig. a) are similar in amplitude and period to the
baseline simulation (dashed line), though offset slightly in position along
the bed. In such cases, the grounding line does not have much (or any)
distance over which it interacts with the retrograde section, thus minimizing
the departure from the baseline simulation where there is no retrograde
section.
In a subset of cases where the retrograde section is located around the
minimum position of the grounding line from the baseline simulation
(including the simulation in Fig. a and where
0.43≲ac≲ 0.7 m yr-1 in Fig. b),
the grounding line retreats onto the retrograde section, remains there for
the duration of the stagnant phase (hundreds to thousands of years) and then
reverses direction and advances onto the prograde slope. We can understand
the mechanism of this behavior by comparing with the static bed case. The
solid line of Fig. shows the transient evolution of a
single simulation with static bed properties (where the final steady state is
marked by an “x” on Fig. a), initialized at x=881 km
and then slowly forced to retreat onto the seaward edge of the retrograde
section (at x0+L=880 km). As we would expect, the marine ice-sheet
instability causes irreversible retreat over 1500 years due to mass loss from
increasing flux through the grounding line. In contrast, when bed properties
are allowed to freely vary (Fig. a), such an
irreversible retreat may not occur because yield stress of the bed is
increasing and the intuition derived from the static bed theory of the marine
ice-sheet instability does not hold. In the second half of the active phase
and into the beginning of the stagnant phase, grounding-line retreat is the
result of thinning of the grounding line, which is itself caused by flux
through the grounding line exceeding the supply of ice advected from
upstream. Following stagnation, however, the rapidly strengthening bed
significantly reduces the mass flux out of the ice stream, slowing the rate
of grounding-line migration that would otherwise manifest as irreversible
retreat across the entire retrograde slope (as in the solid line of
Fig. ). During this slow retreat, a reservoir of excess
ice mass accumulates upstream, driving an increasing gradient in ice
thickness and driving stress, which eventually exceeds the yield strength of
partially frozen till and causes slow sliding in the grounding zone. This
slow sliding leads to initial ice thinning at the grounding line, reversal of
the direction of migration and advance back onto the downstream prograde
slope. At this point enough meltwater is produced through frictional heating
(and insulation provided by the thick ice) that the ice stream activates
further upstream, delivering significant mass to the grounding line, leading
to thickening and rapid advance.
Active ice-stream grounding line reversing direction of migration on a retrograde slope
When there is a shallow retrograde section located around the maximum
grounding-line position (Fig. b), the active ice
stream advances onto the retrograde section, persists for a few centuries,
then reverses direction and retreats to the prograde slope. The oscillations
in ice-stream grounding-line position are also amplified from the baseline
simulation with no retrograde section (also where
0.27≲ac≲ 0.32 m yr-1 in Fig. b).
Transient evolution of ice-stream grounding-line position with bed
properties kept static (τc=0 everywhere). x0=800 km;
L=80 km; bxr=2×10-4. The solid line is
initialized from x0=881 km and ac=0.33 m yr-1,
decreasing to 0.32 m yr-1 over the first 300 years. The dashed line is
initialized from x0=799 km and ac=0.58 m yr-1,
increasing to 0.59 m yr-1 over the first 300 years.
To explain the mechanism behind this persistence on the retrograde section,
we again draw a comparison with the static bed case from
Sect. . In a transient simulation (dashed line
Fig. ), an ice stream with static bed properties is
initiated at x=799 km and gently forced onto the landward edge of the
retrograde section (at x0=800 km). As flux through the grounding line
decreases, the grounding line thickens and advances over the course of
150 years until reaching prograde bed topography. In contrast, when bed
properties are allowed to freely vary, high ice velocity during the initial
part of the active phase delivers significant ice flux from upstream to the
grounding line, leading to thickening and advance. Simultaneously, the flux
from the upstream portion of the ice stream is too high to maintain a
steady state, leading to an “overshoot” where the upstream portion of the
ice stream thins, leading to increased vertical heat conduction and till
strengthening. Over a matter of decades in the latter part of the active
phase, this till strengthening leads to stagnating velocities starting as a
deactivation wave
at the ice
divide. As the grounding line advances onto the retrograde section, the
upstream reservoir of ice is exhausted and thinning occurs at the grounding
line as grounding-line flux exceeds supply from upstream. Then, when the
deactivation wave reaches the grounding line, flux through the grounding line
shuts down and ice at the grounding line begins to thicken, which briefly
wins out over unstable advance over the retrograde slope and leads to a
reversal in the direction of grounding-line migration. The retrograde section
is important, as it causes the grounding line to advance further than it
would in the absence of a retrograde section, effectively amplifying thermal
oscillations. If the retrograde section is steepened, there is an
acceleration in speed of unstable grounding-line advance related to the
marine ice-sheet instability, which can overcome changes in ice flow due to
thermal oscillations (see discussion in Sect. ).
The case illustrated in this section is qualitatively similar to the
persistence of a stagnant ice-stream grounding line on a retrograde section
discussed in Sect. . The only difference here is that the
retrograde section causes a larger advance during activation than would be
expected from thermal oscillations alone (in the baseline experiment with no
prograde slope). Nonetheless, the grounding line of any ice stream undergoing
thermal oscillations will repeatedly reverse direction, even in the absence
of a retrograde slope (see dashed line in Fig. b for
baseline simulation). When the ice stream has built up or depleted a large
reservoir of ice and is furthest from balancing mass accumulation and
grounding-line discharge, a change in ice flow may also cause thickening or
thinning at the grounding line and thus change the direction of migration. In
the cases highlighted in Fig. a and b the retrograde
sections do not change the ice-stream behavior qualitatively, but they may change
the amplitude and period of thermal oscillations. In the following two
sections we highlight cases where retrograde topography changes the dynamics
of the ice stream in a fundamental, qualitative way.
Suppression of thermal oscillations by a retrograde slope
A steep retrograde section located around the maximum grounding-line position
of the baseline simulation can completely suppress thermal oscillations
(Fig. c), leading to a steady-streaming state with
no temporal variability in ice flow
and a grounding line positioned off the retrograde slope.
To explain the suppression of ice-stream oscillations, we can examine how the
retrograde slope interacts with the competing feedbacks that cause thermal
oscillations. When the ice stream activates, there is an internal positive
feedback between frictional heating, meltwater production and till weakening.
There is also an internal negative feedback, where the grounding line
advances, the ice stream thins and the rate of vertical heat conduction at
the bed increases. In the baseline simulation without the retrograde section,
the positive feedback initially dominates, causing the “overshoot” in
horizontal velocity, ice thinning, significant heat loss due to vertical heat
conduction, meltwater freezing and eventually stagnation. By introducing a
retrograde section that is sufficiently long or steep, we increase the extent
of grounding-line advance during activation and the associated internal
negative feedback, which counteracts the internal positive feedback of
frictional heating. The result is that till weakens less during activation
than in the baseline simulation, the ice stream achieves lower peak
horizontal velocity and does not overshoot. Vertical heat conduction at the
bed is compensated by geothermal heat flux and frictional heating, causing
the ice stream to reach a steady state where the grounding-line position is
advanced further than the retrograde section.
Alternatively, suppression of thermal oscillations by a sufficiently long or
steep retrograde section can be explained in terms of the stability criterion
for thermal oscillations derived in , which
predicts that (everything else being equal) a longer ice stream enhances the
importance of a constant geothermal heat flux relative to fluctuations in
vertical conductive heat loss. By forcing the grounding line to advance, the
retrograde section diminishes the amplitude of variations in vertical
conductive heat loss, making it more likely that the basal heat budget will
come into balance and the ice stream will reach a steady-streaming state.
Reduction of thermal oscillation amplitude by a retrograde slope
When the retrograde section is both long and steep, the amplitude of
grounding-line oscillations is reduced from the baseline simulation with no
retrograde section (Fig. d). These reduced-amplitude
oscillations (which also occur where 0.5≲ac≲ 0.8 m yr-1
in Fig. c) are completely limited to a
range beyond the seaward end of the retrograde section (x0+L).
We can explain the reduction in thermal oscillations amplitude by starting
from the suppressed state described in Sect. . The long,
steep retrograde slope forces the ice stream to advance and eventually (after
transient evolution) settle down to a steady-streaming state similar to the
behavior of suppressed oscillations. The difference here is that the
retrograde section is sufficiently high (>100 m) due to its steepness and
the ice thickness above that retrograde slope is correspondingly thin
(compared to the baseline scenario). Thus, it is not the fact that there is a
retrograde section per se but rather that this section of the bed is raised
relative to the background prograde topography. Over the retrograde section,
this leads to anomalously high vertical conductive heat loss from the bed,
freezing of meltwater and strengthening of till. As till strengthens,
velocity decreases and longitudinal stresses rapidly spread this signal
upstream and downstream to the grounding line as a deactivation wave
. Subsequently,
accumulation causes thickening of stagnant ice and leads to reactivation,
within a span of 100–200 years. This behavior is the same as typical thermal
oscillations, with the only difference being that fluctuations in till state
are restricted to a short portion of the bed (∼ 200 km in this
example) mostly downstream of the retrograde slope. During the remainder of
the ∼ 1000 years of the reduced thermal oscillation cycle, the ice
stream is near the same balance as in the suppressed regime and the grounding
line is nearly stationary.
A parameter space picture of retrograde section modification of ice-stream oscillations
The four panels of Fig. map thermal oscillation
amplitude as a function of retrograde section length (L) and bed slope
(bxr), for x0=700 km (panels a, b) and x0=800 km
(panels c, d), and initializing simulations with either a low accumulation
rate (ac=0.1 m yr-1) that starts on the landward side of
the retrograde section (panels a, c) or high accumulation rate (ac=0.8 m yr-1) that starts on the seaward side of the retrograde section
(panels b, d). In all simulations, accumulation rate is then slowly ramped to
ac=0.3 m yr-1 over 30 kyr (with all other parameters
values held constant as listed in Tables 1 and 2). This parameter space
picture captures the most significant modifications of simple thermal
oscillatory behavior by retrograde bed topography. Longer or steeper
retrograde sections than specified in the above ranges are not included
because they may peak above sea level, conflicting with the model assumption
that the bed is always below sea level at the grounding line. The parameter
range spanned in Fig. is sufficient for mapping the
behavior regimes described in the preceding sections.
Amplitude of grounding-line migration associated with thermal
oscillations as a function of length (L) and slope (bxr) of
retrograde section. Simulations with zero oscillation amplitude are shaded in
grey. “x” markers indicate simulations where the grounding line is on a
retrograde slope at its minimum position during the stagnant phase. Circle
markers indicate simulations where the grounding line is on a retrograde
slope at its maximum position during the active phase. Panels (a)
and (b) have a retrograde section starting at x0=700 km.
Panels (c) and (d) have a retrograde section starting at
x0=800 km. Panels (a) and (c) have initial ice-stream
grounding-line position landward of retrograde section at xg(t=0)=640 km. Panels (b) and (d) have initial ice-stream
grounding-line position seaward of retrograde section at xg(t=0)=1150 km. In the baseline simulation the grounding-line position
oscillations are between 690 and 815 km.
Retrograde sections located far upstream of the range of grounding-line
migration associated with thermal oscillations in the baseline simulation
(x0+L<690 km) or of short length (L<80 km) have a minimal impact
on ice-stream behavior. However, when the retrograde section is located near
the minimum grounding-line position the grounding line may persist for
centuries on the retrograde section during ice-stream stagnation (marked by
white “x” marks in Fig. ). Such behavior occurs
for a wide range of bxr, indicating that even strong retrograde
slopes cannot prevent the reversal of grounding-line migration during the
onset of ice-stream activation. There is no evidence that the marine
ice-sheet instability plays any significant role for ice streams which
stagnate on or near sufficiently small retrograde sections (L<50 km),
regardless of the steepness of the retrograde slope.
Shallow retrograde sections located around the maximum grounding-line
position result in prolonged intervals of several centuries where the
grounding line of an active ice stream advances onto and then retreats from a
retrograde section (such cases are marked by white circles in
Fig. ). These long, shallow retrograde sections may
also amplify thermal oscillations by forcing the grounding line to advance
through the same process involved with the marine ice-sheet instability. Such
behavior is limited to relatively shallow retrograde slopes, indicating that
the marine ice-sheet instability plays a more important role here and
prevents such behavior entirely for steeper retrograde slopes.
When the retrograde section is located around the maximum grounding-line
position and is sufficiently steep, thermal oscillations are suppressed
completely. In this case, the steep retrograde section causes the grounding
line to advance sufficiently quickly that ice flow changes associated with
thermal oscillations are not able to reverse the direction of grounding-line
migration. The result is that the ice stream attains a steady state
grounding-line position seaward of the retrograde section.
At even steeper slopes, thermal oscillations reappear, but are reduced in
amplitude relative to the baseline simulation and are restricted to the bed
downstream of the retrograde section. However, there is still a wide gap in
oscillation amplitude between parameter regimes where oscillations are
suppressed and those where it is reduced entirely. A small increase in slope
(bxr) of the retrograde section (e.g., along a transect where
L=150 km in Fig. a and b) results in a transition
from steady state to finite-amplitude oscillations in grounding-line
position. We have yet to find any evidence that the thermal mechanism is
capable of producing oscillations of arbitrarily weak amplitude
.
Hysteresis behavior associated with retrograde slopes is important because it
imprints ice-stream history on future behavior and so must be considered when
spinning up numerical ice-stream models. In Sect. we showed
that for sufficiently steep retrograde slopes the grounding line either
oscillates on the landward side of the retrograde slope, attains a steady
state, or undergoes a limit cycle oscillation on the seaward side with, at
most, small excursions onto the retrograde slope. The hysteresis shown in
Fig. c for this case is analogous to that predicted by the
marine ice-sheet instability for fixed bed properties. We also see evidence
for such hysteresis in a narrow range of retrograde sections with
intermediate bxr and L. Simulations with retrograde sections of
intermediate length (80≲L≲ 140 km) and located near the
minimum position of grounding-line oscillations retain oscillatory behavior
when initialized seaward of the retrograde section, but not always when
initialized from landward. Additionally, simulations with retrograde sections
of intermediate slope (0.0002≲L≲ 0.0004) and located near
the maximum position of grounding-line oscillations retain oscillatory
behavior when initialized landward of the retrograde section, but not always
when initialized from seaward.
Conclusions
There are two main conclusions to be drawn from the results of this study.
First, we have demonstrated that grounding-line position and behavior change
significantly as a result of spatiotemporal variations in bed properties that
arise as a part of unforced ice flow variability. While ice streams which
have a hard bed or a permanently saturated soft bed may not necessarily
undergo large internal variations in basal shear stress, in a broad parameter
regime relevant to observed ice streams, the possibility of a
dynamically evolving soft bed must be considered. As we have shown, models
which do not include dynamically varying bed properties are not able to
simulate a wide array of ice-stream behaviors.
Second, ice-stream grounding lines on or near retrograde slopes may exhibit
behaviors which are not predicted by existing theories for grounding-line
stability or internal ice-stream variability. Ice streams which exhibit
internal variability in bed properties may persist on retrograde slopes for
centuries and reverse their direction of migration on such slopes. The marine
ice-sheet instability is not currently equipped to explain changes in ice
flux through the grounding line associated with time-dependent bed
properties, which complicates how observations of grounding-line migration
are interpreted. The grounding line of an ice stream retreating onto a
section of retrograde slope may continue to retreat irreversibly or may
pause for centuries during stagnation before re-advancing onto the prograde
slope. The latter possibility may explain why the grounding line of the
currently stagnant Kamb Ice Stream rests on a retrograde slope
without evidence for ongoing migration
. Of course, as we discuss below, ice shelf
buttressing may alternately be the dominant process responsible for
stabilizing the grounding line of Kamb Ice Stream. Determining which
processes influence the stability and variability of grounding-line position
is critical to making accurate long-term predictions of ice-stream flux and
grounding-line position. However, such a task is made more difficult by the
short observational records of grounding-line position for many ice streams,
which extend (at most) only a few decades. To make more progress in
contextualizing existing observations and predicting future ice-stream
behavior, numerical models are necessary. To admit the widest range of
possible ice-stream behavior, ice-stream modeling studies must include
dynamic variation in bed properties or convincingly argue that such variation
is not applicable to a particular ice stream.
There are 3-D thermomechanical ice-stream models which include more processes
than the simple flow-line model utilized in this study. The simplicity of this
flow-line model enables exploration of the parameter space, process-level
understanding of dynamical behavior and comparison to theories of
grounding-line stability and ice-stream variability. Other factors, such as
strong ice shelf buttressing, are known to play an important role in
grounding-line stability
and in suppressing thermal oscillations .
Consequently, these processes may dominate the dynamics under some
circumstances. However, to the extent that oscillations in velocity and
grounding-line position occur in some ice streams – and observations show
that they do
– the dynamics discussed in this study are potentially relevant both to
models and observations. Though we have neglected internal strain heating
from this model, simulations with strain heating included indicate that
vertical and longitudinal strains of horizontal velocity contribute
negligibly to the overall heat budget of the ice stream, even during periods
of strong longitudinal strain such as activation. The model used in this
study is laterally integrated over the ice stream to form a longitudinal
flow line, and so lateral shear heating should correspondingly be spread over
the entire width of the ice stream (though shear is zero at the actual
centerline). In this case, shear heating contributes less than 1 % to the
heat budget. However, if we instead considered a flow line through the shear
margin (though this in conflict with other assumptions made in model
formulation), then lateral shear heating is non-negligible (10 %).
Nonetheless, in order to draw definitive conclusions about specific ice
streams, more complete 3-D ice-stream modeling is needed, which takes into
account variations in bed properties simulated in this study and also details
not captured by a flow-line model, such as fully dynamic ice shelf
buttressing, cross-stream variations in basal topography, lateral advection
and shear margin migration .
We do not challenge the notion that an unbuttressed ice stream with basal
shear stress that is only a function of sliding velocity cannot have a stable
steady state grounding line on a retrograde bed slope (excluding other
physical processes). However, there are limitations in trying to fit all
observed ice-sheet dynamics to this simple version of the marine ice-sheet
instability. Indeed, recent studies (reviewed in the introduction) have shown
that an array of potentially important processes not included in the
canonical formulation of the marine ice-sheet instability may inhibit it.
Other studies
have also shown that forced simulations of unstable grounding-line migration
over retrograde slopes are highly sensitive to assumptions about bed
properties. The simulations presented here show the types of grounding-line
behavior that are possible once we step away from the limiting confines of
the classical marine ice-sheet instability.
A new generation of observations
and inverse models
has provided indications that subglacial
hydrology may provide a stabilizing feedback in situations where there is
high basal shear stress near the grounding line of an ice stream under
significant external forcing, such as ocean melting. Other studies
e.g., have shown that patches of high basal
traction can travel along ice streams on forcing-relevant timescales.
However, the paucity of direct observations of bed conditions over most of
West Antarctica, including the presence (or lack) of plastic till and
meltwater and their time-dependent evolution, is still a major obstacle to
making accurate predictions of ice-stream evolution. Future studies must
continue to find new ways to make observations of bed properties, which can
be incorporated into models with dynamically evolving subglacial hydrology.
Such an approach would enhance dynamical understanding of ice-sheet stability
and improve predictions of Antarctic Ice Sheet change.
Code availability
All model results used to produce the figures in this study and flow-line
model code are available from the corresponding author upon request.
Acknowledgements
This work has been supported by the NSF grant AGS-1303604 (Alexander A. Robel
and Eli Tziperman). Eli Tziperman thanks the Weizmann Institute for its
hospitality during parts of this work. Alexander A. Robel has been supported
by the NSF Graduate Research Fellowship. Christian Schoof has been supported
by NSERC Discovery Grant no. 357193-12. Edited
by: R. Bingham Reviewed by: two anonymous referees
References
Alley, R. B., Anandakrishnan, S., Dupont, T. K., Parizek, B. R., and Pollard,
D.: Effect of sedimentation on ice-sheet grounding-line stability, Science,
315, 1838–1841, 2007.
Bamber, J., Vaughan, D., and Joughin, I.: Widespread complex flow in the
interior of the Antarctic ice sheet, Science, 287, 1248–1250, 2000.
Brinkerhoff, D. and Johnson, J.: Dynamics of thermally induced ice streams
simulated with a higher-order flow model, J. Geophys. Res.-Earth, 120,
1743–1770, 2015.
Catania, G., Hulbe, C., Conway, H., Scambos, T., and Raymond, C. F.:
Variability in the mass flux of the Ross ice streams, West Antarctica, over
the last millennium, J. Glaciol., 58, 741–752, 2012.
Christianson, K., Parizek, B. R., Alley, R. B., Horgan, H. J., Jacobel,
R. W., Anandakrishnan, S., Keisling, B. A., Craig, B. D., and Muto, A.: Ice
sheet grounding zone stabilization due to till compaction, Geophys. Res.
Lett. 40, 5406–5411, 2013.
Chugunov, V. A. and Wilchinsky, A. V.: Modelling of a marine glacier and
ice-sheet–ice shelf transition zone based on asymptotic analysis, Ann.
Glaciol., 23, 59–67, 1996.
Clarke, G.: Thermal Regulation of Glacier Surging, J. Glaciol., 16,
231–250, 1976.
Docquier, D., Pollard, D., and Pattyn, F.: Thwaites Glacier grounding-line
retreat: influence of width and buttressing parameterizations, J. Glaciol.,
60, 305–313, 2014.Durand, G., Gagliardini, O., De Fleurian, B., Zwinger, T., and Le Meur, E.:
Marine ice sheet dynamics: Hysteresis and neutral equilibrium, J. Geophys.
Res.-Earth, 114, F03009, 10.1029/2008JF001170, 2009.
Echelmeyer, K., Harrison, W., Larsen, C., and Mitchell, J.: The role of the
margins in the dynamics of an active ice stream, J. Glaciol., 40,
527–538, 1994.Favier, L., Durand, G., Cornford, S., Gudmundsson, G., Gagliardini, O.,
Gillet-Chaulet, F., Zwinger, T., Payne, A., and Le Brocq, A.: Retreat of Pine
Island Glacier controlled by marine ice-sheet instability, Nat. Climate
Change, 4, 117–121, 10.1038/nclimate2094, 2014.Fowler, A.: A theory of glacial surges, J. Geophys. Res.-Solid, 92,
9111–9120, 10.1029/JB092iB09p09111, 1987.
Fowler, A.: Mathematical geoscience, Springer Science & Business Media,
2011.
Fowler, A. and Schiavi, E.: A theory of ice-sheet surges, J. Glaciol.,
44, 104–118, 1998.
Fowler, A., Murray, T., and Ng, F.: Thermally controlled glacier surging,
J. Glaciol., 47, 527–538, 2001.
Fried, M., Hulbe, C., and Fahnestock, M.: Grounding-line dynamics and margin
lakes, Ann. Glaciol., 55, 87–96, 2014.Goldberg, D., Holland, D., and Schoof, C.: Grounding line movement and ice
shelf buttressing in marine ice sheets, J. Geophys. Res., 114, F04026,
10.1029/2008JF001227, 2009.
Gomez, N., Mitrovica, J. X., Huybers, P., and Clark, P. U.: Sea level as a
stabilizing factor for marine-ice-sheet grounding lines, Nat. Geosci., 3,
850–853, 2010.Gomez, N., Pollard, D., Mitrovica, J. X., Huybers, P., and Clark, P. U.:
Evolution of a coupled marine ice sheet–sea level model, J. Geophys.
Res.-Earth, 117, F01013, 10.1029/2011JF002128, 2012.Gudmundsson, G. H., Krug, J., Durand, G., Favier, L., and Gagliardini, O.:
The stability of grounding lines on retrograde slopes, The Cryosphere, 6,
1497–1505, 10.5194/tc-6-1497-2012, 2012.
Haseloff, M., Schoof, C., and Gagliardini, O.: A boundary layer model for ice
stream margins, J. Fluid Mech., 781, 353–387, 2015.
Hindmarsh, R. C.: An observationally validated theory of viscous flow
dynamics at the ice-shelf calving front, J. Glaciol., 58, 375–387, 2012.Horgan, H. and Anandakrishnan, S.: Static grounding lines and dynamic ice
streams: Evidence from the Siple Coast, West Antarctica, Geophys. Res.
Lett., 33, L18502, 10.1029/2006GL027091, 2006.
Jamieson, S., Vieli, A., Livingstone, S., Cofaigh, C., Stokes, C.,
Hillenbrand, C., and Dowdeswell, J.: Ice-stream stability on a reverse bed
slope, Nat. Geosci., 5, 799–802, 2012.
Jamieson, S. S., Vieli, A., Cofaigh, C. Ó., Stokes, C. R., Livingstone,
S. J., and Hillenbrand, C.-D.: Understanding controls on rapid ice-stream
retreat during the last deglaciation of Marguerite Bay, Antarctica, using a
numerical model, J. Geophys. Res.-Earth, 119, 247–263, 2014.
Joughin, I. and Tulaczyk, S.: Positive mass balance of the Ross ice stream,
West Antarctica, Science, 295, 476–452, 2002.
Joughin, I., Tulaczyk, S., Bamber, J. L., Blankenship, D., Holt, J. W.,
Scambos, T., and Vaughan, D. G.: Basal conditions for Pine Island and
Thwaites Glaciers, West Antarctica, determined using satellite and airborne
data, J. Glaciol., 55, 245–257, 2009.Joughin, I., Smith, B. E., and Holland, D. M.: Sensitivity of 21st century
sea level to ocean-induced thinning of Pine Island Glacier, Antarctica,
Geophys. Res. Lett., 37, L20502, 10.1029/2010GL044819, 2010.
Joughin, I., Smith, B. E., and Medley, B.: Marine ice sheet collapse
potentially under way for the Thwaites Glacier Basin, West Antarctica,
Science, 344, 735–738, 2014.Katz, R. and Worster, M.: Stability of ice-sheet grounding lines, P. Roy.
Soc. Lond. A Mat., 466, 1597–1620, 10.1098/rspa.2009.0434, 2010.
MacAyeal, D.: Binge/Purge Oscillations of the Laurentide Ice Sheet as a Cause
of the North Atlantic Heinrich Events, Paleoceanography, 8, 775–784, 1993.
Murphy, J. M., Sexton, D. M., Barnett, D. N., Jones, G. S., Webb, M. J.,
Collins, M., and Stainforth, D. A.: Quantification of modelling uncertainties
in a large ensemble of climate change simulations, Nature, 430, 768–772,
2004.
Nowicki, S. and Wingham, D.: Conditions for a steady ice sheet–ice shelf
junction, Earth Planet. Sc. Lett., 265, 246–255, 2008.
Oerlemans, J.: Glacial cycles and ice-sheet modelling, Climatic Change, 4,
353–374, 1982.
Parizek, B., Christianson, K., Anandakrishnan, S., Alley, R., Walker, R.,
Edwards, R., Wolfe, D., Bertini, G., Rinehart, S., Bindschadler, R., and
Nowicki, S. M. J.: Dynamic (in) stability of Thwaites Glacier, West
Antarctica, J. Geophysical Res.-Earth, 118, 638–655, 2013.Pattyn, F., Schoof, C., Perichon, L., Hindmarsh, R. C. A., Bueler, E., de
Fleurian, B., Durand, G., Gagliardini, O., Gladstone, R., Goldberg, D.,
Gudmundsson, G. H., Huybrechts, P., Lee, V., Nick, F. M., Payne, A. J.,
Pollard, D., Rybak, O., Saito, F., and Vieli, A.: Results of the Marine Ice
Sheet Model Intercomparison Project, MISMIP, The Cryosphere, 6, 573–588,
10.5194/tc-6-573-2012, 2012.
Pritchard, H. D., Arthern, R. J., Vaughan, D. G., and Edwards, L. A.:
Extensive dynamic thinning on the margins of the Greenland and Antarctic ice
sheets, Nature, 461, 971–975, 2009.
Retzlaff, R. and Bentley, C.: Timing of stagnation of Ice Stream C, West
Antarctica, from short pulse radar studies of buried surface crevasses, J.
Glaciol., 39, 553–561, 1993.Rignot, E., Bamber, J., van den Broeke, M., Davis, C., Li, Y., van de Berg,
W., and van Meijgaard, E.: Recent Antarctic ice mass loss from radar
interferometry and regional climate modeling, Nat. Geosci., 1,
106–110, 10.1038/ngeo102, 2008.
Robel, A., DeGiuli, E., Schoof, C., and Tziperman, E.: Dynamics of ice
stream temporal variability: Modes, scales and hysteresis, J. Geophys. Res.,
118, 925–936, 2013.
Robel, A., Schoof, C., and Tziperman, E.: Rapid grounding line migration
induced by internal ice stream variability, J. Geophys. Res., 119,
2430–2447, 2014.
Robin, G. D. Q.: Ice movement and temperature distribution in glaciers and
ice sheets, J. Glaciol., 2, 523–532, 1955.
Robison, R. A., Huppert, H. E., and Worster, M.: Dynamics of viscous
grounding lines, J. Fluid Mech., 648, 363–380, 2010.Sayag, R. and Tziperman, E.: Spatiotemporal dynamics of ice streams due to a
triple valued sliding law, J. Fluid Mech., 640, 483–505,
doi:10.1017/S0022112009991406, 2009.Sayag, R. and Tziperman, E.: Interaction and variability patterns of ice
streams under a triple-valued sliding law and a non-Newtonian rheology, J.
Geophys. Res., 116, F01009, 10.1029/2010JF001839, 2011.
Schmeltz, M., Rignot, E., Dupont, T. K., and Macayeal, D. R.: Sensitivity of
Pine Island Glacier, West Antarctica, to changes in ice-shelf and basal
conditions: a model study, J. Glaciol., 48, 552–558, 2002.Schoof, C.: Marine ice-sheet dynamics. Part 1. The case of rapid sliding,
J. Fluid Mech., 573, 27–55, 10.1017/S0022112006003570, 2007a.Schoof, C.: Ice sheet grounding line dynamics: Steady states, stability, and
hysteresis, J. Geophys. Res.-Earth, 112, F03S28,
10.1029/2006JF000664, 2007b.
Schoof, C.: Marine ice sheet stability, J. Fluid Mech., 698, 62–72, 2012.
Schoof, C. and Hewitt, I.: Ice-sheet dynamics, Annu. Rev. Fluid Mech., 45,
217–239, 2013.
Schroeder, D. M., Blankenship, D. D., and Young, D. A.: Evidence for a water
system transition beneath Thwaites Glacier, West Antarctica, P. Natl. Acad.
Sci. USA, 110, 12225–12228, 2013.
Sergienko, O. V. and Hindmarsh, R. C.: Regular patterns in frictional
resistance of ice-stream beds seen by surface data inversion, Science, 342,
1086–1089, 2013.
Shumskiy, P. and Krass, M.: Mathematical models of ice shelves, J. Glaciol.,
17, 419–432, 1976.
Smith, A. M., Jordan, T. A., Ferraccioli, F., and Bingham, R. G.: Influence
of subglacial conditions on ice stream dynamics: Seismic and potential field
data from Pine Island Glacier, West Antarctica, J. Geophys. Res.-Sol. Ea.,
118, 1471–1482, 2013.
Tsai, V. C., Stewart, A. L., and Thompson, A. F.: Marine ice-sheet profiles
and stability under Coulomb basal conditions, J. Glaciol., 61, 205–215,
2015.Tulaczyk, S., Kamb, W., and Engelhardt, H.: Basal mechanics of Ice Stream B,
West Antarctica 1. Till mechanics, J. Geophys. Res.-Sol. Ea., 105,
463–481, 10.1029/1999JB900329, 2000a.Tulaczyk, S., Kamb, W., and Engelhardt, H.: Basal mechanics of Ice Stream B,
West Antarctica 2. Undrained plastic bed model, J. Geophys. Res.-Sol. Ea.,
105, 483–494, 10.1029/1999JB900328, 2000b.Weertman, J.: Stability of the junction of an ice sheet and an ice shelf, J.
Glaciol., 13, 3–11, 1974.
Wilchinsky, A. V.: Studying ice sheet stability using the method of
separation of variables, Geophys. Astro. Fluid, 94, 15–45, 2001.
Wilchinsky, A. V.: Linear stability analysis of an ice sheet interacting with
the ocean, J. Glaciol., 55, 13–20, 2009.
Wolovick, M. J., Creyts, T. T., Buck, W. R., and Bell, R. E.: Traveling
slippery patches produce thickness-scale folds in ice sheets, Geophys. Res.
Lett., 41, 8895–8901, 2014.