Introduction
Glacial surging is characterized by rapid speedup of ice flow and abrupt
increase in ice discharge. Surging of glaciers found in the mountainous
regions of the Earth is observed to occur on a decadal to centennial
timescale, depending on regional conditions
e.g.,.
Repeated activation and stagnation of ice streams which drain the Siple Coast
region of the West Antarctic Ice Sheet
e.g., alter the flow
pattern and mass balance of this part the ice sheet on a centennial timescale . During glacial periods,
quasi-periodic, large-scale surging of the Laurentide Ice Sheet likely led to
massive iceberg calving into the ocean on a millennial timescale
. These so-called Heinrich events
are associated with
substantial freshening of the North Atlantic and reduction of the Atlantic
meridional overturning circulation and are
connected to abrupt climate changes on a global scale
.
Periodic oscillations of ice-sheet growth and surging have been suggested
either to be of unforced type and thus originating from ice-internal
mechanisms “binge–purge” oscillations, see or to be
driven externally, i.e., by climate forcing
e.g.,.
Numerical modeling studies that investigated ice-sheet-intrinsic surging
include the demonstration of creep instability and
hydraulic runaway as possible main feedbacks that
drive unforced surging, the application to the Laurentide Ice Sheet to
simulate its quasi-periodic surging
,
the simulation of (cyclic) ice streaming and stagnation reminiscent of the
flow variability of the Siple Coast ice streams
and the investigation of ice-stream oscillations in interaction with bed
topography under the influence of ice-shelf buttressing
. Model complexity ranges from the consideration of a
simple slab of ice e.g., to the solution of the
full Stokes equations to simulate real-world problems using satellite data
. Limitations to several of these studies include
the restriction to the flow-line case (only one horizontal dimension
considered), the prescription of a strongly idealized bed geometry (flat bed
or inclined plane) and the use of simplified parameterizations of
ice-internal, lateral and basal stresses (e.g., basal sliding chosen to be
proportional to the driving stress).
Here we apply a channel-type bed geometry to a three-dimensional
state-of-the-art ice-sheet model to simulate the cyclic surging of a marine
ice-sheet-shelf system. In particular, and in contrast to several previous
studies, our simulations use a sliding law that is based on the stress
balance of the ice, and thereby has stress boundary conditions, and
incorporates the effect of overburden pressure and basal meltwater on the
bed strength in nonlinear fashion. In other words, the computed sliding
velocity is not a direct parameterization through the local basal conditions
but results from solving the non-local shallow-shelf approximation (SSA) of the
stress balance and at the same time, in combination with the basal
properties, determines the basal stresses. Our model includes a minimal
version of a subglacier, i.e., a basal till layer underlying the ice which
interacts with the ice sheet through meltwater exchange, an interaction
which is crucial to model unforced cyclic ice-sheet growth and surge. The
nature of the chosen three-dimensional topographic setup allows to simulate
complex ice flow and inherently emerging ice-shelf buttressing. Analyzing the
modeled surge cycle, we identify competing fundamental mechanisms that
underlie successive ice buildup, surge and stabilization. These mechanisms
are visualized by the means of feedback loops. We also investigate conditions
that lead to the damping of the oscillations in our model and explore how
different sliding laws affect ice-flow characteristics. Eventually we discuss
our results and conclude.
In a previous study, oscillatory surge behavior has been investigated with an
earlier version of the model used here . In
contrast to the marine ice-sheet-shelf system modeled here, they simulated a
qualitatively very different type of an ice body, i.e., a land-terminating
glacier. The main differences regarding the models (linear vs. nonlinear
friction law, transport of basal till water) as well as the experimental
setups (bed entirely above sea level vs. marine bed trough, different
boundary conditions) are detailed in the methods section and are picked up in
the discussion section. Results from include the
occurrence of high- and low-frequency oscillations as well as steady, fast ice
flow in the partially sampled parameter space, spanned by two sliding-law
parameters, i.e., the till friction angle and the sliding-law exponent. The
frequency and magnitude of the surge cycle were also found to be sensitive to
variations in the climate forcing (surface temperature and mass balance).
Physical constants and model parameters
Parameter
Value
Unit
Physical meaning
n
3
Exponent in Glen's flow law
q
1/3
Basal friction exponent
u0
1
ms-1
Scaling parameter for basal velocity in the sliding law
ϕ
10
∘
Till friction angle
N0
1000
Pa
Reference effective pressure
δ
0.02
Parameter determining effective overburden pressure δPo
e0
0.69
Reference void ratio at N0
Cc
0.12
Till compressibility
Wtillmax
2
m
Maximum water in till
Cd
0.001
myr-1
Till drainage rate
ρi
918
kgm-3
Ice density
ρw
1000
kgm-3
Freshwater density
ρsw
1028
kgm-3
Seawater density
g
9.18
ms-2
Gravitational acceleration
Lx
700
km
Length of right-hand half of the symmetric domain
Ly
160
km
Width of domain entering Eq. 4 of
fc
16
km
Characteristic width of channel side walls entering Eq. 4 of
dc
500
m
Depth of bed trough compared with side walls entering Eq. 4 of
wc
24
km
Half-width of bed trough entering Eq. 4 of
xcf
640
km
Position of fixed calving front in right-hand half of domain
a
0.3
myr-1
Surface accumulation rate
G
70
mWm-2
Geothermal heat flux
Ts
-20
∘C
Surface temperature of the ice
Methods
Model
We use the open-source Parallel Ice Sheet Model PISM; see
http://www.pism-docs.org;,
version stable07 (https://github.com/pism/pism/). The
thermomechanically coupled model applies a superposition of the shallow-ice
approximation SIA; and the SSA
of the Stokes stress balance
. In particular, the SSA allows for stress
transmission across the grounding line and thus accounts for the buttressing
effect of laterally confined ice shelves on the upstream grounded regions
. The ice rheology is determined by Glen's
flow law . An energy-conserving enthalpy formulation
of the thermodynamics in particular allows for an advanced calculation of the
basal melt rate for polythermal ice . The model
applies a linear interpolation of the freely evolving grounding line and
accordingly interpolated basal friction and uses one-sided differences in
the driving stress close to the grounding line .
A nonlinear Weertman-type sliding law is chosen to calculate the basal shear
stress τb, based on the sliding velocity of the ice
ub with a sliding exponent q=1/3 as used in several
previous studies
e.g.,:
τb=-τcubu0qub1-q.
Here τc is the till yield stress , which
evolves in time with changing basal till water content and overburden
pressure (see below). For simplicity we set the velocity scaling parameter
u0 to 1 m s-1 (unit of the sliding velocity calculated in the
model, as chosen in, e.g., ;
;
). Note that ub
results from solving the non-local SSA stress balance
Eq. 17 in which τb appears as
one of the terms that balance the driving stress. This implementation of
basal sliding is substantially different (and introduces more complexity)
compared to several models that have previously been used in attempt to model
cyclic surging, where ub is a local function of
τb, the latter often given by the negative of the
driving stress at the ice base
e.g.,.
The till yield stress in Eq. () is determined by a Mohr–Coulomb
model :
τc=tan(ϕ)N0δPoN0s10e0Cc(1-s),
which accounts for the effect of evolving ice thickness H, the associated
change in overburden pressure Po=ρigH on the
basal till, and the amount of water stored in the till Wtill. Here
s=Wtill/Wmax is the fraction of the water layer
thickness in the till with respect to a fixed maximum layer thickness
Wmax . All other parameters are
prescribed and are constant in space and time (adopted from ; see
Table for a full list of parameters, their naming and
values). Note that there is an upper bound to the yield
stress enforced in the model, which is determined by the overburden pressure,
i.e., τc,max=tan(ϕ)Po for details
seeSect. 3.2. Compared to ,
where the yield stress is linear in the product of the overburden pressure
and the till water fraction (τc∼(1-s)Po; see their
Eqs. 2 and 4), the yield-stress formula applied here is an exponential
function of s (Eq. ). This allows for a nonlinear evolution of
the till yield stress during drainage or fill-up of the basal water layer.
(a) Bed topography prescribed in the experiments (color bar)
with contours of grounding line and calving front during buildup (gray) and
surge (red; see corresponding circles in Fig. ). Note that throughout
this study we focus on the right-hand half of the symmetric model domain as
shown here (symmetry axis at x=0). Dotted lines mark locations of the cross
sections shown in the other two panels. (b) Cross section in
x direction along the centerline of the model domain. Profiles of the ice
sheet (straight lines) and its velocity (dashed) are shown for the buildup
phase (gray) and during surge (red); bed topography is in black. (c)
Cross section in y direction across the model domain at x=350 km. Same
colors as in panel (b). Red and blue squares on the centerline
denote locations of two point measurements for which time series are shown in
Figs. and .
The subglacial model is a slightly modified version of the undrained plastic
bed model of , as described in
Sect. 3. The term undrained refers to the fact
that this model does not account for horizontal transport of meltwater
stored in the basal till and thus meltwater is produced and consumed only
locally (in contrast to , where there is also
horizontal diffusion of till water; see their Eq. 1). The evolution equation
for the till-stored water thickness, Wtill, is a function of the
local basal melt rate m (positive for melting, negative for refreezing):
∂Wtill∂t=mρw-Cd.
The drainage rate parameter Cd allows for drainage of the till in
the absence of water input. The water layer thickness is bounded (0≤Wtill≤Wtillmax) to avoid unreasonably strong
filling of the till with meltwater. Meltwater which exceeds
Wtillmax is not conserved.
Schematic visualizing the three main feedback mechanisms, each of
them dominating one of the three subsequent phases of slow ice buildup
(gray), abrupt surging (red) and stabilization (blue), forming a full surge
cycle. The sign next to an arrow pointing from variable A to B indicates
whether a small increase in variable A leads to an increase (+) or decrease
(-) in variable B. According to this convention one can deduce from
counting the negative links of a full loop whether this loop describes an
amplifying (positive) or stabilizing (negative) feedback. An even number of
negative links indicates a positive feedback loop (large +) whereas an odd
number of negative links indicates a negative feedback loop (large -).
Experimental setup
The three-dimensional setup is designed to model a marine ice sheet, which
drains through a bed trough, feeding a bay-shaped ice shelf which calves into
the ocean. The idealized bed topography (Fig. 1) is a superposition of two
components: the bed component in x direction, bx(x)=-150m-0.84×10-3|x|, is an inclined plane, sloping down towards the
ocean (Fig. b). The component in y direction, by(y), has
channel-shaped form (Fig. c) and is a widened version of the one
used in the MISMIP+ experiments here with adjusted parameters for
domain width and channel side-wall width; see
Table . The superposition of both components
yields a bed trough which is symmetric in both x and y directions
(symmetry axes x=0 and y=0). While the main ice flow is in x direction
(from the interior through the bed trough towards the ocean), there is also a
flow component in y direction, i.e., from the channel's lateral ridges down
into the trough. Resulting convergent flow and associated horizontal shearing
enable the emergence of ice-shelf buttressing, which leads to a
grounding-line position further downstream than in the absence of an ice
shelf. Ice is cut off from the ice shelf and thus calved into the ocean beyond
a fixed position (Fig. a). There exist more sophisticated methods
to represent calving in a numerical model
e.g.,,
leading to more realistic calving behavior and calving-front geometry. Our
simple approach of prescribing a calving front (fixed in space and time) far
enough from the region of the confined ice shelf makes sure that the calving
front does not interact with grounding-line migration in the course of the
surge cycle. Due to the symmetry of the setup we only consider the right-hand
half of the domain throughout our analysis.
Compared to , who model a land-based glacier that
rests on a bed entirely above sea level and exhibits an unconfined
tongue-shaped (convex) ice-stream terminus (see their Fig. 2), here we
simulate a marine ice-sheet-shelf system. This allows to model oscillatory
flow of a topographically confined ice stream which exhibits a concave
grounding line, being buttressed by the downstream bay-shaped ice shelf
(Fig. ). Our model domain is larger by a factor of about 6 and 2
in the x and y directions, respectively, resulting in a 500 km long ice
sheet, compared to the glacier of about 100 km length modeled in
.
Time series of the main variables which characterize the feedback
loops of growth, surge and stabilization in Fig. .
(a) Ice thickness H, (b) till water thickness
Wtill, (c) basal yield stress τc,
(d) velocity and ice flux (orange) and (e) iceberg calving
rate. Except for the calving rate, data shown is averaged over the area of
grounded ice. The calving rate has been smoothed with a 200-year moving
window. The right-hand side of each panel shows a zoom into a full cycle
(highlighted in gray). Colored circles in panel (a) show the points
in time chosen to be representative for the phases of buildup (gray), surge
(red) and stabilization (blue).
Surface mass balance, surface ice temperature and geothermal heat flux are
assumed to be constant and spatially uniform in contrast towhere
surface mass balance and temperature are parameterized by the bedrock
elevation, prescribing an accumulation/ablation zone.
There is no melting beneath the ice shelf. Glacial isostatic adjustment is
not accounted for in the experiments. The simulations are initiated with a
block of ice from which an ice-sheet-shelf system evolves while ice flow,
basal mechanics and till-stored water content adjust. This spinup lasts a few
thousand years and thus we focus on the time after this phase.
The model is run using finite differences and a regular grid of 1 km
horizontal resolution. An initial examination of the flow field reveals that
the SIA velocities are small compared to the SSA velocities in our
simulation. Despite this fact, considering the SIA in the simulations in
particular allows for the representation of a three-dimensional temperature
field.
Results
Cyclic surging
For the given set of parameters (Table ) the ice-sheet-shelf system
takes on an oscillatory equilibrium of continuously alternating phases of
surge and growth. This unforced behavior has been illustratively described as
“binge–purge” mechanism by using a minimal model of ice
(thermo-)dynamics and geometry (slab of ice, with thickness and
temperature being spatially uniform along the horizontal
axis; ). This mechanism has been further investigated in later
studies, including models of higher complexity in terms of ice dynamics as
well as setup geometry e.g., ice thickness, basal hydrology and other
properties evolving non-uniformly in space and time; application to a
three-dimensional, Laurentide-characteristic bed topography; coupling to a
climate model; see for
instance.
Here we basically describe the same mechanism, focusing on the competing
internal feedback mechanisms that dictate the binge–purge cycle and that
affect the ice dynamics on different timescales (Fig. ), based on
results from a sophisticated, three-dimensional ice-sheet model. At the same
time our results reveal the possibility of cyclic growth and surge of an
inherently buttressed marine ice-sheet-shelf system.
On the slow timescale, the ice sheet tends to grow toward an equilibrium
thickness, which is determined by the balance between snowfall and ice flux
(velocity). If this equilibrium ice thickness is too large to be sustained by
the basal conditions, this buildup (negative gray feedback loop in
Fig. ) is interrupted by an abrupt surge event with a rapid,
self-enforcing speedup of the ice flow (positive feedback loop in red). The
associated large-scale ice discharge into the ocean eventually leads to a
stabilization of the shrunken ice-sheet-shelf system (negative feedback loop
in blue), which again tends to restore a balance thickness before a new surge
event kicks in.
Fields of (a) basal melt rate m, (b) strain
heating, (c) till water thickness Wtill and
(d) velocity for a representative snapshot for each of the three
phases of buildup, surge and stabilization (as denoted by the colored
circles in Fig. ). Thick black contours mark the grounding line and
calving front. Bed topography shown by thin gray contours.
Positive feedback of creep instability, which fosters rapid ice
streaming through the bed trough in addition to the positive feedback of
ice-flow acceleration visualized in Fig. .
At the beginning of the modeled surge cycle the negative feedback loop of
slowing down ice growth is dominant: the basal till water content drops close
to zero and basal friction is high, allowing gradual thickening of the ice
sheet (Figs. , and ). The thickening causes
an increase in basal meltwater production due to the lowering of the
pressure melting point at the ice base. The increasing water content in the
basal till attenuates further increase in basal friction (which still
increases due to the effect of ice thickening, i.e., growing overburden
pressure Po; see Eq. ), leading to an increase in ice
discharge and thus reducing further thickening. In the absence of any other
mechanisms, the ice sheet would hence eventually reach a steady state as ice thickening
would approach zero.
However, the continuous accumulation of water in the subglacial till during
the slow buildup initiates a surge event before the equilibrium thickness is
reached. The self-enforcing feedback of rapid ice speedup becomes dominant:
lowered friction at the well-lubricated ice-sheet base leads to an
acceleration of ice flow through the bed trough (Figs.
and ). In turn, this causes an increase in strain and frictional
heating due to enhanced shearing inside the ice sheet and sliding of the ice
over the bed, respectively (Figs. and ). The
resulting additional meltwater production further lubricates the ice base,
leading to even more speedup termed “hydraulic runaway”
by. Inside the bed trough, the previously relatively
stagnant ice flow has entered a state of rapid ice streaming (velocities at
several km yr-1; Figs. a, b and d). The ice
streaming is additionally fostered by the effect of strain heating at the
side margins of the trough (Fig. ): faster flow causes stronger
shearing of the ice, resulting in more heat production, which in turn softens
the ice, allowing for more shearing and thus flow acceleration so
called “creep instability”;see positive feedback loop in our
Fig. .
The ice streaming inside the bed trough leads to enhanced downstream
advection from the ice sheet's thick interior into the ice shelf, manifesting
a pronounced peak in ice discharge and iceberg calving
(Fig. d and e, respectively). The associated damping feedback between ice velocity
(ice flux) and thickness (blue stabilization loop in Fig. )
counteracts the self-enforcing feedback between ice velocity and till water
(red surge loop). On the long term, this discharge-related thinning of the
ice sheet leads to the end of the surge as meltwater production decreases,
basal friction increases and ice flow decelerates. When the ice sheet has
become too thin to maintain insulation of its base from the cold atmosphere
then the basal melt rate drops. The associated decrease in till water now is
amplified through the same mechanism that was responsible for the till water
increase during the surge phase (red loop) and the ice stream shuts down. At
some point basal refreezing sets in, consuming further water from the till
layer (Fig. ). As the water content in the drained till drops
close to zero and thus bed friction quickly increases, the ice sheet can
build up again. The period duration of a whole surge cycle is about
1800 years, from which the slow buildup phase takes about 80 %.
We would like to note that the domain-averaged yield stresses resulting from
our simulations (order of ∼ 100 kPa; see Fig. c) are
relatively large compared to values from in situ and laboratory experiments
order of ∼ 1 to ∼ 10 kPa; see Table 7.5
in. In our experiments the highest values occur during
the buildup phase, which is when the water content in the till is very close
to zero (s≈0), for which Eq. () yields
τc(s=0)∼105kPa. Though in the model τc is limited by
the overburden pressure, the maximum possible value in our experiments is
still on the order of ∼103 kPa assuming an ice thickness of
1000 m; for a visualization see Fig. 1 in. Such large
values occur predominantly in the regions outside of the bed channel and in
the thick interior of the ice sheet where the basal till layer is
continuously dry and ice flow is stagnant, biasing the domain average towards
high values. In contrast, inside the lubricated bed channel the simulated
yield stresses are much lower, especially during the phases of ice streaming
(on the order of ∼10 kPa; see Fig. A1c), lying within the observatory
range. This is in accordance with the fact that the observational values were
inferred from till samples stemming from regions of relatively fast
ice(-stream) flow
e.g.,.
Time series of (a) ice thickness H, (b) till
water thickness Wtill and (c) ice velocity (all averaged
over area of grounded ice) for different values of the till friction angle
(ϕ≤10∘). Between ϕ=10 (default case) and ϕ=8 there
is a transition from maintained cyclic surging to damped surging.
Surge damping
Varying the bed strength in our simulations, we find that surging is
maintained in a cyclic manner (oscillatory equilibrium) only if the bedrock
roughness allows the evolution of an ice sheet of medium thickness. For
rather slippery basal conditions, realized by low values of the till friction
angle (ϕ≤8∘) and thus rather thin ice sheets, surging occurs
initially but then is damped such that on the long term the ice sheet reaches
a non-oscillating stable equilibrium state (Fig. ). Decreasing the
value of ϕ within this regime leads to faster damping and a shorter
cycle duration. For sufficiently lubricated (thin) ice sheets no surging
takes place at all. In contrast to the case of maintained cyclic surging, the
ice flow enters a state of continuous streaming at velocities of several
100 m yr-1 with stable till water thickness (Fig. b and c).
Time series analogous to Fig. , here for ϕ≥10∘. For relatively large values of ϕ there is a transition
from maintained cyclic surging to damped surging.
Time series of ice thickness H for (a) an ice sheet in
oscillatory equilibrium (ϕ=10∘) which is perturbed by a decrease
of ϕ and (b) an ice sheet in stable equilibrium
(ϕ=8∘) which is perturbed by an increase of ϕ. In order to
bring the stable ice sheet into the regime of maintained surging ϕ has
to be increased substantially (ϕ=8∘→20∘), whereas it has to be lowered only slightly
(ϕ=10∘→8∘) to stabilize the
surging ice sheet.
Time series of (a) ice thickness H, (b) till
water thickness Wtill and (c) ice velocity (all averaged
over area of grounded ice) for different values of the surface accumulation
a. With decreasing a (default case in gray) the surge magnitude decreases
and the cycle duration increases such that for sufficiently low accumulation
surging is not existent.
(a) Surge-cycle duration and (b) mean grounded ice
mass for the q-ϕ parameter space. Each colored rectangle represents a
simulation characterized by either oscillatory surging (white circles),
damped surging (triangles) or stable equilibrium (squares). White rectangles
with an “x” denote parameter combinations for which no grounded ice forms
inside the bed trough and are thus not considered in the analysis. Since the
simulations of stable ice flow do not exhibit periodicity by definition the
associated rectangles in panel (a) are colored in gray. The default
simulation with parameters of q=1/3 and ϕ=10∘ is
highlighted by a purple circle (see Figs. and , gray
curve in Figs. , , , a and
).
Vice versa, increasing the friction angle towards large values yields rougher
beds, promoting the evolution of thicker ice sheets which surge at larger
magnitude. The surge frequency first increases (between ϕ=10 and
ϕ=30∘) before decreasing again (Fig. ). For
sufficiently strong beds with ϕ≥60∘ (and thus comparatively
thick ice sheets) initial surging is damped, similarly to the case of
relatively low values of ϕ discussed above. Consequently, surging is
maintained only in a regime of medium bed strength (medium values of ϕ)
that promotes ice sheets of medium thickness. Damped surging occurs on both
ends of this regime (above an upper and below a lower critical threshold of
ϕ), i.e., for relatively strong and weak beds.
We investigate changes in the ice-flow characteristics close to the lower
regime boundary in response to a small modification of the basal roughness.
For this purpose we perturb the above equilibrium ice sheets of oscillatory
(ϕ=10∘) and non-oscillatory (ϕ=8∘) type by
decreasing/increasing the value of ϕ. Our results show that when the
friction is lowered from ϕ=10∘ to values of ϕ≤8∘
then the originally surging ice sheet undergoes damping and eventually enters a stable
steady state (Fig. a). Hence, the flow
characteristics of the perturbed ice sheet are more or less the same as in
the spinup experiments when using the same values of the friction angle
(compare Figs. a and ). In contrast, when the
system is perturbed in the other direction, i.e., increasing the friction angle from
ϕ=8∘ (Fig. b), maintained surging only
occurs for values of ϕ≥20∘ (compared to ϕ=10∘ for
the case of ice-sheet spinup). For lower values of ϕ the ice flow starts
to oscillate initially but then goes back into a state of stable flow at the
same velocity as before, whereas now ice-sheet thickness and till water
content are larger (both increasing for increasing ϕ). Thus, the ice
sheet in stable equilibrium requires a comparatively large perturbation of
the basal conditions in order to turn into a state of maintained surging. In
contrast, a small perturbation is sufficient to bring the continuously
oscillating ice sheet into a stable steady state.
The finding from above, that thin ice bodies are less likely to surge than
ice sheets of medium thickness, is supported by additional experiments with
reduced surface accumulation a. According to these simulations, lower
accumulation results in thinner ice sheets, a weaker surge amplitude and a
longer surge-cycle duration (Fig. ). The longer cycle duration can
be explained by the fact that less snowfall causes the ice sheet to take
longer to grow thick enough to trigger a surge event. At the same time the
formation of basal till water during buildup takes longer and the kick-off
of the surge event requires a smaller amount of till water (and thus a
thinner ice body). Below a threshold of a fifth of the default value
(a=0.075 m yr-1) a rather thin steady-state ice sheet forms and
surging is nonexistent.
Role of basal sliding law
The above results show cyclic or damped surging for a confined set of
parameter values (default surface accumulation a and till friction angle
ϕ given in Table 1 are only slightly varied). These simulations use a
particular nonlinear sliding law, determined by a basal sliding exponent of
q=1/3 (Eq. ). In general, this exponent can range from q=0
(purely plastic sliding law) to q=1 (linear sliding law). To explore the
influence of the basal sliding law on the ice-flow behavior, we conduct
further simulations, sampling q between 0 and 1 at an interval of 1/12.
For each applied parameter value of q the till friction angle ϕ
(Eq. ) is varied between 5 and 85∘, spanning a wide range
from relatively slippery to very rough bed conditions, respectively. The
resulting q-ϕ parameter space is explored in terms of surge-cycle
duration and ice-sheet volume (Fig. ). Due to the large number of
simulations, the experiments are carried out on a grid of 5 km horizontal
resolution (in contrast to 1 km used in the default simulations).
The results show that (damped) surging occurs in a range from q=1/12 to
q=3/4 (circles and triangles in Fig. ). Within this regime
larger values of q correspond to higher friction angles ϕ; i.e., going
towards a more linear friction law requires a rougher bed in order to observe
(damped) surging. Maintained surging occurs in a smaller range, i.e., from
q=1/6 to q=5/12. This regime is embedded such that the transition
from the oscillatory state into the stable regime in most cases leads through
the damped regime. Generally, decreasing q or increasing ϕ yields a
longer period duration of the surge cycle while the mean grounded ice mass
increases. This can be explained by considering the relevant acting stresses:
a larger value of ϕ leads to a larger magnitude of the basal yield
stress (Eq. ) and thus a stronger basal shear stress
(Eq. ). In the SSA the driving stress (due
to surface slope of the ice sheet) is balanced by a combination of the
membrane stresses (responsible for ice-flow acceleration) and the basal shear
stresses see Eq. 17 and the following paragraph in.
An increase in basal shear thus slows down ice speedup, promoting a longer
period of ice-sheet buildup and larger ice-sheet thickness. Decreasing the
exponent q leads to the same results because here the magnitude of the
basal shear stress increases as well. This becomes evident from
Eq. (),
where the fraction (|ub|/u0)q increases with decreasing
q since |ub|/u0<1. The duration of the surge cycle
ranges from about 1800 to 2700 years for maintained surging (mean
≈ 2200 years) and from 800 to 5700 years for damped surging (mean
≈ 2600 years).
For the particular cases of purely plastic (q=0) or linear sliding (q=1)
no surging occurs in our simulations, which is independent of ϕ. In the
vicinity of q=0 most of the experiments produce a stable and rather thick
ice sheet (squares in Fig. ), whereas around q=1 the ice sheets
become very thin (a few 10 m of thickness). In some cases these very thin
ice sheets do not have any grounded ice inside the bed channel and thus lack
comparability (marked by an “x” in Fig. ). In general, very
small values of ϕ cause continuous streaming of a rather thin ice sheet
on a slippery bed, whereas large ϕ values lead to rough basal conditions
allowing the evolution of a comparatively thick steady-state ice sheet
(Fig. ). Thus, only those ice sheets which are not too large or
small show surging behavior. This confirms and generalizes our specific
results from Sect. that there is a thickness regime in which
surging occurs whereas too-thin or too-thick ice sheets reach a stable
equilibrium.
Discussion and conclusions
We model the cyclic surging of a three-dimensional, inherently buttressed,
marine ice-sheet-shelf system (Fig. ). Periodically alternating
ice growth and surge are unforced and emerge from interactions between the
dynamics of ice flow (evolution of velocity, internal and basal stresses, ice
thickness), its thermodynamics (heat conduction, strain and basal frictional
heating, meltwater production) and the subglacier (meltwater storage and
drainage).
We identify three consecutive phases throughout the surge cycle (ice
buildup, surge and stabilization), each characterized by a dominating
feedback mechanism which we visualize in a feedback-loop scheme
(Fig. ). These feedbacks of ice thickening slowdown, rapid ice
speedup and discharge, and decelerating ice thinning (Figs.
and ) can explain central processes that likely prevailed during
repeated large-scale surging of the Laurentide Ice Sheet and the associated
Heinrich events of global-scale impact. During the surge phase mainly the
process of hydraulic runaway positive feedback between basal meltwater production and flow acceleration; is in effect. It
is complemented by creep instability positive feedback between strain
heating and ice deformation;, which additionally promotes
rapid ice streaming (Figs. and ). The modeled cyclic
alternation of ice streaming and stagnation provides a simple example of
ice-stream shutdown and re-activation, a phenomenon which is characteristic
for the dynamics of some of the Siple Coast outlets in West Antarctica.
Our results suggest that medium-sized ice sheets are more susceptible to
cyclic surging than thin or thick ones. We find a transition from
surge to non-surge behavior (surge damping) of the ice flow when
decreasing/increasing the thickness of the surging ice body in our
simulations, realized by applying lower/larger basal roughness or surface
mass balance (Figs. and ) or by a variation of the
friction exponent in the sliding law (Fig. ). This is consistent
with the existence of a critical minimum ice thickness found by
. According to their results, exceeding this thickness
threshold enables the occurrence of creep instability, potentially leading to
rapid surging. Furthermore, our results reveal that an ice sheet in stable
equilibrium requires a comparatively large perturbation of the basal
conditions in order to turn into a state of maintained surging, whereas a
small perturbation is sufficient to bring the continuously oscillating ice
sheet into a stable steady state (Fig. ).
Compared to the observed interval of about 7000 years at which Heinrich
events re-occurred during the last glacial period , our
modeled surge-cycle period of ∼ 2000 years is much shorter. This is not
surprising given that our idealized model setup on a synthetic bed geometry
is not designed and the parameters are not tuned to represent conditions that
prevailed for the prehistoric Laurentide Ice Sheet. Thus, we refer to studies
designed to model this ice sheet when it comes to the proper representation
of the characteristic surge frequency of Heinrich events
e.g.,.
Our model results are closer to results from conceptual studies which also
use an idealized geometry
e.g.,. These
studies all yield a surge-cycle duration of ∼ 1000–2000 years despite
considerable differences in degree of physical approximations,
parameterizations and complexity in setup geometry. However, all of them use
a Weertman-type, stress-balance-based sliding law (Eq. ) and are
based on the same (though individually modified) subglacial model
, suggesting that both have a strong
imprint on the surge-cycle duration.
Conducting a parameter study that explores the q-ϕ space reveals that
both decreasing the sliding exponent q and increasing the friction angle
ϕ lead to an increase of the surge-cycle duration (Fig. ).
The dependence of the cycle duration on q is in accordance with results
from , who used a previous version of PISM and a
qualitatively different topographic setup (see Sect. 2 and below). By
sampling the parameter space between q=0 and q=0.3 they were able to
model maintained oscillation, including
the case of purely plastic basal sliding (q=0 in Eq. ), which in
our simulations only exists for q ranging between 1/6 and 5/12.
High-frequency oscillations with a period duration of ∼ 100 years as
found in their experiments (in addition to the “low-frequency” cycle
duration of ∼ 1000 years) are not present in our simulations.
Differences in the results compared to the PISM study of
are a combination of differences in (1) the model
versions, (2) the experimental setups and (3) the choice of model parameters.
The main difference between the two model versions is given by the applied
friction law, i.e., being linearly vs. nonlinearly determined by the
overburden pressure and the amount of water in the till layer (see Sect. 2).
One could expect that this substantial difference would reflect in the timescales of the modeled surge cycles. Though the absolute cycle durations
differ significantly between the two studies, the relative timing of the
surge initiation and the relative duration of the surge event (both with
respect to the full cycle duration) are very similar (surge phase takes about
∼ 20 % of the full cycle).
The different experimental setups, in particular the prescribed bed
topographies of different character, yield ice bodies of qualitatively very
different characteristics, i.e, an unbuttressed, tongue-shaped,
land-terminating glacier vs. an inherently buttressed marine ice-sheet-shelf
system (the latter being much larger in horizontal extent). Buttressing
increases the period duration of the surge cycle and decreases the magnitude
of grounding-line migration as shown in . Consistently,
we obtain a longer cycle duration (about a factor of 2) and a smaller
magnitude of grounding-line migration (relative to the ice-sheet length)
compared to . Maximum sliding velocities during
the surge phase are on the same order of magnitude
(|ub|∼ 1000 m yr-1) but still substantially
faster in our simulations. This is facilitated by the complete saturation of
the till layer in almost the entire bed trough and a temporary loss of
buttressing due to the advance of the central portion of the grounding line
in the course of the surge (compare our Figs. ,
and to Figs. 6 and 7 in ). Our
results confirm that complex boundary conditions e.g., spatially
varying surface temperature and accumulation, as applied
in are not a requirement in order to obtain cyclic
surging.
Note that for the conducted q-ϕ parameter study our choice of the
velocity scaling parameter u0 (Table 1) leads to a large spread of
possible magnitudes of the basal shear stress, ranging from
|τb|∼100 kPa for q=0 to
|τb|∼1 Pa for q=1 (calculated for typical values
of τc∼100 kPa and |ub|∼1000 m yr-1).
Choosing a parameter value on the order of magnitude of the sliding velocity
e.g., u0∼1000 m yr-1 or u0∼100 m yr-1, as
done in would result in a much more confined
|τb| value and thus facilitate the comparison of the
results. However, scanning the q-ϕ parameter space with values of u0
on these orders of magnitude revealed that in the given setup no surge-type
oscillations occur but a stable steady-state ice sheet emerges. This confirms
that the differences in the results between the two PISM studies can only be
attributed to the combined effect of model version, experimental setup and
parameter values.
The surface accumulation is found to be a further parameter with strong
influence on the surge-cycle duration in our simulations. Less snowfall leads
to a longer duration of the surge cycle (Fig. ) since the ice
sheet takes longer to grow thick enough to trigger a surge event. This
correlation between surface accumulation and surge frequency is also found in
other studies modeling surge events
e.g.,. However, a decrease of the
surge magnitude with decreasing snowfall as found in our simulations is not
present in these studies, whereas this behavior is consistent with results
from the PISM study of , who find a decrease in
surge amplitude when increasing the equilibrium line altitude. One essential
difference between the corresponding applied models and PISM is that these
models parameterize the sliding velocity through the local basal conditions whereas
in PISM the sliding velocity results from solving the non-local shallow-shelf
approximation of the stress balance. Besides several other differences in
model type and geometric setup, this might also be the cause why a variation
of basal sliding does not affect the period duration of a surge cycle in
these studies, contrary to our findings.
Several other parameters in our model likely have an effect on the occurrence
of surging and its dynamics (e.g., the overburden-pressure fraction δ
in Eq. , the till drainage rate Cd in Eq.
as well as surface temperature, geothermal heat flux and bed slope). However,
further investigation of the parameter dependency of the surging behavior
e.g., as done for surface temperature and geothermal heat flux
in is beyond the scope of this study. In fact, it aims to
report on the realization of cyclic surging/ice streaming of an
ice-sheet-shelf system in the Parallel Ice Sheet Model based on suitable
model components and a justified set of parameters.