Mass loss from the Antarctic Ice Sheet is the main source
of uncertainty in projections of future sea-level rise, with important
implications for coastal regions worldwide. Central to ongoing and future
changes is the marine ice sheet instability: once a critical threshold, or
tipping point, is crossed, ice internal dynamics can drive a self-sustaining
retreat committing a glacier to irreversible, rapid and substantial ice
loss. This process might have already been triggered in the Amundsen Sea
region, where Pine Island and Thwaites glaciers dominate the current mass
loss from Antarctica, but modelling and observational techniques have not
been able to establish this rigorously, leading to divergent views on the
future mass loss of the West Antarctic Ice Sheet. Here, we aim at closing
this knowledge gap by conducting a systematic investigation of the stability
regime of Pine Island Glacier. To this end we show that early warning
indicators in model simulations robustly detect the onset of the marine ice
sheet instability. We are thereby able to identify three distinct tipping
points in response to increases in ocean-induced melt. The third and final
event, triggered by an ocean warming of approximately 1.2 ∘C from
the steady-state model configuration, leads to a retreat of the entire
glacier that could initiate a collapse of the West Antarctic Ice Sheet.
Introduction
The West Antarctic Ice Sheet (WAIS) is regarded as a tipping element in the
Earth's climate system, defined as a major component of the Earth system
susceptible to tipping-point behaviour (Lenton et al., 2008). Its collapse,
potentially driven by the marine ice sheet instability (MISI; Feldmann and
Levermann, 2015), would result in over 3 m of sea-level rise (Fretwell et al.,
2013). Key to MISI are the conditions at the grounding line – the
transition across which grounded ice begins to float on the ocean forming
ice shelves. In a steady state, ice flux across the grounding line balances
the surface accumulation upstream. If grounding-line retreat causes
grounding-line flux to increase and this is not balanced by a corresponding
increase in accumulation, the net mass balance is negative and retreat will
continue (Weertman, 1974; Schoof, 2007). Conversely, grounding-line advance
leading to an increase in accumulation greater than the change in flux will
lead to a continued advance. In this regime, a small perturbation in the
system can result in the system crossing a tipping point, beyond which a
positive feedback propels the system to a contrasting state (Fig. 1c). A
complex range of factors can either cause or suppress MISI (Haseloff and Sergienko,
2018; Pegler, 2018; O'Leary et al., 2013; Gomez et al., 2010; Robel et al., 2016), and the
difficulties in predicting this behaviour are a major source of uncertainty
for future sea-level-rise projections (Church et al., 2013; Bamber et al., 2019;
Oppenheimer et al., 2019; Robel et al., 2019).
Possible range of behaviours for a system state (e.g. ice flux) in
response to perturbations in a control parameter (e.g. ocean temperature). A
system can (a) respond to a perturbation in a linear way that is directly
recoverable with a reversal of the forcing; (b) have a large response to a
small perturbation that is still directly recoverable; (c) have a large
response to a small perturbation that is irreversible (hysteresis
behaviour); and (d) have a large response that is irreversible for any
change in the tested range of the control parameter, a behaviour we refer to
as permanently irreversible (no recovery possible even if forcing is
reversed well below the initial level). Tipping points are crossed only in
panels (c) and (d) and are indicated by x1, x2 and x3. Panel (c) also shows a transient response in which the system state lags behind
changes in the control parameter as is the case for ice sheets and thus
crosses the irreversible system state at a later point, xt.
One area of particular concern is the Amundsen Sea region. Pine Island (PIG)
and Thwaites glaciers, the two largest glaciers in the area, are believed to
be particularly vulnerable to MISI (Favier et al., 2014; Rignot et al., 2014).
Palaeo-records and observational records of PIG show a history of retreat, driven by
both natural and anthropogenic variability in ocean forcing (Jenkins et al., 2018;
Holland et al., 2019). One possible MISI-driven retreat might have happened when
PIG unpinned from a submarine ridge in the 1940s (Jenkins et al., 2010; Smith et al.,
2016). Recent modelling studies indicate that a larger-scale MISI event may
now be underway for both Pine Island and Thwaites glaciers that would lead
to substantial and sustained mass loss throughout the coming centuries
(Favier et al., 2014; Jenkins et al., 2016; Joughin et al., 2010). Being able to identify a MISI-driven retreat and differentiate this from a retreat where a tipping point
has not been crossed is vital information for projections of future
sea-level rise. One of the major hurdles in determining whether a tipping
point has been crossed is that currently this necessitates time-consuming
steady-state simulations to calculate the hysteresis behaviour of an
identified period of retreat (e.g. Garbe et al., 2020). An alternative methodology
that can be applied directly to transient simulations as a post-processing
step would therefore be useful to the ice sheet modelling community. Such
tools based on early warning indicators are presented in this paper.
The tipping behaviour of MISI is an example of a saddle-node (or fold)
bifurcation in which three equilibria exist: an upper and lower stable
branch and a middle unstable branch (Fig. 1c; Schoof, 2012). Starting on the
upper stable branch, perturbing the system beyond a tipping point (x1
in Fig. 1c) will induce a qualitative shift to the lower and contrasting
stable state. Importantly (and in contrast to a system such as that shown in
Fig. 1a and b), in order to restore conditions to the state prior to a
collapse it is not sufficient to simply reverse the forcing to its previous
value. Instead, the forcing must be taken back further (to point x2),
which in some cases may be far beyond the parameter range that triggered the
initial collapse. This type of behaviour is known as hysteresis. A large
change in response to a small forcing is not necessarily indicative of
hysteresis, as shown in Fig. 1b. Tipping points are crossed in both Fig. 1c
and Fig. 1d, and both cases are often referred to as irreversible, although
the two are distinct in that only Fig. 1d is irreversible for any change in
the tested range of the control parameter. Hereafter we will refer to the
former as irreversible, in line with previous studies, and the latter as
permanently irreversible, to differentiate the two. Diagnosing whether a
tipping point has been crossed without some prior knowledge of the system is
not generally possible without reversing the forcing to see if hysteresis
has occurred. An alternative approach to identify tipping points is based on
a process known as critical slowing down, which is known to precede saddle-node bifurcations of
this type (Wissel, 1984; van Nes and Scheffer, 2007; Dakos et al., 2008; Scheffer
et al., 2009). Critical slowing down is a general feature of non-linear systems
and refers to an increase in the time a system takes to recover from
perturbations as a tipping point is approached (Wissel, 1984). We will
explore both hysteresis and critical slowing down as indicators of tipping
points in our model simulations.
In Sect. 2, we explain critical slowing down and early warning indicators
in the context of MISI. We then map out the stability regime of PIG
using numerical model simulations. We force the model with a slowly
increasing ocean melt rate and identify three periods of rapid retreat with
the methodology explained in Sect. 3.1. Using statistical tools from
dynamical systems theory we find critical slowing down preceding each of
these retreat events and go on to demonstrate that these are indeed tipping
points in Sect. 4. This is confirmed by analysing the hysteresis behaviour
of the glacier, showing the existence of unstable grounding-line positions.
To our knowledge, this is the first time that the stability regime of PIG
has been investigated in this detail and the first time that tipping-point
indicators have been applied to ice sheet model simulations. Our results
reveal the existence of multiple tipping points leading to the collapse of
PIG, rather than one single event, that when crossed could easily be
misidentified as simply periods of rapid retreat, with the irreversible and
the self-sustained aspect of the retreat being missed.
Critical slowing down and early warning indicators
As certain classes of complex systems approach a tipping point, they show
early warning signals (i.e. specific changes in system behaviour as detailed
below) which can allow us to anticipate or even predict the onset of a
tipping event by means of statistical tools called early warning indicators
(EWIs; Wissel, 1984). Early warning signals have been found to precede, for
example, collapse of the thermohaline circulation (Held and Kleinen, 2014;
Lenton, 2011), onset of epileptic seizures (Litt et al., 2001; McSharry and
Tarassenko, 2003), crashes in financial markets (May et al., 2008; Diks et al., 2018),
onset of glacial terminations (Lenton, 2011) and wildlife population
collapses (Scheffer et al., 2001). Although most commonly used to detect the onset
of saddle-node bifurcations, of which MISI is an example, they are not
strictly limited to bifurcations of this type and have, for example, also
been successfully used to indicate the onset of Hopf bifurcations (Chisholm
and Filotas, 2009).
Critical slowing down preceding the marine ice sheet instability
Critical slowing down is one example of an early warning signal that has
been used in the past for both model output and observational records such
as palaeoclimate data, with the aim of detecting an approaching bifurcation
(Held and Kleinen, 2004; Livina and Lenton, 2007; Dakos et al., 2008; Lenton et al., 2009, 2012b). Critical slowing down is so called because, as a non-linear
system is gradually forced towards a bifurcation, that system will become
more “sluggish” in its response to perturbations (see middle panel of Fig. 2). This can be shown mathematically, because the dominant eigenvalue of the
system tends to zero as a bifurcation point is approached (Wissel, 1984)
or, equivalently, the recovery time (i.e. the time it takes for a system to
return to a steady state after small perturbations) tends to infinity. The
response time of a glacier to external forcing has also been shown
analytically to increase as a MISI bifurcation is approached (Robel et al., 2018).
While critical slowing down is a general characteristic behaviour of the dynamics
underlying MISI, the question remains whether it can be reliably
detected in the context of a complex glacier where many other processes are
at play.
Critical slowing down can serve as an indicator that the system is
approaching a tipping point. This can be understood conceptually using the
common “ball on a slope” analogy (middle panel), where the ball represents
the system state and minima are stable equilibrium states. Two example cases
are superimposed onto their corresponding positions in the hysteresis plot
of MISI shown by the green line and equivalent to Fig. 1c. The processed
model results demonstrate how critical slowing down manifests itself, as
shown in the blue and red panels at the sides. If the system is far from a
tipping point (blue case), the system state (which is in this study the
grounding-line flux qGL, upper left panel) recovers quickly from
perturbations in the control parameter (which is here the basal melt
variability). This means that from one measurement (at time t) to the next
(at time t+1) the grounding-line flux changes rapidly and has a low lag-1
auto-correlation (lower left panel). Conversely, close to a tipping point
(red case), critical slowing down manifests and the system state responds
more slowly to perturbations in the control parameter (upper right panel).
Since the state variable is changing more slowly, successive measurements are
more similar, resulting in a higher lag-1 auto-correlation (lower right
panel).
As a first step to addressing this question, we model MISI in an idealised
flow line setup of a marine ice sheet. In this setup, we determine the change
in recovery time before a tipping point directly through multiple stepwise
perturbations of the control parameter (Appendix A). Our setup closely
resembles the MISMIP experiments (Pattyn et al., 2012), and indeed hints of
critical slowing down can be identified in that paper (Fig. 2 in Pattyn et al.,
2012). The results in Appendix A show that critical slowing down is easily
identified preceding both MISI-driven advance and retreat bifurcations. This
demonstrates that there is at least the potential that critical slowing down
could be found in a less simplified modelling framework. This is not clear a
priori, and, for example, adding noise to the bed topography reduces
the ability to identify early warning, as detailed in the Appendix.
Identifying critical slowing down in this stepwise perturbation manner is
appealing because it directly extracts the change in response time that we
are searching for; however it is not practical for a realistic model forcing
which would not normally take the form of a step function. A more general
approach, which we adopt for our simulation of PIG, is to use EWIs to
analyse the recovery time of the system as it is forced with natural
variability.
Early warning indicators
As the field of EWIs has expanded, more methods have been developed for
extracting critical-slowing-down information from model results and
observational records. These methods seek to approximate the system recovery
time from some measure of the system state. The challenge is that, for most
real-world applications, natural forcing does not take the form of a step
function and the system is continuously perturbed and so cannot return to a
true steady state. However, if the recovery time of a system is indeed
increasing, the response to a continual stochastic forcing could be detected
as a tendency for each measurement of the system state to become
increasingly similar to the previous measurement, sometimes referred to as
an increase in “memory” of small perturbations. This is shown conceptually
and with examples extracted from our PIG model in Fig. 2. One common way to
measure this effect is by sampling the data at discrete time intervals and
calculating the lag-1 auto-correlation, i.e. the correlation between values
that are one time interval apart (examples given in Fig. 2). This measure,
which we refer to hereafter as the ACF indicator, should increase as a tipping point is
approached (Dakos et al., 2008; Ives, 1995). Since recovery time tends to infinity
as the bifurcation is approached, successive system states should become
more and more similar and the ACF indicator should tend to 1. This
threshold of an indicator, corresponding to the point when a tipping point
would be crossed as predicted by theory, is referred to hereafter as the
critical value. An alternative measure that also seeks to identify changes
in recovery time is to use the detrended fluctuation analysis algorithm
(Livina and Lenton, 2007; Lenton et al., 2012a, b). This first
calculates the mean-centred cumulative sum of the time series, splits the
result into epochs of length n which are detrended and then calculates the
rms F(n) for each epoch. This is repeated for epochs of different length, and
finally an exponent α can be fitted in log–log space such that
Fn∝nα. This exponent yields information on
the self-correlation of the original time series, whereby a value of 0.5
corresponds to uncorrelated white noise and greater values indicate
increasing “memory” up to a maximum of 1.5. To aid comparison with the ACF
indicator and following Livina and Lenton (2007), we rescale the exponent so
that it reaches a critical value of 1 and call this the DFA indicator. These indicators
can be supported by analysing the variance of the system state. Variance can
be shown to increase as a tipping point is approached, since perturbations
to the system decay more slowly and thus large shifts from the mean state
will persist for longer (Scheffer et al., 2009). No critical value exists in this
case, but a persistent positive trend in variance serves as additional
evidence that a tipping point is being approached.
Methods
We conduct a quasi-steady modelling experiment whereby we subject PIG to
slowly increasing rates of basal melt beneath its adjacent ice shelf (Fig. 3). Conducting a transient simulation with an evolving basal melt that
exactly tracks the equilibrium curve (Fig. 1c) is not computationally
feasible or necessary for our purposes. Thus, we adopt this quasi-steady
modelling approach in which the forcing increases slowly enough that it
approximates the steady-state behaviour but faster than the long response
timescales the glacier would require to be truly in equilibrium.
Quasi-steady-state experiments have previously been successfully applied to
identify the tipping point of the Greenland Ice Sheet with respect to the
melt–elevation feedback (Robinson et al., 2012) and to identify hysteresis of the
Antarctic Ice Sheet (Garbe et al., 2020). In Garbe et al. (2020) it was shown that such
transient experiments enable identification of hysteresis behaviour, while
the exact shape of the curve must be mapped out with equilibrium
simulations. We accompany the quasi-steady simulations with simulations that
run to a steady state for constant values of the control parameter at
discrete values (these simulations continue until the change in ice volume
is approximately equal to zero). We use basal melt rate as the control
parameter, i.e. the parameter that we will change to drive the system
towards a tipping point. We make this choice since erosion of ice shelves by
the intrusion of warm ocean currents is widely accepted as the mechanism
responsible for the considerable changes currently observed in this region
(Shepherd et al., 2004; Rignot et al., 2014; Rignot, 1998; Joughin et al., 2010; Park et al., 2013;
Gudmundsson et al., 2019). Sub-ice-shelf melt rates are increased linearly (with
additional variability as explained below) from a value that generates a
steady state for the present-day glacier configuration. Based on the
numerical experiments we then evaluate EWIs to test for critical slowing
down.
Marine ice sheet instability events for Pine Island Glacier. Shown
are (a) grounding-line positions before and after the three MISI-driven
glacier collapses with (b) a zoom-in on the initial events (coloured lines).
The colour map indicates initially modelled ice velocity, and the model domain
boundary is indicated by a dashed black contour in panel (a). Panels (c) and
(d) show a transect through the main trunk of PIG, calculated as an average
of properties between the two dashed magenta lines in (b). The vertical
section along the transect is shown (c) at the initial steady state where
fluxes (Qin and Qout) are in balance and (d) during a MISI event
where retreat causes an increase in Qout, pushing the glacier to be out
of balance and leading to further retreat.
Model description
All simulations use the community Úa ice-flow model (Gudmundsson et al., 2012;
Gudmundsson, 2013, 2020), which solves the dynamical equations
for ice flow in the shallow-ice-stream approximation (SSTREAM or SSA;
Hutter, 1983). Bedrock geometry for the PIG domain is a combination of the
R topo2 dataset (Schaffer et al., 2016) and, where available, an updated bathymetry
of the Amundsen Sea embayment (Millan et al., 2017). Surface ice topography is from
CryoSat-2 altimetry (Slater et al., 2018). Depth-averaged ice density is calculated
using a meteoric ice density of 917 kg m-3 together with firn depths
obtained from the RACMO2.1 firn densification model (Ligtenberg et al., 2011). Snow
accumulation is a climatological record obtained from RACMO2.1 and constant
in time (Lenaerts et al., 2012).
Viscous ice deformation is described by the Glen–Steinemann flow law
ε˙=AτEn with exponent n=3, and basal motion is
modelled using a Weertman sliding law ub=Cτbm with exponent
m=3. The constitutive law and the sliding law use spatially varying
parameters for the ice rate factor (A) and basal slipperiness (C),
respectively, to initialise the model with present-day ice velocities. These
are obtained via optimisation methods using satellite observations of
surface ice velocity from the Landsat 8 dataset (Scambos et al., 2016; Fahnestock
et al., 2016). An optimal solution is obtained by minimising a cost function that
includes both the misfit between observed and modelled velocities and
regularisation terms. An additional term in the cost function penalises
initial rates of ice thickness change in order to ensure that these are
close to zero at the start of simulations. This approach helps to provide a
steady-state configuration of PIG from which we can conduct our perturbation
experiments.
The Úa model solves the system of equations with the finite-element
method on an unstructured mesh, generated with MESH2D (Engwirda et al., 2014). The
mesh remains fixed throughout the simulation to avoid contaminating the time
series with errors resulting from remapping fields onto a new mesh. The mesh
is refined in regions of high strain rate gradients and fast ice flow as well as
around the grounding line. The region of grounding-line mesh refinement, in
which the average element size is ∼ 750 m, extends upstream
sufficiently far so that the grounding line always remains within this
region until after the final MISI collapse.
Basal melt rates are calculated using a widely used, local quadratic
dependency on thermal forcing:
M=fγTρwcpρiLi2T0-TfT0-Tf,
where γT is the constant heat exchange velocity, ρw is seawater density, cp is the specific heat capacity of water,
ρi is ice density, Li is the latent heat of fusion of ice,
T0 is the thermal forcing and Tf is the freezing temperature
(Favier et al., 2019). Melt rates are only applied beneath fully floating elements
to ensure that no melting can possibly occur upstream of the grounding line
(Seroussi and Morlinghem, 2018). The initial melt rate factor (f) is
chosen such that the model finds a steady state with a grounding line
approximately coincident with its position as given in Bedmap2 (Fretwell et al., 2013).
This melt rate factor is the aforementioned control parameter that drives
changes in the model, some of which may be identifiable as tipping points.
To effectively extract information about the system's recovery time using
the statistical methods outlined in Sect. 2, we need to perturb the model in
a way that has some measurable impact on the system state. A slow and
monotonically increasing forcing would make our chosen approach impractical
and is arguably as unrealistic as a stepwise perturbation. We therefore add
natural variability to the linearly increasing melt rate factor (f). There
is strong evidence that the inferred and observed changes in PIG over the
last century can be linked to changes in thermocline depth of the Amundsen
Sea shelf, which in turn is influenced by an atmospheric Rossby wave train
originating in the Pacific Ocean (Jenkins et al., 2018). Following Jenkins et al. (2018),
we use a ∼ 130-year time series of central tropical Pacific
sea surface temperature anomalies as a proxy for relevant variability in our
melt rate forcing. We create an autoregressive (AR) model-based surrogate
from this time series using the Yule–Walker method to fit the AR model and
minimum description length to determine the maximum order of the model. This
new surrogate time series has the same decadal variability that would be
expected for the melting beneath PIG and can be extended to any length
required. As shown in more detail below, by superimposing this signal onto
the linearly increasing melt rate factor we ensure that the system response
contains sufficient variability to extract information about critical
slowing down and thereby enable the calculation of EWIs. Furthermore, using
natural variability enables us to test the versatility of EWIs if they were
to be applied directly to observations.
Detecting critical slowing down
We have already established the control parameter for our model, but another
important decision to make is what model output should be used as a measure
of the system state. One choice could be changes in ice volume, since they can
be related to sea-level rise and ice sheet model simulations tend to focus
on this result. However, ice volume varies very smoothly over time, making
it difficult to detect changes in the system recovery time. Instead, we use
the integrated grounding-line flux, which shows greater temporal variability
and whose change is directly related to the MISI mechanism. As with other
studies of this type, the model output is processed prior to the calculation
of EWIs. This consists of aggregating the output (i.e. data binning) to
remove variability with a frequency higher than that directly relevant to
the internal ice dynamics considered here and thus not related to the system
recovery time, together with detrending to remove non-stationarities
(detrending is included in the DFA algorithm and therefore not required
before calculation of the DFA indicator). Detrending was carried out using a
Gaussian kernel smoothing function that has been shown to perform better
than linear detrending (Lenton et al., 2012a). A smoothing bandwidth was selected
that removed long-term trends without overfitting the model time series.
Indicators are calculated over a moving window with a length of 300 years.
The optimal window length is further discussed in Sect. 4.3.
From the processed time series, we calculate three different EWIs:
Critical slowing down is measurable as an increase in the state variable
auto-correlation. We measure this here using the lag-1 auto-correlation
function (Dakos et al., 2008; Scheffer et al., 2009; Held and Kleinen, 2004) applied to
the grounding-line flux over a 300-year moving window preceding each tipping
point (ACF indicator).
Similarly, DFA (Peng et al., 1994) measures increasing auto-correlation in a time
series and we apply this with the same moving-window approach.
An additional consequence of critical slowing down is that variance will
increase as a tipping point is approached (Scheffer et al., 2009). We calculate
variance of grounding-line flux for each moving window, and this can be used
in conjunction with other indicators to increase robustness.
As described in Sect. 2, recovery time should tend to infinity as a tipping
point is approached. This corresponds to the ACF and scaled DFA indicators
reaching a critical value of 1. In practice, for a complex model there are
a wide variety of reasons why a tipping point might be crossed before the
EWI reaches a critical value. For example, this can be a result of
variability in the control variable pushing the system over a tipping point
despite its long-term mean still being some distance from its critical
value. For this reason, most studies adopt an alternative approach of
looking for a consistent increase in the EWIs in the run up to a tipping
event. This is often measured by calculating the nonparametric Kendall's τ
coefficient, a measure of the ranking or ordering of a variable, which equals
1 if the indicator is monotonically increasing with time (Dakos et al., 2008;
Kendall, 1948). This single value enables a simple interpretation of our
results, since 0<τ≤1 means the EWI is tending to increase with
time, suggesting an imminent tipping point. The closer to 1 the calculated
τ coefficient is, the greater the tendency for an indicator to be
increasing with time, and conversely a τ coefficient close to zero suggests
no clear tendency for an indicator to be changing with time. We present our
results in terms of both aforementioned criteria: whether an EWI reaches a
critical value preceding the tipping point and whether the EWI is
consistently increasing for a period of time before the tipping point.
Results
The quasi-equilibrium simulation shows three potential tipping points with
respect to the applied melt (Fig. 4). Upon crossing each threshold,
indicated by the numbered blue dots in Fig. 4, PIG undergoes periods of not
only rapid but also (as we show below) self-sustained and irreversible mass
loss. At this stage, relying only on a record of changes in ice volume
resulting from an increasing forcing (solid black line in Fig. 4), one can
only speculate that these are indeed tipping points and more analysis is
necessary to confirm this hypothesis, and we will address this point in
Sect. 4.2. The last of the three events causes a permanently irreversible
collapse within the entire model domain (Fig. 4a). We focus our results on
these three major changes in the glacier configuration and ignore any
possible smaller tipping points that do not result in significant grounding-line retreat or changes in ice volume. We increase basal melt rates
gradually and in a quasi-steady-state manner to ensure that successive
retreat events can be isolated and their effects do not overlap during the
simulation. A more rapidly increasing forcing could lead to one tipping
point cascading into the next and result in three individual tipping points
being misinterpreted as only one event.
Change in system state in terms of sea-level equivalent ice volume
as a function of the control parameter, which is the melt rate at the
ice–ocean interface. (a) The model is run forward with a slowly increasing
basal melt rate (solid black line) and shows three distinct tipping points
(blue dots). From the start of the transient simulation to the third tipping
point is approximately 10 kyr. The steady states for a given melt rate in
both an advance and retreat configuration are plotted as dashed grey lines,
with details shown in panel (b). Arrows indicate the direction of the
hysteresis. Panel (b) focuses on the model response before the larger
tipping point (event 3) and shows the three windows that we analyse for
early warning indicators as shaded red boxes (Fig. 5). Circle and square
symbols represent steady-state configurations for a given forcing, and the
dashed grey line is a linear interpolation between these points. Each step
in melt rate for the steady-state runs from ∼ 10 to
∼ 30 m a-1 is approximately equivalent to 0.4 m a-1
of basal melting, or 250 years in the transient simulation. The lower branch
in panel (a) represents a simulation starting from the PIG configuration
after the third major retreat event and reverses the basal melt rate factor
to its lowest value, showing no recovery in ice volume.
Grounding-line positions before each of these retreat events and after the
final collapse are shown in Fig. 3. Events 1 and 2 each contribute
approximately 20 mm of sea-level rise, while event 3, which arises after
slightly more than doubling current melt rates, contributes approximately
100 mm. The actual sea-level rise that would result from this third and
largest event is likely to be larger since in our simulation the effects
stop at the domain boundary and in reality neighbouring drainage basins
would be affected.
Early warning for the marine ice sheet instability
The three periods of MISI-driven retreat, each following the crossing of an
associated tipping point, can be identified clearly using EWIs (Fig. 5). The
ACF indicator increases and tends to 1 as the tipping points are
approached (Fig. 5a–c), indicating a tendency towards an infinitely long recovery
time as predicted by theory. We calculate Kendall's τ coefficient to
identify trends in the indicator, with a value of 1 representing a
monotonic increase in the indicator with time. The positive Kendall's τ
coefficient shows that in all three cases, the lag-1 auto-correlation
increases before the onset of unstable retreat. Furthermore, the ACF
indicator reaches a critical value of 1 relatively close in time to when
the MISI event begins.
EWIs for the marine ice sheet instability in Pine Island Glacier.
Each panel shows the EWIs preceding each of the three MISI tipping-point events
marked in Fig. 3b, along with the linear trend extrapolated to the point in
the simulation when the respective tipping event occurs. Increasing trends
in all indicators are shown by a positive Kendall's τ coefficient which
measures the correlation between each indicator and time at between -1 and 1.
These findings are supported by the DFA indicator, described in Sect. 2. As
with the ACF indicator, Kendall's τ coefficient is positive and the DFA
indicator trends towards a critical value of 1 as each tipping point is
approached. We show the change in normalised variance calculated over each
time window, and in all cases this increases ahead of the tipping points
being crossed with a positive Kendall's τ coefficient. The increase in
variance gives greater confidence to the findings of the other two EWIs,
although variance cannot be used directly to predict when that threshold
will be crossed since it does not approach a critical value before a tipping
point is crossed.
Hysteresis of Pine Island Glacier
In order to verify that we have correctly identified tipping points using
the EWIs, we run the model to a steady state for a given melt rate to search
for hysteresis loops that indicate the presence of unstable grounding-line
positions. These simulations start from either the initial model setup
(advance steady state) or the configuration just prior to the final tipping
point (retreat steady state). Each of these simulations samples a discrete
mean melt factor between these two states that is held constant (but with
the addition of the same natural variability as in the forward simulations)
and is run forward in time until the modelled ice volume reaches a steady
state. The first two tipping events show relatively small but clearly
identifiable hysteresis loops (Fig. 4b), for which recovery of the grounding-line position requires reversing the forcing beyond the point at which
retreat was triggered (i.e. as shown in Fig. 1c). The third event marks the
onset of an almost complete collapse of PIG (Fig. 4a). Unlike the previous
two, this collapse cannot be reversed to regrow the glacier for any value of
the control parameter. This is an example of a permanently irreversible
tipping point, as shown in Fig. 1d. Note that this permanent irreversibility
is only true for the glacier modelled in isolation and by expanding the
domain it would presumably be possible for other catchments that may not
have collapsed to enable this glacier to regrow.
Robustness of the indicators
We carry out several tests to assess the robustness of the EWIs and their
sensitivity to the pre-processing carried out on the model output prior to
calculating each indicator. Two parameters in this processing step are the
bin size into which data are aggregated and the bandwidth of the smoothing
kernel that removes long-term trends in the time series. To check that the
increasing trends in our indicators are a robust feature of our results,
regardless of these choices, we conducted a sensitivity analysis. The
parameters were varied by ±50 %, and the indicators were recalculated
for each resulting time series. As before, we assess the utility of an
indicator by whether it shows an increasing trend before each tipping point,
as measured by a positive Kendall's τ coefficient. The results of this
sensitivity analysis are presented for each MISI event in Fig. 6. Kendall's τ coefficient is positive for all tested combinations of parameters and all
MISI events, although MISI event 2 is particularly insensitive to these
parameter choices, whereas the spread in Kendall's τ coefficient is greater
for the other two events.
Sensitivity analysis for the ACF and DFA indicators. Each
occurrence is Kendall's τ coefficient for a different choice of
filtering bandwidth and data aggregation. The solid red and blue lines show
Kendall's τ coefficient for the DFA and ACF indicators, respectively, as
calculated for the choice of parameters used in Fig. 5.
In general, critical slowing down will only occur close to a tipping point.
Determining how close to a tipping point a system must be in order to
anticipate the approaching critical transition, i.e. the prediction radius,
is an important question and also informs the selection of palaeo-records
that could be used to detect an upcoming MISI event. The results presented
above are for a window size of 300 years (i.e. a record length of 600 years), which is the shortest window size for which the DFA indicator
provides a clear prediction for all tipping events. We explored the
prediction radius of our model by calculating Kendall's τ for the ACF and
DFA indicators and the variance for a range of window lengths; see Fig. 7.
For the main tipping event, preceded by the longest stable period, the
indicators gradually lose their ability to anticipate a tipping event. This
is shown by Kendall's τ values approaching zero or in some cases
becoming negative, meaning that the EWIs are not robustly increasing before
each tipping point as a result of more data being included further from the
bifurcation. The same is true for the two smaller tipping events, but the
drop-off is quicker such that the indicators break down for window lengths
> 500 years. These results suggest that the prediction radius is
relatively small, and thus window sizes that are too large, which hence include data
far from a tipping point, become less useful for the application of EWIs.
The effect of window length on the predictive power of EWIs for
MISI. The three panels show the change in Kendall's τ coefficient as
calculated for each indicator versus window length for MISI events 1, 2 and
3 (panels a, b and c, respectively).
In addition to a sensitivity analysis, it is important to check that trends
in the calculated indicators are statistically significant and not the
result of random fluctuations. We follow the method originally proposed by
Dakos et al. (2012) and produce surrogate datasets from the model time series that
have many of the same properties but should not contain any critical-slowing-down trends. We generate 1000 of these datasets using an autoregressive
AR(1) process-based surrogate. For each of these datasets we calculate the
ACF and DFA indicators and variance in the same way as with the model time
series and then estimate the trend with values of Kendall's τ coefficient.
We calculate the probability of our results being a result of chance for
each indicator and for all three combined as the proportion of cases for
which the surrogate dataset was found to have a higher correlation than the
model time series. We find that P<0.1 in all but one instance for
the ACF and DFA indicators but variance trends were generally less
significant (Table 1). However, the combined probability that all three
indicators would be equally positive as a result of chance was less than
0.02 for the first MISI event and less than 0.005 for the second and third
events.
Probability of Kendall's τ correlation for each indicator
being a result of chance. A total of 1000 surrogate time series of the state
variable are generated, and the indicators and Kendall's τ correlations
are calculated for each one. The probability of Kendall's τ value is then the
fraction of these surrogate time series with a higher correlation
coefficient. The total probability is the fraction of surrogates for which
all three indicators have a higher correlation coefficient than is observed
in the original model time series.
The indicators we have tested provide early warning of tipping points as
they are approached in our transient simulation with gradually increasing
melt rates. Tipping points driven by MISI represent potential “high-impact” shifts in the Earth climate system, since they may lead to
considerable changes in the configuration of the Antarctic Ice Sheet that
are effectively irreversible on human timescales. Computational models are
frequently used to forecast future changes in the Antarctic Ice Sheet in
response to various greenhouse gas emission and warming scenarios. Predictive
studies of this kind sometimes label periods of rapid retreat as “unstable”
without further analysis of the type performed here (e.g. Joughin et al., 2014; Ritz
et al., 2015; Favier et al., 2014) or avoid making this diagnosis altogether (DeConto and
Pollard, 2016). Here, we have demonstrated that EWIs robustly approach
critical thresholds preceding tipping points driven by MISI. Our results
show that EWIs can be used as a method to identify instabilities without the
need for the aforementioned modelling approach based on computationally
expensive equilibrium simulations.
It is important to clearly understand what critical threshold is identified
by the EWIs. In Fig. 4 the simulated steady states show the crossing of the
tipping point earlier than identified by the indicators in the transient
simulation. Since the timescales of ice flow are longer than the forcing
timescale, the ice sheet system modelled here does not evolve along the
steady-state branch (as shown schematically in Fig. 1c). Relaxation to a
steady state takes centuries to millennia in the simulations. This means
that while technically the critical value of the control parameter (basal
melt rate) might have already been crossed, the glacier could return to its
previous state in the transient simulation at that point if the basal melt
rate was reduced below the critical threshold. This is true until the system
state variable crosses its critical value (point xt in Fig. 1c) – and
this is the point identified by the EWIs. This complication in interpreting
EWIs is inherent to ice dynamics because of its long response timescales.
We find that both the ACF and the DFA indicators not only increase as a tipping
point is approached, as shown by positive Kendall coefficients, but also
generally approach the critical value of 1, although with varying degrees
of precision (Fig. 5). This enhances their predictive power, since by
extending a positive trend line it is possible to approximate what value of
the control parameter will eventually cause a tipping point to be crossed.
While our experiments in Appendix A showed that critical slowing down can
accurately predict onset of tipping points in an idealised setup, applying
this method to a more complex case study may fail, and in this context our
finding that these indicators largely retain their predictive power is very
encouraging. One area of additional complexity in our model of PIG compared
to the setup in Appendix A is the bed geometry, which is obtained from
observations and so is much less smooth than the synthetic retrograde bed used
in the MISMIP experiments. We explored how the addition of “bumpiness” to
bed geometry affects the performance of EWIs and found that it reduces how
clearly we can resolve the change in response time (Appendix A). This effect
may account for the fact that EWIs do not precisely reach a value of 1 at
the bifurcation point, but confirming this would require further testing.
There are several important caveats to the use of EWIs as presented here.
Firstly, and as explained above, the tipping point identified is that of the
transient system not in a steady state. Although the transient behaviour is
arguably of greater societal relevance and an ice sheet is unlikely to ever
truly be in a steady state, this is an important distinction to make.
Secondly, the predictive power of this method decreases as the distance to
tipping increases and must eventually break down altogether. This effect can
be clearly seen in Fig. 7 as Kendall's τ coefficient decreases with
increasing window length. Thirdly, there is a risk of so-called “false
alarms” and “missed alarms” (Lenton, 2011). False alarms, whereby a positive
trend in an indicator that is incorrectly interpreted as a tipping point
being imminent, can occur for a wide variety of reasons. First and foremost,
interpreting EWIs requires robust statistical analysis and judicious data
processing to ensure that the response time being measured is that of the
critical mode (Lenton, 2011). It is possible that rising auto-correlation is
a result of other processes, and using more than one indicator together with
changes in variance can help mitigate this risk (Ditlevsen and Johnsen, 2010). It is
also possible for a tipping point to be crossed with no apparent warning
i.e. missed alarms. This could happen if the internal variability in a
system is high so that it changes state before a bifurcation point is
reached or similarly if the forcing is too sudden. This last point is
particularly pertinent, since we intentionally perturb our model slowly and
do not explore how a change in forcing rate affects the performance of our
chosen EWIs. Increasing the forcing rate might present further difficulties
in identifying tipping points by leading to multiple tipping points being
crossed coincidentally. Since in our methodology the control parameter is
not held constant after a tipping point is crossed, then if that parameter
changes sufficiently during the time it takes for one tipping event to
conclude, it might reach a second threshold while the first event is still
underway, disguising the fact that two distinct tipping points have been
crossed. Changing the control parameter very slowly alleviates this issue,
since it will only have altered slightly during the time it takes for a
tipping event to happen. This issue, along with the related issue of
cascading tipping points, is one that we try to avoid in our experiments to
simplify the analysis but is known to influence EWI performance (Dakos et al.,
2015; Brock and Carpenter, 2010). Despite our use of a very slow forcing
rate, it is possible that more than three tipping points exist in our model
configuration. Finding all possible tipping points would necessitate
infinitesimally small changes in the control parameter, in either the
steady-state or the transient simulations, greatly increasing computational cost
but with little benefit in terms of detecting tipping events that constitute
substantial mass loss.
In this paper we have presented an application of EWIs to model output to
anticipate tipping points. This is a useful approach in and of itself, since
it could be used in model studies to detect bifurcations in the system with
minimal computational expense or to check whether a model might be on a
trajectory to cross a tipping point at some point in time beyond the
simulation. Alternatively, it may be possible to use this method on
observational data, palaeo-records or some combination thereof. This raises
the question of what data might qualify as useful for the application of
EWIs, which can be broken down further into (1) the type of data needed and
(2) the length of record necessary. As mentioned previously, ice volume or
related measures of an ice sheet's size do not show sufficient variability
for information on the recovery time to be extracted. Ice speed however can
change significantly over very short timescales; for example many ice
streams show large variability over timescales as short as tidal periods
(Anandakrishnan and Alley, 1997; Gudmundsson, 2006; Minchew et al., 2017). Ice flux was
chosen in this study since it is closely related to the MISI mechanism and
because flux is proportional to velocity, but it is possible that other
metrics related to ice velocity might also exhibit critical slowing down in
a similar way. With regards to record length, we find in this study that
early warning of tipping points becomes less reliable (with a low or even
negative Kendall's τ coefficient) for a moving-window size shorter than
200–300 years. However, this does not mean that this represents the minimum
window size in general and is likely sensitive to a number of the choices in
our methodology. For example, this value is likely to be sensitive to the
rate of forcing applied to the system. In the limiting case of a forcing
rate approaching zero, the necessary window length must increase since EWIs
are only expected to work relatively close to the tipping point. Both of
these points require further study in order to establish suitable datasets
for prediction of MISI onset.
Conclusions
Conducting quasi-steady numerical experiments, whereby the underside of the
PIG ice shelf is forced with a slowly increasing ocean-induced melt, we have
established the existence of at least three distinct tipping points.
Crossing each tipping point initiates periods of irreversible and
self-sustained retreat of the grounding line (MISI) with significant
contributions to global sea-level rise. The tipping points are identified
through critical slowing down, a general behavioural characteristic of non-linear systems as they
approach a tipping point. EWIs have been successfully applied to detect
critical slowing down in other complex systems. We show here that they
robustly detect the onset of the marine ice sheet instability in the
simulations of the realistic PIG configuration which is promising for
application of early warning to further cryospheric systems and beyond.
While the possibility of PIG undergoing unstable retreat has been raised and
discussed previously, this is to our knowledge the first time the stability
regime of PIG has been mapped out in this fashion. The first and second
tipping events are relatively small and could be missed without careful
analysis of model results but nevertheless are important in that they lead
to considerable sea-level rise and would require a large reversal in ocean
conditions for recovery. The third and final tipping point is crossed
with an increase in sub-shelf melt rates equivalent to a +1.2 ∘C
increase in ocean temperatures from initial conditions and leads to a
complete collapse of PIG. Long-term warming and shoaling trends in
Circumpolar Deep Water (Holland et al., 2019), in combination with changing wind
patterns in the Amundsen Sea (Turner et al., 2017), can expose the PIG ice shelf to
warmer waters for longer periods of time and make temperature changes of
this magnitude increasingly likely.
Flow line experiments
MISI has been a major focus of modelling efforts within the
glaciological community in recent years. In an effort to assess how ice-flow
models capture this behaviour, a model inter-comparison experiment was
performed to calculate the hysteresis loop of advance and retreat of a
marine ice sheet on a retrograde slope, known as MISMIP experiment 3
(referred to as EXP 3 hereafter; Pattyn et al., 2012). As a first step to
establishing whether critical slowing down can be observed prior to
MISI, we undertook a slightly modified version of this experiment using the
Úa ice-flow model (Gudmundsson, 2012, 2013; see Methods). In
our modified experiment, the marine ice sheet is forced towards tipping
points through step perturbations in the control parameter as before but
with smaller steps and the additional constraint that the model must be in
a steady state after each perturbation before moving on to the next. In this
experiment the chosen control parameter is the ice rate factor, a parameter
linked to ice viscosity and temperature.
Following each perturbation in the ice rate factor, we analyse the e-folding
relaxation time (TR) of the state variable (in this case, grounding-line position) to directly extract the recovery time of the model as it
approaches each tipping point (both advance and retreat). Theory predicts
that TR→∞ close to a tipping point and that the point at which
TR-2 (as plotted versus the control parameter) reaches 0 thus
identifies the critical value of the control parameter, beyond which a
tipping point is crossed (Wissel, 1984). We show this plot for both the
advance and retreat scenarios of EXP 3 in Fig. A1. In both cases the
relaxation time increases (TR-2 decreases) as predicted by theory,
even far from the tipping point. A linear fit through the last six
perturbations yields a good agreement with theory and accurately predicts
the critical value of the control parameter when compared to the analytical
solution (red arrows in Fig. A1) given by Schoof (2007). Critical slowing
down still occurs outside of this range (equivalent to a change in ice
temperature of > 5 ∘C), but using these more distant
points to forecast the tipping point would yield a less accurate prediction.
These results therefore provide some insight into how far from the tipping
point we can expect the predicted linear response.
Results of EXP 3, showing change in grounding-line position with time
resulting from step perturbations in the ice rate factor (a). The
calculated inverse relaxation time for each corresponding step change in
rate factor in both the advance (square symbols) and retreat (circular
symbols) phase is shown in panel (b). The dashed line in panel (b) is a line of
best fit, calculated for the five steps in rate factor that preceded the
advance or retreat MISI phase. Red arrows indicate the rate factors for
which the analytical solution predicts a MISI event, and black arrows show
the direction of the forcing towards each tipping point.
One major simplification in this idealised experiment is the bed geometry,
which is synthetic and arguably unrealistically smooth. To test whether the
addition of bumpiness to the bed affects how accurately the critical value
of the control parameter can be predicted, we conducted further experiments
in which the bed was made successively less smooth. One simple but flexible
way to generate the desired roughness is to add Perlin noise to the bed.
Perlin noise is a commonly used method in terrain generation that adds noise
at a number of levels with successively smaller wavelengths and amplitudes.
The number of levels is denoted by the octave; the rate at which each octave
changes frequency is the lacunarity, and the rate at which each octave
changes amplitude is the persistence. We made the common choice of a
lacunarity greater than 1 and a persistence less than one, meaning that
each octave adds noise of a higher frequency and lower amplitude. For a
starting octave amplitude of 25 m the difference between the analytical
solution and linear fit is less than 1 %, but this grows to
∼ 5 % with an amplitude of 50 m (note the change in height
from peak to trough in the retrograde region of the smooth bed is
∼ 120 m). This suggests that more realistic bed geometries with
increased roughness might make the task of predicting tipping points more
challenging than it is in this simplified case.
Code availability
The source code of the Úa ice-flow model is available from https://github.com/ghilmarg/UaSource (last access: 30 June 2020, 10.5281/zenodo.3706624, Gudmundsson, 2020), and raw
model output is available from the authors upon request.
Author contributions
SHRR and RR conceived the study, SHRR conducted the modelling experiments;
JFD contributed to the statistical analysis and surrogate time series; JDR
provided an initial model setup. SHRR and RR wrote the manuscript with
contributions from all authors.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We are grateful to the anonymous reviewer, Alexander Robel and the editor Pippa Whitehouse for their comments, which have greatly improved our paper.
Financial support
This research has been supported by the Natural Environment Research Council (grant nos. NE/L013770/1 and NE/S006745/1), the Deutsche Forschungsgemeinschaft (grant no. WI4556/3-1) and Horizon 2020 (TiPACCs (grant no. 820575)).
Review statement
This paper was edited by Pippa Whitehouse and reviewed by Alexander Robel and one anonymous referee.
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