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False alarms: How early warning signals falsely predict abrupt sea ice loss



Uncovering universal early warning signals for critical transitions has become a coveted goal in diverse scientific disciplines, ranging from climate science to financial mathematics. There has been a flurry of recent research proposing such signals, with increasing autocorrelation and increasing variance being among the most widely discussed candidates. A number of studies have suggested that increasing autocorrelation alone may suffice to signal an impending transition, although some others have questioned this. Here, we consider variance and autocorrelation in the context of sea ice loss in an idealized model of the global climate system. The model features no bifurcation, nor increased rate of retreat, as the ice disappears. Nonetheless, the autocorrelation of summer sea ice area is found to increase in a global warming scenario. The variance, by contrast, decreases. A simple physical mechanism is proposed to explain the occurrence of increasing autocorrelation but not variance when there is no approaching bifurcation. Additionally, a similar mechanism is shown to allow an increase in both indicators with no physically attainable bifurcation. This implies that relying on autocorrelation and variance as early warning signals can raise false alarms in the climate system, warning of ``tipping points'' that are not actually there.
Geophysical Research Letters
False alarms: How early warning signals falsely
predict abrupt sea ice loss
Till J. W. Wagner
and Ian Eisenman
Scripps Institution of Oceanography, University of California at San Diego, La Jolla, California, USA
Abstract Uncovering universal early warning signals for critical transitions has become a coveted goal
in diverse scientific disciplines, ranging from climate science to financial mathematics. There has been a
flurry of recent research proposing such signals, with increasing autocorrelation and increasing variance
being among the most widely discussed candidates. A number of studies have suggested that increasing
autocorrelation alone may suffice to signal an impending transition, although some others have questioned
this. Here we consider variance and autocorrelation in the context of sea ice loss in an idealized model of the
global climate system. The model features no bifurcation, nor increased rate of retreat, as the ice disappears.
Nonetheless, the autocorrelation of summer sea ice area is found to increase in a global warming scenario.
The variance, by contrast, decreases. A simple physical mechanism is proposed to explain the occurrence of
increasing autocorrelation but not variance when there is no approaching bifurcation. Additionally, a similar
mechanism is shown to allow an increase in both indicators with no physically attainable bifurcation. This
implies that relying on autocorrelation and variance as early warning signals can raise false alarms in the
climate system, warning of “tipping points” that are not actually there.
1. Introduction
The notion of critical slow down (CSD) as giving rise to “universal” early warning signals (EWS) for critical
transitions has received much attention in recent years. The great appeal of such indicators is their system
independence: any dynamical system featuring certain critical transitions, such as bifurcations, is expected
to undergo CSD as it gradually approaches a transition point, regardless of whether the system is a financial
market, an ecosystem, or an aspect of Earth’s climate. A generic EWS would ideally warn in advance of an
impending catastrophic shift without requiring detailed knowledge of the dynamics of the system. CSD can
have a number of measurable effects in observational time series. Two of the most commonly discussed ones
are (i) the amplification in stochastic fluctuations around the dynamical equilibrium, which manifests as an
increase in variance and (ii) increased autocorrelation, which is related to slower response times to stochastic
perturbations. Both increased variance and increased autocorrelation have been considered as potential indi-
cators of approaching a critical transition in numerous studies [Scheffer et al., 2009, 2012]: applications range
from predicting short-term stock market returns [LeBaron, 1992] to market crashes [Hong and Stein, 2003],
from simple theoretical ecological models [Wissel, 1984] to ecosystem-wide field experiments [Carpenter
and Brock, 2006], and from paleoproxy time series and idealized climate model results [Dakosetal., 2008; Held
and Kleinen, 2004; Kleinen et al., 2003; Lenton et al., 2012] to modern-day satellite observations [Livina and
Lenton, 2013]. A number of studies have suggested that increasing autocorrelation alone may suffice to signal
an impending transition [Dakosetal., 2008; Lenton et al., 2012].
However, the exact conditions under which CSD can be deduced from observational time series are disputed
in some recent studies [Boettiger and Hastings, 2012a, 2012b; Dakos et al., 2012], and it has been suggested
that rising autocorrelation alone does not indicate CSD [Ditlevsen and Johnsen, 2010]. Furthermore, CSD has
been suggested to have limitations as a generic predictor of critical transitions (see review in Dakosetal.
[2015], occasionally allowing “missed alarms” (not predicting an impending critical transition [Boettiger and
Hastings, 2013]) as well as “false alarms” (erroneously predicting a critical transition where there is none [Kefi
et al., 2013]). These failures can be due to a data set being ill-suited to detect CSD, or they can be a result of
conceptually misunderstanding how CSD is linked to the dynamics of a system [Dakosetal., 2015]. Here we
are concerned with the latter case.
Key Points:
Rising autocorrelation, a common
indicator of abrupt change, can raise
false alarms for sea ice
Changes in effective heat capacity,
rather than bifurcations, dominate the
autocorrelation signal
Rising autocorrelation is not a
universal indicator for abrupt change
in physical systems
Correspondence to:
T. J. W. Wagner,
Wagner, T. J. W., and I. Eisenman
(2015), False alarms: How early
warning signals falsely predict abrupt
sea ice loss, Geophys. Res. Lett.,
42, doi:10.1002/2015GL066297.
Received 22 SEP 2015
Accepted 16 NOV 2015
Accepted article online 24 NOV 2015
©2015. American Geophysical Union.
All Rights Reserved.
Geophysical Research Letters 10.1002/2015GL066297
Among systems that may undergo a bifurcation or “tipping point,” Arctic sea ice has been the subject of ardent
recent research [e.g., Winton, 2006; Lenton et al., 2008; Notz, 2009; Eisenmanand Wettlaufer, 2009; Lenton, 2012],
and EWS have been considered in this context [Livina and Lenton, 2013; S. Bathiany et al., Trends in sea ice
variability on the way to an ice-free Arctic, submitted to The Cryosphere Discussions, 2015]. The hypothesized
tipping point is usually attributed to the ice-albedo feedback, which leads to a loss of stability during sea ice
retreat. In a previous paper, we showed that low-order climate models often feature spurious bifurcations
which disappear when relevant leading-order physical processes are included [Wagner and Eisenman, 2015,
hereinafter WE15]. The somewhat more complex climate model introduced in WE15 supports the findings of
general circulation models (GCMs), which predict a gradual loss of Arctic sea ice, without crossing a bifurca-
tion, in contrast to many low-order climate models. Here we use the WE15 model to investigate how variance
and autocorrelation evolve under global warming.
2. Simulated Sea Ice Loss During Global Warming
The model represents a zonally uniform aquaplanet with a slab ocean mixed layer that includes sea ice. It
evolves the seasonally varying surface temperature and sea ice thickness as functions of latitude. The state of
the system is given by the surface enthalpy, E(t, x), which contains information about both surface tempera-
ture and ice thickness, with time t, x sin , and latitude . Specifically, in ice-covered regions, E =−L
h, with
ice thickness
h and latent heat of fusion L
; in ice-free regions, E=c
T. Here c
is the heat capacity of the ocean
mixed layer, and T is the surface temperature measured in terms of the departure from the melting point. We
simulate natural variability by adding a weather-like stochastic forcing [Hasselmann, 1976] to the deterministic
model of WE15. The model can be summarized by a single stochastic partial differential equation:
= aS
A + BT
+ D
+ F
+ F
+ N
. (1)
The net energy flux on the right-hand side consists of seasonally varying solar radiation, S(t, x), scaled by a
coalbedo that depends on the solar zenith angle as well as on the presence of ice, a(x, E); a representation of
outgoing longwave radiation (OLR) that is linearized in the surface temperature, A+BT, with A and B constants;
meridional heat transport in the atmosphere and ocean, represented as diffusion, D
T; upward heat flux
from the ocean below, F
; climate forcing F, which can be varied to represent changing greenhouse gas levels;
and weather noise
N, which is stochastic forcing with a persistence timescale of 1 week. Further details of the
model formulation are given in Appendix A.
We start the model simulations from a spun-up state with F = 0, which corresponds to preindustrial forcing
levels and features a perennial ice cover in high latitudes. In order to capture the full dynamic range of the
systemfrom snowball earth to an ice-free polewe perform an ensemble of realizations for two sets of
simulations: (i) warming runs in which F is gradually increased until a perennially ice-free state is reached and
(ii) cooling runs in which
F is gradually decreased until the planet is completely ice covered. We define A
as the
summer (September) Arctic sea ice area (not to be confused with the constant A). We focus here on A
, as this
quantity typically receives the most widespread attention. Figure 1a shows the evolution of
as F is ramped
up (red) or down (blue). We compute the variance,
, and lag-1 autocorrelation, ,ofA
(see Appendix B for
details). The variations of these two indicators with changing forcing F are shown in Figures 1b and 1c.
3. Successful Early Warning For Snowball Earth Bifurcation
The left-hand side of Figure 1a illustrates a bifurcation that is typically found in climate models: the “snowball
Earth instability,” which is driven by the ice-albedo feedback [e.g., Pierrehumbert et al., 2011]. Starting from
a partial ice cover and cooling the model beyond F =−12 Wm
leads to an abrupt transition to a fully
glaciated Earth (blue curves in Figure 1). Figures 1b and 1c illustrate how both variance and autocorrelation
increase as the instability draws near: both EWS indicators accurately give early warning here that the cooling
system is approaching a bifurcation.
Geophysical Research Letters 10.1002/2015GL066297
Figure 1. Sea ice area and CSD indicators in a model of global climate and sea ice. (a) Evolution of September Arctic sea
ice area
, with climate forcing F (lower horizontal axis) and time t (upper horizontal axis). Five realizations of warming
and cooling from the 1000-run ensemble are shown (faint red and blue), as well as warming and cooling simulations
with no added noise (dark red and blue). Inset: Simulated hysteresis loop of the model with no added noise, with a
schematic indication of the unstable state (black dash); arrows indicate warming and cooling trajectories. (b) Variance
of the time series in Figure 1a, computed using a 100 year running window (black bar). The variance is plotted above
the value of
F at the center of the window. The dashed vertical line marks the point where the first realization becomes
ice-free in September. (c) As in Figure 1b but for lag-1 autocorrelation. See Appendix B for details. Faint red curves show
for running windows that contain values of A
= 0.
4. False Alarm From Rising Autocorrelation
We next consider global warming simulations (red curves in Figure 1), which have steady ice loss until the
Arctic becomes icefree in September. The variance decreases monotonically with
F over the entire range plot-
ted in Figure 1b. Note that some GCMs simulate an increase in variance of September Arctic sea ice area
under global warming, while others simulate a decrease [Goosse et al., 2009]. The autocorrelation in Figure 1c,
however, exhibits a marked increase as ice-free conditions are approached.
A rise in autocorrelation alone, without an accompanying rise in variance, is often considered as an EWS
of an approaching abrupt transition [e.g., Dakos et al., 2008; Lenton et al., 2012]. Hence, with a limited time
series, for example ending at
t = 60 years (F = 3 Wm
), the results in Figure 1c would be interpreted to
imply an approaching sudden loss of the remaining sea ice. This would be a false alarm: when F continues to
increase, there is no bifurcation nor even an increased rate of retreat as A
reaches zero (Figure 1a).
Note that the statistical behavior is qualitatively the same when considering winter (March) or annual mean
sea ice area (not shown). For March, September, and annual mean sea ice volume, however, both variance
Geophysical Research Letters 10.1002/2015GL066297
Figure 2. Simulated polar temperature, T
, and CSD indicators. As in Figure 1 but for September polar temperature
instead of ice area.
and autocorrelation decrease monotonically with increasing F. This raises the question whether volume may
be a better suited variable than area for assessing the stability of the Arctic sea ice cover. Bathiany et al.
(submitted manuscript) consider ice thickness and volume in models that abruptly lose winter ice. They focus
on single-column sea ice models which feature a bifurcation associated with this loss, in contrast with the
model considered here which has no such bifurcation (see WE15). Consistent with earlier work [Moon and
Wettlaufer, 2011; Eisenman, 2012], the authors find that the response time lengthens in these models before
the abrupt winter ice loss. However, they find that this EWS occurs in an impractically narrow range of the
parameter space, whereas changes in autocorrelation attributed to physical processes such as those explored
here occur in much of the parameter space.
Figure 2 gives the time series of the September temperature at the pole, T
. It behaves similarly to A
, with the
variance decreasing and the autocorrelation increasing under warming. Since
is defined at a single location,
this allows for the possibility that spatial variability is not necessary to explain this behavior. We make use of
this in the following section.
5. Mechanism For Rising Autocorrelation
What physical mechanism gives rise to the increase in autocorrelation under warming? WE15 found that
meridional heat transport and seasonal variations act to essentially remove the effect of nonlinearity from
Geophysical Research Letters 10.1002/2015GL066297
albedo changes. Hence, the removal of the heat transport term (setting D = 0) as well as variations in the solar
forcing and albedo from the model (1) may plausibly have compensating effects, leaving the results qual-
itatively unaffected. With these terms removed, there is no spatial dependence. The influence of sea ice
thermodynamic growth (which relates
T to E in the model) is still a source of complexity, but with no sea-
sonal cycle, we can crudely approximate this as a change in the effective heat capacity associated with T.
This effective heat capacity includes latent heat effects associated with ice melt and growth. Using the WE15
model with D = 0 and constant aS gives a timescale for the approach to equilibrium, , which is 5 times larger
for ice-free conditions than for ice. Specifically,
1 year for perennial ice near the transition to seasonal
ice and 5 years for conditions that are ice-free all year; note that these timescales differ somewhat from
previous single-column model results [Moon and Wettlaufer, 2011; Eisenman, 2012, Bathiany et al., submitted
manuscript] due to slightly different parameter values and the suppression here of the ice-albedo feedback.
We are then left with the stochastic differential equation:
= aS −(A + BT)+F
+ F + N, (2)
with a jump in the effective heat capacity
c such that c(T < 0)=c
5 and c(T
. We take the stochastic
forcing, N, to be white noise of intensity
, which is a further simplification compared to the reddened noise
used in (1). In this case (2) represents a linear Ornstein-Uhlenbeck process of intensity
c(T), which
recovers from perturbations on a timescale of = c(T)∕B.
We numerically integrate (2), gradually ramping
F such that the equilibrium temperature increases through
zero (see Appendix C). The equilibrium value of T varies linearly with the control parameter F, with no bifur-
cation or accelerated transition occurring as the forcing is increased (Figure 3a). The only nonlinearity is an
increase in the effective heat capacity associated with the transition from sea ice to open ocean. The indi-
(T) and (T) are computed as in the previous section. Away from the transition at T = 0, analytic
estimates of variance and autocorrelation are readily calculated. The fluctuation-dissipation theorem implies
2 =
, such that the variance decreases when T rises above the freezing point
(Figure 3b). Note that for a typical Ornstein-Uhlenbeck process with constant c,
, and the inverse rela-
tion found here is due to the c dependence of the noise amplitude. The lag-1 autocorrelation with a sampling
period of
Δt = 1 year can be shown to be (T)=exp(−Δt), and hence, it increases when T rises above
the freezing point (Figure 3c). The changes in both and
occur gradually due to the noise causing T to
fluctuate above and below 0 for a range of
F values.
We consider the September polar temperature
in the full model (1) as a point of contact with T in the sim-
plified equation (2). Figure 3 resembles the WE15 model results in Figure 2, suggesting a physical explanation
for the mixed EWS behavior: the increase in effective heat capacity when sea ice is replaced with open ocean
causes the autocorrelation to increase while the variance decreases.
Hence, in some climate scenarios the autocorrelation increases when there is no approaching bifurcation. A
slowdown in variability associated with changes in the relevant heat capacity of the system may also occur
in other components of the climate system such as the Pacific Decadal Oscillation [e.g., Boulton and Lenton,
2015]. Next, we consider a climate scenario in which both autocorrelation and variance increase without a
physically attainable bifurcation.
6. Mechanism For Increase in Both Autocorrelation and Variance
We adjust (2) by allowing for a gradual change in the climate feedbacks, represented by B. For simplicity, here
we hold c = c
constant. As a simple physical example of this, we examine changes in the Planck feedback
due to the nonlinearity of the Stefan-Boltzmann relationship, replacing A + BT with 
. Here
+ T is
the absolute temperature, with melting point
is the Stefan-Boltzmann constant; and we set the effective
surfaceemissivity due to the atmosphere to = A∕(
)=0.65. We note that there are many other examples
of changing climate feedback strengths, including the ice-albedo feedback. In the resulting system,
= aS 
+ F
+ F + N, (3)
Geophysical Research Letters 10.1002/2015GL066297
Figure 3. Temperature and CSD indicators for a simple model with a jump in heat capacity. (a) One realization
of the stochastic warming simulation (faint red), as well as the noise-free solution (dashed). (b) Variance and
(c) autocorrelation, respectively, as in previous figures, with analytic solutions (dashed lines) for
T < 0 and T
Note that the forcing range is shifted compared to previous figures (see Appendix C).
the recovery timescale is approximately c
), which can be derived by linearizing the system
about a given value of
. We integrate equation (3) and compute
T) and (
T) as above (see Appendix C).
Figure 4 shows that both variance and autocorrelation go up as
F decreases and the climate cools.
Interpreted in the context of EWS, these increases raise a false alarm: no abrupt transition will occur as the
system (3) cools (although a bifurcation would occur if the system could cool across absolute zero). The state
of the system can be seen as located in a single potential well, centered around the equilibrium state set by
F.AsF decreases, the stabilizing Planck feedback becomes weaker. This leads to a widening of the potential
well, causing larger and longer-lasting responses to perturbations.
7. Conclusions
Taken together, the present results imply that using variance and autocorrelation as EWS may raise false alarms
during Arctic sea ice retreat, warning of bifurcations that are not actually there. A rise in autocorrelation alone
has previously been argued to be a sufficient EWS of an approaching “tipping point” [Dakosetal., 2012; Lenton
et al., 2012] or of system acceleration [Kefietal., 2013]. We instead find that such a rise can occur due to changes
in effective heat capacity when there is no acceleration in the sea ice decline. Our result is in agreement with
Geophysical Research Letters 10.1002/2015GL066297
Figure 4. Temperature and CSD indicators for a simple model undergoing cooling with varying Planck feedback.
Panels are as in Figure 3, with time
t, rather than F, shown on the horizontal axis; note that here F decreases with t.
the suggestion that an increase in autocorrelation needs to be accompanied by an increase in variance to
function as an EWS [Ditlevsen and Johnsen, 2010]. However, we also find that changing climate feedbacks
can lead to an increase in both variance and autocorrelation in a system with no physically attainable critical
transition. Hence, an increase in one or both of the two indicators is not sufficient to determine the approach
of an abrupt transition without knowledge of the underlying dynamics.
Appendix A: Energy Balance-Sea Ice Model
The parameters in the model (1) are set to their default values from WE15, with the exception of A, which is
set to a value 3 W m
lower than in WE15. F = 0 in the present model then corresponds approximately to
preindustrial rather than modern conditions, allowing simulations with increasing F to include observed sea
ice retreat. Here we ramp
F up by 15 W m
over 300 years in the warming simulations and ramp F down by
over 400 years in the cooling simulations.
Unlike in WE15, we force the present model with stochastic noise, N. The noise at each time step i is computed
= N
, with
representing a random draw from a Gaussian distribution with mean
zero and variance
. The noise is slightly reddened with an autocorrelation coefficient of exp(−Δt
where the correlation time of
= 1 week is set to be long enough to allow a relatively large Δt. It is further
similar to the few-day timescale of observed Arctic surface temperature persistence [e.g., Walsh et al.,
2005]. We use
Δt = 0.3 days. To test whether this time step size is sufficiently converged, we compute two
500-member ensembles of 100 year simulations with constant F = 0: one ensemble with Δt = 0.03 days and
Geophysical Research Letters 10.1002/2015GL066297
one with Δt = 0.3 days. Considering the time series of A
in each realization, the difference in the
ensemble-mean value of
between the two ensembles is 0.22; for it is 0.0071. These differences are also
partially due to the different realizations of noise in the two ensembles. Both of these are considerably smaller
than the signals discussed in Figures 1b and 1c. The noise intensity is chosen such that the resulting Arctic
sea ice extent variability is similar to the observations. In addition to the simulations with spatially uniform
noise, we considered a smaller ensemble of simulations with spatially varying noise that was set to be auto-
correlated in space with a length scale of 10
latitude. Initial results suggest a similar qualitative result for
Figure 1 in both cases, although further work is merited. In the present study, we limit our discussion here to
the simpler case of spatially uniform noise.
The simulated ice area is computed as
1 x
, where A
= 255 × 10
is the surface area of
a hemisphere of the earth and
is the location of the ice edge. Note that there is very little nonlinear recti-
fication of the noise in this model, which would result in the stochastic ensemble mean deviating noticeably
from the deterministic simulation. The stochastic ensemble mean in Figure 1a never deviates from the deter-
ministic simulation by more than 1 × 10
. These deviations visually resemble noise, implying that a
larger number of realizations would be needed to construct a sufficiently precise stochastic ensemble mean
for these purposes. This finding stands in contrast to previous studies of single-column sea ice models, where
there is typically a high level of nonlinear rectification [e.g., Eisenman, 2012]. Further details regarding the
model described in (1) are given in WE15.
Appendix B: Computation of
and 𝝆
All figures are generated using ensembles of 1000 realizations. This ensemble size is chosen to allow sufficient
convergence of
in model (1). To determine whether is sufficiently converged, we consider the slope of the
curve in Figure 1c in the range
F =[4, 5] Wm
. We generate 200 overlapping ensembles that each have 500
realizations by randomly picking sets of 500 runs (with replacement) from the full ensemble of 1000 runs. For
each of these 500-member ensembles we generate data such as in Figure 1c and compute the linear trend
in the specified range using ordinary least squares regression. We find that 190 of the 200 ensembles have
positive trends, which suggests that the rise in would be significant at the 95% level with an ensemble size
of 500 and that it is significant above this level with an ensemble size of 1000.
Anomalies are computed by subtracting the ensemble mean from individual runs. For all models,
and are
computed from the anomalies using a running window length of 100 years. The results depend somewhat
on the window length. For example, the increase in
in Figure 1c is robust for window lengths in the range
of roughly 50 to 200 years, and in Figure 2c the increase in
is robust down to window lengths of 10 years.
The variance
is readily computed using the ensemble mean within each window. The computation of the
autocorrelation, however, is not as straightforward. To obtain
, we compute the correlation between the
value at each time within the window in the full ensemble and the value at the previous time. In other words,
we generate an array of values at the current time as
, , x
, , x
, , x
, and we also
generate an array of values at the previous time as X
, , x
, , x
, , x
. Here the first
index represents the ensemble member, with
m = 1000 being the total number of members, and the second
index represents time, with the window spanning from time index
j to time index j + L (where L = 100). The
autocorrelation is computed as the correlation between
and X
Appendix C: Integration of Simple Models
In the simpler model (2), the WE15 annual mean polar ice-free values for shortwave radiation are used,
S=180 Wm
and a = 0.6. Values of c
, B, and F
are as in the full model (1). We set the noise amplitude,
a value 2 times larger than the full model (1). The OLR is set to A = 136 Wm
to compensate for the colder
pole due to the lack of horizontal diffusion. The model is integrated over 1000 years with
F ramping linearly
from 18.5 to 28.5 W m
, using a time step of Δt = 0.01 year,anda1yearsampling period is used in the com-
putation of
and . Figure 3 shows the 300 year integration period corresponding to F =[22, 25] Wm
We note that there is an increase in fluctuations near the transition from T< 0 to T
0. This shows up as a
slight increase in variance before it decreases. This effect is due to stochastic runs randomly getting stuck” in
the T
0 regime when the equilibrium state is still T < 0, analogous to flickering between the two states of a
double potential well, thereby increasing the variance.
Geophysical Research Letters 10.1002/2015GL066297
In the simpler model (3), we take the annual mean global mean incident shortwave radiation from WE15,
S = 340 Wm
, and we use the equatorial ice-free coalbedo, a = 0.7. We set
to a value 20 times larger than
in the full model (1). F is decreased from 15 to 15 W m
over 1000 years. We use an integration time step of
Δt = 0.01 year, and we use a sampling period of 1 year to compute
and , as for model (2).
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We thank John Wettlaufer, Sebastian
Bathiany, Dirk Notz, Vasilis Dakos,
and Peter Ditlevsen for their helpful
comments on an earlier version of
this paper. This work was supported
by ONR grant N00014-13-1-0469.
Code to numerically solve the model
described in (1) is available at

Supplementary resource (1)

... EWS can occur when there is no transition (e.g., Wagner & Eisenman, 2015). ...
... For this reason, it is advised to assess if multiple indicators, such as variance and autocorrelation, increase instead of relying on a single indicator (e.g., Ditlevsen & Johnsen, 2010). However, increases in multiple indicators without critical transitions are still possible, and can thus lead to false alarms (e.g., Boettiger & Hastings, 2012a;Wagner & Eisenman, 2015). ...
... Data Theory. In sharp contrast to their reputation of being generic or model-agnostic, we have seen that the occurrence of early warning signals depends on the specifics of the system under study; there are systems that show critical slowing down even though they do not exhibit critical transitions (Kéfi et al., 2013;Wagner & Eisenman, 2015), and there are (a potentially large class of) systems which show critical transitions but no critical slowing down (Hastings & Wysham, 2010). Even if the target system falls into the class of systems that show critical slowing down before critical transitions, early warning signals may only be observable in a small number of variables of the system (Boerlijst et al., 2013;Patterson et al., 2021). ...
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Many real-world systems can exhibit tipping points and multiple stable states, creating the potential for sudden and difficult to reverse transitions into a less desirable regime. The theory of dynamical systems points to the existence of generic early warning signals that may precede these so-called critical transitions. Recently, psychologists have begun to conceptualize mental disorders such as depression as an alternative stable state, and suggested that early warning signals based on the phenomenon of critical slowing down might be useful for predicting transitions into depression and other psychiatric disorders. Harnessing the potential of early warning signals requires us to understand their limitations as well as the factors influencing their performance in practice. In this article, we (a) review limitations of early warning signals based on critical slowing down to better understand when they can and cannot occur, and (b) study the conditions under which early warning signals may anticipate critical transitions in online-monitoring settings by simulating from a bistable dynamical system, varying crucial features such as sampling frequency, noise intensity, and speed of approaching the tipping point. We find that, in sharp contrast to their reputation of being generic or model-agnostic, whether early warning signals occur or not strongly depends on the specifics of the system. We also find that they are very sensitive to noise, potentially limiting their utility in practical applications. We discuss the implications of our findings and provide suggestions and recommendations for future research. (PsycInfo Database Record (c) 2022 APA, all rights reserved).
... According to this simplified figure, empirical formulas for calculating the heat-loss coefficient at the top of the collector (U t ) are proposed as Equation (14) [38][39][40]. When the average temperature of the hot plate is between T a and 200 • C, the error of the formulas is no more than 0.3 W/m 2 K [41]. ...
... According to this simplified figure, empirical formulas for calculating the heatloss coefficient at the top of the collector (Ut) are proposed as Equation (14) [38][39][40]. When the average temperature of the hot plate is between Ta and 200 °C, the error of the formulas is no more than 0.3 W/m 2 K [41]. ...
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The vanadium-titanium black ceramic (VTBC) coating on all-ceramic solar collectors has both high absorptance (0.94) and high emissivity (90%). However, the thermal conductivity of ceramic is very low (1.256 W/mK). To improve the heat collection efficiency of VTBC solar collectors, this paper establishes a mathematical model based on the energy-conservation relationships under steady-state conditions and creates a corresponding computer program. Key parameters for VTBC solar collectors include the heat-removal factor, effective transmittance-absorptance product for the absorber, total heat loss coefficient, etc. Then, via experimental testing, this study proposes a reference model for domestic VTBC solar collectors in a cold location (η = 0.89 − 2.20Tm*). Last, this work analyzes the influences of fin design and transparent cover design on VTBC solar collectors individually, using the created computer program. Results show that the most effective optimization method is to increase the transmittance of the transparent cover. By increasing the transmittance from 0.93 to 0.96, this study creates an optimized VTBC solar collector theoretical model (η = 0.92 − 2.20Tm*).
... Experimental evidence suggests that EWSs sometimes precede abrupt ecosystem shifts in aquatic systems, both in marine (13) and freshwater (14) environments, and sometimes by more than a decade in the latter case, but not always (15). In some cases, EWS may not act as an alarm system for abrupt shifts because of the underlying system dynamics, characteristics of the forcing (such as strong seasonality or the existence of multiple drivers), and characteristics of the data [such as measuring the wrong variables or at insufficient time resolution (16)] or may even be false alarms (17). Nonetheless, when EWSs do presage abrupt ecological shifts, there is the opportunity to plan for, or even avoid crossing, a critical transition. ...
Marine microbial communities sustain ocean food webs and mediate global elemental cycles. These communities will change with climate; these changes can be gradual or foreseeable but likely have much more substantial consequences when sudden and unpredictable. In a complex virtual marine microbial ecosystem, we find that climate change–driven shifts over the 21st century are often abrupt, large in amplitude and extent, and unpredictable using standard early warning signals. Phytoplankton with unique resource needs, especially fast-growing species such as diatoms, are more prone to abrupt shifts. Abrupt shifts in biomass, productivity, and community structure are concentrated in Atlantic and Pacific subtropics. Abrupt changes in environmental variables such as temperature and nutrients rarely precede these ecosystem shifts, indicating that rapid community restructuring can occur in response to gradual environmental changes, particularly in nutrient supply rate ratios.
... The variance of a process can increase as a response to weakening feedbacks, but there are also mechanisms that reduce the variance while increasing the persistence. For example, when the heat capacity or inertia of a system changes [18,23,24]. To gain insight into how increased temporal autocorrelation in the environmental variability might affect rate-induced tipping, we study the effect of time-correlation in isolation. ...
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Recently it has been show that in some ecosystems fast rates of change of environmental drivers may trigger a critical transition, whereas change of the same magnitude but at slower rates would not. So far, few studies describe this phenomenon of rate-induced tipping, while it is important to understand this phenomenon in the light of the ongoing rapid environmental change. Here, we demonstrate rate-induced tipping in a simple model of cyanobacteria with realistic parameter settings. We explain graphically that there is a range of initial conditions at which a gradual increase in environmental conditions can cause a collapse of the population, but only if the change is fast enough. In addition, we show that a pulse in the environmental conditions can cause a temporary collapse, but that is dependent on both the rate and the duration of the pulse. Furthermore, we study whether the autocorrelation of stochastic environmental conditions can influence the probability of inducing rate-tipping. As both the rate of environmental change, and autocorrelation of the environmental variability are increasing in parts of the climate, the probability for rate-induced tipping to occur is likely to increase. Our results imply that, even though the identification of rate sensitive ecosystems in the real world will be challenging, we should incorporate critical rates of change in our ecosystem assessments and management.
... Vitally, tipping points are not always observable or predictable, with potential for seemingly healthy systems to be in the process of a 'domino effect' in which collapse is inevitable (Scheffer et al. 2012). Conversely, modelling inaccuracies can raise a false alarm for collapse in actually healthy systems (Wagner and Eisenman 2015). ...
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Background: The 2019-20 Australian bushfire season was the most environmentally detrimental bushfire season on record. The extreme heat and drought exposed normally fire-resistant communities to uncharacteristically dry fuel loads and abnormally high severity burning. In eastern Australia this included mesic rainforest environments (including the World Heritage listed Gondwana Rainforests of Australia), which are often highly sensitive to fire, contain high biodiversity values, are critical habitat for threatened species, embody distinct endemism, provide valuable ecosystem services and are critical for terrestrial carbon storage. East coast rainforests are also highly fragmented, with less than half of pre-European levels remaining. Increases in fire frequency and intensity associated with climate change may threaten these already fragmented rainforest environments. This study considers the efficacy of rainforest refugia under a heightened bushfire climate, via spatial analysis of burn extent, burn severity and topographic characteristics for rainforests during the 2019-20 bushfire season within the Eastern Australian Temperate and Subtropical Forests Conservation Management Zone. Results: Burn severity, vegetation and elevation datasets were merged and analysed across mid-eastern Australia. A significant portion of rainforest was fire affected across the study area (~17%), with ~5% burnt to a high or very high severity. Elevation, topographic position (i.e. valleys), slope and aspect all contributed to maintaining rainforest fire refugia. The study resulted in a mapping product that can be utilised by researchers and protected area managers to locate and assess burnt rainforest in mid-eastern Australia. Conclusions: This study enables the identification of rainforest fire refugia and threatened rainforest communities for future research and conservation efforts in eastern Australia. The results also demonstrate the potential of climate change to enact widespread rainforest declines, with potentially dire consequences for biodiversity and ecosystem services. This event and recurrent fire events may enact positive climate feedback systems by enabling pyrophytic vegetation expansion and converting rainforest carbon pools into a carbon source.
Previous studies have used coupled climate model simulations with perturbed sea ice covers to assess the impact of future Arctic sea ice loss. The results of these studies suggest that Arctic sea ice loss will cause substantial climate impacts both in the Arctic and beyond. The approaches used in these simulations can be broadly categorized into three methodologies: adding a ghost flux to the sea ice module, nudging, and modifying the surface albedo. Here we show that all three methodologies ultimately add heat to the Arctic in order to melt the sea ice, and that this artificial heating causes a spurious warming signal that is added to the warming that occurs due to sea ice loss alone. We illustrate this using an idealized climate model, which provides a preliminary rough estimate of the effect. In this model, the annual-mean warming due to sea ice loss alone can be directly calculated. We compare this with the warming that would be attributed to sea ice loss using each of the three methodologies in the idealized model. The results suggest that each methodology substantially overestimates the warming due to sea ice loss alone, overestimating the surface warming throughout the Northern Hemisphere by a factor of 1.5–2 in the idealized model. Hence these results suggest that previous coupled climate modeling studies have overestimated the climate response to sea ice loss.
Arctic surface warming under greenhouse gas forcing peaks in early winter and reaches its minimum during summer in both observations and model projections. Many mechanisms have been proposed to explain this seasonal asymmetry, but disentangling these processes remains a challenge in the interpretation of general circulation model (GCM) experiments. To isolate these mechanisms, we use an idealized single-column sea ice model (SCM) which captures the seasonal pattern of Arctic warming. SCM experiments demonstrate that as sea ice melts and exposes open ocean, the accompanying increase in effective surface heat capacity can alone produce the observed pattern of peak early winter warming by slowing the seasonal heating and cooling rate, thus delaying the phase and reducing the amplitude of the seasonal cycle of surface temperature. To investigate warming seasonality in more complex models, we perform GCM experiments that individually isolate sea-ice albedo and thermodynamic effects under CO2 forcing. These also show a key role for the effective heat capacity of sea ice in promoting seasonal asymmetry through suppressing summer warming, in addition to precluding summer climatological inversions and a positive summer lapse-rate feedback. Peak winter warming in GCM experiments is further supported by a positive winter lapse-rate feedback that persists with only the albedo effects of sea-ice loss prescribed, due to cold initial surface temperatures and strong surface-trapped warming. While many factors support peak early winter warming as Arctic sea ice declines, these results highlight changes in effective surface heat capacity as a central mechanism contributing to this seasonality.
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The geological record shows that abrupt changes in the Earth system can occur on timescales short enough to challenge the capacity of human societies to adapt to environmental pressures. In many cases, abrupt changes arise from slow changes in one component of the Earth system that eventually pass a critical threshold, or tipping point, after which impacts cascade through coupled climate–ecological–social systems. The chance of detecting abrupt changes and tipping points increases with the length of observations. The geological record provides the only long-term information we have on the conditions and processes that can drive physical, ecological and social systems into new states or organizational structures that may be irreversible within human time frames. Here, we use well-documented abrupt changes of the past 30 kyr to illustrate how their impacts cascade through the Earth system. We review useful indicators of upcoming abrupt changes, or early warning signals, and provide a perspective on the contributions of palaeoclimate science to the understanding of abrupt changes in the Earth system. A synthesis of intervals of rapid climatic change evident in the geological record reveals some of the Earth system processes and tipping points that could lead to similar events in the future.
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Empirical diagnosis of stability has received considerable attention, often focused on variance metrics for early warning signals of abrupt system change or delicate techniques measuring Lyapunov spectra. The theoretical foundation for the popular early warning signal approach has been limited to relatively simple system changes such as bifurcating fixed points where variability is extrinsic to the steady state. We offer a novel measurement of stability that applies in wide ranging systems that contain variability in both internal steady state dynamics and in response to external perturbations. Utilizing connections between stability, dissipation, and phase space flow, we show that stability correlates with temporal asymmetry in a measure of phase space flow contraction. Our method is general as it reveals stability variation independent of assumptions about the nature of system variability or attractor shape. After showing efficacy in a variety of model systems, we apply our technique for measuring stability to monthly returns of the S&P 500 index in the time periods surrounding the global stock market crash of October 1987. Market stability is shown to be higher in the several years preceding and subsequent to the 1987 market crash. We anticipate our technique will have wide applicability in climate, ecological, financial, and social systems where stability is a pressing concern.
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Significance Sea surface temperature (SST) variations in the North Pacific have triggered past abrupt changes in fisheries and other ecosystems. We have discovered that over the last century, fluctuations of North Pacific SSTs have become less frequent and longer-lived. This “reddening” behavior can also be seen in the dominant pattern of climate variability in the region, known as the Pacific Decadal Oscillation index. This fundamental change in climate variability has important implications for ecosystems in the region. It implies that over the last century, ecosystems have become prone to undergoing larger climate-triggered abrupt shifts. Hence our discovery of changing climate variability could have contributed to the large magnitude of well-known abrupt changes in North Pacific ecosystems in 1977 and 1989.
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In the vicinity of tipping points-or more precisely bifurcation points-ecosystems recover slowly from small perturbations. Such slowness may be interpreted as a sign of low resilience in the sense that the ecosystem could easily be tipped through a critical transition into a contrasting state. Indicators of this phenomenon of 'critical slowing down (CSD)' include a rise in temporal correlation and variance. Such indicators of CSD can provide an early warning signal of a nearby tipping point. Or, they may offer a possibility to rank reefs, lakes or other ecosystems according to their resilience. The fact that CSD may happen across a wide range of complex ecosystems close to tipping points implies a powerful generality. However, indicators of CSD are not manifested in all cases where regime shifts occur. This is because not all regime shifts are associated with tipping points. Here, we review the exploding literature about this issue to provide guidance on what to expect and what not to expect when it comes to the CSD-based early warning signals for critical transitions.
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Record lows in Arctic sea ice extent have been making frequent headlines in recent years. The change in albedo when sea ice is replaced by open water introduces a nonlinearity that has sparked an ongoing debate about the stability of the Arctic sea ice cover and the possibility of Arctic “tipping points.” Previous studies identified instabilities for a shrinking ice cover in two types of idealized climate models: (i) annual-mean latitudinally varying diffusive energy balance models (EBMs) and (ii) seasonally varying single-column models (SCMs). The instabilities in these low-order models stand in contrast with results from comprehensive global climate models (GCMs), which typically do not simulate any such instability. To help bridge the gap between low-order models and GCMs, an idealized model is developed that includes both latitudinal and seasonal variations. The model reduces to a standard EBM or SCM as limiting cases in the parameter space, thus reconciling the two previous lines of research. It is found that the stability of the ice cover vastly increases with the inclusion of spatial communication via meridional heat transport or a seasonal cycle in solar forcing, being most stable when both are included. If the associated parameters are set to values that correspond to the current climate, the ice retreat is reversible and there is no instability when the climate is warmed. The two parameters have to be reduced by at least a factor of 3 for instability to occur. This implies that the sea ice cover may be substantially more stable than has been suggested in previous idealized modeling studies.
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Ecosystem responses to external changes can surprise us by their abruptness and irreversibility. Models have helped identifying indicators of impending catastrophic shifts, referred to as 'generic early warning signals'. These indicators are linked to a phenomenon known as 'critical slowing down' which describes the fact that the recovery rate of a system after a perturbation decreases when the system approaches a bifurcation — such as the classical fold bifurcation associated to catastrophic shifts. However, contrary to what has sometimes been suggested in the literature, a decrease in recovery rate cannot be considered as specific to approaching catastrophic shifts. Here, we analyze the behavior of early warning signals based on critical slowing down in systems approaching a range of catastrophic and non-catastrophic situations. Our results show that slowing down generally happens in situations where a system is becoming increasingly sensitive to external perturbations, independently of whether the impeding change is catastrophic or not. These results highlight that indicators specific to catastrophic shifts are still lacking. More importantly, they also imply that in systems where we have no reason to expect catastrophic transitions, slowing down may still be used in a more general sense as a warning signal for a potential decrease in stability.
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The contrast in surface albedo between sea ice and open ocean suggests the possibility of an unstable climate state flanked by two separate stable climate states. Previous studies using idealized single-column models and comprehensive climate models have considered the possibility of abrupt thresholds during sea ice retreat associated with such multiple states, and they have produced a wide range of results. When the climate is warmed such that the summer minimum Arctic sea ice cover reaches zero, some models smoothly transition to seasonally ice-free conditions, others discontinuously transition to seasonally ice-free conditions, and others discontinuously transition to annually ice-free conditions. Among the models that simulate a continuous transition to seasonally ice-free conditions, further warming causes some to smoothly lose the remaining wintertime-only sea ice cover and others to discontinuously lose it. Here, we use a toy model representing the essential physics of thermodynamic sea ice in a single column to investigate the factors controlling which of these scenarios occurs. All of the scenarios are shown to be possible in the toy model when the parameters are varied, and physical mechanisms giving rise to each scenario are investigated. We find that parameter shifts that make ice thicker or open ocean warmer under a given climate forcing make models less prone to stable seasonally ice-free conditions and more prone to bistability and hence bifurcations. The results are used to interpret differences in simulated sea ice stability in comprehensive climate models.
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A reply to Drake (2013) "Early warning signals of stochastic switching"
A stochastic model of climate variability is considered in which slow changes of climate are explained as the integral response to continuous random excitation by short period “weather” disturbances. The coupled ocean-atmosphere-cryosphere-land system is divided into a rapidly varying “weather” system (essentially the atmosphere) and a slowly responding “climate” system (the ocean, cryosphere, land vegetation, etc.). In the usual Statistical Dynamical Model (SDM) only the average transport effects of the rapidly varying weather components are parameterised in the climate system. The resultant prognostic equations are deterministic, and climate variability can normally arise only through variable external conditions. The essential feature of stochastic climate models is that the non-averaged “weather” components are also retained. They appear formally as random forcing terms. The climate system, acting as an integrator of this short-period excitation, exhibits the same random-walk response characteristics as large particles interacting with an ensemble of much smaller particles in the analogous Brownian motion problem. The model predicts “red” variance spectra, in qualitative agreement with observations. The evolution of the climate probability distribution is described by a Fokker-Planck equation, in which the effect of the random weather excitation is represented by diffusion terms. Without stabilising feedback, the model predicts a continuous increase in climate variability, in analogy with the continuous, unbounded dispersion of particles in Brownian motion (or in a homogeneous turbulent fluid). Stabilising feedback yields a statistically stationary climate probability distribution. Feedback also results in a finite degree of climate predictability, but for a stationary climate the predictability is limited to maximal skill parameters of order 0.5.
In the Earth's history, periods of relatively stable climate have often been interrupted by sharp transitions to a contrasting state. One explanation for such events of abrupt change is that they happened when the earth system reached a critical tipping point. However, this remains hard to prove for events in the remote past, and it is even more difficult to predict if and when we might reach a tipping point for abrupt climate change in the future. Here, we analyze eight ancient abrupt climate shifts and show that they were all preceded by a characteristic slowing down of the fluctuations starting well before the actual shift. Such slowing down, measured as increased autocorrelation, can be mathematically shown to be a hallmark of tipping points. Therefore, our results imply independent empirical evidence for the idea that past abrupt shifts were associated with the passing of critical thresholds. Because the mechanism causing slowing down is fundamentally inherent to tipping points, it follows that our way to detect slowing down might be used as a universal early warning signal for upcoming catastrophic change. Because tipping points in ecosystems and other complex systems are notoriously hard to predict in other ways, this is a promising perspective.
Daily maximum and minimum temperature data from 54 stations in Alaska and northern Canada are used to evaluate temporal changes in the variance and autocorrelation of the daily anomalies from 1950–2000. To the extent that the variance and autocorrelation capture variability and anomaly persistence, the results provide a quantitative counterpart to reports from Arctic communities that weather is becoming increasingly variable and unpredictable. Although the results show indications of increased variance at some locations in some seasons, at least as many other stations show no such trends or show trends opposite to the hypothesized increase in variance and decrease in persistence of temperature anomalies. These findings apply to northern subsets as well as to the entire sample of stations. However, the frequency of extreme anomalies relative to single‐month means does show a modest but steady increase from the 1950s to the 1990s. The absence of a more generally apparent increase in variance and decrease in predictability is most likely attributable to either: (a) the choice of temperature as a measure of weather, (b) our use of simple statistical measures of variability and persistence, or (c) the role of technological and social changes in shaping perceptions of weather variability and predictability.