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Geophysical Research Letters

False alarms: How early warning signals falsely

predict abrupt sea ice loss

Till J. W. Wagner

1

and Ian Eisenman

1

1

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.

RESEARCH LETTER

10.1002/2015GL066297

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,

tjwagner@ucsd.edu

Citation:

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.

WAGNER AND EISENMAN FALSE ALARMS DURING SEA ICE LOSS 1

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

f

h, with

ice thickness

h and latent heat of fusion L

f

; in ice-free regions, E=c

w

T. Here c

w

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:

E

t

= aS

solar

−

(

A + BT

)

OLR

+ D∇

2

x

T

transport

+ F

b

ocean

heating

+ F

climate

forcing

+ N

noise

. (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∇

2

x

T; upward heat ﬂux

from the ocean below, F

b

; 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

system—from snowball earth to an ice-free pole—we 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

i

as the

summer (September) Arctic sea ice area (not to be confused with the constant A). We focus here on A

i

, as this

quantity typically receives the most widespread attention. Figure 1a shows the evolution of

A

i

as F is ramped

up (red) or down (blue). We compute the variance,

2

, and lag-1 autocorrelation, ,ofA

i

(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

−2

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.

WAGNER AND EISENMAN FALSE ALARMS DURING SEA ICE LOSS 2

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

A

i

, 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

and

2

for running windows that contain values of A

i

= 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

−2

), 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

i

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

WAGNER AND EISENMAN FALSE ALARMS DURING SEA ICE LOSS 3

Geophysical Research Letters 10.1002/2015GL066297

Figure 2. Simulated polar temperature, T

p

, 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

p

. It behaves similarly to A

i

, with the

variance decreasing and the autocorrelation increasing under warming. Since

T

p

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

WAGNER AND EISENMAN FALSE ALARMS DURING SEA ICE LOSS 4

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:

c

dT

dt

= aS −(A + BT)+F

b

+ F + N, (2)

with a jump in the eﬀective heat capacity

c such that c(T < 0)=c

w

∕5 and c(T

>

0)=c

w

. We take the stochastic

forcing, N, to be white noise of intensity

1

, 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

OU

=

1

∕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-

cators

2

(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

(T)=

2

OU

∕2 =

2

1

∕2B

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,

2

∝ , 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

2

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

T

p

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

w

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

S

T

4

. Here

T ≡

T

m

+ T is

the absolute temperature, with melting point

T

m

;

S

is the Stefan-Boltzmann constant; and we set the eﬀective

surfaceemissivity due to the atmosphere to = A∕(

S

T

4

m

)=0.65. We note that there are many other examples

of changing climate feedback strengths, including the ice-albedo feedback. In the resulting system,

c

w

d

T

dt

= aS −

S

T

4

+ F

b

+ F + N, (3)

WAGNER AND EISENMAN FALSE ALARMS DURING SEA ICE LOSS 5

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

>

0.

Note that the forcing range is shifted compared to previous ﬁgures (see Appendix C).

the recovery timescale is approximately ≡ c

w

∕(4

S

T

3

), which can be derived by linearizing the system

about a given value of

T

. We integrate equation (3) and compute

2

(

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

WAGNER AND EISENMAN FALSE ALARMS DURING SEA ICE LOSS 6

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

−2

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

−2

over 300 years in the warming simulations and ramp F down by

20Wm

−2

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

as

N

i

= N

i−1

+

√

1 −

2

0

, with

0

representing a random draw from a Gaussian distribution with mean

zero and variance

2

0

. The noise is slightly reddened with an autocorrelation coeﬃcient of ≡ exp(−Δt∕

r

),

where the correlation time of

r

= 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

WAGNER AND EISENMAN FALSE ALARMS DURING SEA ICE LOSS 7

Geophysical Research Letters 10.1002/2015GL066297

one with Δt = 0.3 days. Considering the time series of A

i

in each realization, the diﬀerence in the

ensemble-mean value of

2

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

A

i

=

(

1 − x

i

)

A

hem

, where A

hem

= 255 × 10

6

km

2

is the surface area of

a hemisphere of the earth and

x

i

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

6

km

2

. 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

𝝈

2

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

−2

. 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,

2

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

2

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

1

≡

[

x

1,j+1

, … , x

1,j+L

, … , x

m,j+1

, … , x

m,j+L

]

, and we also

generate an array of values at the previous time as X

2

≡

[

x

1,j

, … , x

1,j+L−1

, … , x

m,j

, … , x

m,j+L−1

]

. 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

X

1

and X

2

.

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

−2

and a = 0.6. Values of c

w

, B, and F

b

are as in the full model (1). We set the noise amplitude,

1

,to

a value 2 times larger than the full model (1). The OLR is set to A = 136 Wm

−2

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

−2

, using a time step of Δt = 0.01 year,anda1yearsampling period is used in the com-

putation of

2

and . Figure 3 shows the 300 year integration period corresponding to F =[22, 25] Wm

−2

.

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.

WAGNER AND EISENMAN FALSE ALARMS DURING SEA ICE LOSS 8

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

−2

, and we use the equatorial ice-free coalbedo, a = 0.7. We set

1

to a value 20 times larger than

in the full model (1). F is decreased from 15 to −15 W m

−2

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

2

and , as for model (2).

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Acknowledgments

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

http://eisenman.ucsd.edu/code.html.

WAGNER AND EISENMAN FALSE ALARMS DURING SEA ICE LOSS 9