Introduction
The Asian Summer Monsoon directly influences over 60 % of the world's
population (Wu et al., 2012) and yet the drivers of past and future
variability remain highly uncertain (Zickfeld et
al., 2005; Levermann et al., 2009). Evidence from radiometrically dated East Asian speleothem
records of past monsoon behaviour (Yuan et al., 2004) suggests that on
millennial timescales, the EASM is driven by a 23 kyr precession cycle
(Kutzbach, 1981; Wang et al., 2008), but also influenced by feedbacks in sea
surface temperatures and changing boundary conditions including Northern
Hemisphere ice volume (An, 2000; Sun et al., 2015). The abrupt nature of the
monsoon behaviour (interpreted as a precipitation proxy from δ18O values from Chinese speleothem records; see Sect. 1.4) in
comparison to the sinusoidal insolation forcing strongly implies that this
response is non-linear (Fig. 1); whilst Northern Hemisphere summer
insolation (NHSI) follows a quasi-sinusoidal cycle, the δ18O
profile in speleothems exhibits a step function, suggesting the presence of
threshold behaviour in the monsoon system (Schewe et al., 2012). Though the
vulnerability of society has clearly changed, future abrupt monsoon shifts,
whether caused by orbital or anthropogenic forcing, are likely to have major
devastating societal impacts (Donges et al., 2015).
(a) Northern Hemisphere summer insolation (NHSI) in June at
30∘ N (Berger and Loutre, 1991; grey), δ18O
speleothem data from Sanbao Cave (Wang et al., 2008; dark blue), (b) δ18O speleothem data from Hulu Cave (Wang et al., 2001);
speleothem MSH (red), MSP (blue) and MSX (yellow), (c) δ18O per
mille benthic carbonate (Lisiecki and Raymo, 2005; proxy for global ice
volume; purple). The grey shaded area indicates the Weak Monsoon Interval prior to Termination II.
A minimum conceptual model of the East Asian Summer Monsoon developed by
Zickfeld et al. (2005), stripped down by Levermann et al. (2009) and updated
by Schewe et al. (2012), shows a non-linear solution structure with
thresholds for switching a monsoon system between “on” or “off” states that
can be defined in terms of atmospheric humidity – in particular,
atmospheric specific humidity over the adjacent ocean (Schewe et al., 2012).
Critically, if specific humidity levels pass below a certain threshold, for
instance, as a result of reduced sea surface temperatures, insufficient
latent heat is produced in the atmospheric column and the monsoon fails.
This moisture-advection feedback allows for the existence of two stable
states, separated by a saddle-node bifurcation (Zickfeld et al., 2005; although interestingly, the conceptual models of Levermann et al. (2009)
and Schewe et al. (2012) are characterized by a single bifurcation point for
switching “off” the monsoon and an arbitrary threshold to switch it back
“on”). Crucially, the presence of a critical threshold at the transition
between the strong and weak regimes of the EASM means that early warning
signals related to “critical slowing down” (Dakos et al., 2008; Lenton et
al., 2012) could be detectable in suitable proxy records.
The aim of this study is twofold: (1) to test whether shifts in the EASM
during the penultimate glacial cycle (Marine Isotope Stage 6) are consistent
with bifurcational tipping points, and (2) if so, is it possible to detect
associated early warning signals? To achieve this, we analyse two δ18O speleothem records from China, and construct a simple model that
we derive directly from these data to test whether we can detect early
warning signals of these transitions.
Detecting early warning signals
We perform “tipping point analysis” on both the δ18O speleothem
records and on multiple simulations derived from our model. This analysis
aims to find early warning signs of impending tipping points that are
characterized by a bifurcation (rather than a noise-induced tipping, induced
by stochastic fluctuations with no change in forcing control, or
rate-dependent tipping, where a system fails to track a continuously
changing quasi-static attractor e.g. Ashwin et al., 2012). These tipping
points can be mathematically detected by looking at the pattern of
fluctuations in the short-term trends of a time series before the transition
takes place. A phenomenon called “critical slowing down” occurs on the
approach to a tipping point, whereby the system takes longer to recover from
small perturbations (Kleinen et al., 2003; Held and Kleinen, 2004; Dakos et
al., 2008). This longer recovery rate causes the intrinsic rates of change
in the system to decrease, which is detected as a short-term increase in the
autocorrelation or “memory” of the time series (Ives, 1995), often
accompanied by an increasing trend in variance (Lenton et al., 2012). It has
been theoretically established that autocorrelation and variance should both
increase together (Ditlevsen and Johnsen, 2010; Thompson and Sieber,
2011). Importantly, it is the increasing trend, rather than the absolute
values of the autocorrelation and variance that indicate critical slowing
down. Detecting the phenomenon of critical slowing down relies on a
timescale separation, whereby the timescale forcing the system is much
slower than the timescale of the system's internal dynamics, which is in
turn much longer than the frequency of data sampling the system (Held and
Kleinen, 2004). Importantly, the monsoon transitions span hundreds of years
(corresponding to several data points), meeting the criterion that the
frequency of sampling is higher than the timescale of the transition of the
system.
Missed alarms
Although efforts have been taken to reduce the chances of type I (incorrect
rejection of a true null hypothesis, otherwise known as a “false positive”)
and type II (failure to reject a false null hypothesis, or “false negative”)
errors by correct pre-processing of data e.g. (Lenton, 2011), totally
eradicating the chances of false positive and false negative results remains
a challenge (Scheffer, 2010; Lenton et al., 2012; Dakos et al., 2014). Type
II errors or “missed alarms”, as discussed in Lenton (2011), may occur when
internal noise levels are such that the system is “tipped” into a different
state prior to reaching the bifurcation point, precluding the detection of
early warning signals. Type I errors are potentially easier to guard against
by employing strict protocols by which to reject a null hypothesis.
Using speleothem δ18O data as a proxy of past monsoon
strength
Highly-resolved (∼ 102 years) and precisely dated
speleothem records of past monsoonal variability are well placed to test for
early warning signals. The use of speleothem-based proxies to reconstruct
patterns of palaeo-monsoon changes has increased rapidly over recent decades
with the development of efficient sampling and dating techniques. However,
there is currently some debate surrounding the climatic interpretation of
Chinese speleothem δ18O records (An et al., 2015), which can be
influenced by competing factors that affect isotope fractionation. The
oxygen isotopic composition of speleothem calcite is widely used to
reconstruct palaeohydrological variations due to the premise that speleothem
calcite δ18O records the stable isotopic content of
precipitation, which has been shown to be inversely correlated with
precipitation amount (Dansgaard, 1964; Lee and Swann, 2010), a relationship
known as the “amount effect”. Although the δ18O of speleothem
calcite in China has traditionally been used as a proxy for the “amount
effect” (Cheng et al., 2006, 2009; Wang et al., 2008, 2009), this has been challenged by other palaeo-wetness proxies,
notably Maher (2008), who argues that speleothems may be influenced by
changes in rainfall source rather than amount. The influence of the Indian
Monsoon has also been proposed as an alternative cause for abrupt monsoon
variations in China (Liu et al., 2006; Pausata et al., 2011), though this
has since been disputed (Wang and Chen, 2012; Liu et al., 2014).
Importantly, however, robust replications of the same δ18O
trends in speleothem records across the wider region suggest they
principally represent changes in the delivery of precipitation δ18O associated with the EASM (Cheng et al., 2009, 2012;
Li et al., 2014; Duan et al., 2014; Liu et al., 2014; Baker et al., 2015).
Specific data requirements are necessary to search for early warning signs
of tipping points in climate systems; not only does the data have to
represent a measure of climate, it also must be of a sufficient length and
resolution to enable the detection of critical slowing down. In addition,
since time series analysis methods require interpolation to equidistant data
points, a relative constant density of data points is important, so that the
interpolation does not skew the data. The speleothem δ18O
records that we have selected to fulfil these criteria, as described in more
detail in Sect. 2.1.
Methods
Data selection
We used the Chinese speleothem sequences from Sanbao Cave (31∘40′ N, 110∘26′ E; Wang et al., 2008) and Hulu Cave
(32∘30′ N, 119∘10′ E; Wang et al., 2001) to search for early warning
signals. Sanbao Cave (speleothem SB11) and Hulu Cave (speleothem MSP) have
two of the highest resolution chronologies in the time period of interest,
with a relatively constant density of data points, providing some of the
best records of Quaternary-scale monsoonal variation. Speleothem δ18O records offer considerable advantages for investigating past
changes in the EASM: their long duration (103–104 years),
high-resolution (∼ 100 years) and precise and absolute-dated
chronologies (typically 1 kyr at 1σ), make them ideal for time
series analysis. Speleothem SB11 has one of the longest, continuous δ18O records in China, and is the only series spanning an entire
glacial cycle without using a spliced record (Wang et al., 2008). Speleothem
MSP has a comparable resolution and density to SB11, though is significantly
shorter. Crucially, the cave systems lie within two regionally distinct
areas (Fig. 2), indicating that parallel changes in δ18O
cannot be explained by local effects.
Map showing the location of Sanbao and Hulu caves.
Searching for bimodality
A visual inspection of a histogram of the speleothem δ18O data
was initially undertaken to determine whether the data are likely to be
bimodal. We then applied a Dip-test of unimodality (Hartigan and Hartigan,
1985) to test whether our data are bimodal. To investigate further the
dynamical origin of the modality of our data we applied non-stationary
potential analysis (Kwasniok, 2013, 2015). A non-stationary
potential model (discussed in more detail in Sect. 2.4) was fitted,
modulated by the solar forcing (NHSI June 30∘ N), covering the
possibility of directly forced transitions as well as noise-induced
transitions with or without stochastic resonance.
Tipping point analysis
A search for early warning signals of a bifurcation at each monsoon
transition was carried out between 224–128 kyr of the Sanbao Cave and Hulu
Cave speleothem records. Stable periods of the Sanbao Cave δ18O
record (e.g. excluding the abrupt transitions) were initially identified
visually and confirmed by subsequent analysis using a climate regime shift
detection method described by Rodionov (2004). Data pre-processing involved
removal of long-term trends using a Gaussian kernel smoothing filter and
interpolation to ensure that the data are equidistant (a necessary assumption
for time series analysis), before the trends in autocorrelation and variance
(using the R functions acf and var respectively) are measured over a sliding
window of half the data length (Lenton et al., 2012). The density of data
points over time do not change significantly in either record and thus the
observed trends in autocorrelation are not an artefact of the data
interpolation. The smoothing bandwidth was chosen such that long-term trends
were removed without overfitting the data. A sensitivity analysis was
undertaken by varying the size of the smoothing bandwidth and sliding window
to ensure the results were robust over a range of parameter choices. The
nonparametric Kendall's tau rank correlation coefficient was applied
(Kendall, 1948; Dakos et al., 2008) to test for statistical dependence for a
sequence of measurements against time, varying between +1 and -1,
describing the sign and strength of any trends in autocorrelation and
variance.
Assessing significance
The results were tested against surrogate time series to ascertain the
significance level of the results found, based on the null hypothesis that
the data are generated by a stationary Gaussian linear stochastic process.
This method for assessing significance of the results is based on Dakos et al. (2012a). The surrogate time series were generated by randomizing the
original data over 1000 permutations, which is sufficient to adequately
estimate the probability distribution of the null model, and destroys the
memory while retaining the amplitude distribution of the original time
series. The autocorrelation and variance for the original and each of the
surrogate time series was computed, and the statistical significance
obtained for the original data by comparing against the frequency
distribution of the trend statistic (Kendall tau values of autocorrelation
and variance) from the surrogate data. Importantly, the Kendall tau values
are calculated relatively, thus when the autocorrelation is destroyed by
randomization, the null model distribution does not change. Higher Kendall
tau values indicate a stronger increasing trend. The 90th and 95th
percentiles provided the 90 and 95 % rejection thresholds (or p values
of 0.1 and 0.05) respectively. According to the fluctuation-dissipation
theorem (Ditlevsen and Johnsen, 2010), both autocorrelation and variance
should increase together on the approach to a bifurcation. Previous tipping
point literature has often used a visual increasing trend of autocorrelation
and variance as indicators of critical slowing down. Although using
surrogate data allows a quantitative assessment of the significance of the
results, there is no consensus on what significance level is necessary to declare the presence of precursors of critical slowing down. To guard
against type I errors, we determine for this study that “statistically
significant” early warning indicators occur with increases in both
autocorrelation and variance with p values < 0.1. We have chosen
this benchmark in line with previous studies using a similar null model that
have described results with p < 0.1 as “robust” (Dakos et al., 2008;
Boulton and Lenton, 2015).
Non-stationary potential analysis
(a) Histogram showing the probability density of the speleothem
data aggregated over 224–128 kyr, (b) bifurcation diagram obtained from
potential model analysis, showing bi-stability and hysteresis. Solid black
lines indicate stable states, dashed line unstable states, and dotted
vertical lines the jumps between the two stable branches. Coloured vertical
lines correspond to the insolation values for which the potential curve is
shown in panel (c); (c) shows how the shape of the potential well changes over
one transition cycle (198–175 kyr) (green long dash = 535 W m-2,
purple short dash = 531 W m-2, blue solid = 491 W m-2, red
dotted = 449 W m-2; for more details see Fig. 10).
To supplement the analysis of the speleothem records and help interpret the
results, a simple stochastic model derived directly from the Sanabo cave
δ18O data was constructed. Non-stationary potential analysis
(Kwasniok, 2013, 2015) is a method for deriving from time series
data a simple dynamical model which is modulated by external factors, here
solar insolation. The technique allows extraction of basic dynamical
mechanisms and to distinguish between competing dynamical explanations.
The dynamics of the monsoon system are conceptually described as
noise-driven motion in a time-dependent potential landscape. The governing
equation is a one-dimensional non-stationary effective Langevin equation:
x˙=-V′(x;t)+ση.
The model variable x is identified with the speleothem δ18O
record, which is a proxy for monsoon strength. The potential function
V(x;t) describes the force field governing the monsoon system. η is a white
Gaussian noise process with zero mean and unit variance, and σ is
the amplitude of the stochastic forcing. The noise term is meant to account
for the influence of unresolved temporal and spatial scales. The potential
landscape is time-dependent, modulated by the solar insolation:
V(x;t)=U(x)+γI(t)x.
The time-independent part of the potential is modelled by a fourth-order
polynomial, allowing for possible bi-stability (Kwasniok and Lohmann,
2009):
U(x)=∑i=14aixi.
I(t) is the insolation forcing and γ is a coupling parameter. The
modulation of the potential is only in the linear term, that is, the
time-independent potential system is subject to the scaled insolation
forcing γI(t). The insolation is represented as a superposition of three
main frequencies as
I(t)=α0+∑i=13αicos2πtTi+βisin2πtTi
with time t measured in kyr. The expansion coefficients αi and
βi are determined by least-squares regression on the insolation
time series over the time interval of the speleothem record. The periods
Ti are found by a search over a grid with mesh size 0.5 kyr. They are,
in order of decreasing contribution αi2+βi2,T1= 23 kyr, T2= 19.5 kyr and T3= 42 kyr.
This yields an excellent approximation of the insolation time series over
the time interval under consideration here.
The potential model covers and allows us to distinguish between two possible
scenarios: (i) in the bifurcation scenario, the monsoon transitions are
directly forced by the insolation, where two states are stable in turn, one
at a time. This corresponds to a fairly large value of γ. (ii)
Alternatively, two stable states could be available at all times with
noise-induced switching between them. This is realized with γ= 0,
giving a stationary potential. The height of the potential barrier
separating the two states could be modulated by the insolation, possibly
giving rise to a stochastic resonance which would explain the high degree of
coherence between the solar forcing and the monsoon transitions. The latter
variant would correspond to a small but non-zero value of γ.
The shape of the potential, as well as the noise level, are estimated
directly from the speleothem data according to the maximum likelihood
principle. We take a two-step approach, combining non-stationary probability
density modelling (Kwasniok, 2013) and dynamical modelling (Kwasniok, 2015).
The shape of the potential is estimated from the probability density of the
data. The quasi-stationary probability density of the potential model is
p(x;t)=Z-1(t)exp-2V(x;t)σ2,
with a time-dependent normalization constant Z(t). The coefficients ai and
the coupling
constant γ are estimated by maximizing the likelihood function
L(x1,…,xN)=∏i=1Np(xn;tn),
as described in Kwasniok (2013). The size of the data set is N= 1288. This
leaves the noise level undetermined as a scaling of the potential with a
constant c and a simultaneous scaling of the noise variance with c keeps the
quasi-stationary probability density unchanged. We set σ= 1 for the
(preliminary) estimation of ai and γ. The noise level is now
determined from the dynamical likelihood function based on the time
evolution of the system (Kwasniok, 2015). The Langevin equation is
discretised according to the Euler-Maruyama scheme:
xn+1=xn-δtnV′(xn;tn)+δtnσηn.
The sampling interval of the data is δtn=tn+1-tn. The
log-likelihood function of the data is
l(x1,…,xN|x0)=-N2log2π-Nlogσ-12∑n=0N-1logδtn+xn+1-xn+δtnV′(xn;tn)2δtnσ2.
The scaling constant c is searched on a grid with mesh size 0.01 and the
log-likelihood maximized, giving the final estimates of all parameters. Both
estimation procedures are applied directly to the unevenly sampled data
without any prior interpolation. We remark that the more natural and simpler
approach of estimating all parameters simultaneously from the dynamical
likelihood (Kwasniok, 2015) here yields a negative leading-order coefficient
a4 and thus the model cannot be integrated over a longer time period
without the trajectory escaping to infinity. This possibly points at
limitations in the degree of validity of the one-dimensional potential
model. Palaeoclimatic records reflect a multitude of complex processes and
any model as simple as Eq. (1) cannot be expected to be more than a
skeleton model used to pinpoint and contrast basic dynamical mechanisms. The
described estimation method guarantees a positive leading-order coefficient
a4 and therefore a globally stable model.
(a) δ18O speleothem data from Sanbao Cave (SB11; blue
line) and NHSI in June at 30∘ N (grey line). Grey hatched areas show
the sections of data selected for tipping point analysis. (b) Autocorrelation
and variance for each period prior to a transition.
Histogram showing frequency distribution of Kendall tau
values from 1000 realizations of a surrogate time series model (described in
Sect. 2.3.1), for Sanbao Cave (a, b) and Hulu Cave (c, d) δ18O data.
The grey dashed lines indicate the 90 % (p < 0.1)
and 95 % (p < 0.05) significance level. Each coloured line denotes
the Kendall tau values for autocorrelation and variance, for each section of
speleothem data analysed (red = 131–156 kyr; yellow = 166–177 kyr; purple
= 180–189 kyr; green = 191–198 kyr; orange = 200–208 kyr; blue
= 214–225 kyr).
It has been suggested that the EASM system responds specifically to
21 July insolation at 65∘ N with a “near-zero phase lag”
(Ruddiman, 2006). However, given that EASM development is affected by both
remote and local insolation forcing (Liu et al., 2006), we use an insolation
latitude local to the Sanbao Cave record, consistent with earlier studies
from this and other speleothem sequences (Wang et al., 2001). Since the
monthly maximum insolation shifts in time with respect to the precession
parameter, the 30∘ N June insolation was used, though we
acknowledge that the insolation changes of 65∘ N 21 July as used
by Wang et al. (2008) are similar with regard to the timing of maxima and
minima. Crucially, immediately prior to Termination II, the Chinese
speleothem data (including Sanbao Cave) record a “Weak Monsoon Interval”
between 135.5 and 129 kyr (Cheng et al., 2009), suggesting a lag of
approximately 6.5 kyrs following Northern Hemisphere summer insolation
(Fig. 1).
Having derived a model from the data, 100 realizations were analysed to test
whether early warning signals could be detected in the model output, using
the methods set out in Sect. 2.3. We initially chose the sampling
resolution of the model outputs to be comparable to the speleothem data
(102 years). Subsequently, the model was manipulated by changing both
the noise level and the sampling resolution in order to explore the effect
of these on the early warning signals in a hypothetical scenario. To enable
a straightforward comparison of the rate of forcing and the sampling
resolution we linearized the solar insolation using the minimum and maximum
values of the solar insolation over the time span of the model (224–128 kyr). This approach was preferred rather than using a sinusoidal forcing
since early warning signals are known to work most effectively when there is
a constant increase in the forcing. To detrend the time series data, we ran
the model without any external noise forcing to obtain the equilibrium
solution to the system, which we then subtracted from the time series, which
did include noise. In addition, we manipulated the noise level of the model
by altering the amplitude of the stochastic forcing (σ in Eq. 1). The time step in the series was reduced so that 6000 time points were
available prior to the bifurcation and to ensure no data from beyond the
tipping point was included in the analysis. Sampling the same time series at
different resolutions allowed us to explore the effect of this on the early
warning signals. When comparing early warning signals for differing sample
steps and noise levels, the same iteration of the model was used to enable a
direct comparison.
Results
Bimodality and non-stationary potential modelling
A histogram of δ18O values suggests there are two modes in the
EASM between 224–128 kyr, as displayed by the double peak structure in
Fig. 3a, supporting a number of studies that observe bimodality in
tropical monsoon systems (Zickfeld et al., 2005; Schewe et al., 2012). We
also apply a Dip-test of unimodality (Hartigan and Hartigan, 1985) and find
that our null hypothesis of unimodality is rejected (D= 0.018, p= 0.0063)
and thus our data are at least bimodal. To investigate further the dynamical
origin of this bimodality we applied non-stationary potential analysis
(Kwasniok, 2013, 2015). This showed a bi-stable structure to the
EASM with hysteresis (Fig. 3b, c), suggesting that abrupt monsoon
transitions may involve underlying bifurcations. The monsoon transitions
appear to be predominantly directly forced by the insolation. There is a
phase in the middle of the transition cycle between the extrema of the
insolation where two stable states are available at the same time but this
phase is too short for noise-induced switches to play a significant role.
We are able to clearly refute from the speleothem data the scenario of
noise-induced switching between two simultaneously available states in
favour of the bifurcation scenario. When fitting a model without solar
insolation forcing (that is, γ= 0) we obtain a stationary
potential with two deep wells and noise-driven switching between them.
However, the pdf-based log-likelihood of Eq. (6) is l= -2149.1 vs.
l=-1943.2 for the model with insolation forcing and the dynamical
log-likelihood of Eq. (8) is l=-353.6 vs. l=-346.6. This
provides very strong evidence for the bifurcation scenario; based on both
likelihood functions, both the Akaike and the Bayesian information criterion
clearly prefer the model with solar insolation forcing. The value of
γ is fairly large and the stationary part of the potential is not
strongly bistable, as evidenced by the shape of the potential given in
Fig. 3, ruling out the stochastic resonance scenario. The uncertainty in
all parameters, including the noise level, is very small, making our model
estimation robust. We tried more complicated models where also the
higher-order terms in the potential are modulated by the insolation rather
than just the linear term or where the solar insolation enters nonlinearly
into the model; the gain in likelihood is found to be rather minor compared
to the gain achieved when adding the modulation in the linear term of the
potential.
Tipping point analysis
We applied tipping point analysis on the Sanbao Cave δ18O
record on each section of data prior to a monsoon transition. Although
autocorrelation and variance do increase prior to some of the abrupt monsoon
transitions (Fig. 4), these increases are not consistent through the
entire record. Surrogate data sets used to test for significance of our
results showed that p values associated with these increases are only
< 0.1 for both autocorrelation and variance (Fig. 5) in one
instance. Although a visual increasing trend has been used in previous
literature as an indicator of critical slowing down, we choose more
selective criteria to guard against the possibility of false positives.
The only section of data prior to a monsoon transition that sees p values of
< 0.1 for the increases in both autocorrelation and variance is for
the data spanning the period 150 to 129 kyr in the Sanbao Cave record,
before Monsoon Termination II (Fig. 6). We find that the Kendall tau value
for autocorrelation has a significance level of p < 0.05 and for
variance a significance level of p < 0.1 (Fig. 5a and b). These
proportional positive trends in both autocorrelation and variance are
consistent with critical slowing down on the approach to a bifurcation
(Ditlevsen and Johnsen, 2010).
Tipping Point analysis on data from Sanbao Cave (Speleothem SB11;
31∘40′ N, 110∘26′ E). (a) Data were smoothed over an
appropriate bandwidth (purple line) to produce data residuals (b) and
analysed over a sliding window (of size between the two grey vertical
lines). The grey vertical line at 131 ka BP indicates the tipping point and
the point up to which the data are analysed. (c) AR(1) values and associated
Kendall tau value, and (d) displays the variance and associated Kendall tau
value.
To test whether the signal is present in other EASM records, we undertook
the same analysis on a second speleothem sequence of comparable age (Fig. 7). We find that speleothem MSP from Hulu Cave (32∘30′ N,
119∘10′ E; Wang et al., 2001) displays a comparable increase in
autocorrelation and variance to speleothem SB11 from Sanbao Cave, though
these do display slightly lower p values (Fig. 5c and d).
Tipping Point analysis on data from Hulu Cave (Speleothem MSP; 32∘30′ N, 119∘10′ E); (a) Data were smoothed over an
appropriate bandwidth (purple line) to produce data residuals (b), and
analysed over a sliding window (of size between the two grey vertical
lines). The grey vertical line at 131 ka BP indicates the tipping point, and
the point up to which the data are analysed. (c) Autocorrelation values and
associated Kendall tau value, and (d) the variance and associated Kendall
tau value.
Contour plots showing a range of window and bandwidth sizes for the
analysis; (a) Sanbao SB11 autocorrelation, (b) Sanbao SB11 variance,
(c) Hulu MSP autocorrelation, (d) Hulu MSP variance. Black stars indicate the
parameters used for the analysis in Figs. 6 and 7.
Furthermore, a sensitivity analysis was performed (results shown for data
preceding the monsoon termination in both speleothem SB11 and MSP, Fig. 8)
to ensure that the results are robust over a range of parameters by running
repeats of the analysis with a range of smoothing bandwidths used to detrend
the original data (5–15 % of the time series length) and sliding window
sizes in which indicators are estimated (25–75 % of the time series
length). The colour contours show how the Kendall tau values change when
using different parameter choices; for the autocorrelation at Sanbao Cave
the Kendall tau values are over 0.8 for the vast majority of smoothing
bandwidth and sliding window sizes (Fig. 8a), indicating a robust
analysis.
Potential model simulations
To help interpret these results we applied our potential model. In the model
we find transitions occur under direct solar insolation forcing when
reaching the end of the stable branches, explaining the high degree of
synchronicity between the transitions and solar forcing. The 100
realizations produced from our potential model, all initialized at the first
data point, appear broadly to follow the path of June insolation at
30∘ N with a small phase lag (Fig. 9). The model simulations
also follow the speleothem palaeodata for all but the monsoon transition at
129 ka BP near Termination II, where the model simulations show no extended
lag with respect to the insolation. Again it has to be kept in mind that the
potential model as a skeleton model can only be expected to qualitatively
reproduce the main features of the data. Actually observing the speleothem
record as a realization of the model will always be highly unlikely with any
model as simple as the present one.
No consistent early warning signals were found in the initial 100 model
simulations during the period 224–128 kyr. In order to detect critical
slowing down on the approach to a bifurcation, the data must capture the
gradual flattening of the potential well. We suggest that early warning
signals were not detected due to a relatively fast rate of forcing compared
to the sampling of the system; this comparatively poor sampling prevents the
gradual flattening of the potential well from being recorded in the data; a
feature common to many palaeoclimate data sets. Figure 10 illustrates the
different flattening of the potential well over a transition cycle during
the glacial period and over the transition cycle at the termination. There
is more visible flattening in the potential at the termination, as seen in
panel c, which is thought to be due to the reduced amplitude of the solar
forcing at the termination. The distinction between these two transition
cycles helps to explain why early warning signals in the form of increasing
autocorrelation and variance are found immediately preceding the
termination, but not for the other monsoon transitions.
Probability range of 100 model simulations, with the June
30∘ N NHSI (in red), and the palaeodata from SB11 (in green).
Potential analysis from the Sanbao δ18O data showing
the changing shape of the potential over (b) a transition cycle during
the glacial period (198–175 kyr); and (c) the transition cycle at the
termination (150–128.5 kyr). The potential is shown for stages of the transition over
high, medium, and low insolation values, as depicted in panel (a).
(a) Example of single realization of the approach to a bifurcation
from our potential model, which has been generated using four different noise
levels (original noise = grey, 0.5 noise = black, 0.2 noise = blue,
0.1 noise = green). Tipping point analysis was applied on each
realization, where the red line depicts the detrending line and the grey
dashed vertical line is the cut-off point where data are analysed up to;
distribution of Kendall tau values for (b) autocorrelation and (c) variance
over increasing sample step and differing noise levels.
To test the effect on the early warning signals of the sampling resolution
of the model, we compared a range of different sampling time steps in the
model (see Sect. 2.4) measuring the Kendall tau values of autocorrelation
and variance over each realization of the model (one realization displayed
in Fig. 11), which demonstrates the effects of increasing the sampling
time step in the model. We found that whereas an increasing sampling time
step produces a steady decrease in the Kendall tau values for
autocorrelation (Fig. 11b), Kendall tau values remain fairly constant for
variance (Fig. 11c), suggesting that the latter is not affected by time
step changes. This supports the contention by Dakos et al. (2012b) that
“high resolution sampling has no effect on the estimate of variance”. In
addition, we manipulated the noise level and found that decreasing the noise
level by a factor of 2 was necessary to identify consistent early warning
signals. This is illustrated in Fig. 11a, where the grey line represents
the noise level as determined by the model, which does not follow a step
transition, and cannot be adequately detrended by the equation derived from
the model. However, once the noise level is sufficiently reduced, early
warning signals (displayed here as high Kendall tau values for
autocorrelation and variance) can be detected.
Discussion
It is important to note here that although the detection of early warning
signals in time series data has been widely used for the detection of
bifurcations in a range of systems (Dakos et al., 2008), there are instances
when critical slowing down cannot be detected and/or recorded prior to a
bifurcation. First is the assumption that the abrupt monsoon shifts are
characterized by a bifurcation, rather than noise-induced tipping or
stochastic resonance. The bifurcation hypothesis is supported by previous
studies (Zickfeld et al., 2005; Levermann et al., 2009; Schewe et al., 2012)
as well as our potential model, which selects a bifurcation as the most
likely scenario (whilst considering noise-induced tipping and stochastic
resonance). In a noise-induced tipping or stochastic resonance scenario, no
early warning signals would be expected since there would be no gradual
change in the stability of the system (Lenton, 2011). Even within the
bifurcation scenario, it is possible that early warning signals may not be
detected due to external dynamics of the system, such as a high level of
stochastic noise, or when there is an insufficient sampling resolution. The
results illustrated in Fig. 11 confirm that early warning signals may not
be detected for bifurcations if the rate of forcing is too fast compared to
the sampling rate, such that the flattening of the potential is poorly
recorded in time series; Fig. 11b clearly illustrates the detrimental
effect of a lower resolution on Kendall tau values, particularly for
autocorrelation. “Missed alarms” may therefore be common in palaeodata where
there is an insufficient sampling resolution to detect the flattening of the
potential; a high sampling resolution is thus recommended to help avoid this
issue. There is more flattening visible in the potential for the monsoon
transition at 129 ka BP (Termination II), which is due to the reduced
amplitude of the orbital forcing at the termination, but it is unclear
whether this is sufficient to explain the early warning signal detected in
the palaeodata. We suggest that additional forcing mechanisms may be driving
the termination e.g. (Caley et al., 2011) which cannot be captured by the
potential model (as evidenced by the trajectory of the data falling outside
the probability range of the potential model; Fig. 9).
One possible reason for the detection of a critical slowing down immediately
prior to the termination (129 ka BP) is a change in the background state of
the climate system. Termination II is preceded by a Weak Monsoon Interval
(WMI) in the EASM at 135.5–129 kyr (Cheng et al., 2009), characterized by
the presence of a longer lag between the change in insolation and the
monsoon transition. The WMI is thought to be linked to migrations in the
Inter-tropical Convergence Zone (ITCZ; Yancheva et al., 2007). Changes in
the latitudinal temperature gradient (Rind, 1998) or planetary wave patterns
(Wunsch, 2006) driven by continental ice volume (Cheng et al., 2009) and/or
sea ice extent (Broccoli et al., 2006) have been suggested to play a role in
causing this shift in the ITCZ. For instance, the cold anomaly associated
with Heinrich event 11 (at 135 ka BP) has been invoked as a possible cause
of the WMI, cooling the North Atlantic and shifting the Polar Front and
Siberian High southwards, forcing an equatorward migration of westerly
airflow across Asia (Broecker et al., 1985; Cheng et al., 2009; Cai et al.,
2015). Such a scenario would have maintained a low thermal gradient between
the land and sea, causing the Weak Monsoon Interval and potentially
suppressing a simple insolation response. The implication is that during the
earlier monsoon transitions in Stage 6, continental ice volume and/or
sea-ice extent was less extensive than during the WMI, allowing the solar
insolation response to dominate.