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NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS
WITHIN THE EARTH’S CLIMATE SYSTEM
JOSÉ A. RIAL 1,ROGERA.PIELKESR.2, MARTIN BENISTON3,
MARTIN CLAUSSEN 4, JOSEP CANADELL 5, PETER COX6, HERMANN HELD4,
NATHALIE DE NOBLET-DUCOUDRÉ 7, RONALD PRINN8,
JAMES F. REYNOLDS9and JOSÉ D. SALAS 10
1Wave Propagation Laboratory, Department of Geological Sciences CB#3315,
University of North Carolina, Chapel Hill, NC 27599-3315, U.S.A.
E-mail: jar@email.unc.edu
2Atmospheric Science Dept., Colorado State University, Fort Collins, CO 80523, U.S.A.
3Dept. of Geosciences, Geography, Univ. of Fribourg, Pérolles, Ch-1700 Fribourg, Switzerland
4Potsdam Institute for Climate Impact Research, Telegrafenberg C4, 14473 Potsdam,
P.O. Box 601203, Potsdam, Germany
5GCP-IPO, Earth Observation Centre, CSIRO, GPO Box 3023, Canberra, ACT 2601, Australia
6Met Office Hadley Centre, London Road, Bracknell, Berkshire RG12 2SY, U.K.
7DSM/LSCE, Laboratoire des Sciences du Climat et de l’Environnement, Unité mixte de Recherche
CEA-CNRS, Bat. 709 Orme des Merisiers, 91191 Gif-sur-Yvette, France
8Dept. of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology,
77 Massachusetts Avenue, Cambridge, MA 02139-4307, U.S.A.
9Department of Biology and Nicholas School of the Environmental and Earth Sciences,
Phytotron Bldg., Science Dr., Box 90340, Duke University, Durham, NC 27708, U.S.A.
10Dept. of Civil Engineering, Colorado State University, Fort Collins, CO 80523, U.S.A.
Abstract. The Earth’s climate system is highly nonlinear: inputs and outputs are not proportional,
change is often episodic and abrupt, rather than slow and gradual, and multiple equilibria are the
norm. While this is widely accepted, there is a relatively poor understanding of the different types of
nonlinearities, how they manifest under various conditions, and whether they reflect a climate system
driven by astronomical forcings, by internal feedbacks, or by a combination of both. In this paper, af-
ter a brief tutorial on the basics of climate nonlinearity, we provide a number of illustrative examples
and highlight key mechanisms that give rise to nonlinear behavior, address scale and methodological
issues, suggest a robust alternative to prediction that is based on using integrated assessments within
the framework of vulnerability studies and, lastly, recommend a number of research priorities and
the establishment of education programs in Earth Systems Science. It is imperative that the Earth’s
climate system research community embraces this nonlinear paradigm if we are to move forward in
the assessment of the human influence on climate.
1. Introduction
Nonlinear phenomena characterize all aspects of global change dynamics, from
the Earth’s climate system to human decision-making (Gallagher and Appenzeller,
1999). Past records of climate change are perhaps the most frequently cited ex-
amples of nonlinear dynamics, especially where certain aspects of climate, e.g.,
Climatic Change 65: 11–38, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
12 JOSÉ A. RIAL ET AL.
the thermohaline circulation of the North Atlantic ocean, suggest the existence
of thresholds, multiple equilibria, and other features that may result in episodes
of rapid change (Stocker and Schmittner, 1997). As described in Kabat et al.
(2003), the Earth’s climate system includes the natural spheres (e.g., atmosphere,
biosphere, hydrosphere and geosphere), the anthrosphere (e.g., economy, society,
culture), and their complex interactions (Schellnhuber, 1998). These interactions
are the main source of nonlinear behavior, and thus one of the main sources of
uncertainty in our attempts to predict the effects of global environmental change.
In sharp contrast to familiar linear physical processes, nonlinear behavior in the
climate results in highly diverse, usually surprising and often counterintuitive ob-
servations, so it is important, before embarking on the discussion of data, that we
agree on a few basic characteristics of nonlinear climate.
1.1. LINEAR AND NONLINEAR SYSTEMS
Even an elementary description of Earth’s climate system must deal with the fact
that it is composed of the above subsystems all interconnected and open, allowing
fluxes of mass, energy and momentum from and to each other (see Figure 1). Since
the Earth itself is a closed system, these fluxes eventually cycle through, so that
outputs re-enter the system to become inputs, creating feedbacks and feedback
chains. Eventually, each subsystem affects the response of every other subsystem
and of the climate as a whole. It is this cross talk among the different parts of
the climate that engenders the disproportionate relations between input and output
typical of a nonlinear system. The phrase ‘the whole is more than the sum of its
parts’ underscores the failure of the principle of superposition in a nonlinear system
such as the climate. In sharp contrast, where superposition is valid the whole is
exactly equal to the sum of its parts. The system is linear and there is no cross talk;
each part behaves as if it were acting alone.
How do we tell when there is nonlinearity in the climate we observe? There
are at least three important observable characteristics that separate linear from
nonlinear systems, all of which are exemplified in the data to be discussed.
(1) While linear systems typically show smooth, regular motion in space and
time that can be described in terms of well-behaved, continuous functions, nonlin-
ear systems often undergo sharp transitions, even in the presence of steady forcing.
These transitions usually result from crossing unstable equilibrium thresholds (e.g.,
abrupt climate change, as described by Alley et al., 2003).
(2) The response of a linear system to small changes in its parameters or to
changes in external forcing is usually smooth and proportionate to the stimulation.
In contrast, nonlinear systems are such that a very small change in some parameters
can cause great qualitative differences in the resulting behavior (chaos) as sug-
gested for instance by fluid dynamic models of atmospheric convection (Lorenz,
1963).
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 13
Figure 1. Structure of CLIMBER-2, an Earth System Model of Intermediate Complexity (EMIC;
Claussen et al., 2002). The model consists of four modules which describe the dynamics of the
climate components atmosphere, ocean, terrestrial vegetation, and inland ice. These components
interact via fluxes of energy, momentum (e.g., wind stress on the ocean), water (e.g., precipitation,
snow, and evaporation), and carbon. Also, the land-surface structure is allowed to change in the case
of changes in vegetation cover or the emergence and melting of inland ice masses, for example.
The interaction between climate components is described in a so-called Soil Vegetation Atmosphere
Transfer Scheme (SVAT). CLIMBER-2 is driven by insolation (which can vary owing to changes
in the Earth orbit or in the solar energy flux), by the geothermal heat flux (which is very small, but
important in the long run for inland ice dynamics), and by changes imposed on the climate system
by human activities (such as land use or emission of greenhouse gases (GHG) and aerosols).
(3) After transients dissipate, an oscillatory linear system’s frequency always
equals that of the forcing, while the spectral response of a nonlinear system to
oscillatory external forcing usually exhibits frequencies not present in the forcing
(such as combination tones), phase and frequency coupling, synchronization and
other indications of nonlinearity often detected in past climate data (e.g., Pisias et
al., 1990).
1.2. CHAOS AND COMPLEXITY
Thus, nonlinearity gives rise to unexpected structures and events in the form of
abrupt transitions across thresholds, unexpected oscillations, and chaos (Kaplan
and Glass, 1995). Actually, the climate system is not only chaotic, it is also
‘complex’ (Rind, 1999), in the sense that it is composed of many parts whose
14 JOSÉ A. RIAL ET AL.
interactions can, through a process still not completely understood (Cowan et al.,
1999), provoke spontaneous self-organization and the emergence of coherent, col-
lective phenomena that can be described only at higher levels than those of the
individual parts (Goldenfeld and Kadanoff, 1999). Therefore, it is useful to estab-
lish for clarity’s sake that chaos and complexity are different aspects of nonlinear
response. Chaos refers to simple systems that exhibit complicated behavior, such
as the intricate time series produced by a dripping faucet, the unpredictable oscil-
lations of a double pendulum, or the random behavior of populations in models of
logistic growth (May, 1976). Conversely, complexity refers to complicated systems
that exhibit simple, so-called emergent behavior. For instance, in the highly com-
plex tectonic-geologic subsystem, the emergent behavior is an earthquake, in the
world economy, a stock market crash, and in the biosphere, a massive extinction.
In the climate system, abrupt climate change is a likely example of unpredictable
emergent behavior. In fact, observations indicate that the climate system is, and
has been for millions of years, riddled with episodes of abrupt change, ranging
form large, sudden global warming episodes (e.g., the end of the last ice age), to
drastic and rapid regional changes in the hydroclimatic cycle, precipitation and
aridity (e.g., the expansion of the Sahara). Because of their obvious importance in
understanding future climate trends, these and other examples of abrupt climate
change are discussed in this paper.
Within the climate system chaotic behavior exhibits sensitive dependence to
initial conditions, confinement and typical aperiodicity. This is to say that tiny dif-
ferences in initial states can exponentially blow up to big differences in later states,
but the values of the relevant variables remain confined within fixed boundaries,
never exactly repeating. In the climate system, and as we shall soon discuss, plau-
sible examples of chaos are ENSO (El Niño, Southern Oscillation) and NAO (North
Atlantic Oscillation). In fact, simple deterministic models that exhibit chaotic
behavior qualitatively reproduce the irregular oscillations of ENSO for strong cou-
pling between ocean and atmosphere (e.g., Tziperman et al., 1994). ENSO may
in fact be chaotic in the sense that the equatorial Pacific climate may flip in a
chaotic way (randomly) from one to another of its three preferred quasi-stable
states (normal, La Niña, El Niño).
1.3. FEEDBACKS AND THRESHOLDS
Although chaotic dynamics and emergent properties may be surmised from data
interpretation and from the comparison of data to models, feedbacks are the only
climate processes whose presence and effects can often be quantified and, in some
cases understood with almost certainty. In this paper we illustrate how the presence
of several types of amplifying (positive) and controlling (negative) feedbacks, some
physical (ice sheet-albedo interaction), some biogeophysical (albedo-vegetation in-
teraction), and some biogeochemical (anthropogenic gases-atmosphere interaction)
can be deduced from observations. Feedbacks are the most likely processes behind
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 15
most of the nonlinearities in the climate. The relatively stable global temperature
and benign climate the earth has enjoyed for billions of years is testimony to the
action of regulating negative feedbacks which balance and neutralize amplifying
(explosive) positive feedbacks continuously (e.g., Watson and Lovelock, 1984). It
is quite likely that such a continuously active regulating feedback mechanism failed
to develop in Venus, leading to the present hellish environment of its surface. We
can then imagine that nature has arranged things in such way that on Earth, and
on the average, the net climate-driving feedback is negative, slightly stronger than
the net positive feedback, at least for small values of some (external or internal)
forcing. It is when the forcing grows to a point in which the positive feedback takes
over that its explosive amplification produces the nonlinear effects that we see in
the data. Thus, a critical threshold may in fact be the point at which the two com-
peting feedback effects are just balanced. Since there are countless feedbacks and
thresholds, rapid amplification of potentially exploding variables becomes highly
probable, and sharp, abrupt climate change should then be the norm, as appears to
be suggested by the past records of climate change. We must emphasize however
that there is as yet no basic understanding of abrupt climate change (Clark et al.,
2002).
1.4. PAPER ORGANIZATION
The goal of this paper is to discuss key issues and questions related to nonlinearity
in the Earth’s climate system and its implications in global climate change research.
First, we discuss examples of nonlinear climate response from observations of
abrupt climate change detected in both pre-historic and recent time series (Exam-
ples 2.1–2.5). Next we discuss models of coupled ocean and atmosphere mediated
by chaotic dynamics (Examples 3.1–3.2). Finally we look at nonlinearities in the
carbon cycle and the effects of biogeochemical feedbacks in models of present and
future climate change (Examples 4.1–4.4).
After the examples, we address scale and methodological issues as related to
some of the challenges in predicting the consequences of human actions on the
Earth’s climate system. For example, given the nearly certain occurrence of sudden
transitions between climate states, is ‘prediction’ per se achievable? We suggest
an alternative – and highly robust – approach using integrated assessments within
the framework of vulnerability studies, the details of which we then discuss and
justify. To conclude we provide a series of recommendations for research priorities,
including elucidating potential sources of nonlinearity, identifying key feedbacks
and linkages in the Earth’s climate system, and establishing Earth Systems Science
programs in order to provide the next generation of scientists a more complete view
on this crucial topic.
16 JOSÉ A. RIAL ET AL.
2. Nonlinearity, Abrupt Climate Change and Feedbacks in Past and Present
Climate Time Series
2.1. THE NONLINEAR PACEMAKER OF THE ICE AGES
Nature has been performing climate experiments for millions of years, and many of
the results are recorded in deep-sea sediments and ice cores (e.g., Cronin, 1999). It
is therefore important that we begin our discussion describing paleoclimate data to
provide a historical perspective. As we shall see, the paleoclimate records suggest
a strongly nonlinear, complex climate system.
The ice ages of the Pleistocene are remarkable quasi-periodic events of past
global climate change. At their peak global mean temperature was over 4◦Clower
than today, and enormous ice sheets several kilometers thick covered most of north-
ern North America and Eurasia. However, the records of the ice ages are far from
understood, mostly because the response of the climate to the presumed forcing
(secular changes in Earth’s orbital eccentricity, spin axis, and precession) appears
to be strongly nonlinear. For instance, it is well known that while the main driving
frequency of the ice ages is about 100 ky (1 ky = 1,000 years) the timing between
consecutive glacial periods has been steadily increasing from ∼80 ky to ∼120 ky
over the last ∼500 ky (Raymo, 1997; Petit et al., 1999). This feature, plus the near
absence of a large response at the strongest eccentricity forcing period (413 ky) and
the presence of significant variance at frequencies not present in the orbital forcing,
are strong evidence of nonlinearity in the climate’s response to orbital forcing
(e.g., Nobes et al., 1991; Ghil, 1994). To explain these nonlinear features, Rial
(1999) introduced the idea that the climate system transforms the astronomically
amplitude-modulated insolation into frequency modulated fluctuations of global
ice mass. This is frequency modulation entirely analogous to the electronic process
by which the frequency of a carrier signal is changed in proportion to the amplitude
of a relatively lower frequency signal, as in FM radio and television broadcasting.
Many well-known properties of FM signals are in fact fully consistent with fea-
tures of the paleoclimate data that have puzzled researchers for years, such as the
above mentioned varying duration of the ice age cycle, the presence of combination
tones of orbital frequencies, and perhaps the most telling, the apparent absence of
spectral power at 413 ky (Imbrie et al., 1993).
Frequency modulation is a phase- and frequency-locking process that transfers
energy from one frequency band into another, and creates new frequencies (called
sidebands) as combination tones of the carrier and the modulating frequencies, and
thus a good example of nonlinear cross talk among the frequencies that make up
the response.
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 17
2.2. THE MID-PLEISTOCENE CLIMATE SWITCH
Around 950 ky ago, a prominent switch in the frequency response of the cli-
mate system to orbital forcing occurred. This phenomenon, usually called the
mid-Pleistocene transition (MPT), resulted in a change from the 41 ky predomi-
nant glaciation period to a new ∼100 ky period, without a corresponding change
in the forcing orbital frequencies, as shown in Figure 2a. Though a number of
explanations have been proposed, the MPT continues to be one of the most puz-
zling examples of the nonlinear character of climate response. Figure 2a clearly
shows that the oscillatory response of the climate switches frequency and ampli-
tude at about 950 ky ago while the forcing is essentially the same throughout.
Mudelsee and Schulz (1997) estimate the ice mass to have increased by about
1.05±0.20 ×1019 kg, equivalent to an ice sheet area expansion of 3.1±0.7×1012
m2and thickness of up to 3 km. Such a large increase in ice extent (and ice topog-
raphy) must have created a new atmospheric circulation pattern and new feedbacks
to maintain the new, unprecedented climatic conditions (longer glaciations and
greater, thicker ice caps). A probable clue to the origin of the MPT is the fact that
around one million years ago the mean long-term trend of the insolation dropped
slightly to a new mean (Berger and Loutre, 1991). The corresponding decrease in
mean global temperature, amplified by feedbacks, could have shifted the climate
system’s sensitivity to forcing at lower frequency. The transformation of a mean
temperature step-like drop into a switch to a much lower resonant frequency is
a clear example of nonlinear response, consistent with the previously mentioned
transformation of amplitude modulation into frequency modulation.
2.3. ABRUPT WARMING EPISODES IN THE PALEOCLIMATE RECORD
Paleoclimate records over many time scales exhibit episodes of rapid, abrupt cli-
mate change, which may be defined as sudden climate transitions occurring at
rates faster than their known or suspected cause (Rahmstorf, 2001). Abrupt cli-
mate change is believed to be the result of instabilities, threshold crossings and
other types of nonlinear behavior of the global climate system (Clark et al., 1999;
Alley et al., 1999; Rahmstorf, 2000), but neither the physical mechanisms involved
nor the nature of the nonlinearities themselves are well understood. Figure 2b
shows selected examples of abrupt climate change in the form of rapid warm-
ing episodes followed by much slower cooling episodes. Each warming/cooling
sequence usually repeats at nearly equal time intervals, giving the time series
a characteristic quasi-periodic saw-tooth appearance that, remarkably, appears at
multiple time scales (as shown in the enlargement) and displays an unclear relation
to astronomical forcing.
Throughout most of the paleoclimate proxy data from sediments and ice cores,
there is a frequent repetition of this same theme; abrupt and fast warming (some-
times lasting only a few decades) followed by much slower cooling. This is a
pattern that, having happened often in the past, will likely happen in the future,
18 JOSÉ A. RIAL ET AL.
Figure 2a. Examples of nonlinearities in the paleoclimate. The mid-Pleistocene transition (MPT).
The global ice volume proxy shows a sudden change in predominant frequency around 950 ky ago.
The top panel shows the data (Site 806), the next two panels show (top trace) the result of filtering the
data with a narrow band-pass filter centered at 100 ky and below it the corresponding astronomical
forcing (insolation) filtered in the same manner. The lower two panels show a similar comparison
but for a filter centered at 41 ky. The longer period records reflect the selective nonlinearity of the
system, as the response to 100 ky forcing is negligible for times earlier than 1 Ma, and after that it
becomes strong, without a corresponding change in the forcing. No similar relation is seen in the
short periods. (Modified from Clark et al., 1999; Mudelsee and Schulz, 1997).
which makes compelling evidence for the urgent need to improve our understand-
ing of the physical processes involved. By itself, Figure 2b already provokes a
number of obvious and stimulating questions, such as, why are warming episodes
generally so much faster than cooling ones (saw-tooth)? How can rapid climate
change be triggered by slow change in orbital parameters? Does self-similarity
of response mean similarity of processes regardless of timescale? What nonlinear
processes are at work? Could the present rate of anthropogenic warming trigger
one of those abrupt, huge warming events of the last ice age?
The Dansgaard–Oeschger (D/O) oscillations of the last glacial shown in Fig-
ures 2b and 3a are among the clearest examples of abrupt warming episodes
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 19
Figure 2b. Samples of climate change across different time scales and proxy records (stable isotopic
ratios) for global temperature and ice volume, including SST (sea surface temperature) deep-sea
sediment (Site 667) and ice cores (Vostok, GRIP). Note the typical saw-tooth shape which, created
by the fast warming/slow cooling sequences, appears to be independent of time scale, displaying an
intriguing self-similarity. Main warming periods are indicated by vertical light gray stripes. Also,
note the close similarity between the temperature oscillations in Greenland and in the sub-tropics
(Bermuda) (data taken from Raymo, 1997; GRIP Project Members, 1993; Petit et al., 1999; Sachs
and Lehman, 1999).
(regional temperature in Greenland increased suddenly by up to 10 ◦Cinjusta
few decades and on multiple occasions). The climate was indeed highly variable
during glacial times and switched abruptly and frequently between cold and warm
modes. Ganopolski and Rahmstorf (2001) proposed the following mechanism. The
present-day climate state is characterized by a warm (switched-on) mode of the
thermohaline circulation (THC) being interpreted as an equilibrium state of the
underlying dynamics. Although a second stable state exists for the present-day
climate (see Figure 3b) representing a mode leading to much colder temperatures
over northern Europe (switched-off THC), a transition between the two has not
occurred during the Holocene because of the relatively large basin of attraction of
20 JOSÉ A. RIAL ET AL.
the warm mode. Quite the contrary, during the last glacial period, a stable (cold)
and a marginally unstable (warm) mode existed for the dynamics of the THC with
a much smaller basin of attraction for the cold mode. Utilizing CLIMBER 2.3,
a climate model of intermediate complexity (whose framework is illustrated in
Figure 1), it can be shown that a relatively small perturbation of the freshwater
input at high latitudes is sufficient to switch the system into the marginally un-
stable mode whose lifetime is of the order of several hundred years. A sinusoidal
modulation with amplitude much smaller than the boundaries F1/F2 in Figure 3b
does not induce switching in the present-day climate but does result in periodic
switching under glacial conditions. Preliminary results indicate that in the pres-
ence of noise, this driving amplitude can be further reduced, resulting in a flipping
behavior typical of the nonlinear effect called stochastic resonance (Ganopolsky
and Rahmstorf, 2001). Finally, the D/O events can be explained if a mild periodic
forcing (of unknown origin) of the THC plus noise is assumed. This external trigger
becomes amplified due to the coexistence of a stable state and a marginally unstable
mode in the THC system. Such coexistence is impossible in a linear system; hence,
nonlinearity is a necessary condition for switching behavior.
2.4. THE ABRUPT DESERTIFICATION OF THE SAHARA
Paleoclimatic reconstructions suggest that during the Holocene climate optimum
(9000–6000 years ago), North Africa was wetter and the Sahara was much smaller
than today (Prentice et al., 2000). Annual grasses and shrubs covered the desert,
and the Sahel reached as far as 23◦N (Claussen et al., 1999), over 500 km north
of its present location. During the Holocene optimum a slightly increased tilt of
the Earth’s spin axis and perihelion in July led to stronger insolation of the North-
ern Hemisphere during summer thereby strengthening the North African summer
monsoon (Kutzbach and Guetter, 1986). However, the North African climate is
sensitive to changes in land surface’s albedo, which can result from vegetation
removal. In fact, Charney and Stone (1975) recognized that high albedo resulting
from vegetation removal can enhance desert expansion by reducing rainfall, which
further reduces vegetation, in a strong, desert-expanding positive biogeophysical
feedback. This mechanism offers a possible explanation for climate changes in
the Sahara and particularly for increased drought in the Sahel and its southward
migration in late Holocene. Actually, when using present-day land-cover as initial
condition, models based solely on atmospheric processes do not yield an increase
in precipitation large enough to lead to a substantial reduction in the Sahara 6000
years ago (Joussaume et al., 1999). However, when feedbacks between atmosphere
and vegetation are incorporated, the models simulate a vegetation distribution in
good agreement with paleobotanic reconstructions (Claussen and Gayler, 1997;
deNoblet-Ducoudre et al., 2000; Doherty et al., 2000). Summarizing, precessional
forcing led to an enhancement of the African monsoon, creating conditions that
were then amplified mainly by atmosphere-vegetation feedbacks, and to a lesser
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 21
Figure 3a. Perhaps the most puzzling feature of recent paleoclimate records, highly relevant to under-
standing future global climate change, is the fast-warming/slow-cooling sequence found in the stable
isotope fluctuations (δ18O) time series of Greenland’s ice cores known as the Dansgaard-Oeschger
(D/O) oscillations (Jouzel et al., 1994; Alley et al., 1999). The D/O typically show very sudden,
6–10 ◦C warming episodes lasting a few centuries or perhaps even a few decades, followed by
millennia of relatively slow cooling. Remarkably, reconstructed sea surface temperatures (SST) in
the tropical Atlantic (Figure 2b) mimic the D/O record in the 30 ka to 60 ka interval, and similar
recordings are found in the subtropical Pacific and tropical Indian oceans. The longest period of
the signal in the inset is a submultiple of the precession forcing and evidence of precession forcing
exists elsewhere in the record (Rial, 2003). The ordinals near selected peaks correspond to numbered
interstadials and YD is the Younger Dryas event (Dansgaard et al., 1993).
22 JOSÉ A. RIAL ET AL.
Figure 3b. Climate (temperature) stability as a function of freshwater input at high latitudes in the
North Atlantic (Modified from Paillard, 2001).
extent by atmosphere-ocean interaction (Ganopolski et al., 1998; Braconnot et al.,
1999). These lead to multiple equilibrium states (Claussen, 1997) with the possibil-
ity of abrupt changes when thresholds are crossed (Brovkin et al., 1998), as shown
in Figure 4 (modified from Claussen et al. (1999) and DeMenocal et al. (2000)).
This figure shows a model simulation of an abrupt decline in precipitation in the
Sahara (20◦N–30◦N and 15◦W–50◦E) around 5,500 years ago that is supported
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 23
Figure 4. Simulation of transient development of precipitation (B), and vegetation fraction (C) as
response to changes in insolation (A depicts insolation changes on average over the northern hemi-
sphere during boreal summer). Results from Claussen et al. (1999) (B, C) are compared with data of
terrigenous material and estimated flux of material in North Atlantic cores off the North African coast
(D) by deMenocal et al. (2000). The figure is reproduced from Figure 2.8, in Kabat et al. (2003), with
permission.
by observations from sediment cores off the North African coast. The rapid change
contrasts markedly with the slow decrease in insolation.
2.5. ABRUPT SHIFTS AND TRENDS OF HYDROCLIMATIC TIME SERIES
Here we illustrate abrupt shifts and trends of hydroclimatic time series that occur
at decadal time scales, as compared to hundreds and thousands of years in the
previous examples. The effect of these changes on the environment and society are
of current concern because of their occurrence during our lifetime. An example
of complex time series with multidecadal trends are the annual flows of the Niger
24 JOSÉ A. RIAL ET AL.
River at Koulikoro (Figure 5) and the outflows from the African equatorial lakes
(Figure 6). As depicted in Figure 5, the Niger River series is characterized by a
slow decaying autocorrelation function, reflecting a long ‘memory’, yet there are
occasional large rapid shifts in the annual flows. Sveinsson et al. (2003) show that it
is possible to simulate statistically similar time series patterns of streamflows that
may occur in the future, and analyze the vulnerability of existing and projected
water supply systems in this region. As evident in the time series outflows from
the equatorial lakes measured at the Mongalla station for the period 1915–1983
(Figure 6), it is not necessary to employ any type of statistical analysis to recognize
that something peculiar happened with the outflow time series around 1962. Some
hydrologists have argued that such a sudden shift in the outflow may have been the
result of the lakes’ operation (e.g., Yevjevich, personal communication). However,
others (e.g., Lamb, 1966, Figure 1) have documented that Lake Victoria levels also
show a similar sudden shift during the same time period. Further analysis showed
that the period 1961–1964 has been the wettest consecutive period for the entire
historical precipitation record (Salas et al., 1981). Quite likely not only extreme
precipitation over the equatorial lakes (e.g., the major water input to Lake Victoria
is from precipitation over the lake itself) but also increases in the catchment runoff
and decreases in the lake evaporation and land evaporation/transpiration (as a re-
sult of increased cloudiness of heavy rainy periods during the same time period)
might have contributed to the occurrence of such significant and abrupt shifts in
the Equatorial lakes levels and lake outflows.
3. Nonlinear Irregular Oscillations and Chaos in Ocean-Atmosphere
Interactions
3.1. NORTH ATLANTIC OSCILLATION AND EL NIÑO/SOUTHERN OSCILLATION
The North Atlantic Oscillation (NAO) is a large-scale alternation of atmospheric
pressure fields (i.e., atmospheric mass) with centers of action near the Icelandic
Low and the Azores High. When sea-level pressure is lower than average in the Ice-
landic low pressure center, it is higher than average near the Azores, and vice-versa;
which can be described as a sort of see-saw oscillating behavior of the system.
Like ENSO (El Niño/Southern Oscillation), the NAO represents one of the most
important modes of decadal-scale variability of the climate system, and accounts
for up to 50% of sea-level pressure variability on both sides of the Atlantic (Hurrell,
1995). The NAO exerts a strong influence on precipitation and temperature on both
the eastern third of North America and western half of Europe, particularly during
winter months, and is responsible for many climatic anomalies (Beniston, 1997;
Hurrell, 1995).
The North Atlantic Oscillation index is computed as a difference of sea-level
pressure between the Azores (or Lisbon, Portugal) and Iceland. It is a measure
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 25
Figure 5. Time series of annual streamflows of the Niger River, Africa for the period 1907–1999
showing a complex pattern of high and low flows. The autocorrelation function shows the effect of
long memory (modified from Sveinsson et al. 2003).
Figure 6. Time series of annual outflows from the African equatorial lakes measured at the Mongalla
station for the period 1915–1983, showing an abrupt shift around 1961 and slow decaying downward
trend (adapted from Salas et al. (1981). With permission).
26 JOSÉ A. RIAL ET AL.
of the strength of zonal flows over the North Atlantic. A positive anomaly of the
NAO index represents a warm phase of the oscillation, with drier and warmer than
average conditions in the southern half of Europe. When the NAO index is strongly
positive, there is a general reduction in atmospheric moisture at high elevations
in the Alps (see e.g., Beniston and Jungo, 2002). Because of the highly positive
nature of the NAO index in the latter part of the 20th century, it is speculated
here that a significant part of the observed warming in the Alps results from shifts
in temperature extremes induced by the behavior of the NAO. These changes are
capable of having profound impacts on snow, hydrology, and mountain vegetation.
ENSO represents a nonlinear interplay of coupled ocean-atmosphere phenom-
ena (e.g., Tziperman, 1994). El Niño is the warm phase of ENSO, whereby a
weakening of the prevailing easterly trade winds in the equatorial Pacific allows
the eastward propagation of warm surface water that normally accumulate to the
west of the Pacific basin. Associated areas of deep convection ‘migrate’ with the
propagation of the warm surface water, which are the principal energy source for
convection. The area of anomalously warm surface water at the peak of an El Niño
episode can reach 30 million km2, roughly 3 times the size of Canada, and con-
sequently the sensible and latent heat exchange at the ocean-atmosphere interface
is sufficient to perturb climatic patterns globally. This perturbation occurs in three
simultaneous steps: vertical transfer of energy, heat and moisture through the deep
convection, horizontal propagation through atmospheric flows at high elevations
and, in time, an ‘overflow’ into the mid-latitude synoptic systems that can rein-
force or weaken surface pressure patterns and deflect the jet streams from their
usual trajectories. The cold phase of ENSO, commonly referred to as La Niña,
occurs sometimes (but not always) at the end of an El Niño event. Anomalously
cold waters invade the tropical Pacific region, and the strength of La Niña can in
some instances reverse the previously discussed anomaly patterns, i.e., by reversing
respective precipitation or drought patterns that occur during an El Niño event.
From a mechanistic point of view ENSO’s irregular oscillations can be under-
stood as those of a low-order chaotic system (Pacific ocean-atmosphere oscillator)
driven by the seasonal cycle (Tziperman et al., 1994). Since chaotic systems are
not totally unpredictable, at least not for the short time scale, it may eventually be
possible to estimate a range of predictability for ENSO within which models can
forecast with precision its short-term evolution.
3.2. THE PACIFIC DECADAL OSCILLATION
One close climate process to ENSO is the Pacific Decadal Oscillation (PDO),
which is an atmosphere-ocean phenomenon associated with persistent, bimodal
climate patterns in the North Pacific Ocean. The PDO is a numerical index based on
sea surface temperatures (SSTs) in a specific region of the North Pacific (Mantua
et al., 1997), which shows sudden shifting patterns with mean levels switching
from positive to negative and vice versa in time scales of about 20–50 years (Salas
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 27
Figure 7. Autocorrelation function and power spectrum obtained for the time series of annual PDO
indices for the period 1900–1999. The time series depicts abrupt shifts in addition to low frequency
variations that appear non-stationary. Most of the power is at periods around 50 years. The spec-
trum also shows clear periodicities at around 5.7 years (adapted from Salas and Pielke (2002). With
permission from John Wiley & Sons, Inc.).
and Pielke, 2002). In Figure 7 the autocorrelation function and spectrum reflect
the effect of a shifting low frequency pattern. Such shifting patterns illustrate the
nonstationarity of the climate system, in that the assumption of the stability of so-
called ‘climate normals’ does not adequately represent the real climate system. In
comparison with ENSO, the physical dynamics associated with the PDO are not
well understood, and the phase of the PDO is generally not predictable, although
it is possible to create scenarios depicting similar shifting PDO patterns using
stochastic methods (Sveinsson et al., 2003).
28 JOSÉ A. RIAL ET AL.
4. Nonlinearity and Feedbacks in the Carbon Cycle
4.1. ATMOSPHERE-CARBON CYCLE NONLINEAR FEEDBACKS
The ocean, vegetation, and soil on the land are currently absorbing about half of the
human emissions of atmospheric carbon dioxide (CO2), which has significant im-
plications for global climate change (Schimel et al., 2000). The processes involved
in CO2uptake by both land and ocean are known to be sensitive to the weather and
atmospheric CO2concentration, as well as other environmental factors, e.g., human
perturbations to the nitrogen cycle (Vitousek et al., 1997). For example, the uptake
of CO2by the ocean depends upon the difference in the CO2concentration across
the ocean-air interface (which tends to increase as atmospheric CO2rises), the sol-
ubility of CO2in seawater (which reduces as temperature rises), and the transport
of CO2to depth in the ocean (which is suppressed by thermal stratification and also
depends on the ocean circulation (Sarmiento et al., 1998)). Likewise, CO2uptake
by plants tends to increase with increasing CO2(depending upon the availability
of nutrients, water, temperature, and other variables, such as ozone concentration)
(Körner, 2000), but the breakdown of soil organic matter (SOM) is a highly non-
linear process, characterized by a number of nonlinear feedback loops involving
plants, microbes, SOM, and nutrient availability. Cheng (1999) characterized one
such loop operating in forests as follows: (i) increasing CO2uptake by plants leads
to an increase in carbon inputs to the rhizosphere (plant roots, soil microorganisms,
soil); (ii) increased soil carbon may or may not stimulate increases in microbial
respiration; (iii) altered rhizosphere respiration may either increase or decrease
SOM decomposition; (iv) changes in SOM decomposition cause changes in soil
nutrient mineralization and immobilization; (v) changes in soil nutrient dynam-
ics affect tree growth; and (vi) changes in tree growth have key implications for
global carbon sequestering, and hence, climate change. Supporting this, Gill et al.
(2002) reported that mineralization rates in soils of a Texas grassland decreased
nonlinearly with increasing CO2, and speculated that such decreases in nitrogen
availability will likely have a detrimental effect on long-term plant productivity
and, ultimately, on ecosystem carbon storage.
4.2. LAND-USE/VEGETATION FEEDBACKS ON THE REGIONAL SCALE
Eastman et al. (2001a,b) have shown that land-use change, grazing, and increased
carbon dioxide can significantly alter the regional climate system in the central
Great Plains of the United States. Figure 8 shows these effects on maximum
and minimum temperature, rainfall, and above ground biomass growth during a
growing season in this region. For example, the effects of enhanced atmospheric
concentrations of CO2on plant growth on a seasonal time scale are shown to
amplify the radiative effect of enhanced atmospheric CO2on the region. The non-
linear effect of vegetation-atmospheric feedback on this scale results in a complex
spatial and temporal pattern of response. Not only is there a teleconnection of
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 29
Figure 8. RAMS/GEMTM nonlinear coupled model results – the seasonal domain-averaged (central
Great Plains) for 210 days during the growing season, contributions to maximum daily temperature,
minimum daily temperature, precipitation, and leaf area index (LAI) due to: f1=natural vegetation,
f2=2×CO2radiation, and f3=2×CO2biology (adapted from Eastman et al. (2001), with
permission from Blackwell Publishing).
atmospheric conditions to locations distant from where the land feedback occurs,
but the landscape at distant locations itself is influenced by the altered weather.
In manipulative vegetation experiments where carbon dioxide concentrations are
arbitrarily increased, for example, this nonlinear feedback between the atmosphere
and land surface is missed since there is no feedback to the regional weather (with
greater vegetation cover resulting in greater summer rainfall and cooler maximum
temperatures.)
4.3. ‘SATURATION’OF THE ATMOSPHERIC OXIDATION PROCESS
The removal of a large number of greenhouse and polluting gases from the at-
mosphere (including all hydrocarbons, carbon monoxide (CO), nitrogen oxides
(NOx), sulfur oxides (SOx)) is accomplished through their reaction in the lower
atmosphere with the hydroxyl free radical (OH). This radical is produced by
processes involving nitrogen oxides, ozone, water vapor and short-wavelength ul-
traviolet radiation, and removed by reactions with the aforementioned gases. All
else being equal, lowering emissions of nitrogen oxides and/or increasing emis-
sions of carbon monoxide and methane, could erode OH levels, and therefore
30 JOSÉ A. RIAL ET AL.
increase the lifetimes of the above greenhouse gases (Thompson and Cicerone,
1986; Prinn et al., 2001). The atmospheric concentration of methane and other
greenhouse gases becomes a nonlinear function of emission rate since the reaction
process itself is dependent on their atmospheric concentrations, which in turn is
dependent on the emissions. Such combinations of emission changes therefore
constitute a positive feedback on climate change.
4.4. METHANE POSITIVE FEEDBACK PROCESSES
Methane is the third most important greenhouse gas (after water vapor and car-
bon dioxide). Two of its sources, or potential sources, exhibit nonlinear behavior.
Methanogens in the wetlands become increasingly active with warming above the
freezing point, while methane clathrate hydrates in submarine and subtundral de-
posits become methane sources above their known stability temperatures (Prinn et
al., 1999; Buffett, 2000). Atmospheric methane is already a significant consumer
of the very chemical (hydroxyl radical OH) which removes it. All else being equal,
increased methane emissions from wetlands and new emissions from clathrates
will therefore lower OH and increase the lifetime (and hence greenhouse forcing)
of methane above its current values (Prather, 1996). Thus, for methane, the two
sources above and the OH sink behave in a way that can constitute a significant
positive feedback on warming.
5. Consequences of a Complex, Nonlinear Earth System
In spite of the necessarily incomplete set of examples discussed above we hope to
have contributed to convey some of the challenges that researchers face in a field
where the dynamics are still being understood. The examples we chose illustrate
the existence of a wide diversity of nonlinear interactions that results in the recog-
nizable variability of climatic processes, but we have only touched the surface. If
spatial domain and long-distance interactions are included, there is much more to
investigate. For example, large-scale atmospheric circulation patterns exert a ma-
jor influence on local weather. Conversely, thunderstorm development exemplifies
how small-scale climate processes can upscale to affect large-scale atmospheric
circulations at long distances from the source of the disturbance (i.e., teleconnec-
tions). Another example is the land-use change in the tropics, which nonlinearly
influences thunderstorm patterns that propagate worldwide (Chase et al., 2000;
Zhao et al., 2000; Pielke, 2001b; Pielke et al., 2002).
On the other hand, our examples lead to an inevitable conclusion: since the
climate system is complex, occasionally chaotic, dominated by abrupt changes and
driven by competing feedbacks with largely unknown thresholds, climate predic-
tion is difficult, if not impracticable. Recall for instance the abrupt D/O warming
events (Figure 3a) of the last ice age, which indicate regional warming of over
10 ◦C in Greenland (about 4 ◦C at the latitude of Bermuda). These natural warming
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 31
events were far stronger – and faster – than anything current GCM work predicts
for the next few centuries. Thus, a reasonable question to ask is: Could present
global warming be just the beginning of one of those natural, abrupt warming
episodes, perhaps exacerbated (or triggered) by anthropogenic CO2emissions?
Since there is no reliable mechanism that explains or predicts the D/O, it is not
clear whether the warming events occur only during an ice age or can also occur
during an interglacial, such as the present. Other limitations in predictive skill for
a variety of environmental issues have been recently discussed in Sarewitz et al.
(2000), so prediction of future environmental change seems daunting, at least at
present.
Hence, it appears that one should not rely on prediction as the primary pol-
icy approach to assess the potential impact of future regional and global climate
change. We argue instead that integrated assessments within the framework of
vulnerability (IAV) offer the best solution, whereby risk assessment and disaster
prevention become the alternative to prediction.
In the Working Group II Report of the IPCC, vulnerability is defined as ‘the
degree to which a system is susceptible to, or unable to cope with, adverse ef-
fects of climate change, including climate variability and extremes’ (McCarthy,
2001). The vulnerability of a particular system is, of course, a function of both the
magnitude and rate of climate change as well as the current state of the system
(i.e., its adaptive capacity). Using this methodology, ‘impact models’ are applied
to assess the spectrum of potential changes in environmental forcings that result in
deleterious effects on a particular system. This quantification of the vulnerability
of a system can provide insight into the relative importance of climate, with respect
to other environmental influences. For example, Vörösmarty et al. (2000) use the
IAV approach to demonstrate that population growth is a much greater threat to
potable water supplies than the IPCC-predicted climate change. Other examples are
reported in Kabat et al. (2003), including the mathematical formalism to investigate
vulnerability.
The value of a vulnerability assessment is that the approach focuses on the
integrated effect of the spectrum of forcings and feedbacks on a system (e.g water
resources). Instead of attempting to predict the future state of a system, the risk
to a resource from all environmental (or other) threats is determined including the
presence of thresholds and their resiliency. Prevention substitutes prediction.
Global and regional projections based on models, the paleoclimatic and paleo-
environmental records, the historical record, and worst-case perturbations of the
historical record can be used to estimate which vulnerabilities have a reasonable
likelihood of occurring and eventually how to cope with them. Examples of the
application of the vulnerability approach, which can be used to assess the resilience
and sensitivity of different countries and cultures to environmental disturbance,
include ‘what if’ scenarios such as:
•The ‘dust bowl’ years of the 1930s were to occur again in the United States;
•The ‘Little Ice Age’ were to reoccur in Western Europe;
32 JOSÉ A. RIAL ET AL.
•An abrupt warming on the scale of the D/O (Figure 3a) was to occur; or
•Major volcanic eruptions similar to Tambora in 1815 were to take place?
The consequences (and, when possible, the probabilities) of these events need to
be assessed in the context of current socioeconomic and cultural conditions.
6. Recommended Research Areas
We have provided examples to illustrate that inputs and outputs within the Earth’s
climate system are not proportionate, that change is often episodic and abrupt – not
gradual and continuous – and multiple equilibria are the norm, not the exception
– consistent with the presumed nonlinear nature of Earth’s climate system. Actu-
ally, that the Earth’s climate system responds nonlinearly to (internal or external)
forcing seems widely accepted. However, what sort of nonlinearities are there, how
strong, and whether driven by astronomical forcing, by internal feedbacks, or by
both is far less clear, and only poorly understood. Given this, it is imperative for
the research community to adopt a research strategy that embraces the nonlinear
climate paradigm by, for instance, learning to identify the symptoms of nonlinearity
in the data, and to use the modern theoretical and practical means (models, data
processing) of diagnosing major climatic threats to society.
Therefore, we have agreed on a list of desirable research strategies – some of
which are specific, employing integrated assessments within the framework of a
vulnerability approach, and some of which are general. The list is not intended to be
exhaustive but hopefully illustrative of the many challenges (and opportunities) fac-
ing the Earth’s climate system research community. Accordingly, we recommend
to
•Explore the limits to climate predictability and search for switches and choke
points (or hot spots) of environmental change and variability.
•Construct models to explain the nonlinear response of the climate system to
changes in insolation forcing due to orbital parameter changes, an objective
best approached from the paleoclimate perspective.
•Improve our vision of the climate’s future through a better understanding of its
history. Paleoclimate and hydroclimate records exhibit abrupt changes in the
form of rapid warming events, the irregular oscillations of ENSO, catastrophic
floods, sustained droughts, and many other nonlinear response characteristics.
Extracting, identifying, categorizing, modeling and understanding these non-
linearities will greatly help our ability to understand the present and future
state of the climate.
•Develop GCMs coupled to low-dimensional energy balance ice sheet/litho-
sphere hybrid models (e.g., Deconto and Pollard, 2003) that can simulate the
interaction between hydrosphere, atmosphere and land over a wide range of
spatial (continental to global) and temporal (centennial, millennia) scales.
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 33
•Understand the global connectivity and variability of ocean-atmosphere cou-
pled phenomena, such as the North Pacific Oscillation (NPO), the Pacific
Decadal Oscillation (PDO), the Arctic Oscillation (AO), the North Atlantic
Oscillation (NAO), and the El Niño/Southern Oscillation (ENSO).
•Promote research to improve techniques that measure directly or indirectly
the spectral variability of the Sun’s irradiance output at decadal and millennial
scales.
•Understand the physics of the ocean thermohaline circulation (THC), whose
collapse may be one important cause of major climatic change in Western
Europe and North America (Rahmstorf, 2000).
•Perform sensitivity experiments with global climate models to evaluate the
response of the climate system to biospheric interactions (including vegetation
dynamics, and the effect associated with the anthropogenic input of carbon
dioxide and nitrogen compounds), the microphysical effects on clouds and
precipitation due to anthropogenic aerosol emissions, and land-use change
including fragmentation of ecosystems. Existing experiments to explore these
effects include Cox et al. (2000), Eastman et al. (2001b), and Pielke (2001a,b).
•Investigate the benefits and risks of large-scale deliberate human intervention
in the climate system. For example, carbon sequestration, associated with
land-management practices could be a strategy to remove CO2from the at-
mosphere. This should include the concurrent effect on water vapor fluxes
into the atmosphere and the net irradiance received at the Earth’s surface (e.g.,
Betts, 2000; Claussen, 2001; Pielke, 2001c). Another example is the effect of
the construction of large-scale water systems and the control of large lakes
such as Lake Victoria and the Great Lakes on regional climate systems.
•Identify locations or regions that are particularly sensitive to or easily im-
pacted by the planetary climate system. The Amazon rain forest and its fluvial
regime (Cox et al., 2000; Werth and Avissar, 2002), Southeast Asia (Chase
et al., 2000), the North Atlantic Ocean (Rahmstorf, 2000), the Arctic Ocean
(Foley et al., 1994), the boreal forest (Bonan et al., 1992), and the Nile River
system are examples of such sensitive locations.
•Investigate in increasing detail, nonlinear interactions involving changes in
biospheric emissions of chemically and radiatively important trace gases,
changes in atmospheric chemistry affecting the lifetimes of these gases, and
resultant changes in radiative forcing. Examples of such investigations using
simplified models include Homes and Ellis (1999) and Prinn et al. (1999).
To conclude, we recommend the development of new educational initiatives on
environmental/climate science. The complexity of the climate system, its myr-
iad of parts, interactions, feedbacks and unsolved mysteries needs researchers
able to transcend their own specialties, jump over and build bridges across ar-
tificial disciplinary boundaries. Hence, a fundamental requirement for the future
environmentalist/climatologist is a firm grasp of the mathematics and physics of
34 JOSÉ A. RIAL ET AL.
nonlinearity and of the methods and goals of interdisciplinary climate science.
We enthusiastically endorse John Lawton’s (2001) call for establishing specific
programs on ‘Earth System Science’ (ESS) at various institutions and universi-
ties, in order to provide upcoming generations of scientists with insight into the
complexity, the interdisciplinary nature and the crucial importance of these themes
for the future of humanity. The greatest challenge is to build a strong research
infrastructure that defines ESS, and as Lawton notes, the greatest barrier at present
is the lack of organizations ready to nurture this new discipline.
Acknowledgements
This paper resulted from a Workshop entitled ‘Nonlinear Responses to Global
Environmental Change: Critical Thresholds and Feedbacks – IGBP Nonlinear
Initiative’, organized by the International Biosphere-Geosphere Program (IGBP)
May 26–27th, 2001, Duke University, Durham, North Carolina. This paper con-
tributes to the new IGBP Nonlinear Initiative and to the efforts of individual core
projects on this topic including BAHC, GCTE, PAGES, and GAIM. Support for
this research includes that obtained from USGS Grant #99CRAG005 and SA
9005CS0014. JAR was partially supported by NSF grant #ATM0241274 (Paleocli-
mate program). We appreciate the detailed comments of an anonymous reviewer,
the editing skills of Dallas J. Staley and the incisive comments of Maya Elkibbi.
References
Alley, R. B., Clark, P. U., Keiwin, L. D., and Webb, R. S.: 1999,‘Making Sense of Millennial Scale
Climate Change’, in Clark, P. U., Webb, R. S., and Keiwin, L. D. (eds.), Mechanisms of Global
Climate Change at Millennial Time Scales,Amer. Geophys. Union, Geophys. Monogr. 112, 385–
394.
Alley, R. B., Marotzke, J., Nordhaus, W. D., Overpeck, J. T., Peteet, D. M., Pielke Jr., R. A., Pier-
Rehumbert, R. T., Rhines, P. B., Stocker, T. F., Talley, L. D., and Wallace, J. M.: 2003, ‘Abrupt
Climate Change’, Science 299, 2005–2010.
Aspen Global Change Institute: 1998, ‘Elements of Change 1997: Session One: Scaling from Site-
Specific Observations to Global Model Grids’.
Beniston, M.: 1997, ‘Variations of Snow Depth and Duration in the Swiss Alps over the Last 50
Years: Links to Changes in Large-Scale Climatic Forcings’, Clim. Change 36, 281–300.
Beniston, M. and Jungo, P.: 2002, ‘Shifts in the Distributions of Pressure, Temperature and Moisture
and Changes in the Typical Weather Patterns in the Alpine Region in Response to the Behavior
of the North Atlantic Oscillation’, Theor. Appl. Climatol. 71, 29–42.
Berger, A. and Loutre, M. F.: 1991, ‘Insolation Values for the Climate of the Last 10 Million of
Years’, Quat.Sci.Rev.10 (4), 297–317.
Betts, R. A.: 2000, ‘Offset of the Potential Carbon Sink from Boreal Forestation by Decreases in
Albedo’, Nature 408, 187–190.
Bonan, G. B., Pollard, D., and Thompson, S. L.: 1992, ‘Effects of Boreal Forest Vegetation on Global
Climate’, Nature 359, 716–718.
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 35
Braconnot, P., Joussaume, S., Marti, O., and de Noblet-Ducoudre, N.: 1999, ‘Synergistic Feedbacks
from Ocean and Vegetation on the African Monsoon Response to Mid-Holocene Insolation’,
Geophys. Res. Lett. 26, 2481–2484.
Brovkin, V., Claussen, M., Petoukhov, V., and Ganopolski, A.: 1998, ‘On the Stability of the
Atmosphere-Vegetation System in the Sahara/Sahel Region’, J. Geophys. Res. 103, 31613–
31624.
Buffett, B. A.: 2000, ‘Clathrate Hydrates’, Ann. Rev. Earth Planet. Sci. 28, 477–508.
Chase, T. N., Pielke, R. A., Kittel, T. G. F., Nemani, R. R., and Running, S. W.: 2000, ‘Simulated
Impacts of Historical Land Cover Changes on Global Climate’, Clim. Dyn. 16, 93–105.
Cherney, J. and Stone, P. H.:1975, ‘Drought in the Sahara: A Biogeophysical Feedback Mechanism’,
Science 187, 434–435.
Cheng, W. X.: 1999, ‘Rhizosphere Feedbacks in Elevated CO2’, Tree Physiol. 19, 313–320.
Clark, P. U., Alley, R. B., and Pollard, D.: 1999, ‘Northern Hemisphere Ice Sheet Influences on
Global Climate Change’, Science 286, 1104–1111.
Clark, P. U., Pisias, N. G., Stocker, T. F., and Weaver, A. J.: 2002, ‘The Role of the Thermohaline
Circulation in Abrupt Climate Change, Nature 415, 863–869.
Claussen, M.: 1997, ‘Modelling Biogeophysical Feedback in the African and Indian Monsoon
Region’, Clim. Dyn. 13, 247–257.
Claussen, M.: 2001, ‘Earth System Models’, in Ehlers, E. and Krafft, T. (eds.), Understanding the
Earth System: Compartments, Processes and Interactions, Springer-Verlag, Heidelberg, pp. 145–
162.
Claussen, M. and Gayler, V.: 1997, ‘The Greening of Sahara during the Mid-Holocene: Results of an
Interactive Atmosphere-Biome Model’, Global Ecol. Biogeog. Lett. 6, 369–377.
Claussen, M., Kubatzki, C., Brovkin, V., Ganopolski, A., Hoelzmann, P., and Pachur, H. J.: 1999,
‘Simulation of an Abrupt Change in Saharan Vegetation at the End of the Mid-Holocene’,
Geophys. Res. Lett. 26, 2037–2040.
Claussen, M., Mysak, L. A., Weaver, A. J., Crucifix, M., Fichefet, T., Loutre, M.-F., Weber, S. L.,
Alcamo, J., Alexeev, V. A., Berger, A., Calov, R., Ganopolski, A., Goosse, H., Lohmann, G.,
Lunkeit, F., Mokhov, I. I., Petoukhov, V., Stone, P., and Wang, Z.: 2002, ‘Earth Systems Models
of Intermediate Complexity: Closing the Gap in the Spectrum of Climate System Models’, Clim.
Dyn. 18, 579–586.
Cowan, G. A., Pines, D., and Meltzer, D.: 1999, Complexity, Metaphors, Models and Reality,Perseus
Books, Santa Fe Institute.
Cox, P. M., Betts, R. A., Jones, C. D., Spall, S. A., and Totterdell, I. J.: 2000, ‘Acceleration of Global
Warming Due to Carbon-Cycle Feedbacks in a Coupled Climate Model’, Nature 408, 184–187.
Cronin, T. M. (1999): Principles of Paleoclimatology, Columbia U. Press, New York.
DeConto, R. and Pollard, D.: 2003, ‘Rapid Cenozoic Glaciation of Antarctica Induced by Declining
Atmospheric CO2’, Nature 421, 245–249. deMenocal, P. B., Ortiz, J., Guilderson, T., Adkins,
J., Sarnthein, M., Baker, L., and Yarusinsky, M.: 2000, ‘Abrupt Onset and Termination of the
African Humid Period: Rapid Climate Response to Gradual Insolation Forcing’, Quat. Sci. Rev.
19, 347–361. de Noblet-Ducoudre, N. and Claussen, M.: 2001, ‘Mid-Holocene Greening of the
Sahara: First Results of the GAIM 6000 Year BP Experiment with Two Asynchronously Coupled
Atmosphere/Biome Models’, Clim. Dyn. 16, 643–659.
Doherty, R., Kutzbach, J., Foley, J., and Pollard, D.: 2000, ‘Fully Coupled Climate/Dynamical
Vegetation Model Simulations over Northern Africa during the Mid-Holocene’, Clim. Dyn. 16,
561–573.
Eastman, J. L., Coughenour, M. B., and Pielke, R. A. Sr.: 2001a, ‘The Effects of CO2and Landscape
Change Using a Coupled Plant and Meteorological Model’, Global Change Biol. 7, 797–815.
Eastman, J. L., Coughenour, M. B., and Pielke, R. A. Sr.: 2001b, ‘Does Grazing Affect Regional
Climate’, J. Hydrometeor. 2, 243–253.
36 JOSÉ A. RIAL ET AL.
Foley, J., Kutzbach, J. E., Coe, M. T., and Levis, S.: 1994, ‘Feedbacks between Climate and Boreal
Forests during the Holocene Epoch’, Nature 371, 52–54.
Gallagher, R. and Appenzeller, T.: 1999, ‘Beyond Reductionism: Introduction to Special Section on
Complex Systems’, Science 284, 79–109.
Ganopolski, A., Kubatzki, C., Claussen, M., Brovkin, V., and Petoukhov, V.: 1998, ‘The Influence of
Vegetation-Atmosphere-Ocean Interaction on Climate during the Mid-Holocene’, Science 280,
1916–1919.
Ganopolski, A. and Rahmstorf, S.: 2001, ‘Rapid Changes of Glacial Climate Simulated in a Coupled
Climate Model’, Nature 409, 153–158.
Ghil, M.: 1994, ‘Cryothermodyamics: The Chaotic Dynamics of Paleoclimate’, Physica D 77, 130–
159.
Gill, R. A., Polley, H. W., Johnson, L. J., Maherali, H., and Jackson, R.: 2002, ‘Nonlinear Grasslands
Responses to Past and Future Atmospheric CO2’, Nature 417, 279–282.
Goldenfeld, N. and Kadanoff, L. P.: 1999, ‘Simple Lessons from Complexity’, Science 284, 87–89.
GRIP Project Members: 1993, ‘Climate Instability during the Last Interglacial Period Recorded in
the GRIP Ice Core’, Nature 364, 203–207.
Homes, K. J. and Ellisa, J. H.: 1999, ‘An Integrated Assessment Modeling Framework for Assessing
Primary and Secondary Impacts from Carbon Dioxide Stabilization Scenarios’, Environ. Model.
Assess. 4, 45–63.
Hurrell, J. W.: 1995, ‘Decadal Trends in the North Atlantic Oscillation Regional Temperatures and
Precipitation’, Science 269, 676–679
Imbrie, J., Berger, A., Boyle, E. A., Clemens, S. C., Duffy, A., Howard, W. R., Kukla, G., Kutzbach,
J., Martinson, D. G., McIntyre, A., Mix, A. C., Molfino, B., Morley, J. J., Peterson, L. C., Pisias,
N. G., Prell, W. L., Raymo, M. E., Shackleton, N. J., and Toggweiler, J. R.: 1993, ‘On the
Structure and Origin of Major Glaciation Cycles 2. The 100,000-Year Cycle’, Paleoceanography
8(6), 699–735.
Joussaume, S., Taylor, K. E., Braconnot, P., Mitchell, J. F. B., Kutzbach, J. E., Harrison, S. P.,
Prentice, I. C., Broccoli, A. J., Abe-Ouchi, A., Bartlein, P. J., Bonfiels, C., Dong, B., Guiot, J.,
Herterich, K., Hewit, C. D., Jolly, D., Kim, J. W., Kislov, A., Kitoh, A., Loutre, M. F., Masson,
V., McAvaney, B., McFarlane, N., deNoblet, N., Peltier, W. R., Peterschmitt, J. Y., Pollard, D.,
Rind, D., Royer, J. F., Schlesinger, M. E., Syktus, J., Thompson, S., Valdes, P., Vettoretti, G.,
Webb, R. S., and Wyputta, U.: 1999, ‘Monsoon Changes for 6000 Years Ago: Results of 18
Simulations from the Paleoclimate Modeling Intercomparison Project (PMIP)’, Geophys. Res.
Lett. 26, 859–862.
Kabat, P., Claussen, M., Dirmeyer, P. A., Gash, J. H. C., Bravo de Guenni, L., Meybeck, M., Pielke
Sr., R. A., Vörösmarty, C. J., Hutjes, R. W. A., and Lütkemeier, S. (eds.): 2003, Vegetation,
Water, Humans and the Climate: A New Perspective on an Interactive System, Springer, Berlin,
Heidelberg, New York, approx. 550 pp., in press.
Kaplan, D. and Glass, L.: 1995, Understanding Nonlinear Dynamics, Springer-Verlag, New York.
Kim, Y. C. and Powers, E. J.: 1978, ‘Digital Bispectral Analysis of Self-Excited Fluctuation Spectra’,
Phys. Fluids 21, 1452–1453.
Körner, C.: 2000, ‘Biosphere Responses to CO2Enrichment’, Ecol. Appl. 10, 1590–1619.
Kutzbach, J. E. and Guetter, P. J.: 1986, ‘The Influence of Changing Orbital Parameters and Surface
Boundary Conditions on Climate Simulations for the Past 18,000 Years’, J. Atmos. Sci. 43, 1726–
1759.
Lawton, J. H.: 2001, ‘Earth System Science’, Science 292, 1965.
Lamb, H. H.: 1966, ‘Climate in the 1960s’, Geogr. J. 132, 183–212.
Lorenz, E. N.: 1963, ‘Deterministic Nonperiodic Flow’, J. Atmos. Sci. 20, 130–141
Mantua, N. J., Hare, S. R., Zhang, Y., Wallace, J. M., and Francis, R. C.: 1997, ‘A Pacific Interdecadal
Climate Oscillation with Impacts on Salmon Production’, Bull. Amer. Meteorol. Soc. 78, 1069–
1079.
NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS 37
May, R. M.: 1976, ‘Simple Mathematical Models with Very Complicated Dynamical Behavior’,
Nature 261, 459–467.
McCarthy, J. J., Canziani, O. F., Leary, N. A., Dokken, D. J., and White, K. S. (eds.): 2001, Climate
Change 2001: Impacts, Adaptation and Vulnerability, Contribution of Working Group II to the
Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC), Cambridge
University Press, Cambridge.
Mudelsee, M. and Schultz, M.: 1997, ‘The Mid-Pleistocene Climate Transition: Onset of the 100 ka
Cycle Lags Ice Volume Build-up by 280 ka’, Earth Planet. Sci. Lett. 151, 117–123.
Nobes, D. C., Bloomer, S. F., Mienert, J., and Westall, F.: 1991, ‘Milankovitch Cycles and Nonlinear
Response in the Quaternary Record in the Atlantic Sector of the South Oceans’, Proceedings
ODP Scientific Results 114, 551–576.
Paillard, D.: 2001, ‘Glacial Hiccups’, Nature 409, 147–148.
Petit, J. R., Jouzel, J., Raynaud, D., and Barkov, N. I.: 1999, ‘Climate and Atmospheric History of
the Past 420,000 Years from the Vostok Ice Core, Antarctica’, Nature 399, 429–436.
Pielke Sr., R. A.: 2001a, ‘Earth System Modeling – An Integrated Assessment Tool for Environ-
mental Studies’, in Matsuno, T. and Kida, H. (eds.), Present and Future of Modeling Global
Environmental Change: Toward Integrated Modeling, Terra Scientific Publishing Company,
Tokyo, Japan, pp. 311–337.
Pielke Sr., R. A.: 2001b, ‘Influence of the Spatial Distribution of Vegetation and Soils on the
Prediction of Cumulus Convective Rainfall’, Rev. Geophys. 39, 151–177.
Pielke Sr., R. A.: 2001c, ‘Carbon Sequestration: The Need for an Integrated Climate System
Approach’, Bull. Amer. Meteor. Soc. 82, 2021.
Pielke Sr., R. A., Marland, G., Betts, R. A., Chase, T. N., Eastman, J. L., Niles, J. O., Niyogi, D., and
Running, S.: 2002, ‘The Influence of Land-Use Change and Landscape Dynamics on the Climate
System-Relevance to Climate Change Policy beyond the Radiative Effect of Greenhouse Gases’,
Phil. Trans. A. Special Theme Issue 360, 1705–1719.
Pisias, N. G., Mix, A. C., and Zahn, R.: 1990, ‘Nonlinear Response in the Global Climate System:
Evidence from Benthic Oxygen Isotopic Record in Core RC13–110’, Paleoceanography 5(2),
147–160.
Prentice, I. C., Jolly, D., and BIOME 6000 members: 2000, ‘Mid-Holocene and Glacial-Maximum
Vegetation Geography of the Northern Continents and Africa’, J. Biogeogr. 27, 507–519.
Prather, M. J.: 1996, ‘Natural Modes and Time Scales in Atmospheric Chemistry: Theory, GWPs for
CH4andCO,andRunawayGrowth’,Geophys. Res. Lett. 23, 2597–2600.
Prinn, R. G., Jacoby, H. D., Sokolov, A., Wang, C., Xiao, X., Yang, Z., Eckaus, R. S., Stone,
P. H., Ellerman, A. D., Melillo, J. M., Fitzmaurice, J., Kicklighter, D. W., Holian, G. L., and
Liu, Y.: 1999, ‘Integrated Global System Model for Climate Policy Assessment: Feedbacks and
Sensitivity Studies’, Clim. Change 41, 469–546.
Prinn, R. G., Huang, J., Weiss, R. F., Cunnold, D. M., Fraser, P. J., Simmonds, P. G., Harth, C.,
Salameh, P., O’Doherty, S., Wang, R. H. J., Porter, L., and Miller, B. R.: 2001, ‘Evidence for
Substantial Variations of Atmospheric Hydroxyl Radicals in the Last Two Decades’, Science
292, 1882–1888.
Rahmstorf, S.: 2000, ‘The Thermohaline Ocean Circulation: A System with Dangerous Thresholds?’,
Clim. Change 46, 247–256.
Rahmstorf, S.: 2001, ‘Abrupt Climate Change’, in Steele, J., Thorpe, S., and Turekian, K. (eds.),
Encyclopedia of Ocean Sciences, Academic Press, London, pp. 1–6.
Raymo, M. E.: 1997, ‘The Timing of Major Climate Terminations’, Paleoceanography 12, 577–585.
Rial, J. A.: 1999, ‘Pacemaking the Ice Ages by Frequency Modulation of Earth’s Orbital Eccentric-
ity’, Science 285, 564–568.
Rial, J. A.: 2003, ‘Abrupt Climate Change: Chaos and Order at Orbital and Millennial Scales’, Glob.
Plan. Change, in press.
Rind, D.: 1999, ‘Complexity and Climate’, Science 284, 105–107.
38 JOSÉ A. RIAL ET AL.
Sachs, J. P. and Lehman, S.: 1999, ‘Subtropical North Atlantic Temperatures 60,000 to 30,000 Years
Ago, Science 286, 756–759.
Salas, J. D. and Pielke Sr., R. A.: 2002, ‘Stochastic Characteristics and Modeling of Hydroclimatic
Processes’, Chapter 32 in Potter, T. and Colman, B. (eds.), Handbook of Weather, Climate, and
Wate r , John Wiley and Sons, in press.
Salas, J. D., Obeysekera, J. T. B., and Boes, D. C.: 1981, in Singh, V. P.(ed.), Modeling of the Equato-
rial Lakes Outflows, in Statistical Analysis of Rainfall and Runoff, Water Resources Publications,
Littleton, CO, pp. 431–440.
Sarewitz, D., Pielke Jr., R. A., and Byerly Jr., R.: 2000, Prediction, Science, Decision Making and
the Future of Nature, Island Press, Washington, DC, p. 405.
Sarmiento, J. L., Hughes, T. M. C., Stouffer, R. J. (et al.): 1999, ‘Simulated Response of the Ocean
Carbon Cycle to Anthropogenic Climate Warming’, Nature 402, 245–249.
Schellnhuber, H. J.: 1999, ‘Earth System Analysis and the Second Copernican Revolution’, Nature
402, C19–C26.
Schimel, D., Melillo, J., Tian, H. Q., McGuire, A. D., Kicklighter, D., Kittel, T., Rosenblum, N.,
Running, S., Thorton, P., Ojima, D., Parton, W., Kelly, R., Sykes, M., Neilson, R., and Rizzo,
B.: 2000, ‘Contribution of Increasing CO2and Climate to Carbon Storage by Ecosystems in the
United States’, Science 287, 2004–2006.
Stocker, T. F. and Schmittner, A.: 1997, ‘Influence of CO2Emission Rates on the Stability of the
Thermohaline Circulation’, Science 388, 862–865.
Sveinsson, O. G., Salas, J. D., Boes, D. C., and Pielke Sr., R. A.: 2003, ‘Modeling of Long Term
Variability of Climatic and Hydrologic Processes’, J. Hydrometeor. 4, 489–505.
Thompson, A. M. and Cicerone, R. J.: 1986, ‘Possible Perturbations to Atmospheric CO, CH4,and
OH’, J. Geophys. Res. 91, 10853–10864.
Tziperman, E., Stone, L., Cane, M. A., and Jarosh, H.: 1994, ‘El Niño Chaos: Overlapping of Reso-
nances between the Seasonal Cycle and the Pacific Ocean-Atmosphere Oscillator’, Science 264,
72–74.
Vitusenko, P. M., Mooney, H. A., Lubchenco, J., and Melillo, J. M.: 1997, ‘Human Domination of
Earth’s Ecosystems’, Science 277, 494–499.
Vörösmarty, C. J. P., Green, P., Salisbury, J., and Lammers, R. B.: 2000, ‘Global Water Resources:
Vulnerability from Climate Change Acid Population Growth’, Science 289, 284–288.
Watson, A. J. and Lovelock, J. E.: 1984, ‘Biological Homeostasis of the Global Rnvironment: The
Parable of Daisyworld’, Tellus 35, 284–289.
Werth, D. and Avissar, R.: 2002, ‘The Local and Global Effects of Amazon Deforestation’, J.
Geophys. Res. 107, D20, 8087, doi: 10.1029/2001JD000717.
Zhao, M., Pitman, A. J., and Chase, T.: 2000, ‘The Impact of Land Cover Change on the Atmospheric
Circulation’, Clim. Dyn. 17, 467–477.
(Received 11 March 2002; in revised form 1 October 2003)
