The fluctuation-dissipation theorem (FDT) has its roots in the classical
theory of many-particle systems in thermodynamic equilibrium. The idea is very
simple: the system’s return to equilibrium will be the same whether the
perturbation that modified its state is due to a small external force or to an
internal, random fluctuation. The FDT thus relates natural and forced
fluctuations of a system (e,g., Kubo, 1957, 1966); it is a cornerstone of
statistical mechanics and has applications in many areas (Marconi  _et al._ ,
2008, and references therein).

We have emphasized already in Sect. III.2 that, even when the climate system
is in a steady state, it is not at all in thermodynamic equilibrium (e.g.,
Ghil, 2019; Lucarini and Ragone, 2011). Still, Leith (1975) showed that FDT
applies to a 2-D or QG turbulent flow with two integral invariants, kinetic
energy $E$ and enstrophy $Z$, under some additional assumptions of normal
distribution of the realizations and stationarity. Soon thereafter, Bell
(1980) showed that the FDT still seemed to work for a highly truncated version
of such a model, even in the presence of dissipative terms that invalidate the
thermodynamic equilibrium assumption.

The FDT has been applied to the output of climate models to predict the
climate response to a step-like increase of the solar irradiance North  _et
al._ (1993) as well as to increases in atmospheric CO2 concentration Cionni
_et al._ (2004); Langen and Alexeev (2005), while Gritsun and Branstator
(2007) and Gritsun  _et al._ (2008) used it to predict the response of an
atmospheric model to localised heating anomalies. Most recently, Cox  _et al._
(2018) tried to reduce the uncertainty in the ECS discussed in Sect. IV.2 by a
systematic FDT application to an ensemble of model outputs, as well as to the
observed instrumental climate variability.

The FDT-based response of the system to perturbations in the above examples
reproduces the actual changes at a good qualitative, rather than strictly
quantitative level. An important limitation of these insightful studies is the
use of a severely simplified version of the FDT that is heuristically
constructed by taking a gaussian approximation for the invariant measure of
the unperturbed system. This approximation amounts to treating the climate as
being in thermodynamic equilibrium; see also Majda  _et al._ (2009), who
specifically address the applicability of the FDT in a reduced phase space.

In order to address the problem of gaussian approximation, Cooper and Haynes
(2011) proposed to construct a kernel-based approximation of the actual
invariant measure of the unperturbed system and then use Eq. (56) below to
construct the system’s Green function. This approach has been applied
successfully in a very low-dimensional system, but its robustness may be
limited by the kernel’s arbitrary cut-off not being able to account for the
smaller scales of the invariant measure’s fine structure.

####  IV.5.3 Ruelle Response Theory

FDT generalizations to systems out of equilibrium have been developed since
the early 1950s (e.g., Kubo, 1966). But a particularly fruitful change in
point of view was provided by D. Ruelle Ruelle (1998, 1999, 2009), who
considered the problem in the setting of dynamical systems theory, rather than
that of statistical mechanics. The former point of view is justified in this
context by the so-called chaotic hypothesis Gallavotti and Cohen (1995), which
states, roughly, that chaotic systems with many degrees of freedom possess a
physically relevant invariant measure, as discussed in Sect. IV.5.1.

It is common in practice to assume that a time-dependent measure
$\mu_{t}(\mathrm{d}x)$ associated with the evolution of the dynamical system
given by Eq. (50) does exist. Still, computing the expectation value of
measurable observables with respect to this measure is in general far from
trivial and requires setting up a large ensemble of initial states in the
Lebesgue measurable set $\mathbf{B}$ mentioned before. Moreover, PBAs and the
physical measures they support only set the stage for predicting the system’s
sensitivity to small changes in the forcing or the dynamics.

Ruelle’s response theory allows one to compute the change in the measure
$\mu(\mathrm{d}x)$ of an autonomous Axiom A system due to weak perturbations
of intensity $\epsilon$ applied to the dynamics, in terms of the unperturbed
system’s properties. The basic idea behind it is that the invariant measure of
such a system, even though supported on a strange attractor, is differentiable
with respect to $\epsilon$. Baladi (2008) reviewed extensions of response
theory and provided a complementary approach to it, while Lucarini (2016)
estimated its radius of convergence in $\epsilon$.

The nonautonomous version of the Ruelle response theory allows one to
calculate the time-dependent measure $\mu_{t}(\mathrm{d}x)$ on the PBA by
computing the time-dependent corrections to it with respect to a reference
state $x(t)=\tilde{x}(t)$. This provides a fairly general formulation for the
climate system’s response to perturbations.

Let us assume that we can write

$\dot{x}=F(x,t)=F(x)+\epsilon X(x,t)$ (52)

where, $\forall t\in\mathbb{R}$ and $\forall
x\in\mathcal{Y}\subset\mathbb{R}^{d}$, $|\epsilon X(x,t)|\ll|F(x)|$. Hence, we
can take $F(x)$ as the background dynamics and $\epsilon X(x,t)$ as a
perturbation. We only treat here the case of deterministic dynamics; see
Lucarini (2012) for stochastic perturbations. As shown by Lucarini  _et al._
(2017), we can restrict our analysis without loss of generality to the
separable case of $F(x,t)=F(x)+\epsilon X(x)T(t)$.

To evaluate the expectation value $\langle\Psi\rangle^{\epsilon}(t)$ of a
measurable observable $\Psi(x)$ with respect to the measure
$\mu_{t}(\mathrm{d}x)$ of the system governed by Eq. (50), one writes:

$\langle\Psi\rangle^{\epsilon}(t)=\int\Psi(x)\mu_{t}(\mathrm{d}x)=\langle\Psi\rangle_{0}+\sum_{j=1}^{\infty}\epsilon^{j}\langle\Psi\rangle_{0}^{(j)}(t);$
(53)

here $\langle\Psi\rangle_{0}=\int\Psi(x)\bar{\mu}(\mathrm{d}x)$ is the
expectation value of $\Psi$ with respect to the SRB invariant measure
$\bar{\mu}(\mathrm{d}x)$ of the autonomous dynamical system $\dot{x}=F(x)$. We
will restrict ourselves here to the linear correction term, which can be
written as:

$\displaystyle\langle\Psi\rangle_{0}^{(1)}(t)$
$\displaystyle={\displaystyle\int\int_{0}^{\infty}}\Lambda
S_{0}^{\tau}\Psi(x)T(t-\tau)\mathrm{d}\tau\bar{\mu}(\mathrm{d}x);$ (54)
$\displaystyle={\displaystyle\int_{0}^{\infty}}G^{(1)}_{\Psi,X}(\tau)T(t-\tau)\mathrm{d}\tau.$

The Green’s function $G^{(1)}_{\Psi,X}(\tau)$ above is given by

$G^{(1)}_{\Psi,X}(\tau)=\int\Theta(\tau)\Lambda
S_{0}^{\tau}\Psi(x)\bar{\mu}(\mathrm{d}x),$ (55)

where $\Lambda(\bullet)={X}\cdot\nabla(\bullet)$ and
$S_{0}^{t}(\bullet)=\exp(t{F}\cdot\nabla)(\bullet)$ is the semigroup of
unperturbed Koopman operators888In discrete-time dynamics given by
$x_{k+1}=g(x_{k})$, the Koopman operator is a linear operator acting on
observables $h:\mathcal{Y}\to\mathbb{R}$ via $Uh(x)=h(f(x))$. In continuous-
time dynamics, like Eq. (50) herein, this operator is replaced by a semigroup
of operators, as in Eq. (55) above; see Budi$v$sić  _et al._ (2012) for a good
review., $(\cdot)$ denotes the inner product in $\mathcal{Y}$, and the
Heaviside distribution $\Theta(\tau)$ enforces causality. If the unperturbed
invariant measure $\mathrm{d}\bar{\mu}(x)$ is smooth with respect to the
standard Lebesgue measure, one has
$\bar{\mu}(\mathrm{d}{x})=\tilde{\mu}(x)\mathrm{d}{x}$, with $\tilde{\mu}(x)$
the density, and the Green’s function can be writen as follows:

$\displaystyle G^{(1)}_{\Psi,X}(\tau)$
$\displaystyle=\Theta(\tau){\displaystyle\int\frac{-\nabla\cdot(\tilde{\mu}(x)X)}{\tilde{\mu}(x)}}S_{0}^{\tau}\Psi(x)\tilde{\mu}(x)\mathrm{d}x$
(56) $\displaystyle=\Theta(\tau)\mathcal{C}(\Phi,S_{0}^{\tau}\Psi);$

here $\Phi={-\left(\nabla\cdot(\tilde{\mu}(x)X)\right)}/{\tilde{\mu}(x)}$ and
$\mathcal{C}(A,S_{0}^{\tau}B)$ is the $\tau$-lagged correlation between the
variables $A$ and $B$, and the average of $\Phi$ vanishes. Note that Eq. (56)
is the appropriate generalization of the FDT for the nonautonomous, out-of-
equilibrium system (52).

Given any specific choice of the forcing’s time dependency $T(t)$ in Eqs.
(52)–(54) and measuring the linear correction term
$\langle\Psi\rangle_{0}^{(1)}(t)$ from a set of experiments, the same
equations allow one to derive the appropriate Green’s function. Therefore,
using the output of a specific set of experiments or of GCM simulations, we
achieve predictive power for any temporal pattern of the forcing $X(x)$.

Consider now the Fourier transform of Eq. (54):

$\langle\Psi\rangle_{0}^{(1)}(\omega)=\chi^{(1)}_{\Psi,X}(\omega)T(\omega),$
(57)

where we have introduced the susceptibility
$\chi^{(1)}_{\Psi,X}(\omega)=\mathcal{F}[G^{(1)}_{\Psi,X}]$, defined as the
Fourier transform of the Green’s function $G^{(1)}_{\Psi,X}(t)$. Under
suitable integrability conditions, the fact that the Green’s function
$G_{\Psi,X}^{(t)}$ is causal is equivalent to saying that its susceptibility
obeys the so-called Kramers-Kronig relations Ruelle (2009); Lucarini and Sarno
(2011); these provide integral constraints that link the real and imaginary
parts, so that
$\chi^{(1)}_{\Psi,X}(\omega)=i\mathcal{P}(1/\omega)\star\chi_{\Psi,X}^{(1)}(\omega)$,
where $i=\sqrt{-1}$, $\star$ indicates the convolution product, and
$\mathcal{P}$ stands for integration by parts. Extensions to the case of
higher-order susceptibilities are also available Lucarini  _et al._ (2005b);
Lucarini (2008a, 2009a); Lucarini and Colangeli (2012).

Instead of studying merely individual trajectories, one can study the
evolution of ensembles of trajectories to obtain probabilistic estimates. The
evolution of the measure $\rho$ driven by the dynamical system $\dot{x}=F(x)$
is described by the transfer or Perron-Frobenius operator
$\mathcal{L}_{0}^{t}$ Baladi (2000); Chekroun  _et al._ (2014); Villani
(2009), which is the adjoint of the Koopman or composition operator
$S_{0}^{t}$; it is defined as follows:

$\int\mathcal{L}_{0}^{t}\rho(x)\Psi(x)\mathrm{d}x=\int\rho(x)S_{0}^{t}\Psi(x)\mathrm{d}x.$
(58)

Assuming that no degenaries are present, the generator of the semigroup
$\\{\mathcal{L}_{0}^{t}\\}$ is the so-called Liouville operator $L$ and it
satisfies $\partial_{t}\rho=-\nabla(\rho F)=L\rho$.

Let $\\{(\sigma_{k},\rho_{k}):k=1,\ldots,\infty\\}$ be the eigenpairs of $L$.
Then the eigenvalues of $\mathcal{L}_{0}^{t}$ are given by
$\\{\exp(\sigma_{k}t)\\}$, with the same eigenvectors $\\{\rho_{k}\\}$.
Correspondingly, the invariant measure $\bar{\mu}$ is the eigenvector having
the null eigenvalue $\sigma_{1}=0$ for $L$ or, $\forall t\geq 0$, unit
eigenvalue of $\mathcal{L}_{0}^{t}$. Note that $\forall k\geq 2$,
$\Re(\sigma_{k})<0$, which implies exponential decay of correlations for all
the system’s smooth observables, with an asymptotic rate that is given by the
largest value of $\Re(\sigma_{k})$, $k\geq 2$. Moreover, the presence of a
small spectral gap between the unit eigenvalue of $\mathcal{L}_{0}^{t}$ and
its other eigenvalues within the unit disk leads to a small radius of
expansion for Ruelle’s perturbative approach Chekroun  _et al._ (2014);
Lucarini (2016).

It turns out that, if one neglects the essential part of the spectrum, the
susceptibility can be written as

$\chi^{(1)}_{\Psi,X}(\omega)=\sum_{k=1}^{\infty}\frac{\alpha_{k}\\{\Psi,X\\}}{\omega-\sigma_{k}},$
(59)

where the factor $\alpha_{k}$ evaluates how the response of the system to the
forcing $X$ for the observable $\Psi$ projects on the eigenvector
$\\{\rho_{k}\\}$. The constants $\pi_{k}=i\sigma_{k}$ are usually referred to
as Ruelle-Pollicott poles Ruelle (1986); Pollicott (1985). Note that Eq. (59)
implies that the Green’s function corresponding to the susceptibility can be
written as a weighted sum of exponential functions, if one neglects possible
degeneracies.999Note that Eq. (59) mirrors the quantum-mechanical expressions
for the electric susceptibility of atoms or molecules. In the latter case, the
summation involves all the pairs of eigenstates of the system’s unperturbed
Hamiltonian operator; in each term, the poles’ imaginary part corresponds to
the energy difference between the pair of considered eigenstates; and the real
part is the so-called line width of the transition, whose inverse is the life
time; finally, the numerator is its so-called dipole strength Cohen-Tannoudji
_et al._ (2007).

####  IV.5.4 Climate Change Prediction via Ruelle Response Theory

A somewhat different approach to constructing the climate response to forcings
focuses on computing it directly from Eq. (55), without relying on the
applicability of the FDT, which fails in certain cases of geophysical
relevance (e.g., Gritsun and Lucarini, 2017). The difficulty in applying this
direct approach lies in the fact that the formula contains contributions from
both stable and unstable directions in the tangent space Ruelle (2009).

Evaluating the contribution of the unstable directions is especially hard;
hence, Abramov and Majda (2007) proposed a blended approach that also uses the
FDT. Using adjoint methods in the direct approach has also yielded promising
results Wang (2013). Faced with the so-called cold-start problem of climate
simulations, Hasselmann  _et al._ (1993) suggested a heuristic approach to
computing a climate model’s Green’s function and applied it to study the
relaxation to steady state of a coupled GCM’s globally averaged surface
temperatures.

Lucarini and associates Lucarini and Sarno (2011); Lucarini  _et al._ (2017);
Ragone  _et al._ (2016) proposed to evaluate the Green’s function using an
experimental but rigorous approach, suggested by standard optics laboratory
practice Lucarini  _et al._ (2005b). The idea is to use a set of carefully
selected probe experiments — typically, step-like increases of the parameter
of interest — to construct the Green’s function and then, exploiting Eq. (55),
use this operator to predict the response of the system to a temporal pattern
of interest for the forcing.

Given a set of forced climate simulations and a background unperturbed one,
such an approach allows one to construct the Green’s functions response
operators for as many observables as desired, global as well as local in
space. Such a toolkit allows one to treat a continuum of scenarios of temporal
patterns forcings, thus providing a general framework for improving climate
change projections given in the form shown in Fig. 18.

The idea is to consider the set of equations describing an unperturbed climate
evolution in the form $\dot{x}=F(x)$, with the vector field $X=X(x)$ in Eq.
(52) as the 3-D radiative forcing associated with the increase of CO2
concentration, and $\epsilon T(t)$ its time modulation. By plugging
$T(t)=\epsilon\Theta(t)$ into Eq. (54), we have, for any climatic observable
$\Psi$,

$\frac{\mathrm{d}}{\mathrm{d}t}\langle\Psi\rangle_{0}^{(1)}(t)=\epsilon
G_{\Psi,[{\rm CO}_{2}]}^{(1)}(t).$ (60)

We estimate $\langle\Psi\rangle_{0}^{(1)}(t)$ by taking the system’s average
of response over an ensemble of initial states and use the previous equation
to derive our estimate of $G_{\Psi,[CO_{2}]}^{(1)}(t)$, by assuming linearity
in the response.

Note that these Green’s functions are specifically related to changes in the
atmospheric CO2 concentration, rather than to a generic radiative forcing.
Indeed, for each climatic variable, the response to changes in the solar
irradiance — e.g., via modulation of the parameter $\mu$ in Eqs. (13) — is
different from the impact of changes in the CO2 concentration, because the
details of the radiative forcing are very different in the two cases. Thus,
geoengineering proposals that aim to reduce solar irradiance by injecting
sulphate aerosols in the stratosphere (e.g., Smith and Wagner, 2018) rely on
flawed scientific reasoning Lucarini (2013); Bódai  _et al._ (2020), as well
as being of dubious practical help Proctor  _et al._ (2018) and hardly
defensible from an ethical perspective Lawrence  _et al._ (2018).

Ragone  _et al._ (2016) derived, moreover, a closed formula relating the TCR
and the ECS. Indeed, with the definition of $TCR(\tau)$ given at the beginning
of Sect. IV.3, one gets

$\displaystyle ECS-TCR(\tau)=INR(\tau)$
$\displaystyle=\Delta\tilde{R}_{2\times{\rm
CO}_{2}}\times\mathcal{P}{\displaystyle\int_{-\infty}^{\infty}}\chi^{(1)}_{T_{S},[{\rm
CO}_{2}]}(\omega^{\prime})$
$\displaystyle{\displaystyle\frac{1+{\displaystyle\frac{\sin(\omega^{\prime}\tau/2)}{\omega^{\prime}\tau/2}}\exp[-i\omega^{\prime}\tau/2]}{\displaystyle{2\pi
i\omega}}}\mathrm{d}\omega^{\prime}.$ (61)

The difference between ECS and TCR is a weighted integral of the
susceptibility, accounting for the contribution of processes and feedbacks
occurring at different time scales. The integral in Eq. (61) yields
$\chi^{(1)}_{T_{S},[{\rm CO}_{2}]}(0)$ if $\tau\rightarrow 0$ (i.e.,
$TCR(0)=0$), decreases with $\tau$, and vanishes in the limit
$\tau\rightarrow\infty$. $INR(\tau)$ is a measure of the system’s inertia at
the time scale $\tau$, due to the overall contribution of the internal
physical processes and characteristic time scales of the relevant climatic
subsystems.

Figure 42 illustrates a climate prediction obtained by applying the response
theory above to the open-source model PLASIM (see http://tiny.cc/zgk0bz), an
atmospheric GCM of intermediate complexity coupled with a mixed-layer ocean
model Fraedrich  _et al._ (2005); Fraedrich (2012). The figure shows a good
agreement between (a) the ensemble average of 200 simulations where the CO2
concentration is increased at the rate of 1% until doubling (black curve), and
then kept constant; and (b) the prediction done by convolving the Green’s
function of Eq. (60) with the time pattern of the forcing (blue curve). Note
that the temperature increase predicted at year 70, when doubling of the CO2
is reached, gives as estimate the $TCR\simeq 4.1$ K (blue curve). The true
value of the $TCR\simeq 4.3$ K is given by the black line. The TCR is indeed
smaller than the $ECS\simeq 4.8$ K, which corresponds to a good approximation
to the temperature increase predicted at year 200.

Figure 42: Comparison between the climate model simulation (black) and
response theory prediction (blue) for a TCR experiment using PLASIM, a GCM of
intermediate complexity. The CO2 concentration was ramped up by 1% per year to
double its initial value. The upper and lower limit of the light-shaded bands
are computed as two standard deviations of the ensemble distribution.
Reproduced with permission from Lucarini  _et al._ (2017).

The power of response theory lies in the fact that — once the Green’s function
of interest has been computed — one can predict the future evolution of
bespoke observables, defined as needed. Furthermore, Lucarini (2018) has shown
that it is possible to use certain classes of observables as surrogate
forcings of other observables, in the spirit of the linear feedback analysis
of Eq. (44). Given the notoriously more difficult prediction of precipitation,
Lucarini  _et al._ (2017) have found that predictions of changes in the
globally averaged precipitation can be as good as those of the temperature.

Proceeding from global predictions to more localized ones, Fig. 43 shows the
outcome of predicting the change in the zonal averages of the surface
temperature. It is clear that response theory does a good job in reproducing
the spatial patterns of temperature change, except for an underestimate of the
temperature change in the high latitudes on the scale of few decades. Indeed,
such mishap is due the strong polar amplification of the warming due to the
ice-albedo feedback, which can only qualitatively captured by a linear
approach like the one used by Lucarini  _et al._ (2017).

Figure 43: Patterns of climate response for the zonal average of the surface
temperature. (a) Prediction via response theory (red and black lines not
relevant in this context). b) Difference between the ensemble average of the
direct numerical simulations and the predictions obtained using the response
theory. Reproduced with permission from Lucarini  _et al._ (2017).

####  IV.5.5 Slow Correlation Decay and Sensitive Parameter Dependence

Equation (59) implies that resonant amplification occurs if the forcing acts a
a frequency $\omega$ that is close to the imaginary part of a subdominant
Ruelle-Pollicott pole $\pi_{k},\;k\geq 2$ that has a small real part, i.e.,
for some $k\geq 2$,

$|\omega-\Im(\pi_{k})|\ll\omega,\quad|\Re(\pi_{k})|\ll 1,$ (62)

because the system’s susceptibility is greatly enhanced at such an $\omega$.
Conditions (62) are easily satisfied for a broadband forcing and a system that
has a small spectral gap. Conversely, as discussed after Eq. (58), the
presence of a small spectral gap is associated with a slow decay of
correlation for smooth observables. We now provide two examples that show how
linear response breaks down for forced systems possessing slow decay of
correlations due to a small spectral gap in the unperturbed dynamics.

The first example is due to Chekroun  _et al._ (2014), who investigated the
response of the highly simplified ENSO model of Jin and Neelin (1993), to
which Jin  _et al._ (1994) added periodic forcing that led to chaotic
behavior. Chekroun and coauthors focussed on how the response changes when one
varies the model parameter $\delta$ that controls the travel time of the
equatorially trapped waves, which play an essential role in the ENSO
mechanism. Chekroun  _et al._ (2014) found that when the spectral gap is
small, the system exhibits rough dependence of its properties with respect to
small modulations of the parameters, since Ruelle’s perturbative expansion
breaks down even for a very small intensity of the forcing.

Figure 44: Rough dependence (red dots) of the skewness of the equatorial ocean
temperature’s distribution with respect to changes in the parameter $\delta$
that controls the travel time of equatorially trapped waves in a simplified
ENSO model; see text for details. Black points refer to nonchaotic windows
where the parameter dependence is smooth; blue points indicate particularly
sharp jumps in the parameter dependence. Reproduced with permission from
Chekroun  _et al._ (2014).

The second example shows the importance of correlation slowdown near saddle-
node bifurcation points in the PLASIM model discussed already in the previous
subsection. This intermediate-complexity model is multistable: its bifurcation
diagram with respect to the solar insolation parameter $\mu$ is very similar
to Fig. 21 of Sect. III.1 see Lucarini  _et al._ (2010a, 2013). PLASIM’s
bifurcation diagram is reproduced in Fig. 45a here, in which the present
climate $S=S_{0}=1360$ Wm-2 is denoted by a bullet $\bullet$ and the
bifurcation points associated with the transition from the warm to the
snowball state ($W-SB$) and from the snowball to the warm state ($SB-W$) are
indicated by solid arrowheads $\blacktriangle$. For comparison with Fig. 21,
note that $\mu=S/S_{0}$ there.

Tantet  _et al._ (2018) investigated PLASIM’s response to changes in the solar
irradiance for $S<S_{0}$. They found that as the $W-SB$ transition nears, the
lag correlation for a large-scale observable — namely the average equatorial
near surface air temperature — decays more and more slowly, as a result of a
narrowing spectral gap. Indeed, near the saddle-node bifurcations, the
response of the system to perturbations is greatly amplified, and, by
definition, becomes singular exactly at the bifurcation point.101010Scheffer
_et al._ (2009) have proposed a set of “warning signals” when a dynamical
system approaches a critical transition. Correlation slowdown is one of the
main ones and it is verified in the present case for the back-to-back saddle-
node bifurcations that make up the backbone of the PLASIM model’s hysteresis
cycle. Colon  _et al._ (2015), though, have found that — in a predator–prey
system, modeled by an ODE system as well as by an agent-based mode — more
complex critical transitions can behave rather differently than in the by-and-
large saddle-node–bound early warning literature.

This second example leads us naturally to the investigation of the climate
system’s global stability properties and of its critical transitions, which
are the the subject of the next section.

a)
b)

Figure 45: Slowdown of correlations near a saddle-node bifurcation point in
the PLASIM model. (a) The model’s bifurcation diagram as a function of the
value of the solar irradiance $S$. The critical transitions Warm-to-Snowball
and Snowball-to-Warm as well as the present-day conditions are indicated.
Reproduced with permission from Lucarini  _et al._ (2010a). (b) Decay of
correlation of the equatorial mean surface temperature for various values of
the solar irradiance $S$. The decay becomes considerably slower as $S$
approaches the critical value of $S\simeq 1265$ Wm-2. Reproduced with
permission from Tantet  _et al._ (2018).

##  V Critical Transitions and Edge States

In Sect. III, we introduced several types of bifurcations that involve
transitions between two or more regimes: saddle-node bifurcations whose
pairing can lead to coexistence of two stable steady states (fixed points);
pitchfork bifurcations that, in the presence of a mirror symmetry, can lead
from one stable symmetric steady state to the coexistence of two steady states
that are mirror images of each other; Hopf bifurcations that lead from a
stable steady state to a stable periodic solution (limit cycle) and then again
from the stable limit cycle to a stable torus on which an infinity of quasi-
periodic solutions live; and nonlocal bifurcations associated with the
existence of homoclinic and heteroclinic orbits that lead on to chaotic
regimes.

Successive bifurcation scenarios (Eckmann, 1981; Ghil and Childress, 1987;
Dijkstra and Ghil, 2005; Dijkstra, 2013) that involve several of these
bifurcations and additional ones lead from solutions with high symmetry in
space and time to successively more complex and chaotic ones. We saw in Sect.
III that such bifurcation scenarios shed considerable light on the
phenomenology of large-scale atmospheric, oceanic and coupled ocean–atmosphere
flows. Finally, in Sects. III.7 and IV.5, we introduced stochastic effects
into the nonlinearly deterministic setting recalled above, and outlined the
role of pullback attractors (PBAs) and of the invariant measures they support
in the theory of nonautonomous and random dynamical systems (NDSs and RDSs).

Recently, the interest in bifurcations in the climate and environmental
sciences has greatly increased due to the introduction of the concept of
tipping points from the social sciences (Gladwell, 2000; Lenton  _et al._ ,
2008). Clearly, a tipping point sounds a lot more threatening than a
bifurcation point, especially when dealing with a hurricane or a dramatic and
irreversible climate change.

Beyond the linguistic effectiveness of the term, it does also generalize the
bifurcation concept in the context of open systems that are modeled
mathematically by NDSs or RDSs. As we saw in Sects. III and II, the climate
system — as well as its subsystems, namely the atmosphere, oceans, biosphere
and cryosphere — are open and exchange time-dependent fluxes of mass, energy
and momentum with each other and with outer space. Hence it is quite
appropriate to consider this generalization. Following Ashwin  _et al._
(2012), one distinguishes among three kinds of tipping points, as shown in
Table 2.

The investigation of systems possessing multiple attractors — which may
include fixed points, limit cycles, strange attractors as well as PBAs or
random attractors — is an active area of research, encompassing mathematics,
as well as the natural and socio-economic sciences. Feudel  _et al._ (2018),
for instance, provide a brief review that introduces a special issue of the
journal Chaos on the topic and sketch several interesting examples.

This section is devoted to providing a general framework for the study of
multistability in the Earth system, to forced transitions between different
regimes when multistability is present, and to the critical transitions taking
place at classical bifurcation and more general tipping points. Recall, for
motivation, two cases of multistablility in the climate system.

First, planet Earth as a whole has boundary conditions that arise from its
space environment in the solar system, which are compatible with at least two
competing, stable regimes: today’s relatively ice-free and warm climate and a
deep freeze or snowball climate; see Sect. III.1. Second, the oceans’ THC is —
over a rather broad range of the parameters that control its heat and
freshwater budget — bistable, with the current active state coexisting with a
greatly reduced or even reversed circulation; see Sect. III.3.

In order to illustrate the main mathematical and physical aspects of the
problem, we choose at first the specific example of coexistence between the
warmer, Holocene-like, and the much colder, snowball state of planet Earth, as
discussed in Sects. III.1 and IV.5.5; see, in particular, Figs. 21 and 45a.

Type |  Cause |  Mechanism
---|---|---
B-Tipping |  Bifurcation-due tipping |  Slow change in a parameter leads to the system’s passage through a classical bifurcation
N-Tipping |  Noise-induced tipping |  Random fluctuations lead to the system’s crossing an attractor basin boundary
R-Tipping |  Rate-induced tipping |  Due to rapid changes in the forcing, the system loses track of a slow change in its attractors
Table 2: Tipping points in open systems; see also Ashwin  _et al._ (2012) and
Fig. 46.

###  V.1 Bistability for Gradient Flows and EBMs

The theory of SDEs and of RDSs provides a comprehensive framework for deriving
the probability of occurrence of coexisting regimes in multistable systems and
for estimating the probability of stochastic-forcing–triggered transitions
between them. A good starting point is gradient flow in the presence of
additive white noise.

The governing SDE in this case is:

$\mathrm{d}x=-\nabla_{x}V(x)\mathrm{d}t+\epsilon\mathrm{d}W,$ (63)

where $x\in\mathbb{R}^{d}$, the potential
$V:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is sufficiently smooth, and
$\mathrm{d}W$ is a vector of $d$ independent increments of Brownian motion,
while $\epsilon$ determines the strength of the noise. The Fokker-Planck
equation associated with Eq. (63) describes the evolution of the PDF
$p_{\epsilon}(x,t)$ of an ensemble of trajectories obeying the SDE.

The stationary solution corresponding to the latter equation’s invariant
measure $\lim_{t\rightarrow\infty}p_{\epsilon}(x,t)=p_{\epsilon}(x)$ is given
by a large-deviation law (Varadhan, 1966):

$p_{\epsilon}(x)=\frac{1}{Z}\exp\left(-\frac{2V(x)}{\epsilon^{2}}\right)$ (64)

where $Z$ is a normalization constant. The local minima of the potential $V$
are the stable fixed points of the deterministic dynamics. One obtains them by
setting $\epsilon=0$ in the SDE (63), and they correspond to the local maxima
of $p$. Thus, for instance, the double-well potential of Fig. 20 corresponds
to a bimodal PDF.

Figure 46: Schematic diagram of B-Tipping: as a control parameter is being
changed slowly, the number of stable equilibria is reduced from two to one.
Reproduced with permission from Lenton  _et al._ (2008).

The mountain pass lemma (e.g., Bisgard, 2015) states that the two minima of
$V$ that give rise to the two stable fixed points have to be separated by an
unstable one of the saddle type, which is a maximum of $V$ in one direction
and a minimum in all the other ones. Such a saddle looks like a mountain pass
on a topographic map, hence the name of the lemma. Ghil and Childress (1987,
Sec. 10.4) discussed its application to 1-D EBMs.

In the limit of weak noise, trajectories starting near a local minimum of $V$
located at $x=x_{1}$ typically wait a long time before moving to the
neighborhood of a different local minimum of $V$. The transitions are most
likely to occur through the lowest energy saddle $x=x_{\rm s}$ that links the
initial basin of attraction to any other basin. Let us call the corresponding
local minimum $x=x_{2}$.

The optimal path linking $x_{1}$ to $x_{\rm s}$ minimizes the Freidlin and
Wentzell (1984) action and is called an instanton. In the weak-noise limit
$\epsilon\rightarrow 0$, the persistence time inside the basin of attraction
of $x=x_{1}$ is

$\bar{\tau}_{\epsilon}\propto\exp\left(\frac{2(V(x_{\rm
s})-V(x_{1}))}{\epsilon^{2}}\right),$ (65)

which is referred to as the Kramers (1940) formula.

Assume now, for simplicity, that we deal with a bistable system, as in Fig. 20
of Sect. III.1. If the potential $V$ depends on a slowly varying control
parameter $\phi$, a B-tipping would involve, for instance, the number of local
minima decreasing from $2$ to $1$, as a local minimum merges with a saddle
point at $\phi=\phi_{\rm c}$. This merging is illustrated in Fig. 46 for a
simple 1-D case, like that of the 0-D EBM governed by Eqs. (13)111111Note that
the number of dimensions in phase space is 1, while that of the model in
physical space is 0..

When the system nears the tipping point, the persistence time in the shallow,
metastable minimum is reduced, because $\Delta V\equiv V_{\rm
s}-V_{1}\rightarrow 0$. Another flag anticipating the tipping point is the
increase of the autocorrelation time $\tau$ in the metastable state, which is
proportional to the inverse of the second derivative of the potential
evaluated at the minimum, $\bar{\tau}_{\epsilon}\propto
1/V^{\prime\prime}_{x_{1}}$.

These two easily observable phenomena are the simplest manifestation of the
slowing-down process Scheffer  _et al._ (2009), which we have discussed
already in Sect. IV.5.5 when describing the findings of Tantet  _et al._
(2018). As indicated in the footnote there, the warning signals of tipping-
point approach have been mostly studied in the B-tipping case for saddle-node
bifurcations. Much remains to be done in this context for more complex
situations that may involve N-tipping, R-tipping or global bifurcations rather
than local ones. For instance, Ditlevsen and Johnsen (2010) analyzed a high-
resolution ice core record and excluded the possibility of interpreting
Dansgaard-Oeschger events as B-tipping by noting the lack of the two early
warning signals mentioned above. Their results leave us with the alternative
of interpreting the associated irregular oscillations (e.g., Boers  _et al._ ,
2018) either as the result of N-tipping for an S-shaped back-to-back saddle-
node bifurcation or as B-tipping for a Hopf bifurcation; see Fig. 29 and its
discussion in Sect. III.5.3 for similar ambiguities in interpreting changes in
atmospheric LFV.

a)
b)

Figure 47: Bistability for the Ghil-Sellers 1-D EBM. (a) Simplified
bifurcation diagram for the Ghil (1976) 1-D EBM; compare Figs. 21 and 45. The
red, blue, and green segments correspond to the globally averaged surface air
temperature for the warm, snowball, and Melancholia state, respectively, as a
function of the relative solar irradiance $\mu$; see also Arcoya  _et al._
(1998). (b) Meridional temperature profile for the warm (red), snowball
(blue), and $M$-state (green) for this model under present-day conditions
($\mu=1$), with the sine of latitude on the $x$-axis; compare Ghil (1976, Fig.
3a). The circles indicate the $M$-state estimate obtained using the edge
tracking method. Adapted with permission from Bódai  _et al._ (2015).

Returning to the right-hand side of the 0-D EBM of Eq. (13a), it can be
written as minus the derivative of a potential $V(T)$, cf. Ghil (1976). The
gain factor $\Lambda$ given in Eq. (47), as well as the ECS, is proportional
to the inverse of the second derivative of the potential $V$ evaluated at the
local minimum defining the reference climate $T_{0}$,
$ECS\propto\Lambda\propto 1/V^{\prime\prime}_{T_{0}}$. Hence, near the
critical transition, a small forcing in the right direction can lead the
system to jump to the other basin of attraction, even in the absence of noise.
Note, though, that, near such a transition, the linear stability analysis
performed in Eqs. (46, 47) is no longer valid, and an accurate quantitative
analysis of climate response requires taking into account the essential
nonlinearity in the problem; see the Zaliapin and Ghil (2010) comments on Roe
and Baker (2007).

Determining the saddle point that potentially connects, via instantons, two
local minima is not at all trivial when looking at high- or even infinite-
dimensional systems. Figure 47 reports the findings of Bódai  _et al._ (2015)
on the Ghil (1976) 1-D EBM. Panel (a) shows the bifurcation diagram of the
model, in which the control variable is the globally averaged surface air
temperature $\bar{T}$, and the tuning parameter is the insolation parameter
$\mu=S/S_{0}$ introduced in Eq. (13b).

Panel (b) shows the zonally averaged temperatures of the 1-D model’s steady
states, for the reference conditions $T=T(x)$ at $\mu=1$. The red and blue
lines represent the two stable solutions — warm ($W$) and snowball ($SB$),
respectively — while the green line is the unstable solution lying in-between.
Note that, at each tipping point, the unstable state comes in contact with a
stable one, and then both disappear, in accordance with the scenario of basin
boundary crisis Ott (2002). In fact, the lack of stability of the steady-state
solution described by the green line is apparent by observing that $\bar{T}$
in Fig. 47a decreases with increasing solar radiation — as opposed to the $W$
and $SB$ states — which is clearly not physical (Ghil, 1976; North  _et al._ ,
1981) and can be loosely interpreted as the result of a negative heat
capacity; see discussion in Bódai  _et al._ (2015).

###  V.2 Finding the Edge States

In one phase space dimension, every first-order system $\dot{x}=f(x;\mu)$ is a
gradient system, since one can always write $V(x;\mu)=-\int
f(\xi,\mu)\mathrm{d}\xi$. This is not so in two or more dimensions and one
might wonder whether it is possible to compute unstable solutions for more
general multistable systems, with no gradient property. Scott  _et al._ (1999)
presented an example of such an unstable stationary solution in a simple box
model of the THC featuring competing stable solutions and no gradient
structure. Moreover, the actual computation of the unstable steady state in
Ghil (1976, Fig. 3a) did not rely on the system’s potential $V(T(x))$ but on
its 1-D character in physical space and on a shooting method to solve the
Sturm-Liouville equation that results when setting $\partial T(x,t)/\partial
t\equiv 0$ in the Ghil-Sellers model. Neither of these approaches Ghil (1976);
Scott  _et al._ (1999) can easily be extended to more general systems.

In the remainder of this section, we discuss a more general paradigm of
multistability for determininistic systems and, subsequently, how this
paradigm may be useful for studying the problem of noise-induced transitions
between competing states, without the simplifying assumption of a gradient, as
in Eq. (63).

Figure 48: Schematic diagram of the edge tracking algorithm proposed by Skufca
_et al._ (2006) and Schneider  _et al._ (2007), as applied to the 1-D EBM of
Fig. 47. Reproduced with permission from Bódai  _et al._ (2015).

One can formally introduce general multistable systems as follows. We consider
a smooth autonomous continuous-time dynamical system acting on a smooth
finite-dimensional compact manifold $\mathcal{M}$, described by a ODE system
of the form ${\dot{{x}}}={F}({x})$. The system is multistable if it possesses
more than one asymptotically stable state, defined by the attractors
$\\{\Omega_{j}:j=1,\ldots,J\\}$. The asymptotic state of an orbit is
determined by its initial state, and the phase space is partitioned between
the basins of attraction $\\{B_{j}\\}$ of the attractors $\\{\Omega_{j}\\}$
and the boundaries $\\{\partial B_{\ell}$: $\ell=1,\ldots,L\\}$ separating
these basins.

The basin boundaries can be strange geometrical objects with codimension
smaller than one. Orbits initialized on the basin boundaries $\\{\partial
B_{\ell}\\}$ are attracted towards invariant saddles $\\{\Pi_{\ell}{{}_{k}}$:
$\ell=1,\ldots,L$, $k=1,\ldots,k_{\ell}\\}$, where we allow for the existence
of $k_{\ell}$ separate saddles in the basin boundary $\partial B_{\ell}$;
these saddles are often referred to as edge states (cf. Skufca  _et al._ ,
2006; Schneider  _et al._ , 2007).121212Note that, strictly speaking, edge
states refer to solutions separating long-lived turbulence and laminar flow,
rather than describing the interface between truly separated attractor basins.
For nongradient flows, the edge states, like the asymptotic states, can
feature chaotic dynamics Grebogi  _et al._ (1983); Robert  _et al._ (2000);
Ott (2002); Vollmer  _et al._ (2009). Lucarini and Bódai (2017, 2019a, 2019b)
refer to the chaotic edge states as Melancholia ($M$) states. Hence, we cannot
expect to find easily edge states and, a fortiori, $M$-states for such general
multistable systems.

The edge tracking algorithm of B. Eckhart, J. A. Yorke and associates Skufca
_et al._ (2006); Schneider  _et al._ (2007) allows one to do so by
constructing a shadowing trajectory that leads an orbit starting on the basin
boundary toward the corresponding edge state, cf. Fig. 48; therein one uses a
bisection method to control the instability associated with the trajectories’
diverging away from the basin boundary. Bódai  _et al._ (2015) first used edge
tracking in a geophysical context to reproduce the unstable solution of Ghil
(1976) with this more easily generalizable approach; compare green line and
circles in Fig. 47b.

Lucarini and Bódai (2017) computed $M$-states for an intermediate complexity
climate model with O($10^{5}$) degrees of freedom that couples PUMA, an
atmospheric primitive equations model Frisius  _et al._ (1998) with a modified
version of the 1-D EBM of Ghil (1976) that acts as a surrogate ocean and
contributes to meridional heat transport. Figure 49 shows the bifurcation
diagram of this climate model, where the $W\rightarrow SB$ tipping point is
located near $\mu=0.98$, while the $SB\rightarrow W$ tipping point is located
near $\mu=1.06$. Just as in the 1-D EBM of Fig. 47, the tipping points are
associated with basin boundary crises Ott (2002).

Over a wide range of $\mu$-values, the edge state features chaotic dynamics
that arises from the atmospheric model’s baroclinic instability; it leads to
weather variability and to a limited predictability horizon. Since this
instability is much faster than the climatic one due to the ice-albedo
feedback, the basin boundary is a fractal set with near-zero codimension, in
agreement with results obtained in low-dimensional cases Grebogi  _et al._
(1983); Lai and Tél (2011). In other words, the basin boundary has almost full
Lebesgue measure. As a result, near the basin boundary there is virtually no
predictability on the asymptotic state of the system, because infinitesimally
close initial states have a high probability of belonging to separate basins
of attraction.

Figure 49: Bifurcation diagram for the intermediate-complexity climate model
in Lucarini and Bódai (2017), drawn for the long-term, globally averaged ocean
temperatures $[\langle T_{S}\rangle]$. Bistability is found for a large range
of values of the control parameter $\mu=S/S_{0}$. The relevant solid lines are
red for the Warm $W$-states, blue for the Snowball $SB$-states, and green for
the Melancholia $M$-states, as in Fig. 47. Reproduced with permission from
Lucarini and Bódai (2019a).

###  V.3 Invariant Measures and Noise-induced Transitions

As mentioned above, transitions across the basin boundaries of competing
attractors are possible and quite likely in the presence of stochastic
perturbations. Noise-induced escape from an attractor has, in fact, long been
studied in the natural sciences (e.g., Hanggi, 1986; Kautz, 1987; Grassberger,
1989).

Let us generalize Eq. (63) as follows:

$\mathrm{d}x={F}({x})\mathrm{d}t+\epsilon{s}({x})\mathrm{d}W;$ (66)

here $x\in\mathbb{R}^{d}$, ${F}({x})\mathrm{d}t$ is the drift term given by a
vector flow field that admits multiple steady states, as discussed in the
previous two subsections, $\mathrm{d}W$ are the increments of a
$d$-dimensional Brownian motion. The volatility matrix
${s}({x})\in\mathbb{R}^{d\times d}$ is such that ${s}({x})^{T}{s}({x})$ is the
covariance matrix of the noise. Finally, the parameter $\epsilon\geq 0$
controls the noise intensity.

Recent extensions of the classical Freidlin-Wentzell Freidlin and Wentzell
(1984) theory Graham  _et al._ (1991); Hamm  _et al._ (1994); Lai and Tél
(2011) yield results that mirror closely those summarized before in the case
of gradient flows with additive noise, cf. Eqs. (64) and (65), given suitable
assumptions on the drift term and the volatility (e.g., Lucarini and Bódai,
2019a, b). In the weak-noise limit given by $\epsilon\rightarrow 0$, the
invariant measure of the system can be written as a large-deviation law.
Formally, one just has to replace $V(x)$ by a general pseudo-potential
$\Phi({x})$, which depends in a nontrivial way on ${F}(x)$ and ${s}({x})$.
Moreover, the constant $Z$ in Eq. (64) is replaced by a function $\zeta(x)$,
which is relatively unimportant because the behavior of the system depends
mostly on the properties of $\Phi(x)$.

Certain general properties of $\Phi({x})$ apply to all choices of the noise
law and, once a noise law is chosen, one can derive how the properties of the
system change as a function of the parameter $\epsilon$. In general,
regardless of the noise law, $\Phi({x})$ has local minima supported on the
attractors $\\{\Omega_{j}:j=1,\ldots,J\\}$ of the deterministic dynamics.
Correspondingly, the invariant measure has local maxima on the attractors.
Moreover, $\Phi({x})$ has a constant value on the support of each edge state
$\Pi_{\ell}$ and each attractor $\Omega_{j}$, respectively.

In the simplest case of just $J=2$ attractors and $L=1$ edge states then, in
the weak-noise limit, transitions away from either attractor basin take place
exponentially more likely along the instanton connecting the corresponding
attractor with the edge state $\Pi_{1}$. Just as for the gradient flows in
Sect. V.1, instantons can be calculated as minimizers of a suitably defined
action Kautz (1987); Grassberger (1989); Kraut and Feudel (2002); Beri  _et
al._ (2005).

Consider now the case that more than one edge state separate a given
$\Omega_{j}$ from the other attractors $\\{\Omega_{k}:k\neq j\\}$. Let us
denote then by $\Pi_{j}$ the edge state for which the value of $\Phi$ is
lowest. In this case, the most probable exit path connects $\Omega_{j}$ with
$\Pi_{j}$, while the other escape channels are basically switched off in the
weak-noise limit.

Note that, if the attractors and the edge states are more complex sets than
isolated points, the instantons connecting them are not unique, because in
principle any point of the attractor can be linked by an instanton to any
point of the edge state, because the pseudo-potential is constant on
$\Omega_{j}$ and $\Pi_{j}$. Indeed, the edge states are the gateways for the
noise-induced escapes from the deterministic basins of attraction, as
illustrated in the bistable example with nongradient flow given below.

Figure 50: Invariant measure of the transitions between the two stable regimes
of the stochastically perturbed climate model of Lucarini and Bódai (2017) via
the $M$-state separating them, for $\mu=1$. Main graph: projection of this
invariant measure on the reduced space $(T_{S},[\Delta T_{S}])$; $W$ attractor
– red dot; $SB$ attractor – blue dot; and $M$-state – green dot. Red and blue
dashed lines plot the $W\rightarrow SB$ and the $SB\rightarrow W$ instanton,
respectively. Top left inset: marginal PDF with respect to $\Delta T_{S}$;
bottom right inset: marginal PDF with respect to $[T_{S}]$; and center right
inset: probability along the two instantons. Reproduced with permission from
Lucarini and Bódai (2019a).

Lucarini and Bódai (2019a) introduced stochastic forcing into the climate
model studied by Lucarini and Bódai (2017) as a fluctuating factor of the form
$1+\epsilon\mathrm{d}W/\mathrm{d}t$, where $\mathrm{d}W$ is a Brownian motion
that modulates the solar insolation parameter $\mu$. This forcing yields a
multiplicative noise law because the energy input into the system depends on
the product of $\mu$ times the co-albedo $(1-\alpha)$, where the albedo
$\alpha=\alpha(T)$ depends explicitly on the surface air temperature
$T=T(x,t))$, which is a state variable of the model; compare with Eq. (13b).

In Fig. 50 we reproduce the estimate by Lucarini and Bódai (2019a) of a 2-D
projection of the invariant measure for $\mu=1$ from a climate simulation of
$\simeq 6.0\times 10^{4}$ yr with fluctuations in the solar insolation of
$1.5\%$ on the scale of $100$ yr. The simulation features 92 $SB\rightarrow W$
and $W\rightarrow SB$ transitions, and the measure is projected onto the
$([T_{S}],\Delta T_{S})$-plane; here $[T_{S}]$ is the globally averaged
surface air temperature, and $\Delta T_{S}$ measures the temperature
difference between the surface temperature in the low- and high-latitude
regions.

The peaks of the PDFs are very close to the $W$ and $SB$ attractors, and the
agreement improves even further when considering the two marginal PDFs (top
left and bottom right insets). Both the $W\rightarrow SB$ and the
$SB\rightarrow W$ instantons can be estimated: their starting and final points
agree remarkably well with the attractors and the $M$-state. The instantons
follow a path of monotonic descent that tracks closely the crests of the PDF,
with the minimum occurring at the $M$-state.

These features are all in excellent agreement with the theoretical predictions
for multistable systems with a generalized pseudo-potential.

Figure 51: Escape from the $W$ attractor near the $W\rightarrow SB$ critical
transition of the stochastically perturbed climate model of Lucarini and Bódai
(2017), for $\mu=0.98$. (a) The expected value of the transition time obeys
Eq. (65) in the weak-noise regime; the diagram is in log–linear coordinates
and $\sigma$ on the $x$-axis is the relative fluctuation of the solar
insolation $\mu$. (b) Estimate of the $W\rightarrow SB$ instanton and, in the
inset, empirical density; both used 50 trajectories that escape to the $SB$
state. Reproduced with permission from Lucarini and Bódai (2019a)

###  V.4 Nearing Critical Transitions

The general framework outlined in the previous subsection is also quite useful
for studying the properties of such systems near a critical transition (e.g.,
Lucarini and Bódai, 2019a). In Fig. 49, it is clear that $\mu=0.98$ is close
to the $W\rightarrow SB$ tipping point. One finds that at low noise
intensities — i.e., at a relative $\mu$-fluctuation smaller or equal to 1.4%
on a centennial scale — it is extremely hard to escape from the $SB$ state.
This finding happens to agree with snowball state simulations that used more
detailed models as well (e.g., Crowley  _et al._ , 2001; Pierrehumbert, 2004;
Ghil, 2019)

Let us then focus on the escape from the $W$-state. Lucarini and Bódai (2019a)
estimated the expected value of the transition time from the $W$ to the $SB$
state using 50 simulations per noise strength value. These values grow
exponentially with the inverse of the square of the parameter $\epsilon$, as
predicted by the theory; see Fig. 51a. Note that the difference between the
pseudo-potential value $\Phi$ at the $M$-state and at the $W$-state is equal
to half the slope of the straight line, in agreement with Eq. (65).

Finally, the escape transition paths from the $W$-state to the $M$-state are
plotted in Fig. 51b. In the weak-noise case of relative $\mu$ fluctuation
smaller than 1% on a centennial scale, the highest densities of these paths
lie quite close to the instanton connecting the $W$ attractor to the
$M$-state, and follow a path of decreasing probability.

These encouraging results suggest that the underlying methodology of edge
tracking and instanton estimation could be applied to observational and
reanalysis data sets. As discussed in Sect. III.5.3 the episodic, or
‘particle’, approach to atmospheric LFV results in a Markov chain of regimes
$\\{R_{j}:j=1,\ldots,J\\}$ that involves preferential transition paths
$\\{\Theta_{j,k}:j=1,\ldots,J\;k=1,\ldots,J\\}$ between them. Heretofore,
these transition paths, as well as the regimes themselves, were estimated by
purely statistical methods; see Ghil  _et al._ (2018) and Fig. 29a herein.
Deloncle  _et al._ (2007) and Kondrashov  _et al._ (2007) applied a random-
forest algorithm Breiman (2001) to find the best real-time predictors of the
next regime in a Markov chain, conditional upon the one currently occupied.
The results were quite satisfying for the QG3 model of Marshall and Molteni
(1993) and encouraging for an observational data set of 55 winters of Northern
Hemisphere 700-hPa geopotential height anomalies (Deloncle  _et al._ , 2007;
Kondrashov  _et al._ , 2007, respectively).

###  V.5 Chaos-to-Chaos Transition

So far, we have seen that critical transitions associated with saddle-node
bifurcations and mild generalizations thereof are fairly well understood by
now. We conclude this section and the main part of the text with a somewhat
more exotic example of chaos-to-chaos transition for a delay differential
equation (DDE) system with and without stochastic perturbations.

We saw in Sect. II.2.1 and III.5.2that ENSO plays a key role in the global
climate on interannual-to-interdecadal scales. Hence, a large number of
relatively simple models thereof exist to better understand its main features.
Important mechanisms involved are air–sea interaction, equatorial wave
dynamics, and radiative forcing by the seasonal cycle Bjerknes (1969). In
particular, the role of the wave dynamics has been captured by introducing one
or two delays into the governing equations of some of the simpler models
(e.g., Ghil  _et al._ , 2008a; Tziperman  _et al._ , 1994b, and references
therein).

Chekroun  _et al._ (2018) studied the PBA of the seasonally forced Tziperman
_et al._ (1994b) model both with and without stochastic perturbations. The
model has two delays, associated with a positive and a negative feedback;
these delays are based on the basin-crossing times of the eastward-traveling
Kelvin waves and the westward-traveling Rossby waves. The control parameter
$a$ is the intensity of the positive feedback and the PBA undergoes a crisis
that consists of a chaos-to-chaos transition; the authors refer to it as a
strange PBA since model behavior is chaotic within it in the purely
deterministic case.

The changes in the invariant, time-dependent measure $\mu_{t}$ supported on
this ENSO model’s PBA are illustrated in Figs. 52a–52d, as a function of the
control parameter $a$. The PBA experiences a critical transition at a value
$a_{\ast}$, as illustrated in Fig. 52, where $h(t)$ is the thermocline depth
anomaly from seasonal depth values at the domain’s eastern boundary, with $t$
in years. Here $a=(1.12+\delta))/180$ and $0.015700<\delta_{\ast}<0.015707$.

The transition in the Kolmogorov-Smirnov metric of the invariant measure’s
dependence on the parameter $a$, i.e. in $\mu_{t}=\mu_{t}(a)$, is quite sharp,
according to Chekroun  _et al._ (2018, Fig. 3; not shown here). The singular
support of the measure is in full agreement with rigorous mathematical
results, as well as with the numerical results Chekroun  _et al._ (2011); Ghil
(2017) on the random attractor of the stochastically perturbed Lorenz (1963)
convection model.

(a) High-variance PBA; $\delta=0$

(b) $\delta=0.01500$

(c) $\delta=0.0157$

(d) Low-variance PBA; $\delta=0.015707$

Figure 52: Embedding of the invariant, time-dependent measure $\mu_{t}$
supported on the PBA associated with the highly idealized ENSO DDE model of
Tziperman  _et al._ (1994b). The embedding is shown within the
$(h(t),h(t+1))$-plane for $a=(1.12+\delta))/180$, $t\simeq 147.64$ yr and,
respectively, (a) $\delta=0.0$; and (b) $\delta=0.01500$; (c) $\delta=0.0157$;
and (d) $\delta=0.015707$. The red curves in the four panels represent the
singular support of the measure. Reproduced with permission from Chekroun  _et
al._ (2018).

The change in the PBA is clearly associated with the population lying towards
the ends of the elongated filaments apparent in Figs. 52a–52d. This population
is due to the occurrence of large warm, El Niño and cold, La Niã events. Thus,
$\mu_{t}(a)$ encripts faithfully the disappearance of such extreme events as
$a\nearrow a_{\ast}$.

Note that Mukhin  _et al._ (2015a, b) have used neural-network methodology to
predict such transitions between two types of chaotic regimes in inverse
models learned from time series simulated by ENSO models of increasing
complexity. The ENSO models were subjected to a linear change in a parameter
meant to represent gradual global warming. Strikingly, transitions between
regimes in the inverse model did occur out of sample, even when the training
interval did not contain both types of behavior.

Returning to the results of Chekroun et al. (2018), they found that perturbing
the Tziperman  _et al._ (1994b) model by small additive noise eliminates the
crisis. The explanation of this numerical observation is tied to the role
played in ENSO dynamics by the interaction between the intrinsic frequency of
the coupled ocean–atmosphere system (e.g., Neelin  _et al._ , 1998) and the
seasonal forcing Ghil  _et al._ (2008a); Jin  _et al._ (1994, 1996); Tziperman
_et al._ (1994b). This interaction induces a Devil’s staircase in model
frequency, which has plausible counterparts in observations Ghil and Robertson
(2000).

As shown by Ghil  _et al._ (2008, Appendix B), a Devil’s staircase step that
corresponds to a rational rotation number can be smoothed out by a
sufficiently intense noise. In fact, the narrower a Devil’s staircase step is,
the less robust is it to noise perturbations, while the wider ones are the
most robust. The effect of noise on the paradigmatic example of such a
staircase, the standard circle map, has been examined in greater depth by
Marangio  _et al._ (2019).

This example is just one step on the long road of using the tools of
nonautonomous and random dynamical systems for a better understanding of major
climatic phenomena and processes. Note, however, that the model, while
relatively simple, is actually infinite-dimensional because of the dependence
of a DDE solution on a full interval of initial values on the real axis.

##  VI Concluding Remarks

The goal of this review was to highlight some of the key physical and
mathematical ingredients that can help address the description, understanding
and prediction of climate variability and climate change. Complementary
aspects of observations, theory and numerics have been taken into
consideration. We have leaned heavily on dynamical systems theory and
nonequilibrium statistical mechanics, and have tried to present a coherent
picture of the time dependence of the climate system, its multiscale nature,
and its multistability.

We have emphasized the complex interplay between intrinsic climate variability
and the climate’s response to perturbations. The topic is, in fact, relevant
for three problems of great scientific relevance: (a) anthropogenic climate
change; (b) coevolution of the Earth’s climate and of the biosphere, and (c)
the quest for life on other planets, along with the habitability of our own
planet.

The presentation also aimed to show the extent to which basic mathematical and
physical tools can help solve the main challenges inherent to the climate
sciences. These challenges cannot be overcome merely by increasing the
resolution of numerical models and including in them more and more physical
and biogeochemical processes. In addition, a balanced interplay of
observations, modeling and theory is definitely needed to achieve the
necessary progress.

This review is far from exhaustive: the authors had to make hard and,
obviously, personal choices on the topics to be covered. We would like to
briefly mention here several additional approaches to the problems at hand
that have been developed in recent years.

  * •

Network Theory (e.g. Barrat  _et al._ , 2008; Newman, 2010) has provided a
novel viewpoint for constructing a parsimonious yet efficient representation
of many complex processes taking place in the Earth system (e.g., Tsonis and
Roebber, 2004; Tsonis  _et al._ , 2006; Donges  _et al._ , 2009). Gozolchiani
_et al._ (2011) and Wang  _et al._ (2013) provide examples of use of networks
for capturing specific climatic processes, while Boers  _et al._ (2014) an
example of applying network theory to climate prediction.

  * •

Extreme Value Theory (EVT) (e.g., Embrechts  _et al._ , 1999; Coles, 2001) has
been used extensively in studying the statistical properties of rare
hydrometeorological events, such as the occurrence of very intense rain or of
extreme temperatures (Katz  _et al._ , 2002; Ghil  _et al._ , 2011, and
references therein). Yet this classical methodology has been almost entirely
neglected in IPCC reports dedicated to the study of extremes, cf. IPCC (2012).
Recently, though, rapid advances in EVT theory for general observables of
chaotic deterministic dynamical systems Holland  _et al._ (2012); Lucarini
_et al._ (2014b, 2016) have led to the derivation of indicators of weather
regimes based on extreme value statistics in the recurrence of atmospheric
fields. In addition to the mere classification of such regimes, as mentioned
at the end of Sect. V.4, this approach allows one to infer higher or lower
instability of these regimes and, hence, their lower or higher predictability
Faranda  _et al._ (2017); Hochman  _et al._ (2019).

  * •

Large Deviation Theory (e.g., Touchette, 2009; Varadhan, 1984) has been first
used in the context of geophysical fluid dynamics for studying the self-
organization and the multistability of turbulent flows and, specifically, of
jet structures Bouchet and Venaille (2012); Bouchet  _et al._ (2014); see also
Sects. V.1 and V.3 herein. More recently, it has been applied successfully in
studying weather and climate extremes. First, it has helped formulate rare
events algorithms able to nudge a climate model towards representing
preferentially the class of extreme events of interest Ragone  _et al._
(2018). Second, it has provided a solid theoretical and numerical basis for
the study of spatially extended or temporally persistent temperature extremes
Gálfi  _et al._ (2019). Large Deviation Theory has also been recently used to
study multiscale and coupled atmosphere-ocean instabilities in a hierarchy of
climate models Vannitsem and Lucarini (2016); De Cruz  _et al._ (2018).

  * •

Detection and Attribution Studies. In a complex system like the climate,
inferring causal relationships among events and phenomena is far from obvious.
Nonetheless, doing so for well-defined weather and climate events is essential
for building simplified models and, at a practical level, for causal
attribution of weather- and climate-related events. This is quite important in
the case of detection and attribution studies of anthropogenic climate change,
especially when aiming to go beyond changes in mean climate properties, such
as globally averaged temperatures, and on to determining to what extent the
occurrence of an individual extreme event — e.g., of a hurricane or an
extended drought — can be attributed to climate change Allen (2003); Adam
(2011).

Methodologies based on the causal counterfactual theory of Pearl (2009) are
being increasingly recognised as a key instrument for providing a more
rigorous basis for detection and attribution studies (e.g., Hannart  _et al._
, 2016a). They also seem better suited for defining reliably the link between
anthropogenic climate forcing and individual events Hannart and Naveau (2018).
Combining these methods with those of data assimilation, discussed herein in
Sect. II.1, appears to be well suited to refine the distinction between the
factual and counterfactual world that separates causation from the lack
thereof (e.g., Hannart  _et al._ , 2016b).

  * •

Beyond Linear Response Theory. As shown in Sect. IV.5, linear response theory
can provide a systematic improvement upon the standard methodology of forward
integration of model ensembles with perturbed initial states and parameters.
While it does apply to systems out of thermodynamic equilibrium, it is still
limited by its linearity to fairly small perturbations in parameters. In Sect.
IV.4, we mentioned that one can use the Wasserstein distance to measure
arbitrary changes between two invariant measures, whether supported on a time-
independent, classical attractor or a time-dependent PBA. One might thus wish
to apply the PBA-based methodologies herein directly to observational data
sets or to the simulations of IPCC-class models, either instead of or in
combination with previously tested statistical methods.

In Sect. V.5, we showed that the PBA of an intermediate, but still infinite-
dimensional ENSO model can undergo a chaos-to-chaos transition that involves
major changes in its invariant measure. Moreover, these changes could be
connected to a physically quite significant change in model behavior, namely
in the number and size of extreme events, i.e. of the largest warm and cold
events. Thus exploring similarly interesting changes in model PBAs and in the
time-dependent invariant measures supported by them appears to be a very
promising road toward a deeper understanding of climate variability and its
interaction with both natural and anthropogenic forcing.

In the text above, we have pointed to several open scientific problems; see
also Ghil (2019) for a historical perspective. An important matter that was
left practically untouched, aside from the brief discussion of ENSO and its
prediction in Sec. III.5.2, is the tropical atmosphere. Half the surface of
the Earth lies in the Tropics, between $30$ degrees N and $30$ degrees S, and
much of humanity lives there. As one gets closer to the Equator, the quasi-
geostrophic approximation collapses and moist processes take on a much greater
importance. Four of the classical references are Palmer (1951); Riehl (1954);
Krishnamurti  _et al._ (1979) and Yanai (1975). The current literature is too
plethoric to marshall here.

More broadly, we have already alluded at the end of Sec. I to the lack of a
proper definition of climate and to the efforts to provide a mathematically
rigorous definition that would clearly distinguish it from weather. We have
also mentioned in Sec. III.5.1 the distinction made by Von Neumann (1960)
between the short-, medium- and long-term prediction problem in the climate
sciences.

At an even higher level of abstraction, one might wish to recall the parallel
drawn by H. Poincaré between the deterministic unpredictability of weather and
that of the position of the planets on the ecliptic (Poincaré, 1902, pp.
69-70). Harking back to these two giants of both physics and mathematics, we
could say that: (i) the astronomical ephemerides are the analog of weather and
of its short-term prediction; (ii) the problem of climate corresponds to that
of the long-term stability of the solar system; and (iii) the secular
variations of the orbital configurations in the solar system are the
appropriate counterpart of the slow evolution of the climate system’s low-
frequency modes of variability Chown (2004).

Making these analogies stick at a level that might help determine more
reliably climate sensitivity to human activities is clearly a worthwhile
effort for the best mathematicians, physicists and climate scientists.

## Acknowledgments

We have benefited immensely from discussions with many colleagues and friends
in the last few years, but we would like to thank especially T. Bódai, N.
Boers, F. Bouchet, M. D. Chekroun, J. I. Díaz, H. A. Dijkstra, D. Faranda, A.
Feigin, U. Feudel, K. Fraedrich, G. Gallavotti, A. Gritsun, A. Groth, D. Holm,
F.-F. Jin, D. Kondrashov, S. Kravtsov, J. Kurths, J. C. McWilliams, J. D.
Neelin, S. Pierini, F. Ragone, D. Ruelle, A. Speranza, H. Swinney, O.
Talagrand, A. Tantet, S. Vannitsem, J. Yorke, and J. Wouters. Two anonymous
reviewers provided constructive comments. The authors acknowledge the
financial support provided by Horizon 2020 through the project TiPES (grant
No. 820970). This activity has been supported by the EIT Climate-KIC. EIT
Climate-KIC is supported by EIT, a body of the European Union. VL also
acknowledges the financial support provided by Horizon 2020 through the
projects CRESCENDO (grant No. 641816) and Blue-Action (grant No. 727852) and
by the Royal Society (grant No. IEC R2 170001).

##  Appendix A Acronyms

This Appendix contains two tables of acronyms used throughout the paper: Table
3 contains the scientific acronyms and Table 4 the institutional ones.

Table 3: Scientific acronyms Acronym | Meaning
---|---
$AS$ | Arakawa-Schubert (parametrization)
$CN$s | Complex networks
$DDE$ | Delay differential equation
$DNS$ | Direct numerical simulation
$EBM$ | Energy Balance Model
$ECS$ | Equilibrium Climate Sensitivity
$ENSO$ | El Niño–Southern Oscillation
$EMS$ | Empirical Mode Reduction
$FDT$ | Fluctuation–Dissipation Theorem
$GCM$ | General Circulation Model
$GCM$ | Global Climate Model
$GFD$ | Geophysical fluid dynamics
$GHGs$ | Greenhouse Gases
$GLE$ | Generalized Langevin equation
$LFV$ | Low Frequency Variability
$MJO$ | Madden-Julian Oscillation
$MOC$ | Meridional overturning circulation
$MZ$ | Mori-Zwanzig
$NAO$ | North Atlantic Oscillation
$NDS$ | Nonautonomous dynamical system
$NWP$ | Numerical Weather Prediction
$NSEs$ | Navier-Stokes Equations
$ODE$ | Ordinary differential equation
$PBA$ | Pullback attractor
$PDE$ | Partial differential equation
$PNA$ | Pacific North American (pattern)
$PSA$ | Pacific South American (pattern)
$QG$ | Quasi-geostrophic (flow, model)
$RDS$ | Random dynamical system
$SDE$ | Stochastic differential equation
$TCR$ | Transient Climate Response
$THC$ | Thermohaline Circulation
Table 4: Institutional acronyms Acronym | Meaning
---|---
$AR$ | Assessment Report
$CMIP$ | Climate Model Intercomparison Project
$ECMWF$ | European Centre for
| Mid-range Weather Forecast
$ENSO$ | El-Niño-Southern Oscillation
$IPCC$ | Intergovernmental Panel on Climate Change
$NCAR$ | National Center for Atmospheric Research
$NCEP$ | National Center for Environmental Prediction
$PCMDI$ | Program for Climate Model
| Diagnostics and Intercomparison
$SPM$ | Summary for Policy Makers
$UNEP$ | United Nations Environmental Programme
$WCRP$ | World Climate Research Programme
$WMO$ | World Meteorological Organization

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