In this special issue contribution, I provide a personal view on the role of bifurcation analysis of climate models in the development of a theory of climate system variability. The state of the art of the methodology is shortly outlined, and the main part of the paper deals with examples of what has been done and what has been learned. In addressing these issues, I will discuss the role of a hierarchy of climate models, concentrate on results for spatially extended (stochastic) models (having many degrees of freedom) and evaluate the importance of these results for a theory of climate system variability.

The climate system, comprised of the atmosphere, ocean, cryosphere,
land and biosphere components, displays variability on a broad range
of temporal and spatial scales. Much information on this variability has
become available from observations (both instrumental and proxy) over
the last decades. Through these observations, many specific phenomena
of variability have been identified, such as the interannual-timescale
El Niño–Southern Oscillation (ENSO) in the equatorial Pacific

In classical meteorology, the weather is defined as the variability on a timescale of a few days, and climate is the “average weather” where usually an averaging time of 30 years is taken. However, such a concept of climate is not very useful as the climate system displays variability over many timescales. Hence, in modern climate dynamics an often used concept is that of the climate system variability which includes the weather and also variability in the ocean, land, biosphere and ice components. Much of this variability in the climate system is intrinsic (or internal), indicating that it would exist even if the insolation from the Sun were constant. Intrinsic variability arises through instabilities, in most cases associated with positive feedback processes. There is also variability through radiative forcing variations associated with the diurnal and seasonal cycle and variations of the Earth's orbit (Milankovitch forcing). If we do not consider human activities to be part of the climate system, then the changes in atmospheric composition due to anthropogenic emissions are also considered as a forcing. The same can be done with lithospheric processes such that volcanic activity is also a forcing component.

Natural climate variability is then
all variability due to natural processes (both intrinsic and forced), and
anthropogenic climate change is only that part due to human
activities. To reliably project future climate change, a thorough
knowledge of the natural variability is required. At the moment, there appear to be two different paradigms of natural climate variability (Fig.

To understand the results of the observations, i.e., to relate them to elementary well-established physical principles, the observations themselves are in most cases not enough and models are needed. Fortunately, a hierarchy of such models, from conceptual ones (capturing only a few elementary processes or scales) to global climate models (which are multi-scale and multi-process representations), is available. Traditionally, climate system modeling is seen as an initial value problem. The model equations are integrated in time from a specific initial condition (or an ensemble of them), and then the transient behavior is analyzed. A subsequent statistical analysis is performed on the results using in general uni- or multivariate statistical methods. Often, parameters in the model are varied to study the sensitivity of the results to physical processes (associated with the parameters) and to determine mechanisms of specific phenomena from the statistical analyses.

Changes in parameters can lead to qualitatively different behavior; for example, oscillatory behavior appears or transitions occur. When relatively strong changes occur under small changes of a parameter, critical conditions associated with so-called tipping behavior may have been crossed.

In particular regarding issues of qualitative changes in model behavior
once parameters are varied, there is a complementary methodology available
from dynamical systems theory, which is targeted to directly compute
the asymptotic (long-time) states (attractors) of the model. In the most simple autonomous models (steady forcing), these attractors are fixed points and periodic orbits. Non-autonomous models are studied through pullback attractor analysis

Sketch of dynamical systems concepts and approaches for the
Taylor–Couette flow (as modified from

A canonical problem of transition behavior in fluid dynamics is the flow
between two concentric cylinders of which only the inner cylinder rotates
with an angular frequency

The main issue addressed in this paper is hence whether such a dynamical systems analysis of models of (parts of) the climate system is useful to understand the variability of this system. In Sect. 2, the model hierarchy is sketched and a short overview will be given of the basic techniques focusing on the application to large-dimensional dynamical systems generated from discretized (stochastic) partial differential equations. In Sect. 3, I will discuss results of studies where dynamical systems analysis has been performed on spatially extended climate models, focusing on what has been done so far and what has been learned. This is followed by Sect. 4, where an outlook is given for the role of dynamical systems analysis in developing an overarching theory of climate variability.

In

Organization of climate models according to the two model traits:
number of processes and number of scales

Any spatially extended climate model consists of a set of conservation
laws, which are formulated as a set of coupled partial differential
equations, that can be written in general form as

When Eq. (

When noise is added, the evolution of the flow can generally be described
by a stochastic differential-algebraic equation of the form

These methods form part of the numerical bifurcation analysis toolbox;
here we are restricted to a single parameter

Suppose that the deterministic part of Eq. (

In the special case that

Numerical bifurcation methodology has been mostly applied to dynamical systems with small

From elementary bifurcation theory

The back-to-back saddle-node bifurcation structure is canonical for tipping points, which we will discuss in Sect. 3.1 below. Although the dynamical system is high-dimensional, the behavior of the system can be dominated by only a few (even only one) positive feedbacks, and hence transitions occur in a low-dimensional space. The Hopf bifurcation is canonical for the occurrence of spontaneous oscillatory behavior associated with one eigenmode of the linearized dynamical system, which is often referred to as the leading mode. A Hopf bifurcation needs the presence of both positive and negative feedbacks; when only a few dominate the dynamical behavior these can be found in high-dimensional systems as discussed in Sect. 3.2. In models where a sequence of Hopf bifurcations occurs, the resulting behavior can in general no longer be described using low-dimensional dynamics. In this case, collective interactions occur and this cannot be captured in a single bifurcation and associated pattern. This case will be discussed in Sect. 3.3 below.

An overview of possible tipping elements in the Earth's system is
given in

For a back-to-back saddle-node bifurcation there are two transition scenarios
possible, called (i) bifurcation tipping and (ii) noise-induced tipping

Numerical bifurcation analyses provided the basis for the stability indicator

For a spatially two-dimensional ocean-only model, the covariance matrices

The canonical bifurcation diagram with the back-to-back saddle node
indicating two stable states (a, c) and an unstable state (b). Bifurcation tipping occurs when the parameter

In high-dimensional climate models, also so-called edge states or Melancholy
states have been computed, for example in a coupled atmospheric sea-ice model investigating ice-covered/ice-free multi-stability

It is remarkable that on interannual-to-multidecadal timescales the variability in sea surface temperature is organized in large-scale patterns (Fig.

Overview of patterns of climate variability (AMO, PDO and ENSO) as
determined in

Numerical bifurcation analysis has been applied to several spatially
extended models, in particular ENSO, PDO and the AMO.

My interpretation of these results is that several of these SST patterns (but not all) appear through a normal mode which destabilizes the mean state through positive feedbacks; the presence of negative feedbacks causes the oscillatory behavior. In this case, the associated Hopf bifurcation (of a spatially extended model) provides both the dominant timescale of variability and its spatial pattern. The elegant structure of leading modes in ocean models and the ZC model was presented in

In the previous two subsections climate system variability phenomena were attributed to low-order dynamics. However, there are many phenomena which are intrinsically caused by the collective interaction of multiple instabilities. Clearly, the role of numerical bifurcation theory becomes quite limited in determining the behavior of these (in general) chaotic
dynamical systems; I briefly describe below two examples.

The cases briefly described above are examples of strongly nonlinear systems, where the nonlinearities occur in the momentum advection and where the mean state is strongly modified through rectification. Of course, there are many more examples of such geophysical systems, in particular on timescales up to interannual both in the ocean (internal waves) and the atmosphere (weather).

In this paper, I have given a short overview of results of studies where continuation methods were applied to spatially extended climate models. My interpretation of these results is that there are climate variability phenomena that can be attributed to low-order behavior; only one or a few spatial patterns are involved, associated with dominant feedbacks. Several of these studies have shown that successive instability behavior can also occur. This leads to a collective interaction between patterns that is eventually responsible for emergent variability in climate models. A summary of the different phenomena based on this distinction is provided in Fig.

Summary of what has been learned from dynamical systems analysis of spatially extended climate models, based on the distinction of low-order phenomena, emergent phenomena through collective interactions and critical transitions. The “hope” is that mechanisms of the phenomena in the green boxes can be determined from numerical bifurcation analysis of intermediate complexity climate models.

This first challenge I see is to better understand processes behind
the background variability which is “red noise like” in

Once the physics of this background are clear, the next challenge is to attribute spatial patterns which rise above it to specific instabilities. Several spatial patterns of SST variability are robust over the model hierarchy. I would interpret this to indicate that these spatial patterns, such as ENSO and the AMO, are due to a single-mode destabilization of the background induced by dominant large-scale feedbacks. These spatial patterns can already be captured in detail in intermediate complexity models, such as the ZC model for ENSO. Capturing the temporal variability involves representation of small-scale processes (noise) and possibly non-normal growth

Other spatial patterns (such as the MAV and WBC variability) arise through a collective interaction of instabilities and hence can only be captured in detail through models high in the hierarchy (representing a multitude of scales). This holds for example for the path variability of the Kuroshio Current, where it is known that the interactions of the barotropic instabilities of the current and the (baroclinic) mesoscale eddy field are important

Apart from the internal variability introduced by single normal (oscillatory) modes and collective phenomena, also clear large-scale tipping phenomena (in the sense of critical transitions) can affect climate variability. The canonical behavior is a back-to-back saddle-node bifurcation appearing generically in conceptual models. It was shown here that for models of the AMOC and MIS, indeed such bifurcation behavior is found in high-dimensional models. Transition behavior hence may occur when critical conditions are crossed or through noise in the multiple equilibrium regime. A third challenge I see is to show that such transitions remain robust once small-scale processes are included; work in this direction has been initiated

All of the results of continuation methods described above were obtained under stationary forcing and this seems disjoint from the real climate system, which is obviously forced by a non-stationary insolation component (on diurnal, seasonal and orbital timescales). For the present-day climate system, there is also the non-stationary anthropogenic component of climate change. A fourth challenge is to understand the relevance of these diurnal and seasonal non-stationary periodic components in natural internal variability on longer timescales. While one may argue that they are irrelevant and are averaged out, few detailed results are available. Probably only on interannual timescales can there be an interaction between the seasonal cycle and internal variability, for example, with the ENSO mode (due to nonlinear resonances). On very large timescales, however, certainly the non-stationary orbital forcing is crucial for the observed variability such as glacial cycles. The modification of natural variability under climate forcing is of course also a challenging issue.

Has the end point been reached of the models for which bifurcation analysis can be applied? Since starting with this endeavor in the early 1990s, I have been repeatedly asked this question. When we showed results for spatially two-dimensional ocean models, we were asked if we could do this for three-dimensional models. When we did, the question was on the application to ocean–atmosphere models. Although there are certainly still interesting details to be investigated in the ocean-only context, I think the main challenge with these models is to develop theory for internal variability in the geological past

The author declares that there is no conflict of interest.

This article is part of the special issue “Centennial issue on nonlinear geophysics: accomplishments of the past, challenges of the future”. It is not associated with a conference.

The author thanks his “partner in crime”, Fred W. Wubs (University of Groningen,
the Netherlands), for the (now

This work was sponsored by the Netherlands Earth System Science Centre (NESSC), financially supported by the Ministry of Education, Culture and Science (OCW (grant no. 024.002.001)).

This paper was edited by Ana M. Mancho and reviewed by two anonymous referees.