The climatic response to time-dependent parameters is revisited from a nonlinear dynamics perspective. Some general trends are identified, based on a generalized stability criterion extending classical stability analysis to account for the presence of time-varying coefficients in the evolution equations of the system's variables. Theoretical predictions are validated by the results of numerical integration of the evolution equations of prototypical systems of relevance in atmospheric and climatic dynamics.

The climatic impact of systematic variations of certain key parameters in
time arising from anthropogenic effects such as increasing

On the other hand, it is widely recognized that the atmosphere and climate are highly nonlinear systems subjected to intricate feedbacks giving rise to a rich variety of complex dynamical behaviors such as self-generated periodicities, deterministic chaos, or transitions between different states (Nicolis and Nicolis, 1987; Dijkstra, 2013). A major advance of nonlinear dynamics has been to show that these behaviors often rest on a limited number of generic, global features independent of details concerning individual processes (Guckenheimer and Holmes, 1983). This suggests that it might be of interest to search for regularities likely to recur across different models and scenarios that could possibly be masked in a detailed full-scale analysis. In this work we revisit the climatic response to time-dependent parameters from such a nonlinear dynamics perspective, extending an early investigation in this direction by the present author (Nicolis, 1988).

The starting point is a set of equations governing the evolution of the atmospheric and climatic variables. We consider a reference state corresponding to a solution of these equations for some particular values of the parameters. We next switch on a systematic variation of these parameters in time and follow the subsequent transient response of the reference state to this forcing. The questions we raise are whether and if so for how long the system will follow passively this variation while remaining in the same branch of states; under what conditions it will jump to a new regime and if so when this transition will occur; and finally, whether states that would otherwise prevail in the absence of parameter variation are altered significantly or missed altogether.

A general formulation for addressing these questions is outlined in Sect. 2, where a generalized stability criterion for remaining or not in the vicinity of the reference state is derived and some general scenarios of subsequent evolution are discussed. In the light of these ideas the response to time-varying parameters is analyzed in Sects. 3 to 5 in situations giving rise to oscillatory behavior, to chaotic behavior and to transitions between simultaneously stable states. The main conclusions are summarized in Sect. 6.

Throughout her career Anna Trevisan managed to combine harmoniously theoretical ideas and tools and large-scale numerical approaches to tackle fundamental problems of concern in atmospheric physics. This paper is dedicated to her memory.

Let

We are interested in situations in which one of these parameters varies
systematically in time as a result of an externally induced forcing of
natural or anthropogenic origin. The particular form of variation we shall
focus on is a slow variation in the form of a ramp,

Following the procedure outlined in the Introduction we consider now a
particular, possibly time-dependent state

Equations (4b) constitute a set of coupled equations with slowly varying
coefficients. Generalizing the time-exponential solutions familiar from
classical stability analysis, we seek solutions of these equations of the WKB
form (Kevorkian and Cole, 1996):

We are now in the position to derive the condition under which the response

This relation, if satisfied, defines a critical time

In what follows these questions will be addressed for selected classes of
systems giving rise to periodic behavior, to chaotic dynamics and to
transitions between simultaneously stable steady states. We stress that the
logic underlying our formulation differs from the one adopted in typical
general circulation model-based experiments (Gregory et al., 2015) in which,
e.g.,

A dynamical system giving rise to sustained oscillations must involve at
least two coupled variables. The onset of oscillatory behavior will occur
through a Hopf bifurcation, in the vicinity of which the Jacobian matrix
associated to the rate functions

Here

Previous studies have shown that as long as

In the context of the present work it will be natural to choose

We choose again as a reference state the steady-state solution

This relation determines a critical time

We now confront these predictions to the results of direct numerical integration of Eqs. (8) with
parameter

Evolution of variable

These results hold for a wide range of values of

As in Fig. 1 but

A question related to the foregoing observations and of interest in the
context of atmospheric and climate dynamics is when a particular variable of
relevance in a system subjected to a systematic time-dependent forcing will
cross for the first time a certain prescribed level. Figure 3 summarizes the
results obtained by numerically integrating Eqs. (8) and (9) for a wide range
of values of the ramp parameter

Instantaneous value of parameter

Chaotic dynamics is ubiquitous in the atmosphere, where it is responsible for
the growth of prediction errors arising from small uncertainties in the
initial conditions (Lorenz, 1984). There are strong arguments supporting the
view that it also underlies a host of large-scale phenomena responsible for
climatic variability (Tsonis, 1992; Essex and McKitrick, 2007). In the
present section we analyze the effect of a systematic time variation of
parameters on a simplified model of thermal convection giving rise to chaotic
behavior due to Lorenz (Lorenz, 1963) in which the velocity and temperature
fields are expanded in Fourier series keeping one Fourier mode for the
vertical component of the velocity (variable

The parameters

Equations (13) have been studied extensively in the literature (Sparrow,
1982). We briefly summarize some results that will be relevant for our
purposes.

(i) The steady state

(ii) Beyond

(iii) At

(iv) Beyond

In what follows it will be natural to consider

Setting

Figure 4 summarizes the results obtained by numerical evaluation of the
integral in Eq. (16). We have set for this purpose

Theoretical estimate of the onset of chaotic solutions versus the
initial value of parameter

Figure 5, to be compared with Fig. 1, depicts the evolution of variable

Related to the foregoing observations is the question when the variable

Time evolution of variable

As in Fig. 5 but with

As in Fig. 3 but for model (13) with

Assuming now that the system has settled in the chaotic regime, we wish to
quantify in some way the effect of the time variation of parameter

Figure 8b depicts the time evolution (again via the dependence on

Ensemble averages

There is ample evidence of large-scale climatic transitions between glacial and interglacial regimes (Berger, 1981). On a shorter timescale transitions between different global circulation patterns associated to the phenomenon of persistent flow regimes at mid-latitudes, also referred to as “blocking” in contrast to the familiar zonal flows, are well documented and constitute one of the principal elements of low-frequency atmospheric variability.

In this section we analyze the effect of systematic time variations of
parameters in the classic three-variable model of the zonal to blocking
transitions that goes back to the pioneering work of Charney (Charney and De
Vore, 1979). The model consists in expanding the stream function

Here

Bifurcation diagram of model (17) as parameter

Figure 9 depicts the bifurcation diagram of model (17) in which the zonally
averaged velocity mode

In what follows we choose

Figure 10 summarizes the results of numerical simulations of the full
Eqs. (17) and (18) for three different initial conditions that in the absence
of time variation of

Time evolution of variable

A second series of numerical simulations is reported in Fig. 11, starting
this time from a state in the vicinity of the lower stable branch. For very
small

As in Fig. 10 but for initial conditions in the vicinity of the
lower stable branch of the bifurcation diagram and three different

A more quantitative explanation, albeit limited to the vicinity of the limit
points, appeals to the fundamental result that in the vicinity of a limit
point bifurcation the dynamics simplifies considerably. Specifically, there
exists a single variable

Setting again

Here Ai and Bi are the Airy functions, the prime denotes the derivative with
respect to the whole argument and

In this work we identified some universal trends underlying the response of a system to systematic changes of parameters in time. Most prominent among them are that, starting with a stable branch of states, transitions to new regimes that would occur in the “static” case of absence of time variation of parameters tend to be delayed; states that in the static case are unstable are temporarily stabilized; and states that in the static case are stable can be skipped altogether. As a corollary, the times at which threshold values are first crossed have been obtained as a function of the rate of increase of the parameters in time.

These conclusions were based on a generalized stability criterion extending
classical stability analysis to account for the presence of time-varying
coefficients in the evolution equations of the system's variables, as well as
on analytic solutions prevailing in the vicinity of transition points. They
were validated by the results of numerical integration of the evolution
equations of prototypical systems of relevance in atmospheric and climate
dynamics giving rise to periodic behavior, to chaotic dynamics and to
transitions between simultaneously stable steady states. As it turned out for
sufficiently small rates

The extended stability analysis followed in this work belongs to the class of
linear response theories, in the sense that it is focussing on the conditions
under which perturbations, initially assumed to be small, will at some stage
start to grow in time. On the other hand it is purely deterministic, as
random external perturbations or intrinsic fluctuations have not been
incorporated into the description. A different class of linear response
theories was recently developed in the climate literature (see, e.g.,
Lucarini, 2012; Nicolis and Nicolis, 2015) in which the change in the
fluctuation properties of a system due to the presence of noise and the
response of the noise-free system to deterministic forcings were linked.
Implicit in these approaches is the existence of a well-defined invariant
probability measure of the reference system with respect to which statistical
averages are carried out. Our analysis suggests that this can be so under the
conditions that the system is operating around a well-defined, single stable
regime, i.e., (a), that the range of variations of the forcing is nested
between two successive bifurcation points; and (b), that the rate

Throughout our approach the time variation of the parameters has been fully and consistently incorporated into the intrinsic time evolution of the system's variables as given by the appropriate rate equations. Our results depend critically on this view of parameter-system co-evolution, a scenario reflecting, we believe, the way a natural system is actually evolving in time. This scenario differs from those adopted in current studies on climatic change based on the integration of large numerical models, where parameters are suddenly set at a different level and the system is subsequently left to relax under these new conditions. It would be interesting to allow for different scenarios beyond the standard ones, closer to our fully dynamical approach, and to test the robustness of the conclusions reached under these different conditions.

Data can be accessed by directly contacting the author. They are not publicly accessible because they have been created for the specific purposes of the present work.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Numerical modeling, predictability and data assimilation in weather, ocean and climate: A special issue honoring the legacy of Anna Trevisan (1946–2016)”. It is not associated with a conference. Edited by: Juan Manuel Lopez Reviewed by: two anonymous referees