Where are the coexisting parallel climates? Large ensemble climate projections from the point of view of chaos theory

We review the recent results of large ensemble climate projections considering them to be the simulations of chaotic systems. The quick spread of an initially localized ensemble in the first weeks after initialization is an appearance of the butterfly effect, illustrating the unpredictability of the dynamics. We show that the growth rate of uncertainty (an analog of the Lyapunov exponent) can be determined right after initialization. The next phase corresponds to a convergence of the no longer localized ensemble to the time-dependent climate attractor and requires a much longer time. After convergence takes place, the ensemble faithfully represents the climate dynamics. Concerning a credible simulation, the observed signal should then wander within the spread of the converged ensemble all the time, i.e., to behave just as any of the ensemble members. As a manifestation of the chaotic-like climate dynamics, one can imagine that beyond the single, observed time-dependent climate, a plethora of parallel climate realizations exists. Converged climate ensembles also define the probability distribution by which the physical quantities of the different climate realizations occur. Large ensemble simulations were shown earlier to be credible in the sense formulated. Here, in addition, an extended credibility condition is given, which requires the ensemble to be a converged ensemble, valid also for low-dimensional models. Interestingly, to the best of our knowledge, no low-order physical or engineering systems subjected to time-dependent forcings are known for which a comparison between simulation and experiment would be available. As illustrative examples, the CESM1-LE climate model and a chaotic pendulum are taken.

"The climate is what you expect, the weather is what you get" /Robert Heinlein, 1973/ "Climate lasts all the time and weather only a few days" /Mark Twain, 1887/ • What is climate (theoretically)?
• Question: if climate is what we "expect", then what is the expectation value, what is the underlying statistics (probability)?

Motivation -The Snapshot attractor view
• Trajectories evolving from the "distant past" • A set to which the system evolves after a long enough time (tc) • Variability: the characteristic size of the attractor • Instantaneously permitted parallel climate realization

CESM1-LE -HadCRUT5
Consequences (credible models -Where are parallel climates at all?) • A model is credible if: 1. observed signal wanders within the band of the ensemble • 2. the climate model must be a converged one!
• Meaning: being credible (globally at least) requires the agreement between simulated and measured quantities • in nonlinear science, we can say that they are present indeed in any credible ensemble simulation, even if not in an individual sense, rather in the form of the probabilities generated by the dynamics.
• It is valid for PAST and FUTURE EQUALLY! • consider using only converged SMILES (Single model initial-condition large ensembles) • "..concept, .. referred to as the "Theory of Parallel Climate Realizations" (Tél et al., 2020)... has enormous implications."C. Deser Earth Future Parallel climates in CESM1-LE and for a driven pendulum (Herein et al., 2023) agreement between simulated and measured quantities -HadCRUT5 The importance of "being" converged • To illustrate convergence we run climate model PlaSim (T21, LSG ocean, historical+Keeling forcing) • converged (grey) and un-converged (blue) ensembles

Convergence time
The lack of convergence • "Where are "parallel climates at all??" -"present indeed in any credible ensemble simulation, even if not in an individual sense, rather in the form of the probabilities generated by the dynamics" •Due to chaos -> "butterfly effect" -none of the trajectories are distinguished •BUT still we have one measured reality only •Credible models: only converged ensembles, in harmony with observations •TRUE EVEN in chaos related experiments!
•Convergence time: at least decades •Climate ensembles should be generated in the "distant past" •Lack of convergence can lead to misleading results • Co-existing "climates": today's Earth, slushball, snowball ?
• Co-existing "climate states" -splitting of the snapshot attractor

Conclusions
Ongoing research / Further prospectives Schematic view of the convergence to snapshot attractor (after Sévellec, F., & Fedorov, A. V. 2015) A conceptual climate model (Lorenz'84) with x as the speed of westerlies (over a hemisphere), and y as the strength of cyclonic activity.The forcing parameter: decrease of the temperature contrast (F) after the onset of the climate change in year 0. (a) 25 years (b) 50 years (c) 85 years after climate change (d) One member (magenta), ensemble average (black) and forcing (orange) after Tél et al., 2020.Illustration of the time-dependence of a snapshot attractor -Lorenz'84 system One trajectory is not "relevant" Consequences (Chaos analogy) CESM1-LE Driven pendulum CESM1-LE Herein et al., 2023 • parallel climate realizations in CESM1-LE (Kay et al., 2015) • Driven pendulum "l Lyapunov exponent"  − / () + /  ( Ω ) ()