.2021 May 27;11(1):11126.

doi: 10.1038/s41598-021-90138-1.

Tipping points induced by parameter drift in an excitable ocean model

Stefano Pierini^{1 2}, Michael Ghil^{3 4}

Affiliations

¹ Department of Science and Technology, Parthenope University of Naples, Centro Direzionale, Isola C4, 80143, Napoli, Italy. stefano.pierini@uniparthenope.it.
² CoNISMa, Rome, Italy. stefano.pierini@uniparthenope.it.
³ Geosciences Department and Laboratoire de Météorologie Dynamique (CNRS and IPSL), École Normale Supérieure and PSL University, Paris, France.
⁴ Atmospheric and Oceanic Sciences Department, University of California at Los Angeles, Los Angeles, CA, USA.

PMID:34045519
PMCID: PMC8159979
DOI: 10.1038/s41598-021-90138-1

Tipping points induced by parameter drift in an excitable ocean model

Stefano Pierini et al. Sci Rep.2021.

.2021 May 27;11(1):11126.

doi: 10.1038/s41598-021-90138-1.

Authors

Stefano Pierini^{1 2}, Michael Ghil^{3 4}

Affiliations

¹ Department of Science and Technology, Parthenope University of Naples, Centro Direzionale, Isola C4, 80143, Napoli, Italy. stefano.pierini@uniparthenope.it.
² CoNISMa, Rome, Italy. stefano.pierini@uniparthenope.it.
³ Geosciences Department and Laboratoire de Météorologie Dynamique (CNRS and IPSL), École Normale Supérieure and PSL University, Paris, France.
⁴ Atmospheric and Oceanic Sciences Department, University of California at Los Angeles, Los Angeles, CA, USA.

PMID:34045519
PMCID: PMC8159979
DOI: 10.1038/s41598-021-90138-1

Erratum in

Author Correction: Tipping points induced by parameter drift in an excitable ocean model.
Pierini S, Ghil M.Pierini S, et al.Sci Rep. 2022 May 18;12(1):8303. doi: 10.1038/s41598-022-12470-4.Sci Rep. 2022.PMID:35585180Free PMC article.No abstract available.

Abstract

Numerous systems in the climate sciences and elsewhere are excitable, exhibiting coexistence of and transitions between a basic and an excited state. We examine the role of tipping between two such states in an excitable low-order ocean model. Ensemble simulations are used to obtain the model's pullback attractor (PBA) and its properties, as a function of a forcing parameter [Formula: see text] and of the steepness [Formula: see text] of a climatological drift in the forcing. The tipping time [Formula: see text] is defined as the time at which the transition to relaxation oscillations (ROs) arises: at constant forcing this occurs at [Formula: see text]. As the steepness [Formula: see text] decreases, [Formula: see text] is delayed and the corresponding forcing amplitude decreases, while remaining always above [Formula: see text]. With periodic perturbations, that amplitude depends solely on [Formula: see text] over a significant range of parameters: this provides an example of rate-induced tipping in an excitable system. Nonlinear resonance occurs for periods comparable to the RO time scale. Coexisting PBAs and total independence from initial states are found for subsets of parameter space. In the broader context of climate dynamics, the parameter drift herein stands for the role of anthropogenic forcing.

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Conflict of interest statement

The authors declare no competing interests.

Figures

**Figure 1**
Definition of the ramp function $R_{τ} (t)$ ; its duration is defined as $τ = t_{2} - t_{1}$ .

**Figure 2**
Typical solutions in the two regimes of the model’s autonomous version, cf. Eq. (1), with $α = β = 0$ in the forcing given by Eq. (2). The constant values $γ$ of the factorG(t) in the forcing are shown in the upper panels by a solid purple line, with (a) $γ = 0.9$ and (c) $γ = 1.2$ . The corresponding model solutions are plotted for $Ψ_{3} (t)$ in the lower panels (b) and (d); see text and Fig. 4 for the initialization of the trajectories in the lower panels.

**Figure 3**
Transition from the excitable regime to the RO regime for a ramp forcing in Eq. (2). (a) Time dependence of the factorG(t) in the forcing for $γ = 0.9 < γ_{c} = 1$ , $α = 0.3$ , $τ = 800$ year, and $β = 0$ ; (b) corresponding response of $Ψ_{3} (t)$ ; see text for the choice of initial states. (c,d) Same as panels (a,b) but for $β = 0.05$ and perturbation period $T = 5$ year.

**Figure 4**
ES subject to the same forcing used in the single forward time integrations of Fig. 3a, b. The filled red circle in the oval indicates the initial state used to initialize the four simulations of Figs. 2 and 3. After an initial transient, it is visually obvious that the ES converges to a stable cylinder-shaped PBA, obtained by the translation in time of an autonomous limit cycle.

**Figure 5**
Dependence of key results on the ramp duration $τ$ in experiments Exp1–Exp6. (a) TP timing $t_{tp}$ versus $τ$ ; when a line portion lies in the grey area the tipping occurs after the end of the ramp. The filled circles on the red line indicate the presence of data; their absence indicates that no TP is reached. (b) Forcing values $G (t_{tp})$ at the TP (for Exp1 and Exp2) or $G_{tp}^{*}$ (for Exp3–Exp6) versus $τ$ ; the filled blue circles $P_{1} - P_{3}$ correspond to the ESs for Exp1 shown in Supplementary Figure S5, while the filled circle $P_{4}$ corresponds to the orange ES of Supplementary Figure S6.

**Figure 6**
Same as Fig. 5b but shown as a function of the ramp steepness $δ$ defined in Eq. (3). The solid black vertical line in the zoomed inset indicates the ramp steepness $δ$ corresponding to the two ESs of Exp1 (blue line) and Exp2 (red line) shown in Fig. 8.

**Figure 7**
Dependence of the number $\bar{C}$ of trajectory clusters on $τ$ for Exp1, with $T_{0} = 50$ year and $r = 0.5$ in Eq. (6). The magenta, green and red bars refer to the ESs shown in Supplementary Figures S5, S6 and S8, respectively.

**Figure 8**
Dependence of the TP on ramp length for two ESs belonging to Exp1 and Exp2 and having the same drift rate $δ$ . (a) Time dependence of the factorG(t) for Exp1 with $τ = 600$ year. (b) $Ψ_{3} (t)$ of the corresponding ES. (c) Time dependence of the factorG(t) for Exp2 with $τ = 835$ year. (d) $Ψ_{3} (t)$ of the corresponding ES.

**Figure 9**
Rate-induced tipping in the presence of periodic perturbations. Same as Fig. 6 but for a limited range of $δ$ -values for the unperturbed (Exp1 and Exp2) and corresponding perturbed (Exp3 and Exp5) cases with $β = 0.025$ and $T = 5$ year.

**Figure 10**
Dependence of the TP on periodT in Exp7 and Exp8. (a) TP timing $t_{tp}$ versus periodT. The horizontal dashed line indicates the value corresponding to Exp1 for $τ = 800$ year. (b) $G_{tp}^{*}$ versusT. Black dashed line as in panel (a), while the gray dashed line corresponds to the critical value of the autonomous system. The red filled circles in both (a) and (b) indicate the value that is common to Exp7 and Exp4.

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References

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Tipping points induced by parameter drift in an excitable ocean model

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Tipping points induced by parameter drift in an excitable ocean model

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