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Over two decades ago, The Intergovernmental Panel on Climate Change introduced the idea of tipping points as "large scale discontinuities in the climate" [1]. While there is no precise mathematical definition of a tipping event, tipping can be thought of as a rapid, and often irreversible, change in the equilibrium state of a dynamical system [2]. In [3] it was proposed that tipping events could be classified according to whether the underlying mathematical mechanism involves, predominantly, a bifurcation (B-tipping), noise induced transitions (N-tipping), or a rate dependent parameter (R-tipping); see also [4, 5, 6, 7]. Classically, B-tipping has been used to study subsystems of the climate which are vulnerable to tipping. However, in B-tipping an implicit assumption is that parameters change slowly relative to the intrinsic timescale set by the deterministic dynamics. The validity of this assumption in climate applications is not clear since anthropogenic change has occurred over a much shorter time span (centuries) than the shortest geological time scale (millions of years) [8]. Moreover, in many climate applications the dynamical system is necessarily stochastic and non-autonomous and thus there is need to study how a combination of these mechanisms can also lead to tipping [3].

In [9] we studied these tipping mechanisms for a low-dimensional model of a tropical cyclone. Specifically, this model couples the dynamics of the tangential wind speed of a cyclone to its inner core moisture with, essentially, the potential energy of the storm, wind shear, and the temperature of the ocean serving as parameters. While classic bifurcations occur as these parameters are varied, we found non-intuitive results when considering rate and noise-induced tipping. Specifically, we found that rapid increases in the potential energy of a storm coupled with sufficient wind shear can destabilize a storm. This result is surprising in that while a slow increase of the potential energy results in a more intense storm, if the energy is increased sufficiently rapid the tangential wind speed cannot increase quickly enough and is dissipated by wind shear. We also found in this model that while the stability of cyclones are exceptionally robust to additive random fluctuations, the presence of noise can lead to the rapid formation of a tropical storm. This consequence of the model results from the existence of a center manifold along which the deterministic dynamics is relatively weak. In this article, we summarize the key techniques used in our analysis.

Because of the destruction that tropical cyclones can cause [10], understanding what mechanisms lead to their formation should be of interest to policy makers, risk analysts, and climate scientists. Indeed, a presumed impact of global climate change is the increase in frequency and intensity of tropical cyclones. For example, in 2019 Hurricane Dorian made landfall in the Bahamas as a category 5 hurricane, sustaining winds over 185 miles per hours and it caused an estimated $7 billion in damages, over 400 dead or missing persons, and immeasurable losses to reef and mangroves, which in turn impacted tourism, the fishing industry, and protection from future storms [11]. As a second example, in 2005, Hurricane Katrina struck the gulf coast of Louisiana and was one of the costliest storms on record, causing over $125 billion, and over 1,800 lives lost despite only sustaining winds of 127 mph upon landfall [12].

Tropical cyclones are axisymetric vortices that form over warm ocean water in which there is a temperature gradient between the warm ocean and cooler lower atmosphere. In these regions, as warm water evaporates, the resulting warm air mass rises and cools rapidly releasing heat through condensation back into the atmosphere. As the warm air rises, an area of low pressure forms and air begins to move from all directions to fill this void. The air in this region swirls from the Coriolis effect and, due to conservation of angular momentum, eventually forms a rotating air mass around the area of low pressure, i.e., the eye of the storm. In this idealized setting, this process can modeled as a Carnot engine in which the maximum potential velocity of the hurricane, \(V_p>0\) can be obtained by equating the kinetic energy with the theoretical maximum power that could be sustained by the storm [13, 14]. However, in realistic storms, wind shear in the form of strong upper level winds can dissipate the storm structure by displacing warm temperatures above the storm eye.

To study the formation of tropical storms, the model we considered was a stochastic perturbation of a dynamical system developed in [15, 16] and is given in dimensionless form by

where we have restricted the dynamics to the physically relevant first quadrant by assuming reflecting boundary conditions on the axes. Here \(v=V/V_p\in [0,1]\) is a dimensionless measure of the tangential velocity \(V\geq 0\) of the storm relative to a maximum potential velocity \(V_p>0\), \(\tau \sim V_p^{-1}t\), \(m\in [0,1]\) is the relative humidity in the core of the storm, \(W_1,W_2\) are standard Wiener processes, \(c=2.2S/V_p\) is a dimensionless measure of wind shear \(S\) relative to \(V_p\), and \(\sigma_1,\sigma_2>0\) are measures of the amplitude of random fluctuations. The dimensionless parameter \(\gamma\) is defined by \(\gamma=(T_A-T_O)/T_O+\kappa\), where \(T_A\), \(T_O\) are the temperatures of the lower atmosphere and upper ocean respectively and \(\kappa\) is a constant, and thus \(\gamma^{-1}\) is a proxy for the temperature of the ocean. In this model, \(m\) serves as a source of energy for the storm and if \(m=1\), i.e., the core is fully saturated, the equilibrium velocity is given by \(V=V_p\). The wind shear plays the role of friction in the system in the sense that it dissipates energy by pulling moisture from the storm and if \(c=0\) we see that \(m=1,v=1\) is a stable fixed point, i.e., the storm reaches its full potential intensity (assuming \(\sigma_1=\sigma_2=0\)).

The deterministic skeleton of Equation (1) contains an asymptotically stable fixed point ** O** (non-storm state) at the origin. The system also exhibits a saddle-node bifurcation in which a stable node

Rate-induced tipping occurs when a quick change of a parameter causes the system to move away from one attractor
to another [2]. However, this change of system behavior happens without the system undergoing a bifurcation; fixed points may move through different basins of attraction over a parameter shift, but never cross a bifurcation curve. In particular, it is the rate of change of the parameters not the necessarily the specific changes in the values of the parameters that governs the tipping. In our work, we assumed a parameter shift \(\Lambda_r\) that varies at a rate \(r>0\), is bi-asymptotically constant, and is monotonically increasing. Note, since ** O** is always a fixed point it follows that regardless of parameter values there can be no rate induced tipping away from

The conditions for rate-induced tipping to occur are actually the same for any dimension: if the initial state satisfies a sufficient condition called *forward threshold unstable* [17]. Essentially this condition implies the initial state at the start of the parameter shift lies in the basin of attraction of a different stable state at the end of the parameter shift. However, we note forward threshold stability is not a necessary condition to prevent rate-induced tipping in systems of higher dimension. Indeed, for systems of dimension \(n>1\), the more refined condition of *inflowing stability* guarantees that rate-induced tipping cannot happen [18].

To consider the possibility of rate-induced tipping aiding in the formation or destabilization of a storm, we needed physical parameters with the ability to rapidly change. In the model, both wind shear and maximum potential velocity meet these requirements, and thus we allowed \(c\) and \(V_p\) to vary with time. As rate-induced tipping is a deterministic mechanism, we set \(\sigma_1=\sigma_2=0\) and considered the dimensional system. To do this, we redefined both \(V_p\) and \(c\) as functions of a parameter shift \(\Lambda_r=\frac{1}{2}(1+\tanh(rt))\). Specifically, we assumed \(V_p,c\) to transition in time between \(V_p^-\) to \(V_p^+\) and \(c^-\) to \(c^+\), respectively, which represents the minimum and maximum values of the parameters. Additionally, we chose parameter shifts such that there are always three fixed points at the start and end of the ramp which we denote ** O**,

In regard to storm destabilization, we used the condition of inflowing stability to show that if either \(V_p(\Lambda_r(\tau))\) or \(c(\Lambda_r(\tau))\) is nonincreasing as a function of \(\tau\), there can be no rate-induced tipping away from the stable storm state ** S**\(^-\) to the non-storm state

A noise-induced tipping event from ** O** to

The observations given above were quantitatively studied in the asymptotic limit \(\sigma_1,\sigma_2\rightarrow 0\) using the Freidlin-Wentzell (FW) theory of large deviations; see [19, 20, 21, 22] for thorough introductions to this topic. To simplify the following exposition, we let \(F=(f,g)\) denote the vector field with components \(f\), \(g\), introduce the matrix

\( \Sigma=\begin{bmatrix} \sigma_1^{-2} & 0\\ 0 & \sigma_2^{-2} \end{bmatrix}, \) |

and define for \(\mathbf{v}_1,\mathbf{v}_2\in \mathbb{R}^2\) the weighted inner product \(\langle \mathbf{v}_1,\mathbf{v}_2\rangle_{\Sigma}=\mathbf{v}_1^{T}\Sigma \mathbf{v}_2\) and the weighted norm \(\|\mathbf{v}_2-\mathbf{v}_1\|_{\Sigma}^2=\langle \mathbf{v}_2-\mathbf{v}_1,\mathbf{v}_2-\mathbf{v}_1\rangle_{\Sigma}\). One key result from FW theory is that in the limit \(\sigma_1,\sigma_2\rightarrow 0\) tipping events concentrate about most probable transition paths \(\Psi^*(s)=(\psi^*_1(s),\psi^*_2(s))\) which minimize a rate functional. Specifically, for Equation (1), the rate functional is for most probable transition paths from ** O** to

\( I[\Psi]=\frac{1}{2}\int_{-\infty}^{\infty}\|\dot{\Psi}-F(\Psi)\|_{\Sigma}^2ds, \) | (2) |

which is defined for sufficiently regular curves satisfying \(\lim_{s\rightarrow -\infty}\Psi(s)=\)** O** and \(\lim_{s\rightarrow \infty}\Psi(s)=\)

\( \mathbb{E}[\tau^*]=\exp(I[\Psi^*])\left(C+O\left(\left\|\sqrt{\Sigma^{-1}}\right\|\right)\right), \) | (3) |

where \(C>0\) is a constant.

To numerically compute minimizers, we used a gradient flow which matches well with Monte-Carlo simulations; see Figure 4. However, through a Legendre transformation \(\mathbf{p}=\Sigma (\dot{\Psi}-F(\Psi))\) the Euler-Lagrange equations corresponding to \(I\) can be expressed in the following Hamiltonian form

\( \begin{aligned} \dot{\Psi}&=F(\Psi)+\Sigma^{-1}\mathbf{p},\\ \dot{\mathbf{p}}&=-\nabla F^T(\Psi)\mathbf{p}, \end{aligned} \) | (4) |

with corresponding Hamiltonian \(H=\frac{1}{2}\|\mathbf{p}\|_{\Sigma^{-1}}^2+\langle F(\Psi), \mathbf{p}\rangle\). Consequently, most probable transition paths can be interpreted as heteroclinic orbits connecting ** O** to

This scaling law identifies the two dimensionless measures of noise strength \(\tilde{\sigma}_1^2=\sigma_1^2/c^7\) and \(\tilde{\sigma}_2^2=\sigma_2^2/c\) which control the tipping time. In particular, in Figure 3 the ratio \(c^7/\sigma_1^2\) is \(O(1)\) explaing why ** O** is particulary susceptible to noise-induced tipping.

In our analysis of tipping in a tropical storm, we considered the various tipping mechanisms to be independent of each other. A natural question is how do various mechanisms couple to induce tipping events and so a natural extension of the analysis presented in our work is consider the interplay of a parameter shift and additive noise. For the specific system we considered, we expect that in tipping away from the stable storm state, there will be an interplay between the rate and noise-induced tipping mechanisms, and the additive noise will lower the critical rate needed for tipping [23, 24]. Additionally, tipping should occur away from the non-storm state, but as there was no rate-induced tipping with this initialization, we must explore how rapidly changing parameters could enhance or inhibit noise induced tipping from ** O**.

[1] T. M. Lenton, J. Rockström, O. Gaffney, S. Rahmstorf, K. Richardson, W. Steffen, and H. J. Schellnhuber, *Climate tipping points—too risky to bet against*, Nature, vol. 575, no. 7784, pp. 592-595, 2019.

[2] P. Ashwin, C. Perryman, and S. Wieczorek, *Parameter shifts for nonautonomous systems in low dimension: bifurcation-and rate-induced tipping*, Nonlinearity, vol. 30, no. 6, p. 2185, 2017.

[3] P. Ashwin, S. Wieczorek, R. Vitolo, and P. Cox, *Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system*, Philosophical Transactions of the Royal Society A, vol. 370, no. 1962, pp. 1166-1184, 2012.

[4] J. Thompson and J. Sieber, *Predicting climate tipping as a noisy bifurcation: a review*, International Journal of Bifurcation and Chaos, vol. 21, no. 02, pp. 399-423, 2011.

[5] T. Lenton, *Early warning of climate tipping points*, Nature Climate Change, vol. 1, no. 4, p. 201, 2011.

[6] L. Halekotte and U. Feudel, *Minimal fatal shocks in multistable complex networks*, Scientific reports, vol. 10, no. 1, pp. 1-13, 2020.

[7] H. Alkhayuon, R. C. Tyson, and S. Wieczorek, *Phase tipping: how cyclic ecosystems respond to contemporary climate*, Proceedings of the Royal Society A, vol. 477, no. 2254, p. 20210059, 2021.

[8] D. J. Wuebbles, D. W. Fahey, K. A. Hibbard, D. J. Dokken, B. C. Stewart, T. K. Maycock, D. J. Wuebbles, D. W. Fahey, K. A. Hibbard, B. DeAngelo, S. Doherty, K. Hayhoe, R. Horton, J. P. Kossin, P. C. Taylor, A. M. Waple, and C. P. Weaver, *Executive summary*, pp. 12-34, Washington, DC, USA: U.S. Global Change Research Program, 2017.

[9] K. Slyman, J. A. Gemmer, N. K. Corak, C. Kiers, and C. K. Jones, *Tipping in a low-dimensional model of a tropical cyclone*, Physica D: Nonlinear Phenomena, vol. 457, p. 133969, 2024.

[10] N. Mori, T. Takemi, Y. Tachikawa, H. Tatano, T. Shimura, T. Tanaka, T. Fujimi, Y. Osakada, A. Webb, and E. Nakakita, *Recent nationwide climate change impact assessments of natural hazards in japan and east asia*, Weather and Climate Extremes, vol. 32, p. 100309, 2021.

[11] C. Dahlgren and K. Sherman, *Preliminary assessment of hurricane dorian’s impact on coral reefs of abaco and grand bahama*, Perry Institute of Marine Science Report to the Government of The Bahamas, 2020.

[12] A. Graumann, T. G. Houston, J. H. Lawrimore, D. H. Levinson, N. Lott, S. McCown, S. Stephens, and D. B. Wuertz, *Hurricane katrina: A climatological perspective: Preliminary report*, US Department of Commerce, National Oceanic and Atmospheric Administration, National Environmental Satellite Data and Information Service, National Climatic Data Center, 2006.

[13] K. Emanuel, *Hurricanes: Tempests in a greenhouse*, Physics Today, vol. 59, no. 8, pp. 74-75, 2006.

[14] K. Emanuel, *Self-stratification of tropical cyclone outflow. part ii: Implications for storm intensification*, Journal of the Atmospheric Sciences, vol. 69, no. 3, pp. 988-996, 2012.

[15] K. Emanuel and F. Zhang, *The role of inner-core moisture in tropical cyclone predictability and practical forecast skill*, Journal of the Atmospheric Sciences, vol. 74, no. 7, pp. 2315-2324, 2017.

[16] K. Emanuel, *A fast intensity simulator for tropical cyclone risk analysis*, Natural Hazards, vol. 88, pp. 779-796, Sept. 2017.

[17] S. Wieczorek, C. Xie, and P. Ashwin, *Rate-induced tipping: Thresholds, edge states and connecting orbits*, Nonlinearity, vol. 36, no. 6, p. 3238, 2023.

[18] C. Kiers and C. K. Jones, *On conditions for rate-induced tipping in multi-dimensional dynamical systems*, Journal of Dynamics and Differential Equations, vol. 32, pp. 483-503, 2020.

[19] M. I. Freidlin and A. D. Wentzell, *Random perturbations of dynamical systems*, Springer Science & Business Media, vol. 260, 2012.

[20] N. Berglund, *Kramers' law: Validity, derivations and generalisations*, Markov Process Relat Fields, vol. 19, no. 3, pp. 459-490, 2013.

[21] E. Forgoston and R. O. Moore, *A primer on noise-induced transitions in applied dynamical systems*, SIAM Rev, vol. 60, no. 4, pp. 969-1009, 2018.

[22] V. M. Gálfi, V. Lucarini, F. Ragone, and J. Wouters, *Applications of large deviation theory in geophysical fluid dynamics and climate science*, La Rivista del Nuovo Cimento, vol. 44, no. 6, pp. 291-363, 2021.

[23] P. Ritchie and J. Sieber, *Early-warning indicators for rate-induced
tipping*, Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 26, no. 9, p. 093116, 2016.

[24] K. Slyman and C. K. Jones, *Rate and noise-induced tipping working in
concert*, Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 33, no. 1, 2023.

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