Seeking more robust early warning signals for climate tipping points: the Ratio of Spectra method (ROSA)

Abstract

Potential tipping points in the Earth System present challenges for society and ecosystems, especially as the global warming thresholds at which these may be triggered remain uncertain. Fortunately, a theory of `critical slowing down' has been developed which could warn of approaching tipping points. Applications of this theory often implicitly assume stationary white-noise forcing, itself requiring a clean separation between forced trends and variability, which is especially difficult under contemporary climate change. This paper proposes a modified method to derive early warning signal in a system, such as the climate, which is forced by time correlated processes. The method looks at the Ratio of Spectra (ROSA) of a system state variable relative to a forcing variable. We demonstrate the ROSA method on an idealised forced dynamical system, before applying it to a particular challenging example from the Earth System: dieback of the Amazon rainforest. We show that ROSA identifies more examples of abrupt transitions in the Amazon than conventional early warning signals in state-of-the-art CMIP6 Earth System Models.

Unveiling Geomagnetic Reversals: Insights From Tipping Points Theory

“…However, if we move to the Fourier space we can relax any hypothesis on the nature of the noise as recently introduced by Clarke et al. (2023) with the new method known as ratio of spectra (ROSA) method. Indeed, Equation can be written as $$${-}i\ \omega \ \hat{x}=-\lambda \ \hat{x}+\hat{\eta }$$$ where $$\hat{\cdot }$$ stands for Fourier transform.…”

“…By taking the squared modulus we have $$$\mathcal{R}(\omega )=\frac{{\hat{x}}^{2}}{{\hat{\eta }}^{2}}=\frac{1}{{\omega }^{2}+{\lambda }^{2}}$$$ such that by knowing the spectrum of the time series x ( t ) and regardless of the nature of the noise, $$\mathcal{R}(\omega )$$ has a universal character. This has a profound implication for reducing a priori assumptions on the nature of the noise that could lead to misleading results (Clarke et al., 2023). Indeed, usually the noise is thought to be representative of the fast or of the unknown dynamics of a given system and, although a standard Wiener process has been widely employed in literature (Benzi et al., 1982; Boers & Rypdal, 2021; Boers et al., 2022; Consolini & De Michelis, 2003), it cannot always reproduced as a Brownian‐like dynamics as in the case of turbulence (Alberti et al., 2023) or random dynamical systems with pullback attractors (Chekroun et al., 2011; Vannitsem et al., 2021).…”

“…The simplest choice for η ( t ) is a white noise process, however in the following we relax this assumption, as recently proposed by Clarke et al. (2023), only requiring that its Fourier Transform exists. If we denote y ± the two stable states of the system and we linearize Equation around them, that is, y ( t ) = y ± + x ( t ), we have $$$\frac{dx}{dt}\approx -\lambda x+\eta (t),$$$ with $$$\lambda \doteq \frac{df(y,\mu )}{dy}{\vert }_{y={y}_{\pm }}$$$ representing the rate at which the system returns to the equilibrium states y ± .…”

“…If λ → 0 an early warning signal of approaching a tipping point ( λ = 0) can be derived. Power spectra are evaluated by using the Welch's method (Clarke et al., 2023; Welch, 1967), while data gaps are linearly interpolated only if less than the 10% of data in each window.…”

“…Furthermore, tipping points often involve complex interactions and feedback loops between multiple components of a system, making their analysis and prediction even more complex (Kuehn, 2011). Nonetheless, significant advances in mathematical modeling, data analysis, and empirical observations have been made to reveal indicators associated with tipping points (Boers et al., 2021; Boettner & Boers, 2022; Clarke et al., 2023).…”