Beyond Critical Slowing Down: Slow Modes, Extreme Tails, and Field Decoherence in Tipping Transitions

Abstract

We study early-warning signals of climate tipping in the metastable stochastic Ghil--Sellers energy balance model. Rather than relying on a single scalar indicator, we analyze the transition through three complementary lenses: reduced Ruelle--Pollicott (RP) resonances, extreme value statistics, and full-field data-adaptive harmonic modes. This distinguishes bulk relaxation, tail excursions, and spatial phase organization as interacting aspects of tipping. First, using a reduced transfer-operator construction for global mean temperature and meridional thermal contrast, we estimate reduced RP resonances and Kolmogorov modes. Near tipping, several dominant decay rates drop and their modes harmonize along a common slow direction. Consequently, Green's functions aligned with this direction acquire coherent delayed-recovery tails and enhanced low-frequency susceptibility. The warning is thus carried by a bundle of slow modes rather than a single spectral gap. Second, Extreme Value Theory reveals that the cold tail of the global mean temperature anomaly becomes less sharply bounded and more persistent near the transition. The shape and extremal indices show an asymmetric organization: cold excursions probing the escape direction become more accessible and clustered. Third, Data-Adaptive Harmonic Mode (DAHM) analysis of the full temperature field shows that near tipping, leading modes still capture the large-scale trend, but fixed-rank reconstruction degrades and the DAHM phase distribution broadens. We interpret this as multivariate phase decoherence: the field retains a coherent transition component while losing sharp latitudinal phase organization. Ultimately, metastable tipping is marked by a joint reorganization of reduced spectral response, extreme-event statistics, and full-field phase coherence.