# Global Stability Properties of the Climate: Melancholia States, Invariant Measures, and Phase Transitions

## TL;DR

This paper investigates the stability of Earth's climate states, focusing on transitions between warm and snowball conditions, using stochastic modeling, invariant measures, and Melancholia states to understand phase transitions and noise-induced shifts.

## Contribution

It introduces a novel method for constructing Melancholia states from simulations and analyzes their role as gateways in noise-induced climate transitions.

## Key findings

- Melancholia states sit at the boundary between competing climates.
- Noise induces transitions between warm and snowball states.
- Phase transition occurs at a critical solar irradiance level.

## Abstract

For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in the past our planet flipped between these two states. The main physical mechanism responsible for such an instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attraction. In the weak noise limit, large deviation laws define the invariant measure, the statistics of escape times, and typical escape paths called instantons. By constructing the instantons empirically, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first-order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we put forward a new method for constructing Melancholia states from direct numerical simulations, black which provides a possible alternative with respect to the edge-tracking algorithm.