EGU General Assembly 2020
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.

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

Valerio Lucarini1,2,3
Valerio Lucarini
  • 1University of Reading, Department of Mathematics and Statistics, Reading, United Kingdom of Great Britain and Northern Ireland (
  • 2Centre for the Mathematics of Planet Earth, Department of Mathematics and Statistics, Reading, United Kingdom
  • 3CEN, University of Hamburg, Hamburg

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 past our planet flipped between these two states. The main physical mechanism responsible for such instability is the ice-albedo feedback. Following an idea developed by Eckhardt and co. for the investigation of multistable turbulent flows, we study the global instability giving rise to the snowball/warm multistability in the climate system by identifying the climatic Melancholia state, a saddle embedded in the boundary between the two basins of attraction of the stable climates. We then introduce random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attractions. In the weak noise limit, large deviation laws define the invariant measure and the statistics of escape times. By empirically constructing the instantons, we show that the Melancholia states are the gateways for the noise-induced transitions in the weak-noise limit. 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 propose a new method for constructing Melancholia states from direct numerical simulations, thus bypassing the need to use the edge-tracking algorithm.


V. Lucarini, T. Bodai, Edge States in the Climate System: Exploring Global Instabilities and Critical Transitions, Nonlinearity 30, R32 (2017)

V. Lucarini, T. Bodai, Transitions across Melancholia States in a Climate Model: Reconciling the Deterministic and Stochastic Points of View, Phys. Rev. Lett. 122,158701 (2019)

How to cite: Lucarini, V.: Global Stability Properties of the Climate: Melancholia States, Invariant Measures, and Phase Transitions, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-9491,, 2020

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  • CC1: Comment on EGU2020-9491, James Annan, 06 May 2020

    Many congrats on the award. I always enjoy your presentations and hope to see the medal lecture next year. A quick question just a little on the edge of your presentation - the Green's function approach to climate scenarios - is it very different from the step function approach of Good et al, and/or were you aware of this work? They don't specifically mention Green's functions but the concept seems similar.


    Clim Dyn
    DOI 10.1007/s00382-012-1571-1

    A step-response approach for predicting and understanding non-linear precipitation changes

    Peter Good • William Ingram • F. Hugo Lambert • Jason A. Lowe • Jonathan M. Gregory •
    Mark J. Webb • Mark A. Ringer • Peili Wu

    • AC1: Reply to CC1, Valerio Lucarini, 06 May 2020


      thanks a lot for your nice words.

      Regarding question on Green functions. Indeed the work by Good et al. you refer to uses a related  approach. In a very recent work we have discussed the difference between what they (and other authors) do and the approach we use, which we follows more directly from statistical mechanics. See: 

      V.  Lembo, V. Lucarini, F. Ragone, Beyond Forcing Scenarios: Predicting Climate Change through Response Operators in a Coupled General Circulation Model, Scientific Reports,  in press (2020) []

      Thanks for your comment!


      • CC3: Reply to AC1, James Annan, 12 May 2020

        Thanks for the link and the clear explanation in the paper :-)

  • CC2: Comment on EGU2020-9491, John Bruun, 06 May 2020
    Dear Valerio
    Great to hear about your work and congratulations again on the Lewis Fry Richardson medal!
    Your framework approach sounds extremely versatile. The breaking time reversal symmetry point. These wave symmetry properties appear in the statistical mechanics of random and coherent wave systems.
    To follow-up your request, here are two papers that show this:
    Delplace, P.; Marston, M. and Venaille, A. (2017), Topological origin of equatorial waves, Science, 358, 1075-1077, 10.1126/science.aan8819.

    Bruun, J.T. and Evangelou, S.N. (2019),  Anderson localization and extreme values in chaotic climate dynamics, .

    It would be great to evaluate such wave interaction properties in your framework: I would anticipate that it could help distinguish autonomous system type.

    Happy to talk through this further. 

    Best John (

    • AC2: Reply to CC2, Valerio Lucarini, 06 May 2020

      Hello John,

      thanks for your comment - I have to look into these topics. Quite interesting!