Attractors and bifurcation diagrams in complex climate models

Abstract

The climate is a complex nonequilibrium dynamical system that relaxes toward a steady state under the continuous input of solar radiation and dissipative mechanisms. The steady state is not necessarily unique. A useful tool to describe the possible steady states under different forcing is the bifurcation diagram, which reveals the regions of multistability, the position of tipping points, and the range of stability of each steady state. However, its construction is highly time consuming in climate models with a dynamical deep ocean, whose relaxation time is of the order of thousand years, or other feedback mechanisms that act on even longer time scales, like continental ice or carbon cycle. Using a coupled setup of the MIT general circulation model, we test two techniques for the construction of bifurcation diagrams with complementary advantages and reduced execution time. The first is based on the introduction of random fluctuations in the forcing and permits to explore a wide part of phase space. The second reconstructs the stable branches using estimates of the internal variability and of the surface energy imbalance on each attractor, and is more precise in finding the position of tipping points.

Alternative climatic steady states for the Permian-Triassic paleogeography

“…Using the method described in Brunetti et al (2019), we perform dozens of simulations by varying the initial conditions 1 , and let the system relax toward a climatic attractor. Second, starting from the attractors obtained at the previous step, we construct the stable branches of the BD by exploring a large range of pCO 2 , using Method II from Brunetti and Ragon (2023). More precisely, we slightly increase or decrease the forcing by ∆pCO 2 = 2-4 ppm at regular 1 Either by 1) using different mean values of oceanic temperature; 2) transiently varying some parameters of internal processes, such as the relative humidity threshold for low cloud formation or atmospheric CO 2 content, in order to perturb the radiative budget at TOA and obtain a different climate trajectory.…”

“…In fact, even by increasing the forcing up to ∼ 500 ppm, we do not find a tipping point when SAT increases toward values larger than 38 • C. Rather, for these values, the internal variability becomes very large, suggesting that the model becomes numerically unstable and is unable to describe such extreme conditions. The limitation is due to the high temperature rather than to the pCO 2 value, since previous works using the MITgcm for an aquaplanet managed to simulate twice larger pCO 2 values, although for lower values of SAT (Brunetti and Ragon, 2023).…”

“…We use Method II described in Brunetti and Ragon (2023) for obtaining the bifurcation diagram. In particular, once the three attractors at 320 ppm are found by scanning a large ensemble of initial conditions (see main text, Section 3.1), the atmospheric pCO 2 content is varied by ∆pCO 2 = 2-4 ppm at regular temporal intervals of ∆t = 100 yr starting from each attractor, in order to construct the respective stable branches (Fig.…”

“…However, our knowledge of deep-time climates is subject to large uncertainties and their numerical modelling needs to consider, in general, a wide range of initial and boundary conditions. In this context, the framework of multistability, where an ensemble of initial conditions is explored to find the possible attractors under the same forcing and boundary conditions, and the following construction of the so-called bifurcation diagram (BD) where the forcing is varied (Brunetti and Ragon, 2023), seems particularly useful to find possible scenarios of tipping mechanisms. Multistability has been observed in the entire hierarchy of climate models, from energy balance models (Budyko, 1969;Sellers, 1969;Ghil, 1976;Abbot et al, 2011), to Earth models of intermediate complexity with the present Earth topography (Lucarini et al, 2010;Boschi et al, 2013), and general circulation models with slab (Popp et al, 2016) or dynamical ocean (Ferreira et al, 2011;Rose, 2015;Popp et al, 2016;Brunetti et al, 2019;Ragon et al, 2022;Zhu and Rose, 2023;Brunetti and Ragon, 2023) using aquaplanet or idealized continental configurations.…”

“…Climatic attractors are complex dynamical objects living in a high-dimensional manifold(Brunetti et al, 2019;Falasca and Bracco, 2022). The projection of their invariant (or natural) measure(Eckmann and Ruelle, 1985) on a given state variable is arbitrary(Faranda et al, 2019;Tél et al, 2020;Brunetti and Ragon, 2023). In the present study, the projection is performed, as commonly done in the literature, in terms of the global mean SAT.…”