Steady states in complex climate models and different methods for the construction of the bifurcation diagram

The climate system can be regarded as a non-equilibrium dynamical system that relaxes toward a steady state under the continuous input of solar radiation and dissipative mechanisms over a multitude of temporal and spatial scales. The steady state is not necessarily unique. A useful tool to describe the possible steady states of the climate system is the bifurcation diagram, where the long-term behaviour of a state variable (like surface air temperature) is plotted as a function of force intensity. This diagram reveals the regions of multi-stability, the position of B-tipping (bifurcation points at critical forcing values giving rise to an abrupt and irreversible climate change), the range of stability of each attractor and the intensity of climate variability needed to induce transitions between states (N-tipping).

The construction of the bifurcation diagram requires to run long simulations from a huge ensemble of initial conditions until convergence to a steady state is attained (standard method). This procedure has prohibitive computational costs in general circulation models of the climate that include deep ocean dynamics relaxing on timescales of the order of thousand years, or other feedback mechanisms with even longer time scales, like continental ice or carbon cycle.

Using a coupled setup of the MIT general circulation model, we propose two techniques that require lower computational costs and show complementary advantages. We test them in a numerical setup that includes deep ocean dynamics and we compare the resulting bifurcation diagram with the one obtained with the standard method. The first technique is based on the introduction of random fluctuations in the forcing and allows one to explore a large part of the phase space. The second, based on the estimate of internal variability and relaxation time, is more precise in finding B-tipping.