Transitions across Melancholia States in a Climate Model: Reconciling the Deterministic and Stochastic Points of View

The Earth is well known to be, in the current astronomical configuration, in a regime where two asymptotic states can be realized. The warm state we live in is in competition with the ice-covered snowball state. The bistability exists as a result of the positive ice-albedo feedback. In a previous investigation performed on a intermediate complexity climate model we identified the unstable climate states (melancholia states) separating the coexisting climates, and studied their dynamical and geometrical properties. The melancholia states are ice covered up to the midlatitudes and attract trajectories initialized on the basin boundary. In this Letter, we study how stochastically perturbing the parameter controlling the intensity of the incoming solar radiation impacts the stability of the climate. We detect transitions between the warm and the snowball state and analyze in detail the properties of the noise-induced escapes from the corresponding basins of attraction. We determine the most probable paths for the transitions and find evidence that the melancholia states act as gateways, similarly to saddle points in an energy landscape.

Figure 1

Bifurcation diagram for the model studied in Ref. [9] for the long term, globally averaged ocean temperature $[⟨T_S⟩]$. Bistability is found for a large range of values of the control parameter $\mu$. Of interest here: red line: warm ($W$) states; blue line: snowball ($SB$) state; green line: melancholia ($M$) states, (constructed via the edge tracking algorithm). The $W\to SB$ ($SB\to W$) tipping point is located at ${\mu }_{W\to SB}\sim 0.965$ (${\mu }_{SB\to W}\sim 1.06$).

Figure 2

Escape times for the $W\to SB$ transitions for various noise strengths. Each dot corresponds to an observed escape time. The slope of the straight line fit is twice the quantity $\mathrm{\Delta }\mathrm{\Phi }$, see Eq. (3). An optimal algorithm for estimating $\mathrm{\Delta }\mathrm{\Phi }$ is reported in Ref. [36].

Figure 3

Main graph: Logarithm of $\tilde{\rho }$ projected onto $(T_S,[\mathrm{\Delta }{T}_S])$ for $\mu =0.98$; $W$ attractor (red dot); $M$ state (green dot). We have used ${\sigma }_{100\mathrm{yr}}=1\%$. The $W\to SB$ instanton (red dashed line) is indicated. Bottom right inset: probability along the instanton.

Figure 4

Main graph: $\rho$ in the projected phase space $(T_S,[\mathrm{\Delta }{T}_S])$. $W$ attractor (red dot), $SB$ attractor (blue dot), $M$ state (green dot) for $\mu =1$. Red (blue) dashed line: $W\to SB$ ($SB\to W$) instanton. We have used ${\sigma }_{100\mathrm{yr}}=1.5\%$. Top left inset: marginal PDF with respect to $\mathrm{\Delta }{T}_S$. Bottom right inset: marginal PDF with respect to $[T_S]$. Center right inset: probability along the two instantons.