Catastrophes and resilience of a zero‐dimensional climate system with ice‐albedo and greenhouse feedback

Abstract

AbstractAn ‘almost trivial’ climate system of geometrical dimension zero is analysed. It consists of the global energy storage which is balanced by the globally averaged net radiation (solar input minus infrared emission) with ice‐albedo and greenhouse feedbacks included. This nonlinear and time‐dependent climate system is formulated as a gradient system of a potential. It can be analysed without explicit time integration according to catastrophe theory, so that the resilience of the system, due to changes of the external parameters and due to variations of the state variable, can be determined. For the ice‐albedo or the greenhouse feedback alone, two equilibrium solutions appear: the stable ones are attractors, characterizing the present day interglacial; the unstable ones define a lower or an upper bound, respectively, for temperature changes to be absorbed by the systems. Beyond thresholds of external parameters (fold catastrophes) no equilibria exist, so that the systems attain the states of ‘deep freeze’ and ‘desert heat’, respectively. The two feedbacks combined lead to three equilibria, the stable interglacial being bounded by two unstable solutions, beyond which initial temperature values tend towards the ‘deep freeze’ or ‘desert heat’. External parameter changes can lead to structural instability occurring at fold lines, which meet at the cusp catastrophe. Outside the fold and cusp catastrophes only unstable equilibria appear, from which initial temperature values tend towards ‘desert heat’ or ‘deep freeze’ situations.