A Simple Dynamical System for Representing Climate Tipping Points with Hysteresis

Abstract. The risk that the climate system may contain tipping points remains a concern. First, a level of global warming may be reached at which relatively small additional warming could cause major parts of the Earth system to transition to a new state. Depending on the location and specific Earth system component, this could disproportionately impact large sectors of society. Second, the Earth system component may exhibit hysteresis effects, and so if global temperatures are subsequently lowered after triggering a jump in state, a return to earlier conditions may not occur until warming is substantially reduced. Earth System Models (ESMs) are numerical frameworks that operate at fine spatial scales, designed to estimate how all components of the climate system will evolve in response to changes in atmospheric greenhouse gas concentrations caused by human activity. Many ESM projections suggest that various parts of the climate system are capable of tipping. Yet ESMs are computationally demanding and have therefore been operated only over a small range of scenarios. Very few "overshoot" simulations with ESMs exist, where climate change is reversed, resulting in limited understanding of hysteresis effects following a tipping event.

Advances in nonlinear mathematics include the development of equation sets, known as dynamical systems, that depend on a bifurcation parameter. These equations can effectively reproduce tipping points, jumps in state and hysteresis as the bifurcation parameter changes. Mapping the broad behaviour of the components of ESM projections onto these simpler models could offer many advantages, including the characterisation of such climate models and a method extrapolate their projections to a wider range of forcing scenarios. The bifurcation parameters in dynamical systems may represent changing forcings, such as an increasing warming level that leads to a tipping event. Progress has been made in mapping components of the Earth system onto large-scale variables for representation as dynamical systems. Most advances to date have focused on understanding whether tipping events can be avoided if systems possess substantial inertia, allowing climate change to temporarily exceed thresholds that might otherwise trigger major nonlinear change. However, potential hysteresis effects in the context of climate change are less well represented in equation form. Achieving such a mathematical formulation requires a dynamic system to describe a climate system component not only at the point of tipping but also for substantial periods before and after. This behaviour corresponds to a bifurcation parameter that first increases and then decreases, with the modelled behaviours differing significantly during the return phase.

To support such necessary developments, we present a parameter-sparse dynamical system model that can exhibit hysteresis following a tipping occurrence, offering the potential for characterising Earth system components with this feature. We place particular emphasis on presenting in full the algebra needed to map known or modelled key attributes of a system that can tip onto the simplified dynamical system equation. We drive the equation with a time-evolving forcing representing an ``overshoot'' trajectory of global warming that exceeds a threshold for potential tipping. Calculations are performed over a range of system inertia values, illustrating a threshold inertia above which full tipping and hysteresis can be avoided. We use scale analysis to relate this threshold to those reported in existing climate research on behaviour near potential tipping points.

We hope that the framework we present offers a simple-to-use equation structure to quantify tipping points in the climate system, including a more complete description of behaviours both before and after tipping, with the latter potentially involving substantial hysteresis.