Abstract
Tipping points in a system have particular significance in the context of climate science. Events such as Amazon rainforest die-back, West African monsoon shift, and Arctic sea ice loss are all thought to have an underlying tipping point, beyond which the system will not recover. Mathematically, tipping can be understood using a dynamical systems analysis of a model, and there are geophysical models which have been found to have a specific mathematical feature which can cause tipping (a saddle-node bifurcation). Empirical data suggests there are more such systems. Systems with this feature will have strongly non-linear behaviour close to the tipping point, which can cause issues for data assimilation methods. This thesis considers the effect of a saddle-node bifurcation in two strands: in the context of real-world tipping events, and; in the context of data assimilation. In the first strand, the effect of successive annual droughts on a carbon cycle model of a forest is considered. The resilience of the forest to these droughts is investigated using knowledge of the dynamics of the model together with a framework to determine the existence and stability of limit cycles, which is developed using a simpler test system with the same underlying dynamical feature, that of a saddle-node bifurcation. The second strand of this thesis considers the effect of tipping on sequential data assimilation schemes. For this, a test model is constructed, and a saddle-node bifurcation provides the mechanism for tipping. The ability of a stochastic ensemble Kalman filter (stochastic EnKF), ensemble square-root filter (EnSRF), ensemble transpose Kalman filter (ETKF), and ensemble adjustment Kalman filter (EAKF) to track the system as it tips is investigated, and associated error estimates for each method are analysed. In addition, the effect of varying levels of observation noise, and varying the likelihood of an observation are explored, to understand conditions under which the error is maximised.