The climate system is multiscale, multidimensional and nonlinear. Here we propose a robust framework for visualizing and analyzing its dynamics, accounting for both dependencies and nonlinearities. At each time t, the system is uniquely described by a state space vector parameterized by N variables and their spatial variability. The dynamics is confined on a manifold with dimension lower than the full state space and a strategy for manifold learning is presented via linear and nonlinear algorithms. We focus on the Tropical Pacific Ocean using a reanalysis as observational proxy (ERA5) and two state-of-the-art models from the CMIP6 catalog, MPI and EC-Earth3.

The analysis spans four variables over two 40 years periods at daily frequency, during historical times and in the SSP585 scenario. The manifold learning step allows for comparing nonlinear contributions as well as the relative role of each variable in the system's dynamics. Instantaneous properties of the high dimensional attractor are then quantified through the local dimension and persistence metrics, recently introduced to the climate community.

These metrics quantify geometrical properties of the manifold and the stability of local motions. Both models underestimate the average dimension and overestimate the potential predictability of Tropical Pacific climate compared to ERA5, which is indicative of common and persistent differences between modelled and observed dynamics. These model's biases are nearly identical during the historical period while diverging in the global warming scenario analyzed.