  The threat of global warming and the demand for reliable climate predictions
pose a formidable challenge being the climate system multiscale,
high-dimensional and nonlinear. Spatiotemporal recurrences of the system hint
to the presence of a low-dimensional manifold containing the high-dimensional
climate trajectory that could make the problem more tractable. Here we argue
that reproducing the geometrical and topological properties of the
low-dimensional attractor should be a key target for models used in climate
projections. In doing so, we propose a general data-driven framework to
characterize the climate attractor and showcase it in the tropical Pacific
ocean using a reanalysis as observational proxy and two state-of-the-art
models. The analysis spans four variables simultaneously over the periods
1979-2019 and 2060-2100. At each time t, the system can be 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 that we characterize through manifold learning algorithms,
both linear and nonlinear. The local geometry and local stability of the
high-dimensional, multi-variable climate attractor are quantified through the
local dimension and persistence metrics. Model biases that hamper climate
predictability are identified and found to be similar in the multivariate
attractor of the two models during the historical period while diverging under
the warming scenario considered. Finally, the relationships between different
sub-spaces (univariate fields), and therefore among climate variables, are
evaluated. The proposed framework provides a comprehensive, physically based,
test for assessing climate feedbacks and opens new avenues for improving their
model representation.
