Climatologists use a hierarchy of models, from simple conceptual ones to global general circulation models. Delay models hold a special place because they compactly describe oscillations with memory, such as El Niño. However, the delay is often introduced ad hoc, based on physical reasoning such as wave travel times. This work stands alongside research on model reduction and extracting patterns from data, but unlike those, it offers a rigorous formal derivation. This fills the gap between intuitive models and full equations.
The authors apply the Mori-Zwanzig formalism to a two-strip El Niño model described by partial differential equations. They first consider the linear version, then introduce nonlinearity through the dependence of the thermocline feedback on temperature. Projection onto a single variable (ocean surface temperature) yields a Langevin equation with a memory term. To compute the orthogonal dynamics, they use the method of characteristics and the pseudo-orthogonal dynamics approximation. In parallel, they apply the exact method of variation of constants for comparison.
A nonlinear delay model is obtained, which contains an additional cubic term compared to the classical Suarez-Schopf model. This term lengthens the oscillation period from 2–3 years to 3.5–4.5 years, which is closer to the observed 4–7 years. The oscillation amplitude decreases, also matching data better. Bifurcation analysis shows that the region of stable oscillations in parameter space narrows. The pseudo-orthogonal dynamics approximation yields a noticeable error, indicating the need for more accurate methods.
The work demonstrates that the Mori-Zwanzig formalism can serve as a basis for systematically deriving delay models in climatology. Unlike neighboring works that either use heuristic models or study general properties of reduction, here the method is applied to a specific geophysical system. This allows the delay parameters to be linked to physical quantities such as wave speed and latitude. Such an approach could be extended to other climate phenomena with wave propagation.
A promising direction is to abandon the delta function for the spatial distribution of wind forcing, which would lead to a distributed delay. One could also include nonlinearities in the thermocline equations to test the robustness of the derivation. Improving the approximation of the orthogonal dynamics remains a key challenge for applying the formalism to more complex models.
The results could improve conceptual models of El Niño and other oscillations in the climate system, as well as justify the use of delay models in related fields such as ecology and economics.
Test the method on models with nonlinear unresolved variables and develop more accurate approximations for the orthogonal dynamics.
The work is directly related to the problem of well-founded model reduction in climatology: how to preserve essential dynamics when going from full equations to conceptual models. It also touches on the origin of nonlinearities in delay models, which are often introduced heuristically.
The delay in the El Niño model corresponds to the time it takes for Rossby waves to cross the Pacific basin, reflect off the western boundary, and return as Kelvin waves. This journey takes about half a year.