# Derivation of Delay Equation Climate Models Using the Mori-Zwanzig Formalism

## TL;DR

This paper uses the Mori-Zwanzig formalism to systematically derive delay models for climate phenomena like ENSO, providing a more justified approach and resulting in models with improved accuracy in replicating observed periods.

## Contribution

It introduces the Mori-Zwanzig formalism to derive delay climate models from PDEs, including methods to approximate unresolved dynamics, and demonstrates its application to ENSO modeling.

## Key findings

- Derived a nonlinear delay model for ENSO with Mori-Zwanzig formalism
- The new delay term increases the ENSO period, aligning better with observations
- Provides a systematic framework for delay model derivation in climate science

## Abstract

Models incorporating delay have been frequently used to understand climate variability phenomena, but often the delay is introduced through an ad-hoc physical reasoning, such as the propagation time of waves. In this paper, the Mori-Zwanzig formalism is introduced as a way to systematically derive delay models from systems of partial differential equations and hence provides a better justification for using these delay-type models. The Mori-Zwanzig technique gives a formal rewriting of the system using a projection onto a set of resolved variables, where the rewritten system contains a memory term. The computation of this memory term requires solving the orthogonal dynamics equation, which represents the unresolved dynamics. For nonlinear systems, it is often not possible to obtain an analytical solution to the orthogonal dynamics and an approximate solution needs to be found. Here, we demonstrate the Mori-Zwanzig technique for a two-strip model of the El Nino Southern Oscillation (ENSO) and explore methods to solve the orthogonal dynamics. The resulting nonlinear delay model contains an additional term compared to previously proposed ad-hoc conceptual models. This new term leads to a larger ENSO period, which is closer to that seen in observations.