A Delay Equation Model for the Atlantic Multidecadal Oscillation

TL;DR

This paper introduces a novel method using the Mori-Zwanzig formalism to derive delay models from PDEs, applied to the Atlantic Multidecadal Oscillation, offering a new approach to understanding climate variability.

Contribution

It presents a new technique to derive delay difference equations from PDE systems, specifically applied to climate models of the Atlantic Multidecadal Oscillation, expanding modeling tools.

Findings

Derived a delay difference model for the Atlantic Multidecadal Oscillation.

Connected the memory term in PDEs to delay terms in difference equations.

Provided error estimates for smoothing approximations of the delay model.

Abstract

A new technique to derive delay models from systems of partial differential equations, based on the Mori-Zwanzig formalism, is used to derive a delay difference equation model for the Atlantic Multidecadal Oscillation. The Mori-Zwanzig formalism gives a rewriting of the original system of equations which contains a memory term. This memory term can be related to a delay term in a resulting delay equation. Here the technique is applied to an idealized, but spatially extended, model of the Atlantic Multidecadal Oscillation. The resulting delay difference model is of a different type than the delay differential model which has been used to describe the El Ni\~no- Southern Oscillation. In addition to this model, which can also be obtained by integration along characteristics, error terms for a smoothing approximation of the model have been derived from the Mori-Zwanzig formalism. Our new method of deriving delay models from spatially extended models has a large potential to use delay models to study a range of climate variability phenomena.