**Published by lead author, MPE PhD Student Swinda Falkena, on 17th February 2021 in Proceedings of the Royal Society A, Mathematical, Physical and Engineering Sciences, the paper can be viewed in full here.**

**Authors: **Swinda K.J. Falkena, Courtney Quinn, Jan Sieber, and Henk A. Dijstra.

**Abstract:**

A new technique to derive delay models from systems of partial differential equations, based on the Mori–Zwanzig (MZ) formalism, is used to derive a delay-difference equation model for the Atlantic Multidecadal Oscillation (AMO). The MZ 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 AMO. The resulting delay-difference model is of a different type than the delay differential model which has been used to describe the El Niño 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 MZ 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.

**Figure 1: **The AMO index for the last 160 years. The index is computed as the deviation of the area-weighted average SST over the North Atlantic. In black, the 12-monthly running mean is shown. Index computed by Enfield *et al.* [5] using the Kaplan SST dataset provided by the NOAA/OAR/ESRL PSD, Boulder, CO, USA (https://www.esrl.noaa.gov/psd/):