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Derivation of delay equation climate models using the Mori-Zwanzig formalism

journal contribution
posted on 2023-05-21, 06:46 authored by Falkena, SKJ, Courtney QuinnCourtney Quinn, Sieber, J, Frank, J, Dijkstra, HA
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 Niño 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.


Publication title

Proceedings of the Royal Society A





Article number







School of Natural Sciences


Royal Soc London

Place of publication

6 Carlton House Terrace, London, England, Sw1Y 5Ag

Rights statement

2019 The Author(s) Published by the Royal Society. All rights reserved.

Repository Status

  • Restricted

Socio-economic Objectives

Climate variability (excl. social impacts); Expanding knowledge in the mathematical sciences