Justification of Delay Equations in Climate Models

Abstract

A new technique to derive delay models from systems of partial differential equations is investigated. This technique is based on the Mori-Zwanzig formalism, which gives a formal rewriting of the system using a projection onto a set of resolved variables. The rewritten system contains a memory term. The computation of this memory term requires the solving of the so-called orthogonal dynamics equation, which represents the unresolved dynamics. Finding an accurate solution to this equation is crucial in the application of the formalism. Here, the new technique is applied to a two-strip model of the El Niño Southern Oscillation. A mathematical derivation of a delay differential model, using the Mori-Zwanzig formalism and an alternative (exact) method based on variation of constants, is given. The derived delay model contains an additional term compared to a previously proposed conceptual model. This new term leads to a higher period of the model, which is closer to that seen in data. The Mori-Zwanzig formalism turns out to be not necessary to arrive at the resulting equations. Furthermore, the technique is applied to a model of the Atlantic Multidecadal Oscillation. This results in a delay difference model for the phenomena. In addition to this result, which can also be obtained by integration along characteristics, error terms for a smoothening approximation of this delay difference system have been derived.