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

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 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.

A delay equation model for the Atlantic Multidecadal Oscillation

“…These two most prominent signals correspond to period doubling of the two delay values which arises naturally from the boundary conditions. There is a smaller peak corresponding to a cycle of approximately 2 3 τ − years. The signals corresponding to 2τ − (53.07) and 2 3 τ − (17.77) year cycles align with the literature regarding possible cycle lengths of the AMO [7,9].…”

“…If the aim of applying the MZ formalism is to obtain a set of reduced-order model equations and not only a numerical result, this method is not suitable. Another approach, relying on integration along characteristics, has been explored by Falkena et al [2] and yielded an exact reduced-order (delay) model for the system studied. Here, we build on this work to see whether delay-type models can be derived for other systems of wave equations.…”

“…For spatially constant coefficients a j and b j and equal damping α in all components, we may also derive the leading orders of (4.18) by integration along wave characteristics. This approach is similar to that taken in [2] and we refer to it as the MoC. The damped free-wave solutions of the two-layer system,…”

“…Delay models have already been used to describe certain climate phenomena, particularly for the El Niño Southern Oscillation (ENSO) and Earth's energy balance [1]. A new method of deriving delay equation models has been proposed [2], allowing for a potential extension of the use of delay models to study other climate phenomena. This method of deriving delay models is based on the Mori-Zwanzig (MZ) formalism, which allows for the reduction of high-dimensional systems to reduced-order models [3].…”

“…These reduced-order models are simpler to study, while still describing the physical processes present in the original high-dimensional model. So far the method in [2] has only been applied to a PDE model of ENSO, for which ad hoc delay models were already proposed [4]. Here, we apply the MZ formalism to a PDE model of the Atlantic Multidecadal Oscillation (AMO) to investigate whether this phenomenon can be described by a delay model as well.…”