Surrounding this work are studies where quantum and computational methods tame complex systems, from balancing on a quantum tightrope to neural-network cartographers that recover invisible laws. Nearby sit works on quantum logic in the brain and fractional vortices in superfluidity. Our article is about something else: how to extract a simple delay equation from a spatially distributed ocean model while preserving the physics. This is not a quantum algorithm or a neural network, but an analytic reduction: we reduce an infinite-dimensional system to a finite one, but with memory. This approach complements the neighbors: they often seek numerical or trainable approximations, whereas here we have an exact symbolic simplification rooted in the wave nature of the phenomenon.
The starting model is a system of partial differential equations for temperature in the two upper layers of the North Atlantic, derived by Sévellec and Huck. In it, Rossby thermal waves travel westward at speeds l+ and l−, reflecting off boundaries with a sign change. The authors first discretized the model on a grid of N points, obtaining a system of ordinary differential equations. They then applied the Mori–Zwanzig formalism: they chose the temperatures at the western boundary (T1 and T2 at x=0) as the resolved variables and projected all other variables to zero. The orthogonal dynamics gave noise and memory. After taking the limit N→∞ (or ϵ=1/N→0), a delay equation emerged: the current boundary temperature is expressed in terms of its past values with delays τ+=1/l+≈2.8 years and τ−=1/l−≈26.7 years. In parallel, the same equation was derived using the method of characteristics, solving the wave equations along propagation lines.
The final model is a system of two coupled delay equations, where the right-hand side contains terms with T1(t−τ+) and T2(t−τ−). Numerically, it reproduces two main periods: about 53 years (twice τ−) and about 5–9 years (twice τ+). Spectral analysis of trajectories shows peaks at frequencies corresponding to 2τ−, 2/3τ−, and 2τ+. Sensitivity to basin width: as latitude changes from 10°N to 70°N, the period 2τ− varies from 87 to 30 years, and 2/3τ− from 29 to 10 years. This agrees with observations: 50–70 years for surface temperature and 20–30 years for subsurface temperature. The error of the smoothing term ϵdT/dt is proportional to the discretization error of the original model: at N=2000 the maximum error amplitude is about 2·10⁻³ °C.
Unlike neighboring works, where simplification is achieved numerically or through learning, here an exact analytic delay equation is obtained, directly linked to wave physics. This is the first demonstration that the Mori–Zwanzig formalism yields a delay model for the AMO, not just for El Niño. It shows that the delay arises from the finite travel time of Rossby thermal waves across the basin. This result supports the hypothesis that Rossby thermal waves are the mechanism behind the AMO and provides formal justification for using conceptual delay models in climatology.
The method can be extended to other climate phenomena with wave dynamics, such as the Pacific Decadal Oscillation or ocean–atmosphere coupling. For nonlinear models, a more accurate approximation of the orthogonal dynamics is needed: the paper shows that the pseudo-orthogonal approximation gives significant error for nonlinear El Niño. The next step is to find a better way to approximate orthogonal dynamics to derive reliable nonlinear delay models. It is also interesting to investigate how adding a background meridional circulation changes the damping of the high-frequency mode.
The work influences physical oceanography and climate modeling by offering a compact way to describe decadal variability. It is also useful for the theory of delay dynamical systems, providing a new example of hierarchical delays.
Test the model on more realistic configurations with background circulation and inhomogeneous parameters. Develop an approximation of orthogonal dynamics for nonlinear systems.
For physical oceanography, a key unresolved problem is the exact mechanism of the Atlantic Multidecadal Oscillation and its predictability. This work links the AMO to specific wave processes and provides a tool for testing hypotheses about the role of circulation. The question remains open as to how well a model with constant coefficients and no background circulation reflects the real ocean, where parameters vary in space and time.
The 26.7-year delay is the time it takes a Rossby thermal wave to cross the North Atlantic from east to west. If the basin were half as wide, the AMO period would shrink to about 30 years, matching observations at high latitudes.