A paper from last week with high press visibility that makes claims to climate¹ applicability is titled: Topology shapes dynamics of higher-order networks

The higher-order Topological Kuramoto dynamics, defined in Eq. (1), entails one linear transformation of the signal induced by a boundary operator, a non-linear transformation due to the application of the sine function, concatenated by another linear transformation induced by another boundary operator. These dynamical transformations are also at the basis of simplicial neural architectures, especially when weighted boundary matrices are adopted.

This may be a significant unifying model as it could resolve the mystery of why neural nets can fit fluid dynamic behaviors effectively without deeper understanding. In concise terms, a weighted sine function acts as a nonlinear mixing term in a NN and serves as the non-linear transformation in the Kuramoto model².

As the paper is vague in exactly how the network of oscillators Kuramoto model precisely applies to climate dynamics, I will try to provide a mapping interpretation. First, I added the F(t) term in the equation, as it’s likely that ocean dynamics such as the SOI standing waves of ENSO respond most strongly during intervals of metastability — what are called the annual predictability barrier months (thought to be in the spring) — where the phase of the collective of oscillators can phase-lock with a wavenumber/winding number³ characteristic of the ocean basin and the driving value of F(t). The key to this is that ENSO transitions are thought to be driven by some external forcing, so it may follow more of an entrainment model than a resonance alone. So that entrainment would better describe a periodically forced Kuramoto model, see this paper.

Alas, the above may sound like a contrived premise that wouldn’t necessarily apply to continuous fluid dynamics. Yet a low-dimension PDE based on Laplace’s Tidal Equations (LTE) can be derived that allows for a similar formulation, thus mimicking a forced Kuramoto model. The LTE simplification reduces to a similar application of a sine function, sin(k F(t)) in this case, which leads to a wrapping or winding in transformed amplitude as the scaling factor k is increased, as shown in the figure below. This is referred to as an LTE modulation.

As in conventional standing modes, there can be multiple k factors, but each k is associated with a spatial wavenumber mode — with the lowest k associated with the standing wave of greatest spatial extent. The following figure is borrowed from Chapter 12 of Mathematical Geoenergy:

The important caveat is what happens during metastable transitions at the annual steps in forcing. Can these transients be ignored? Or perhaps the transitions are assigned to topological phase slips⁴. As a starting point, we can apply an ansatz whereby fast step transitions with gradually sloped plateaus as a forcing function provides a sufficient premise as a working model — see the figure below. The PDE closure rationale for this is that second derivative and higher orders vanish as long as the slopes are sufficiently smooth on the forcing plateau.

A recent paper “Turbulence in the tropical stratosphere, equatorial Kelvin waves, and the quasi-biennial oscillation” — which applies to the toroidal ENSO-equivalent of the stratosphere, the QBO — may be applicable in further the understanding. The QBO nominally shows a similar step/plateau cyclical time-series profile but with less phase wrapping required in the model forcing response. The insight to be gained here is that the fast metastable step transitions is accommodated by turbulence that appears to be contained enough so that the larger energy cascade, i.e. the global wave reversal, remains unimpeded.

“We show that the turbulent fraction of the tropical stratosphere is strongly modulated by the quasi-biennial oscillation (QBO). Turbulence is enhanced during the QBO phase shifts, and the atmosphere is most turbulent right before the QBO phase switches from negative to positive, where turbulent instabilities typically occur within specific phases of Kelvin waves. Turbulence is less common when the QBO phase is well established, and the atmosphere is least turbulent during the negative phase of the QBO. The turbulent fraction of the equatorial lower stratosphere varies over a factor of ten depending on QBO phase. This relationship provides a robust observational constraint on the multiscale dynamics within this region”

And note the suppression of turbulence on the plateau where the phase of QBO is locked in. Thus, the ENSO and QBO have potentially a similar physical fluid dynamics foundation in spite of a vast difference in fluid characteristics, i.e. liquid vs gas.

---

As stated in past blog posts, the substantiation of the LTE model will be accomplished through various cross-validation techniques, such as temporal cross-validation and domain cross-validation.

Temporal Cross-Validation

Consider NINO4, where we apply an LTE modulation and train on 3 different intervals: 1880-1925, 1925-1980, and 1980-2024, and check for stationarity of parameters across the orthogonal span. In each case, the yellow shaded areas are test/validation intervals.

In each interval the LTE modulation trains (with identical forcing shown in the middle) to approximately the same set of values, which can be seen in the lower right inset in each panel, with values below. Perhaps only the high frequency LTE modulation in the 1925-1980 interval differs.

In terms of cross-validation, this is impressive as it does not show artifacts due to overfitting.

Domain Cross-Validation

A recent paper claims that the Atlantic Ocean AMO and the Indian Ocean IOD East may show a relationship: Interdecadal modulation of Ningaloo Niño/Niña strength in the Southeast Indian Ocean by the Atlantic Multidecadal Oscillation

The idea here is to check for similarity in the common mode forcing between AMO and IOD-East using a small window between 1940-1945 as a temporal cross-validation check.

The annual impulsed tidal forcing profiles are identical, but the LTE modulations differ

The training parameters are here : https://gist.github.com/pukpr/87223b66cdeaf9e5bff150d1d79663c6?permalink_comment_id=5464680#gistcomment-5464680

Summary

The Nature Physics paper implies a unification that extends to high-order network models, topological insulators with winding numbers, neural networks, and fluid dynamics and thus climate cycles. I think that’s what is generating some of the hype, yet curiously a paper from 2018 “Topological Origin of Equatorial Waves” is not mentioned in the context of climate. I mentioned that paper in Mathematical Energy since it provided insight into what may be happening with respect to wave dynamics. The field of topological insulators spreads across many domains of condensed matter and solid-state physics, so that it’s highly likely that all the thinking has yet to be unified.

Footnotes

1. Climate connection may be more in terms of teleconnection, as teleconnections are the network graph edges that may be part of the global climate model. For example, teleconnections across different ocean basins. Yet, a GCM is also a network mesh or tesselation of connected volumes within an ocean basin, used in solving 3D fluid dynamics problems, so a similar synchronization or entrainment can happen on this smaller scale. ↩︎
2. The book Nonlinear Dynamics and Chaos by Steven Strogatz was revised with Chapter 13 describing the Kuramoto model. “The Kuramoto model is an icon of nonlinear dynamics. Introduced in 1975 by the Japanese physicist Yoshiki Kuramoto, his elegant model is one of the rare examples of a high-dimensional nonlinear system that can be solved by elementary means.” ↩︎
3. A winding number (also known as Chern number) is defined for topological insulators as a phase advancement (also known as Berry phase) with respect to 2π, whereas a wavenumber does the same for Fourier series. See https://physics.stackexchange.com/questions/391389/relation-of-berry-phase-and-winding-number ↩︎
4. In the context of the Kuramoto model, phase slips refer to events where the phase difference between two oscillators increases by 2π (or a multiple of 2π) over time. This phenomenon occurs when the coupling between oscillators is not strong enough to maintain synchronization, leading to a temporary or permanent loss of phase locking. Perhaps similar to he concept of a breaking wave, as both phenomena involve a loss of coherence — phase slips lead to a loss of phase coherence in coupled oscillators, while breaking waves lead to a loss of spatial coherence in the wave profile. ↩︎