A Digital Twin of ENSO

The idea of a digital twin is relatively new in terms of coinage of terms, but the essential idea has been around for decades. In the past, a digital twin was referred to as a virtual simulation of a specific system, encoded via a programming language. In the case of a system that was previously built, the virtual simulation emulated all the behaviors and characteristics of that system, only operated on a computer, with any necessary interactive controls and displays provided on a console, either real or virtual. A widely known example of a VS is that of a flight simulator, which in historical terms was the industrial forerunner to today’s virtual reality. A virtual simulation could also be used during the design of the system, with the finished digital twin providing a blueprint for the actual synthesis of the end-product. This approach has also been practiced for decades, both in the electronics industry via logic synthesis of integrated circuits from a hardware description language and with physical products via 3D printing from CAD models.

So there is nothing groundbreaking in announcing a workshop on Digital Twins in Atmospheric, Climate, and Sustainability Science, apart from the possibility that the virtual models and simulations would be accurate and high-fidelity representation of the systems under study. Therein lies the rub: unless any of these researchers claim that they can duplicate known climate behaviors precisely, they likely should not be called a digital twin, as the simulation results may not agree against the measured climate indices — except on a probabilistic basis. And to even get to that point requires significant computational resources. I would offer that the only reliable digital twins are in tidal simulations for the ocean and sunrise and sunset times for the atmosphere** — for both of these behaviors, highly precise timetables can be created with minimal computing time. The reason for this is that the eternal forcings are known and the results are deterministic and suitable for calibration.

This points to the premise that if we have any chance of creating a digital twin for some climate behavior, it will have to derive from some known external forcing combined with a predictable physics-based response. So that anything chaotic or random in either the forcing or (especially) response will mean that a digital twin would have little of the fidelity of the real system. For unpredictable forcings, there may be value in generating potential scenarios or projections, but consider that doing this for tidal projections is only of value for hypothetical situations, since the forcing is actually highly predictable and will never deviate from the norm.

Now, consider this passage hyping “dazzling success”, from a book [1] promoted in a recent tweet:

where the claim is that the null hypothesis is that “the atmosphere forces the ocean but with little feedback from the ocean”. This means that according to the consensus that nothing is predictable, since the atmosphere certainly isn’t, and any link between ocean, atmosphere, and land is subject to causal ambiguity.

But with all that said, IMO there is hope in making progress towards a digital twin of ENSO.

Start with as a null hypothesis that the annual and lunar cycle acts as the null hypothesis for forcing. By suggesting this is as null, any competing hypothesis will need to do better at predicting behavior. This null hypothesis certainly works as a causal mechanism for tides, so it seems like a valid first-order premise for the ocean’s thermocline.

The tidal model for ENSO is described elsewhere on this blog and in Mathematical Geoenergy, so the description won’t be repeated in this post. The starting point is to get an accurate representation of tidal forces, in this case calibrated from length-of-day (LOD) measurements since 1962 [2].

This set of tidal factors was generated from a multiple linear regression best fit (in BLUE) and that used in the ENSO model in RED. Any difference is due to the slight perturbation of the factors during the fitting process.

Next, to get as narrow an ENSO representation as possible, the region 140W-150W longitude +/-10 latitude was selected, directly north of the Tahiti SOI node shown below. The 20th Century Reanalysis database generate the raw time-series, which included annual harmonics.

Since this ENSO time-series has a strong annual component, an annual/semi-annual factor was fit alongside the LTE modulation of the semi-annual impulsed tidal factors (middle panel below). For an aggressive fit from scratch, assuming a fundamental LTE modulation and harmonics thereof, and relatively narrow training intervals (similar to what one would do with tidal analysis), this results:

The out-of-bound evaluation/test intervals of the model shows striking agreement to the ENSO time-series, with the highlighted regions in bright yellow showing the few cases of uncorrelated mapping. This is also clear from the inset showing windowed correlation across the entire time series. This cross-validating agreement is essentially statistically impossible to achieve unless long-range order, i.e. temporal coherence, exists in the behavior. And this order is achieved by a well-characterized stationary forcing that is responded to by an inferred nonlinear fluid dynamics solution to Laplace’s tidal equations (see Chapter 12). The minimal degrees-of-freedom in the solution arise during the fitting process and presumably define the standing wave modes of the ENSO dipole along with the higher-mode harmonics that map to the oceanic basin boundary conditions (all modes are harmonics of the principal or fundamental standing-wave mode).

These alternative “dazzling” results imply that ENSO is a stationary process, forced by external annual and tidal factors, similar to understanding of conventional ocean tides but on a longer time-scale.

Now, reflecting back on the theme of this post, this cross-validation results suggests that this LTE model is a suitable digital twin of ENSO behavior. It is not too complicated (taking minutes to fit on a standard PC), uses calibrated forcing data, and deterministically cross-validates to regions not tainted by the training interval. Someone else will need to generate a better model to replace a tidal model as the null hypothesis for ENSO behavior.

** This is stretching it as an atmospheric behavior, yet I can’t think of a case of anything in the atmosphere that is highly predictable, apart from consistent seasonal swings along mean values.


  1. S-P. Xie, Coupled Atmosphere-Ocean Dynamics: From El Nino to Climate Change. (Elsevier, 2022) https://doi.org/10.1016/C2021-0-02107-6
  2. Yu, N., Liu, H., Chen, G., Chen, W., Ray, J., Wen, H., & Chao, N.. Analysis of relationships between ENSO events and atmospheric angular momentum variations. Earth and Space Science, 8 (2021), e2021EA002030. https://doi.org/10.1029/2021EA002030
    (for recent research on the relationship between ENSO/SOI, LOD, tides, and atmospheric angular momentum)

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