# Limits of Predictability?

A decade-old research article on modeling equatorial waves includes this introductory passage:

“Nonlinear aspects plays a major role in the understanding of fluid flows. The distinctive fact that in nonlinear problems cause and effect are not proportional opens up the possibility that a small variation in an input quantity causes a considerable change in the response of the system. Often this type of complication causes nonlinear problems to elude exact treatment. “

https://doi.org/10.1029/2012JC007879

From my experience if it is relatively easy to generate a fit to data via a nonlinear model then it also may be easy to diverge from the fit with a small structural perturbation, or to come up with an alternative fit with a different set of parameters. This makes it difficult to establish an iron-clad cross-validation.

This doesn’t mean we don’t keep trying. Applying the dLOD calibration approach to an applied forcing, we can model ENSO via the NINO34 climate index across the available data range (in YELLOW) in the figure below (parameters here)

The lower right box is a modulo-2π reduction of the tidal forcing as an input to the sinusoidal LTE modulation, using the decline rate (per month) as the divisor. Why this works so well per month in contrast to per year (where an annual cycle would make sense) is not clear. It is also fascinating in that this is a form of amplitude aliasing analogous to the frequency aliasing that also applies a modulo-2π folding reduction to the tidal periods less than the Nyquist monthly sampling criteria. There may be a time-amplitude duality or Lagrangian particle-relabeling in operation that has at its central core the trivial solutions of Navier-Stokes or Euler differential equations when all segments of forcing are flat or have a linear slope. Trivial in the sense that when a forcing is flat or has a 1st-order slope, the 2nd derivatives due to divergence in the differential equations vanish (quasi-static). This means that only the discontinuities, which occur concurrently with the annual ENSO predictability barrier, need to be treated carefully (the modulo-2π folding could be a topological Berry phase jump?). Yet, if these transitions are enhanced by metastable interface instabilities as during thermocline turn-over then the differential equation conditions could be transiently relaxed via a vanishing density difference. Much happens during a turn-over, but it doesn’t last long, perhaps indicating a geometric phase. MV Berry also discusses phase changes in the context of amphidromic tidal singularities here.

Suffice to say that the topological properties of reduced dimension volumes and at interfaces remain mysterious. The main takeaway is that a working NINO34-fitted ENSO model is produced, and if not here then somewhere else a machine-learning algorithm will discover it.

The key next step is to apply the same tidal forcing to an AMO model, taking care not to change the tidal factors enough to produce a highly sensitive nonlinear response in the LTE model. So we retain an excluded interval from training (in YELLOW below) and only adjust the LTE parameters for the region surrounding this zone during the fitting process (parameters here).

The cross-validation agreement is breathtakingly good in the excluded (out-of-band) training interval. There is zero cross-correlation between the NINO34 and AMO time-series to begin with so that this is likely revealing the true emergent characteristics of a tidally forced mechanism.

As usual all the introductory work is covered in Mathematical Geoenergy

A community peer-review contributed to a recent QBO article is here and PDF here. The same question applies to QBO as ENSO or AMO: is it possible to predict future behavior? Is the QBO model less sensitive to input since the nonlinear aspect is weaker?

Added several weeks later: This monograph PDF available “Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects”. Ignoring higher-order time derivatives is key to solving LTE.

Note the cite to Billy Kessler

## 3 thoughts on “Limits of Predictability?”

1. Gary Moore

The limits of predictability: a philosophical perspective from an above average Joe. (hubris intended). I cannot accurately predict anything until I understand everything.
I appreciate the way you continue to push the boundaries of understanding.
Thanks Pukite

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2. Superoscillations and the quantum potential, MV Berry

https://iopscience.iop.org/article/10.1088/1361-6404/abc5fd/pdf

This new ψ satisfies the free-space Helmholtz equation (2.6) (with k0 = 1). AB physics is incorporated by ψ being non-singlevalued if the quantum flux is non-integer: under continuation around a circuit of the flux line, ψ acquires a phase shift −2πα. In this representation, kK(r) is simply the phase gradient as in (2.7). Gauge invariance implies [19] that AB observables in quantum physics must be periodic in the flux α. This is true of the wave (4.6) but is not true for the AB wave (4.3); although (4.3) is singlevalued under continuation around the flux line, the phase of this wave shifts by 2πα as α increases by unity. However, the kinetic momentum kK(r) is periodic in α, and so it is in principle observable; it is a gauge-invariant vector, which was the reason for studying its streamlines [16, 18]. However, the wave equation (4.1) also describes the classical physics of waves in a vortex flow, with α representing the strength of the vortex, in a medium flowing much more slowly than the wave [20]. For such cases, the phase, although not periodic in α, is directly observable, and it was observed for water waves encountering a bathtub vortex [20] (for an extension of this idea to fast flows, see [21]).

A superoscillation is like a delta-function or step change

5. Concluding remarks
The connection reported here, between zeros of the quantum potential and the boundaries of superoscillatory regions, can be seen in a wider context that would be useful in graduatelevel physics teaching: the intertwining of historical threads linking old ideas with current research. As I have explained elsewhere [22, 23], there is a sense in which the germ of the Madelung–Bohm representation of waves, incorporating the quantum potential, can be traced back to Isaac Newton’s speculations about edge diffraction at the beginning of the 18th century.
Superoscillations are especially strong near phase singularities, which were first discovered by William Whewell in the early 19th century in the patterns of tides in the world’s oceans[22, 24, 25], and are currently being extensively explored and applied, especially in optics[26, 27]. The modern exploration of superoscillations was triggered by Yakir Aharonov’s ‘weak measurements’ in quantum physics [9, 28], although there were anticipations in radar theory in world war II [29, 30]

https://iopscience.iop.org/article/10.1088/1367-2630/ac2bd7/pdf
Semiclassical superoscillations: interference, evanescence, post-WKB, MV Berry

Sean Carroll podcast with M.M. https://www.preposterousuniverse.com/podcast/2019/09/23/65-michael-mann-on-why-our-climate-is-changing-and-how-we-know/

MM: But under certain conditions, when the north-south temperature variations are just right, the atmosphere behaves as a wave guide, there’s basically a wall at sort of the subtropical end and at the subpolar end and these Rossby waves are funneled through a wave guide with minimum loss of energy. And it turns out the mathematics to solve for the dispersion relation, it requires the use of the same WKB approximation that was used in quantum mechanics in the early 20th century, yeah.

Roadmap on superoscillations, Berry, Aharanov, others
https://iopscience.iop.org/article/10.1088/2040-8986/ab0191/meta

The second early context was phase singularities (=wave vortices, nodal points and lines, or wave dislocations), understood as topologically stable features of waves of all kinds [5]. Around a circuit of such a singular point P in the plane, the phase changes by 2π; so, close to P, the local phase gradient can be arbitrarily larger than any of the wavevectors in the Fourier superposition representing the wave. Therefore, all band-limited waves, in particular monochromatic ones, are superoscillatory near their phase singularities. This understanding emerged belatedly, in 2007; since then, research in superoscillations and phase singularities have merged. Dennis calculated [6] that superoscillations in waves are unexpectedly common: for random monochromatic light in the plane (e.g. speckle patterns), 1/3 of the area is superoscillatory, with similar fractions for these ‘natural superoscillations’ in more dimensions. A monochromatic superoscillatory wave is illustrated in figures 1 and 2.

interferometric phase

https://www.sciencedirect.com/science/article/pii/S0375960122006351
Generalizations of Berry phase and differentiation of purified state and thermal vacuum of mixed states

Instead of decomposing the density matrix to obtain a matrix-valued phase factor, a geometrical phase is directly assigned to a mixed state after parallel transport by an analogue of the optical process of the Mach-Zehnder interferometer. Hence, the geometrical phase generated in this way is often referred to as the interferometric phase. This approach can also be cast into a formalism by rewriting a mixed state as a purified state. The interferometric phase has been generalized to non-unitary processes [46], [47], [48], [49], [50], but the transformations are still on the system only.

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3. Implications from this paper — use NINO4 when possible?

A Nonlinear Cause for the seasonal Predictability Barrier of SST
anomaly in the 2 tropical Pacific

Click to access 8b1ed053f271d2322cadb0c682a8386f.pdf

The Sample Entropy increases when there is more self-similarity within a time-series

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