Mach-Zehnder

Pacific North American index (PNA)

/ Paul Pukite (@whut) / 1 Comment
Map showing PNA tri-pole pressure patterns for February 2016 (left) and February 2019 (right), highlighting variations in average 500-mb height in meters with labels for high and low pressure areas.

This index goes back to 1950. The claim is :

“ As a pattern of internal climate variability, the state of the PNA occasionally changes without a clear and identifiable cause. This reduces the predictability of the PNA and can complicate long-range seasonal weather forecasts. Predictability of the PNA is limited to roughly within 10 days“

YET, if we apply the universal latent manifold identified via LTE analysis, and invoke a winding scalogram with respect to the data, we find strong continuous ridges indicating a stationary non-autonomous modulation is occurring and is likely causing the PNA cycles. Shown below is the winding scalogram, with the bright yellow indicating integer windings of the fundamental winding at 0.2065, the strongest being 9×9\times of the fundamental (compare this to a strong 6×6\times for the Baltic MSL and SSL)

A left panel showing winding scalograms with mean spectrum and wind number indicators, alongside two right panels depicting 'Data' and 'Model' windings against forcing, represented in color gradients across different years.

The physical realization of this is the excellent cross-validated fit using the LTE mapping described in Mathematical Geoenergy, Chapter 121 shown below. The region from 1950 to 2001 was used for training, and the region after 2016 was used for predictive tuning, i.e. training produced a multiple regressed set of coefficients that projected into the future based on the manifold. After the training completed, the out-of-band interval shown by the dashed line below was used as a predictive test (the classic train/validate/test CV approach).

Top graph displaying a time series plot for site #pna, showing both the model and data over the years.
Line graph comparing the power spectrum of data and model over frequency, displaying log2 power on the y-axis and frequency in cycles/year on the x-axis. The graph includes a legend for Data, Model, AR1 red-noise floor (mean), and AR1 floor (p99).

The test interval does show a drop in correlation but that can be expected given the erratic nature of the PNA cycle. This is a cyclic pattern that has eluded climate scientists essentially forever, as it barely exceeds red noise in its power spectrum (right). Yet a handful of winding transformations (with one strong winding, see below) about the manifold, identified by the winding spectrogram, and fitted to the PNA data is able to map the pattern more than good enough to pass a crucial cross-validation test. This is not just predicting past 10 days (as the quote at the top of this post claims), but potentially for years.

Graph showing composite sinusoid comparison, plotting composite amplitude (a.u.) against terms over manifold range. The line represents the fitted sinusoid with a legend indicating 'PNA'.

How can that be? In other words, how can this have gone unnoticed over decades of analysis? To understand that fully, consider that this LTE formulation falls in the same category as Mach-Zehnder modulation2. The complexity of the transfer function in M-Z modulation is of sufficient nonlinearity that it has been used as a practical encryption scheme for analog signals3. The gist of this is that if you don’t know the decryption key, you don’t have a chance of decoding a M-Z encoded signal. But if you do have it, the decryption is concise and fast. There are infinitely many encryption keys possible, so it’s not practical to enumerate and try to discover their value in any systematic way. Applying NVIDIA-powered AI algorithms are of no help here either, as the search space is much too large to explore.

World map displaying ocean tidal patterns and amplitudes, with color gradients indicating varying tide heights measured in centimeters.
from Ocean and shelf tides – Coastal Wiki

But remember that in our context, nature can narrow down the encryption key possibilities — there aren’t infinitely many to choose from, but instead the ones that make sense are governed by physics and so can be formulated and straightforwardly evaluated. That’s why the LOD-calibrated (from a set of tidal factors) manifold worked — it was essentially the first one that I considered4 and have been tweaking for some time, long before the advent of convenient LLM tools. It also makes sense in the case of PNA — the geospatial dynamics of the tri-pole pressure pattern should remind one of amphidromic tidal patterns in the oceans (see right). These are due to the faster conventional diurnal tides, but they hint at how the long-period tidal factors impact the massive inertia of the subsurface waters — as that’s where the source and sink for tropospheric heat resides.

So that explains why this research may succeed — the answer was never really overlooked nor hidden in conventional signal processing artifacts. It was actually uncovered by an educated guess at what the decryption key was. All these ocean indices are falling like dominoes — NINO4, AMO, PDO, any MSL site, etc can be modelled with the LOD-calibrated manifold. Will show the Southern Annular Mode index next go round.

Footnotes

  1. Pukite P., D. Challou, and D.Coyne, Mathematical Geoenergy, (Wiley, 2018), https://agupubs.onlinelibrary.wiley.com/doi/book/10.1002/9781119434351 ↩︎
  2. Pukite, Paul. “Nonlinear Differential Equations with external forcing.” ICLR 2020 Workshop on Integration of Deep Neural Models and Differential Equations. openreview.net/pdf?id=XqOseg0L9Q ↩︎
  3. Takiguchi, K. and Ishihara, W., “Encryption of 300 GHz-band wireless signal using optical vector modulator-type encoder”, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 2026, vol. 13895, Art. no. 1389506. doi:10.1117/12.3078963;
    Vaidman, L. (1995). “Cryptographic scheme based on a Mach-Zehnder interferometer.” Physical Review Letters, 75(18), 3206. Physical Review Letters. ↩︎
  4. Pukite, P.R. “Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models.” AGU Fall Meeting, 2017. https://doi.org/10.1002/essoar.10500568.1 ↩︎

Is the equatorial LTE solution a 1-D gyre?

/ Paul Pukite (@whut) / 8 Comments

The analytical solution to Laplace’s Tidal Equation along the 1-D equatorial wave guide not only appears odd, but it acts odd, showing a Mach-Zehnder-like modulation which can be quite severe. It essentially boils down to a sinusoidal modulation of a forcing, belonging to a class of non-autonomous functions. The standing wave comes about from deriving a separable spatial component.

sin( f(t) ) sin(kx)

The underlying structure of the solution shouldn’t be surprising, since as with Mach-Zehnder, it’s fundamentally related to a path integral formulation known from mathematical physics. As derived via quantum mechanics (originally by Feynman), one temporally integrates an energy Hamiltonian over a path allowing the wave function to interfere with itself over all possible wavenumber (k) and spatial states (x).

Because of the imaginary value i in the exponential, the result is a sinusoidal modulation of some (potentially complicated) function. Of course, the collective behavior of the ocean is not a quantum mechanical result applied to fluid dynamics, yet the topology of the equatorial waveguide can drive it to appear as one, see the breakthrough paper “Topological Origin of Equatorial Waves” for a rationale. (The curvature of the spherical earth can also provide a sinusoidal basis due to a trigonometric projection of tidal forces, but this is rather weak — not expanding far beyond a first-order expansion in the Taylor’s series)

Moreover, the rather strong interference may have a physical interpretation beyond the derived mathematical interpretation. In the past, I have described the modulation as wave breaking, in that the maximum excursions of the inner function f(t) are folded non-linearly into itself via the limiting sinusoidal wrapper. This is shown in the figure below for progressively increasing modulation.

(click on image to expand)

In the figure above, I added an extra dimension (roughly implying a toroidal waveguide) which allows one to visualize the wave breaking, which otherwise would show as a progressively more rapid up-and-down oscillation in one dimension.

Perhaps coincidentally (or perhaps not) this kind of sinusoidal modulation also occurs in heuristic models of the double-gyre structure that often appears in fluid mechanics. In the excerpt below, note the sin(f(t)) formulation.

(from https://npg.copernicus.org/preprints/npg-2022-1/npg-2022-1.pdf )

The interesting characteristic of the structure lies in the more rapid cyclic variations near the edge of the gyre, which can be seen in an animation (Jupyter notebook code here).

Whether the equivalent of a double-gyre is occurring via the model of the LTE 1-D equatorial waveguide is not clear, but the evidence of double-gyre wavetrains (Lagrangian coherent structures,  Kelvin–Helmholtz instabilities), occurring along the equatorial Pacific is abundantly clear through the appearance of tropical instability (TIW) wavetrains.

These so-called coherent structures may be difficult to isolate for the time being, especially if they involve subtle interfaces such as thermocline boundaries :

“For instance, boundaries of coherent material eddies embracing and transporting volumes of ocean water with different salinity or temperature are notoriously difficult to identify.”

Mercator analysis does show higher levels of waveguide modulation, so perhaps this will be better discriminated over time (see figure below with the long wavelength ENSO dipole superimposed along with the faster TIW wavenumbers in dashed line, with the double-gyre pairing in green + dark purple), and something akin to a 1-D gyre structure will become a valid description of what’s happening along the thermocline. In other words, the wave-breaking modulation due to the LTE modulation is essentially the same as the vortex gyre mapped into a 1-D waveguide.