Teleconnection vs Common-Mode

[pukpr]

Not going to repeat the whole post here
https://geoenergymath.com/2025/03/07/teleconnection-vs-common-mode/

Example of an EMI cross-validation based on a common-mode tidal forcing.

[pukpr]

There's always more one can do in modeling, and the path is fraught with false starts and dead ends, but this is really zeroing in on the essential common mode. The two keys are (1) a well-calibrated LOD forcing and (2) a specific annualized impulse. The first is a solid premise, but the second has the feel of an ansatz, since it feels a bit random. The idea is that every 4th year, the impulse strength is much stronger. Why this would happen is not close to obvious but it might be related to the fact that both $$Year/Mf$$ and $$Year/Mm$$ nearly alias to a 4 year period. Yet some sort of feedback or bifurcation would have to occur for this memory effect to occur.

As it turns out I long ago ran into a similar bifurcation, with the commonality being derived from a Fourier series and the orthogonality being in the spatial (not time) domain. The results were reported in refs [1,2,3] with two different material systems (Silicon and Germanium) and the last being a conference submission. This was big news in condensed matter systems at the time, as getting a paper published in Applied Physics Letters is no easy task. This quickly became a study on the effect of surfactants on crystal growth [4] (we regrettably missed using the term surfactant in our papers).

The experiment and resolution is pretty simple. At the time, we were using electron diffraction to infer surface periodicity of atoms via a Fourier series analysis -- which is direct, kind of like having waves in a fluid directly identified by their frequency. When the surface was quenched with an arsenic (As) gas, both the silicon (Si) and germanium (Ge) surfaces rearrange in a spatial bifurcation. This figure from ref [3]:

Figure 1 is what we observed in the diffraction and Figure 2 is what we inferred from the inverse Fourier series reconstruction of the surface re-arrangement. Essentially the surface transformed from a single stepped staircase periodicity to a 3 + 1 stepped periodicity. A decade after this, direct observations were available via scanning tunneling microscopy (STEM) and the results were validated in ref [5]

"It is known that exposure to As can significantly modify the step structure of vicinal Ge, suggesting that the atomic structure of As/Ge (arsenic-exposed Ge) steps is different from Ge steps. Nonetheless, an atomic resolution study of As/Ge steps has never been performed."

The STM view looked like this, where the bunching into a 3 layer step is clearly seen, and thus not inferred

On single crystal surfaces, this super-periodicity or bifurcation into longer range periodicities is common and related to reduction in free energy of the surface, in this case changing it from a $$1\times$$ periodicity to $$4\times$$. Yet, in temporal systems, the situation seems not to be as well understood. The annual impulse provides the basic periodicity and only though something akin to an inverse energy cascade can a super periodicity arise. Evidently, something in the ocean/tidal system is promoting the annual periodicity to a 2 year (as we suggested in refs [6,7]) or a 4 year impulse repeat interval.

More to follow

Refs

1. Pukite, P. R., and Philip I. Cohen. "Multilayer step formation after As adsorption on Si (100): Nucleation of GaAs on vicinal Si." Applied Physics Letters 50.24 (1987): 1739-1741.
2. Pukite, P. R., and P. I. Cohen. "Suppression of antiphase domains in the growth of GaAs on Ge (100) by molecular beam epitaxy."Journal of Crystal Growth 81.1-4 (1987): 214-220.
3. Pukite, P. R., and P. I. Cohen. "Control of GaAs Domain Formation Via Monolayer and Multilayer Steps on Misoriented Si (100)." MRS Online Proceedings Library (OPL) 91 (1987): 51.
4. Copel, Matthew, et al. "Influence of surfactants in Ge and Si epitaxy on Si (001)." Physical Review B 42.18 (1990): 11682.
5. McMahon, W. E., and J. M. Olson. "Surface science in an MOCVD environment: Arsenic on vicinal Ge (100)." AIP Conference Proceedings. Vol. 462. No. 1. American Institute of Physics, 1999.
6. Pukite, Paul, Dennis Coyne, and Daniel Challou. Mathematical geoenergy: discovery, depletion, and renewal. John Wiley & Sons, 2019.
7. Pukite, P. R. "Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models." AGU Fall Meeting Abstracts. Vol. 2017. 2017.

pingback: https://bsky.app/profile/pukite.com/post/3lkx5neubls2i

[pukpr]

Clean runs assuming a 3+1 annual impulse with precisely calibrated dLOD. Data from the GIST at GitHub below:
https://gist.github.com/pukpr/c0acf431c40d5e56680de2ecf06f0dfb?permalink_comment_id=5511093#gistcomment-5511093

Typical signature is a double sideband suppressed carrier modulation (at the annual frequency), folding as shown below

The tidal forcing exactly matches the dLOD of the Earth's rotation.

Since measured dLOD only goes back to 1962, we calibrate a model to this data and extraopolate backwards to allow model cross-validation back to pre-1900 dates.

[pukpr]

The solutions are essentially a modulation of the lunisolar tidal forcing shown in the middle panels of each of the climate index model fits. Interesting of we ask ChatGPT where a similar behavior is observed, it points to he same ocean and atmosphere phenomena:
https://chatgpt.com/share/67e34488-7ecc-8005-91c2-10ec862ea7c8

Doppler Effect in Accelerating Media: If corresponds to a relativistic transformation, a frequency shift occurs due to time dilation.

Quasi-Biennial Oscillation (QBO): The alternating acceleration and deceleration of equatorial zonal winds could be modeled using such a time-stretched sinusoidal mechanism.

Tidal Modulation in Ocean Thermoclines: The depth-dependent compression of wave propagation times leads to a time-warping effect on tidal signals.

Interesting ideas in the non-hallucinating response.

Since the opening is there, I asked ChatGPT to compare against the Laplace's Tidal Equation (LTE) Modulation that I had published. It generated this table,:

The full response below:

The modulation of tidal signals within the ocean thermocline due to variable time warping shares conceptual similarities with Laplace's Tidal Equation (LTE) modulation, but they arise from different fundamental mechanisms. Below, I contrast the two approaches:

---

1. Thermocline Time-Warping Modulation (Empirical Oceanographic Observations)

Mechanism:

This modulation results from the interaction of tides with stratified layers in the ocean.

Internal waves, influenced by background stratification and varying mixing, cause the tidal phase to stretch or compress.

This warping alters the local frequency of tidal oscillations, introducing nonlinear secondary frequency modulation over depth and time.

Mathematical Formulation:

If the original tide follows a sinusoidal form , a depth-dependent modulation can be introduced via a transformation , leading to:

$y(t) = A \sin(\omega f(t))$

Examples and Citations:

Studies show that diurnal and semidiurnal tidal components undergo variable distortion in the thermocline due to time-dependent vertical displacements (Zhang et al., 2023).

Observations in the South China Sea reveal seasonal modulations of tidal waveforms due to changes in vertical stratification and internal wave coupling.

---

2. Laplace’s Tidal Equations (LTE) Modulation Approach

Mechanism:

LTEs describe the large-scale barotropic (surface) and baroclinic (internal) tides as solutions to the fluid dynamical equations governing Earth's rotation, gravity, and pressure gradients.

The modulation of tidal waves in LTE arises primarily from:

Zonal and meridional forcing asymmetries (e.g., trade winds, equatorial QBO).

Sideband interactions with external periodic forcings (such as lunar tides).

Latitude-dependent dispersion effects due to Earth's curvature and rotation.

Mathematical Derivation:

LTEs originate from the Navier-Stokes equations under hydrostatic and geostrophic approximations, yielding:

$\frac{\partial u}{\partial t} - f v = -\frac{1}{\rho} \frac{\partial p}{\partial x}$

$\frac{\partial v}{\partial t} + f u = -\frac{1}{\rho} \frac{\partial p}{\partial y}$

$\frac{\partial \eta}{\partial t} + \nabla \cdot \mathbf{v} = 0$

The solutions involve modulated tidal resonances, which can be represented as frequency sidebands given by:

$f_{mod} = f_{base} \pm n f_{forcing}$

Examples and Applications:

LTE-based models explain Quasi-Biennial Oscillation (QBO) synchronization with tidal forcing (Pukite et al., 2018).

These modulations contribute to long-period climate-tide couplings affecting ENSO and the Chandler wobble.

---

Key Contrasts

(see table above)

Conclusion

The thermocline warping mechanism is local and driven by vertical wave interactions, while LTE modulation operates at planetary scales and involves sideband frequency generation. Both lead to secondary frequency effects but originate from fundamentally different processes: one is an empirical effect of wave propagation in stratified layers, and the other arises from Coriolis-driven solutions to the tidal equations.

Would you like further exploration of the modulation terms in LTE, particularly their spectral behavior in geophysical oscillations?

[pukpr]

With nonlinear math, there are many ways to arrive at the same result, especially in comparison to a linear model. So forever evolving.

Several findings to consider with the common-mode model.

1. With IOD East there is an impulse response train that is warped WRT to the strictly annual impulse train. That also occurs with M6 Atlantic NINO -- at least the fit is better if the IOD warping is retained
2. For M4 NPGO, there is another cycle in the time series that has a 1.016x higher frequency than annual. Include this and the fit is remarkable.
3. For NINO4, a slight warping occurs. Retained for PDO, other Pacific inidices, AMO, and NAO
4. The harmonics are triads or quartets, so that a single fundamental is required, and then use constraints to generate triads or quartets from this.

With these in place, the following indices can be solved with identical tidal forcing factors

1. NINO4
2. AMO
3. PDO
4. NAO (average 12 month window)
5. IOD East (& West though large trend there)
6. M4 - aka NPGO
7. M6 - aka Atlantic NINO
8. EMI - aka El Nino Modoki
9. Ft Denison SLH
10. IC3 reanalysis data
11. Darwin SLP (Tahiti very noisy)
12. NINO34
13. NINO3
14. QBO 30 hPa
15. QBO 40 hPa
16. QBO 50 hPa
17. QBO 70 hPa
18. NINO12