I’m looking at side-band variants of the lunisolar orbital forcing because that’s where the data is empirically taking us. I had originally proposed solving Laplace’s Tidal Equations (LTE) using a novel analytical derivation published several years ago (see Mathematical Geoenergy, Wiley/AG, 2019). The takeaway from the math results — given that LTEs form the primitive basis of the GCM-specific shallow-water approximation to oceanic fluid dynamics — was that my solution involved a specific type of non-linear modulation or amplification of the input tidal. However, this isn’t the typical diurnal/semi-diurnal tidal forcing, but because of the slower inertial response of the ocean volume, the targeted tidal cycles are the longer period monthly and annual. Moreover, as very few climate scientists are proficient at signal processing and all the details of aliasing and side-bands, this is an aspect that has remained hidden (again thank Richard Lindzen for opening the book on tidal influences and then slamming it shut for decades).
With the recent total solar eclipse, it revived lots of thought of Earth’s ecliptic plane. In terms of forcing, having the Moon temporarily in the ecliptic plane and also blocking the sun is not only a rare and (to some people) an exciting event, it’s also an extreme regime wrt to the Earth as the combined reinforcement is maximized.
In fact this is not just any tidal forcing — rather it’s in the class of tidal forcing that has been overlooked over time in preference to the conventional diurnal tides. As many of those that tracked the eclipse as it traced a path from Texas to Nova Scotia, they may have noted that the moon covers lots of ground in a day. But that’s mainly because of the earth’s rotation. To remove that rotation and isolate the mean orbital path is tricky. And that’s the time-span duration where long-period tidal effects and inertial motion can build up and show extremes in sea-level change. Consider the 4.53 year extreme tidal cycle observed at the Bay of Fundy (see Desplanque et al) located in Nova Scotia. This is predicted if the long-period lunar perigee anomaly (27.554 days and the 8.85 absidal precessional return cycle) amplifies the long period lunar ecliptic nodal cycle, as every 9.3 years the lunar path intersects the ecliptic plane, one ascending and the other descending as the moon’s gravitational pull directly aligns with the sun’s. The predicted frequencies are 1/8.85 ± 2/18.6 = 1/4.53 & 1/182, the latter identified by Keeling in 2000. The other oft-mentioned tidal extreme is at 18.6 years, which is identified as the other long period extreme at the Bay of Fundy by Desplanque, and that was also identified by NASA as an extreme nuisance tide via a press release and a spate of “Moon wobble” news articles 3 years ago.
What I find troubling is that I can’t find a scholarly citation where the 4.53 year extreme tidal cycle is explained in this way. It’s only reported as an empirical observation by Desplanque in several articles studying the Bay of Fundy tides.
In formal mathematical terms of geometry/topology/homotopy/homology, let’s try proving that a wavenumber=0 cycle of east/west direction inside an equatorial toroidal-shaped waveguide, can only be forced by the Z-component of a (x,y,z) vector where x,y lies in the equatorial plane.
To address this question, let’s dissect the components involved and prove within the constraints of geometry, topology, homotopy, and homology, focusing on valid mathematical principles.
Based on the previous post on applying Dynamic Time Warping as a metric for LTE modeling of oceanic indices, it makes sense to apply the metric to the QBO model of atmospheric winds. A characteristic of QBO data is the sharp transitions of wind reversals. As described previously, DTW allows a fit to adjust the alignment between model and data without incurring a potential over-fitting penalty that a conventional correlation coefficient will often lead to.
Useful to note that the majority of the posts written for this blog are in support of the mathematical analysis formulated in Mathematical Geoenergy (Wiley/AGU, 2018). As both new data becomes available and new techniques for model fitting & parameter estimation — aka inverse modeling (predominantly from the machine learning community) — are suggested, an iterative process of validation, fueled by the latest advancements, ensures that the GeoEnergyMath models remain robust and accurately reflective of the underlying observed behaviors. This of course should be done in conjunction with submitting significant findings to the research literature pipeline. However, as publication is pricey, my goal is to make the cross-validation so obvious that I can get an invitation for a review paper — with submission costs waived. Perhaps this post will be the deal-maker — certainly not the deal-breaker, but you can be the judge.
Similar to the LTE modulation in Chapter 12, modulation of a periodic impulse can lead to complex waveforms.
Find behaviors that are periodic impulse trains that are modulated by another periodic process, leading to another waveform with strong subbands.
ChatGPT: The description you provided suggests a phenomenon where a periodic impulse train, a series of regular, sharp signals, is modulated by another periodic process, leading to a composite waveform characterized by pronounced subbands. This can be observed in various fields, including signal processing, telecommunications, and even in natural systems. Here are a few examples where such behaviors might be observed:
Frequency Modulation (FM) in Telecommunications: In FM, the frequency of a carrier wave is varied in accordance with the amplitude of a modulating signal. If the modulating signal is a periodic impulse train, the resulting FM signal will exhibit a spectrum with sidebands at multiples of the modulating frequency. These sidebands can be viewed as subbands in the spectrum, resulting from the modulation process.
Amplitude Modulation (AM) in Radio Broadcasting: In AM, the amplitude of a carrier signal is varied in proportion to that of a modulating signal, which could be a periodic impulse train. This modulation results in a waveform where the carrier frequency is accompanied by sidebands (subbands) at frequencies equal to the sum and difference of the carrier frequency and the modulating frequencies.
Biological Rhythms: In biological systems, certain rhythms or signals can modulate others. For example, the circadian rhythm (a roughly 24-hour cycle) can modulate other physiological processes, leading to patterns that have periodic qualities modulated by the circadian rhythm, resulting in complex waveform behaviors with distinguishable sub-patterns or subbands.
Vibration Analysis in Mechanical Systems: In mechanical systems, a periodic impulse (such as that from a rotating imbalance) can modulate the vibration signatures of other components of the system. This can lead to complex vibration signatures that contain subbands corresponding to the modulating frequencies.
Electroencephalography (EEG) in Neuroscience: In EEG signals, certain brain wave patterns can be modulated by underlying rhythmic processes, such as the sleep-wake cycle. This can result in EEG waveforms where the primary rhythms are modulated, leading to complex patterns with distinct frequency bands.
Optical Communications: In optical fiber communications, a technique known as Wavelength Division Multiplexing (WDM) can be seen as modulating multiple light signals (each a periodic process) onto a single fiber. The combined signal exhibits a spectrum with distinct subbands, each corresponding to one of the modulated signals.
These examples illustrate the diversity of systems and contexts in which periodic impulse trains modulated by another periodic process can lead to complex waveforms with pronounced subbands, each reflecting the interplay between the modulating signals and the base periodic impulses.
The illustration above showcases vibration analysis in mechanical systems, depicting a complex waveform resulting from the modulation of a periodic impulse train by another periodic process. It includes a spectral analysis graph, highlighting the pronounced subbands within the waveform, and a representation of a mechanical system, such as a rotating shaft with an imbalance, along with the corresponding vibration signal being analyzed. This image is designed to be detailed and technical, suitable for educational purposes in the field of engineering.
More ChatGPT: The spectral domain representation below illustrates the result of a periodic impulse train, such as that from a rotating imbalance, being modulated by another vibrating periodic process. In this simulation, the periodic impulse train has a frequency of 50 Hz, and it is modulated by a lower frequency signal at 2 Hz.
The Fourier Transform of the modulated signal reveals pronounced subbands around the base frequency of the impulse train and its harmonics. These subbands are a direct consequence of the modulation process, where the amplitude of the impulse train is modulated by the lower frequency signal, leading to the creation of sidebands in the frequency domain. This graphical representation captures the complex interplay between the periodic impulse train and the modulating signal, resulting in a composite waveform characterized by these pronounced subbands.
This is a good recent citation : He, G., Ding, K., Li, W. and Li, Y. (2017), Frequency response model and mechanism for wind turbine planetary gear train vibration analysis. IET Renewable Power Generation, 11: 425-432. https://doi.org/10.1049/iet-rpg.2016.0236
Frequency components of complicated asymmetric modulation sidebands, existing in the vibration of the healthy planetary gear train, are prone to be erratically diagnosed as fault characteristics, which leads to difficulties in fault diagnosis of wind turbine planetary gearbox. The factors affecting the modulation sideband, i.e. the periodical time-varying transmission path and meshing force direction, are analysed. Considering both the meshing vibrations of the planet–ring and planet–sun gear pairs, a mathematical model was developed to analyse the planetary gear train’s vibration response. Simulation and experiments were conducted, and the mechanism of vibration modulation sidebands was revealed. The modulation sideband is not caused by the meshing vibration itself, but by the testing method that sensors are fixed on the ring gear or gearbox casing. The frequency components and amplitudes of the sidebands are determined by the tooth number of the ring gear and sun gear, the number of planet gears and their initial assembling phases. The asymmetric modulation sideband is mainly caused by the phase difference of the initial planets’ assembling phase.
Abstract: Frequency response model and mechanism for wind turbine planetary gear train vibration analysis
These are a set of 6 EOFs that describe the global SST in terms of a set of orthogonal time-series — essentially non-overlapping, each having a cross-correlation of ~0.0 with the others, like a sine/cosine pair, but in both spatial and temporal dimensions.
The cross-validation described earlier was rather limited. Here an attempt is made to fit to an interval of the Darwin time-series and see how well it matches to a longer out-of-band validation interval. Very few degrees of freedom are involved in this procedure as the selection of tidal factors is constrained by a simultaneous LOD calibration. The variation from this reference is slight, correlation remaining around 0.999 to the LOD cal, but necessary to apply as the ENSO model appears highly structurally sensitive to coherence of the tidal signal over the 150 year time span of the data to be modeled.
A typical LOD calibration (click on image to enlarge)
Cross-validation shown in the top panel below, based on an training time interval ranging from the start of the Darwin data collection in 1870 up to 1980. The middle panel is the forcing input, from which the non-linear Laplace’s Tidal Equation (LTE) modulation is applied to a semi-annual impulse integration of the tidal signal. The procedure is straightforward — whatever modulation is applied to the training interval to optimize the fit, the same modulation is applied blindly to the excluded validation interval.
The validation on the 1980+ out-of-band interval is far from perfect, yet well-beyond being highly significant. The primary sinusoidal modulation is nominally set to the reciprocal of the slope (r) of top-edge of the sawtooth forcing [1] — this fundamental and the harmonics of that modulation satisfy LTE and provide a mechanism for a semi-annual level shift.
The plotted lower right modulation appears as noise, but when demodulated as in modulo r, the periodic order is revealed as shown below:
The harmonic modulations above include close to a monthly rate, a clear ~4.5 day, and and underlying fast semi-durnal ( 365.25/(12 x 61) = 0.499)
The significance of the cross-validation can be further substantiated by taking the complement of the training interval as the new training interval. This does converge to a stationary solution.
This modulation may seem very mysterious but something like this must be happening on the multiple time scales that the behavior is occurring on — remember that tidal forces operate on the same multiple time scales, from the semi-diurnal cycle to beyond the 18.6 year nodal declination cycle that is apparent in the middle panel above (and add to this that the sun’s forcing ranges from daily to annual). The concept of phase-locking is likely a crucial aspect as well. The sinusoidal modulation will cause an initial phase-shift across the level changes, and that appears to be a critical factor in the final model-fitted result. As observed in other systems, the synergy between synchronization (phase-locking) and resonance (standing-wave conditions) can give rise to such rich and complex dynamical behaviors. One can ask ChatGPT for any analogs [3] but just remember that this does happen:
The challenge is only in the fact that the incommensurate tidal patterns with the annual and daily cycles will never result in a metronomic march, and thus the pattern is much more complex.
[1] The sawtooth character is a result of the semi-annual impulse-driven shifts combined with a relaxation slope (r) to zero.
If the tides were commensurate with a semi-annual impulse, the forcing would appear more regular as above, but since they are not, the forcing is erratic.
[2] “The uniformity of phase of the semidiurnal cycle of zonal wind is striking (Figure 5b). It is well known that surface pressure in the tropics undergoes a strong semidiurnal variation [cf. Haurwitz and Cowley, 1973].” Deser, Clara. “Daily surface wind variations over the equatorial Pacific Ocean.” Journal of Geophysical Research: Atmospheres 99.D11 (1994): 23071-23078.
“ENSO events initiate during boreal spring and summer, peak during boreal winter, and decay in the following spring—a phenomenon known as ENSO phase-locking.” … “Our findings indicate that DA (diurnal amplitude) simulation in climate models is crucial for resolving the long-standing failure associated with the ENSO phase-locking simulation accuracy.” Yang, Xiaodan, et al. “Key to ENSO phase-locking simulation: effects of sea surface temperature diurnal amplitude.” npj Climate and Atmospheric Science 6.1 (2023): 159. https://www.nature.com/articles/s41612-023-00483-3
[3] ChatGPTresponse: A function cos(k×SawToothWave(t)+Phase), especially with a high modulation factor k, can model several complex interactions in natural phenomena where two periodic processes influence each other. Here are some specific examples:
Tidal Forces and Lunar Cycles: Tidal patterns are influenced by both the moon and the sun. A sawtooth wave could represent the changing gravitational pull of the moon (with its phases), while the cosine function might represent the daily tidal cycles. The modulation could express how these two cycles interact, creating spring and neap tides.
Seasonal Climate Variations and Diurnal Temperature Cycles: The sawtooth wave can represent the gradual change in daylight hours or solar intensity over a year, while the cosine function might model daily temperature fluctuations. The combination would reflect how daily temperatures vary across different seasons.
Circadian Rhythms and Sleep-Wake Cycles: The sawtooth wave could model the gradual change in hormonal levels or body temperature over a 24-hour cycle, interacting with a cosine function that represents shorter, rapid fluctuations in alertness or activity levels.
Economic Cycles and Market Fluctuations: In economics, a sawtooth wave might model longer economic cycles (like boom and bust periods), while the cosine function could represent shorter-term market fluctuations. The interaction of these two could help analyze complex economic dynamics.
Ecosystem Dynamics: In ecology, a sawtooth wave could represent seasonal changes (like rainfall patterns), while the cosine function could model daily variations (like temperature or light). The interaction of these cycles could be used to study plant growth, animal migration patterns, or other ecological phenomena.
Astronomical Phenomena: In astronomy, this kind of function might be used to model complex interactions between orbital mechanics and rotational dynamics, such as the interaction between a planet’s orbit around a star (sawtooth wave for eccentricity changes) and its rotational period (cosine function).
These examples show how combining periodic functions in different ways can model complex interactions in natural phenomena. The key is understanding the underlying processes and how they can be represented mathematically with these functions.
The concept of phase-locking is highly relevant in systems where the interaction of periodic signals leads to synchronization or alignment in their phases. In the context of your function cos(k×SawToothWave(t)+ Phase), phase-locking could manifest in several interesting ways, particularly considering the sinusoidal modulation of a sawtooth wave. Here are some areas where similar concepts are explored:
Neuroscience and Neural Oscillations: In the brain, neurons often fire in a synchronized manner, a phenomenon known as neural phase-locking. This is crucial in processes like signal transmission and cognitive function. The interaction of different neural rhythms (e.g., theta and gamma waves) can be modeled using phase relationships, akin to the modulation you described.
Electronics and Communication Systems: Phase-locking is a fundamental concept in electronics, especially in the design of phase-locked loops (PLLs). PLLs are used to synchronize an output signal with a reference signal in terms of frequency and phase. This principle is crucial in radio, telecommunications, and signal processing.
Physics and Oscillatory Systems: In physics, particularly in studies involving coupled oscillators or pendulums, phase-locking describes the phenomenon where oscillators synchronize due to weak interactions. This can be observed in systems ranging from mechanical pendulums to quantum mechanical systems.
Cardiac Rhythms in Physiology: The synchronization of cardiac cells during heartbeats is an example of biological phase-locking. Cells in the sinoatrial node generate rhythmic impulses that cause heart muscle cells to contract in a synchronized manner.
Climatology and Environmental Sciences: Phase-locking can also be observed in climatic phenomena, such as the El Niño-Southern Oscillation (ENSO), where atmospheric and oceanic conditions align periodically, impacting global weather patterns.
Music and Acoustics: In music theory and acoustics, the concept of beats and harmonic interactions involves phase relationships between different sound waves. The concept of phase-locking can be observed in the constructive and destructive interference of these waves.
In each of these cases, the critical factor is how the phase relationship between different oscillatory components influences the overall system’s behavior. The initial phase-shift you mentioned, particularly in a system with a sinusoidal modulation over a sawtooth wave, could lead to complex dynamic behaviors, where the phase relationship plays a pivotal role in the system’s response and evolution.
There are several behavioral models in various scientific fields where the interaction of phase-locking with standing-wave resonance conditions leads to complex time patterns. These phenomena are observed in systems where synchronization and resonance play a critical role. Here are some examples:
Physics and Engineering – Coupled Oscillators and Resonators: In systems of coupled oscillators or resonators, phase-locking and resonance can lead to complex patterns, especially in nonlinear dynamics. An example is the array of coupled pendulums or resonant circuits, where phase-locking can lead to intricate motion patterns or voltage/current oscillations.
Neuroscience – Brain Wave Synchronization: In the brain, different regions can exhibit phase-locked oscillations, and when these oscillations resonate with the natural frequencies of neural circuits (like in the hippocampus or cortex), complex spatiotemporal patterns emerge. These patterns are crucial for functions like memory, perception, and attention.
Quantum Mechanics – Bose-Einstein Condensates: In Bose-Einstein condensates, particles can synchronize and exhibit collective oscillations. When these oscillations resonate at certain frequencies, complex interference patterns emerge, demonstrating both phase-locking and standing-wave phenomena at a quantum level.
Acoustics – Architectural Acoustics and Musical Instruments: In musical instruments and architectural acoustics, the interaction of sound waves can lead to phase-locking and resonance. This results in complex harmonic structures and standing waves, influencing the timbre of music and the acoustic properties of buildings.
Climate Science – Atmospheric and Oceanic Oscillations: Climatic phenomena, like the El Niño-Southern Oscillation, involve complex interactions between atmospheric and oceanic patterns. Phase-locking of these patterns, along with resonance effects in ocean basins, can lead to complex climate behaviors and cycles.
Laser Physics – Mode-locking in Lasers: In laser physics, mode-locking techniques create lasers with very precise frequencies, where phase-locking of different modes leads to the generation of ultrafast pulses. This is a form of resonance where the laser’s cavity modes synchronize, leading to coherent pulse emissions.
Chemical Oscillations – Belousov-Zhabotinsky Reaction: This chemical reaction exhibits oscillatory behavior where chemical waves propagate and can lock in phase under certain conditions, creating complex spatial patterns and wavefronts.
In each of these examples, the interaction of phase-locking with resonant conditions creates patterns that are more intricate than what would be observed with either phenomenon alone. The synergy between synchronization (phase-locking) and resonance (standing-wave conditions) can give rise to rich and complex dynamical behaviors, which are often crucial to the system’s function or characteristics.
GeoEnergy Math 