The Perplexity response:
Continue readingForcing selection
LOD
The mystery of El Nino
Google Gemini makes predictions on the next El Nino when prompted — scroll to the end.
https://gemini.google.com/app/8d7c963c197e05c0
Continue readingThere has ALWAYS been stratification in the ocean via the primary thermocline. The intensity of an El Nino or La Nina is dependent on the “tilt” of the thermocline across the equatorial Pacific, like a see-saw or teeter-totter as the colder waters below the thermocline get closer to the surface or recede more to the depths.
The only mystery is to what provokes the motion. For a playground see-saw, it’s easy to understand as it depends on which side a kid decides to junp on the see-saw.
For the ocean, the explanation is less facile than that, explain.
Pairing of solar and lunar factors
A number of the Earth’s geophysical behaviors characterized by cycles have both a solar and lunar basis. For the ubiquitous ocean tides, the magnitude of each factor are roughly the same — rationalized by the fact that even though the sun is much more massive than the moon, it’s much further away.
However, there are several behaviors that even though they have a clear solar forcing, lack a lunar counterpart. These include the Earth’s fast wobble, the equatorial SAO/QBO, ENSO, and others. The following table summarizes how these gaps in causation are closed, with the missing lunar explanation bolded. Unless otherwise noted by a link, the detailed analysis is found in the text Mathematical Geoenergy.
| Geophysical Behavior | Solar Forcing | Lunar Forcing |
|---|---|---|
| Conventional Ocean Tides | Solar diurnal tide (S1), solar semidiurnal (S2) | Lunar diurnal tide (O1), lunar semidiurnal (M2), |
| Length of Day (LOD) Variations | Annual, semi-annual | Monthly, fortnightly, 9-day, weekly |
| Long-Period Tides | Solar annual variations (Sa), solar semi-annual (Ssa) | Fortnightly (Mf), monthly (Mm, Msm), mixed harmonics |
| Chandler Wobble | Annual wobble | 433 day cycle caused by draconic stroboscopic effect |
| Quasi-Biennial Oscillation (QBO) | Semi-Annual Oscillation (SAO) above QBO in altitude | 28-month caused by draconic stroboscopic effect |
| El Niño–Southern Oscillation (ENSO) | Seasonal impulse acts as carrier and spring unpredictability barrier | Erratic cycling caused by draconic + other tidal factors per stroboscopic effect |
| Eclipse events | Sun-Moon alignment (draconic cycle critical) | Sun-Moon alignment (draconic cycle critical) |
| Other Climate Indices and MSL | Strong annual modulation and trigger | Similar to ENSO, see https://github.com/pukpr/GEM-LTE |
| Milankovitch Cycles | Eccentricity, obliquity, and precession | Axial drift in precessional cycle |
| Regression of nodes (nutation) | Controlled +/- about the Earth-Sun ecliptic plane | Draconic & tropical define an 18.6 year beat in nodal crossings |
| Atmospheric ringing | Daily atmospheric tides | Fortnightly modulation https://geoenergymath.com/the-just-so-story-narrative/ |
| Seasonal Climate | Annual tilted orbit around the sun | – |
| Daily Climate | Earth’s rotation rate | – |
| Anthropogenic Global Warming | – | – |
| Seismic Activity | (sporadic stochastic trigger) | (sporadic stochastic trigger) |
| Geomagnetic, Geothermal, etc | ? | ? |
The most familiar periodic factors – the daily and seasonal cycles – being primarily radiative processes obviously have no lunar counterpart.
And climate science itself is currently preoccupied with the prospect of anthropogenic global warming/climate change, which has little connection to the sun or moon, so the significance of the connections shown is largely muted by louder voices.
References:
- Mathematical Geoenergy, 2019 (in BOLD)
- Cartwright & Edden, Tidal Generation studies
- Various oceanography & geodesy literature
- Stroboscopic effect — these researchers were close but made the mistake of comparing to a sunspot cycle
Lunar Torque Controls All
The truly massive scale in the motion of fluids and solids on Earth arises from orbital interactions with our spinning planet. The most obvious of these, such as the daily and seasonal cycles, are taken for granted. Others, such as ocean tides, have more complicated mechanisms than the ordinary person realizes (e.g. ask someone to explain why there are 2 tidal cycles per day). There are also less well-known motions, such as the variation in the Earth’s rotation rate of nominally 360° per day, which is called the delta in Length of Day (LOD), and in the slight annual wobble in the Earth’s rotation axis. Nevertheless, each one of these is technically well-characterized and models of the motion include a quantitative mapping to the orbital cycles of the Sun, Moon, and Earth. This is represented in the directed graph below, where the BLUE ovals indicate behaviors that are fundamentally understood and modeled via tables of orbital factors.
(rendered by GraphViz)
However, those ovals highlighted in GRAY are nowhere near being well-understood in spite of being at least empirically well-characterized via years of measurements. Further, what is (IMO) astonishing is the lack of research interest in modeling these massive behaviors as a result of the same orbital mechanisms as that which causes tides, seasons, and the variations in LOD. In fact, everything tagged in the chart is essentially a behavior relating to an inertial response to something. That something, as reported in the Earth sciences literature, is only vaguely described — and never as a tidal or tidal/annual interaction.
I don’t see how it’s possible to overlook such an obvious causal connection. Why would the forcing that causes a massive behavior such as tides suddenly stop having a connection to other related inertial behaviors? The answers I find in the research literature are essentially that “someone looked in the past and found no correlation” [1].
Continue readingProof for allowed modes of an ideal QBO
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.
Continue readingDynamic Time Warping
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.
Continue readingFull Wave Forcing
MSet
Global Models of Sea Surface Temperature
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.
Continue readingThe Power of Darwin (part 2)
Continuation of the model described in part 1.
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)
LTE Modulation Amplitude Phase Harmonic
3.95901009601, 0.10819059771, 2.56829482810 0 -- slow LTE modulation
1.34461504256, 0.12014470401, 0.28639994030 0 -- slow LTE modulation
-20.01129999289, 0.11320535624, 2.58186128147 1 -- monthly fundamental
-140.07909995021, 0.49935565041, 2.12022069445 7 -- strong 4.5 day
-1220.6892995660, 0.95817753106, -2.88519906135 61 -- strong semi-diurnalThe 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] ChatGPT response: 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.
https://chat.openai.com/share/c9a9d58b-e5db-466f-b369-0b51ccff7458
Unified Model of Earth Dynamics
Lorenz turned out to be a chaotic dead-end in understanding Earth dynamics. Instead we need a new unified model of solid liquid dynamics focusing on symmetries of the rotating earth, applying equations of solid bodies & fluid dynamics. See Mathematical Geoenergy (Wiley, 2018).
Should have made this diagram long ago: here’s the ChatGPT4 prompt with the diagramming plugin.
Graph
Ocean Tides and dLOD have always been well-understood, largely because the mapping to lunar+solar cycles is so obvious. And the latter is getting better all the time — consider recent hi-res LOD measurements with a ring laser interferometer, pulling in diurnal tidal cycles with much better temporal resolution.
That’s the first stage of unification (yellow boxes above) — next do the other boxes (CW, QBO, ENSO, AMO, PDO, etc) as described in the book and on this blog, while calibrating to tides and LOD, and that becomes a cross-validated unified model.
Annotated 10/11/2023
ontological classification according to wavenumber kx, ky, kz and fluid/solid.
Added so would not lose it — highlighted tidal factor is non-standard
