First evaluation of Gemini Advanced 1.5 Pro with Deep Research. Logged in with trial subscription and gave this as an initial prompt. Saved the results to a Google Docs file, and then created the following PDF. Note the top-level focus on this blog and published citations in the scope of Chapter 12 of Mathematical Geoenergy, even though the chapter was not directly cited — only via an embedded cite in a submitted ESD Ideas article, reference 1. Interesting that the Lin & Qian paper not cited.
Prompt: Explain tidal forcing behind ENSO using derivations based on reduced effective gravity on equatorial thermocline.
A climate science breakthrough likely won’t be on some massive computation but on a novel formulation that exposes some fundamental pattern (perhaps discovered by deep mining during a machine learning exercise). Over 10 years ago, I wrote on a blog post on how one can extract the ENSO signal by doing simple signal processing on a sea-level height (SLH) tidal time-series — in this case, at Fort Denison located in Sydney harbor.
The formulation/trick is to take the difference between the SLH reading and that from 2 years (24 months) prior, described here
The rationale for this 24 month difference is likely related to the sloshing of the ocean triggered on an annual basis. I think this is a pattern that any ML exercise would find with very little effort. After all, it didn’t take me that long to find it. But the point is that the ML configuration has to be open and flexible enough to be able to search, generate, and test for the same formulation. IOW, it may not find it if the configuration, perhaps focused on computationally massive PDEs, is too narrow. That was my comment to a RC post on applying machine learning to climate science, see the following link and subsequent quote:
“ML-based parameterizations have to work well for thousands of years of simulations, and thus need to be very stable (no random glitches or periodic blow-ups) (harder than you might think). Bias corrections based on historical observations might not generalize correctly in the future.”
This same issue arises when using ML to simulate PDEs. The solution is to analytically calculate what the stability condition(s) is (are), then at each timestep to add some numerical diffusion that nudges the solution towards satisfying the stability condition(s). I imagine this same technique could be used for ML-based parametrizations.
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.
The cyan background represents behaviors that have a longitudinal dependence (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].
Dimensionality reduction of chaos by feedbacks and periodic forcing is a source of natural climate change, by P. Salmon, Climate Dynamics (2024)
Bottom line is that a forcing will tend to reduce chaos by creating a pattern to follow, thus the terminology of “forced response”. This has implications for climate prediction. The first few sentences of the abstract set the stage:
The role of chaos in the climate system has been dismissed as high dimensional turbulence and noise, with minimal impact on long-term climate change. However theory and experiment show that chaotic systems can be reduced or “controlled” from high to low dimensionality by periodic forcings and internal feedbacks. High dimensional chaos is somewhat featureless. Conversely low dimensional borderline chaos generates pattern such as oscillation, and is more widespread in climate than is generally recognised. Thus, oceanic oscillations such as the Pacific Decadal and Atlantic Multidecadal Oscillations are generated by dimensionality reduction under the effect of known feedbacks. Annual periodic forcing entrains the El Niño Southern Oscillation.
In Chapters 11 and 12 in Pukite, P., Coyne, D., & Challou, D. (2019). Mathematical Geoenergy. John Wiley & Sons, I cited forcing as a chaos reducer:
“It is well known that a periodic forcing can reduce the erratic fluctuations and uncertainty of a near‐chaotic response function (Osipov et al., 2007; Wang, Yang, Zhou, 2013).“
But that’s just a motivator. Tides are the key, acting primarily on the subsurface thermocline. Salmon’s figure comparing the AMO to Barents sea subsurface temperature is substantiating in terms of linking two separated regions by something more than a nebulous “teleconnection”.
Likely every ocean index has a common-mode mechanism. The tidal forcing by itself is close to providing an external synchronizing source, but requires what I refer to as a LTE modulation to zero in on the exact forced response. Read the previous blog post to get a feel how this works:
As Salmon notes, it’s known at some level that an annual/seasonal impulse is entraining or synchronizing ENSO, and also likely PDO and AMO. The top guns at NASA JPL point out that the main lunisolar terms are at monthly, 206 day, annual, 3 year, and 6 year periods, and this is what is used to model the forcing, see the following two charts
Now note how the middle panel in each of the following modeled climate indices does not change markedly. The most challenging aspect is the inherent structural sensitivity of the manifold1 mapping involved in LTE modulation. As the Darwin fit shows, the cross-validation is better than it may appear, as the out-of-band interval does not take much of a nudge to become synchronized with the data. Note also that the multidecadal nature of an index such as AMO may be ephemeral — the yellow cross-validation band does show valleys in what appears to be a longer multidecadal trend, capturing the long-period variations in the tides when modulated by an annual impulse – biennial in this case.
1 The term manifold has an interesting etymology. From the phonetics, it is close to pronounced as “many fold”, which is precisely what’s happening here — the LTE modulation can fold over the forcing input many times in proportion to the mode of the standing wave produced. So that a higher standing wave will have “many folds” in contrast to the lowest standing wave model. At the limit, the QBO with an ostensibly wavenumber=0 mode will have no folds and will be to first-order a pass-through linear amplification of the forcing, but with likely higher modes mixed in to give the time-series some character.
In Mathematical Geoenergy, Chapter 12, a biennially-impulsed lunar forcing is suggested as a mechanism to drive ENSO. The current thinking is that this lunar forcing should be common across all the oceanic indices, including AMO for the Atlantic, IOD for the Indian, and PDO for the non-equatorial north Pacific. The global temperature extreme of the last year had too many simultaneous concurrences among the indices for this not to be taken seriously.
NINO34
PDO
AMO
IOD – East
IOD-West
Each one of these uses a nearly identical annual-impulsed tidal forcing (shown as the middle green panel in each), with a 5-year window providing a cross-validation interval. So many possibilities are available with cross-validation since the tidal factors are essentially invariantly fixed over all the climate indices.
The approach follows 3 steps as shown below
The first step is to generate the long-period tidal forcing. I go into an explanation of the tidal factors selected in a Real Climate comment here.
Then apply the lagged response of an annual impulse, in this case alternating in sign every other year, which generates the middle panel in the flow chart schematic (and the middle panel in the indexed models above).
Finally, the Laplace’s Tidal Equation (LTE) modulation is applied, with the lower right corner inset showing the variation among indices. This is where the variability occurs — the best approach is to pick a slow fundamental modulation and generate only integer harmonics of this fundamental. So, what happens is that different harmonics are emphasized depending on the oceanic index chosen, corresponding to the waveguide structure of the ocean basin and what standing waves are maximally resonant or amplified.
Note that for a dipole behavior such as ENSO, the LTE modulation will be mirror-inverses for the maximally extreme locations, in this case Darwin and Tahiti
A machine learning application is free to scrape the following GIST GitHub site for model fitting artifacts.
Another analysis that involved a recursively cycled fit between AMO and PDO. It switched fitting AMO for 2.5 minutes and then PDO for 2.5 minutes, cycling 50 times. This created a common forcing with an optimally shared fit, forcing baselined to PDO.
PDO
AMO
NINO34
IOD-East
IOD-West
Darwin
Tahiti
The table above shows the LTE modulation factors for Darwin and Tahiti model fits. The highlighted blocks show the phase of the modulation, which should have a difference of π radians for a perfect dipole and higher harmonics associated with it. (The K0 wavenumber = 0 has no phase, but just a sign). Of the modes that are shared 1, 45, 23, 36, 18, 39, 44, the average phase is 3.09, close to π (and K0 switches sign).
Contrast to the IOD East/West dipole. Only the K0 (wavenumber=0) shows a reversal in sign. The LTE modulation terms are within 1 radian of each other, indicating much less of a dipole behavior on those terms. It’s possible that these sites don’t span a true dipole, either by its nature or from siting of the measurements.
Cross-validating a large interval span on PDO
using CC
using DTW metric, which pulls out more of the annual/semi-annual signal
adding a 3rd harmonic
Complement of the fitting interval, note the spectral composition maintains the same harmonics, indicating that the structure mapped to is stationary in the sense that the tidal pattern is not changing and the LTE modulation is largely fixed.
This is the resolved tidal forcing, finer than the annual impulse sampling used on the models above.
Below can see the primary 27.5545 lunar anomalistic cycle, mixed with the draconic 27.2122/13.606 cycle to create the 6/3 year modulation and the 206 day perigee-syzygy cycle (or 412 full cycle, as 206 includes antipodal full moon or new moon orientation).
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).
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.
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.
