Reminded by a 20-year anniversary post at RealClimate.org, that I’ve been blogging for 20 years + 6 months on topics of fossil fuel depletion + climate change. The starting point was at a BlogSpot blog I created in May 2004, where the first post set the stage:
Click on the above to go to the complete archives (almost daily posts) until I transitioned to WordPress and what became the present blog. After 2011, my blogging pace slowed down considerably as I started to write in more in more technical terms. Eventually the most interesting and novel posts were filtered down to a set that would eventually become the contents of Mathematical Geoenergy : Discovery, Depletion, and Renewal, published in late 2018/early 2019 by Wiley with an AGU imprint.
The arc that my BlogSpot/WordPress blogging activity followed occupies somewhat of a mirror universe to that of RealClimate. I initially started out with an oil depletion focus and by incrementally understanding the massive inertia that our FF-dependent society had developed, it placed the climate science aspect into a different perspective and context. After realizing that CO2 did not like to sequester, it became obvious that not much could be done to mitigate the impact of gradually increasing GHG levels, and that it would evolve into a slow-moving train wreck. That’s part of the reason why I focused more on research into natural climate variability. In contrast, RealClimate (and all the other climate blogs) continued to concentrate on man-made climate change. At this point, my climate fluid dynamics understanding is at some alternate reality level, see the last post, still very interesting but lacking any critical acceptance (no debunking either, which keeps it alive and potentially valid).
The oil depletion aspect more-or-less spun off into the PeakOilBarrel.com blog [*] maintained by my co-author Dennis Coyne. That’s like watching a slow-moving train wreck as well, but Dennis does an excellent job of keeping the suspense up with all the details in the technical modeling. Most of the predictions regarding peak oil that we published in 2018 are panning out.
As a parting thought, the RealClimate hindsight post touched on how AI will impact information flow going forward. Having worked on AI knowledgebases for environmental modeling during the LLM-precursor stage circa 2010-2013, I can attest that it will only get better. At the time, we were under the impression that knowledge used for modeling should be semantically correct and unambiguous (with potentially a formal representation and organization, see figure below), and so developed approaches for that here and here (long report form).
As it turned out, lack of correctness is just a stage, and AI users/customers are satisfied to get close-enough for many tasks. Eventually, the LLM robots will gradually clean up the sources of knowledge and converge more to semantic correctness. Same will happen with climate models as machine learning by the big guns at Google, NVIDIA, and Huawei will eventually discover what we have found in this blog over the course of 20+ years.
Note: [*] In some ways the PeakOilBarrel.com blog is a continuation of the shuttered TheOilDrum.com blog, which closed shop in 2013 for mysterious reasons.
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].
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).
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.
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.
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.
Annotated10/11/2023
ontological classification according to wavenumber kx, ky, kz and fluid/solid.
Added so would not lose it — highlighted tidal factor is non-standard
“Because the distance effect annual cycle (from perihelion to perihelion, otherwise called the anomalistic year, 365.259636 d (ref. 6)) is slightly longer than the tilt effect annual cycle (from equinox to equinox, otherwise known as the tropical year, 365.242189 d (ref. 6)), the LOP increases over time. A complete revolution of the LOP is the precession cycle, about 22,000 yr (ref. 38). “
The claim is that the exact timing and strength of the annual cycle extreme is important in initiating an El Nino or La Nina cycle. as the Longitude of Perihelion (LOP) moves over time, so that when this aligns with the solstice/equinox events, the ENSO behavior will mirror the strength of the forcing.
This is not a 22,000 year cycle (as stated its actuallly closer to 21,000), but it’s possible that if tidal cycles play a part of the forcing, then a much shorter paleo cycle may emerge. Consider that the Mf tidal cycle of 13.66 days will generate a cycle of ~3.81 years when non-linearly modulated against the tropical year (seasonal cycle), but ~3.795 years modulated against the perigean or anomalistic year (nearest approach to the sun).
The difference between the two will reinforce every 783 years, which is close to the 800 year cycle. Aliasing and sideband frequency calculations reveal these patterns.
The idea of a digital twin is relatively new in terms of coinage of terms, but the essential idea has been around for decades. In the past, a digital twin was referred to as a virtual simulation of a specific system, encoded via a programming language. In the case of a system that was previously built, the virtual simulation emulated all the behaviors and characteristics of that system, only operated on a computer, with any necessary interactive controls and displays provided on a console, either real or virtual. A widely known example of a VS is that of a flight simulator, which in historical terms was the industrial forerunner to today’s virtual reality. A virtual simulation could also be used during the design of the system, with the finished digital twin providing a blueprint for the actual synthesis of the end-product. This approach has also been practiced for decades, both in the electronics industry via logic synthesis of integrated circuits from a hardware description language and with physical products via 3D printing from CAD models.
