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].
As a starting point, let’s take just two paired examples from the above chart : (1) the connection between the Chandler Wobble and Annual Wobble, and (2) that between QBO and SAO.
Chandler Wobble and Annual Wobble
The annual wobble in the Earth’s rotation is obviously the result of the external factor of the Sun generating a change in the inertial moment of the Earth via such mechanisms of seasonal ice/glacier formation and the torque that the Sun applies to the Earth’s rotational moment of inertia. That can’t really be disputed as it would be a pure coincidence that anything else would produce an annual cycle. This is a case of both plausibility and parsimony. As controlled experiments on the Earth are not possible to validate such a mechanism, the plausibility of an annual mechanism matches well with the parsimony of having the cycles precisely match. In other words, the Earth has to wobble at least somewhat because of the Sun’s annual forcing, and the observational results confirm that. This is the same kind of argument as with ocean tides — no controlled experiment possible but years of observation results as confirmation.
Yet, for the Chandler Wobble, which is 433 days instead of 365 days, no such obviously plausible and parsimonious model exists …. unless a lunar torque is taken into account properly. Which is what has to happen (i.e. an additional wobble) if such a plausible external torque is applied besides that of the Sun — and that it matches exactly as theoretically predicted is parsimonious to boot. This is described fully in Chapter 13 of Mathematical Geoenergy and more recently here & here. However, this is still not accepted in the world of geophysics, where novel ideas are accepted slowly, see my attempts here.
QBO and SAO
The semi-annual oscillation (SAO) in equatorial upper stratospheric winds is empirically shown to have an exact semi-annual periodicity, or a pair of complete reversing cycles per year (2 west-east and 2 east-west half cycles per year). Like the annual wobble described above, the plausibility and parsimony of having a causal basis in the Sun’s seasonal pattern in cycling between the hemispheres with 4 distinct phases per year is overwhelming (i,.e. equator to North Pole, North Pole to equator, equator to South Pole, and South Pole to equator will match each phase reversal 1-to-1). That’s why SAO is colored BLUE in the above chart, as it’s cause is readily inferred.
Yet, for the quasi-biennial oscillation (QBO), which has a ~2.37y cycle and exists right below the SAO in altitude, no such obviously plausible and parsimonious model exists …. unless again a lunar torque is taken into account properly. So, if another external torque is considered besides that of the Sun, the Moon as a candidate must have at least some impact as a plausible forcing, and like for the Chandler wobble the theoretical prediction matches exactly. This is described fully in Chapter 11 of Mathematical Geoenergy and more recently here & here. This idea was also quashed at the same time I offered the Chandler Wobble as part of a unified model here. (So much for unification as a means of improving parsimony)
That’s the start of how unified forces can expand our understanding of global behaviors. They are unified in the sense that no new forces are needed, they just need to be applied correctly for the domain under study — which strongly suggests that climate scientists are still overlooking a lunar mechanism — see this PubPeer QBO review.
Furthermore, an even greater unification can be achieved if the massive subsurface ocean behaviors (such as ENSO) can also be tied in to the same external something. There is no question that tidal forces interact with the subsurface thermocline, with a sensitivity that is orders of magnitude higher than that of surface tides. The math of geophysical fluid dynamics is key here since the lunar tidal cycles do not pattern match as readily as they do for surface tides.
All the grayed oceanic behaviors in the above chart (ENSO, AMO, PDO, IOD, NAO) are candidates for modeling since they each exhibit massive inertial shifts that have to at least have some origin in tidal cycles. Like that for QBO and the Chandler Wobble, there is too much evidence for it not to have an impact. Consider first the following two correlations from the chart above: LOD & AAM and ENSO & AAM.
LOD & AAM
Most of the variations comprising LOD is due to lunar and annual cycles acting on the solid Earth, but there is a remnant that comes from the fluid portion of the lithosphere, either ocean or atmosphere, which doesn’t have as directly coupled a response. The atmospheric angular momentum (AAM) estimates the delta in the atmosphere based on measuring the averaged total wind speed — data here for global AAM (GLAAM). The plot below (from https://usclivar.org/working-groups/mjo/science/mjo-atmospheric-angular-momentum-length-of-day) show how closely residual LOD matches AAM over an interval, which substantiates the notion that angular momentum is shared and exchanged between the solid body and atmosphere.
ENSO & AAM
Now with the dynamic correlation between LOD and AAM established, we can evaluate how much may plausibly come from the ocean, which is the primary source of the energy required to drive kinetic wind variability. As one can see below, that yes indeed much of the variability is correlated with an ENSO index (in this case NINO34) with possibly the GLAAM lagging by one month.
So, this sets up the premise that likely the ocean is responding to tidal forcing, but in a potentially more non-linear manner than the linear response of the solid-body response portion of LOD. IOW, the pattern matching may emerge after a non-linear mapping from forcing to response.
The strong likelihood of lunar forces interacting non-linearly with the annual signal is reinforced when one considers how easily satellite side-bands emerge from signal processing of an ENSO time-series. In the frequency spectrum shown below, folding about the twice the annual frequency reveals that some unknown modulating signal is interacting with an annual signal. The mapping is too aligned for this signal to be random, and it will just take some digging to identify the pattern of the modulation — the first guess being of the principal lunar tidal cycles.
The approach then to take is to systematically evaluate how possible tidal forcings with a non-linear fluid dynamics response can model the observations made in various ocean basins, including ENSO (equatorial Pacific), AMO (Atlantic Ocean), IOD (Indian Ocean), PDO (northern Pacific), and NAO (northern Atlantic). Ideally, the tidal forcings should be similar and only the standing-wave boundary conditions of the basins will provide a discrimination in the time-series. This is described fully in Chapter 12 of Mathematical Geoenergy
The following sections present models for AMO, PDO, IOD, ENSO, NAO followed by AAM. Each of the ocean indices is cross-validated with a window between the years 1900 to 1905 applied as a test interval. Afterwards, the forcings are compared against each other for commonality
AMO
The Atlantic Multidecadal Oscillation (AMO) occupies a basin that appears to be remarkably sensitive to gradual changes in tidal forcing, resulting in the multidecadal variation on top of the more rapid cycling. The Laplace’s Tidal Equation (LTE) modulation shown is an aspect of the non-linear mapping as described in Chapter 12.
PDO
The Pacific Decadal Oscillation (PDO) occupies a basin that compared to AMO is less sensitive to gradual changes in tidal forcing, but a long-term variation is still evident.
IOD
The Indian Ocean Dipole (IOD) is split into 2 lobes, East and West. They each show an upward trend, which may be an AGW factor.
ENSO (NINO34)
The El Nino Southern Oscillation (ENSO) is represented by the NINO34 time-series. The time-series shows little decadal variation in contrast to AMO.
NAO
The North Atlantic Oscillation (NAO) has a much higher frequency content, perhaps as a result of a greater atmospheric sensitivity. This is perhaps the poorest correlation due to the complexity, yet the cross-validation interval is promising.
AAM
The Global Atmospheric Angular Momentum (GLAAM) is hindered by a shorter interval for fitting, and so the cross-validation interval is shifted to [1965-1970].
If the input tidal forcings for each of these indices is compared, no clear outliers or systematic variation is observed. The factors were allowed to vary as part of the fitting process, yet they appear to remain grouped together with perhaps a random scatter, with the weakest factors showing the greatest percentage variation. The summed set of factors for each oceanic index shows a relative standard deviation of 1%.
MJO & ENSO
The final aspect of the chart at the top, is the correlation between ENSO and MJO, which is described here. The Madden-Julian Oscillation (MJO) is a unique index in that it appears as a traveling wave as opposed to the standing wave of ENSO. The traveling wave circles around the world.
The comparison below demonstrates that MJO may be a time-lagged variant of ENSO, with the SOI of ENSO leading the MJO 140° phase by 21 days.
This is consistent with the following narrative description provided by the climate.gov site
“The S.S. ENSO cruise ship and MJO speedboat making their way across the “harbor” of the tropical Pacific Ocean. The cruise ship represents the stationary ENSO pattern creating steady, rolling waves. The speedboat represents the rapidly moving MJO travelling through the waves created by the S.S. ENSO, altering the wake that the MJO speedboat is producing. The floating person represents the United States that feels the impact from the both the cruise ship’s wake and the speedboat’s modified wake. “
The collective theme is that if one considers that lunar forcing is an important factor for certain well-characterized geophysical behaviors, then there’s little reason to believe that they won’t be a factor for the less well-understood behaviors such as QBO, Chandler wobble, and various oceanic dipoles. The key is to consider non-linear mechanisms that match the observations [2].
Additionally, it’s instructive to point out how much the exchange of angular momentum between the Earth, Moon, and Sun provide a common unifying conservation mechanism controlling the behavior of each of the phenomena shown in the top chart (excepting the seasons). Apart from the non-linear fluid dynamics, angular momentum applies Newton’s laws at a basic level.
Consider again the fundamental behavior of tides and how that can modify sea level height (SLH), not only at a daily/diurnal period but at longer periods, and how that reveals itself. In a post from 10 years ago, I wrote how one can extract the ENSO signal by doing simple signal processing on a SLH tidal time-series — in this case, at Fort Denison located in Sydney harbor [3]. This interaction is added to the top-level figure below.
The obvious premise is that since SLH cycles are largely tidal, why can’t we infer that all the longer SLH cycles have a tidal origin as well? And since we have shown that ENSO is tidally forced, by association longer-term SLH variations correlated to ENSO will therefore be tidal as a root cause, The formulation/trick is to take the difference between the SLH reading and that from 2 years (24 months) prior, described here:
With some averaging to reduce noise, the delayed differential (DD) SLH time series lines up very well with an ENSO index such as SOI or NINO4.
The arrows above the instances of where the pattern matching fails. So, if the region between1915-1920 is excluded from an ENSO-like model fit as follows., we can then use that to cross-validate against NINO4
then to NINO4
One can see that the cross-validation largely corrects the non-trained test interval to match the NINO4 data. There is virtually no difference in the tidal forcing and only slight changes to the 7 LTE parameters, with the primary LTE modulation parameter fixed at a value of 37.45 +/- 0.03 and similar relative amplitude compared to the other LTE factors (which vary more)
I couldn’t find any other SLH-ENSO match in the literature using the same delay differencing formulation, and ChatGPT only finds my post with the exact details: https://chatgpt.com/share/677ef333-3a6c-8005-974b-7ccc4840d32c. A recent paper in GPL on seasonal variability of SLH [4] misses the ENSO connection entirely, as no mention in the online paper.
The rationale for applying a 24-month difference is likely related to the sloshing of the ocean triggered in conjunction with the lunar forcing amplitude on an annual basis [5]. A 12-month difference also works to some degree — part of this is due to that it’s a decent way to remove annual variability, but that’s not the entire rationale. The LTE modulation, especially along the equator, where ENSO predominates is also modeled as a delayed differential — and this will remove the longer-term modulation of anything much greater than 2 years, i.e. when f = 1 as shown below.
ENSO = LTE(time) – f · LTE(time-2)
Yet, for AMO, PDO, and IOD (all nom-equatorial indices), this delay differencing is much less of a factor (f = 0) because the multidecadal variation in the climate indices is important. In fact, if the delay differential is applied to the climate indices, they tend to match up to each other and to the raw ENSO (NINO4 in this case) index.
This is further evidence that the lunar forcing model (and incidentally the 2-year feedback as mentioned in Mathematical Geoenergy) is a unifying feature in understanding seemingly disparate geophysical behaviors.
References
[1] Often cited are the negative findings of Richard Lindzen, an atmospheric tide expert, when it concerns to a causal connection to tidal forces. He stated the following as absence of evidence for cycles that would otherwise readily be apparent; but not realizing that non-linear effects would obscure the mapping by shifting the frequencies via aliasing.
- “For oscillations of tidal periods the nature of the forcing is clear” –
Lindzen, Richard D. “Planetary waves on beta-planes.” Monthly Weather Review 95.7 (1967): 441-451.
- ” .. it is unlikely that lunar periods could be produced by anything other than the lunar tidal potential” –
Lindzen, Richard S., and Siu-shung Hong. “Effects of mean winds and horizontal temperature gradients on solar and lunar semidiurnal tides in the atmosphere.” Journal of the atmospheric sciences 31.5 (1974): 1421-1446.
[2] Pukite, P., Coyne, D., & Challou, D. (2019). Mathematical Geoenergy: Discovery, Depletion, and Renewal (Vol. 241). John Wiley & Sons.
[3] Google maps: Fort Denison, Sydney Harbor
SLH time-series data are split into 2 Ft. Denison overlapping sets (for some unknown reason) at the PSMSL site:
https://psmsl.org/data/obtaining/stations/65.php 1886 – 1993
https://psmsl.org/data/obtaining/stations/196.php 1914 – 2023
These overlap nicely so an extended time series is possible. The region of disagreement between 1930-1935 was replaced by 1886-1993. In the chart below, 2 year differences, SLH(t) – SLH(t-2), in levels are shown
Incidentally, this is the location used by climate change deniers to show how sea level does not change … guess how? Answer: They take the pictures of Ft. Denison (see photo at right) at a lower tide to compensate for an overall raising level and then claim no secular rise over 140 years.
[4] Yang, J., & Chen, K. (2025). Profound changes in the seasonal cycle of sea level along the United States Mid-Atlantic Coast. Geophysical Research Letters, 52, e2024GL112273. https://doi.org/10.1029/2024GL112273
[5] Discussion at RealClimate https://www.realclimate.org/index.php/archives/2024/12/ai-caramba/#comment-828825. I assert: “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 PDEs, is too narrow.”
Other indices
PNA (Pacific North America)
NPGO (North Pacific Gyre Oscillation)
EMI (El Nino Modoki Index)
prompt: “Top to bottom, every dynamical geophysical process on the Earth responds to the gravitational torque of the moon and the gravitational torque and unequal heating provided by the sun. It starts with the domination of the gravitational forces on surface ocean tides and continues through to the domination on the differential changes in the Earth’s rotation rate due these same tidal forces. It continues on to every known cyclic geophysical behavior.“
https://chatgpt.com/share/672a1901-4b48-8005-bc4f-656c2205a647
Looking at symbolic regression aspects
https://github.com/MilesCranmer/PySR/discussions/568#discussioncomment-11122833
Model data for indices
AMO
-0.00000000000 :offset:
-0.05994021733 :bg:
-0.06319691195 :impA:
3.68579777947 :impB:
2.80233498010 :impC:
-0.12342363196:delA:
0.91649908741:delB:
0.00959050219:asym:
0.32325397549:ann1:
2.88060742584:ann2:
0.12677544903:sem1:
-1.47787996278:sem2:
-0.00007218559:year:
-0.10370534736:IR:
0.02049632899 :mA:
0.00120369471 :mP:
-0.47826938150 :shiftT:
0.30935948722 :init:
—- Tidal —-
27.32166155400, -2.53256938026, -64.40952633726, 1, -1253210, 2.02102769668380E-04
27.21222081500, 0.19536832916, -2.04306591017, 2, 14508, 1.33739927797268E-03
1095.17502233364, 0.18540204186, -4.93385895367, 3, 181378, 1.02162405535151E-04
346.61613995903, -0.03431851865, -2.74945221582, 4, -5237, 6.68020756457130E-04
173.30806997952, 0.29575909264, -14.03296284287, 5, 26788, 1.09998339712384E-03
13.63341568476, 0.25028196976, -18.56917097092, 6, 96, 1.27536054918252E-01
13.66083077700, -2.14271041643, -1.33601415374, 7, -778, 3.15864689649859E-01
13.60611040750, -0.21448938071, -61.87561302196, 8, -2340, 9.57478917081662E-03
27.55454988600, -1.28154198104, 8.73256722287, 9, -1596, 8.56364901331059E-02
13.77727494300, -0.15723698083, -0.61181997695, 10, -1276, 1.33743330759702E-02
27.10365333842, 0.06738187750, -3.99137203556, 11, 2992, 2.17928150447984E-03
-411.77887747408, 0.01664810323, 2.57763637009, 12, 7834, 2.09824536021795E-04
27.09267692660, -0.04829145814, -2.10033708026, 13, -1654, 3.10843568311545E-03
-205.88943873704, -0.10921036647, 2.94477474861, 14, -15739, 6.98301918830809E-04
13.69115772864, -0.07146589059, 2.39724535791, 15, -2472, 3.01288414361810E-03
9.10846048884, -0.00639759528, 6.16115357630, 16, -217, 5.44762246983538E-03
27.44323926226, -0.15838086315, -2.89994384817, 17, -3552, 4.58825378044224E-03
27.66676714572, -0.17673739937, 6.47681166312, 18, -2544, 7.23042231444426E-03
2190.35004466729, -0.23503726605, 0.14422425671, 19, -42683, 5.51950580521193E-04
26.87829334283, -0.04669089003, -9.53640897537, 20, -5727, 8.29786129130903E-04
9.13295078376, 0.39875540178, 1.55308373610, 21, 366, 8.55829600504843E-02
9.12068919638, -0.04242227089, 3.53442577241, 22, -223, 3.44509453473477E-02
6.83041538850, 0.03369465422, -1.01204648784, 23, 3433, 9.53751477200680E-04
9.10722051800, -0.03234233795, 5.39266631273, 24, -890, 4.09226346158531E-03
—- LTE —-
0.05979273556 :trend:
0.00047198787 :accel:
-0.66364644876 :K0:
112.89598974477 :level:
-3.03360212333, 0.51877492368, 0.73786871570 7
-0.43988927641, 1.32228073762, -2.31281903400 1
-16.71579250345, 0.59518157469, -1.05190059225 38
-17.59557105626, 0.71349326369, 1.70632936533 40
-36.51080994175, 0.85180116360, 0.43294772376 83
-24.63379947877, 0.57867026281, -1.09325137941 56
-43.54903836426, 0.96139937892, 0.29744313602 99
-17.15568177986, 0.43581289593, -2.94021319303 39
CC 0.7314006315 0.8285491958 2 1
PS C:UsersppgithubpukprlteMod-main-scratchrun> ../backup MZ_AMO_mf4-7hpt843-dtw-fewer-tidal3dtw–cc
From <https://gist.github.com/pukpr/e562138af3a9da937a3fb6955685c98f> PDO
0.00000000000 :offset:
-0.04516033782 :bg:
-0.04984433980 :impA:
-26.22379150965 :impB:
2.96284727411 :impC:
-0.11922311790:delA:
0.91647454694:delB:
0.00969425616:asym:
0.03519891167:ann1:
-1.74070752476:ann2:
-0.02316645559:sem1:
-4.91081809743:sem2:
-0.00007468286:year:
-0.13563941782:IR:
0.02467578154 :mA:
-0.00118562397 :mP:
-0.46607785688 :shiftT:
0.29687007041 :init:
—- Tidal —-
27.32166155400, -2.53062452787, -64.44250878617, 1, -1252247, 2.02102769668380E-04
27.21222081500, 0.15642555034, -2.17921272566, 2, 11596, 1.33739927797268E-03
1095.17502233364, 0.21969993615, -5.24463935814, 3, 214950, 1.02162405535151E-04
346.61613771009, -0.02985508423, -2.91520116531, 4, -4569, 6.68020756457130E-04
173.30806885505, 0.28241949162, -13.97410474991, 5, 25575, 1.09998339712384E-03
13.63341568476, 0.24771866795, -18.59675666466, 6, 94, 1.27536054918252E-01
13.66083077700, -2.15653373233, -1.33594307322, 7, -783, 3.15864689649859E-01
13.60611040750, -0.21388423887, -61.87874562449, 8, -2334, 9.57478917081662E-03
27.55454988600, -1.28598734584, 8.77434156186, 9, -1602, 8.56364901331059E-02
13.77727494300, -0.12513681786, -0.91263238847, 10, -1036, 1.33743330759702E-02
27.10365333842, 0.05810638778, -4.22550952477, 11, 2566, 2.17928150447984E-03
-411.77887430007, 0.01542841766, 1.62141119987, 12, 7253, 2.09824536021795E-04
27.09267692660, -0.08305048271, -1.88312982030, 13, -2772, 3.10843568311545E-03
-205.88943715003, -0.14177007161, 2.94238140496, 14, -20402, 6.98301918830809E-04
13.69115772864, -0.05983807599, 2.60618287031, 15, -2086, 3.01288414361810E-03
9.10846048884, -0.00693985341, 3.03625515988, 16, -227, 5.44762246983538E-03
27.44323926226, -0.17780363447, -2.63976045185, 17, -3975, 4.58825378044224E-03
27.66676714572, -0.16721641584, 6.67096750167, 18, -2413, 7.23042231444426E-03
2190.35004466729, -0.23776015880, 0.20698662570, 19, -43176, 5.51950580521193E-04
26.87829334283, -0.05387545723, -8.96682455256, 20, -6593, 8.29786129130903E-04
9.13295078376, 0.39928887802, 1.55153759441, 21, 367, 8.55829600504843E-02
9.12068919638, -0.04882734201, 3.55412674483, 22, -242, 3.44509453473477E-02
6.83041538850, 0.05588265057, -1.08534945410, 23, 5759, 9.53751477200680E-04
9.10722051800, -0.01855686555, 5.48726784782, 24, -553, 4.09226346158531E-03
—- LTE —-
0.01604601777 :trend:
0.00007068708 :accel:
-0.01874038041 :K0:
-30.71038508055 :level:
1.53349989208, 0.56262967909, -0.62951220221-3
-0.43922810843, 0.32083428736, 1.49292657813 1
-11.41993081907, 0.32016419817, 2.76403243963 26
-32.06365191509, 0.75245854089, 0.18973067728 73
-12.73761514435, 0.50863741228, -1.31719853726 29
-4.39228108426, 0.35568662293, 2.59292767461 10
-4.83150919269, 0.32547238273, 0.29173725300 11
-28.11059893926, 0.50819831553, 0.41397268527 64
CC 0.4714520341 0.7534052084 6 1
PS C:UsersppgithubpukprlteMod-main-scratchrun> ../backup MZ_AMO_mf4-7hpt843-dtw-fewer-tidal3dtw_pdo3dtwCCpt75
From <https://gist.github.com/pukpr/e562138af3a9da937a3fb6955685c98f> IOD-E
0.00000000000 :offset:
-0.00340131891 :bg:
-0.06083035390 :impA:
-36.34831563286 :impB:
2.92550389333 :impC:
-0.11971011752:delA:
0.91647454694:delB:
0.00969385742:asym:
0.03913345007:ann1:
-3.80723470775:ann2:
-0.11715720490:sem1:
0.87908218286:sem2:
-0.13547832660:IR:
0.02592492572 :mA:
-0.00118554827 :mP:
0.00000048871 :shiftT:
0.28926834907 :init:
—- Tidal —-
27.32166155400, -2.51567141624, -64.25914605946, 1, -1244849, 2.02102769668380E-04
27.21222081500, 0.12672638148, -2.51081528931, 2, 9376, 1.33739927797268E-03
1095.17502233364, 0.16958721198, -5.17161244193, 3, 165898, 1.02162405535151E-04
346.61613221592, -0.03054816284, -3.04566176294, 4, -4673, 6.68020756457130E-04
173.30806610796, 0.27247273168, -13.61464323568, 5, 24671, 1.09998339712384E-03
13.63341568476, 0.25331635755, -18.60092293518, 6, 99, 1.27536054918252E-01
13.66083077700, -2.16484356811, -1.34717230786, 7, -785, 3.15864689649859E-01
13.60611040750, -0.21035933282, -61.90678363465, 8, -2297, 9.57478917081662E-03
27.55454988600, -1.20642957152, 8.77902797052, 9, -1509, 8.56364901331059E-02
13.77727494300, -0.08719246376, -1.11280182667, 10, -752, 1.33743330759702E-02
27.10365333842, 0.06901327017, -4.02316890142, 11, 3067, 2.17928150447984E-03
-411.77886654595, 0.01491349736, 3.64272328061, 12, 7008, 2.09824536021795E-04
27.09267692660, -0.03760665924, -1.67759676683, 13, -1310, 3.10843568311545E-03
-205.88943327297, -0.15222661061, -3.15357009275, 14, -21900, 6.98301918830809E-04
13.69115772864, -0.06738379666, 2.94232347959, 15, -2337, 3.01288414361810E-03
9.10846048884, -0.00284566696, -4.07437761050, 16, -152, 5.44762246983538E-03
27.44323926226, -0.18574574530, -2.51784863419, 17, -4148, 4.58825378044224E-03
27.66676714572, -0.15488519384, 6.61040234343, 18, -2242, 7.23042231444426E-03
2190.35004466729, -0.26604665601, 0.27107030426, 19, -48301, 5.51950580521193E-04
26.87829334283, -0.04309292636, -11.10709517404, 20, -5293, 8.29786129130903E-04
9.13295078376, 0.40602391078, 1.55132975142, 21, 374, 8.55829600504843E-02
9.12068919638, -0.04988863818, 3.08957920953, 22, -245, 3.44509453473477E-02
6.83041538850, 0.04751289026, 6.61274724140, 23, 4882, 9.53751477200680E-04
9.10722051800, -0.00241189470, 5.83303094528, 24, -159, 4.09226346158531E-03
—- LTE —-
0.00069737167 :trend:
0.00003374325 :accel:
0.00364812671 :K0:
0.89105752510 :level:
1.52701032258, 0.07299621404, -0.89502075607 2
1.00114797790, 0.03877679976, -2.62012958485 1
15.01721966850, 0.04740570710, -2.92041086543 15
38.04362316019, 0.08525528364, 2.92854770809 38
90.10331801097, 0.08051511112, 1.25893726012 90
43.04936304969, 0.16020516118, 2.98478666198 43
53.06084282868, 0.08961750878, -2.38763971825 53
51.05854687288, 0.05699323235, -1.61016804494 51
12.01377573480, 0.07732756766, 1.32017564850 12
26.02984742539, 0.04376414491, 2.98517568109 26
DTW 0.8079031537 0.7833386989 1 1
PS C:UsersppgithubpukprlteMod-main-scratchrun> $env:CLIMATE_INDEX=”iod_e_1880.dat”
PS C:UsersppgithubpukprlteMod-main-scratchrun> ../restore MZ_AMO_mf4-7hpt843-dtw-fewer-tidal3dtw_pdo3dtwCC-n4f_dtw_iode2cc2dtw
From <https://gist.github.com/pukpr/e562138af3a9da937a3fb6955685c98f>
IOD-W
0.00000000000 :offset:
-0.00000024909 :bg:
-0.01975821597 :impA:
-96.72401868012 :impB:
2.87995256418 :impC:
-0.11971507016:delA:
0.91647454694:delB:
0.00969385742:asym:
0.07858206506:ann1:
-1.52412556489:ann2:
0.03453772579:sem1:
1.45146842375:sem2:
0.02588617365 :mA:
-0.00118596071 :mP:
-0.17980319567 :shiftT:
0.28576763553 :init:
—- Tidal —-
27.32166155400, -2.50488734063, -64.26347325186, 1, -1239513, 2.02102769668380E-04
27.21222081500, 0.13582035472, -2.40962448412, 2, 10056, 1.33739927797268E-03
1095.17502233364, 0.16885305474, -5.23040910775, 3, 165179, 1.02162405535151E-04
346.61613283807, -0.03118634736, -3.00863641309, 4, -4768, 6.68020756457130E-04
173.30806641903, 0.28120932838, -13.59541848023, 5, 25465, 1.09998339712384E-03
13.63341568476, 0.25526777635, -18.59076375920, 6, 100, 1.27536054918252E-01
13.66083077700, -2.17293931783, -1.34460667404, 7, -788, 3.15864689649859E-01
13.60611040750, -0.23133087931, -61.84321861696, 8, -2516, 9.57478917081662E-03
27.55454988600, -1.19300815773, 8.75821577495, 9, -1493, 8.56364901331059E-02
13.77727494300, -0.09024210264, -1.12833144375, 10, -775, 1.33743330759702E-02
27.10365333842, 0.05620595110, -4.16502874806, 11, 2479, 2.17928150447984E-03
-411.77886742400, 0.01275630186, -4.05928761982, 12, 5980, 2.09824536021795E-04
27.09267692660, -0.01455053471, -0.57017122613, 13, -568, 3.10843568311545E-03
-205.88943371200, -0.17115351076, 3.16868283322, 14, -24610, 6.98301918830809E-04
13.69115772864, -0.06444120644, 3.04408214935, 15, -2239, 3.01288414361810E-03
9.10846048884, -0.00703202215, -3.15985956496, 16, -229, 5.44762246983538E-03
27.44323926226, -0.17234454724, -2.53082639620, 17, -3856, 4.58825378044224E-03
27.66676714572, -0.14723569460, 6.67602588532, 18, -2136, 7.23042231444426E-03
2190.35004466729, -0.26057731853, 0.27052809291, 19, -47310, 5.51950580521193E-04
26.87829334283, -0.06081585566, -11.15881540057, 20, -7429, 8.29786129130903E-04
9.13295078376, 0.40610643956, 1.54950290715, 21, 375, 8.55829600504843E-02
9.12068919638, -0.05152479403, 3.06651799102, 22, -250, 3.44509453473477E-02
6.83041538850, 0.04161556846, 5.91740819301, 23, 4263, 9.53751477200680E-04
9.10722051800, 0.00138340722, -6.44642390781, 24, -66, 4.09226346158531E-03
—- LTE —-
0.00021988820 :trend:
0.00005489242 :accel:
0.03086173638 :K0:
-0.32684274059 :level:
1.39130487013, 0.02996125472, -0.64087188043 1
0.99909167009, 0.01317391401, 1.56432612734 1
92.91552531837, 0.04679770529, -0.60170645282 93
37.96548346342, 0.12840097471, 2.42576202158 38
33.96911678306, 0.06607517711, 1.64012624656 34
42.96094181387, 0.07794146967, 2.47865575137 43
52.95185851477, 0.08501610669, -2.93028235325 53
59.94550020540, 0.04129640706, 1.28417307222 60
11.98910004108, 0.05699315341, 0.66251089405 12
32.97002511297, 0.04841326113, -2.95377816024 33
DTW 0.8865981674 0.8624643549 2 1
PS C:UsersppgithubpukprlteMod-main-scratchrun> $env:CLIMATE_INDEX=”iod_w_1880.dat”
PS C:UsersppgithubpukprlteMod-main-scratchrun> ../restore MZ_AMO_mf4-7hpt843-dtw-fewer-tidal3dtw_pdo3dtwCC-n4f_dtw_iode2cc2dtw-iodw-dtw2
From <https://gist.github.com/pukpr/e562138af3a9da937a3fb6955685c98f> NINO34
0.00000000000 :offset:
0.12172886586 :bg:
-0.06469284236 :impA:
-28.21466736882 :impB:
2.34382485155 :impC:
-0.11971628224:delA:
0.91647454694:delB:
0.00969385742:asym:
0.07353286697:ann1:
-1.55951682491:ann2:
0.01894080900:sem1:
0.04489997990:sem2:
-0.13547832660:IR:
0.02613756960 :mA:
-0.00118715544 :mP:
-0.54725360949 :shiftT:
0.28634744098 :init:
—- Tidal —-
27.32166155400, -2.50694614192, -64.26298892960, 1, -1240531, 2.02102769668380E-04
27.21222081500, 0.13781601731, -2.54505868011, 2, 10205, 1.33739927797268E-03
1095.17502233364, 0.18302007320, -5.28306585912, 3, 179046, 1.02162405535151E-04
346.61613257865, -0.02884898784, -3.06396650751, 4, -4419, 6.68020756457130E-04
173.30806628933, 0.27068429491, -13.60322230251, 5, 24508, 1.09998339712384E-03
13.63341568476, 0.25180256402, -18.58509677202, 6, 97, 1.27536054918252E-01
13.66083077700, -2.15473694011, -1.34667633705, 7, -782, 3.15864689649859E-01
13.60611040750, -0.21428501316, -62.01979369409, 8, -2338, 9.57478917081662E-03
27.55454988600, -1.20942584096, 8.77658621425, 9, -1512, 8.56364901331059E-02
13.77727494300, -0.13080341506, -0.85426932214, 10, -1078, 1.33743330759702E-02
27.10365333842, 0.05368328962, -3.97054488967, 11, 2363, 2.17928150447984E-03
-411.77886705788, 0.00549238281, 3.09730980841, 12, 2518, 2.09824536021795E-04
27.09267692660, -0.03799766810, -1.92327808566, 13, -1322, 3.10843568311545E-03
-205.88943352894, -0.12632329668, 3.10191661439, 14, -18190, 6.98301918830809E-04
13.69115772864, -0.06400641534, 2.59592359791, 15, -2224, 3.01288414361810E-03
9.10846048884, -0.01171746846, -2.70368981342, 16, -315, 5.44762246983538E-03
27.44323926226, -0.17390178761, -2.48853685136, 17, -3890, 4.58825378044224E-03
27.66676714572, -0.15478083412, 6.65335186997, 18, -2241, 7.23042231444426E-03
2190.35004466729, -0.25849990438, 0.27111199822, 19, -46934, 5.51950580521193E-04
26.87829334283, -0.05512321234, -10.95313621354, 20, -6743, 8.29786129130903E-04
9.13295078376, 0.40581873999, 1.55207874801, 21, 374, 8.55829600504843E-02
9.12068919638, -0.04550895063, 3.12924363391, 22, -232, 3.44509453473477E-02
6.83041538850, 0.07505325823, -6.15814890297, 23, 7769, 9.53751477200680E-04
9.10722051800, -0.00539490486, 4.87571157437, 24, -232, 4.09226346158531E-03
—- LTE —-
0.01537903936 :trend:
0.00013382687 :accel:
-0.14834086206 :K0:
28.80370615285 :level:
1.35675362986, 0.18381771529, -1.68904674392 1
1.00090338691, 0.16525062431, 1.47599220998 1
60.05420321456, 0.11922192381, -0.70363822921 60
8.00722709528, 0.24951876821, -2.43462246972 8
87.07859466112, 0.13692152511, -1.32263703401 87
43.03884563710, 0.52956041155, -2.39895181155 43
53.04787950620, 0.31550950495, -0.17093435539 53
23.02077789892, 0.17457455157, -1.83320738166 23
49.04426595856, 0.10197491439, -2.13745816788 49
45.04065241092, 0.15745726664, 1.87861526681 45
CC 0.4036547039 0.7066712791 1 1
PS C:UsersppgithubpukprlteMod-main-scratchrun> ../backup MZ_AMO_mf4-7hpt843-dtw-fewer-tidal3dtw_pdo3dtwCC-n4f_dtw_n34
PS C:UsersppgithubpukprlteMod-main-scratchrun> $env:CLIMATE_INDEX=”nino34_knmi_1880.dat”
From <https://gist.github.com/pukpr/e562138af3a9da937a3fb6955685c98f>
NAO
0.00000000000 :offset:
0.00000000000 :bg:
-0.08700993232 :impA:
-362.33335612822 :impB:
2.54138674706 :impC:
-0.12011047994:delA:
0.90902554415:delB:
0.00984273631:asym:
-0.37469623598:ann1:
-3.13611187164:ann2:
-0.15605060568:sem1:
1.16487966340:sem2:
-0.13547582520:IR:
0.02813950334 :mA:
-0.00118960648 :mP:
0.33434379767 :shiftT:
0.28148427038 :init:
—- Tidal —-
27.32166155400, -2.53947967447, -64.27302822540, 1, -1256629, 2.02102769668380E-04
27.21222081500, 0.12174050940, -2.52214853474, 2, 9003, 1.33739927797268E-03
1095.17502233364, 0.14754253883, -5.18949626545, 3, 144320, 1.02162405535151E-04
346.61613089385, -0.02638273872, -2.78102587862, 4, -4049, 6.68020756457130E-04
173.30806544692, 0.27826685703, -13.59739719918, 5, 25197, 1.09998339712384E-03
13.63341568476, 0.24541340178, -18.54947481133, 6, 92, 1.27536054918252E-01
13.66083077700, -2.19713830114, -1.34035823682, 7, -796, 3.15864689649859E-01
13.60611040750, -0.22627456323, -68.15827245348, 8, -2463, 9.57478917081662E-03
27.55454988600, -1.22640600883, 8.75059193658, 9, -1532, 8.56364901331059E-02
13.77727494300, -0.10208288216, -1.21758856906, 10, -863, 1.33743330759702E-02
27.10365333842, 0.05201910208, -3.95785607829, 11, 2287, 2.17928150447984E-03
-411.77886468006, 0.01433012478, 2.32142686478, 12, 6730, 2.09824536021795E-04
27.09267692660, 0.00000359447, 0.00186526367, 13, -100, 3.10843568311545E-03
-205.88943234003, -0.18661086283, 3.16438299032, 14, -26824, 6.98301918830809E-04
13.69115772864, -0.02853042882, 3.45129982909, 15, -1047, 3.01288414361810E-03
9.10846048884, -0.01034802961, -2.65192542982, 16, -290, 5.44762246983538E-03
27.44323926226, -0.17999037219, -2.48188015077, 17, -4023, 4.58825378044224E-03
27.66676714572, -0.15642278272, 6.60514672441, 18, -2263, 7.23042231444426E-03
2190.35004466729, -0.28685659442, 0.27075362104, 19, -52071, 5.51950580521193E-04
26.87829334283, -0.05347401711, -11.28622903648, 20, -6544, 8.29786129130903E-04
9.13295078376, 0.40694660893, 1.54757369663, 21, 375, 8.55829600504843E-02
9.12068919638, -0.05197338309, 2.92370477141, 22, -251, 3.44509453473477E-02
6.83041538850, 0.03477487270, 6.44374266222, 23, 3546, 9.53751477200680E-04
9.10722051800, 0.00952413256, -8.68829647094, 24, 133, 4.09226346158531E-03
—- LTE —-
0.00624477425 :trend:
-11.77041019155 :level:
5.37824617507, 0.22731212344, 0.77630091358 6
0.96974762021, 0.11246663896, 1.95557433740 1
73.70081913614, 0.17056404489, -2.88456895615 76
42.66889528934, 0.24692069502, -2.24549084529 44
68.85208103507, 0.17404047334, 1.34736523969 71
65.94283817444, 0.35935044464, 0.07525726890 68
32.97141908722, 0.26692332240, -1.68314180522 34
33.94116670743, 0.12501239853, -2.98448940709 35
76.61006199677, 0.21033309001, 0.90719810749 79
39.75965242871, 0.14367654984, 1.77439913697 41
DTW 0.5349850466 0.6106786413 8 1
PS C:UsersppgithubpukprlteMod-main-scratchrun> $env:CLIMATE_INDEX=”nao_median5_1880.dat”
PS C:UsersppgithubpukprlteMod-main-scratchrun> ../restore MZ_AMO_mf4-7hpt843-dtw-fewer-tidal3dtw_pdo3dtwCC-n4f_dtw_iode2cc2dtw-iodw-dtw2-nao
From <https://gist.github.com/pukpr/e562138af3a9da937a3fb6955685c98f> GLAAM
-0.00000000000 :offset:
-0.17480663867 :bg:
3.00925598373 :impA:
1.74954852031 :impB:
2.70469411057 :impC:
-0.00000001488:delA:
0.41648277006:delB:
-0.12344674643:asym:
-1.40797293381:ann1:
3.47948536679:ann2:
1.06749055425:sem1:
0.92258930620:sem2:
-0.00007018551:year:
-0.10370534736:IR:
0.01823279389 :mA:
0.00121103613 :mP:
-0.27657623956 :shiftT:
0.30582857565 :init:
—- Tidal —-
27.32166155400, -2.53218643880, -64.41814647642, 1, -1253020, 2.02102769668380E-04
27.21222081500, 0.22178845296, -2.14554907753, 2, 16484, 1.33739927797268E-03
1095.17502233364, 0.17543401548, -4.67024247251, 3, 171621, 1.02162405535151E-04
346.61614176023, -0.03752612469, -2.67102572426, 4, -5718, 6.68020756457130E-04
173.30807088011, 0.30417842687, -14.02428878129, 5, 27553, 1.09998339712384E-03
13.63341568476, 0.26957512529, -18.35865529596, 6, 111, 1.27536054918252E-01
13.66083077700, -2.13857113820, -1.33697987110, 7, -777, 3.15864689649859E-01
13.60611040750, -0.21524306251, -61.97757723148, 8, -2348, 9.57478917081662E-03
27.55454988600, -1.23586382098, 8.74224891627, 9, -1543, 8.56364901331059E-02
13.77727494300, -0.20848705156, -0.62671487582, 10, -1659, 1.33743330759702E-02
27.10365333842, 0.07995645795, -4.01378766210, 11, 3569, 2.17928150447984E-03
-411.77888001617, 0.03341934982, 1.69764287859, 12, 15827, 2.09824536021795E-04
27.09267692660, -0.02977003277, -1.86352368107, 13, -1058, 3.10843568311545E-03
-205.88944000809, -0.10183588518, 2.80732234269, 14, -14683, 6.98301918830809E-04
13.69115772864, -0.07574093201, 2.33908257668, 15, -2614, 3.01288414361810E-03
9.10846048884, -0.00725611524, -5.69246217810, 16, -233, 5.44762246983538E-03
27.44323926226, -0.18683864564, -2.96713601846, 17, -4172, 4.58825378044224E-03
27.66676714572, -0.16896935454, 6.48822613251, 18, -2437, 7.23042231444426E-03
2190.35004466729, -0.26822510773, 0.07157403949, 19, -48696, 5.51950580521193E-04
26.87829334283, -0.01078077267, -9.36726055213, 20, -1399, 8.29786129130903E-04
9.13295078376, 0.39708870202, 1.54696844924, 21, 364, 8.55829600504843E-02
9.12068919638, -0.04579748475, 3.50341277058, 22, -233, 3.44509453473477E-02
6.83041538850, 0.04621671918, -0.79939933235, 23, 4746, 9.53751477200680E-04
9.10722051800, -0.04510925094, 5.32777248261, 24, -1202, 4.09226346158531E-03
—- LTE —-
0.00000000000 :trend:
0.00000000000 :accel:
-15.44561171735 :K0:
7.99735508774 :level:
0.43690920162, 11.62097786256, -1.66819445333 1
36.26346373443, 11.42446576316, 1.84014870403 83
44.56473856520, 3.60810657402, 3.02251500301 102
1.31072760486, 2.09640538231, 0.05115776960 3
CC 0.7131493931 0.7667316804 2 1
PS C:UsersppgithubpukprlteMod-main-scratchrun> ../restore MZ_AMO_mf4-7hpt843-dtw-fewer-tidal3dtw–ccplus2_glaam6
From <https://gist.github.com/pukpr/e562138af3a9da937a3fb6955685c98f>
https://www.researchgate.net/profile/Jinyu-Wang-35/publication/385561149_Insights_of_Dynamic_Forcing_Effects_of_MJO_on_ENSO_from_a_Shallow_Water_Model/links/672ae27d5852dd723caaf86a/Insights-of-Dynamic-Forcing-Effects-of-MJO-on-ENSO-from-a-Shallow-Water-Model.pdf
Insights of Dynamic Forcing Effects of MJO on ENSO from a Shallow Water Model
“The Madden–Julian oscillation (MJO) is believed to be able to trigger El Niño–
Southern Oscillation (ENSO) events and influence our understanding of the fundamental nature of ENSO. In this study, spatiotemporally filtered forcing experiments are implemented on an anomalous shallow water model. The results show that MJO is more important for improving the simulation of surface zonal currents rather than the sea surface heights and tends to contribute to the onset of El Niño events rather than La Niña events through triggering strong intraseasonal oceanic Kelvin waves.”
so wrong, causality reversed
https://www.threads.net/@therollingstones/post/DCW5KmOOFEx
“You have the sun, you have the moon, you have the air that you breathe – and you have the Rolling Stones!” – Keith Richards
Thanks to gravity for the rolling assist
https://ams.confex.com/ams/105ANNUAL/meetingapp.cgi/Paper/456689
Low dimensional chaos explains ENSO asymmetry and period
Abstract
Fundamental features of the El Niño-Southern Oscillation (ENSO) phenomenon, such as its 4 year period and its asymmetric temporal evolution, remain poorly understood despite decades of research. We study the chaotic dynamics of ENSO by reconstructing its phase space using high frequency sea-surface temperature observations. We find that ENSO’s strange attractor can be characterized by a set of unstable periodic orbits, with periods of 1-4 years. By visiting these orbits the system rapidly explores the warm region of the phase space (El Niño to La Niña transitions) and slowly visits the cold region of the phase space (La Niña to El Niño transitions), which leads to the attractor to curl in a specific way. Based on this result we propose that ENSO dynamics can be modeled with a periodically forced Takens-Bogdanov bifurcation. The asymmetric oscillation obtained with the Takens-Bogdanov bifurcation mathematically models the thermocline coupling and the periodic forcing the influence of the annual cycle. This low-dimensional chaotic system explains the occurrence of multi-year La Niñas and the mean period of 4 years of the oscillation. The proposed dynamics is consistent with previous studies, which show that the annual cycle can synchronize with the slow decay of La Niña, which is in turn controlled by the nonlinearity of the thermocline feedback. We find that the majority of the state-of-the-art CMIP models present an accurate ENSO dynamics, which we show by analyzing the topology of their ENSO phase space and comparing it to the observed. Differences among models are associated with the persistence of the unstable periodic orbits and the interplay between chaotic behavior and noise, which characterizes the complexity of the resulting signal. These different behaviors can be modeled with the same low dimensional model under slightly different forcing parameters, thus, the proposed underlying dynamics is appropriate to describe all model behaviors.
4 year => Mf~3.8y, Mm~3.9y
“On the spatial double peak of the 2023–2024 El Niño event”
https://www.nature.com/articles/s43247-024-01870-1
El Niño–Global Atmospheric Oscillation as the main mode of interannual climate variability
In [76], it was shown that these oceanic baroclinic waves form a single dynamical
system whose average period is 4 years (the effective period varies between 1.5 and 7 years).
This quadrennial quasi-stationary wave (QSW) is coupled to an annual QSW formed by
a first baroclinic, fourth meridional mode equatorial Rossby wave. The annual QSW is
resonantly forced by easterlies, while the quadrennial QSW is partly forced by ENSO,
which is a component of the GAO.
Long Wave Resonance in Tropical Oceans and Implications on Climate: the Atlantic Ocean
Pingback: 20yrs of blogging in hindsight | GeoEnergy Math
Pingback: QBO Metrics | GeoEnergy Math
Pingback: Tidal Gauge Differential | GeoEnergy Math
Pingback: Difference Model Fitting | GeoEnergy Math
Pingback: Topology shapes climate dynamics | GeoEnergy Math