An interesting Nature paper “Seasonal overturn and stratification changes drive deep-water warming in one of Earth’s largest lakes” focusing on Lake Michigan
Note the strong impulse at the thermocline that occurs on an annual basis coinciding to an overturning event, panel B below.
This is likely the same instability that occurs along the equatorial Pacific ocean thermocline as the differences in density become smaller, and the gravitational tidal force at that moment provides an impulse to slosh the interface, leading to ENSO events.
A simple premise, yet barely considered, except in Mathematical Geoenergy, chapter 12.
In a thesis Hydrodynamic modelling of Lake Ontario, it was mentioned that “displacement of
water masses leads to rhythmic oscillations in the entire lake. These long waves or seiches have wavelengths of the same order of magnitude as the basin dimensions. Seiches are reflected at the lake boundaries and combine into standing wave patterns on the thermocline [18]”. The cited book on limnology is instructive.
Available on Google Books, likely a better description of thermocline dynamics than you will find in an oceanography textbbook.
]]>This is a powerful technique on its own as it is used frequently (and depended on) in machine learning models to eliminate poorly performing trials. But it gains even more importance when new data for validation will take years to collect. In particular, consider the arduous process of collecting fresh data for El Nino Southern Oscillation, which will take decades to generate sufficient statistical significance for validation.
So, what’s necessary in the short term is substantiation of a model’s potential validity. Nothing else will work as a substitute, as controlled experiments are not possible for domains as large as the Earth’s climate. Cross-validation remains the best bet.
As a practical aside, CV is not for the faint-of-heart, since anyone doing cross-validation will get accused of cheating (what they apparently refer to as researcher Degrees Of Freedom). Well, of course the “unexplored” data is there for anyone to see, so everyone is in the same boat when it comes to avoiding tainting the results or priming the pump, so to speak. Yet, this paranoia is strong enough that critics may use the rDOF excuse to completely ignore cross-validation results (see link above for example of me trying to get any interest in cross-validation of a Chandler wobble model — taint goes both ways apparently, in this case as prejudice i.e. biased pre-judging as to the modeler’s intent).
An effective yet non-controversial example is cross-validation of a delta Length-of-Day (dLOD) model. The LOD data is from Paris IERS and is transformed into an acceleration by taking the differential, thus dLOD. A cross-validation can easily be generated by taking any interval in the time series, fitting that to an appropriate model of the geophysics, and then extrapolating over the rest of the length (which stretches from 1962 to the current day). To do that, first we need to select the physical factors that will act to modify the angular momentum of the Earth’s rotation — these factors are simply the tidal torques as generated by the moon and sun, tabulated by R.D. Ray in “Long-period tidal variations in the length of day” (see Table 3, column labeled V_{0}/g for forcing values, with amplitude scaled by frequency since this is a differential LOD).
The strongest 30 tidal factors are arranged as a Fourier series and then optimally fit using a maximum linear regression algorithm (source code). The results (raw fit here) are shown below for 3 orthogonal training intervals, each approximately 20 years each (1962-1980, 1980-2000, and 2000-present).
It appears that Ray may have evaluated his predicted forcing values against similar LOD data, as the cross-validation agreement is very good across the board. He writes:
“The only realistic test of this new tidal LOD model is to examine how well it removes tidal energy from real LOD measurements. To test that we use the SPACE2008 time series of Earth rotation variations produced by Ratcliff and Gross [2010] from various types of space-geodetic measurements. Their method employs a specially designed Kalman filter to combine disparate types of measurements and to produce a time series with daily sampling interval; see also Gross et al. [1998]. After computing and subtracting the new LOD tidal model from the SPACE2008 data, we examine the residual spectrum for the possible presence of peaks at known tidal frequencies.”
This is an excellent example of effective cross-validation as the model is essentially stationary across the entire interval, indicating that the tidal factors represent the actual torque controlling the fastest cycling in the Earth’s LOD. Yes, it is possible that Ray applied his “researcher degrees of freedom” to further calibrate his tabulated tidal factors against this data, but it doesn’t detract from the excellent stationarity of the model itself. So as with a conventional tidal analysis, it doesn’t matter if a tidal model is calibrated via historical data cross-validation or from future data, as the foundational model has withstood the test of time as well as being internally self-consistent.
With that as a blueprint, we now enter the realm of non-linear model fitting a la ENSO. The relevant steps are reviewed in a recent blog post, which starts from the same set of tidal factors calibrated from dLOD data as described above. Keep in mind that allowing a researcher at least a few degrees of freedom to experiment with is the same as allowing them insight and educated guesses. Research would never advance without allowing flexibility when treading into unknown waters. So, the insight is to seed the initial model fitting with a few nonlinear modulation factors that represent the possible standing-wave modes of ENSO — one low-frequency modulation, and a high-frequency modulation set as a 7 & 14x harmonic of the fundamental, representing Tropical Instability Waves. The result of fitting the LTE model (using the GEM software) is shown below with the excluded-from-training intervals shown. Since these intervals were considered pristine from the point of view of the randomly mutating fitting process, any correlation between the model and the data within these intervals should be considered significant.
Click on the image’s link to magnify and get a sense on how well the model works in the highlighted validation intervals. There are essentially the 3 standing wave nodes (lower left), and the dLOD starting tidal factors (upper right) that are gradually varied to arrive at the final fit. It’s not perfect, but enough of the peaks and valleys align that not much additionally fitting is needed to model the data with a high correlation-coefficient across the entire time-series. As an example, including the pre-1880 ENSO data (which is somewhat iffy apart from the late-1870’s El Nino peak) generates this fit:
This “from scratch” cross-validation differs from the alternate approach of fitting to the entire interval and then excluding a portion before refitting, which is more suspect to bias (even though it can also show the effects of over-fitting as demonstrated in this experiment).
The difficulty in this scratch process, in contrast to the properties of the underlying dLOD model, is that the non-linear transformation required of LTE is much more structurally sensitive than the linear transformation of pure harmonic tidal analysis. For example, harmonic tidal analysis is essentially:
so changes in k or F(t) reflect as a linear scaling in the output of f(t).
Whereas with the non-linear LTE model
so that changes in k or F(t) can cause f(t) to swing wildly in both positive and negative direction.
The bottom-line is that the cross-validation results can’t be denied, but other researchers need to be involved in the process to improve the model enough to make it production quality. The LTE fitting software on GitHub is fast (it takes just a few minutes for the model to start aligning) and with a faster multi-CPU core (say try a 128 core) it would appeal to the scientific computing enthusiasts. As I designed the software to use all cores available, the speed-up would be nearly linear with number of cores. Could be down to minutes for a full fit, and thus very amenable to rapid turnaround experimentation.
]]>A highly esteemed climate scientist, Isaac Held, even participated and voiced his opinion on how feasible that would be. Eventually the forum decided to concentrate on the topic of modeling El Nino cycles, starting out with a burst of enthusiasm. The independent track I took on the forum was relatively idiosyncratic, yet I thought it held promise and eventually published the model in the monograph Mathematical Geoenergy (AGU/Wiley) in late 2018. The forum is nearly dead now, but there is recent thread on “Physicists predict Earth will become a chaotic world”. Have we learned nothing after 10 years?
My model assumes that El Nino/La Nina cycles are not chaotic or random, which is still probably considered blasphemous. In contrast to what’s in the monograph, the model has simplified, and a feasible solution can be mapped to data within minutes. The basic idea remains the same, explained in 3 parts.
The fitting process is to let all the parameters to vary slightly and so I use the equivalent of a gradient descent algorithm to guide the solution. The impulse month is seeded along with starting guesses for the two slowest wavenumbers. Another MLR algorithm is embedded to estimate the amplitude and phases required.
The multi-processing software is at https://github.com/pukpr/GeoEnergyMath/wiki
Recently it has taken mere minutes to arrive at a viable model fit to the ENSO data (the ENS ONI monthly data (1850 – Jul 2022)), starting with the initially calibrated dLOD factors. Each of the tidal factors is modified slightly but the correlation coefficient is still at 0.99 of the starting dLOD.
Even with that, the only way to make a convincing argument is to apply cross-validation during the fitting process. A training interval is used during the fitting and the model is extrapolated as a check once the training error is minimized. Even though the model is structurally sensitive, it does not show wild over-fitting errors. This is explainable as only a handful of degrees of freedom are available.
So this demonstrates that the behavior is stationary and definitely not chaotic, only obscured by the non-linear modulation applied to tidally forced waveform.
]]>Amazing number of harmonics
https://www.sciencedirect.com/science/article/pii/B9780128215128000177
]]>The most important mechanism for turbulence production in equatorial parallel shear flows is the inflectional instability, which operates at local maxima of the mean shear profile (Smyth and Carpenter, 2019). In the presence of stable stratification, inflectional instability is damped, but it may yet grow, provided that the minimum value of Ri is less than critical. In this case, the process is termed Kelvin–Helmholtz (KH) instability.
The (nearly) common forcing
with the applied LTE of a 180° phase difference
leads to adequately fitted models to the respective time series
The fact that the fundamental (and 7th harmonic) are aligned between ENSO and AMO strongly suggest that the standing-wave wavenumbers are not governed by the basin geometry but are more of a global characteristic that remains coherent across the land masses. The Atlantic basin has a smaller width than the Pacific so intuitively one might have predicted unique wavenumbers that would fit within the bounding coastlines, but this is perhaps not the case.
Instead, the LTE modulation wraps around the earth and produces an anti-phase relationship in keeping with the approximately 180° longitudinal difference between the Atlantic and Pacific.
Any additional phase shift ϕ can also easily produce the anomalously large multidecadal variations in the AMO due to the biasing properties of the sinusoidal LTE modulation.
Just a matter of time until machine-learning algorithms start discovering these patterns. But, alas, they may not know how to deal with the findings
NdGT has a point — you do see the earth’s shadow moving across the moon, but once covered, a #lunarEclipse just looks like a duller moon (similar “new moons” are also observed like clockwork and thus take the excitement out of it). Yet the alignment of tidal forces does a number on the Earth’s climate that is totally cryptic and thus overlooked. Perhaps old Dr. Neil would find more interesting tying lunar cycles to climate indices such as ENSO and the Indian Ocean Dipole? It’s all based on geophysical fluid dynamics. Oh, and a bonus — discriminate on the variability of IOD and there’s the underlying AGW trend!
BTW, a key to this IOD model fit is to apply dual annual impulses, one for each monsoon season, summer and winter. Whereas, ENSO only has the spring predictability barrier.
Added 5/17/2022:
Several flavors of the IOD time-series available, referred to as the Dipole Mode Index (DMI). I originally used the JAMSTEC data-set : https://www.jamstec.go.jp/aplinfo/sintexf/e/iod/dipole_mode_index.html
The Climate Explorer sets are also available
A combination of the two above give the following:
Interesting about the IOD dipole mode index in contrast to ENSO indices such as NINO34 is in the number of inflection points, in the time-series. Qualitatively, by comparing the two visually, the IOD DMI appears twice as busy and thus should be correspondingly much more difficult to fit.
Yet, with the application of the semi-annual impulse, the details of the index are modelled nearly as well, with the cross-validation of the extrapolated intervals encouraging (excepting for the powerful El Nino event of 1879 not captured in the IOD model).
As with all these models, the tidal forcing input was allowed to vary from the dLOD calibration, primarily so the iterative fitting procedure could more easily not get stuck in local minima.
This is the model fit to dLOD alone
This is the dLOD model when optimizing the IOD DMI correlation with an iterated LTE modulation — i.e. both tidal forcing and LTE modulation were allowed to vary during the fitting process.
Subtle differences in the tidal factor distribution, as the correlation coefficient drops from 0.97 to ~0.93.
Some variation due to the harmonic of Mm. Mf vs Msf are cumulatively the same as they differ by the semi-annual influence of the sun. Mt’ (prime) is Mm and Mf’ interaction. Some additional forcing from the monthly Draconic and Tropical factors which is entirely plausible given the asymmetry of north vs south hemisphere.
The semi-annual impulse is really the breakthrough in this analysis, as that may be the key to modeling the faster cycling, much like it is used to model the noisy-appearing NOA and AO indices, as shown in a previous post. In retrospect, those are also impressive fits featuring a dominant LTE modulation. Will likely revisit the NOA and AO indices with the added knowledge of a dLOD calibrated IOD model.
]]>The premise of the paper is that the ocean will show modulation of mixing with a cycle of ~18 years corresponding to the 18.6-year lunar declination cycle. That may indeed be the case, but it likely pales in comparison to the other so-called long-period tidal cycles. In particular, every ~2 weeks the moon makes a complete north-south-north declination cycle that likely has a huge impact on the climate as it sloshes the subsurface thermocline (cite the paper by Lin & Qian^{1}). Unfortunately, this much shorter cycle is not directly observed in the observational data, making it a challenge to determine how the pattern manifests itself. In the following, I will describe how this is accomplished, referring to the complete derivation found in Chapter 12 of Mathematical Geoenergy^{2}.
Consider that the 2-week lunar declination cycle is observed very clearly in the Earth’s rotational speed, measured in terms of small transient changes in the length of day (LOD). From the IERS site, we can plot the differential LOD (dLOD) and fit to the known tidal factors, leaving a clean closed-form signal that one can use as a forcing function to evaluate the ocean response, in this case comparing it to the well-defined ENSO climate index.
The 18.6-year nodal cycle can be seen in the modulation of the cyclic dLOD data. At a higher resolution, the comparison is as follows:
To do that, we first make the assumption that the tidal cycle is modulated on an annual cycle, corresponding to the well-known “spring predictability barrier”. So, by integrating a sequence of May impulses against the value of the tidal forcing at that point, the following time series is generated.
Obviously, this does not match the ENSO NINO34 signal, but assuming that the subsurface response is non-linear (derivation in cite #2 below) and creates standing wave-modes based on the geometry of the ocean basin, then one can use a suitable transformation to potentially extract the pattern. The best approach based on the solution to the shallow-water wave model (i.e. Laplace’s Tidal Equations) is to map the input forcing (graph above) to the output corresponding to the NINO34 index, using a Fourier series expansion.
The result is the Laplace’s Tidal Equation (LTE) modulation spectra, shown below in a particular cross-validation configuration. Here, the NINO34 data is split into 2 halves, one time-series taken from 1870-1945 and the second from 1945-2020. The spectra were calculated individually and then multiplied point-by-point to identify long-lived stationary standing-wave nodes in the modulation. Thus, it isolates modulations that are common to each interval.
This is a log-plot, so the peak excursions shown are statistically significant and so can be modeled by a handful of quantifiable standing-wave modulations. The lowest wavenumber modulations are associated with the ENSO dipole modes and the higher wavenumber modulations are potentially associated with tropical instability waves (TIW)^{2}.
As a final step, by applying this set of modulations to the lunisolar forcing (the blue chart above), a fit to the NINO34 time-series results. The chart shown below is a very good fit and can be cross-validated via several approaches^{10}.
The mix of incommensurate tidal factors, the annual impulse, and a nonlinear response function is what causes the highly erratic nature of the ENSO waveform. It is neither chaotic nor random, as some researchers claim but instead is deterministically tied to the tidal and annual cycles, much like conventional tidal cycles have proven over the course of time.
To further quantify the decomposition of the tidal factors that force both the dLOD and the sloshing ENSO response, the paper by Ray and Erofeeva is vital^{8}. When trying to understand the assignment of frequencies, note that after the annual impulse is applied, the known tidal factors corresponding to such tidal factors labelled Mf, Mm, etc get shifted from normal positions due to signal aliasing (see chart below in gray). This is a confusing factor to those who have not encountered aliasing before. As an example, the long-term modulation (>100 years) displayed in the blue chart above is due to the aliased 9.133 day Mt tidal factor, which almost synchronizes with the annual cycle, but the amount it is off leads to a gradual modulation in the forcing — so overall confusing in that a 9 day cycle could cause multidecadal changes.
Ding & Chao^{9} provide an independent analysis of LOD that provides a good cross-check to the non-aliased cross-factors. It may be possible to use lunar ephemeris data to calibrate the forcing but that adds degrees-of-freedom that could lead to over-fitting ^{10}.
The reason that Lin & Qian were not able to further substantiate their claim of tidal forcing lies in that they could not associate the seasonal aliasing and a nonlinear mapping against their observations, only able to demonstrate the cause and effect of tidal forcing on the thermocline and thereby ruling out wind forcing. Other sources to cite are “Topological origin of equatorial waves” ^{4} and “Solar System Dynamics and Multiyear Droughts of the Western USA” ^{5}, the latter discussing the impact of axial torques on the climate. Researchers at NASA JPL including J.H. Shirley, C. Perigaud^{6}, and S.L. Marcus^{7} have touched on the LOD, lunar, ENSO connection over the years.
Bottom-line take aways :
A recent citation to use: Pukite, Paul. “Nonlinear long-period tidal forcing with application to ENSO, QBO, and Chandler wobble.” EGU General Assembly Conference Abstracts. 2021. https://ui.adsabs.harvard.edu/abs/2021EGUGA..2310515P
References
The non-linearity will act to fold (or break) the signal, a la Mach-Zehnder modulation, to destroy the long period and then the superposition will not necessarily lead to a predictable constructive interference. This also may make physical sense in that the Mf factor is a horizontal transverse (latitudinal) forcing applied to the equatorial toroidal (equatoroidal?) waveguide while the Mm factor is vertical. In other words, lunar perigee forcing (Mm) is orthogonal to lunar declination (Mf) forcing for the equatoroid and so the LTE modulation may in fact be independent in the two dimensions and thus the two resultants superposed as magnitudes — or at least that’s my latest ansatz.
Before breaking down the analysis, the chart below shows the initial excellent fit, and which converged surprisingly quickly.
This achieves a correlation coefficient of >0.9, which is quite good considering the low DOF afforded by the applied parameters. Individually, the models based on Mf and Mm alone are adequate but prone to overfitting
These two, each with a correlation coefficient of between 0.6 and 0.7, when combined give a much higher as shown in the first graph.
The presumptuous finality of this fit can be considered in the context of how constrained the tidal factors turned out to be. In 2018 at the AGU, we presented the model by constraining the tidal factors against known lunar ephemeris and also against the factors contributing to the Earth’s delta length-of-day (LOD). This is also described in the book and in a previous blog post.
Note that this agreement was not initially constrained but that the iterative solver fit arrived at the solution on its own accord, driven by the best fit to ENSO. There is a slight shift in alignment of perhaps a 1/2 a year, but this may be due to the solver not being able to move quickly to a better solution.
What does the forcing agreement look like on a finer temporal resolution scale? See below.
Ansatz that !
]]>A hypothesis based on LTE modulation is that decadal behavior arises from the shallowest modulation mode, and one that corresponds to even symmetry (i.e. cos not sin). So for a model that was originally fit to an ENSO time-series, it is anticipated that the modulation trending to a more even symmetry will reveal less rapid fluctuations — or in other words for an even f(x) = f(-x) symmetry there will be less difference between positive and negative excursions for a well-balanced symmetric input time-series. This should then exaggerate longer term fluctuations, such as in PDO. And for odd f(x) = -f(-x) symmetry it will exaggerate shorter term fluctuations leading to more spikiness, such as in ENSO.
To evaluate this hypothesis, all one needs to do is plot the fitted LTE modulation for + and – excursions along the same axis, as shown below
Removing the higher frequency LTE modulation, the even symmetry for PDO is even more apparent, while the underlying primary modulation frequency is shared by ENSO and PDO (expected, as same ocean basin implies same standing wave modes).
In addition to substantiating the premise of even vs odd non-autonomous modulation, this finding is perhaps revealing something even more fundamental about the LTE solution for PDO. As the high degree of even-symmetry (correlation coefficient > 0.92) implies, some symmetric physical mechanism or boundary condition must be driving the PDO behavior. So that positive (+) and negative (-) tidal forcing excursions have identical impacts off the equator (for PDO) but slightly asymmetrical on the equator (for ENSO). Indeed, this could be a foundational topological requirement or perhaps one that conserves energy and angular momentum at higher latitudes.
This even-symmetry finding is also a surprising result from a statistical signal processing and machine learning perspective, as the LTE modulation was not constrained to be even-symmetry during the fitting process. So not only could it reveal a natural symmetry as discussed in the previous paragraph, but it may lead to a leaner and more parsimonious modeling procedure since the number of degrees of freedom (DOF) can be reduced. As all the modulation phases need to be aligned it may actually cut the number of DOF in half!
As an addendum, the forcing applied on a monthly level is primarily the Mm tidal factor, with additional factors creating an 18-year Saros modulation, and 6-year sub-modulation.
This is the spectral representation when integrated with respect to an annual impulse.
The PDO data is from https://climexp.knmi.nl/getindices.cgi?WMO=UWData/pdo_ersst&STATION=PDO_ERSST based on ERSST anomalies.
UPDATE 4/25/2002
Making the PDO modulation perfectly even-symmetric by enforcing LTO(+) = LTO(-), this is the result
This has a weak 6 year repeat forcing pattern superimposed. An alternate fit with a 4.4-year repeat forcing pattern is below
Example of PDO (northern Pacific blob) and ENSO (la Nina conditions)
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