Limnology 101

I doubt many climate scientists have taken a class in limnology, the study of freshwater lakes. I have as an elective science course in college. They likely have missed the insight of thinking about the thermocline and how in dimictic upper-latitude lakes the entire lake overturns twice a year as the imbalance of densities due to differential heating or cooling causes a buoyancy instability.

An interesting Nature paper “Seasonal overturn and stratification changes drive deep-water warming in one of Earth’s largest lakes” focusing on Lake Michigan

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Cross-validation

Cross-validation is essentially the ability to predict the characteristics of an unexplored region based on a model of an explored region. The explored region is often used as a training interval to test or validate model applicability on the unexplored interval. If some fraction of the expected characteristics appears in the unexplored region when the model is extrapolated to that interval, some degree of validation is granted to the model.

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.

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What happened to simple models?

It’s been over 10 years since a thread on simplifying climate models was posted on the Azimuth Project forum => https://forum.azimuthproject.org/discussion/981/the-fruit-fly-of-climate-models

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.

(click images to expand)
  • Tidal Forcing. A long-period tidal forcing is generated, best done by fitting the strongest tidal factors (mapped by R.D.Ray) to the dLOD (delta Length-of-Day) data from the Paris Observatory IERS. With a multiple linear regression (MLR) algorithm, that takes less than a second and fits a sine-wave series model to the data with a correlation coefficient higher than 0.99.

Integrated Response Forcing
  • Annual Trigger Barrier. A semi-annual (+/-) excursion impulse train is constructed, which acts as a sample-and-hold input driver when multiplied by the tidal forcing above. A single parameter controls the slight decay of the sampled value, i.e. the hold or integrated response. Another parameter allows for a slight asymmetry in the + excursions, and the – excursions that occur 6 months later. This creates the erratic pseudo-square wave structure shown above. A monthly time-series is sufficient, with the chosen impulse month the most critical parameter.

  • Fluid Dynamics Modulation. The Laplace’s Tidal Equations are solved along the equator via an ansatz, which essentially creates a non-linear sin(F(t)) mapping corresponding to standing-wave modes spanning the Pacific Ocean basin. A slow-mode mode(s) is used to characterize the main ENSO dipole, and a faster mode(s) to characterize the tropical instability waves. For each mode, a wavenumber, amplitude, and phase is required to calculate the modulation.

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.

LTE modulation (low)
LTE modulation (high)

So this demonstrates that the behavior is stationary and definitely not chaotic, only obscured by the non-linear modulation applied to tidally forced waveform.

Notes on Sloshing

Informative article on development of cusped waveforms and Kelvin-Helmholtz instabilities

An experimental study of two-layer liquid sloshing under pitch excitations, Physics of Fluids · May 2022
DOI: 10.1063/5.0093716

Amazing number of harmonics

Related to Tropical Instability Waves along the equator

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. 

ENSO and AMO, a standing wave phase change?

Using as few independent parameters as possible, the difference in characterizing the temporal behavior of ENSO and AMO may amount to a standing-wave phase change. Noted earlier that ENSO and AMO can be derived from a common lunisolar forcing — and have now found that the LTE modulation is not that fundamentally different between the two.

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.

  • ENSO ~ sin (k F(t))
  • AMO ~ sin (k F(t) + π + ϕ)

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

Lunar Eclipse Unremarkable

Neil deGrasse Tyson calls it

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.

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Response to the lunar cycle

Paper under review “The modelled climatic response to the 18.6-year lunar nodal cycle and its role in decadal temperature trends”

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 & Qian1). 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 Geoenergy2.

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:

As the high-resolution dLOD measurement only goes back to 1962, but the ENSO NINO34 time-series goes back to 1870 3, the closed form model fit can be used across the entire duration of a climate index (assuming that tidal factors are stationary).

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 approaches10.

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 vital8. 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 & Chao9 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 USA5, the latter discussing the impact of axial torques on the climate. Researchers at NASA JPL including J.H. Shirley, C. Perigaud6, and S.L. Marcus7 have touched on the LOD, lunar, ENSO connection over the years.

Bottom-line take aways :

  1. Tidal factors are numerous so a measure such as dLOD is critical for calibrating the forcing.
  2. Use the knowledge of a seasonal impulse, a la the spring predictability barrier, to advantage, while considering the temporal aliasing that it will cause.
  3. The solution to the geophysical fluid dynamics produces a non-linear response, so clever transform techniques such as Fourier series are useful to isolate the pattern.

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

  1. Lin, J., Qian, T. Switch Between El Nino and La Nina is Caused by Subsurface Ocean Waves Likely Driven by Lunar Tidal ForcingSci Rep 9, 13106 (2019).
  2. Pukite, P., Coyne, D., Challou, D. (2018). Mathematical Geoenergy: Discovery, Depletion, and Renewal.  AGU/Wiley. DOI:10.1002/9781119434351
  3. https://psl.noaa.gov/gcos_wgsp/Timeseries/Data/nino34.long.anom.data
  4. Delplace, Pierre, J. B. Marston, and Antoine Venaille. “Topological origin of equatorial waves.” Science 358.6366 (2017): 1075-1077. https://arxiv.org/pdf/1702.07583.pdf
  5. Shirley, James H. “Solar System Dynamics and Multiyear Droughts of the Western USA.” arXiv preprint arXiv:2112.02186 (2021).
  6. Perigaud, Claire. “Importance of the Earth-Moon system for reducing uncertainties in climate modelling and monitoring.” (2009).
    Perigaud, Claire. “ENSO Modulated by Lunar Cycles” (2009). https://www.aviso.altimetry.fr/fileadmin/documents/OSTST/2009/poster/Perigaud.pdf
  7. Marcus, S. L. (2016). “Does an Intrinsic Source Generate a Shared Low-Frequency Signature in Earth’s Climate and Rotation Rate?,” Earth Interactions20(4), 1-14. Retrieved May 12, 2022, from https://journals.ametsoc.org/view/journals/eint/20/4/ei-d-15-0014.1.xml
  8. Ray, Richard D., and Svetlana Y. Erofeeva. “Long‐period tidal variations in the length of day.” Journal of Geophysical Research: Solid Earth 119.2 (2014): 1498-1509. https://ir.library.oregonstate.edu/downloads/gx41mk60z
  9. Ding, H., & Chao, B. F. (2018). Application of stabilized AR-z spectrum in harmonic analysis for geophysics. Journal of Geophysical Research: Solid Earth, 123, 8249– 8259. https://doi.org/10.1029/2018JB015890
  10. Pukite, P. R., D. Coyne, and D. J. Challou. “Ephemeris calibration of Laplace’s tidal equation model for ENSO.” AGU Fall Meeting Abstracts. Vol. 2018. 2018. https://www.proquest.com/openview/1397f7741d3f81e0b26fb16b0c35da75/1

Mf vs Mm

In an earlier post, the observation was that ENSO models may not be unique due to the numerous possibilities provided by nonlinear math. This was supported by the fact that a tidal forcing model based on the Mf (13.66 day) tidal factor worked equally as well as a Mm (27.55 day) factor. This was not surprising considering that the aliasing against an annual impulse gave a similar repeat cycle — 3.8 years versus 3.9 years. But I have also observed that mixing the two in a linear fashion did not improve the fit much at all, as the difference created a long interference cycle which isn’t observed in the ENSO time series data. But then thinking in terms of the nonlinear modulation required, it may be that the two factors can be combined after the LTE solution is applied.

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PDO is even, ENSO is odd

As the quality of the tidally-forced ENSO model improves, it’s instructive to evaluate its common-mode mechanism against other oceanic indices. So this is a re-evaluation of the Pacific Decadal Oscillation (PDO), in the context of non-autonomous solutions such as generated via LTE modulation. In particular, in this note we will clearly delineate the subtle distinction that arises when comparing ENSO and PDO. As background, it’s been frequently observed and reported that the PDO shows a resemblance to ENSO (a correlation coefficient between 0.5 and 0.6), but also demonstrates a longer multiyear behavior than the 3-7 year fluctuating period of ENSO, hence the decadal modifier.

ENSO Model
PDO Model — identical forcing to ENSO (cross-validated in upper panel)

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.

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