Atmospheric Science

I don’t immediately trust the research published by highly cited atmospheric scientists. By my count many of them seem more keen on presenting their personal views rather than advancing the field. Off the top of my head, Richard Lindzen, Murry Salby, Roy Spencer, Tim Dunkerton, Roger Pielke, Cliff Mass, Judith Curry are all highly cited but come across as political and/or religious zealots. One guy on the list, Dunkerton, is also a racist, who happened to make the Washington Post twice : “Physicist ousted from research post after sending offensive tweet to Hispanic meteorologist” and “Atmospheric scientist loses honor, membership over ethics violation“. Awful stuff and he hasn’t stopped spouting off on Twitter.

Granted that Dunkerton says dumb stuff on Twitter but his highly cited research is also off-base. That’s IMO only because recent papers by others in the field of atmospheric science do continue to cite his ideas as primary, if not authoritative. For example, from a recently published paper “The Gravity Wave Activity during Two Recent QBO Disruptions Revealed by U.S. High-Resolution Radiosonde Data”, citations 1 & 12 both refer to Dunkerton, and specifically to his belief that the QBO period is a property of the atmospheric medium itself

Straight-forward to debunk this Dunkerton theory since the length of the cycle directly above the QBO layer is semi-annual and thus not a property of the medium but of the semi-annual nodal forcing frequency. If we make the obvious connection to the other nodal forcing — that of the moon — then we find the QBO period is fixed to 28 months. I have been highlighting this connection to the authors of new QBO papers under community review, often with some subsequent feedback provided such as here: . Though not visible yet in the comments, I received some personal correspondence that showed that the authors under peer-review are taking the idea seriously and attempting to duplicate the calculations. They seem to be methodical in their approach, asking for clarification and further instructions where they couldn’t follow the formulation. They know about the GitHub software, so hopefully that will be of some help.

In contrast, Dunkerton also knows about my approach but responds in an inscrutable (if not condescending) way. Makes you wonder if scientists such as Dunkerton and Lindzen are bitter and taking out their frustrations via the media. Based on their doggedness, they may in fact be intentionally trying to impede progress in climate science by taking contrarian stances. In my experience, the top scientists in other research disciplines don’t act this way. YMMV

Limits of Predictability?

A decade-old research article on modeling equatorial waves includes this introductory passage:

“Nonlinear aspects plays a major role in the understanding of fluid flows. The distinctive fact that in nonlinear problems cause and effect are not proportional opens up the possibility that a small variation in an input quantity causes a considerable change in the response of the system. Often this type of complication causes nonlinear problems to elude exact treatment. “

From my experience if it is relatively easy to generate a fit to data via a nonlinear model then it also may be easy to diverge from the fit with a small structural perturbation, or to come up with an alternative fit with a different set of parameters. This makes it difficult to establish an iron-clad cross-validation.

This doesn’t mean we don’t keep trying. Applying the dLOD calibration approach to an applied forcing, we can model ENSO via the NINO34 climate index across the available data range (in YELLOW) in the figure below (parameters here)

The lower right box is a modulo-2π reduction of the tidal forcing as an input to the sinusoidal LTE modulation, using the decline rate (per month) as the divisor. Why this works so well per month in contrast to per year (where an annual cycle would make sense) is not clear. It is also fascinating in that this is a form of amplitude aliasing analogous to the frequency aliasing that also applies a modulo-2π folding reduction to the tidal periods less than the Nyquist monthly sampling criteria. There may be a time-amplitude duality or Lagrangian particle-relabeling in operation that has at its central core the trivial solutions of Navier-Stokes or Euler differential equations when all segments of forcing are flat or have a linear slope. Trivial in the sense that when a forcing is flat or has a 1st-order slope, the 2nd derivatives due to divergence in the differential equations vanish (quasi-static). This means that only the discontinuities, which occur concurrently with the annual ENSO predictability barrier, need to be treated carefully (the modulo-2π folding could be a topological Berry phase jump?). Yet, if these transitions are enhanced by metastable interface instabilities as during thermocline turn-over then the differential equation conditions could be transiently relaxed via a vanishing density difference. Much happens during a turn-over, but it doesn’t last long, perhaps indicating a geometric phase. MV Berry also discusses phase changes in the context of amphidromic tidal singularities here.

Suffice to say that the topological properties of reduced dimension volumes and at interfaces remain mysterious. The main takeaway is that a working NINO34-fitted ENSO model is produced, and if not here then somewhere else a machine-learning algorithm will discover it.

The key next step is to apply the same tidal forcing to an AMO model, taking care not to change the tidal factors enough to produce a highly sensitive nonlinear response in the LTE model. So we retain an excluded interval from training (in YELLOW below) and only adjust the LTE parameters for the region surrounding this zone during the fitting process (parameters here).

The cross-validation agreement is breathtakingly good in the excluded (out-of-band) training interval. There is zero cross-correlation between the NINO34 and AMO time-series to begin with so that this is likely revealing the true emergent characteristics of a tidally forced mechanism.

As usual all the introductory work is covered in Mathematical Geoenergy

A community peer-review contributed to a recent QBO article is here and PDF here. The same question applies to QBO as ENSO or AMO: is it possible to predict future behavior? Is the QBO model less sensitive to input since the nonlinear aspect is weaker?

Gerstner waves

An exact solution for equatorially trapped waves
Adrian Constantin, JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, C05029, doi:10.1029/2012JC007879, 2012

Nonlinear aspects plays a major role in the understanding of fluid flows. The distinctive fact that in nonlinear problems cause and effect are not proportional opens up the possibility that a small variation in an input quantity causes a considerable change in the response of the system. Often this type of complication causes nonlinear problems to elude exact treatment. A good illustration of this feature is the fact that there is only one known explicit exact solution of the (nonlinear) governing equations for periodic two-dimensional traveling gravity water waves. This solution was first found in a homogeneous fluid by Gerstner

These are trochoidal waves

Even within the context of gravity waves explored in the references mentioned above, a vertical wall is not allowable. This drawback is of special relevance in a geophysical context since [cf. Fedorov and Brown, 2009] the Equator works like a natural boundary and equatorially trapped waves, eastward propagating and symmetric about the Equator, are known to exist. By the 1980s, the scientific community came to realize that these waves are one of the key factors in explaining the El Niño phenomenon (see also the discussion in Cushman-Roisin and Beckers [2011]).

modulo-2π and Berry phase


It turns out that the Darwin location of the Southern Oscillation Index (SOI) dipole is brilliantly easy to behaviorally model on it’s own.

The input forcing is calibrated to the differential length-of-day (LOD) with a correlation coefficient of 0.9997, and only a few terms are required to capture the standing-wave modes corresponding to the ENSO dipole.

So which curve below is the time-series data of atmospheric pressure at Darwin and which is the Laplace’s Tidal Equation (LTE) model calibrated from dLOD measurements?

  • (bottom, red) = ?
  • (top, blue) = ??

As a bonus, the couple of years outside of the training interval are extrapolated from the model. This shouldn’t be hard for climate scientists, …. or is it still too difficult?

If that isn’t enough to discriminate between the two, the power spectra of the LTE mapping to model and to data is shown below. This identifies a couple of the lower frequency modulations as strong peaks and a few weaker higher harmonic peaks that sharpen the model’s detail. This shows that the data’s behavior possesses a high amount of order not apparent in the time series.

Poll on Twitter =>

Why isn’t the Tahiti time-series included since that would provide additional signal discrimination via a differential measurement as one should be the complement of the other? It should accentuate the signal and remove noise (and any common-mode behavior) if the Darwin and Tahiti are perfect anti-nodes for all standing-wave modes. However, it appears that only the main ENSO standing-wave mode is balanced in all modes.

In that case, the Darwin set alone works well. Mastodon

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 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 =>

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

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

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