Annual and Biennial Modulations

Here are a couple of items I had in comments in this blog or recovered from elsewhere.

(1) There is an interesting highly nonlinear spiked peak around November in chlorophyl phytoplankton at higher latitudes, which matches the position of the forcing modulation used in the ENSO model (-0.123 years).

Phenological Responses to ENSO in the Global Oceans
Surveys in Geophysics, 2017, Volume 38, Number 1, Page 277
M.-F. Racault, S. Sathyendranath, N. Menon

Is this a response or a nodal forcing mechanism?

(2) In another paper, scientists analyzing historical ENSO records used a CGM to support some of their observations. What they do not adequately explain is the strength of the biennial period in their simulation results.


Evolution and forcing mechanisms of El Niño over the past 21,000 years
Zhengyu Liu, Zhengyao Lu, Xinyu Wen, B. L. Otto-Bliesner, A. Timmermann, K. M. Cobb, Nature 515, 550–553 (27 November 2014) doi:10.1038/nature13963

Intro section

“To understand ENSO’s evolution during the past 21 kyr, we analyse the baseline transient simulation (TRACE) conducted with the Community Climate System model version 3 (CCSM3). This simulation uses the complete set of realistic climate forcings: orbital, greenhouse gases, continental ice sheets and meltwater discharge (Fig. 1a, d and Methods). TRACE has been shown to replicate many key features of the global climate evolution”

Methods section

“Model ENSO.ENSO simulated by the model for the present day shows many realistic features, although the ENSO period tends to be biased towards quasi-biannual, as opposed to a broader 2–7-year peak in the observation38. The ENSO mode resembles the SST mode29 and propagates westwards as in many CGCMs. In the past 21 kyr, the preferred period of model ENSO remains at quasi-biannual, with the power spectrum changing only modestly with time (Extended Data Fig. 1).”

This is interesting in that they mischaracterize the periodicity as biannual, which also means semiannual, or twice a year, but that it is clearly biennial in the chart, which is defined as once every two years. That could be just a typo not caught during proof-reading. Yet the peak is sharply centered around a two-year fundamental period, which is the interesting aspect.

I replot that curve below to show the symmetry around the 2-year period. Drawing a Lorentzian curve around that frequency and linearizing the axis makes it symmetric.


A Lorentzian or Cauchy often results as the frequency response of a driven damped harmonic oscillator. If that is the case, what the simulation shows may be the result of forcing comprised of different stimuli frequencies collected over time and the accompanying frequency response (e.g. a Bode plot). Yet this implies that the characteristic frequency or eigenvalue would need to be 2 years. But why would a characteristic frequency just happen to align with a period so close to 2 years? The 2 year period is likely not an eigenvalue of the properties of a damped harmonic oscillator, e.g. a spring constant and a damper, but more than likely connected to a period doubling based on the annual cycle. This is what Faraday and Rayleigh observed and then explained in the earliest liquid sloshing experiments [1]

Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe,Springer Science & Business Media, 2006

This is a spectrum of the biennial modulated ENSO data of the last 130 years. Pairs show up +/- around the 2 year central period. These are all related to aliased lunar cycles, when associated with a sharp biennial modulation.  This is a more detailed view showing the phase response as well.


The strongest factors above are shown as pluses on the historical model below, these are not to scale in the vertical but positioned along the horizontal to highlight the symmetry:


So the actual situation is that the measured ENSO power spectrum over the modern instrumental record is not a smooth Lorentizian but a discrete set of frequencies that are balanced +/- around the central biennial frequency. A question is whether these discrete frequencies are fixed over time. Based on what I found with ENSO proxy data it appears that it may indeed be fixed, at least for spans of 100's of years --


[1] Rayleigh, Lord. "XXXIII. On maintained vibrations." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 15.94 (1883): 229-235.

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