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The ENSO phenomena is thought to be a coupled interaction between the Pacific Ocean and the atmosphere along with potentially other geophysical effects.

My idea is that resonances cascade starting with the strongest biennial and annual signals. These strong forcings then couple to the most compliant medium -- the low-density fluid of the stratosphere -- and create the first resonance, that of the QBO. And then this resonance couples to the higher inertial and less compliant medium of the ocean, and thus creating the ENSO resonance.

The figure below shows a simple model of a SemiAnnual Oscillation, found in the very-thin upper stratosphere, and how that transforms into a Quasi-biennial oscillation found in the lower stratosphere. At the highest altitude the response function remains the forcing but as it moves lower, the resonance gradually takes over as the fluid response.

The contour is slanted as I added a linear phase shift as the waveform moves down the atmosphere. That's the conceptual model that is trying to capture the following observed behavior. • Options
1.

This shows that the contour level at a stratospheric altitude of 20 hPa contains a significant biennial or semiannual modulation. To detect this, the second-derivative was processed through a machine-learning exercise. The modulation is of a Mathieu equation, whereby the strength of the biennial sinusoidal scaling is proportional to the signal itself.

The full differential equation for the 20 hPa QBO features a characteristic response frequency, and both annual and biennial forcing factors with modulation. There is a long-term jitter in the period which appears to correspond to a TSI variation similar to that which impacts ENSO .

The waveforms are more rounded than a typical sinusoid, which I model by taking the square-root of the waveform. The wave equation solves for the contained energy in the oscillation, while QBO is a measure of wind velocity, which makes it the square root of energy. The correlation coefficient is 0.88, with the residual shown in the lower panel.

The next step is to propagate this lower to the troposphere where it becomes a staged forcing for ENSO.

 White, Warren B., and Zhengyu Liu. "Non‐linear alignment of El Nino to the 11‐yr solar cycle." Geophysical Research Letters 35.19 (2008).

Comment Source:This shows that the contour level at a stratospheric altitude of 20 hPa contains a significant biennial or semiannual modulation. ![modulation](https://imageshack.com/i/pcoHyeZ6g) To detect this, the second-derivative was processed through a machine-learning exercise. The modulation is of a Mathieu equation, whereby the strength of the biennial sinusoidal scaling is proportional to the signal itself. The full differential equation for the 20 hPa QBO features a characteristic response frequency, and both annual and biennial forcing factors with modulation. ![QBO fit](http://imageshack.com/a/img909/8920/mhfplY.gif) There is a long-term jitter in the period which appears to correspond to a TSI variation similar to that which impacts ENSO . The waveforms are more rounded than a typical sinusoid, which I model by taking the square-root of the waveform. The wave equation solves for the contained energy in the oscillation, while QBO is a measure of wind velocity, which makes it the square root of energy. The correlation coefficient is 0.88, with the residual shown in the lower panel. The next step is to propagate this lower to the troposphere where it becomes a staged forcing for ENSO.  White, Warren B., and Zhengyu Liu. "Non‐linear alignment of El Nino to the 11‐yr solar cycle." Geophysical Research Letters 35.19 (2008).
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2.

Some of the phase change as the QBO descends in altitude is due to an asymmetric drag of the winds in one direction as the earth's surface interacts with the motion.

If the wave equation is modulated with a signed value scaled with the magnitude, it will result in an oblong asymmetry in the sinusoidal waveform. This is most clearly shown in the phase plot below. If this is applied to the QBO fit in #2 whereby the drag factor is increased as the altitude is decreased, here is the result: The energy transmitted by the drag is partially absorbed by the ocean surface, resulting in a forcing input to the ENSO dynamics. The knee in the QBO profile between 30 and 50 hPa shown in #2 is where this drag starts to make its impact.

That explains the cascading effect of the semiannual and annual forcing at high altitudes into a longer period forcing at the ocean's surface, where it is closer to the natural resonance of 4¼ year period.

0.5 & 1.0 year periods --> QBO of 2⅓ year period --> ENSO of 4¼ quasi-period

The general idea is that forcing periods that are closer to the natural resonance are more effective in producing an oscillation. Thus, a staged set of forcing conditions will cascade the effect to where it can make a difference in the resultant ocean dynamics.

Comment Source:Some of the phase change as the QBO descends in altitude is due to an asymmetric drag of the winds in one direction as the earth's surface interacts with the motion. If the wave equation is modulated with a signed value scaled with the magnitude, it will result in an oblong asymmetry in the sinusoidal waveform. This is most clearly shown in the phase plot below. ![asymmetricWave](http://imageshack.com/a/img905/9488/nlRl0a.gif) If this is applied to the QBO fit in #2 whereby the drag factor is increased as the altitude is decreased, here is the result: ![qbo drag](http://imageshack.com/a/img912/3157/WzEufM.gif) The energy transmitted by the drag is partially absorbed by the ocean surface, resulting in a forcing input to the ENSO dynamics. The knee in the QBO profile between 30 and 50 hPa shown in #2 is where this drag starts to make its impact. That explains the cascading effect of the semiannual and annual forcing at high altitudes into a longer period forcing at the ocean's surface, where it is closer to the natural resonance of 4¼ year period. 0.5 & 1.0 year periods --> QBO of 2⅓ year period --> ENSO of 4¼ quasi-period The general idea is that forcing periods that are closer to the natural resonance are more effective in producing an oscillation. Thus, a staged set of forcing conditions will cascade the effect to where it can make a difference in the resultant ocean dynamics.
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3.

This  is a detailed simulation of the SAO and QBO, where one can definitely see a mix of underlying frequencies, and the drag at lower altitudes that effectively breaks up the periodicity. But this is also a full GCM which I have no interest in solving, but rather in extracting the salient characteristics. Sometimes it is helpful to string together words that you think describes what is happening. After putting together cascade and resonance in different combinations, these are papers I found:

This paper  mentions stochastic cascade

And this treatise  mentions resonance cascade. But this loses me, where the analogy is of driving through Central Park! S. Schirber, E. Manzini, and M. J. Alexander, “A convection‐based gravity wave parameterization in a general circulation model: Implementation and improvements on the QBO,” Journal of Advances in Modeling Earth Systems, vol. 6, no. 1, pp. 264–279, 2014.

S. Verrier, M. Crépon, and S. Thiria, “Scaling and stochastic cascade properties of NEMO oceanic simulations and their potential value for GCM evaluation and downscaling,” Journal of Geophysical Research: Oceans, vol. 119, no. 9, pp. 6444–6460, 2014.

E. Kartashova, V. Lvov, S. Nazarenko, and I. Procaccia, “Towards a Theory of Discrete and Mesoscopic Wave Turbulence,” month, 2010. http://www.risc.jku.at/publications/download/risc_3967/Meso-2010-arxiv.pdf

Comment Source:This  is a detailed simulation of the SAO and QBO, where one can definitely see a mix of underlying frequencies, and the drag at lower altitudes that effectively breaks up the periodicity. But this is also a full GCM which I have no interest in solving, but rather in extracting the salient characteristics. ![sao](http://imageshack.com/a/img673/6172/a5JRFg.gif) Sometimes it is helpful to string together words that you think describes what is happening. After putting together cascade and resonance in different combinations, these are papers I found: This paper  mentions stochastic cascade And this treatise  mentions resonance cascade. ![mso2](http://imageshack.com/a/img661/4515/iFaJA4.gif) But this loses me, where the analogy is of driving through Central Park! ![mso1](http://imageshack.com/a/img633/5479/AuQ8CE.gif) S. Schirber, E. Manzini, and M. J. Alexander, “A convection‐based gravity wave parameterization in a general circulation model: Implementation and improvements on the QBO,” Journal of Advances in Modeling Earth Systems, vol. 6, no. 1, pp. 264–279, 2014. S. Verrier, M. Crépon, and S. Thiria, “Scaling and stochastic cascade properties of NEMO oceanic simulations and their potential value for GCM evaluation and downscaling,” Journal of Geophysical Research: Oceans, vol. 119, no. 9, pp. 6444–6460, 2014. E. Kartashova, V. Lvov, S. Nazarenko, and I. Procaccia, “Towards a Theory of Discrete and Mesoscopic Wave Turbulence,” month, 2010. <http://www.risc.jku.at/publications/download/risc_3967/Meso-2010-arxiv.pdf>
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4.

This paper is kind of a missing link in the research on ocean-atmosphere interactions:

McWilliams, J. C., and P. R. Gent. "A coupled air and sea model for the tropical Pacific." Journal of the Atmospheric Sciences 35.6 (1978): 962-989.

McWilliams developed a simple model for ocean sloshing that appears very promising as applied to El Nino (see Section 6). As background he notes the potential complexity:

... a blind extensive search for connections would be prohibitively expensive with such models due to the disparity between time scales in the two media. and the number of physical processes which, on a global scale, might be important.

and then scaling back to this

... we feel it is also important to develop alternative models with a more deterministic bias. This paper presents such a model, a "thought model," that is, an ad hoc set of equations, in contrast to equations with first-principle antecedents (e.g., the Navier-Stokes equations).

The odd aspect of this paper and what makes it a missing link is that McWilliams never referenced it again. He is a big name in chaos and climate circles and still going strong. Did he disown it? Beats me.

But what stands out to me is that what was perhaps difficult back in 1978 due to the primitive nature of computers, is now feasible. Yet, it appears that few have revisited the path set forth by McWilliams.

The following paper is interesting, as it extends what it calls "prototype coupled models" that McWilliams initially adopted.

Xian-Chun, Zhou, Yao Jing-Sun, and Mo Jia-Qi. "Asymptotic solving method for sea–air coupled oscillator ENSO model." Chin. Phys. B 13 (2012): 28.

They couple the SST (partial atmosphere forcing) with the thermocline depth as two interacting equations.

$T'(t) = C \cdot T(t) + D \cdot h(t) - \epsilon \cdot f(T(t))$

$h'(t) = - E \cdot T(t) + R_h \cdot h(t) - \epsilon \cdot g(h(t))$

This will obviously oscillate for an underdamped selection of parameters, and then they add a perturbation scaled by $\epsilon$.

They end the paper by suggesting

"And we can furnish weather forecast more accurately."

High hopes indeed.

Comment Source:This paper is kind of a missing link in the research on ocean-atmosphere interactions: McWilliams, J. C., and P. R. Gent. ["A coupled air and sea model for the tropical Pacific."](http://journals.ametsoc.org/doi/pdf/10.1175/1520-0469%281978%29035%3C0962%3AACAASM%3E2.0.CO%3B2) Journal of the Atmospheric Sciences 35.6 (1978): 962-989. McWilliams developed a simple model for ocean sloshing that appears very promising as applied to El Nino (see Section 6). As background he notes the potential complexity: > ... a blind extensive search for connections would be prohibitively expensive with such models due to the disparity between time scales in the two media. and the number of physical processes which, on a global scale, might be important. and then scaling back to this >... we feel it is also important to develop alternative models with a more deterministic bias. This paper presents such a model, a "thought model," that is, an ad hoc set of equations, in contrast to equations with first-principle antecedents (e.g., the Navier-Stokes equations). The odd aspect of this paper and what makes it a missing link is that McWilliams never referenced it again. He is a big name in chaos and climate circles and still going strong. Did he disown it? Beats me. But what stands out to me is that what was perhaps difficult back in 1978 due to the primitive nature of computers, is now feasible. Yet, it appears that few have revisited the path set forth by McWilliams. The following paper is interesting, as it extends what it calls "prototype coupled models" that McWilliams initially adopted. Xian-Chun, Zhou, Yao Jing-Sun, and Mo Jia-Qi. ["Asymptotic solving method for sea–air coupled oscillator ENSO model."](http://cpb.iphy.ac.cn/EN/article/showArticleFile.do?attachType=PDF&id=45745) Chin. Phys. B 13 (2012): 28. They couple the SST (partial atmosphere forcing) with the thermocline depth as two interacting equations. $T'(t) = C \cdot T(t) + D \cdot h(t) - \epsilon \cdot f(T(t))$ $h'(t) = - E \cdot T(t) + R_h \cdot h(t) - \epsilon \cdot g(h(t))$ This will obviously oscillate for an underdamped selection of parameters, and then they add a perturbation scaled by $\epsilon$. They end the paper by suggesting > "And we can furnish weather forecast more accurately." High hopes indeed.
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5.

I got this reference from an AGW denier who suggested this as "an alternative explanation" for ENSO.

K. Stein, N. Schneider, A. Timmermann, and F.-F. Jin, “Seasonal Synchronization of ENSO Events in a Linear Stochastic Model,” Journal of Climate, vol. 23, no. 21, pp. 5629–5643, Nov. 2010.

What do they say is the linear stochastic model?

$\frac{\partial T}{\partial t} = 2 \gamma(t) T + \omega(t)H + \xi(t)$

$\frac{\partial H}{\partial t} = - R T$

T is the surface temperature and H is the thermocline depth, with a random forcing $\xi$

This is essentially a variation of the Hill/Mathieu equation once again , where $\omega$ is the time varying part, but with a damping term also time varying with $\gamma$.

If the seasonal synchronization starts with the seasonal forcing on the QBO and then propagates downward to the ocean surface, then this could be a fundamental formulation for describing the dynamics.

Once you start looking, this 2nd-order differential equation seems to pop up everywhere, but no one seems to want to try to solve or fit it to the ENSO dynamics ... except us of course. Isn't that curious?

Comment Source:I got this reference from an AGW denier who suggested this as "an alternative explanation" for ENSO. K. Stein, N. Schneider, A. Timmermann, and F.-F. Jin, “Seasonal Synchronization of ENSO Events in a Linear Stochastic Model,” Journal of Climate, vol. 23, no. 21, pp. 5629–5643, Nov. 2010. What do they say is the linear stochastic model? $\frac{\partial T}{\partial t} = 2 \gamma(t) T + \omega(t)H + \xi(t)$ $\frac{\partial H}{\partial t} = - R T$ *T* is the surface temperature and *H* is the thermocline depth, with a random forcing $\xi$ This is essentially a variation of the Hill/Mathieu equation once again , where $\omega$ is the time varying part, but with a damping term also time varying with $\gamma$. If the seasonal synchronization starts with the seasonal forcing on the QBO and then propagates downward to the ocean surface, then this could be a fundamental formulation for describing the dynamics. Once you start looking, this 2nd-order differential equation seems to pop up everywhere, but no one seems to want to try to solve or fit it to the ENSO dynamics ... except us of course. Isn't that curious?
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6.

The reference in comment #6 is co-authored by Axel Timmerman who is featured in this article He has the equation for ENSO but he doesn't try solving it. That's the real mystery -- I would like to understand why this equation pops up so often but never gets evaluated as a first-order approximation.

--

In doing quality-control in submitted research papers, ARXIV applies a "sophisticated machine learning logistic classifier". It would be interesting if we could apply that kind of machine-learning to find wave equations that are proposed in various research articles to see what the consensus is.

Machine learning to discover consensus in model formulations instead of in the data, that is an intriguing idea

Comment Source:The reference in comment #6 is co-authored by Axel Timmerman who is featured in [this article](http://www.reportingclimatescience.com/news-stories/article/warming-pacific-drives-global-temperatures.html) ![axel](http://imageshack.com/a/img910/1101/8RZiJR.gif) He has the equation for ENSO but he doesn't try solving it. That's the real mystery -- I would like to understand why this equation pops up so often but never gets evaluated as a first-order approximation. -- In doing quality-control in submitted research papers, ARXIV applies a ["sophisticated machine learning logistic classifier"](http://news.sciencemag.org/scientific-community/2014/12/study-massive-preprint-archive-hints-geography-plagiarism). It would be interesting if we could apply that kind of machine-learning to find wave equations that are proposed in various research articles to see what the consensus is. Machine learning to discover consensus in model formulations instead of in the data, that is an intriguing idea
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7.

Similar to the ENSO resonance but on a much smaller scale is the amazing tidal resonance found in places such as the Bay of Fundy.

The reason that the height differential of the tide is so high at this location is because the harbor has a natural resonance frequency that is fairly close to the 12.42 hr M2 tidal period. http://www.tide-forecast.com/locations/Hopewell-Cape-New-Brunswick/tides/latest The governing equation is again the wave equation. In this paper , it is written in the frequency domain

$\nabla \cdot ( C C_g \nabla \phi) + \frac{C_g \omega^2}{C} \phi = 0$

With very little viscous drag, as the tidal frequency gets close to the characteristic frequency of the harbor location, it sets up a strong resonance that amplifies the tidal effect. The beat frequency of the envelope of the response is essentially defined by the difference of the characteristic frequency and the tidal frequency.

 Xing, Xiuying, Jiin-Jen Lee, and Fredric Raichlen. "Harbor Resonance: A Comparison of Field Measurements to Numerical Results." Coastal Engineering Proceedings 1.32 (2011): currents-42.

Comment Source:Similar to the ENSO resonance but on a much smaller scale is the amazing tidal resonance found in places such as the Bay of Fundy. The reason that the height differential of the tide is so high at this location is because the harbor has a natural resonance frequency that is fairly close to the 12.42 hr M2 tidal period. ![pic](http://imageshack.com/a/img537/6085/f55Gxg.gif) <http://www.tide-forecast.com/locations/Hopewell-Cape-New-Brunswick/tides/latest> ![hopewell](http://imageshack.com/a/img661/2104/gK1jPH.gif) The governing equation is again the wave equation. In this paper , it is written in the frequency domain $\nabla \cdot ( C C_g \nabla \phi) + \frac{C_g \omega^2}{C} \phi = 0$ With very little viscous drag, as the tidal frequency gets close to the characteristic frequency of the harbor location, it sets up a strong resonance that amplifies the tidal effect. The beat frequency of the envelope of the response is essentially defined by the difference of the characteristic frequency and the tidal frequency.  Xing, Xiuying, Jiin-Jen Lee, and Fredric Raichlen. "Harbor Resonance: A Comparison of Field Measurements to Numerical Results." Coastal Engineering Proceedings 1.32 (2011): currents-42.