As one of the ongoing projects -- understanding El Nino behavior -- I thought I would review the status of my ENSO model. It is in a relatively stable configuration but I occasionally revisit it and tweak the parameters.

The model essentially emulates a 2nd-order wave equation as described by Allan Clarke [1]. The characteristic frequency $\omega$ has a period of approximately 4.25 years. This is modulated by variations due to TSI and forced by QBO wind variations, angular forcing variations characterized by the Chandler wobble effect, and by TSI. The result is a differential formulation with the characteristics of a Mathieu or Hill equation . This formulation is well-known in the hydrodynamics literature describing sloshing of liquid volumes [2][3].

$ f''(t) + (\omega^2 + k TSI(t)) f(t) = QBO(t) + TSI(t) + CW(t) $

To solve this equation, I tried to create closed-form expressions for the known forcings and then applied the expressions to solve the differential equation using Mathematica. The number above each chart is the correlation coefficient multiplied by 100. These formulations are pseudo-periodic so that the tweaking I applied is to best emulate the experimentally measured oscillations, without going overboard in fidelity. The curve labeled SOIM is the model of the ENSO SOI data.

![SOIM](http://imageshack.com/a/img661/3316/N7hLzP.gif)

As an example of a tweak, the TSI curve was broken into two pieces at 1980. Before that point in time, the 22-year period cycle was more evident and after that the 11-year was stronger. The CW correlation coefficient seems low because the period appears to exhibit a phase reversal before 1930. Whether or not this is a real physical transition, I am not sure, so I left it alone.

The million dollar question is whether such an overall fit is possible just by chance selection of these factors. Each one of the factors, QBO [4], CW [5], and TSI [] is suggested as important in ENSO behavior mechanics in the literature.

I wrote up the paper [here on ARXIV](http://arxiv.org/abs/1411.0815). I submitted this to a journal but it got rejected w/o review.

[1] A. J. Clarke, S. Van Gorder, and G. Colantuono, “Wind stress curl and ENSO discharge/recharge in the equatorial Pacific,” Journal of physical oceanography, vol. 37, no. 4, pp. 1077–1091, 2007.

[2] O. M. Faltinsen and A. N. Timokha, Sloshing. Cambridge University Press, 2009.

[3] J. B. Frandsen, “Sloshing motions in excited tanks,” Journal of Computational Physics, vol. 196, no. 1, pp. 53–87, 2004.

[4] W. M. Gray, J. D. Sheaffer, and J. A. Knaff, “Inﬂuence of the stratospheric QBO on ENSO variability,” J. Meteor: Soc. Japan, vol. 70, pp. 975–995, 1992.

[5] R. S. Gross, “The excitation of the Chandler wobble,” Geophysical Research Letters, vol. 27, no. 15, pp. 2329–2332, 2000.

[66] L. Kuai, R.-L. Shia, X. Jiang, K. K. Tung, and Y. L. Yung, “Modulation of the period of the quasi-biennial oscillation by the solar cycle,” Journal of the Atmospheric Sciences, vol. 66, no. 8, pp. 2418–2428, 2009.

The model essentially emulates a 2nd-order wave equation as described by Allan Clarke [1]. The characteristic frequency $\omega$ has a period of approximately 4.25 years. This is modulated by variations due to TSI and forced by QBO wind variations, angular forcing variations characterized by the Chandler wobble effect, and by TSI. The result is a differential formulation with the characteristics of a Mathieu or Hill equation . This formulation is well-known in the hydrodynamics literature describing sloshing of liquid volumes [2][3].

$ f''(t) + (\omega^2 + k TSI(t)) f(t) = QBO(t) + TSI(t) + CW(t) $

To solve this equation, I tried to create closed-form expressions for the known forcings and then applied the expressions to solve the differential equation using Mathematica. The number above each chart is the correlation coefficient multiplied by 100. These formulations are pseudo-periodic so that the tweaking I applied is to best emulate the experimentally measured oscillations, without going overboard in fidelity. The curve labeled SOIM is the model of the ENSO SOI data.

![SOIM](http://imageshack.com/a/img661/3316/N7hLzP.gif)

As an example of a tweak, the TSI curve was broken into two pieces at 1980. Before that point in time, the 22-year period cycle was more evident and after that the 11-year was stronger. The CW correlation coefficient seems low because the period appears to exhibit a phase reversal before 1930. Whether or not this is a real physical transition, I am not sure, so I left it alone.

The million dollar question is whether such an overall fit is possible just by chance selection of these factors. Each one of the factors, QBO [4], CW [5], and TSI [] is suggested as important in ENSO behavior mechanics in the literature.

I wrote up the paper [here on ARXIV](http://arxiv.org/abs/1411.0815). I submitted this to a journal but it got rejected w/o review.

[1] A. J. Clarke, S. Van Gorder, and G. Colantuono, “Wind stress curl and ENSO discharge/recharge in the equatorial Pacific,” Journal of physical oceanography, vol. 37, no. 4, pp. 1077–1091, 2007.

[2] O. M. Faltinsen and A. N. Timokha, Sloshing. Cambridge University Press, 2009.

[3] J. B. Frandsen, “Sloshing motions in excited tanks,” Journal of Computational Physics, vol. 196, no. 1, pp. 53–87, 2004.

[4] W. M. Gray, J. D. Sheaffer, and J. A. Knaff, “Inﬂuence of the stratospheric QBO on ENSO variability,” J. Meteor: Soc. Japan, vol. 70, pp. 975–995, 1992.

[5] R. S. Gross, “The excitation of the Chandler wobble,” Geophysical Research Letters, vol. 27, no. 15, pp. 2329–2332, 2000.

[66] L. Kuai, R.-L. Shia, X. Jiang, K. K. Tung, and Y. L. Yung, “Modulation of the period of the quasi-biennial oscillation by the solar cycle,” Journal of the Atmospheric Sciences, vol. 66, no. 8, pp. 2418–2428, 2009.