nad said on 9/3/2016:

> "I would like to add that the year, where the temperature anomalies do seem not to follow a similar biannual pattern as the QBO is (apart from the years >2011) the year 1971. In this year there was the volcano Teneguia."

A biennial cycle seems to be an underlying behavior for ENSO.

I'm satisfied with the physical model for QBO, as I can derive it [mathematically from first principles](http://contextearth.com/2016/08/23/qbo-model-final-stretch/) starting from Laplace's tidal equations. The key to a successful derivation is that the important measure is the *acceleration* of the QBO, and not the wind *velocity*. The final result is obtained by solving a Sturm-Liouville equation with a surprising simple analytical formulation. The acceleration can then be tied directly to a gravitational forcing with period related to the draconic lunar month of 27.212 days. When the acceleration is integrated to a velocity, the model fit to the data is impressive. There isn't anything close to this in the QBO research literature, starting way back to Lindzen. Some have hinted for evidence of atmospheric tides, but that discussion mainly concerns solar tides.

Where I am at with the ENSO model is [perhaps as impressive in terms of a fit](http://contextearth.com/2016/05/12/deterministically-locked-on-the-enso-model/), but the explanation is more tenuous. I start with a Mathieu wave equation, which is a standard formulation for modeling sloshing of liquid volumes. The thermocline within the equatorial Pacific ocean is very sensitive to angular momentum variations as the density differences between the warmer surface layer and colder layer can cause it to slosh longitudinally.

What I find is that two primary angular momentum cycles of 14 years and 6.5 years (related to the Chandler wobble) reproduce the general profile of the ENSO time series -- but only if an underlying biennial modulation is simultaneously applied. That's the tenuous explanation as a *strictly biennial cycle* is a metastable phenomena; little motivates such a cycle apart from a possible period doubling of the annual cycle due to a nonlinear bifurcation. That metastability may have driven a biennial phase inversion which occurred during the period 1980-1996, timed to a Pacific ocean climate shift.

Yet driving this model to second-order is a remarkable fit of the lunar tidal frequencies to the ENSO residual, which fills in the detail of the El Nino and La Nino peaks and valleys that the primary angular momentum variations missed. The nonlinear aliasing of the lunar cycles can actually mimic a biennial modulation of a few fundamental frequencies, which are clearly expressed as Fourier components within the ENSO time-series, [as shown here](http://contextearth.com/2016/06/10/pukites-model-of-enso/).

Taken together, the common forcing frequencies discovered between the QBO and ENSO suggest that even though the QBO model has the best theoretical basis, the ENSO model will not be far behind. A couple of years have passed since the start of this thread and I can honestly say that the explanation is becoming more concise.

> "I would like to add that the year, where the temperature anomalies do seem not to follow a similar biannual pattern as the QBO is (apart from the years >2011) the year 1971. In this year there was the volcano Teneguia."

A biennial cycle seems to be an underlying behavior for ENSO.

I'm satisfied with the physical model for QBO, as I can derive it [mathematically from first principles](http://contextearth.com/2016/08/23/qbo-model-final-stretch/) starting from Laplace's tidal equations. The key to a successful derivation is that the important measure is the *acceleration* of the QBO, and not the wind *velocity*. The final result is obtained by solving a Sturm-Liouville equation with a surprising simple analytical formulation. The acceleration can then be tied directly to a gravitational forcing with period related to the draconic lunar month of 27.212 days. When the acceleration is integrated to a velocity, the model fit to the data is impressive. There isn't anything close to this in the QBO research literature, starting way back to Lindzen. Some have hinted for evidence of atmospheric tides, but that discussion mainly concerns solar tides.

Where I am at with the ENSO model is [perhaps as impressive in terms of a fit](http://contextearth.com/2016/05/12/deterministically-locked-on-the-enso-model/), but the explanation is more tenuous. I start with a Mathieu wave equation, which is a standard formulation for modeling sloshing of liquid volumes. The thermocline within the equatorial Pacific ocean is very sensitive to angular momentum variations as the density differences between the warmer surface layer and colder layer can cause it to slosh longitudinally.

What I find is that two primary angular momentum cycles of 14 years and 6.5 years (related to the Chandler wobble) reproduce the general profile of the ENSO time series -- but only if an underlying biennial modulation is simultaneously applied. That's the tenuous explanation as a *strictly biennial cycle* is a metastable phenomena; little motivates such a cycle apart from a possible period doubling of the annual cycle due to a nonlinear bifurcation. That metastability may have driven a biennial phase inversion which occurred during the period 1980-1996, timed to a Pacific ocean climate shift.

Yet driving this model to second-order is a remarkable fit of the lunar tidal frequencies to the ENSO residual, which fills in the detail of the El Nino and La Nino peaks and valleys that the primary angular momentum variations missed. The nonlinear aliasing of the lunar cycles can actually mimic a biennial modulation of a few fundamental frequencies, which are clearly expressed as Fourier components within the ENSO time-series, [as shown here](http://contextearth.com/2016/06/10/pukites-model-of-enso/).

Taken together, the common forcing frequencies discovered between the QBO and ENSO suggest that even though the QBO model has the best theoretical basis, the ENSO model will not be far behind. A couple of years have passed since the start of this thread and I can honestly say that the explanation is becoming more concise.