Dave said:

> "Here is a science team that addresses ENSO Chaos via Lyapunov Exponent analysis (no "punting"):"

In case you are unaware, it's known that calculating a Lyapunov Exponent can only be done on a mathematical formulation, i,e. on a hypothetical model of the empirical data. It can't be done on the actual data because it requires a perturbation of the internal variables to be able to gauge output sensitivity to a change in the input. Since one does not know what these internal variables are, it's all just a guess as to whether the actual behavior is chaotic. It's basically a chicken-and-egg problem,

So any Lyapunov Exponent you see quoted is with respect to a mathematical model and it does not really show anything but a characterization of how much the modeler thinks the resulting behavior is chaotic.

I don't think my model is chaotic as it does have a repeat period (however long that may be) and so the Lyaponuv Exponent holds little relevance. Moreover, it's possible to show that the measured ENSO time-series contains long term coherence. This is related to the annual spring behavior which synchronizes the behavior. The following intra-spectral cross-correlation demonstrates clearly that the physical response is based on an annual impulse. There is a symmetry in the spectrum about the 0.5/yr frequency value.

![](https://imagizer.imageshack.com/v2/1242x470q90/r/922/YmyFPN.png)

Likely not chaotic

You say:

> " The ENSO Elephant is not yet fully grasped."

Dave, That's an argument of Appeal to Authority mixed in with Appeal To Complexity. If you want to pick apart some aspect of the model, then do that.

You say:

> "[Lin & Qian 2019] offers partial Lunar Tidal Forcing as a refinement to previous models, as they take pains to explain."

It appears that the discussion along this track will continue.

> "Dear Dr. Pukite,

>Thank you very much for your comments. It’s great to know that a mathematical model of tidally forced ENSO is being developed. Hope it could make great long-lead forecasts. I’m editing an AGU monograph in which I’ll emphasize the importance of tidal forcing (attached).

>Cheers,

>Jialin Lin"

And a Russian team (Serykh & Sonechkin) is also looking at this as well

["Interrelations Between Temperature Variations in Oceanic Depths and the Global Atmospheric Oscillation"](https://link.springer.com/article/10.1007/s00024-020-02615-9)

They think it is somehow associated with the pole tide, which is connected to the Chandler wobble, bringing it full circle.

Look at the upper left of the curve and the spectral spike at ~0.85/yr labelled Chandler Wobble.

![](https://imagizer.imageshack.com/img923/7461/Fj2DqK.png)

ENSO doesn't have wavenumber 0 symmetry so the regional tidal factors should be more of a factor than the Chandler wobble forcing. The Russians are essentially where I was about 5 years ago, when I spotted some correlation

https://forum.azimuthproject.org/discussion/comment/14538/#Comment_14538

The same Russian team is aware of our work as you can see Serykh is commenting here:

https://esd.copernicus.org/preprints/esd-2020-74/#discussion

> "Here is a science team that addresses ENSO Chaos via Lyapunov Exponent analysis (no "punting"):"

In case you are unaware, it's known that calculating a Lyapunov Exponent can only be done on a mathematical formulation, i,e. on a hypothetical model of the empirical data. It can't be done on the actual data because it requires a perturbation of the internal variables to be able to gauge output sensitivity to a change in the input. Since one does not know what these internal variables are, it's all just a guess as to whether the actual behavior is chaotic. It's basically a chicken-and-egg problem,

So any Lyapunov Exponent you see quoted is with respect to a mathematical model and it does not really show anything but a characterization of how much the modeler thinks the resulting behavior is chaotic.

I don't think my model is chaotic as it does have a repeat period (however long that may be) and so the Lyaponuv Exponent holds little relevance. Moreover, it's possible to show that the measured ENSO time-series contains long term coherence. This is related to the annual spring behavior which synchronizes the behavior. The following intra-spectral cross-correlation demonstrates clearly that the physical response is based on an annual impulse. There is a symmetry in the spectrum about the 0.5/yr frequency value.

![](https://imagizer.imageshack.com/v2/1242x470q90/r/922/YmyFPN.png)

Likely not chaotic

You say:

> " The ENSO Elephant is not yet fully grasped."

Dave, That's an argument of Appeal to Authority mixed in with Appeal To Complexity. If you want to pick apart some aspect of the model, then do that.

You say:

> "[Lin & Qian 2019] offers partial Lunar Tidal Forcing as a refinement to previous models, as they take pains to explain."

It appears that the discussion along this track will continue.

> "Dear Dr. Pukite,

>Thank you very much for your comments. It’s great to know that a mathematical model of tidally forced ENSO is being developed. Hope it could make great long-lead forecasts. I’m editing an AGU monograph in which I’ll emphasize the importance of tidal forcing (attached).

>Cheers,

>Jialin Lin"

And a Russian team (Serykh & Sonechkin) is also looking at this as well

["Interrelations Between Temperature Variations in Oceanic Depths and the Global Atmospheric Oscillation"](https://link.springer.com/article/10.1007/s00024-020-02615-9)

They think it is somehow associated with the pole tide, which is connected to the Chandler wobble, bringing it full circle.

Look at the upper left of the curve and the spectral spike at ~0.85/yr labelled Chandler Wobble.

![](https://imagizer.imageshack.com/img923/7461/Fj2DqK.png)

ENSO doesn't have wavenumber 0 symmetry so the regional tidal factors should be more of a factor than the Chandler wobble forcing. The Russians are essentially where I was about 5 years ago, when I spotted some correlation

https://forum.azimuthproject.org/discussion/comment/14538/#Comment_14538

The same Russian team is aware of our work as you can see Serykh is commenting here:

https://esd.copernicus.org/preprints/esd-2020-74/#discussion