A recent climate science review paper covers "Inferring causation from time series in Earth system sciences", Runge et al 2019, Nature Communications

Unfortunately they do not describe common-mode factor causation. I took one of their figures and added that class of mechanism in yellow


Common-mode factors are very common and well-known to experimentalists and trouble-shooters. In the figure, a mechanism linking two regions, which is often classified as a teleconnection may actually be common-mode -- in this case a correlation due to a shared lunar forcing that may tie the regions together.

So let's try a particular geospatial correlation. If we take the left region and shift it to the ENSO region in the Pacific and the right region to AMO in the Atlantic, we can evaluate the common mode between these two oceanic indices.

These two indices on first glance show no time correlation (even with lag shifts applied) -- the scatter plot correlation looks like a blob.


But what happens when we apply the same forcing used in modeling ENSO and then apply that to an AMO model? The Laplace's Tidal Equation (LTE) formulation is essentially normalized to *sin (E f(t) + θ)* for ENSO and *sin (A f(t) + φ)* for AMO, where the *E* and *A* factors calibrate the distinct standing wave number boundary conditions for the two regions, and *f(t)* is the common forcing.

Because of the way that the iteration works in fitting the model to the data (need to fiddle the parameters to avoid getting stuck at local minima) the two forcings aren't precisely the same, R=0.96, but can be considered to be virtually aligned.


The fitted models to the ENSO and AMO with common forcing are below


So the distinction between the two oscillating dipoles resides completely in the LTE sinusoidal modulation applied (the *E* and *A* factors), which differs for the two by a non-trivial amount. In other words, this is a single degree of freedom adjustment corresponding to a global wavenumber of 5 for ENSO and 4 for AMO.


The causative mechanism is thus simply a scaling in LTE space accounting for the difference in the geometry of the Atlantic and Pacific basins for a common-mode lunisolar forcing. As quoted in the previous post

> "physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones."

This is as natural and elegant as it can get -- just a twist added to conventional tidal analysis

N.B. If you think it odd that a single parameter modification can change completely the character of a solution, consider the case of how band structures in materials can change completely with slight lattice transformations. This has a related explanation in terms of Brillouin zone folding. Solid state physicists treat this complication as a cost of doing business