David wrote:

> There’s no trace of a plausible explanation for why cooperativity would foreshadow El Niño the next year. Of course, if it’s a real empirical discovery, then the explanation could come later.

I agree that this is very much worth thinking about. But we shouldn't necessarily expect the explanation would appear in this particular paper.

It seems quite plausible to me that correlations between different locations increase as we approach a widespread event like an El Niño. In statistical mechanics we think a lot about 2-point functions - covariances between the value of some field $F$ at one point $i$ and another point $j$:

$$C_{i,j} = \langle F_i F_j \rangle - \langle F_i \rangle \langle F_j \rangle$$

2-point functions typically decay exponentially as the distance between the points $i$ and $j$ increases. However, as our system approaches a phase transition, e.g. as a solid approaches its melting point, its 2-point functions decay more slowly, and right at the phase transition they often show power-law decay.

In other words: when something dramatic is on the brink of happening, the system displays a lot of correlation between distant locations.

_Does the start of an El Niño act in this way?_ That seems like a good question.

The network guys seem to be hoping something like this is true. But unlike people in stat mech, they're studying the 2-point function by drawing a graph where there's an edge between any two sites where the 2-point function is "big" (according to some specific criterion). They're hoping this graph will change character right before the start of an El Niño.

A fun alternative would be to directly study the 2-point function, see how it decays with distance, and see if the approach of an El Niño changes the decay rate.

Now by 2-point function I either mean the _time-delayed_ cross-covariance function David mentioned:

$$C_{i,j}^{t}(-\tau) = \langle T_i(t) T_j(t - \tau) \rangle - \langle T_i(t) \rangle \langle T_j(t - \tau) \rangle$$

or else, more simply, the version with no time delay:

$$C_{i,j}^{t} = \langle T_i(t) T_j(t) \rangle - \langle T_i(t) \rangle \langle T_j(t) \rangle$$