>I'm not sure what Nad is wondering about, but here's a guess. She might be puzzled that we're calling
$$ \langle A B \rangle - \langle A \rangle \langle B \rangle $$
the covariance between $A$ and $B$. She might be more used to
$$ \langle A - \langle A \rangle \rangle \; \langle B - \langle B \rangle \rangle $$
However, these are equal! (Fun exercise.)
No I think they are not equal. That is I haven't performed the calculation but I am pretty sure that they are not equal - I think they are only equal if you can treat the average as a number with respect to averaging, but given the above definition of an average, together with the time dependence
this doesn't seem to be the case.
I don't think it's a fun exercise
to keep track of summands. Moreover as said it's not fully clear which definition of an average of an average they use and maybe this equality is more or less just a typo and the computer programm uses the usual definition of covariance anyway so apart from the typo there would be no big problem, regardless whether the equality holds or not. Or they wanted to have a different definition of a correlation. Finally alone typing all steps of the calculation would probably take me at least half an hour. I don't have that much time for these kind of things and I mostly made a comment that I see a problem here because it could eventually be the case that they want to use their predictions for real life applications. But then I don't really know what happens in case of El Nino warnings.