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Some examples of annular modes include ENSO, NAM, and SAM.

I think one of the major mysteries of them is how they formed to begin with.

I'm just curious - do we have any idea of whether or not these annular modes existed in paleoclimates, particularly around the time of Pangaea? Do we have the computational power to predict them?

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    $\begingroup$ Depends if you trust the models ;-) $\endgroup$ – gerrit Apr 15 '14 at 19:33
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Since you're asking about Pangaea time, i would assume you're talking about the Permian (from 299Ma to 252 Ma).

We have a good idea of what Permian paleogeography was, a fair idea of how Permian oceans circulated globally (see one of your previous questions), enough data to produce credible models on how was the climate at the time (see e. g. Gibbs et al. 2002), even some idea on what atmospheric circulation (see e. g. Tibor & Montanez 2002) was like during that Period. So in theory i don't see any major hurdles to model these kind of oscillations (keep in mind that i am not a climate model specialist).

Now in my opinion, the big issue would be the data to back the model on, and particularly the time resolution needed to study these kind of oscillations: ENSO, PDO, NAM, SAM etc are annual to decadal oscillations (1-10yr).
A resolution "commonly"-achieved so far in detecting climatic oscillations in the Permian fossil record is for cycles of ca. 20kyr (i. e. 20 000 years; Milankovitch precession cycle). The kind of sedimentation rates and sediment quality needed to be able to record annual layers are extremely rare for such older sediments.

With that in mind, it's not impossible to find such sediments, in fact Anderson (1982) detailed the 260kyr-long continuous, annual varved Castille and Bell Canyon Permian evaporitic Formations, but even in such a high-resolution formation the shorter-length oscillation detected was still at ca. 180yr so way above that of any of the annular modes you mentioned.

References:
Anderon, R. Y., 1982. A long geoclimatic record from the Permian. Journal of Geophysical Research : Oceans, 87: 7285-7294.
Gibbs, M. T., McAllister Rees, P., Kutzbach, J. E., Ziegler, A. M., Behling, P. J., Rowley, D. B., 2002. Simulations of Permian Climate and Comparisons with Climate-Sensitive Sediments. Journal of Geology, 110: 33-55.
Tibor, N. J., Montanez, I. P., 2002. Shifts in late Paleozoic atmospheric circulation over western equatorial Pangea: Insights from pedogenic mineral $\delta^{18}O$ compositions. Geology, 30: 1127-1130.

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