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This index has been mentioned elsewhere: http://forum.azimuthproject.org/discussion/169/el-nino-southern-oscillation-enso/?Focus=11994#Comment_11994
I ran a Eureqa machine learning dispersion analysis on the MEI numbers and found some rather predictable periods.
The numbers that pop up in the solution space are frequencies of 6.3 rads/year and 12.6 rads/year, which are the annual and biannual signals
$ f''(t) = -12.6 f+ 6.3 f \sin(2 \pi t) - 6.3 f \sin(4 \pi t) + 13.22 \sin(4 \pi t) - 9.922 \sin(12.84 t) $
Because of the Mathieu modulation in the 2nd-order DiffEq and a forcing signal that is slightly off the biannual frequency, the result is a very erratic waveform with no clearly discernible period. The biannual forcing is interacting with a modulation that is annual and biannual, which makes it very sensitive to phase changes.
According to the documentation, MEI is a heavily processed index that goes through a data reduction technique known as principal components analysis (PCA).
"Here we attempt to monitor ENSO by basing the Multivariate ENSO Index (MEI) on the six main observed variables over the tropical Pacific. These six variables are: sea-level pressure (P), zonal (U) and meridional (V) components of the surface wind, sea surface temperature (S), surface air temperature (A), and total cloudiness fraction of the sky (C). These observations have been collected and published in ICOADS for many years. The MEI is computed separately for each of twelve sliding bi-monthly seasons (Dec/Jan, Jan/Feb,..., Nov/Dec). After spatially filtering the individual fields into clusters (Wolter, 1987), the MEI is calculated as the first unrotated Principal Component (PC) of all six observed fields combined. This is accomplished by normalizing the total variance of each field first, and then performing the extraction of the first PC on the co-variance matrix of the combined fields (Wolter and Timlin, 1993). In order to keep the MEI comparable, all seasonal values are standardized with respect to each season and to the 1950-93 reference period. "
PCA may be removing other frequencies, as only one principal component is retained.