%Variance explained by each EOF

1. MJO MetricsCombined EOF’s using 20-100 day filtered OLR, u850, and u200 averaged between 15°N-15°S Prior to computing EOFs, each equatorially-averaged field is normalized by the square-root of the zonal mean of the temporal variance at each longitudinal point. The normalizations are (OLR, u850, u200): 8.64 Wm-2, 1.18 ms-1, 3.34 ms-1 (all seasons); 9.67 Wm-2, 1.26 ms-1, 3.58 ms-1 (southern summer); 7.51 Wm-2, 1.09 ms-1, 3.09 ms-1 (northern summer). Data used was January 1980 to December 2002. Winds were NCEP/NCAR Reanalysis. Southern Summer = November to April; Northern Summer = May to October

2. note that eof2 and eof1 swapped for n. summer to be consistent with other seasons %Variance explained by each EOF

3. Coherence squared and phase between PC1 and PC2 from each EOF calculation (The input data was band-pass filtered for 20-100 days, so the cross-spectrum outside this range is meaningless.) All Seasons S. Summer N. Summer Cross-spectra of the all-season PCs computed separately for southern summer and northern summer. Note: These cross-spectra were computed by applying FFTs to a long time-series composed of stringing-together all the 6-month segments.

4. Project the anomaly data onto the EOFs computed with the band-pass filtered data. The anomalies have the long-term mean and 3 harmonics of the seasonal cycle removed. Compute spectra for ALL SEASONS. That is, project all seasons of data onto the season-specific EOFs, and compute spectra using all months of the year. Spectra computed on 6-month segments of data, padded with zeroes to 256 days. Projection onto All Season EOFs Projection onto S. Summer EOFs Projection onto N. Summer EOFs The EOFs act as a highly selective filter for the frequencies of the MJO Note that the projected PCs will no longer have unit standard deviation. In fact, the variance of the PCs computed from the projection onto the northern summer EOFs is greater because the normalization factors used for northern summer are less.

5. Project the anomaly data onto the EOFs computed with the band-pass filtered data. The anomalies have the long-term mean and 3 harmonics of the seasonal cycle removed. Compute spectra for SOUTHERN SUMMER only. Spectra computed on 6-month segments of data, padded with zeroes to 256 days. Projection onto All Season EOFs Projection onto S. Summer EOFs Note that the projected PCs will no longer have unit standard deviation, and the different normalization factors cause the projected PCs to have different amplitudes.

6. Project the anomaly data onto the EOFs computed with the band-pass fitlered data. The anomalies have the long-term mean and 3 harmonics of the seasonal cycle removed. Compute spectra for NORTHERN SUMMER only. Spectra computed on 6-month segments of data, padded with zeroes to 256 days. Projection onto All Season EOFs Projection onto N. Summer EOFs Note that the projected PCs will no longer have unit standard deviation, and the different normalization factors cause the projected PCs to have different amplitudes.

7. Create a “synthesized” MJO OLR field through 3-d regression with the projected PCs. That is, compute the 3-d regression between (20-100 day filtered) OLR at each grid point and PC1 and PC2 from the different EOF analyses. This regression is seasonally-varying, that is, a different regression is formed for each month of the year. There is little difference in the MJO-OLR variance in JJAS associated with the all season EOF analysis compared to the northern summer only EOFs. a) Variance of synthesized MJO OLR field (from All Season EOFs) - JJAS b) Variance of synthesized MJO OLR field (from N. Summer EOFs) - JJAS

8. Create a “synthesized” MJO OLR field through 3-d regression with the projected PCs. That is, compute the 3-d regression between (20-100 day filtered) OLR at each grid point and PC1 and PC2 from the different EOF analyses. This regression is seasonally-varying, that is, a different regression is formed for each month of the year. There is little difference in the MJO-OLR variance in DJFM associated with the all season EOF analysis compared to the southern summer only EOFs: All season EOFs are adequate. a) Variance of synthesized MJO OLR field (from All Season EOFs) - DJFM b) Variance of synthesized MJO OLR field (from S. Summer EOFs) - JJAS

9. Conclusion Even though the spatial structure of the seasonally-specific EOFs differ slightly from the all-season EOFs (especially over the E. Pacific), virtually the same variability is captured by the all-season EOFs. Furthermore, the all season EOFs resolve the distinct behavior of the MJO off of the equator in N. Summer and S. Summer via regression or composites for the two seasons separately (see attached composites of OLR and for DJFM and MJ from Wheeler and Hendon 2004). Hence, for ease of understanding and simplicity, we recommend computation of combined EOFs using all seasons of data. That is, we recommend using the all season EOFs for this MJO metric. We also see no good reason for varying the latitudinal averaging for different seasons. Even though the OLR signal of the MJO shifts into the summer hemisphere, the strongest zonal wind signal tends to shift into the winter hemisphere. Hence, if your intent is to capture variability associated with the MJO, we recommend averaging 15°S to 15°N.

10. “Projected” PCs computed when anomaly data is projected onto the EOFs of the filtered data PCs from the all season EOF calculation on 20-100 day filtered data Lag correlations between PC1 and itself, and with PC2. Cross-spectra between PC1 and PC2. Note that they are identical in the 20-100 day band.

11. Fig. 8 from Wheeler and Hendon. Composite OLR and 850HPa winds for DJF based on all season EOFs

12. Fig. 9 from Wheeler and Hendon. Composite OLR and 850HPa winds for MJ based on all season EOFs