1. Predictability: How can we predict the climate decades into the future when we can’t even predict the weather for next week?
Predictability of the first kind: Predict the future based on initial conditions, with boundary conditions constant. This is limited by the chaotic nature of the atmosphere, which is a physical system with built-in instabilities, in vertical convection (e.g., thunderstorms) and horizontal motion (e.g., baroclinic instability - development of low pressure systems, such as hurricanes and Nor’Easters).

2. Consider a prediction using the above equation of the future state of the variable X, say the surface air temperature. The subscript n indicates the time, say the day, and a is a constant representing the physics of the climate system. X for any day is a times its value on the previous day minus X squared on the previous day.
With such a simple equation, it should be possible to predict X indefinitely into the future. Right?

3. Let’s assume that a is exactly 3
Let’s assume that a is exactly and that a prediction with three decimal places is the exact solution.
Then let’s consider three types of errors: imprecise knowledge of the physics of the climate system, imprecise initial conditions, and rounding due to limited computer resources.
This example is from Edward Lorenz.

6. Predictability: How can we predict the climate decades into the future when we can’t even predict the weather for next week?
Predictability of the second kind: Predict the future based on boundary conditions, independent of initial conditions. If there are slowly-varying (with respect to the atmospheric predictability limit of 2-3 weeks) boundary conditions (e.g., greenhouse gases, stratospheric aerosols, sea surface temperatures, soil moisture, snow cover) that can be predicted, then the envelope of the weather can be predicted. [The first two examples are external to the climate system, and the last three are internal.]

7. NOAA Medium Range Forecasts

8. https://climatedataguide.ucar.edu/climate-data/era-interim

9. ECMWF forecast skill

10. MJO forecast from Dee et al. (2011).
Correlation of > 0.6 has skill.
Dee, D. P., et al., 2011: The ERA-Interim reanalysis: configuration and performance of the data assimilation system. Q. J. R. Meteorol. Soc., 137, 553–597. DOI: /qj.828

11. NOAA MJO forecasts
(1) Reanalysis 2 (R2) is used since CFS operational forecast utilizes the R2 as initialization data.
(2) The indices for the latest 4 days are calculated using NCEP GDAS (Global Data Analysis System). 
(3) The MJO definition used here is identical to the Matt Wheeler's (Wheeler and Hendon 2004),  i.e., to represent the MJO, the first two EOFs of combined fields of OLR, u850 and u200 are used.  The followings are some details of the forecast models.
(4) CFS operational:  this is a 2003 version and two member ensemble mean is used.
(5) GFS offline: this runs exactly the same as CFS operational model (e.g. the same R2 initial data) except that air-sea interaction is not allowed.  Four member ensemble mean is used.
(6) GFS operational:  the model keeps being updated.  Model climatology from the GFS offline model is used.  The 11-member ensemble mean is used.
(7) AR: Autoregressive time series model.
(8) PCRLAG: Lagged multiple linear regression.
For details please contact to Kyong-Hwan Seo

12. NOAA MJO forecast from GEFS model

13. NOAA MJO forecast from statistical models

14. NOAA MJO forecast from GEFS model

15. NOAA MJO forecast from statistical models

16. ERA-Interim
Persistence
Dee, D. P., et al., 2011: The ERA-Interim reanalysis: configuration and performance of the data assimilation system. Q. J. R. Meteorol. Soc., 137, 553–597. DOI: /qj.828

17. CONSTRUCTED ANALOG METHOD, Huug van den Dool, http://www. cpc. ncep

18. CONSTRUCTED ANALOG METHOD, Huug van den Dool, http://www. cpc. ncep

19. CONSTRUCTED ANALOG METHOD, Huug van den Dool, http://www. cpc. ncep