Connections between decadal changes in the tropical Pacific mean state and El Ni¿o--Southern Oscillation (ENSO) decadal modulation are examined using three runs of a coupled general circulation model (CGCM). The differences between the three simulations, i.e., a standard coupled model (one AGCM is coupled to a single OGCM) and two interactive ensemble models (six or twelve AGCMs are coupled to a single OGCM), are confined to the amplitude of internal atmospheric variability. All three simulations have the same tropical Pacific basin scale SST mode that dominates the low-frequency variability, which is identified by the first EOF mode. This low-frequency mode is largely independent of the simulated ENSO and is neither a residual of the decadal ENSO modulation nor does it produce any low-frequency modulation of ENSO. The analysis presented here suggests that this low-frequency mode is stochastically driven by atmosphere noise. There are, however, low-frequency changes in the tropical Pacific mean state that are connected to ENSO decadal modulation. The mean state associated with ENSO amplitude has different structures in the SST and wind stress anomalies than does the dominant tropical Pacific mean state identified by the first EOF. The tropical Pacific mean state that is unambiguously associated with ENSO decadal modulation, which is remarkably similar to the second EOF SST mode in two interactive ensemble models, but is difficult to detect in a standard coupled model simulation. These results argue that there is a component of ENSO variability that cannot be explained by a linear, damped and stochastically forced process.