The relation among sea surface temperature (SST) cooling in the southeastern Indian Ocean (SEIO), oceanic Rossby waves, and the seasonal onset of the Madden-Julian oscillation (MJO) is examined for the period 1993-2012. A westward propagation of the annual downwelling Rossby waves occurs in the south Indian Ocean for most of the years. However, their amplitude and phase speed vary interannually. Positive SST anomalies (SSTA) migrate concurrently with the Rossby waves but are followed by a wide-spread cold SST area in the SEIO from boreal summer to fall. Although SEIO cooling tends to persist for a longer period until November during positive Indian Ocean Dipole (IOD) and/or El Niño years, it occurs irrespective of the IOD. Convection related to the MJO events during boreal winter propagates from the Indian Ocean to the Pacific only after SEIO cooling is terminated. A correlation analysis indicates that negative SSTA during SEIO cooling are confined to the Southern Hemisphere, but their influence on convection reaches north of the equator via the excitation of local circulations over the eastern Indian Ocean and the tropical western Pacific. The resulting southerly surface wind anomalies may advect dry air from the south of the equator to the north and suppress atmospheric convection around the equator. Thus, SEIO cooling tends to prevent intraseasonal convection from propagating eastward to the Pacific. By briefly analyzing the process of SEIO cooling, the SST variability in the SEIO from boreal summer to fall is found to correlate well with zonal advection and surface heat flux. Zonal advection, in turn, is connected with the strength of westward currents associated with the Rossby waves. An understanding of SEIO upper-ocean processes can contribute to predict the seasonal onset of an MJO sequence.