On the slow mode of a simple air-sea coupled model: The sensitivity to the zonal phase difference between the SST and the atmospheric heating

bn eigen analysis of the simple air-sea coupled model is carried out to understand the coupled mechanism with the slow time scale (so called 'slow mode') related to the zonal phase difference between the SST and atmospheric heating anomalies. Both frequency and instability of the slow mode are very sensitive to the zonal phase difference between the SST and atmospheric heating anomalies. When the positive atmospheric heating is located between one-quarter wavelength west and one-quarter wavelength east of the positive SST anomaly, the slow mode becomes the unstable eastward-propagating mode. The further eastward shifting of the atmospheric heating results in damping of the eastward-propagating mode resembling the forced Kelvin mode. On the other hand, the slow mode becomes the unstable westward-propagating mode when atmospheric heating lies farther west than one-quarter wavelength west from the SST anomaly. This unstable westward-propagating mode resembles the forced Rossby mode. SST changes in the eastward- and westward-propagating slow modes are mainly induced by the changes of the oceanic height and the zonal advection by the current, respectively. The slow mode in the fast-wave limit and the non-rotating system is also discussed.