It is well known that the oscillations with various frequencies and wavenumbers can interact via some nonlinear mechanisms, such as the resonant interaction. The question proposed in this study is whether the oscillations with different temporal and spatial scales can interact with each other and lead to instabilities in both the ocean and the atmosphere in a linear framework. With simple but widely used dynamic equations in the ocean and the atmosphere, the possibility is examined. The uncoupled ocean and atmosphere are stable. When the ocean and the atmosphere are coupled, instabilities evolve. If the periods and wavelengths are required to be the same, unstable waves are generated due to the resonance. The interactions between different scales can also lead to instabilities. As a response to large-scale eastward atmospheric waves, there are meso-scale oceanic waves, one eastward and two westward. One of the westward oceanic waves increases during its propagation and is expected to lead to instabilities in the west of an ocean basin. The meso-scale oceanic waves can also feedback to large-scale atmospheric waves. There is one eastward-propagating wave, which decays in an intraseasonal time scale. For the latter mechanism, the internal interactions in the ocean and the atmosphere are also critical processes for the ocean-atmosphere interactions. Such coupling pattern is qualitatively comparable to the SST-MJO interaction, in which the large-scale MJOs propagate eastward in the atmosphere decaying over the eastern Pacific Ocean and the meso-scale oceanic properties propagate westward leading to instabilities in the western Indian Ocean. Of course, the analytical solutions cannot simulate the real SST-MJO interactions and they cannot quantitatively compare with the observations, either. More detailed and realistic model experiments are needed for further test of the application of this mechanism in the future.

We propose that there are probably two types of ocean-atmosphere interactions. One, the traditional one, is like a piston interaction. The spatial and temporal scales of the oscillations both in the ocean and the atmosphere are nearly the same. Instabilities are mainly triggered by the resonance, viz. the equal of the forcing scales and the intrinsic scales. The other one, which is discussed in this study, is like a gear interaction. The spatial and temporal scales of the oscillations in the ocean and the atmosphere are distinct from each other. However, instabilities can be triggered when the combination of the temporal and spatial scales of the forcing matches the combination of the intrinsic oscillations, viz. when the dispersion relation of the oceanic and the atmospheric oscillations match each other.