A scatterring theory for the interaction of tropical and extratropical waves

We study the excitation of Kelvin waves by extratropical Rossby wave activity in the presence of shear. Recent observational evidence indicates that extratropical wave activity is associated with equatorial convection even in regions where there is a "critical line" to Rossby wave propagation at upper levels. In some cases the wave activity propagates into low latitudes but not as far as the convectively active regions, exciting convectively coupled Kelvin waves. In other cases the extratropical wave activity is confined to the storm track, far removed from the tropics, yet equatorial Kelvin activity is still excited.

Using multiple scale asymptotic methods, Majda and Biello developed simplified equations for the interaction of equatorial Rossby waves and barotropic Rossby waves with a significant midlatitude projection. Using similar methods, we show how vertical and horizontal shear can lead to the coupling of extratropical Rossby waves with convectively coupled Kelvin waves. The simplified equations have a mix of wave-mean flow and wave-wave interaction due to the dispersionless nature of equatorial long waves. We demonstrate some solutions to these equations and discuss their relevance to the question of observed Kelvin wave excitation. We also show how this problem can be viewed as the scattering of ``free'' midlatitude Rossby waves off of ``bound'' equatorial Kelvin and Rossby waves.