Analytical Solutions of Initial Value Problem of Diabatic Rossby Waves

The initial value problem of diabatic Rossby waves is analytically solved in a vertical-zonal two-dimensional quasi-geostrophic system on an f-plane. The given basic state is in thermal-wind-balance, and dry baroclinic instability is excluded. The diabatic heating (or cooling) is proportional to the vertical velocity on the middle level and generates potential vorticity (PV) disturbances on the lower and upper levels of the atmosphere. The PV disturbance propagates eastward (westward) on the lower (upper) level relative to the fluid, while the prescribed basic zonal flow advects the PV disturbance toward the opposite direction. For a zonal wavelength shorter than a marginal one, which becomes shorter as the heating becomes stronger, the advection effect is dominant, and consequently, the lower and upper PV disturbances move downstream and away from each other without effective interaction or growth. On the other hand, for a zonal wavelength longer than a marginal one, the propagation effect is dominant, and consequently, the lower and upper PV disturbances tend to move upstream and away from each other. However, this leads to strong mutual interactions between the lower and upper PV disturbances. As a result, the PV disturbances grow exponentially and attain a phase-locked upshear-tilted vertical structure. The behaviour of PV disturbances can be qualitatively explained on the basis of PV thinking.