The stability characteristics of normal mode perturbations on idealized basic states that have meridional potential vorticity (PV) gradients that are zero in the troposphere, very large at the tropopause, and order beta in the stratosphere are checked. The results are compared to the corresponding models that have a lid at the tropopause. The dispersion relations and the vertical structures of the modes are similar in the two models, thus confirming the relevance of the Eady problem to unbounded atmospheres. The effect of replacing the lid with a more realistic tropopause is to complicate the interaction of tropopause and surface waves, such as to inhibit phase locking for a range of wavenumbers. This causes the short-wave cutoff of the Eady model to move to longer waves. Also, there is a slight destabilization of the long waves, which have large amplitudes in the stratosphere. The effect of gradually changing the tropospheric PV gradients from zero (Eady-type profile) to beta (Green-type profile) on the stability of normal modes is checked. The dispersion relations show a smooth transition from the Green profiles to the Eady profiles, and a short-wave cutoff is gradually formed. Finally, the possibility of neutralizing the atmosphere through the short-wave cutoff of the Eady model by lifting the tropopause while keeping PV gradients zero in the troposphere is examined. It is found that instability depends on some minimal amount of tunneling of waves between the surface and the tropopause. The amount of tunneling depends on the vertical integral of N in the troposphere. It is necessary for the vertical integral of N to increase for the short-wave cutoff to move to longer waves. For reasonable Brunt-Vaisala frequency profiles, lifting the tropopause causes the short-wave cutoff to move to longer wavelengths, but the details are sensitive to boundary values of N2 and wind shear.
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