In the first chapter, we extend previous analysis of dissipative modes derived using a wave radiation boundary condition at the tropopause by considering the effects of relaxing hydrostatic balance, finding agreement in the hydrostatic limit. The absence of hydrostatic balance on shorter horizontal length scales introduces a singular perturbation of the problem which corresponds to the emergence of a novel barotropic mode. The frequency of this barotropic mode provides a limit to the frequency and decay rate of the high horizontal wavenumber baroclinic modes, thereby introducing a scale selective wave drag. The decay of the baroclinic modes can be characterized by the angle of the wavefront. A damping operator for the bulk equations is proposed, so that the damped rigid lid solutions approximate this decay.

In the second chapter, we investigate the effect of restricting the sign of the phase velocity of waves with small wavenumber, nonzero frequency and dispersion in one spatial dimension using multiscale asymptotics. We consider two examples of long waves with these characteristics. The first investigates breather solutions to the Sine-Gordon equation used in quantum field theory. The second considers the Yanai wave observed in the equatorial atmosphere. Ultimately, we find that this restriction of direction changes the description of the waves in a subtle way, but does not produce any new physics.

In the the third chapter, we investigate the effects of meridional circulation on the equatorial waves. We find that relaxing the long wave scaling used in previous work no longer guarantees stability, and find a necessary condition for instability.

In the fourth chapter, we investigate the effects of a zonally moving off-equatorial forcing on equatorial waves. We find a formula that describes the magnitude of a Kelvin response, as well as a mechanism for such excitations via the interaction of barotropic and baroclinic waves.