Parabolic equations for gravity and acousto-gravity waves

Parabolic equations for gravity and acousto-gravity waves are derived and implemented. The wave equations for these problems contain singularities at depths at which the buoyancy frequency equals the forcing frequency. One of the advantages of the parabolic equation solution is that it is easy to avoid numerical problems associated with the singularities. Some problems involve an infinite number of propagating modes. This artifact of neglecting viscosity is handled by including stability constraints in the rational approximations used in the implementation of the parabolic equation. The parabolic equation is tested for idealized problems involving surface, internal, and interface gravity waves. Parabolic equation solutions are also presented for range-dependent problems involving internal waves in the ocean and acousto-gravity waves in the atmosphere.