Linear dynamics of stably-neutrally stratified fluid consisting of the stably stratified upper layer and the homogeneous lower layer is studied with and without rotation. The density and other fields are continuous at the interface between the layers. A special feature of this configuration is existence of the wave mode related to the homogeneous layer. In non-rotating fluid this is the homogeneous layer vortex mode characterized by a stationary three-dimensional velocity field confined to the lower layer. In the presence of rotation, the mode turns into the gyroscopic waves. Besides the mode, the wave spectrum contains internal waves and the zero frequency horizontal vortex mode with zero vertical velocity. In non-rotating fluid, the vertical velocity consists of the dispersive internal waves and of a steady component in the homogeneous layer. With increasing time the internal waves decay at a fixed point because of dispersion, and the vertical velocity decays in the upper layer and becomes stationary in the lower layer. A non-stationary boundary layer develops near the interface in the stratified layer at large times. In rotating fluid we examined the wave spectrum not using the traditional and hydrostatic approximations, and found the spectrum consists of the super-inertial internal waves, the sub-inertial gyroscopic waves and the sub- and super-inertial internal inertio-gravity waves. In the case of strong stratification f/N << 1(f is the inertial frequency and N is the stratified layer buoyancy frequency) and for the long wave scales f2/N2 << H/L << 1(H and L are the fluid depth and the horizontal scale), the internal and the super-inertial inertio-gravity waves freely penetrate into the lower layer, and the gyroscopic waves are localized in the lower layer and are close to the inertial oscillations. Any long-wave field of the vertical velocity is split into the internal waves, and the inertial oscillations (long gyroscopic waves) confined to the lower layer. With time, the internal waves decay because of dispersion, and the vertical velocity goes to zero in the upper layer and in the lower layer only the inertial oscillations remain.
Reznik, Gregory M.. 2013. "Linear dynamics of a stably-neutrally stratified ocean." Journal of Marine Research 71, (4). https://elischolar.library.yale.edu/journal_of_marine_research/373