Transition modes of rotating shallow water waves in a channel

Normal modes of shallow water waves in a channel wherein the Coriolis parameter and the depth vary in the spanwise direction are investigated based on the conservation of the number of zeros in an eigenfunction. As a result, it is generally shown that the condition for transition modes (Kelvin modes and mixed Rossby-gravity modes) to exist, besides Rossby and Poincaré modes, is determined only by boundary conditions. A Kelvin mode is interpreted as a modification of a Kelvin wave or a boundary wave along a closed boundary, and a mixed Rossby-gravity mode as a modification of an inertial oscillation or a boundary wave along an open boundary. Transition modes appearing in edge and continental-shelf waves, equatorial waves and free oscillations over a sphere are systematically understood by applying the theory in this paper.