There are two types of normal modes in a uniform channel: (1) Those defined at a constant along-channel wavenumber k, for which the eigenvalues are the frequencies ωa (k) and the eigenmodes are Kelvin, Poincaré, and Rossby waves, and (2) Those defined at a fixed ω, for which the eigenvalues are ka (ω) and the eigenmodes are Kelvin and Poincaré/Rossby waves. The first set is useful in the initial value problem, whereas the second one is applied to the study of forced solutions and the wave-scattering phenomena. Orthogonality conditions are derived for both types of normal modes and shown to be related to the existence of adjoint systems whose normal modes coincide with those of the direct system. The corresponding inner products are similar, but not exactly equal, to the energy density and flux, respectively. They do not define a metric, and, consequently, there is not a property of the wave field equal to the sum of the real nonnegative contributions of each mode (except in the first case and without friction). A numerical approximation that has the required orthogonality properties is presented and used to give examples of forced solutions, such as Taylor's problem with wind forcing and topography.
All Science Journal Classification (ASJC) codes
- Geochemistry and Petrology
- Earth and Planetary Sciences (miscellaneous)
- Space and Planetary Science
- Atmospheric Science
- Astronomy and Astrophysics