Ocean channel modes

There are two types of normal modes in a uniform channel: (1) Those defined at a constant along-channel wavenumber Ic, for which the eigenvalues are t he frequencies omega(a) (k) and the eigenmodes are Kelvin, Poincare, and Ro ssby waves, and (2) Those defined at a fixed omega, for which the eigenvalu es are k(a) (omega) and the eigenmodes are Kelvin and Poincare/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 phe nomena. Orthogonality conditions are derived for both types of normal modes and shown to be related to the existence of adjoint systems whose normal m odes coincide with those of the direct system. The corresponding inner prod ucts are similar, but not exactly equal, to the energy density and flux, re spectively. 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 contrib utions of each mode (except in the first case and without friction). A nume rical approximation that has the required orthogonality properties is prese nted and used to give examples of forced solutions, such as Taylor's proble m with wind forcing and topography.