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The free Rossby wave (RW) solutions in an ocean with a straight coast when the offshore wavenumber of incident ( l 1 ) and reflected ( l 2 ) wave are equal or complex are discussed. If l 1 = l 2 the energy streams along the coast and a uniformly valid solution cannot be found; if l 1,2 are complex it yields the sum of an exponentially decaying and growing (away from the coast) Rossby wave. The channel does not admit these solutions as free modes. If the wavenumber vectors of the RWs are perpendicular to the coast, the boundary condition of no normal flow is trivially satisfied and the value of the streamfunction does not need to vanish at the coast. A solution that satisfies Kelvin's theorem of time-independent circulation at the coast is proposed. The forced RW solutions when the ocean's forcing is a single Fourier component are studied. If the forcing is resonant, i.e. a free Rossby wave (RW), the linear response will depend critically on whether the wave carries energy perpendicular to the channel or not. In the first case, the amplitude of the response is linear in the direction normal to the channel, y , and in the second it has a parabolic profile in y . Examples of these solutions are shown for channels with parameters resembling the Mozambique Channel, the Tasman Sea, the Denmark Strait and the English Channel. The solutions for the single coast are unbounded, except when the forcing is a RW trapped against the coast. If the forcing is non-resonant, exponentially decaying or trapped RWs could be excited in the coast and both the exponentially “decaying” and exponentially “growing” RW could be excited in the channel.

We study the weakly nonlinear self-interaction of a single Rossby mode in a straight channel of arbitrary orientation on a β -plane. As in the problem of reflection from a single wall [Graef, F., 1993. First order resonance in the reflection of baroclinic Rossby waves. J. Fluid Mech., 251, 515–532], resonance is possible only if 0 α |≤1/3, where α is the angle between east and the direction along the channel. However, in contrast to the problem of a single wall, the straightforward expansion yields a uniformly valid solution in the flow domain in the resonant case, so there is no need of using multiple scales. The three waves (two belonging to the mode and the excited one), although in resonance, do not exchange energy in time due to the additional constraint on the motion imposed by the boundary conditions at the channel walls. A full solution to the first order equations is given in both the resonant and non-resonant cases. We also inquire if resonance is possible in the self-interaction of a Rossby mode in a gulf. Since a mode in a gulf can be physically conceived as the reflection of a channel mode impinging at the gulf's head, the above necessary condition for resonance in the channel holds for the gulf. However there is another possibility of resonant interaction for which the condition 0 α |≤1/3 applies. Therefore, in a gulf is easier to find resonance.

In this study we investigate which conditions are needed to assure uniqueness of solutions of the barotropic QG model, and, how these conditions are related to the conservation laws of mass, vorticity and energy. Uniqueness and conservation laws are analyzed for a simply connected domain (a closed basin) and for a doubly connected domain (a periodic or re-entrant channel). For the multiply connected domain we find, besides the model proposed by McWilliams in 1977, another consistent model whose solutions satisfy the conservation laws. The additional conditions in our model are: (a) the sum of the circulations around all closed solid walls is time-independent; (b) the value of the QG streamfunction (Φ) is the same at all closed walls. McWilliams' model and ours are not equivalent. As a simple application we study the free Rossby normal modes in a channel. For a non-zonal channel on a β-plane there are solutions (modes) that are independent of the coordinate along the channel. These are used to compare the modes and frequencies obtained from three different, but well posed, models. Solutions that are independent of the along-channel coordinate do not exist for the planetary geostrophic and for the shallow water equations on a β-plane.