A one-layer reduced-gravity model in the unbounded equatorial β plane is the simplest way to study nonlinear effects in a tropical ocean. The free evolution of the system is constrained by the existence of several conserved quantities, namely, potential vorticity, zonal momentum, zonal pseudomomentum and energy; only the last two are quadratic, to lowest order, in the departure from the equilibrium state. The eigenfunctions of the linear problem are found to span an orthogonal and complete basis; this is used to expand the dynamical variables, without making any assumption on their magnitude. Thus, the state of the system is fully described, at any time, by the set of expansion amplitudes; their evolution is controlled by a system of equations (with only quadratic nonlinearity) which are an exact representation of the original ones. A straightforward formula is obtained for the evaluation of the coupling coefficients. As a first example for the use of this formalism, the Kelvin modes self-interac... Abstract A one-layer reduced-gravity model in the unbounded equatorial β plane is the simplest way to study nonlinear effects in a tropical ocean. The free evolution of the system is constrained by the existence of several conserved quantities, namely, potential vorticity, zonal momentum, zonal pseudomomentum and energy; only the last two are quadratic, to lowest order, in the departure from the equilibrium state. The eigenfunctions of the linear problem are found to span an orthogonal and complete basis; this is used to expand the dynamical variables, without making any assumption on their magnitude. Thus, the state of the system is fully described, at any time, by the set of expansion amplitudes; their evolution is controlled by a system of equations (with only quadratic nonlinearity) which are an exact representation of the original ones. A straightforward formula is obtained for the evaluation of the coupling coefficients. As a first example for the use of this formalism, the Kelvin modes self-interac...