| The Kelvin wave is the slowest eastward propagating eigenmode of Laplace's Tidal Equation. It is widely observed in both the ocean and the atmosphere. On the sphere, in the absence of mean currents, the Kelvin wave depends on two parameters: the zonal wavenumber s [always an integer] and Lamb's parameter epsilon. First, we derive an asymptotic approximation for the linear Kelvin wave valid in the limit s2+e >> 1, which generalizes the usual "equatorial wave" limit that epsilon → infinity for fixed s. Then for the weakly nonlinear Kelvin wave we derive the analytic solution of the traveling Kelvin wave for small epsilon and amplitude with a perturbation method. For the strongly nonlinear Kelvin wave, through numerical computations using a Fourier/Newton iteration/continuation method, we show that for sufficiently small amplitude, there are Kelvin traveling waves (cnoidal waves); as the amplitude increases, the branch of traveling waves terminates in a so-called "corner wave" with a discontinuous first derivative. All waves larger than the corner wave evolve to fronts and break.;On the equatorial beta-plane, Kelvin waves are nondispersive without a background mean. To obtain the traveling wave solution, we include a jet symmetric about the equator. We show that the linear Kelvin waves have much more complicated structures and phase speeds than the Kelvin wave with a resting background. In longitude, the nonlinear traveling waves also terminate in a "corner wave". In latitude, as the wave amplitude increases, the waves narrow for a westward jet but widen for an eastward jet. Phase speeds are largely determined by the linear Kelvin waves' dynamics; nonlinearity only increases the phase speeds by a few percent.;Tropical Instability Waves (TIWs) are prominent westward propagating intraseasonal oscillations observed in both equatorial Pacific and Atlantic oceans. How the nonlinearity of the TIWs affects the development of the instabilities is studied through solving the linear stability problem and high resolution time dependent numerical simulation. We show that neutral Yanai waves with periods about 15-22 days emerge near the equator when the unstable TIWs centered near 5°N grow into fully nonlinear vortices which explains the coexistence of two different types of TIWs observed during the TIW season. The strength of the Yanai waves is sensitive to the instability of the initial mean flow and the external forcing. |
