Equatorial Solitary Waves. Part I: Rossby Solitons

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Abstract
Using the method of multiple scales, I show that long, weakly nonlinear, equatorial Rossby waves are governed by either the Korteweg-deVries (KDV) equation (symmetric modes of odd mode number n) or the modified Korteweg-deVries (MKDV) equation. From the same localized initial conditions, the nonlinear and corresponding linearized waves evolve very differently. When nonlinear effects are neglected, the whole solution is an oscillatory wavetrain which decays algebraically in time so that the asymptotic solution as t→∝ is everywhere zero. The nonlinear solution consists of two parts: solitary waves plus an oscillatory tail. The solitary waves are horizontally localized disturbances in which nonlinearity and dispersion balance to create a wave of permanent form. The solitary waves are important because 1) they have no linear counterpart and 2) they are the sole asymptotic solution as t→∝. The oscillatory wavetrain, which lags behind and is well-separated from the solitary waves for large time, dies o... Abstract Using the method of multiple scales, I show that long, weakly nonlinear, equatorial Rossby waves are governed by either the Korteweg-deVries (KDV) equation (symmetric modes of odd mode number n) or the modified Korteweg-deVries (MKDV) equation. From the same localized initial conditions, the nonlinear and corresponding linearized waves evolve very differently. When nonlinear effects are neglected, the whole solution is an oscillatory wavetrain which decays algebraically in time so that the asymptotic solution as t→∝ is everywhere zero. The nonlinear solution consists of two parts: solitary waves plus an oscillatory tail. The solitary waves are horizontally localized disturbances in which nonlinearity and dispersion balance to create a wave of permanent form. The solitary waves are important because 1) they have no linear counterpart and 2) they are the sole asymptotic solution as t→∝. The oscillatory wavetrain, which lags behind and is well-separated from the solitary waves for large time, dies o...

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