Rossby solitary waves embedded in equatorial jets

Extending our previous work on nonlinear Kelvin traveling waves, we have numerically calculated Rossby solitary waves in idealized equatorial jets in the nonlinear shallow water (1-1/2 layer) model. The calculations are difficult because the shear induces resonances between the Rossby soliton and a plethora of other equatorial modes: this is both a physical and a computational complication. We discuss insights from the theory of weakly nonlocal solitary waves and hyperasymptotic perturbation theory.

We solved the nonlinear eigenvalue problem using a two-dimensional spectral Galerkin method to discretize the partial differential equations into a large system of algebraic equations which were solved by Newton's iteration with Armijo line search; continuation in amplitude supplied the initialization at each amplitude.

The shear-embedded vortex solitons qualitatively resemble mature Tropical Instability Vortices (TIVs), much more closely than to the shear-free analytic KdV solitary waves of earlier work of Boyd's, but the quantitative agreement is not close.