Geometry-controlled dispersion and Kelvin–Rossby coupling of equatorial waves forced by tropical cyclones

Equatorially trapped Kelvin waves rapidly transmit cyclone-forced thermocline and sea-level anomalies across tropical basins, yet their phase speed and dispersive tails vary from event to event as the post-storm waveguide evolves. We develop a geometry-explicit reduced model that converts a slowly varying thermocline imprint into directly testable kinematics without tuning. Starting from the wind-forced equatorial shallow-water equations on a $\beta$ plane, we reduce the dynamics to a variable-coefficient perturbed Korteweg--de Vries equation for the Kelvin-mode envelope. Closed Kelvin projections provide the coefficients: the leading speed shift is a Kelvin-weighted meridional mean displacement, and the dispersive coefficient is a nonnegative quadratic functional of the meridionally symmetric (even) displacement spectrum. These relations motivate amplitude-normalized shape diagnostics and an observable-first falsification protocol based on ridge travel-time anomalies and trailing-tail metrics, both computable from reconstructed geometry prior to any envelope integration. We further supply regression benchmarks for ridge-based phase-shift estimation. When the Kelvin-only closure fails, a minimal Kelvin--Rossby extension uses the antisymmetric projection to parameterize potential leakage. The resulting coefficient map offers an interpretable bridge from post-storm thermocline geometry to basin-scale timing and weak-dispersion propensity in tropical ocean dynamics.