Weakly nonlinear internal waves by tidal flow over a ridge in a shear current

Abstract

We extend Thorpe successive approximation expansion to the forced, tide locked internal wave generation problem of Lamb and Dunphy, where a barotropic tide over a ridge radiates a discrete spectrum of Taylor Goldstein eigenmodes in a steady shear current. At second order in topographic steepness, each mode forces a bound second harmonic via diagonal mode pair kernels derived in closed form; both kernels vanish for uniform flow, recovering Thorpe limit. At third order, solvability of the resonant fundamental yields a nonlinear wavenumber correctionthe forced counterpart of Thorpe phase speed shift, with tidal frequency fixed. Computations at the parameters of Lamb and Dunphy show that the corrections are small but systematic, with the bound harmonic distorting the displacement profile modestly and concentrated in the surface shear layer for downstream modes. The corrections peak at modes 2 to 3, grow quadratically with ridge height and current strength, and amount to a nonlinear slow-down of every mode, explaining from within the theory why the linear discrete spectrum model agreed so well with fully nonlinear simulations. A pycnocline case with stratification colocated with the shear layer amplifies the nonlinear corrections substantially, showing that the uniform stratification verdict is not generic and that the corrections are controlled by where N2 sits relative to the shear.