Long solitary internal waves in stable stratifications

Abstract

Observations of internal solitary waves over an antarctic ice shelf (Rees and Rottman, 1994) demonstrate that even large amplitude disturbances have wavelengths that are bounded by simple heuristic arguments following from the Scorer parameter based on linear theory for wave trapping. Classical weak nonlinear theories that have been applied to stable stratifications all begin with perturbations of simple long waves, with corrections for weak nonlinearity and dispersion resulting in nonlinear wave equations (Korteweg-deVries (KdV) or Benjamin-Davis-Ono) that admit localized propagating solutions. It is shown that these theories are apparently inappropriate when the Scorer parameter, which gives the lowest wavenumber that does not radiate vertically, is positive. In this paper, a new nonlinear evolution equation is derived for an arbitrary wave packet thus including one bounded below by the Scorer parameter. The new theory shows that solitary internal waves excited in high Richardson number waveguides are predicted to have a halfwidth inversely proportional to the Scorer parameter, in agreement with atmospheric observations. A localized analytic solution for the new wave equation is demonstrated, and its soliton-like properties are demonstrated by numerical simulation.