Ship Waves and Lee Waves

Abstract

Three-dimensional internal trapped lee wave modes produced by an isolated obstacle in a stratified fluid are shown to have dynamics analogous to surface ship waves on water of finite depth. Two models which allow for vertical trapping of wave energy are treated in detail: 1) uniform upstream flow and stratification bounded above by a rigid lid; and 2) a semi-infinite fluid of uniform stability, with wind velocity increasing exponentially with height. This second model is taken as representing the atmosphere. Unlike the surface ship wave, both of these models allow for an infinity of wave modes, a finite number (possibly zero) of which both have transverse and diverging systems, the remainder of the infinite set consisting only of diverging waves. Each mode is contained within a characteristic wedge angle, and each mode amplitude is a function of height. Pursuing lines of analysis similar to these established for the ship wave problem, we have produced formal asymptotic solutions to our models. However, because of the limitations of the approximations and the infinity of modes in the solution, these formal solutions have extremely limited quantitative value. Therefore, we have developed time-dependent numerical models for both surface ship waves and internal and atmospheric ship waves and present a variety of results.