An application of Mathematica to internal wave generation

David Vasholz

Johns Hopkins University Applied Physics Laboratory

Abstract

This work investigates properties of an internal wave wake generated by a solid body moving uniformly at constant depth through a stratified fluid.  When analyzed using the basic equations of motion, the generation of  internal waves is accompanied by the generation of copious amounts of algebra that is impractical to perform by hand.  This work will demonstrate a Mathematica notebook consisting of over 1000 Mathematica input statements (surrounded by explanatory material) that takes the reader from the five fundamental equations (incompressibility, conservation of mass, and the three components of the Navier-Stokes equations) through a formal description of stratified wakes for an arbitrary buoyancy frequency profile.  The "hydrogen atom" of stratified wakes (which occurs when the buoyancy frequency is uniform in depth) is discussed in detail, revealing a rich mathematical structure and illustrating how  the depth-based Froude number emerges as a fundamental quantity that produces phase-like transitions as it passes through a value of unity. 
  
Work supported by ONR