We study the dynamics of two{layer, stratified shallow water flows. This is a model in which two scenarios for eventual mixing of stratified flows (shear-instability and internal breaking waves) are, in principle, possible. We find that unforced flows cannot reach the threshold of shear-instability, at least without breaking first. This is a fully nonlinear stability result for a model of stratified, sheared flow. Mathematically, for 2X2 autonomous systems of mixed type, a criterium is found deciding whether the elliptic domain is reachable {smoothly{ from hyperbolic initial conditions. If the characteristic fields depend smoothly on the system's Riemann invariants, then the elliptic domain is unattainable. Otherwise, there are hyperbolic initial conditions that will lead to incursions into the elliptic domain, and the development of the associated instability.

Original languageEnglish
Pages (from-to)427-442
Number of pages16
JournalCommunications in Mathematical Sciences
Issue number3
Publication statusPublished - 31 Dec 2004

Bibliographical note

Funding Information:
Acknowledgments. The work of C. Turner and F. Menzaque was partially supported by a grant from the Fundación ANTORCHAS, and the work of Milewski and Tabak by a grant from the Division of Mathematical Sciences at the National Science Foundation.

Publisher Copyright:
© 2004 International Press


  • Shallow water
  • Shear instability
  • Systems of mixed type
  • Two-layer flows

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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