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Topological effects on vorticity evolution in confined stratified fluids

Published online by Cambridge University Press:  03 July 2015

R. Camassa ,
G. Falqui ,
G. Ortenzi  and
M. Pedroni
R. Camassa
Affiliation:
University of North Carolina at Chapel Hill, Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, Chapel Hill, NC 27599, USA
G. Falqui
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Milano, I-20125, Italy
G. Ortenzi*
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Milano, I-20125, Italy Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione, Università di Bergamo, Dalmine (BG), I-24044, Italy
M. Pedroni
Affiliation:
Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione, Università di Bergamo, Dalmine (BG), I-24044, Italy
*
†Email address for correspondence: giovanni.ortenzi@unimib.it
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Abstract

For a stratified incompressible Euler fluid under gravity confined by rigid boundaries, sources of vorticity are classified with the aim of isolating those which are sensitive to the topological configurations of density isopycnals, for both layered and continuous density variations. The simplest case of a two-layer fluid is studied first. This shows explicitly that topological sources of vorticity are present whenever the interface intersects horizontal boundaries. Accordingly, the topological separation of the fluid domain due to the interface–boundary intersections can contribute additional terms to the vorticity balance equation. This phenomenon is reminiscent of Klein’s ‘Kaffeelöffel’ thought-experiment for a homogeneous fluid (Klein, Z. Math. Phys., vol. 59, 1910, pp. 259–262), and it is essentially independent of the vorticity generation induced by the baroclinic term in the bulk of the fluid. In fact, the two-layer case is generalized to show that for the continuously stratified case topological vorticity sources are generically present whenever density varies along horizontal boundaries. The topological sources are expressed explicitly in terms of local contour integrals of the pressure along the intersection curves of isopycnals with domain boundaries, and their effects on vorticity evolution are encoded by an appropriate vector, termed here the ‘topological vorticity’.

JFM classification

Mathematical Foundations: Mathematical Foundations Geophysical and Geological Flows: Stratified flows Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 776 , 10 August 2015 , pp. 109 - 136
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© 2015 Cambridge University Press 

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