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Transient dynamics by continuous-spectrum perturbations in stratified shear flows

Published online by Cambridge University Press:  07 February 2013

R. Camassa  and
C. Viotti
R. Camassa
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
C. Viotti*
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
*
†Email address for correspondence: [email protected]
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Abstract

The transient dynamics of the linearized Euler–Boussinesq equations governing parallel stratified shear flows is presented and analysed. Solutions are expressed as integral superpositions of generalized eigenfunctions associated with the continuous-spectrum component of the Taylor–Goldstein linear stability operator, and reveal intrinsic dynamics not captured by its discrete-spectrum counterpart. It is shown how continuous-spectrum perturbations are generally characterized by non-normal energy growth and decay with algebraic asymptotic behaviour in either time or space. This behaviour is captured by explicit long-time/far-field expressions from rigorous asymptotic analysis, and it is illustrated with direct numerical simulations of the whole (non-Boussinesq) stratified Euler system. These results can be helpful in understanding recent numerical observations for parallel and non-parallel perturbed stratified shear flows.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Instability Geophysical and Geological Flows: Stratified flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 717 , 25 February 2013 , R5
Copyright
©2013 Cambridge University Press

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