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dc.contributor.authorRiedinger, X
dc.contributor.authorGilbert, Andrew D.
dc.date.accessioned2015-11-27T11:31:26Z
dc.date.issued2014-06-23
dc.description.abstractIn this study a linear stability analysis of shallow-water flows is undertaken for a representative Froude number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}F=3.5$F=3.5 . The focus is on monotonic base flow profiles $U$U without an inflection point, in order to study critical layer instability (CLI) and its interaction with radiative instability (RI). First the dispersion relation is presented for the piecewise linear profile studied numerically by Satomura (J. Meterol. Soc. Japan, vol. 59, 1981, pp. 148–167) and using WKBJ analysis an interpretation given of mode branches, resonances and radiative instability. In particular surface gravity (SG) waves can resonate with a limit mode (LM) (or Rayleigh wave), localised near the discontinuity in shear in the flow; in this piecewise profile there is no critical layer. The piecewise linear profile is then continuously modified in a family of nonlinear profiles, to show the effect of the vorticity gradient $Q^{\prime } = - U^{\prime \prime }$Q ′ =−U ′′ on the nature of the modes. Some modes remain as modes and others turn into quasi-modes (QM), linked to Landau damping of disturbances to the flow, depending on the sign of the vorticity gradient at the critical point. Thus an interpretation of critical layer instability for continuous profiles is given, as the remnant of the resonance with the LM. Numerical results and WKBJ analysis of critical layer instability and radiative instability for more general smooth profiles are provided. A link is made between growth rate formulae obtained by considering wave momentum and those found via the WKBJ approximation. Finally the competition between the stabilising effect of vorticity gradients in a critical layer and the destabilising effect of radiation (radiative instability) is studied.en_GB
dc.identifier.citationVol. 751, pp. 539 - 569en_GB
dc.identifier.doihttp://dx.doi.org/10.1017/jfm.2014.303
dc.identifier.urihttp://hdl.handle.net/10871/18803
dc.language.isoenen_GB
dc.publisherCambridge University Pressen_GB
dc.relation.urlhttp://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9289397&fileId=S0022112014003036en_GB
dc.rightsCopyright © 2014 Cambridge University Pressen_GB
dc.subjectshallow water flowsen_GB
dc.subjectfree shear layersen_GB
dc.subjectcritical layersen_GB
dc.titleCritical layer and radiative instabilities in shallow-water shear flowsen_GB
dc.typeArticleen_GB
dc.date.available2015-11-27T11:31:26Z
dc.identifier.issn0022-1120
dc.identifier.journalJournal of Fluid Mechanicsen_GB


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