Skip to content

Research at St Andrews

Imperfect bifurcation for the quasi-geostrophic shallow-water equations

Research output: Contribution to journalArticle

Open Access Status

  • Embargoed (until 12/10/19)

Abstract

We study analytical and numerical aspects of the bifurcation diagram of simply connected rotating vortex patch equilibria for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations are a generalisation of the Euler equations and contain an additional parameter, the Rossby deformation length ε−1, which enters into the relation between the stream function and (potential) vorticity. The Euler equations are recovered in the limit ε→0. We prove, close to circular (Rankine) vortices, the persistence of the bifurcation diagram for arbitrary Rossby deformation length. However we show that the two-fold branch, corresponding to Kirchhoff ellipses for the Euler equations, is never connected even for small values ε, and indeed is split into a countable set of disjoint connected branches. Accurate numerical calculations of the global structure of the bifurcation diagram and of the limiting equilibrium states are also presented to complement the mathematical analysis.
Close

Details

Original languageEnglish
Pages (from-to)1853-1915
Number of pages63
JournalArchive for Rational Mechanics and Analysis
Volume231
Issue number3
Early online date12 Oct 2018
DOIs
StateE-pub ahead of print - 12 Oct 2018

Discover related content
Find related publications, people, projects and more using interactive charts.

View graph of relations

Related by author

  1. On the regularity of the Green-Naghdi equations for a rotating shallow fluid layer

    Dritschel, D. G. & Jalali, M. R. 19 Feb 2019 In : Journal of Fluid Mechanics. 865, p. 100-136

    Research output: Contribution to journalArticle

  2. Scale-invariant singularity of the surface quasigeostrophic patch

    Scott, R. K. & Dritschel, D. G. 28 Jan 2019 In : Journal of Fluid Mechanics. 863, 12 p., R2

    Research output: Contribution to journalArticle

  3. Circulation conservation and vortex breakup in magnetohydrodynamics at low magnetic Prandtl number

    Dritschel, D. G., Diamond, P. H. & Tobias, S. M. 25 Dec 2018 In : Journal of Fluid Mechanics. 857, p. 38-60

    Research output: Contribution to journalArticle

ID: 255838518