Formal stability of circular vortices

1992 ◽  
Vol 242 ◽  
pp. 249-278 ◽  
Author(s):  
R. C. Kloosterziel ◽  
G. F. Carnevale
Keyword(s):  
Angular Momentum ◽  
Linear Combination ◽  
Linear Stability ◽  
Second Variation ◽  
Piecewise Constant ◽  
Constant Vorticity ◽  
Formal Stability ◽  
Stability Regime ◽  
The Second Variation ◽  
Equivalent Expressions

The second variation of a linear combination of energy and angular momentum is used to investigate the formal stability of circular vortices. The analysis proceeds entirely in terms of Lagrangian displacements to overcome problems that otherwise arise when one attempts to use Arnol'd's Eulerian formalism. Specific attention is paid to the simplest possible model of an isolated vortex consisting of a core of constant vorticity surrounded by a ring of oppositely signed vorticity. We prove that the linear stability regime for this vortex coincides with the formal stability regime. The fact that there are formally stable isolated vortices could imply that there are provable nonlinearly stable isolated vortices. The method can be applied to more complicated vortices consisting of many nested rings of piecewise-constant vorticity. The equivalent expressions for continuous vorticity distributions are also derived.

scholarly journals Variational formulations of steady rotational equatorial waves

2020 ◽  
Vol 10 (1) ◽  
pp. 534-547
Author(s):  
Jifeng Chu ◽  
Joachim Escher
Keyword(s):  
Linear Stability ◽  
Stream Function ◽  
Water Waves ◽  
Variational Formulation ◽  
Energy Functional ◽  
Equatorial Waves ◽  
Second Variation ◽  
Governing Equations ◽  
The Second Variation ◽  
Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

The second variation and neighboring optimal feedback guidance for multiple burn orbit transfers

1995 ◽  
Author(s):  
C-HGoodson, Chuang, , Troy D ◽  
Laura Ledsinger ◽  
John Hanson
Keyword(s):  
Second Variation ◽  
Optimal Feedback ◽  
The Second Variation

Integral criteria for closed surfaces with constant mean curvature

2007 ◽  
Vol 463 (2081) ◽  
pp. 1199-1210
Author(s):  
Luca Guzzardi ◽  
Epifanio G Virga
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Closed Surfaces ◽  
Area Functional ◽  
Symmetric Surface ◽  
The Second Variation ◽  
Integral Identities

We propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three-dimensional Euclidean space. These criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements. We obtain from them a new proof of the old result by Delaunay, to the effect that the sphere is the only closed axis-symmetric surface.

scholarly journals Autoparallel Deviation in the Geometry of Lyra

1960 ◽  
Vol 3 (3) ◽  
pp. 263-271 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Calculus Of Variations ◽  
Finsler Manifold ◽  
Geometrical Approach ◽  
Second Variation ◽  
Geodesic Deviation ◽  
The Difference ◽  
The Second Variation

One of the fruitful tools for examining the properties of a Riemannian manifold is the study of “geodesic deviation”. The manner in which a vector, representing the displacement between points on two neighbouring geodesies, behaves gives an indication of the difference between the manifold and an Euclidean space. The study is essentially a geometrical approach to the second variation of the lengthintegral in the calculus of variations [1]. Similar considerations apply in the geometry of Lyra [2] but as we shall see, appropriate analytical modifications must be made. The approach given here is modelled after that of Rund [3] which was originally designed to deal with a Finsler manifold but which applies equally well to the present case.

scholarly journals The second variation of nonorientable minimal submanifolds

1997 ◽  
Vol 349 (8) ◽  
pp. 3093-3104 ◽  
Author(s):  
Marty Ross
Keyword(s):  
Minimal Submanifolds ◽  
Second Variation ◽  
The Second Variation

On the Second Variation of a Simple Integral with Movable End Points

1949 ◽  
Vol s1-24 (4) ◽  
pp. 241-247 ◽  
Author(s):  
E. T. Davies
Keyword(s):  
Second Variation ◽  
End Points ◽  
The Second Variation
Sign in / Sign up

Export Citation Format

Share Document