On the stability of hydromagnetic flow

1986 ◽  
Vol 35 (1) ◽  
pp. 145-150
Author(s):  
R. J. Lucas
Keyword(s):  
Steady Flow ◽  
Linear Stability ◽  
Normal Modes ◽  
Sufficient Conditions ◽  
Hydromagnetic Flow ◽  
Special Cases ◽  
The Stability

The linear stability of steady flow of an inhomogeneous, incompressible hydromagnetic fluid is considered. Circle theorems which provide bounds on the complex eigenfrequencies of the unstable normal modes are obtained. Sufficient conditions for stability follow in a number of special cases.

Notes on the stability problem for inhomogeneous equilibria

1983 ◽  
Vol 29 (2) ◽  
pp. 275-286 ◽  
Author(s):  
K. R. Symon
Keyword(s):  
Growth Rate ◽  
Normal Mode ◽  
Stability Problem ◽  
Normal Modes ◽  
Electromagnetic Mode ◽  
Spectral Matrix ◽  
First Order ◽  
Real Frequency ◽  
Special Cases ◽  
The Stability

Several conclusions regarding the stability of inhomogeneous Vlasov equilibria are drawn from earlier work. A technique is presented for generating first-order formulae for the change δω in the frequency of any normal mode, when a parameter λ characterizing the equilibrium is changed slightly. Several applications are given, including a first-order calculation of the growth rate or damping of an electromagnetic mode due to the presence of plasma. A condition is derived for the existence of a normal mode with real frequency. When there are ignorable co-ordinates, the normal modes can be written in the form of waves propagating in the ignorable directions. The character of the modes depends on certain symmetries in the dynamic spectral matrix. Special cases arise when the orbits can be approximated in certain ways.

scholarly journals A framework for linear stability analysis of finite-area vortices

2013 ◽  
Vol 469 (2152) ◽  
pp. 20120709 ◽  
Author(s):  
Alan Elcrat ◽  
Bartosz Protas
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Singular Integrals ◽  
Numerical Approach ◽  
Perturbation Equation ◽  
Constant Vorticity ◽  
Special Cases ◽  
Euler Flows ◽  
The Stability

In this investigation, we revisit the question of linear stability analysis of two-dimensional steady Euler flows characterized by the presence of compact regions with constant vorticity embedded in a potential flow. We give a complete derivation of the linearized perturbation equation which, recognizing that the underlying equilibrium problem is of a free-boundary type, is carried out systematically using methods of shape-differential calculus. Particular attention is given to the proper linearization of contour integrals describing vortex induction. The thus obtained perturbation equation is validated by analytically deducing from it stability analyses of the circular vortex, originally due to Kelvin, and of the elliptic vortex, originally due to Love, as special cases. We also propose and validate a spectrally accurate numerical approach to the solution of the stability problem for vortices of general shape in which all singular integrals are evaluated analytically.

scholarly journals A note on the stability of steady inviscid helical gas flows

1978 ◽  
Vol 89 (3) ◽  
pp. 401-411 ◽  
Author(s):  
Knut S. Eckhoff ◽  
Leiv Storesletten
Keyword(s):  
Linear Stability ◽  
Richardson Number ◽  
Sufficient Conditions ◽  
Expansion Method ◽  
Necessary Condition ◽  
Gas Flows ◽  
Wave Expansion ◽  
The Stability

A necessary condition for linear stability of steady inviscid helical gas flows is found by the generalized progressing-wave expansion method. The criterion obtained is compared with the known Richardson number criteria giving sufficient conditions for stability.

Stability and instability of solitary waves of Korteweg-de Vries type

1987 ◽  
Vol 411 (1841) ◽  
pp. 395-412 ◽  
Keyword(s):  
Solitary Waves ◽  
Sufficient Conditions ◽  
Solitary Wave Solutions ◽  
Weakly Nonlinear ◽  
Wave Solutions ◽  
Stability And Instability ◽  
Special Cases ◽  
De Vries ◽  
The Stability ◽  
Necessary And Sufficient

Considered herein are the stability and instability properties of solitary-wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear, dispersive long waves. Special cases for which our analysis is decisive include equations of the Korteweg-de Vries and Benjamin-Ono type. Necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable.

scholarly journals Some necessary and sufficient conditions for stability of three-layer difference schemes

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 705-713
Author(s):  
Vanja Vukoslavcevic
Keyword(s):  
Sufficient Conditions ◽  
Difference Schemes ◽  
Necessary And Sufficient Conditions ◽  
Special Cases ◽  
The Stability ◽  
Necessary And Sufficient

This paper investigates two classes of three-layer difference schemes with weights in the form ?y?t,n+??2y?tt,n+?1Ayn-1+(E-?1-?2)Ayn+?2Ayn+1 = ?n and ?y?t,n+??2y?tt,n+A(?1yn-1+(E-?1-?2)yn+?2yn+1) = ?n. It obtains some sufficient conditions for the stability in a defined norm and, also, in special cases we achieve conditions for stability which do not depend on the choice of norm.

Dynamics of vorticity defects in shear

1997 ◽  
Vol 333 ◽  
pp. 197-230 ◽  
Author(s):  
N. J. BALMFORTH ◽  
D. DEL-CASTILLO-NEGRETE ◽  
W. R. YOUNG
Keyword(s):  
Normal Modes ◽  
Sufficient Conditions ◽  
Linear Stability Theory ◽  
Initial Value ◽  
Complete Form ◽  
Critical Layers ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Secondary Instabilities ◽  
Expansion Scheme

Matched asymptotic expansions are used to obtain a reduced description of the nonlinear and viscous evolution of small, localized vorticity defects embedded in a Couette flow. This vorticity defect approximation is similar to the Vlasov equation, and to other reduced descriptions used to study forced Rossby wave critical layers and their secondary instabilities. The linear stability theory of the vorticity defect approximation is developed in a concise and complete form. The dispersion relations for the normal modes of both inviscid and viscous defects are obtained explicitly. The Nyquist method is used to obtain necessary and sufficient conditions for instability, and to understand qualitatively how changes in the basic state alter the stability properties. The linear initial value problem is solved explicitly with Laplace transforms; the resulting solutions exhibit the transient growth and eventual decay of the streamfunction associated with the continuous spectrum. The expansion scheme can be generalized to handle vorticity defects in non-Couette, but monotonic, velocity profiles.

scholarly journals Symmetric Stability of Compressible Zonal Flows on a Generalized Equatorial β Plane

2008 ◽  
Vol 65 (6) ◽  
pp. 1927-1940 ◽  
Author(s):  
Mark D. Fruman ◽  
Theodore G. Shepherd
Keyword(s):  
Angular Momentum ◽  
Linear Stability ◽  
Euler Equations ◽  
Coriolis Force ◽  
Sufficient Conditions ◽  
Zonal Flow ◽  
Positive Definiteness ◽  
Planetary Rotation ◽  
The Stability ◽  
Invariant Functional

Abstract Sufficient conditions are derived for the linear stability with respect to zonally symmetric perturbations of a steady zonal solution to the nonhydrostatic compressible Euler equations on an equatorial β plane, including a leading order representation of the Coriolis force terms due to the poleward component of the planetary rotation vector. A version of the energy–Casimir method of stability proof is applied: an invariant functional of the Euler equations linearized about the equilibrium zonal flow is found, and positive definiteness of the functional is shown to imply linear stability of the equilibrium. It is shown that an equilibrium is stable if the potential vorticity has the same sign as latitude and the Rayleigh centrifugal stability condition that absolute angular momentum increase toward the equator on surfaces of constant pressure is satisfied. The result generalizes earlier results for hydrostatic and incompressible systems and for systems that do not account for the nontraditional Coriolis force terms. The stability of particular equilibrium zonal velocity, entropy, and density fields is assessed. A notable case in which the effect of the nontraditional Coriolis force is decisive is the instability of an angular momentum profile that decreases away from the equator but is flatter than quadratic in latitude, despite its satisfying both the centrifugal and convective stability conditions.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

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