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.

Stability of Vlasov equilibria. Part 1. General theory

1982 ◽  
Vol 27 (1) ◽  
pp. 13-24 ◽  
Author(s):  
K. R. Symon ◽  
C. E. Seyler ◽  
H. R. Lewis
Keyword(s):  
Distribution Function ◽  
General Theory ◽  
Linear Stability ◽  
Vlasov Equation ◽  
Stability Problem ◽  
Dispersion Matrix ◽  
Initial Value ◽  
Spectral Matrix ◽  
Concise Representation ◽  
The Stability

We present a general formulation for treating the linear stability of inhomogeneous plasmas for which at least one species is described by the Vlasov equation. Use of Poisson bracket notation and expansion of the perturbation distribution function in terms of eigenfunctions of the unperturbed Liouville operator leads to a concise representation of the stability problem in terms of a symmetric dispersion functional. A dispersion matrix is derived which characterizes the solutions of the linearized initial-value problem. The dispersion matrix is then expressed in terms of a dynamic spectral matrix which characterizes the properties of the unperturbed orbits, in so far as they are relevant to the linear stability of the system.

scholarly journals Spectral Analysis of Multidimensional Stochastic Geophysical Models with an Application to Decadal ENSO Variability

2011 ◽  
Vol 68 (1) ◽  
pp. 13-25 ◽  
Author(s):  
Richard Kleeman
Keyword(s):  
Normal Mode ◽  
Linear Models ◽  
Decadal Variability ◽  
Normal Modes ◽  
Spectral Character ◽  
Stochastic Forcing ◽  
Spectral Matrix ◽  
Enso Variability ◽  
Cross Spectrum ◽  
The Cross

Abstract Simple linear models with additive stochastic forcing have been rather successful in explaining the observed spectrum of important climate variables. Motivated by this, the authors analyze the spectral character of such a general stochastic system of finite dimension. The spectral matrix is derived in the case that the spectrum is a linear combination of dynamical variables and their stochastic forcings. It is found that the most convenient basis for analysis is provided by the normal modes. In general the spectrum consists of two pieces. The first “diagonal” piece is a symmetric Lorentzian curve centered on the normal mode frequencies with breadth and strength determined by the normal mode dissipation. The second cross-spectrum piece derives usually from the coherency of the stochastic forcing of two different normal modes. The cross-spectrum is smaller in magnitude than the corresponding two diagonal pieces. This relative magnitude is controlled by the Wiener coherency, which is equal to the magnitude of the correlation of the stochastic forcings of different normal modes. This new analysis framework is studied in detail for the ENSO case for which a two-dimensional stochastically forced oscillator has been previously suggested as a minimal model. It is found that the observed spectrum is rather easily reproduced given appropriate dissipation. Further, it is found that the cross-spectrum results in a phase-dependent enhancement or suppression of frequencies smaller than the dominant ENSO frequency. This therefore provides a new mechanism for decadal ENSO variability. Since the cross-spectrum is phase dependent, the decadal variability generated has a distinctive spatial character. The significance of the cross-spectrum depends on the Wiener coherency, which in turn depends on the statistics of the stochastic forcing.

On Estimates for Growth Rates of Unstable Azimuthal Disturbances in the Stability Problem of Swirling Flows with Radius- Dependent Density

2017 ◽  
Vol 13 (3) ◽  
pp. 1-12
Author(s):  
Halle Dattu Malai Subbiah
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Stability Problem ◽  
Variable Density ◽  
Swirling Flows ◽  
Wave Number ◽  
Two Dimensional ◽  
Azimuthal Wave Number ◽  
The Stability ◽  
Dependent Density

Estimates for the growth rate of unstable two-dimensional disturbances to swirling flows with variable density are obtained and as a consequence it is proved that the growth rate tends to zero as the azimuthal wave number tends to infinity for two classes of basic flows.

Gaseous unimolecular reactions: Theory of the effects of pressure and of vibrational degeneracy

1953 ◽  
Vol 246 (906) ◽  
pp. 57-80 ◽  
Keyword(s):  
High Pressure ◽  
Classical Model ◽  
Normal Modes ◽  
Log P ◽  
First Order ◽  
Special Cases ◽  
Gas Molecule ◽  
Modes Of Vibration ◽  
Degenerate Modes ◽  
Internal Potential

A theory which gave the high-pressure unimolecular reaction rate as K 8 = v exp ( — E 0 /kT) is extended to find the decline of rate with pressure; the gas molecule is again a classical vibrating system which dissociates at a critical extension of an internal co-ordinate. The general rate K is found to be approximately... where n is the effective number of normal modes of vibration; d is proportional to pT~^n, but depends also on the molecular structure and size. For n < 13, this integral is tabulated, and the pressures at which the rate declines from first order are estimated. The pressure tends to decrease as n increases; for E 0 /k T ~ 40, it is estimated that only molecules with six or more atoms should show rates approaching KCX) at normal pressures. The table of K/K;a is not carried as far as the ‘bimolecular’ range, but a precise technique is developed for this region. The theory is compared with Kassel’s classical theory of a molecule of s ‘oscillators’. The lowpressure activation energy, and the shape of the curve of log K against log p, are similar in the two theories if n = 2s — 1; the absolute values of p for a given rate are also roughly comparable. Two results are proved, for the present severely classical model, concerning special cases. (i) A pair or triplet of degenerate modes with equal frequencies counts as one in assessing ‘n’ for the general rate K. (ii) If the dissociation co-ordinate q relates atoms ml, and mx is replaced by an isotope m*, the high-pressure rate changes in the ratio d{m 1 (m*+ m 2 )/m^(m 1 +m 2 )}; for this, the internal potential energy V need not be quadratic, nor need q be isolated in V from other co-ordinates.

On the Sensitivity of Non-Generic Bifurcation of Non-Linear Normal Modes

2007 ◽  
Author(s):  
C. H. Pak ◽  
Y. S. Choi
Keyword(s):  
Normal Mode ◽  
Natural Frequencies ◽  
Normal Modes ◽  
New Normal ◽  
Perturbed System ◽  
Saddle Node ◽  
Non Linear ◽  
The Difference ◽  
Generic Bifurcation ◽  
The Stability

It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r = 1, 3, 5 ·······. Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears and generic bifurcation appear in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.

On the stability of large-amplitude vortices in a continuously stratified fluid on the f-plane

1998 ◽  
Vol 355 ◽  
pp. 139-162 ◽  
Author(s):  
E. S. BENILOV ◽  
D. BROUTMAN ◽  
E. P. KUZNETSOVA
Keyword(s):  
Growth Rate ◽  
Boundary Value Problem ◽  
Large Amplitude ◽  
Boundary Value ◽  
Normal Modes ◽  
Large Displacement ◽  
Primitive Equations ◽  
Rossby Number ◽  
Spatial Coordinates ◽  
The Stability

The stability of continuously stratified vortices with large displacement of isopycnal surfaces on the f-plane is examined both analytically and numerically. Using an appropriate asymptotic set of equations, we demonstrated that sufficiently large vortices (i.e. those with small values of the Rossby number) are unstable. Remarkably, the growth rate of the unstable disturbance is a function of the spatial coordinates. At the same time, the corresponding boundary-value problem for normal modes has no smooth square-integrable solutions, which would normally be regarded as stability.We conclude that (potentially) stable vortices can be found only among ageostrophic vortices. Since this assumption cannot be verified analytically due to complexity of the primitive equations, we verify it numerically for the particular case of two-layer stratification.

scholarly journals Linear electrostatic gyrokinetics for electron–positron plasmas

2018 ◽  
Vol 84 (6) ◽  
Author(s):  
D. Kennedy ◽  
A. Mishchenko ◽  
P. Xanthopoulos ◽  
P. Helander
Keyword(s):  
Growth Rate ◽  
Negative Charge ◽  
Charge Carriers ◽  
Mass Ratio ◽  
Flux Surface ◽  
Gene Code ◽  
Real Frequency ◽  
Hydrogen Plasmas ◽  
Electron Positron ◽  
The Stability

Gyrokinetic stability of plasmas in different magnetic geometries is studied numerically using the GENE code. We examine the stability of plasmas, varying the mass ratio between the positive and negative charge carriers, from conventional hydrogen plasmas through to electron–positron plasmas. Stability is studied for prescribed temperature and density gradients in different magnetic geometries: (i) An axisymmetric, circular flux surface, low$\unicode[STIX]{x1D6FD}$(tokamak) configuration. (ii) A non-axisymmetric quasi-isodynamic (optimised stellarator) configuration using the geometry of the stellarator Wendelstein 7-X. We also present the analytic theory of trapped particle modes in electron–positron plasmas. We found similar behaviour of the growth rate and real frequency compared to previous studies on the tokamak case. We are able to identify two distinct regimes dominated by modes propagating in the electron diamagnetic direction and modes propagating in the ion/positron diamagnetic direction, depending on the mass ratio. In both the tokamak and the stellarator case we observe that the real frequency tends to zero as the mass ratio approaches unity and are able to explain this using gyrokinetic theory.

Nonlinear Normal Modes in the Presence of Internal Resonances: An Energy-Based Approach

1995 ◽  
Author(s):  
Melvin E. King ◽  
Alexander F. Vakakis
Keyword(s):  
Normal Mode ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Weakly Nonlinear ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

Abstract In this work, modifications to existing energy-based nonlinear normal mode (NNM) methodologies are developed in order to investigate internal resonances. A formulation for computing resonant NNMs is developed for discrete, or discretized for continuous systems, sets of weakly nonlinear equations with uncoupled linear terms (i.e systems in modal, or canonical, form). By considering a canonical framework, internal resonance conditions are immediately recognized by identifying commensurable linearized natural frequencies. Additionally, the canonical formulation allows for a single (linearized modal) coordinate to parameterize all other (modal) coordinates during a resonant modal response. Energy-based NNM methodologies are then applied to the canonical equations and asymptotic solutions are sought. In order to account for the resonant modal interactions, it will be shown that high-order terms in the O(1) solutions must be considered. Two applications (‘3:1’ resonances in a two-degree-of-freedom system and ‘3:1’ resonance in a hinged-clamped beam) are then considered by which to demonstrate the application of the resonant NNM methodology. Resonant normal mode solutions are obtained and the stability characteristics of these computed modes are considered. It is shown that for some responses, nonlinear modal relations do not exist in the context of physical coordinates and thus the transformation to canonical coordinates is necessary in order to define appropriate NNM relations.

Metastability and dynamic modes in magnetic island chains

2021 ◽  
Author(s):  
G M Wysin
Keyword(s):  
Stability Problem ◽  
Normal Modes ◽  
Dipolar Interactions ◽  
Magnetic Islands ◽  
Magnetic Island ◽  
One Dimensional ◽  
Dipole Interactions ◽  
The Stability ◽  
Stability Limits ◽  
Infinite Range

Abstract The uniform states of a model for one-dimensional chains of thin magnetic islands on a nonmagnetic substrate coupled via dipolar interactions are described here. Magnetic islands oriented with their long axes perpendicular to the chain direction are assumed, whose shape anisotropy imposes a preference for the dipoles to point perpendicular to the chain. The competition between anisotropy and dipolar interactions leads to three types of uniform states of distinctly different symmetries, including metastable transverse or remanent states, transverse antiferromagnetic states, and longitudinal states where all dipoles align with the chain direction. The stability limits and normal modes of oscillation are found for all three types of states, even including infinite range dipole interactions. The normal mode frequencies are shown to be determined from the eigenvalues of the stability problem.

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.

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