On the Stability of Axisymmetric MHD Modes

1988 ◽  
Vol 43 (12) ◽  
pp. 1009-1016 ◽  
Author(s):  
D. Lortz
Keyword(s):  
Equatorial Plane ◽  
Stability Problem ◽  
Plasma Boundary ◽  
Test Functions ◽  
Shift Condition ◽  
Ideal Mhd ◽  
Mhd Equilibria ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Free Plasma

The stability of axisymmetric ideal MHD equilibria which are symmetric with respect to the equatorial plane is considered. It is found that for external axisymmetric modes which are antisymmetric with respect to the equatorial plane and for profiles such that the current density vanishes at the free plasma boundary the stability problem reduces to a classical interior-exterior scalar eigenvalue problem. Because of the separation property the resulting stability condition is necessary and sufficient and is thus more stringent than criteria derived by choosing special test functions, e.g. the vertical shift condition.

Computation of Tokamak Equilibria With Steady Flow

1987 ◽  
Vol 42 (10) ◽  
pp. 1154-1166 ◽  
Author(s):  
W. Kerner ◽  
S. Tokuda
Keyword(s):  
Steady Flow ◽  
Stationary Flow ◽  
Plasma Boundary ◽  
Ideal Mhd ◽  
Mhd Equilibria ◽  
Free Plasma

The equations for ideal MHD equilibria with stationary flow are re-examined and addressed as numerically applied to tokamak configurations with a free plasma boundary. Both the isothermal (purely toroidal flow) and the poloidal flow cases are treated. Experiment-relevant states with steady flow (so far only in the toroidal direction) are computed by the modified SELENE40 code.

Some Structural Properties of Systolic Tree Automata

1989 ◽  
Vol 12 (4) ◽  
pp. 571-585
Author(s):  
E. Fachini ◽  
A. Maggiolo Schettini ◽  
G. Resta ◽  
D. Sangiorgi
Keyword(s):  
Structural Properties ◽  
Stability Problem ◽  
Sufficient Condition ◽  
Tree Automata ◽  
Necessary And Sufficient Condition ◽  
The Stability ◽  
Necessary And Sufficient

We prove that the classes of languages accepted by systolic automata over t-ary trees (t-STA) are always either equal or incomparable if one varies t. We introduce systolic tree automata with base (T(b)-STA), a subclass of STA with interesting properties of modularity, and we give a necessary and sufficient condition for the equivalence between a T(b)-STA and a t-STA, for a given base b. Finally, we show that the stability problem for T(b)-ST A is decidible.

MONODROMY AND STABILITY FOR NILPOTENT CRITICAL POINTS

2005 ◽  
Vol 15 (04) ◽  
pp. 1253-1265 ◽  
Author(s):  
M. J. ÁLVAREZ ◽  
A. GASULL
Keyword(s):  
Critical Points ◽  
Limit Cycles ◽  
Stability Problem ◽  
Short Proof ◽  
Sufficient Conditions ◽  
Planar Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Lyapunov Constants

We give a new and short proof of the characterization of monodromic nilpotent critical points. We also calculate the first generalized Lyapunov constants in order to solve the stability problem. We apply the results to several families of planar systems obtaining necessary and sufficient conditions for having a center. Our method also allows us to generate limit cycles from the origin.

Almost sure stability of maxima

1973 ◽  
Vol 10 (2) ◽  
pp. 387-401 ◽  
Author(s):  
Sidney I. Resnick ◽  
R. J. Tomkins
Keyword(s):  
Markov Chain ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Almost Sure Stability ◽  
Normalizing Constants ◽  
The Stability ◽  
Necessary And Sufficient

For random variables {Xn, n ≧ 1} unbounded above set Mn = max {X1, X2, …, Xn}. When do normalizing constants bn exist such that Mn/bn→ 1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for all and in this case bn ∼ F–1 (1 – 1/n) Necessary and sufficient conditions for lim supn→∞, Mn/bn = l > 1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn = 1 a.s. except when l = 1. When the Xn are r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

Stability Analysis of Switched Positive Systems with Delays

2011 ◽  
Vol 48-49 ◽  
pp. 1093-1096
Author(s):  
Xiu Yong Ding ◽  
Lan Shu ◽  
Chang Cheng Xiang ◽  
Xiu Liu
Keyword(s):  
Stability Analysis ◽  
Lyapunov Function ◽  
Discrete Time ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
System Stability ◽  
Positive Systems ◽  
The Stability ◽  
Switched Positive Systems ◽  
Necessary And Sufficient

This brief investigates the stability problem of discrete-time switched positive systems with delays, and establishes some necessary and sufficient conditions for the existence of a switched copositive Lyapunov function(SCLF) for such systems. It turns out that the size of the delays does not affect the stability of these systems. In other words, system stability is completely determined by the system matrices.

An energy criterion for the linear stability of conservative flows

2000 ◽  
Vol 402 ◽  
pp. 329-348 ◽  
Author(s):  
P. A. DAVIDSON
Keyword(s):  
Variational Principle ◽  
Linear Stability ◽  
Energy Criterion ◽  
Conservative System ◽  
Base Flow ◽  
Recirculating Flow ◽  
Ideal Mhd ◽  
Mhd Equilibria ◽  
Euler Flows ◽  
The Stability

We investigate the linear stability of inviscid flows which are subject to a conservative body force. This includes a broad range of familiar conservative systems, such as ideal MHD, natural convection, flows driven by electrostatic forces and axisymmetric, swirling, recirculating flow. We provide a simple, unified, linear stability criterion valid for any conservative system. In particular, we establish a principle of maximum action of the formformula herewhere η is the Lagrangian displacement,e is a measure of the disturbance energy, T and V are the kinetic and potential energies, and L is the Lagrangian. Here d represents a variation of the type normally associated with Hamilton's principle, in which the particle trajectories are perturbed in such a way that the time of flight for each particle remains the same. (In practice this may be achieved by advecting the streamlines of the base flow in a frozen-in manner.) A simple test for stability is that e is positive definite and this is achieved if L(η) is a maximum at equilibrium. This captures many familiar criteria, such as Rayleigh's circulation criterion, the Rayleigh–Taylor criterion for stratified fluids, Bernstein's principle for magnetostatics, Frieman & Rotenberg's stability test for ideal MHD equilibria, and Arnold's variational principle applied to Euler flows and to ideal MHD. There are three advantages to our test: (i) d2T(η) has a particularly simple quadratic form so the test is easy to apply; (ii) the test is universal and applies to any conservative system; and (iii) unlike other energy principles, such as the energy-Casimir method or the Kelvin–Arnold variational principle, there is no need to identify all of the integral invariants of the flow as a precursor to performing the stability analysis. We end by looking at the particular case of MHD equilibria. Here we note that when u and B are co-linear there exists a broad range of stable steady flows. Moreover, their stability may be assessed by examining the stability of an equivalent magnetostatic equilibrium. When u and B are non-parallel, however, the flow invariably violates the energy criterion and so could, but need not, be unstable. In such cases we identify one mode in which the Lagrangian displacement grows linearly in time. This is reminiscent of the short-wavelength instability of non-Beltrami Euler flows.

Nonlinear Evolution of Internal Ideal MHD Modes Near the Boundary of Marginal Stability

1982 ◽  
Vol 37 (8) ◽  
pp. 816-829
Author(s):  
E. Rebhan
Keyword(s):  
Equations Of Motion ◽  
Stability Limit ◽  
Mhd Equations ◽  
Marginal Stability ◽  
Unstable State ◽  
Stable Regime ◽  
Ideal Mhd ◽  
Mhd Equilibria ◽  
The Stability ◽  
Marginal Mode

A family of ideal MHD equilibria is considered introducing the concept of a driving parameter λ the increase of which beyond a certain threshold λ0 drives the plasma from a linearly stable to an unstable state. Using reductive perturbation theory, the nonlinear ideal MHD equations of motion are expanded in the neighbourhood of λ0 with respect to a small parameter ε. An appropriate scaling for the expansions is derived from the linear eigenmode problem. Integrability conditions for the reduced nonlinear equations yield nonlinear amplitude equations for the marginal mode. Nonlinearly, the instabilities are either oscillations about bifurcating equilibria, or they are explosive. In the latter case, the stability limit depends on the amplitude of the perturbation and is shifted into the linearly stable regime. Generally bifurcation of dynamically connected equilibria is observed at λ0

On Rayleigh’s Stability Criterion for Couette Flow

1993 ◽  
Vol 48 (5-6) ◽  
pp. 703-704 ◽  
Author(s):  
D. Lortz
Keyword(s):  
Kinetic Energy ◽  
Couette Flow ◽  
Ideal Fluid ◽  
Stability Criterion ◽  
Stability Problem ◽  
Total Kinetic Energy ◽  
Coaxial Cylinders ◽  
Circular Flow ◽  
The Stability ◽  
Necessary And Sufficient

Abstract The stability problem of a stationary circular flow of an ideal fluid between two coaxial cylinders is considered. It is shown that Rayleigh's circulation criterion is necessary and sufficient for the total kinetic energy of an axisymmetric disturbance to be bounded in time.

Almost sure stability of maxima

1973 ◽  
Vol 10 (02) ◽  
pp. 387-401 ◽  
Author(s):  
Sidney I. Resnick ◽  
R. J. Tomkins
Keyword(s):  
Markov Chain ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Almost Sure Stability ◽  
Normalizing Constants ◽  
The Stability ◽  
Necessary And Sufficient

For random variables {Xn, n≧ 1} unbounded above setMn= max {X1,X2, …,Xn}. When do normalizing constantsbnexist such thatMn/bn→1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for alland in this casebn∼F–1(1 – 1/n) Necessary and sufficient conditions for lim supn→∞,Mn/bn= l >1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn= 1 a.s. except whenl= 1. When theXnare r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

scholarly journals Energy Principle for Dissipative Two-fluid Plasmas

1978 ◽  
Vol 33 (3) ◽  
pp. 257-260 ◽  
Author(s):  
H. Tasso
Keyword(s):  
Tokamak Plasmas ◽  
Test Functions ◽  
Energy Principle ◽  
Tearing Modes ◽  
Lagrangian Form ◽  
Fluid Theory ◽  
The Stability ◽  
Necessary And Sufficient ◽  
The Magnetic Field ◽  
Two Fluid

AbstractAn energy principle for the dissipative two-fluid theory in Lagrangian form is given. It represents a necessary and sufficient condition for stability allowing the use of test functions. It is exact for two-dimensional disturbances but is still correct in terms of perturbation theory for long wavelengths along the magnetic field. This may well find application in tokamak plasmas. A discussion of the general case and its relation to the stability of flows in hydrodynamics is given. This energy principle may be applied for estimating the magnitude of residual tearing modes in tokamaks.

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