Ballooning and drift-Alfven modes in tokamaks

1991 ◽  
Vol 69 (7) ◽  
pp. 864-872
Author(s):  
A. Hirose ◽  
S. Roy Choudhury ◽  
O. Ishihara
Keyword(s):  
Dispersion Relation ◽  
Stability Boundary ◽  
Order Unity ◽  
Poloidal Beta ◽  
Ballooning Mode ◽  
The Stability

The magnetohydrodynamic (MHD) ballooning mode in tokamaks was analyzed in the compressible limit. The stability boundary becomes strongly dependent on the toroidicity parameter, εn. This was confirmed by a semilocal kinetic dispersion relation that incorporates kinetic resonances of both ions and electrons. The stability boundary found from the kinetic dispersion relation is insensitive to the degree of plasma compressibility. A nonideal MHD analysis revealed a marginally stable drift-Alfven mode described by [Formula: see text] when βθ (poloidal beta) is of order unity.

Convection in a rotating annulus uniformly heated from below

1971 ◽  
Vol 46 (1) ◽  
pp. 65-81 ◽  
Author(s):  
Robert P. Davies-Jones ◽  
Peter A. Gilman
Keyword(s):  
Thermal Stresses ◽  
Stability Boundary ◽  
Second Order ◽  
Infinite Plane ◽  
Reynolds Stresses ◽  
Order Unity ◽  
Prandtl Numbers ◽  
The Stability ◽  
Boussinesq Fluid ◽  
Outer Half

We present a linear stability analysis, to second order in initial amplitude, of Bénard convection of a Boussinesq fluid in a thin rotating annulus for modest Taylor numbers T ([les ] 104). The work is motivated in part by the desire to study further a mechanism for maintaining, through horizontal Reynolds stresses induced in the convection, the sun's ‘equatorial acceleration’, which has been demonstrated for a rotating convecting spherical shell by Busse & Durney. The annulus is assumed to have stress free, perfectly conducting top and bottom (which allows separation of the equations) and non-conducting non-slip sides. A laboratory experiment which fulfills these conditions (except perhaps the free bottom) is being developed with H. Snyder.We study primarily annuli with gap-width to depth ratios a of order unity. The close, non-slip side-walls produce a number of effects not present in the infinite plane case, including overstability at high Prandtl numbers P, and multiple minima in Rayleigh number R on the stability boundary. The latter may give rise to vacillation. The eigenfunctions for stationary convection for a = 2, T [lsim ] 2000 clearly show momentum of the same sense as the rotation is transported from the inner to the outer half of the annulus, corresponding to transport toward equatorial latitudes on the sphere. The complete second-order solutions for the induced circulations indeed give faster rotation in the outer half, except for large P (> 102), in which case thermal stresses dominate. At all P, this differential rotation is qualitatively a thermal wind. Overstable convective cells, and stationary cells at higher T, induce more complicated differential rotations.

Wave breakdown in stratified shear flows

1977 ◽  
Vol 79 (3) ◽  
pp. 481-497 ◽  
Author(s):  
M. T. Landahl ◽  
W. O. Criminale
Keyword(s):  
Shear Flows ◽  
Flow Instability ◽  
Stability Boundary ◽  
Small Scale ◽  
Mean Flow ◽  
Order Unity ◽  
Stable Regime ◽  
Density Interface ◽  
Density Interfaces ◽  
The Stability

The wave-mechanical condition (Landahl 1972) for breakdown of an unsteady laminar flow into strong small-scale secondary instabilities is applied to some simple stratified inviscid shear flows. The cases considered have one or two discrete density interfaces and simple discontinuous or continuous velocity profiles. A primary wavelike disturbance to such a flow produces a perturbation velocity that is discontinuous at a density interface. The resulting instantaneous system, defined as the sum of the mean flow and the primary oscillation, develops a local secondary shear-flow instability that has a group velocity equal to the arithmetic mean of the instantaneous velocities on the two sides of the interface. According to the breakdown criterion, the disturbed flow will become critical whenever this velocity reaches a value equal to the phase velocity of the primary wave. The calculations show that for a single density interface breakdown may occur for low to moderate wave amplitudes in a fairly narrow range of Richardson numbers on the stable side of the stability boundary. On the other hand, in the unstable regime maximum wave slopes of order unity may be reached before breakdown occurs; this conclusion is in qualitative agreement with experiments. When the system includes two density interfaces, it is found that there exists a range of high Richardson numbers far into the stable regime for which breakdown may take place even for very small and zero wave interface deflexions.

Tertiary solutions and their stability in rotating plane Couette flow

1998 ◽  
Vol 358 ◽  
pp. 357-378 ◽  
Author(s):  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Stability Boundary ◽  
Streamwise Vortex ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Plane Couette Flow ◽  
Streamwise Direction ◽  
The Stability ◽  
Oscillatory Instabilities

The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

scholarly journals On Minimum-B Stabilization of Electrostatic Drift Instabilities

1966 ◽  
Vol 21 (11) ◽  
pp. 1953-1959 ◽  
Author(s):  
R. Saison ◽  
H. K. Wimmel
Keyword(s):  
Dispersion Relation ◽  
Particle Distribution ◽  
Inhomogeneous Plasma ◽  
Distribution Functions ◽  
Loss Cone ◽  
First Order ◽  
Stabilization Theorem ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Limiting Case

A check is made of a stabilization theorem of ROSENBLUTH and KRALL (Phys. Fluids 8, 1004 [1965]) according to which an inhomogeneous plasma in a minimum-B field (β ≪ 1) should be stable with respect to electrostatic drift instabilities when the particle distribution functions satisfy a condition given by TAYLOR, i. e. when f0 = f(W, μ) and ∂f/∂W < 0 Although the dispersion relation of ROSENBLUTH and KRALL is confirmed to first order in the gyroradii and in ε ≡ d ln B/dx z the stabilization theorem is refuted, as also is the validity of the stability criterion used by ROSEN-BLUTH and KRALL, ⟨j·E⟩ ≧ 0 for all real ω. In the case ωpi ≫ | Ωi | equilibria are given which satisfy the condition of TAYLOR and are nevertheless unstable. For instability it is necessary to have a non-monotonic ν ⊥ distribution; the instabilities involved are thus loss-cone unstable drift waves. In the spatially homogeneous limiting case the instability persists as a pure loss cone instability with Re[ω] =0. A necessary and sufficient condition for stability is D (ω =∞, k,…) ≦ k2 for all k, the dispersion relation being written in the form D (ω, k, K,...) = k2+K2. In the case ωpi ≪ | Ωi | adherence to the condition given by TAYLOR guarantees stability.

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

scholarly journals Flexural-Torsional Vibration of Thin-Walled Beams Subjected to Combined Initial Axial Load and End Bending Moment: Application to the Design of Saw Tooth Blades

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Van Binh Phung ◽  
Anh Tuan Nguyen ◽  
Hoang Minh Dang ◽  
Thanh-Phong Dao ◽  
V. N. Duc
Keyword(s):  
Boundary Conditions ◽  
Axial Load ◽  
Stability Boundary ◽  
Bending Moment ◽  
Element Analysis ◽  
The Finite Element Analysis ◽  
Rayleigh Method ◽  
Thin Walled ◽  
The Stability ◽  
Fixed End

The present paper analyzes the vibration issue of thin-walled beams under combined initial axial load and end moment in two cases with different boundary conditions, specifically the simply supported-end and the laterally fixed-end boundary conditions. The analytical expressions for the first natural frequencies of thin-walled beams were derived by two methods that are a method based on the existence of the roots theorem of differential equation systems and the Rayleigh method. In particular, the stability boundary of a beam can be determined directly from its first natural frequency expression. The analytical results are in good agreement with those from the finite element analysis software ANSYS Mechanical APDL. The research results obtained here are useful for those creating tooth blade designs of innovative frame saw machines.

Double-diffusive instabilities in a vertical slot

1999 ◽  
Vol 392 ◽  
pp. 213-232 ◽  
Author(s):  
OLIVER S. KERR ◽  
KIT YEE TANG
Keyword(s):  
Temperature Difference ◽  
Salinity Gradient ◽  
Stability Boundary ◽  
Large Temperature ◽  
Vertical Slot ◽  
Stability And Instability ◽  
Heat Diffusivity ◽  
Prandtl Numbers ◽  
The Stability ◽  
Double Diffusive

A fluid stably stratified by a salinity gradient and enclosed between two vertical boundaries can become unstable when it is subjected to a temperature difference between the walls. The linear stability of such a fluid in a vertical slot is investigated. Errors in earlier results are found, confirming recent results of Young & Rosner (1998). Four different asymptotic regimes on the stability boundary are identified. One of these, the limit of a strong salinity gradient, has previously been analysed. The analyses of the separate asymptotic limits of weak salinity gradient, large temperature difference and small wavenumber are also given. These four cases make up much of the total boundary between stability and instability for double-diffusive instabilities in a vertical slot, and so most of this boundary can be mapped out for general Prandtl numbers and salt/heat diffusivity ratios using these results.

Instability of Heated Panels in Supersonic Air Flow

2008 ◽  
Vol 33-37 ◽  
pp. 1101-1108
Author(s):  
Zhi Chun Yang ◽  
Wei Xia
Keyword(s):  
Stability Analysis ◽  
Stability Boundary ◽  
Boundary Curve ◽  
Static Stability ◽  
Static Deformation ◽  
Limit Cycle Oscillation ◽  
Aerodynamic Load ◽  
Aeroelastic System ◽  
Aerodynamic Pressure ◽  
The Stability

An investigation on the stability of heated panels in supersonic airflow is performed. The nonlinear aeroelastic model for a two-dimensional panel is established using Galerkin method and the thermal effect on the panel stiffness is also considered. The quasi-steady piston theory is employed to calculate the aerodynamic load on the panel. The static and dynamic stabilities for flat panels are studied using Lyapunov indirect method and the stability boundary curve is obtained. The static deformation of a post-buckled panel is then calculated and the local stability of the post-buckling equilibrium is analyzed. The limit cycle oscillation of the post-buckled panel is simulated in time domain. The results show that a two-mode model is suitable for panel static stability analysis and static deformation calculation; but more than four modes are required for dynamic stability analysis. The effects of temperature elevation and dimensionless parameters related to panel length/thickness ratio, material density and Mach number on the stability of heated panel are studied. It is found that panel flutter may occur at relatively low aerodynamic pressure when several stable equilibria exist for the aeroelastic system of heated panel.

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