Frequency-domain instability criteria for time- varying and nonlinear systems.
Frequency domain instability criteria generated from stability criteria for time varying and nonlinear feedback problems
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Mathematical Instability Criteria for Elastic Structures
Theoretical paper discusses physical significance of vanishing of hyperbolic coefficients in equations of elastodynamics. Paper presents generalized approach to structural elastodynamics as part of continuing effort to develop mathematical stability criteria for structures and simulate postinstabilty behavior of elastics in general.
Zak, M.↗
The Generation of Smooth High Speed Solar Wind from Plume-Interplume Mixing
Plumes and rays are magnetic field aligned density striations in coronal holes with different values of plasma beta. The overall plasma beta is very small in the low corona but exceeds unity beyond 15-20 solar radius. High speed solar wind reported beyond 0.3 AU is relatively smooth and uniform and known to originate from the much filamented coronal hole. Thus the obvious question is how to generate a smooth solar wind from seemingly filamentary structure. Hence one has to find a mechanism to substantiate this apparent observed (Ulysses) phenomenon. To do this we model plumes as jets (or wakes) of plasma emitted from the solar surface. The shear between a jet and its ambient is known to become unstable to the MHD Kelvin-Helmholtz ("KH") instability if the Alfven Mach number of the jet is greater than one and the uniform external magnetic field is small. Starting with a simple configuration we consider a jet of half thickness R, having uniform density and uniform internal magnetic field. The external medium has also a uniform density and uniform magnetic field. The jet is perturbed at the boundary with a linear amplitude and fixed frequency. We simulate the coronal jet using the 3D ZEUS code. The first results indicate the slab jet is unstable to the MHD KH instability at 5-10 solar radius for some angle of wave propagation. The propagating instability may smooth the filamented flow. It may also produce the entrained Alfvenic fluctuations observed by Ulysses in the high speed wind. We are at present determining the parameters which induce large growth rate. This may clarify the mystery behind the emergence of fast smooth solar wind from very filamentary structures in coronal holes. Also, using the dispersion relation already available for such a flow we obtain some general description of the instability criteria for the KH instability at a jet interface.
Parhi, Shyamsundar↗
A linear MHD instability analysis of solar mass ejections with gravitation
The linear MHD instability of a cylindrical plasma is used to investigate the origin of solar mass ejections, and the dispersion relation is solved numerically. The initial plasma-flow velocity is found to have a significant effect on the instability criteria and growth rate, and the instability growth-rate is shown to be larger in cases where plasma flow exists, relative to the static case. Results suggest that the plasma column may break into small pieces. Assuming a thin-tube approximation, gravity is found to have little effect on the instability of quasi-horizontal ejection, but to have considerable effect on the vertical ejection. In considering the gravitational force, an exact analytical solution is found for the vertical case, while asymptotic solutions are given for the horizontal and oblique cases.
Song, M. T.↗
Destabilization of binary mixing layer in supercritical conditions
Compressible mixing layer instabilities are of importance to a wide range of environmental and industrial applications. Past studies have focused on either ideal-gas or real-fluid thermodynamic regimes of single-species mixing layers. However, mixing layers of binary mixtures at supercritical conditions, commonly encountered in fuel injection systems, introduce additional complexities due to the added compositional degree-of-freedom. Moreover, the effect of strong variations in thermodynamic response functions across the Widom line on the binary mixing layer stability remains poorly understood. Thus, the objective of this study is to examine the coupling between the hydrodynamic instability and the real-fluid thermodynamics across the Widom line and its effects on the overall binary mixing layer dynamics. To this end, we develop a linear stability analysis of the full binary-species compressible transport equations coupled with the PC-SAFT equation of state. Analysis shows the existence of a novel instability mechanism that arises from juxtapositioning of the Widom-line transition and the hydrodynamic inflection point. This novel thermodynamically induced instability mechanism has the net effect of destabilizing the binary mixing layer at lowering supercritical conditions towards the critical pressure point. This is in contrast to previous stability analyses of supercritical single-species mixing layers, where increasing pressure destabilizes the flow due to its effect on reducing the density stratification. The discovered thermodynamically induced instability mechanism of binary mixing flows highlights the need for an extension of classical instability criteria to incorporate the effect of strong variations in the thermodynamic response functions across the Widom line on mixing layer instability.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Theory of the magnetothermal instability in coronal plasma flows
The theory of the magnetothermal instability (MTI) is revisited through the lens of the stability of uniform systems. The linear stability analysis includes flow advection and Nernst transport. The instability criteria derived distinguish between the convective and the absolute nature of the perturbation growth. It is proven that, in the region where the Nernst and plasma blowoff velocities cancel, the MTI can be absolute and wave-packet perturbations grow in situ. This instability is mediated by the internal feedback between the Biermann battery and Righi–Leduc terms. Here, the analysis is extended to derive the dispersion relation for short-wavelength perturbations developing in nonuniform profiles with the application to coronal plasmas. It is found that the condition for MTI requires the net B-field convection velocity to be small at the isothermal sonic section, and the plasma conditions in this section govern the dynamics of the instability. Analysis of hydro-equivalent implosions suggests that unstable perturbations undergo more e-foldings of growth in larger-size targets.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
On the Stability Analysis of Astrophysical Cooling Functions
To model the temperature evolution of optically thin astrophysical environments at MHD scales, radiative and collisional cooling rates are typically either pretabulated or fit into a functional form and then input into MHD codes as a radiative loss function. Thermal balance requires estimates of the analogous heating rates, which are harder to calculate, and due to uncertainties in the underlying dissipative heating processes these rates are often simply parameterized. The resulting net cooling function defines an equilibrium curve that varies with density and temperature. Such cooling functions can make the gas prone to thermal instability (TI), which will cause departures from equilibrium. There has been no systematic study of thermally unstable parameter space for nonequilibrium states. Motivated by our recent finding that there is a related linear instability, catastrophic cooling instability, that can dominate over TI, here we carry out such a study. We show that Balbus instability criteria for TI can be used to define a critical cooling rate, Λc, that permits a nonequilibrium analysis of cooling functions through the mapping of TI zones. We furthermore illustrate how thermal conduction modifies the shape of TI zones. Upon applying a Λc-based stability analysis to coronal loop simulations, we find that loops undergoing periodic episodes of coronal rain formation are linearly unstable to catastrophic cooling instability, while TI is stabilized by thermal conduction.
Amanda Stricklan↗
On the Stability Analysis of Astrophysical Cooling Functions
To model the temperature evolution of optically thin astrophysical environments at MHD scales, radiative and collisional cooling rates are typically either pretabulated or fit into a functional form and then input into MHD codes as a radiative loss function. Thermal balance requires estimates of the analogous heating rates, which are harder to calculate, and due to uncertainties in the underlying dissipative heating processes these rates are often simply parameterized. The resulting net cooling function defines an equilibrium curve that varies with density and temperature. Such cooling functions can make the gas prone to thermal instability (TI), which will cause departures from equilibrium. There has been no systematic study of thermally unstable parameter space for nonequilibrium states. Motivated by our recent finding that there is a related linear instability, catastrophic cooling instability, that can dominate over TI, here we carry out such a study. We show that Balbus instability criteria for TI can be used to define a critical cooling rate, Λ c , that permits a nonequilibrium analysis of cooling functions through the mapping of TI zones. We furthermore illustrate how thermal conduction modifies the shape of TI zones. Upon applying a Λ c -based stability analysis to coronal loop simulations, we find that loops undergoing periodic episodes of coronal rain formation are linearly unstable to catastrophic cooling instability, while TI is stabilized by thermal conduction.
79 ASTRONOMY AND ASTROPHYSICS↗
Stability instability, invertibility and causality
Feedback systems stability treated by functional analysis, describing procedures for generating instability criteria
Willems, J. C.↗
High-frequency electrostatic waves in the magnetosphere.
High-frequency electrostatic microinstabilities in magnetospheric plasmas are considered in detail. Rather special plasma parameters are found to be required to match the theoretical wave spectrum with satellite observations in the magnetosphere. In particular, it is necessary to have a cold and a warm species of electrons such that (1) the warm component has an anomalous velocity distribution function that is nonmonotonic in the perpendicular component of velocity and is the source of free energy driving the instabilities, (2) the density ratio of the cold component to the hot component is greater than about 0.01, and (3) the temperature ratio of the two components for cases of high particle density is no less than 0.1. These requirements and the corresponding instability criteria are satisfied only in the trapping region; this is also the region in which the waves are most frequently observed. The range of unstable wavelengths and an estimate of the diffusion coefficient are also obtained. The wave are found to induce strong diffusion in velocity space for low-energy electrons during periods of moderate wave amplitude.
Young, T. S. T.↗
Ordinary electromagnetic mode instability
The instability of the ordinary electromagnetic mode propagating perpendicular to an external magnetic field is studied for a single-species plasma with ring velocity distribution. The marginal instability boundaries for both the purely growing mode and the propagating growing modes are calculated from the instability criteria. The dispersion characteristics for various sets of plasma parameters are also given. The typical growth rates are of the order of the cyclotron frequency.
Cheng, C. Z.↗
Ordinary electromagnetic mode instability
The instability of the ordinary electromagnetic mode propagating perpendicular to an external magnetic field is analyzed for a single-species plasma with ring velocity distribution. The marginal instability boundaries for both the purely growing mode and the propagating growing modes are calculated from the instability criteria. The dispersion characteristics are derived for a cold plasma and a very strong static magnetic field. The point is determined at which the real parts of the complex frequency band are centered. The typical growth rates are found to be of the order of the cyclotron frequency and can be enhanced by increasing the ratio of parallel kinetic pressure to magnetic pressure.
Cheng, C.-Z.↗
DESAP 1: A structural design program with stress and displacement constraints. Volume 1: Theoretical and user's manual
A finite element program is presented for computer-automated, minimum weight design of elastic structures with constraints on stresses (including local instability criteria) and displacements. Volume 1 of the report contains the theoretical and user's manual of the program. Sample problems and the listing of the program are included in Volumes 2 and 3. The element subroutines are organized so as to facilitate additions and changes by the user. As a result, a relatively minor programming effort would be required to make DESAP 1 into a special purpose program to handle the user's specific design requirements and failure criteria.
Kiusalaas, J.↗
DESAP 2: A structural design program with stress and buckling constraints. Volume 1: Theoretical and user's manual
DESAP 2 is described as a finite element program for computer-automated, minimum weight design of elastic structures with constraints on stresses (including local instability criteria) and buckling loads. No limits are placed on the number of load conditions for stress-constrained design, but only one of these load conditions can be chosen as the potential buckling load. A substantial portion of DESAP 2, particularly the analysis of the prebuckling state, is derived from the SOLID SAP finite element program. The stress-constrained design is based on the classical stress ratio method, which drives the design towards a fully stressed state. The constraints on the buckling load are handled by solving the appropriate optimality criterion by successive iterations. During each iteration, the element sizes determined by the stress ratio method are used as the minimum size constraints. The element subroutines are organized in a manner that permits the user to make additions and changes with a minimal programming effort. Consequently, DESAP 2 can readily be changed into a special-purpose program to handle the user's specific design requirements and failure criteria.
Kiusalaas, J.↗
Structural design with stress and displacement constraints
DESAPI program synthesizes linear elastic structures under static loads. Its design objective is finding element sizes that minimize total weight without changing layout structure. Primary constraints are upper limits on stresses and displacements prescribed as yield and loacl instability criteria. Program is written in FORTRAN IV for batch execution and is implemented on IBM 360 computer.
Kiusalaas, J.↗
Interplanetary gas. XXVI - On the reconnection of magnetic fields in cometary ionospheres at interplanetary sector boundary crossings
The reconnection process in the cometary ionosphere believed responsible for the disconnecting plasma tails phenomenon is studied through the basic equations of reconnection theory and current sheet instability criteria. It is proposed that reconnection occurs when the interplanetary magnetic fields incident on a comet that has gone just past a sector boundary are pressed into the fields captured from the previous sector. The fields are of opposite polarity, and the previously captured fields constitute the 'roots' of the plasma tail. An estimated duration of reconnection during a disconnection event (DE) of 0.75 days is used along with estimates of other cometary parameters to construct fusion region dimensions and resistivity with the adopted time scale.
Niedner, M. B., Jr.↗
Computational methods for probability of instability calculations
This paper summarizes the development of the methods and a computer program to compute the probability of instability of a dynamic system than can be represented by a system of second-order ordinary linear differential equations. Two instability criteria based upon the roots of the characteristics equation or Routh-Hurwitz test functions are investigated. Computational methods based on system reliability analysis methods and importance sampling concepts are proposed to perform efficient probabilistic analysis. Numerical examples are provided to demonstrate the methods.
Wu, Y.-T.↗
Probabilistic methods for rotordynamics analysis
This paper summarizes the development of the methods and a computer program to compute the probability of instability of dynamic systems that can be represented by a system of second-order ordinary linear differential equations. Two instability criteria based upon the eigenvalues or Routh-Hurwitz test functions are investigated. Computational methods based on a fast probability integration concept and an efficient adaptive importance sampling method are proposed to perform efficient probabilistic analysis. A numerical example is provided to demonstrate the methods.
Wu, Y.-T.↗