Kind Of Instability Research Articles

The instabilities of finite-amplitude barotropic Rossby waves

The stability of a plane Rossby wave in a homogeneous fluid is considered. When the two-dimensional equation which governs fluid flow on a beta-plane is linearized in the disturbance stream function, a partial differential equation with a periodic coefficient results. Substitution of a solution dictated by the Floquet theory leads to a determinant equation, and it may be shown from its symmetry properties that disturbances to a Rossby wave may be of only two types: (i) neutrally stable modes not necessarily contiguous to a stability boundary and (ii) a pair of temporally unstable waves, one growing and the other decaying.The determinant is solved numerically for the neutral-stability boundaries and curves of constant disturbance growth rate; two distinct types of instability emerge. The first is the parametric instability, which renders all waves unstable, and is shown to be asymptotic to the classical nonlinear resonant interaction in the limit of vanishing basic-state amplitude. The details of the disturbance frequency bifurcation for zero-amplitude basic-state waves are presented, and calculations for waves with eastward and westward group velocities are made and discussed in the context of Rhines’ (1975) results for waves and turbulence on a beta-plane. In addition, a second type of instability is computed which is separate and distinct from the parametric instability. The very limited evidence presented suggests that this second kind of instability may possess characteristics which are identifiable in part with the shearing of the fluid by the large-amplitude basic state and in part with the overturning of the ambient vorticity gradient.

Experiments in ocean circulation modelling

Abstract In a laboratory model ocean, fluid in a rotating tank of varying depth is subjected to “wind-stress”, For a certain range of the parameters, Ekman number E and Rossby number R, a homogeneous fluid displays steady, westward intensified flow. For the same range of E and R, a two-layer fluid can have baroclinic instabilities. The parameter range for the various kinds of instabilities is mapped in a regime diagram. The northward transport in the western boundary current is measured as it varies with Rossby number for both homogeneous and two-layer fluid.

Excitation of Alfvén waves by trapped alpha particles in a tokamak reactor

Alfvén wave instability in a tokamak caused by trapped alpha particles with non-monotonic energy distribution is investigated theoretically. It is shown that this kind of instability is associated with perturbations, the transverse wave-length of which is of the order of the banana width of trapped alpha particles. It occurs during resonance between the Alfvén wave frequency and the bounce frequency of trapped alpha particles. The instability growth rate is of the order of the oscillation frequency multiplied by the ratio of the alpha-particle density to the plasma density.

Stabilization of superconductors for use in magnets

There are two main causes of degraded performance in superconducting magnets, magnetic instability or flux jumping and mechanical instability. Cryostatically stabilized conductors are able to cope with both kinds of instability but finely subdivided conductors are only stable against flux jumping. It is suggested that the inclusion of a small proportion of helium in the magnet windings could provide sufficient transient stability to enable finely subdivided conductors to work well in the presence of mechanical instabilities.

Discussion on the formation of crenulation cleavage

D r S. H. T reagus writes: Cosgrove (1976) has attempted to explain the variety in form of crenulation cleavage in terms of two main processes: instabilities of different types initiating in a stressed multilayer or fabric, and the development of these structures into zones or planes by mineral migration. These form the two main sections to his paper. It is the first section, that concerned with the buckling behaviour of anisotropic materials, which I wish to criticise. The theory presented by Dr Cosgrove, and the basis of the first section of his paper, is Biot’s (1964) analysis of instabilities in the confined multilayer. The multilayer can be considered statistically as an anisotropic homogeneous material ( Biot 1965 p. 186, Cobbold et al. 1971 ). Biot showed that two kinds of instability were solutions to the general displacement equation, and were governed by particular conditions relating to initial stress and elastic moduli ( Cosgrove 1976 equations 7-10). The first type of instability is internal sinusoidal buckling, and the second Biot illustrates as a plane of slip or kinking. Geologists ( Cobbold et al. 1971 , Johnson 1970 , Cosgrove 1976 ) have used Biot’s theory to explain the initiation of two distinct structures, buckles and kink bands, in compressed multilayered rocks. It will be shown that in his recent paper Dr Cosgrove has used the two instability solutions incorrectly to explain structures in a variety of types of anisotropic material. Cosgrove (1976) considers three orthotropic models of anisotropy, in the form of fine layering, at 0°, 45° and 90°

Nature of one kind of instability of spin detonation

Nature of one kind of instability of spin detonation

A Stable Variant of the Secant Method for Solving Nonlinear Equations

The usual successive secant method for solving systems of nonlinear equations suffers from two kinds of instabilities. First the formulas used to update the current approximation to the inverse Jacobian are numerically unstable. Second, the directions of search for a solution may collapse into a proper affine subspace, resulting at best in slowed convergence and at worst in complete failure of the algorithm. In this report it is shown how the numerical instabilities can be avoided by working with factorizations of matrices appearing in the algorithm. Moreover, these factorizations can be used to detect and remedy degeneracies among the directions.

Effect of solute dispersion on thermal convection in a porous medium layer, 2

In some situations associated with geothermal activity, groundwater motions are affected by convection currents due to large temperature gradients. In such cases, usually saline hot water is located in the deep layers of the aquifer from which salt and heat are transferred to the upper layers. In part 1 of this study the parameters of the two‐dimensional flow field stability were determined. In this paper, further analysis of the phenomenon in three dimensions is presented. It was found that the convection cells have the shape of rolls whose axes are perpendicular to the steady state flow velocity. However, there is also a possibility of overstability of the flow field caused by rolls whose axes are parallel to the steady state velocity. The parameters of these two kinds of instability were determined in this study.

Stability of laminar flow of liquid in vertical layers with free convection

We give the results of experimental and theoretical investigations of the stability of a laminar flow of liquid in a vertical layer. The experimental investigations qualitatively confirm the theoretical solutions and indicate the existence of various kinds of instability.

Stability of bubbles subjected to an in-plane field

A discussion is given of the consequences for the stability and behavior of bubbles and strip domains of an additional magnetic field which is parallel to the plane of the platelet. Apart from the instability of bubble domains, leading to a collapse or a running out into strip domains, a second kind of instability is discussed. This is due to the fact that inside a domain, the magnetization becomes unstable when a suitable in-plane field is applied and will be rotated to a direction parallel to the magnetization outside the domains.

On Non-Geostrophic Baroclinic Stability: Part III. The Momentum and Heat Transports

Abstract The results of Parts I and II are used to calculate the transports of heat and momentum that accompany growing baroclinic instabilities in Eady's model. The transports are calculated for both the conventional (“geostrophic”) kind of baroclinic instability and for symmetric instability, without any restriction on the stratification, as measured by the Richardson number. The transports are calculated consistently to second order in the amplitude expansion of stability theory, so that the transports are the sum of an eddy transport term and a mean transport term. The results show that both kinds of instability always transport heat upward and poleward, and always transport zonal momentum downward. Under geostrophic conditions the horizontal transport of zonal momentum depends on the horizontal shear of the basic flow. This shear is neglected in Eady's model so the horizontal momentum transports calculated here only contain the non-geostrophic contribution to the transport. The results show that this...

The effect of boundary conditions on current oscillations in n-type germanium at 77°K

Recent experimental work on current instabilities and related properties of n-type germanium in high electric fields shows several kinds of instabilities. The purpose of this paper is to show that the differences can be explained by assuming different boundary conditions at the cathode. The method of analysis follows that of Conwell.

Baroclinic stability under non-hydrostatic conditions

Eady's model for the stability of a thermal wind in an inviscid, stratified, rotating system is modified to allow for deviations from hydrostatic equilibrium. The stability properties of the flow are uniquely determined by two parameters: the Richardson number Ri, and the ratio of the aspect ratio to the Rossby number δ. The latter parameter may be taken as a measure of the deviations from hydrostatic equilibrium (δ = 0 in Eady's model). It is found that such deviations decrease the growth rates of all three kinds of instability which can occur in this problem: ‘geostrophic’ baroclinic instability, symmetric instability, and Kelvin–Helmholtz instability. The unstable wavelengths for ‘geostrophic’ and Kelvin–Helmholtz instability are increased for finite values of δ, while the unstable wavelengths for symmetric instability are unaffected. The ‘non-hydrostatic’ effects (δ ≠ 0) are significant for symmetric and Kelvin–Helmholtz instability when δ [gsim ] 1, but not for ‘geostrophic’ instability unless δ [Gt ] 1. Consequently, the first two types of instability tend to be suppressed relative to ‘geostrophic’ instability by ‘non-hydrostatic’ conditions. Figure 3 summarizes the different instability régimes that can occur. In laboratory experiments symmetric instability can be studied best when δ [lsim ] 1, while Kelvin–Helmholtz instability can be studied best when δ [Lt ] 1.

Non-Ohmic Conduction in Germanium under a Strong Magnetic Field

A distinct non-ohmic current-voltage characteristics in n -type germanium containing 10 20 to 10 22 impurity atoms/m 3 under a strong impulsive magnetic field was found at liquid hydrogen temperatures. The effect was also observed in p -type germanium with appropriate impurity concentrations for the transverse magnetic field stronger than 1 Wb/m 2 . Essential mechanism underlying this effect would be a “transverse breakdown” introduced by M. Toda et al. In the present case, however, “transverse impact ionization” should be more appropriate than “transverse breakdown”. Necessary conditions for the occurrence of this non-ohmic effect are (1) the existence of the strong Hall field and (2) neutralized donors or acceptors in germanium crystals. Some peculiar phenomena associated with this effect i.e. , a kind of instability in electric current, negative differential drift mobility and the anomalous electric field dependence of the Hall angle were also observed.

On the Occurrence of Oscillations around the Steady State in Systems of Chemical Reactions far from Equilibrium

The onset of oscillations around the steady state in systems of chemical reactions far from equilibrium is studied in certain specific cases. It is demonstrated that the appearance of oscillations is not followed by any kind of instability but comes progressively, and on modifying the thermodynamic constraints, continuously.

On the Instability of Crossed-field Electron Beams in Cylindrical Geometries†

Abstract In a previous work the authors investigated the oscillations of the electron cloud in heavily cut-off split-anode small-cathode magnetrons whoso segments were connected by a variable resistor. It was found that the simultaneously radiated oscillations could be divided into two groups originating in two spatially separated regions of the cloud. In the present paper it is shown that, in accordance with the recently developed theory of the instability of crossed-field election beams, it can be reasonably assumed that two different kinds of instabilities are responsible for the oscillations considered. The first one is the resistive wall instability of the near-anode layers. The second one seems to be the diocotron instability which probably takes place only in a thin layer of enhanced electron concentration, situated in the inner region of the cloud.

Collision Controlled Stability of a Plasma Diode†

ABSTRACT Investigation of a plasma diode, in which the electron motion is impeded by Coulomb scattering, proves quite fruitful in spite of the involved current-voltage response. Deduction of a characteristic equation containing information as to the stability of the system is carried out for several interesting states. Among these, the weakly damped case turns out to be tractable, loading to a quartic characteristic equation which evidently embodies two kinds of instabilities. By analysing the critical and overly damped regions where the organized oscillations are suppressed, it is found that an instability is yet possible. The make-up of the reduced (quadratic) characteristic equation becomes sufficiently transparent to identify random-like oscillations of thermal origin; in fact, on passage to the collision-free or cold plasma, these fluctuations no longer prevail. Stable d.c. performance of a plasma diode can be maintained where there is enough damping and large enough currents are drawn. We surmise th...

Modes of instability in spiral flow between rotating cylinders

A study of the stability of flow between concentric cylinders, with the inner one rotating, distinguished three kinds of instabilities: the familiar axisymmetric mode, an azimuthal mode with the predicted exp i(θ—ωt) angular dependence, and a completely non-symmetric instability which apparently arises from the interaction of the other two. The effect of small axial flow upon all of these modes was to give an approximately parabolic dependence of the critical Taylor number on the axial flow rate. In the case of the axisymmetric mode, agreement with the theory of Krueger & Di Prima (1964) was found to be excellent.

Transversal plasma wave instabilities in two counterstreaming plasmas without external fields

Some aspects of the theory of longitudinal and transversal waves in a collisionless nonrelativistic plasma are treated in this paper. A dispersion relation for multicomponent plasmas is derived from the linearized Boltzmann-Vlasov equation using the full set of Maxwell's equations without an external field. The velocity distributions of the plasma streams are assumed to be Maxwellian. For two counterstreaming plasmas it is shown that there exist transversal instabilities for all counterstreaming velocities whereas the well known two stream instabilities only exist for velocities greater than a critical velocity. Exact solutions for the onset of the instabilities can be given. This kind of instability may occur for any nonisotropic velocity distribution in a collisionless plasma.