Waves of finite amplitude in a hot plasma

A study is made of a plane shock wave of arbitrary strength propagating in a hot rarefied plasma across the magnetic field. The question of the propagation of nonstationary waves of finite but small amplitude under these conditions is examined.Fairly detailed studies have been made of waves of finite amplitude in a cold rarefied plasma. The profile of such waves is formed as the result of nonlinear and dispersion effects, the dispersion effects being caused by electron inertia and plasma anisotropy. If the gas-kinetic pressure of the plasma is taken into account, then dispersion effects appear which are associated with the fact that the Larmor radius of the ions is finite. Stationary waves of small but finite amplitude propagating across the magnetic field in a hot plasma (when the gas-kinetic pressure p is comparable with the magnetic pressure H²/87) have been treated in [1, 2]. In [1] an isolated rarefaction wave was found in a hot plasma, instead of the compression wave characteristic of a cold plasma, and a qualitative picture of the shock wave structure was given. In [2] a study was made of a small-amplitude shock wave with the finite size of the ion Larmor radius taken into account. The present paper investigates the structure of shock waves of arbitrary strength which propagate across the magnetic field in a fairly hot rarefied plasma, and also examines nonstationary waves of finite but small amplitude excited in a plasma by a magnetic piston acting over a limited time interval.Notation p gas-kinetic pressure - H magnetic field - u, v macroscopic velocities along the x and y axes - density - m_e(m_i) mass of electron (ion) - plasma conductivity - _H ion-cyclotron frequency - V_A Alfvèn velocity - c velocity of light - adiabatic exponent - V specific volume - _0e(_0i) electron (ion) plasma frequency - S₀ velocity of sound. In conclusion the author thanks R. Z. Sagdeev and N. N. Yanenko for discussing the paper, and also R. N, Makarov for helping with the numerical computations.     

Hydrodynamic models as applied to the investigation of magnetic plasma stability

In the present paper magnetohydrodynamic models are employed to investigate the stability of an inhomogeneous magnetic plasma with respect to perturbations in which the electric field may be regarded as a potential field (rot E 0). A hydrodynamic model, actually an extension of the well-known Chew-Goldberg er-Low model [1], is used to investigate motions transverse to a strong magnetic field in a collisionless plasma. The total viscous stress tensor is given; this includes, together with magnetic viscosity, the so-called inertial viscosity.Ordinary two-fluid hydrodynamics is used in the case of strong collisions=. It is shown that the collisional viscosity leads to flute-type instability in the case when, collisions being neglected, the flute mode is stabilized by a finite Larmor radius. A treatment is also given of the case when epithermal high-frequency oscillations (not leading immediately to anomalous diffusion) cause instability in the low-frequency (drift) oscillations in a manner similar to the collisional electron viscosity, leading to anomalous diffusion.Notation f particle distribution function - E electric field component - H₀ magnetic field - density - V particle velocity - e charge - m, M electron and ion mass - _i, _e ion and electron cyclotron frequencies - viscous stress tensor - P pressure - r_i Larmor radius - P pressure tensor - t time - frequency - T temperature - collision frequency - collision time - j current density - _i, _e ion and electron drift frequencies - k_x, k_y, k_z wave-vector components - n₀ particle density - g acceleration due to gravity. The authors are grateful to A. A. Galeev for valuable discussion.     

Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - p pressure in the-phase, N/m² - p intrinsic phase average pressure for the-phase, N/m² - p – p , spatial deviation of the pressure in the-phase, N/m² - r ₀ radius of the averaging volume, m - t time, s - v velocity vector for the-phase, m/s - v phase average velocity vector for the-phase, m/s - v intrinsic phase average velocity vector for the-phase, m/s - v – v , spatial deviation of the velocity vector for the-phase, m/s - V averaging volume, m³ - V volume of the-phase contained within the averaging volume, m³ Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m² - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s     

Approximate First Integrals of a Galaxy Model

First-order approximate first integrals (conserved quantities)of a Hamiltonian dynamical system with two degrees of freedomwhich arises in the modeling of central part of a deformed galaxy [1] havebeen obtained based on the approximate Noether symmetries for resonances₁=₂, ₁=2₂ and 2₁=3₂. Furthermore,KAM curves have been obtained analytically and they have been compared with thenumerical ones on the Poincaré surface of section.     

Nonlinear analysis of the forced response of a beam with three mode interaction

An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,_n . Three mode interaction (₂ 3₁ and _{3 1} + 2₂) is considered and its influence on the response is studied. The case of two mode interaction (₂ 3₁) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.     

Harmonic solutions of micropolar elastodynamics

The equations of micropolar elastodynamics are considered for an unbounded continuum subjected to a body force and a body couple. These act harmonically with the same real frequency , but with individual arbitrary spatial distributions. Over a harmonic state, the displacement and microrotation are related to two radiation conditioned harmonic vectors, each acquiring three eigenvalue contributions, assuming a noncritical -frequency. Altogether, four distinct eigenvalues are admissible. If ²<2² ₀, ₀ being a frequency parameter of the continuum, two of these are real while two are purely imaginary. But if ²<2² ₀, then all admissible eigenvalues are real. Each eigenvalue contribution resolves into a series of Hankel and Bessel functions coupled to Hankel type transforms of: (i) spherical integrals which, in turn, can be expanded via spherical harmonics for the 3-dimensional problem, (ii) circular integrals for the 2-dimensional problem. Axisymmetric and spherically symmetric results are deduced in 3-dimensions. Asymptotic solutions are also established; they disclose long-range formation of radially attenuated spherical (or circular) waves propagating with, generally, anisotropic amplitudes but, invariably, isotropic eikonals.If, in the absence of a body couple, a body force acts radially in 3-dimensions with a spherically symmetric strength, then the elastic displacement behaves likewise while the microrotation vanishes identically. Another application is made to a 2-dimensional problem for a 1 × 3 source system of body force plus body couple without longitudinal variation but with magnitudes symmetric about a longitudinal axis.As approaches a certain critical frequency , dependent solely on the continuum, at least two eigenvalues approach the same value. The phenomenon is explored for a continuum consistent with ²<2² ₀ and under the hypothesis ²<2² ₀. All admissible eigenvalues are then real throughout an -neighbourhood of . Here, two associated eigenvalue contributions behave singularly. Nevertheless, their essential singularities cancel out within the relevant combination. Examination of a far-field suggests that critical frequency attainment sets off a slow instability in the 2-dimensional configuration. In the 3-dimensional configuration, however, it preserves stability and eliminates radial attenuation; an exact solution is formulated for this case.     

The retardation of a fluid in a cylinder of low height

The experimental study of specific types of hydrodynamic instability is often carried out from observations of the behavior of a fluid twisting in vessels of various shapes [1, 2]. If the fluid and the vessel are not rotating as a single whole, transverse circulation arises in such a system together with the principal azimuthal motion (see, for example, [3–7]). The classical problem of the retardation of a fluid in an axisymmetric vessel in its full formulation has so far not been solved, since at large Reynolds numbers the initial stage of the process is of a turbulent nature. The most interesting case for technical applications is considered in [8, 9], in which the problem is solved on the assumption of large Reynolds numbers and small Rossby numbers (¦/¦ 1), i.e., under conditions in which the fluid, which at the initial moment is in a state of rapid rigid rotation, experiences thereafter the effect of an envelope which has received a small increase in its angular velocity . The present study considers another limiting case, which corresponds to the viscous stage in the retardation of a fluid in a cylinder of low height after it has been instantaneously stopped, i.e., the problem is solved on the assumption of small Reynolds numbers for ¦/¦=1. A simple formula is obtained which describes the evolution of the tranverse circulation. The results are compared with experiment.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 19–25, January–February, 1985.The authors wish to thank A. M. Obukhov for suggesting the subject, E. M. Zhukhovitskii for constructive discussions, and N. Yu. Polunin for help in the layout of the illustrative material.     

Shock heating of plasma in a rapidly increasing magnetic field

The experimental excitation of intense collisionless shock waves (M 5) with subsequent plasma compression by the magnetic field of a shock coil is described. A magnetic plug > 20 kOe is produced in 100 × 10^–9 sec by a current generator, a long line with 250-kV water insulation and a characteristic impedance of l At an initial deuterium-plasma density of 2 × 10¹⁴ cm^–3, shock waves with a front width of 20c/₀₃and a velocity of 5 × 10₇ cm/sec are recorded. The ion energy after the accumulation, determined from the neutron yield, turns out to be 2 ke V. Axial shock waves excited by the plasma flow beneath the shock coil are observed.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, Vol. 11, No. 2, pp. 28–38, March–April, 1970.The authors thank G. I. Budker and R. Z. Sagdeev for formulating the problem, R. I. Soloukhin for interest in the study, and S. P. Shalamov for construction of the apparatus.     

Evolution of Domains of Attraction of a Forced Beam with Two-Mode Interaction

When a dissipative dynamical system has multiple attractors, it is a task to determine and recognize the global domain of attraction of each attractor. In this paper we study the global behavior of a forced hinged-clamped beam with two-mode interaction. The equation of motion of the beam is reduced to four first-order autonomous ordinary differential equations. The system has an internal resonance condition of 2 31, where 1 and 2 denote natural frequencies of the first and second modes, respectively. When the excitation frequency is near 1, the system can have three equilibrium solutions, among which two are asymptotically stable and one is unstable. We examine how the domains of attraction of two stable equilibrium solutions evolve as the forcing frequency is varied across jump points. By using a special plane which contains all equilibrium points, called the principal plane, the global domains of attraction can be discussed more effectively. Results show that knowledge of this evolution helps us better understand the jump phenomenon.     

Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentv , which is defined on the scale of measurement support, and a zero mean sub-field-scale componentv _s , which fluctuates randomly on scales smaller than. Without loss of generality, we work formally with unconditional statistics ofv _s and conditional statistics ofv . We then require that, within this (or other selected) working framework,v _s andv be mutually uncorrelated. This holds whenever the correlation scale ofv is large in comparison to that ofv _s . The formalism leads to an integro-differential equation for the conditional mean total concentration c which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration cc. We solve the former, and evaluate the latter, for mildly fluctuatingv by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformv . These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments c and cc of total concentrationc, which are associated with the scale below, cannot be used to estimate the field-scale concentrationc directly. To do so, a spatial average over the field measurement scale is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofc and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.     

The dynamic performance of the Weissenberg rheogoniometer III. Large amplitude oscillatory shearing — harmonic analysis

The harmonic content of the nonlinear dynamic behaviour of 1% polyacrylamide in 50% glycerol/water was studied using a standard Model R 18 Weissenberg Rheogoniometer. The Fourier analysis of the Oscillation Input and Torsion Head motions was performed using a Digital Transfer Function Analyser.In the absence of fluid inertia effects and when the amplitude of the (fundamental) Oscillation Input motion _I is much greater than the amplitudes of the Fourier components of the Torsion Head motion _Tn empirical nonlinear dynamic rheological propertiesG _n (, ⁰),G _n (, ⁰) and/or _n (, ⁰), _n (, ⁰) may be evaluated without a-priori-knowledge of a rheological constitutive equation. A detailed derivation of the basic equations involved is presented.Cone and plate data for the third harmonic storage modulus (dynamic rigidity)G ₃ (, ⁰), loss modulusG ₃ (, ⁰) and loss angle ₃ (, ⁰) are presented for the frequency range 3.14 × 10^–2 1.25 × 10² rad/s at two strain amplitudes, _CP ⁰ = 2.27 and 4.03. Composite cone and plate and parallel plates data for both the third and fifth harmonic dynamic viscosities ₃ (, ⁰), _S (, ⁰) and dynamic rigiditiesG ₃ (, ⁰),G ₅ (, ⁰) are presented for strain amplitudes in the ranges 1.10 _CP ⁰ 4.03 and 1.80 _PP ⁰ 36 for a single frequency, = 3.14 × 10^–1 rad/s. Good agreement was obtained between the results from both geometries and the absence of significant fluid inertia effects was confirmed by the superposition of the data for different gap widths.     

Numerical calculation of creep compliance from dynamic data for linear viscoelastic materials

Summary Numerical formulae are given for calculation of creep compliance from the known course of the storage and loss compliance with frequency for linear viscoelastic materials. These formulae involve values of the storage compliance and/or loss compliance at frequencies which are equally spaced on a logarithmic frequency scale. The ratio between successive frequencies corresponds to a factor of two.A method is introduced by which bounds for the relative error of those formulae can be derived. These bounds depend on the value of the damping, tan, at the angular frequency, ₀, at which the calculation is performed. The lower this damping, the easier is the calculation of the creep compliance. This calculation involves either the value of the storage compliance at a frequency ₀ = 1/t, and the values of the loss compliance in a rather narrow frequency region around ₀; or the value of the storage compliance at frequency ₀, the value of the loss compliance at frequency ₀/2, and the derivative of the storage compliance with respect to the logarithm of frequency in a frequency region around ₀.

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Zusammenfassung Numerische Formeln werden gegeben, die die Berechnung der Kriechfunktion aus der dynamischen Nachgiebigkeit ermöglichen. In diesen Formeln treten Werte der Speicher- bzw. Verlustkomponente der dynamischen Nachgiebigkeit auf, die bei logarithmisch äquidistanten Frequenzen gemessen wurden. Das Verhältnis zweier aufeinanderfolgender Frequenzen entspricht stets einem Faktor 2.Für alle Formeln werden obere und untere Schranken für den relativen Fehler abgeleitet. Diese Schranken hängen vom Werte der Dämpfung (tan) ab, die bei der Kreisfrequenz ₀ auftritt, für die die Berechnung erfolgt. Die Berechnung der Kriechfunktion ist desto leichter, je niedriger der Wert der Dämpfung ist. Zu dieser Berechnung benötigt man entweder den Wert der Speicherkomponente der dynamischen Nachgiebigkeit bei der Kreisfrequenz ₀ = 1/t und die Werte der Verlustkomponente der dynamischen Nachgiebigkeit in einem ziemlich engen Frequenzintervall um ₀; oder den Wert der Speicherkomponente bei der Kreisfrequenz ₀, den Wert der Verlustkomponente bei der Kreisfrequenz ₀/2 und den Wert der logarithmischen Frequenzableitung der Speicherkomponente in einem Frequenzintervall um ₀.

     

Conversion of longitudinal waves into electromagnetic radiation in a slowly varying plasma under W.K.B. approximation

This paper investigates the conversion of a dispersive longitudinal oscillation into reflected and transmitted electromagnetic radiation fields in slowly varying unmagnetized warm fluid plasmas, using W.K.B. approximations. The expressions for the power of the transmitted and reflected electromagnetic radiations, generated by electron acoustic waves, have also been obtained. It is shown that this conversion process becomes most efficient under certain conditions.

Nomenclature

In § 2 H magnetic field - H ₁ - u electron fluid velocity - k _t wave number of the transverse wave - k ₁ wave number of the longitudinal wave in electron fluid - m electronic mass - N ₀ number density of electrons in the unperturbed state - N perturbation in the electron number density - p perturbation in the electron fluid pressure - v _e adiabatic sound velocity of the electron fluid - K _t ² c ^{2 2}–e² - K ₁ ² v _e ^{2 2}–e² - wave frequency - _e electron plasma frequency - 1– _e ² / ² - c velocity of light in vacuum In § 3 K ₀ wave number in the 0X direction - K ₁ ² K ₁ ² –K ₀ ² - K ₂ ² K _t ² –K ₀ ² - K ₃ K ₁–K ₂ - K ₄ K ₁+K ₂ - K ₅ (K ₁ K ₂)^1/2 See Appendix A - A ₁ pressure amplitude of the reflected part of the incident wave - B ₁ pressure amplitude of the transmitted part of the incident wave - L characteristic length of variation ofN ₀ - e _x unit vector along 0X - e _z unit vector along 0Z In § 4 S _t Poynting flux of the transverse electromagnetic radiation - S _tZ ^/t Average of the transmitted part of the poynting flux along 0Z over the time period 2/ - S _tZ ^/r Average of the reflected part of the poynting flux along 0Z over the time period 2/ In § 5 S ₁ Energy flux carried by the longitudinal pressure wave - S _1Z ^/t Average of the transmitted part ofS ₁ along 0Z over the time period 2/     

Flow visualization and analysis of aerodynamically detuned supersonic rotors

The effect of aerodynamic detuning on the supersonic steady and unsteady blade passage flow field is experimentally investigated on a free surface water table by means of color Schlieren and shadowgraph flow visualization techniques. Two aerodynamic detuning mechanisms are considered: (1) alternate circumferential spacing of adjacent airfoils; (2) the replacement of alternate airfoils with splitters. The steady flow visualization demonstrates the significant effect of aerodynamic detuning on the passage flow field and, in particular, the shock wave-airfoil surface intersection locations. The unsteady flow visualization studies show the importance of the interblade phase angle. A mathematical model is also described and utilized to demonstrate the enhanced aeroelastic stability associated with the altered cascade passage shock wave structure due to these aerodynamic detuning mechanisms.List of symbols a dimensionless perturbation sonic velocity - C airfoil chord - I _x mass moment of inertia - k reduced frequency, k = C/2 - K spring constant - M _R dimensionless unsteady aerodynamic moment - M Mach number - P _split split splitter circumferential spacing - P _start splitter leading edge location - R reference full chord airfoil - R _s reference splitter - Sp _chord splitter chord length - u dimensionless perturbation chordwise velocity - v dimensionless perturbation normal velocity - amplitude of oscillation - interblade phase angle - level of aerodynamic detuning - undamped natural torsional frequency - ₀ reference frequency - flutter frequency     

The linearization of certain class of nonlinear simultaneous equation set containing amplitude and phase frequency characteristics and application

A class of complex function of rational fraction type is frequently used to describe the dynamical properties of systems. It is however quite difficult to establish a mathematical model of this type on the basis of amplitude and phase frequency data collected from experiments conducted on the related physical system. Since the erection of mathematical model G(j) would involve the solution of a set of nonlinear simultaneous equations with the unknown coefficients a_is and b_is(i=0, 1, ..., m, ..., n) in G(j). Up to now, these nonlinear equations have been considered to be very difficult to solve directly. In spite of the fact there are special computer programmes in certain software packages available to tackle this problem, it is by no means an easy task due to the complex procedures involved in picking up a set of initial values that should be close enough to the exact solutions. This paper proposes a simplified method of linearizing these nonlinear equations set so that direct solution is possible. The method can also be applied to systems with factors of (j) andej0 in G(j). An illustration by a workable example is furnished at the end of this paper to show its versatility.     

Primary and secondary bifurcation in baroclinic instability

In modelling atmospheric flows the baroclinic instability of the flow in a differentially heated rotating annulus plays a central role. This paper deals with an experimental study using LDV and flow visualization techniques. Usually the temperature difference, T, was kept fixed while the angular velocity, , was varied. On crossing the stability boundary, the primary bifurcation, the basic flow gives way to a baroclinic wave flow. For a given annulus geometry the wave number, m, of the first wave pattern was found to be uniquely defined by T. The measured critical values of , _crit, agree reasonably well with those obtained by other authors. On increasing above _crit the wave number changed, this process showing hysteresis. The situation might indicate secondary bifurcation phenomena. Flow visualization using aluminium particles shows surface flow details.This paper is dedicated to Prof. Dr. K. Gersten on the occasion of his 60th birthday     

Nonlinear theory of the propagation of pulses of electromagnetic waves through a plasma boundary

A solution is obtained for the problem of the propagation of electromagnetic waves of arbitrary form through a plasma boundary on condition that the length of the wave train is much greater than the wave length. A solution is found both for the case of a wide spectrum of width much greater than the plasma frequency ₀, as well as for a narrow spectrum. The results obtained enable us to draw conclusions about the time and space variation of the shape of electromagnetic pulses in a plasma.The passage of high frequency electromagnetic waves through a plasma is similar to that of a beam of charged particles [1, 2]. This is associated with the fact that decay processes are similar to Cerenkov radiation effects. The dynamics of the development of transverse wave instabilities in a uniform Isotropic plasma were studied in [2] assuming that the wave phase behaves stochastically. It was calculated here that instabilities develop quite differently in the case of a wide frequency spectrum than in the case of a narrow monochromatic spectrum. If we can speak of transverse quanta diffusion effects in the field of the generated longitudinal quanta in the first case, and if the resulting effects are closely similar to the nonlinear effects arising when beam instability develops [3, 4], then the development of instabilities in the case of a narrow spectrum leads to the appearance of red satellites in the transverse wave spectrum differing from the basic frequency by a quantity ₀ (=1, 2, 3,...). In this case the development of the instability corresponds to a tendency for a plateau over the satellites to appear.Attention should however be drawn to the fact that the dynamics of instability development in a semibounded plasma may be quite different. This is associated first with the different values of group velocities of transverse and longitudinal waves, and what is also important, with the effect of longitudinal wave accumulation in the boundary region if the length of the wave train is sufficiently large. The treatment of a similar problem for beam instabilities in paper [5] showed that a narrow transition layer may arise with a transverse wave energy density greatly in excess of the energy density of the injected beam. In what follows we examine the part played by boundary effects in the passage of pulses of electromagnetic waves through the boundary of the plasma. The cases of both narrow and wide spectra are considered. We note that in the case of narrow spectra the wave train must necessarily be greatly in excess of ^–1, and the effects of the accumulation of oscillations will be appreciable.The phases of both transverse waves, and also generated longitudinal waves are assumed to be stochastic quantities. The boundary effects which have been treated may be applied both in the generation of longitudinal waves necessary for the effective acceleration of particles in a plasma as well as in the modulation and alteration of the initial transverse wave spectrum. It should also be stressed that these effects which have been considered could be applied for turbulent plasma diagnostics, as has already been pointed out in [2].The authors are grateful to Ya. B. Fainberg, M. S. Rabinovich, I. S. Danilkin, and M. D. Raizer for their interest in the paper and for valuable criticisms.     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

Normal forms for random diffeomorphisms

Given a dynamical system (,, ,) and a random diffeomorphism (): ^{d d} with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h() to make the random diffeomorphism ()=h()^–1() h() as simple as possible, preferably linear. The linearization D(, 0)=:A() generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptof-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.     

Particle-in-a-box study of the evolution of an ion-acoustic shock wave

In an analysis of a one-dimensional numerical model of a nonisothermal plasma it is shown that an ion-acoustic shock wave of subcritical amplitude separates a soliton from the shock front after the reversing stage. This process is accompanied by turbulent flow behind the front and by trapping of ions in potential wells. The numerical particle-in-a-box method is being used widely to study plasma phenomena. One field in which this method has been found fruitful is in the study of a nonisothermal plasma, characterized by an ion-acoustic wave branch.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 3–5, May–June, 1971.The authors thank R. Z. Sagdeev for support and interest in this study.