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Dec 1979

Volume 22, Issue 12, pp. 2259-2434

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Cavitation of a bubble in an inviscid compressible liquid, with comparisons to a viscous incompressible liquid

G. J. Lastman and R. A. Wentzell

Phys. Fluids 22, 2259 (1979); http://dx.doi.org/10.1063/1.862534 (8 pages) | Cited 4 times

Online Publication Date: 6 August 2008

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A model of a bubble, consisting of an adiabatic gas surrounded by an inviscid compressible liquid with surface tension at the liquid‐gas interface, is studied in order to explain experimental results not explained (or not adequately explained) by the viscous incompressible liquid model. The inviscid compressible liquid model is used to predict various characteristics of cavitating and noncavitating bubbles. In particular, compressibility accounts for the rapid damping of bubble oscillations after the cavitation phase; an upper bound is determined for the pressure produced by cavitation; the sharp pressure spike at the initiation of cavitation is predicted; the threshold bubble radius for the onset of cavitation is estimated; and the range of bubble radii is estimated, for which the tension pulse is reproduced for certain noncavitating conditions. In many instances comparisons are made between the viscous incompressible liquid model and the inviscid compressible liquid model.
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43.25.Yw Nonlinear acoustics of bubbly liquids
47.55.dp Cavitation and boiling

Deceleration of a rotating disk in a viscous fluid

Layne T. Watson and Chang‐Yi Wang

Phys. Fluids 22, 2267 (1979); http://dx.doi.org/10.1063/1.862535 (3 pages) | Cited 11 times

Online Publication Date: 6 August 2008

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A disk rotating in a viscous fluid decelerates with an angular velocity inversely proportional to time. It is found that the unsteady Navier–Stokes equations admit similarity solutions which depend on a nondimensional parameter S =α/Ω0, measuring unsteadiness. The resulting set of nonlinear ordinary differential equations is then integrated numerically. The special case of S =−1.606 699 corresponds to the decay of rotation of a free, massless disk in a viscous fluid.
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47.15.-x Laminar flows
47.15.Cb Laminar boundary layers
47.32.Ef Rotating and swirling flows

Surface tension restoring forces on gravity waves in narrow channels

David Heckerman, Steven Garrett, Gary A. Williams, and Patrick Weidman

Phys. Fluids 22, 2270 (1979); http://dx.doi.org/10.1063/1.862536 (7 pages) | Cited 2 times

Online Publication Date: 6 August 2008

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Surface tension restoring forces are investigated for low amplitude gravity waves propagating in narrow channels. Liquids which do not wet the container walls experience a nonuniform displacement of the meniscus as the wave passes. The variation of surface curvature leads to a surface tension force which increases the velocity of the gravity wave. The effect is substantial in experiments with water in Plexiglass channels. The dependence on channel width and contact angle has been investigated, and agreement is found with a simple theoretical model. Addition of a wetting agent to the water eliminates the effect, decreasing the velocity to within 1% of the classical dispersion relation.
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47.35.-i Hydrodynamic waves
68.03.Cd Surface tension and related phenomena
47.60.-i Flow phenomena in quasi-one-dimensional systems

Relations between point sources buoyant convection phenomena

D. J. Shlien

Phys. Fluids 22, 2277 (1979); http://dx.doi.org/10.1063/1.862537 (7 pages) | Cited 7 times

Online Publication Date: 6 August 2008

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The gross convection velocities of various simple point source convective phenomena in an unstratified stationary fluid can be related to each other through empirical constants by using Turner’s model of the starting plume cap. Considered here are ’’weak’’ (Rayleigh number approaching zero), laminar, and turbulent regimes of the phenomena, the thermal, the starting plume cap, and the steady plume. Thus, by determining an empirical constant a, for either the thermal or the starting plume cap, the convection velocity of the other phenomenon can be estimated. Here, a is defined in terms of the thermal, azt2/ktAm, where zt is the thermal height above the buoyancy source at time t, k is a molecular diffusivity, A is the effective Rayleigh number, and m=1/2, 1, or 2 depending on the flow regime. The dependence of a on the Prandtl number Pr is presently unknown. For experimental verification of the model, new measurements of the laminar and of the turbulent starting plume are presented. The constant a, determined independently for the thermal and for the starting plume cap was within 20% of the mean value of 0.038 for the laminar case, while for the turbulent case the constant was within 13% of the mean value of 4.2. The new experimental data further reveals that a change in Prandtl number from 21 to 7 results in a 50% increase in a for the laminar case.
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47.27.T- Turbulent transport processes
47.15.-x Laminar flows
92.60.hk Convection, turbulence, and diffusion

Total dispersion of a scalar quantity in turbulent flow

R. Chevray and K. S. Venkataramani

Phys. Fluids 22, 2284 (1979); http://dx.doi.org/10.1063/1.862538 (5 pages) | Cited 1 time

Online Publication Date: 6 August 2008

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A general approach to the problem of the dispersion of a scalar quantity in a turbulent flow is given. The essential feature of this formulation is the consideration of the motion of isoconcentration surfaces. Use of this continuum kinematic concept, while preserving the exact results for both the limiting cases of zero molecular diffusivity and no fluid motion, enables the study of the dispersion of scalar quantities, like heat, without the use of physical concepts which are inconsistent with the continuum field description. It is shown that the effect of the interaction of the molecular and turbulent diffusion processes is such that the total dispersion of the scalar is less than the sum of the individual dispersions due to fluid motion alone and the direct effect of molecular diffusivity. Of the two terms arising from the interaction, the first one obtained by a simple approximation to the position field is identical to that obtained by an entirely different method. The additional term which was not found in the earlier studies varies as the square of the time. A preliminary extension to the dispersion of a specie of negligible molecular diffusivity undergoing a first‐order chemical reaction shows that this method can also be extended to other related problems.
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47.27.T- Turbulent transport processes
47.10.-g General theory in fluid dynamics

On soliton amplification

S. Leibovich and J. D. Randall

Phys. Fluids 22, 2289 (1979); http://dx.doi.org/10.1063/1.862539 (7 pages) | Cited 6 times

Online Publication Date: 6 August 2008

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A modified Korteweg–de Vries equation that permits wave amplification or damping is considered. A ’’terminal similariry’’ solution is identified for large times in amplified systems. This asymptotic result is confirmed by a series of numerical computations.
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47.35.-i Hydrodynamic waves

Confinement of a plasma column by ablation fronts from cold gas plugs

Wolfgang Liese, Boye Ahlborn, and Bruce Armstrong

Phys. Fluids 22, 2296 (1979); http://dx.doi.org/10.1063/1.862540 (4 pages) | Cited 3 times

Online Publication Date: 6 August 2008

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A cylindrical plasma column of about 1–2 eV temperature and ne ≈ 1016 cm−3 is confined and compressed in the axial direction by an ablation front supported with a specific power in the range W1 = 108–5×109 W cm/g. Pressure and particle velocity are measured as a function of W1 and compared with model predictions.
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52.30.-q Plasma dynamics and flow
52.58.-c Other confinement methods

Self‐similar expansion of a plasma into a vacuum

Patrick Mora and R. Pellat

Phys. Fluids 22, 2300 (1979); http://dx.doi.org/10.1063/1.862541 (5 pages) | Cited 65 times

Online Publication Date: 6 August 2008

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The Boltzmann equilibrium for electrons has been an adhoc assumption of previous work about self‐similar plasma expansion in vacuum. Using Boltzmann’s relation, one finds a linear electrostatic potential as a solution of the self‐similar expansion. It is first shown that in such a potential, the electron behavior is exactly solvable. As a result, one finds that the exact electron density differs substantially from the Boltzmann equilibrium for moderate values of the self‐similar parameter. The complete calculations of the self‐similar expansion, without any assumption about the equation of state of the electrons, is presented. It is shown, in particular, that half the electron density depletion in logarithmic units is compensated for by the self‐consistent modification of the electric potential. Finally, the effect is analyzed in terms of energy exchange between the electrons and the ions in the expansion.
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52.30.-q Plasma dynamics and flow
52.50.Jm Plasma production and heating by laser beams (laser-foil, laser-cluster, etc.)
52.20.Dq Particle orbits

Theory of strongly turbulent two‐dimensional convection of low‐pressure plasma

R. N. Sudan and Michael J. Keskinen

Phys. Fluids 22, 2305 (1979); http://dx.doi.org/10.1063/1.862533 (10 pages) | Cited 24 times

Online Publication Date: 6 August 2008

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The ’’direct interaction approximation’’ of Kraichnan as modified by Kadomtsev is employed to develop a strong two‐dimensional turbulence theory which predicts both nonlinear frequency broadening and a power law for the spectrum of a convecting plasma. These results are compared both with experimental observations and numerical simulations.
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52.35.Ra Plasma turbulence
52.65.-y Plasma simulation
94.20.Vv Ionospheric disturbances, irregularities, and storms
94.20.dg E region

Kinetic theory of evolution of anisotropic plasmas

Young‐ping Pao

Phys. Fluids 22, 2315 (1979); http://dx.doi.org/10.1063/1.862542 (9 pages) | Cited 1 time

Online Publication Date: 6 August 2008

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A kinetic theory is constructed to evaluate the evolution and motion of an anisotropic axisymmetric plasma caused by collisional effects and pressure anisotropy. The relationship to neoclassical transport theory is also discussed.
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52.25.Dg Plasma kinetic equations
52.25.Fi Transport properties
52.55.Dy General theory and basic studies of plasma lifetime, particle and heat loss, energy balance, field structure, etc.
52.55.Fa Tokamaks, spherical tokamaks
52.55.Hc Stellarators, torsatrons, heliacs, bumpy tori, and other toroidal confinement devices

Vlasov‐fluid stability of a rigidly rotating theta pinch

C. E. Seyler

Phys. Fluids 22, 2324 (1979); http://dx.doi.org/10.1063/1.862543 (7 pages) | Cited 31 times

Online Publication Date: 6 August 2008

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The stability of a rigidly rotating theta pinch is investigated using the Vlasov‐fluid model. The main differences between the present and the asymptotic theory are due to the effect of resonant ions present in the exact Vlasov‐fliud equations. An analytical explanation of the resonant ion effects is given.
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52.55.Ez Theta pinch
52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)
52.30.-q Plasma dynamics and flow

Parametric instabilities and ion heating in a two ion‐species plasma

C. C. Lin, A. T. Lin, and H. Okuda

Phys. Fluids 22, 2331 (1979); http://dx.doi.org/10.1063/1.862544 (3 pages) | Cited 6 times

Online Publication Date: 6 August 2008

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Parametric instabilities and the associated ion heating in the presence of two‐ion species have been studied using a one‐dimensional, quasi‐neutral particle simulation model. At an early stage the heating is due to trapping or resonance heating of the decay waves. When the average Larmor radius reaches half the excited electrostatic wavelength, the heating becomes stochastic.
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52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)
52.50.Gj Plasma heating by particle beams
52.65.-y Plasma simulation

Production and flow of plasma in ion beams

E. B. Hooper, O. A. Anderson, and P. A. Willmann

Phys. Fluids 22, 2334 (1979); http://dx.doi.org/10.1063/1.862545 (12 pages) | Cited 6 times

Online Publication Date: 6 August 2008

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Steady plasma flow is analyzed for the case where a positive‐ or negative‐ion beam penetrates and ionizes background gas (or vapor). The work of previous authors is extended to treat nonuniform beam densities that fall to low values at the transverse plasma boundary, and to include gas densities that are nonuniform along the beam axis. These nonuniformities give a two‐dimensional plasma flow problem that is analyzed using a fluid model. The effect of unusually low gas density on the plasma flow is discussed; for negative‐ion beams, the electron density becomes much smaller than the beam density and space‐charge neutralization is maintained by a reduction in plasma flow. When there is a large local gas density, as in a gas cell, the resulting high electron density is also localized to the same region by the nature of the flow.
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52.30.-q Plasma dynamics and flow
29.27.Eg Beam handling; beam transport
29.25.Bx Electron sources
52.40.Mj Particle beam interactions in plasmas

Axial eigenmodes for long −λ waves in plasmas bounded by sheaths

Francis F. Chen

Phys. Fluids 22, 2346 (1979); http://dx.doi.org/10.1063/1.862546 (13 pages) | Cited 8 times

Online Publication Date: 6 August 2008

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Waves such as drift waves and lower hybrid oscillations in a plasma are sensitive to the degree to which charge neutrality can be maintained by electron flow along the magnetic field. When the plasma is bounded axially, the sheath conditions on the end plates determine the parallel wavelength. It is found that the nature of the bounded modes depends on whether the motion of electrons is resistive or inertial. If it is resistive, the sheath matching conditions can be satisfied by standing waves with the proper wavelength. If it is inertial, pure standing waves are not possible; there must also be a variation of phase along B. Application is made to two‐ion hybrid waves in connection with isotope separation.
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52.35.Fp Electrostatic waves and oscillations (e.g., ion-acoustic waves)
52.35.Kt Drift waves
52.40.Kh Plasma sheaths
52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)

Shape stability for a plasma contained within a standing electromagnetic field

Donald L. Ensley

Phys. Fluids 22, 2359 (1979); http://dx.doi.org/10.1063/1.862547 (5 pages)

Online Publication Date: 6 August 2008

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For an over‐dense plasma located within a resonant cavity and surrounded by standing electromagnetic radiation, it is shown that shape stability depends only on the ratio of the mean plasma radius (ra) to the wavelength of the surrounding radiation. For an initial cavity mode wavenumber k10, stability occurs for k10 ra≲π. Becausetheplasma‐surfaceoscillation frequency νλ is shown to scale linearly with λ (the order of the surface wave), νλ remains small compared with the electromagnetic field frequency for stable plasmas. These results are of significance in designing inertial fusion experiments in which intense electromagnetic fields are used for heating and compressing a dense plasma.
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52.35.Hr Electromagnetic waves (e.g., electron-cyclotron, Whistler, Bernstein, upper hybrid, lower hybrid)
52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)

Stabilization of trapped‐electron shear‐Alfvén instabilities by temperature gradient

David W. Ross, Swadesh M. Mahajan, R. D. Hazeltine, and H. R. Strauss

Phys. Fluids 22, 2364 (1979); http://dx.doi.org/10.1063/1.862548 (3 pages) | Cited 3 times

Online Publication Date: 6 August 2008

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Localized shear‐Alfvén modes with large m numbers are shown, numerically, to be strongly damped by the collisionless electron response in the presence of a temperature gradient. The trapped‐electron drift‐tearing instability is stabilized by this effect in a tokamak unless the inverse aspect ratio exceeds a critical value, typically between 0.1 and 0.2. Analytical models demonstrate the scaling of these results with plasma parameters.
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52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)
52.35.Kt Drift waves

Wave‐particle interaction in the late beam‐plasma instability

K. B. Freese, J. E. Walsh, and John Lohr

Phys. Fluids 22, 2367 (1979); http://dx.doi.org/10.1063/1.862549 (10 pages) | Cited 5 times

Online Publication Date: 6 August 2008

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Experiments are reported on the beam‐plasma instability at times after saturation when the electron distribution function exhibits a high energy tail. The results show that high energy electrons are produced by a velocity resonant wave‐particle interaction. The experiments were formed on quite different devices at Dartmouth College and The University of Texas; however, the results from the two laboratories are consistent.
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52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)
52.40.Mj Particle beam interactions in plasmas
52.50.Gj Plasma heating by particle beams

Diocotron instability of a relativistic hollow electron beam

Han S. Uhm and John G. Siambis

Phys. Fluids 22, 2377 (1979); http://dx.doi.org/10.1063/1.862550 (5 pages) | Cited 15 times

Online Publication Date: 6 August 2008

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The diocotron stability properties of a relativistic hollow electron beam are investigated within the framework of the macroscopic, cold fluid model, in which the electron motion is assumed to be laminar. The stability analysis is carried out for the case of a sharp boundary density profile, including the important influence of relativistic effects and fractional charge neutralization on the stability behavior. It is found that the growth rate of instability is proportional to 1/γb2, thereby being reduced considerably by increasing the electron energy γbmc2. Moreover, the fractional charge neutralization plays a significant role in the stability behavior, particularly for the relativistic electron beam.
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52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)
52.27.Ny Relativistic plasmas
52.40.Db Electromagnetic (nonlaser) radiation interactions with plasma
52.35.Hr Electromagnetic waves (e.g., electron-cyclotron, Whistler, Bernstein, upper hybrid, lower hybrid)

Linear and nonlinear theory of ballooning trapped electron mode

André Rogister and Günter Hasselberg

Phys. Fluids 22, 2382 (1979); http://dx.doi.org/10.1063/1.862551 (7 pages) | Cited 7 times

Online Publication Date: 6 August 2008

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The linear and nonlinear theories of the trapped electron mode are developed in the limit where the modes overlap many rational surfaces. In cases where shear stabilization is annihilated by toroidal coupling, it is found that the radial width of the mode increases with increasing level of turbulence until ion Landau damping balances the destabilizing effect of trapped electrons. Knowledge of the turbulence level enables the evaluation of the anomalous particle and energy transport. The diffusion coefficient scales with temperature, magnetic field, and density as the superbanana diffusion coefficient due to field asymmetries. Radial localization of the destabilizing trapped electron term caused by the difference between the pitch of the magnetic field and that of the mode results in a reduction of the linear growth rate of order Δ/Xt, where Δ is the spacing between rational surfaces and Xt is the radial wavelength.
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52.35.-g Waves, oscillations, and instabilities in plasmas and intense beams
52.35.Kt Drift waves
52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)

Poloidal mode ballooning of trapped electron instability

S. R. Pandey and David W. Ross

Phys. Fluids 22, 2389 (1979); http://dx.doi.org/10.1063/1.862552 (5 pages) | Cited 3 times

Online Publication Date: 6 August 2008

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The effect of the magnetic moment dependence of the bounce averaged toroidal drift velocity on the eigenvalues and poloidal structure of the trapped electron mode is investigated in the radially local model. It is found that eigenvalues from the magnetic moment dependent dispersion relation deviate significantly from those of the constant drift velocity approximation for negative and large positive shear values. However, for shear ranging from rq−1dq/dr=0.5 to 1.5, the trapped electron magnetic drift frequency can be approximated by ωdew (0.4 + 0.8rq−1dq/dr), where ωde is a constant, w is the particle kinetic energy, and q is the safety factor.
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52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)
52.55.Fa Tokamaks, spherical tokamaks
52.55.Hc Stellarators, torsatrons, heliacs, bumpy tori, and other toroidal confinement devices

Observations of a highly collisional toroidal plasma equilibrium

A. W. Allen, J. A. Antoniades, and G. C. Goldenbaum

Phys. Fluids 22, 2394 (1979); http://dx.doi.org/10.1063/1.862553 (6 pages)

Online Publication Date: 6 August 2008

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The time evolution of temperature and density profiles in a finite beta, noncircular cross section (b/a=2) toroidal plasma (R0/a=2) has been measured. Measurements show ion temperatures decreasing as the density is increasing along the major radius at the z=0 midplane. For these conditions, the electron temperature remains approximately uniform over the cross section. A recent theoretical model for highly collisional circular cross section plasma predicts qualitatively similar temperature and density profiles which are different from those for the less collisional Pfirsch–Schlüter regime.
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52.55.Fa Tokamaks, spherical tokamaks
52.55.Hc Stellarators, torsatrons, heliacs, bumpy tori, and other toroidal confinement devices
52.25.Kn Thermodynamics of plasmas
52.50.Lp Plasma production and heating by shock waves and compression

High beta stellarator stability theory

Michael J. Schmidt

Phys. Fluids 22, 2400 (1979); http://dx.doi.org/10.1063/1.862554 (8 pages) | Cited 2 times

Online Publication Date: 6 August 2008

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An analytic study of the stability of a straight diffuse high beta stellarator with arbitrary wall corrugation is presented. It is found that a solvability condition for the equilibrium of such a plasma is intimately related to a sufficient condition for stability. The results of the calculation suggest that if the equilibrium model is valid, then all high beta stellarators are unstable.
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52.55.Dy General theory and basic studies of plasma lifetime, particle and heat loss, energy balance, field structure, etc.
52.55.Fa Tokamaks, spherical tokamaks
52.55.Hc Stellarators, torsatrons, heliacs, bumpy tori, and other toroidal confinement devices
52.55.Pi Fusion products effects (e.g., alpha-particles, etc.), fast particle effects

Ponderomotive effects and magnetic field generation in radiation plasma interaction

Patrick Mora and R. Pellat

Phys. Fluids 22, 2408 (1979); http://dx.doi.org/10.1063/1.862555 (10 pages) | Cited 57 times

Online Publication Date: 6 August 2008

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A kinetic theory of ponderomotive effects and magnetic field generation is presented. For a comparison with standard fluid theories, explicit equations for momentum and energy conservations are established. As an example, it is shown that collisions do not lead to the usual resistive drag. In the important case of magnetic field generation via resonant absorption, three regimes with different behavior are found. In the collision‐dominated regime of the parameters, the main source of the magnetic field is the off‐diagonal terms of the particle pressure tensor. An intermediate regime appears when the nonlinear effects are collisionless and the amplitude of electromagnetic field due to the resonance absorption mechanism is limited by the plasma resistivity. In the collisionless regime, the validity of published results is extended by a self‐consistent nonlinear calculation. As a result, the distribution function of electrons may include suprathermal tails, and the magnetic field generated does not appear explicitly in the formula.
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52.40.Db Electromagnetic (nonlaser) radiation interactions with plasma
52.25.Dg Plasma kinetic equations
52.30.-q Plasma dynamics and flow
52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)

Transport and radiation studies of belt pinches and high‐beta tokamaks

R. N. Byrne and C. K. Chu

Phys. Fluids 22, 2418 (1979); http://dx.doi.org/10.1063/1.862532 (6 pages) | Cited 1 time

Online Publication Date: 6 August 2008

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Transport and radiation effects on eliptical cross‐section pinch machines have been performed using the G2m code. Profile effects are shown to be small for machines of the general parameters of Torus I at Columbia and significant for those like Torus II. An explanation of toroidal paramagnetism is advanced.
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52.55.Fa Tokamaks, spherical tokamaks
52.55.Hc Stellarators, torsatrons, heliacs, bumpy tori, and other toroidal confinement devices
52.25.Fi Transport properties
52.65.-y Plasma simulation

Magnetosheath effects on cylindrical Langmuir probes

Edward P. Szuszczewicz and Peter Z. Takacs

Phys. Fluids 22, 2424 (1979); http://dx.doi.org/10.1063/1.862556 (6 pages) | Cited 20 times

Online Publication Date: 6 August 2008

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The response of cylindrical Langmuir probes in magnetoplasmas is studied from a perspective which focuses on the relative magnitudes of Larmor radius and sheath size. The approach results in a classification for magnetic field effects which involves not only the magnetic field strength but also the plasma parameters of density, temperature, and the applied probe potential. It is specifically shown that a 0.25 G field can have similar effects on the current collection properties of the probe in an ionospheric plasma (Ne≈106 cm−3) as a 30 kG field would have in a hot, dense laboratory plasma (Ne≈1015 cm−3). The classifications are found to agree with new experimental results collected in an ionospheric plasma. The data also show: (a) the effects of probe orientation on electron current collection from magnetoplasmas; (b) that these effects can be important even when the electron Larmor radius is larger than the radius of the probe; and (c) that substantial magnetic field effects occur when the probe’s sheath is comparable to or greater than the Larmor radius.
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52.70.Ds Electric and magnetic measurements
52.40.Kh Plasma sheaths
52.25.Kn Thermodynamics of plasmas
94.30.cj Magnetosheath
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