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Jan 1986

Volume 29, Issue 1, pp. 3-341

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A model for estimating transport quantities in two‐phase materials

Andreas Acrivos and Eric Chang

Phys. Fluids 29, 3 (1986); http://dx.doi.org/10.1063/1.866018 (2 pages) | Cited 22 times

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A novel technique is presented for estimating the various transport quantities in two‐phase materials of random structure by extending, in an approximate manner, the effective continuum description of such systems. The method is then applied to three representative examples and is found to give results in excellent agreement with those obtained from exact calculations.
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05.70.Ce Thermodynamic functions and equations of state

A nonlinear stability theorem for baroclinic quasigeostrophic flow

Gordon E. Swaters

Phys. Fluids 29, 5 (1986); http://dx.doi.org/10.1063/1.865953 (2 pages) | Cited 10 times

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The baroclinic quasigeostrophic equations describe the essential dynamics of large‐scale, low‐frequency atmospheric ocean flow. A nonlinear stability theorem is given based on a convexity argument of Arnold [Am. Math. Soc. Transl. 19, 267 (1969)], complementing a linear analysis by Blumen [J. Atmos. Sci. 25, 929 (1968)]. An a priori estimate bounding the growth of perturbation is derived.
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47.20.Ky Nonlinearity, bifurcation, and symmetry breaking
47.35.-i Hydrodynamic waves
92.10.Hm Ocean waves and oscillations
92.60.Gn Winds and their effects

Hamiltonian formulation of the baroclinic quasigeostrophic fluid equations

Darryl D. Holm

Phys. Fluids 29, 7 (1986); http://dx.doi.org/10.1063/1.865956 (2 pages) | Cited 15 times

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A Hamiltonian formulation using a noncanonical Poisson bracket is presented for the nonlinear dynamics of baroclinic quasigeostrophic fluid flow. Nonlinear integral invariants for the system are found to be in the kernel of the noncanonical Poisson bracket. A novel feature of this formulation is that dynamical boundary conditions are generated from the constrained energy Hamiltonian, via the noncanonical Poisson bracket.
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47.55.Hd Stratified flows
92.10.Hm Ocean waves and oscillations
92.60.Gn Winds and their effects

Thermal nonequilibrium hypersonic shock layer near the stagnation point

S. M. Yen

Phys. Fluids 29, 9 (1986); http://dx.doi.org/10.1063/1.865958 (2 pages) | Cited 1 time

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At high altitudes, the condition in the shock layer near the stagnation point deviates significantly from thermal equilibrium. This condition is comparable to that in a relatively strong shock wave. The shock layer is, therefore, a Knudsen layer. Together with the strong shock wave preceding the shock layer, the hypersonic flow problem near a stagnation point is that of a triple Knudsen layer.
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05.20.Dd Kinetic theory
05.70.Ln Nonequilibrium and irreversible thermodynamics
47.40.Ki Supersonic and hypersonic flows
52.35.Tc Shock waves and discontinuities

Thermal equilibrium of a cryogenic magnetized pure electron plasma

Daniel H. E. Dubin and T. M. O’Neil

Phys. Fluids 29, 11 (1986); http://dx.doi.org/10.1063/1.865987 (3 pages) | Cited 12 times

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The thermal equilibrium correlation properties of a magnetically confined pure electron plasma (McPEP) are related to those of a one‐component plasma (OCP). The N‐particle spatial distribution ρs and the Helmholtz free energy F are evaluated for the McPEP to O(λ2d/a2), where λd is the thermal de Broglie wavelength and a is an interparticle spacing. The electron gyromotion is allowed to be fully quantized while the guiding center motion is quasiclassical. The distribution ρs is shown to be identical to that of a classical OCP with a slightly modified potential. To O(λ2d/a2) this modification does not affect that part of F which caused by correlations, as long as certain requirements concerning the size of the plasma are met. This theory is motivated by a current series of experiments that involve the cooling of a magnetically confined pure electron plasma to the cryogenic temperature range.
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52.25.Fi Transport properties
05.30.Fk Fermion systems and electron gas

Transport processes in fractals. VI. Stokes flow through Sierpinski carpets

P. M. Adler

Phys. Fluids 29, 15 (1986); http://dx.doi.org/10.1063/1.865971 (8 pages) | Cited 11 times

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The Stokes flow of a Newtonian fluid is calculated inside a porous medium that is spatially periodic, the unit cell being a fractal named a Sierpinski carpet. Complete results are given for the longitudinal permeability. A scaling argument and complete numerical calculations provide two exponents of the power law that differ by only 2% when the construction stage is large; in this limit, the scaling argument provides the same result as the classical Carman equation. The agreement between these two results may be fortuitous and thus has to be considered with caution. Various comments and extensions to three‐dimensional media such as the Menger sponge are also presented.
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47.56.+r Flows through porous media
47.15.-x Laminar flows

Wavenumber selection in large‐amplitude axisymmetric convection

Jeffrey C. Buell and Ivan Catton

Phys. Fluids 29, 23 (1986); http://dx.doi.org/10.1063/1.865980 (8 pages) | Cited 19 times

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Unique wavenumbers are calculated for axisymmetric Rayleigh–Bénard convection as a function of the Rayleigh number (R) up to the second critical value for several different Prandtl numbers. The analysis assumes slightly bent rolls (large radius of curvature) and that there exists a horizontal pressure gradient strong enough to force the net mean flow induced by curvature to be zero. The assumptions are satisfied for axisymmetric convection in a large aspect ratio cylinder (however, this may not be the only case). Manneville and Piquemal [Phys. Rev. A 28, 1774 (1983)] found the initial slope of the selected wavenumber with respect to Rayleigh number (using an analytic solution valid for small amplitude solutions) and our calculations agree with theirs. This initial slope is sensitive to the Prandtl number (P), but at moderate to large R the selected wavenumber is approximately independent of P when P>3. For smaller P larger wavenumbers are found, but this does not contradict any available experimental evidence.
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47.27.T- Turbulent transport processes
47.20.-k Flow instabilities
02.60.-x Numerical approximation and analysis

Internal solitary waves and their head‐on collision. II

Rida M. Mirie and C. H. Su

Phys. Fluids 29, 31 (1986); http://dx.doi.org/10.1063/1.865944 (7 pages) | Cited 3 times

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The head‐on collision between two modified Korteweg–de Vries (MKdV) solitary waves is investigated where cubic and quadratic nonlinearities balance dispersion. These waves propagate at the interface of an inviscid two‐fluid system where the ratio of the fluids densities is comparable to the square of the ratio of depths. The third‐order perturbation solution is obtained and it is found that the collision is inelastic because of a dispersive wave train generated behind each emergent solitary wave.
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47.35.-i Hydrodynamic waves
47.55.Hd Stratified flows
68.03.-g Gas-liquid and vacuum-liquid interfaces
68.05.-n Liquid-liquid interfaces

Calculation of swirling jets with a Reynolds stress closure

M. M. Gibson and B. A. Younis

Phys. Fluids 29, 38 (1986); http://dx.doi.org/10.1063/1.865951 (11 pages) | Cited 40 times

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The transport equations for the Reynolds stresses are closed by modeling the turbulence and mean‐strain parts of the pressure‐strain‐rate correlation. The model constants are determined from simple relationships deduced from measurements in rectilinear and longitudinally curved shear flows. It is found that the effects of complex strain fields are more correctly predicted when the influence of the mean‐strain part is reduced from levels indicated by rapid distortion theory, and the turbulence part is adjusted to conform approximately with the measured rates of return to isotropy. The case of the swirling jet is used to illustrate the improved performance of the model.
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47.27.N- Wall-bounded shear flow turbulence
47.27.T- Turbulent transport processes
47.32.Ef Rotating and swirling flows
47.27.W- Boundary-free shear flow turbulence

Lyapunov stability of relativistic fluids and plasmas

Darryl D. Holm and Boris A. Kupershmidt

Phys. Fluids 29, 49 (1986); http://dx.doi.org/10.1063/1.865952 (20 pages) | Cited 13 times

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Lyapunov stability of relativistic ideal fluid and plasma equilibria is studied analytically using the energy‐Casimir method. Two‐ and three‐dimensional relativistic equilibria in a fixed bounded domain are investigated within the framework of the macroscopic multifluid plasma model. Linearized Lyapunov stability conditions and stability norms are given, accounting for warm‐plasma effects as well as relativistic and electromagnetic effects. The resulting Lyapunov stability conditions are compared to spectral stability analyses for relativistic cold plasmas in various examples and special cases, including (1) non‐neutral electron flow in a planar diode and (2) circularly symmetric plasma flow enclosed in a coaxial waveguide. These linearized stability results can be extended readily to nonlinear Lyapunov stability conditions for finite‐amplitude perturbations by employing standard convexity arguments for the Lyapunov functions given here. The relativistic stability conditions are shown to reduce either to their nonrelativistic counterparts or to trivial identities in the nonrelativistic limit.
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47.75.+f Relativistic fluid dynamics
47.20.-k Flow instabilities
52.27.Ny Relativistic plasmas

Ambipolarons: Solitary wave solutions for the radial electric field in a plasma

D. E. Hastings, R. D. Hazeltine, and P. J. Morrison

Phys. Fluids 29, 69 (1986); http://dx.doi.org/10.1063/1.865954 (7 pages) | Cited 15 times

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The ambipolar radial electric field in a nonaxisymmetric plasma can be described by a nonlinear diffusion equation. This equation is shown to possess solitary wave solutions. A model nonlinear diffusion equation with a cubic nonlinearity is studied. An explicit analytic step‐like form for the solitary wave is found. It is shown that the solitary wave solutions are linearly stable against all but translational perturbations. Collisions of these solitary waves are studied and three possible final states are found: two diverging solitary waves, two stationary solitary waves, or two converging solitary waves leading to annihilation.
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52.25.Fi Transport properties
52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)
52.35.Sb Solitons; BGK modes

Two‐point theory of current‐driven, ion‐cyclotron turbulence

T. Chiueh and P. H. Diamond

Phys. Fluids 29, 76 (1986); http://dx.doi.org/10.1063/1.865955 (21 pages) | Cited 3 times

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An analytical theory of current‐driven, ion‐cyclotron turbulence that treats incoherent phase‐space density granulations (clumps) is presented. In contrast to previous investigations it focuses on the physically relevant regime of weak collective dissipation, where waves and clumps coexist. The threshold current for nonlinear instability is calculated and is found to deviate from the linear threshold by up to 7%. A necessary condition for the existence of stationary wave‐clump turbulence is derived and shown to be analogous to the test‐particle model, fluctuation–dissipation theorem result. The structure of three‐dimensional magnetized clumps is characterized. It is proposed that nonlinear instability is saturated by collective dissipation due to ion‐wave scattering. For this wave‐clump turbulence regime, it is found that the fluctuation level (eφ/Te)rms≤0.1, and that the modification of anomalous resistivity to levels predicted by conventional nonlinear wave theories is moderate. In marked contrast to the quasilinear prediction, it is also shown that ion heating significantly exceeds electron heating, and that a drag force on electrons sustains a steady current and is set up to counteract the accelerating electric field along field lines.
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52.35.Ra Plasma turbulence
52.25.Gj Fluctuation and chaos phenomena
52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)
52.35.Qz Microinstabilities (ion-acoustic, two-stream, loss-cone, beam-plasma, drift, ion- or electron-cyclotron, etc.)

Electrostatic electron surface modes on a plasma–vacuum interface of finite width

Roger D. Jones

Phys. Fluids 29, 97 (1986); http://dx.doi.org/10.1063/1.865957 (3 pages) | Cited 3 times

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The electron surface mode dispersion relation, including Landau damping, is obtained for a vacuum–plasma interface. Unlike previous work, the interface is permitted to have a finite width and no wall boundary conditions are assumed. When the density gradient scale length Ln is large compared with a Debye length k−1D and small compared with a surface mode wavelength 2πk−1, then the mode frequency is ω=(ωp/21/2) (1+kLn/6), and the Landau damping rate is γ =−[6/(2π)1/2]ωp/(kDLn). These expressions are much different than the comparable expressions for a wall‐confined plasma.
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52.35.Fp Electrostatic waves and oscillations (e.g., ion-acoustic waves)
52.40.Hf Plasma-material interactions; boundary layer effects

Collisional effects on coherent nonlinear wave–particle interactions at cyclotron harmonics

M. D. Carter, J. D. Callen, D. B. Batchelor, and R. C. Goldfinger

Phys. Fluids 29, 100 (1986); http://dx.doi.org/10.1063/1.865985 (10 pages) | Cited 14 times

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When particle orbits are trapped very near a cyclotron harmonic resonance, the quasilinear concept of weakly perturbed, uncorrelated passages through resonance breaks down, and nonlinear effects become important. In numerical as well as analytic studies, it is demonstrated that relativistic detuning of the resonance can be important for electrons even at low initial energies (∼20 eV) and that coupling to perturbed parallel motion can lead to strong interactions for values of the turning point where the wave frequency differs from a harmonic multiple of the bounce‐averaged gyrofrequency by an integral multiple of the bounce frequency. The resultant motion is described by large periodic energy excursions for which small‐angle Coulomb collisions or other randomization processes are required to realize net heating. Analytic formulas are derived describing the energy excursion behavior and heating in a mildly relativistic limit. Also, a Monte Carlo numerical model of the collisional effects on the orbits has been employed to study electron heating at the second‐harmonic cyclotron resonance and to test the analytic results. In certain regimes of collisionality, a strong enhancement over quasilinear heating has been found.
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52.20.Dq Particle orbits
52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)
52.50.Gj Plasma heating by particle beams

The dielectric function for the Balescu–Lenard–Poisson kinetic equations

J. R. Jasperse and B. Basu

Phys. Fluids 29, 110 (1986); http://dx.doi.org/10.1063/1.865986 (12 pages) | Cited 7 times

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By using the propagator expansion method applied to an electon–ion plasma near thermal equilibrium, a closed‐form solution is found for the high‐frequency, collisional dielectric function in the electrostatic approximation to the first order in the plasma parameter when the Balescu–Lenard collision operator [Phys. Fluids 3, 52 (1960); Ann. Phys. (N.Y.) 3, 390 (1960)] is used to describe electron–electron and electron–ion collisions. The Balescu–Lenard dielectric function is shown to be an entire function of the complex frequency variable ω. Since an exact solution for the collisional propagator for the Balescu–Lenard problem is probably impossible, these results illustrate the usefulness of the propagator expansion method as a way of obtaining the dielectric function for collisional plasmas. A comparison is made between the Balescu–Lenard result for the plasma conductivity as the wave vector k → 0 and the Guernsey result, obtained by Oberman, Ron, and Dawson [Phys. Fluids 5, 1514 (1962)]. By solving the Balescu–Lenard dispersion relation in the long wavelength approximation, a formula is obtained for the total damping rate for Langmuir waves Γk, which is the sum of the collisionless (Landau) part γLk and the collisional part γνk. A numerical solution of the Balescu–Lenard dispersion relation has also been performed, and the analytical and numerical results for the damping rates are compared at long wavelengths. Comparisons of the Balescu–Lenard damping rate to the quantum mechanical result obtained by Dubois, Gilinsky, and Kivelson [Phys. Rev. Lett. 8, 419 (1962)] and to other results are also made.
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52.25.Mq Dielectric properties
52.20.Fs Electron collisions
52.25.Dg Plasma kinetic equations

Intrinsic diffraction pattern inside the resonance cone

H. de Feraudy

Phys. Fluids 29, 122 (1986); http://dx.doi.org/10.1063/1.865988 (6 pages) | Cited 3 times

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The oscillatory pattern of the electrostatic potential radiated in a hot magnetoplasma at frequencies ω<min (ωp, ωc) is examined in terms of the intrinsic diffraction phenomenon: ωp and ωc are, respectively, the electron plasma and cyclotron frequencies. It is shown that the potential maxima are located on equally spaced cones parallel to the resonance cone. The spacing of these cones, measured along the ambient magnetic field, is simply related to k∥I, the parallel component of the largest unattenuated wave vector. The consistency of these features with experimental observations is shown. A method of measurement of the electron temperature is derived.
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52.40.Fd Plasma interactions with antennas; plasma-filled waveguides
52.35.Fp Electrostatic waves and oscillations (e.g., ion-acoustic waves)

Modulational instability of obliquely modulated ion‐acoustic waves in a two‐ion plasma

R. S. Chhabra and S. R. Sharma

Phys. Fluids 29, 128 (1986); http://dx.doi.org/10.1063/1.865989 (5 pages) | Cited 19 times

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Using the KBM perturbation technique, the stability of oblique modulation of ion‐acoustic waves in a two‐ion plasma is studied. It is found that the presence of a small amount of lighter ion impurities significantly changes the instability domain in the ω‐ϕ plane. The effect of the concentration and mass of impurity ions on the modulational instability is discussed in detail. The threshold amplitude for instability and nonlinear frequency shift of the wave are also calculated. The predictions of the theory are found to be in good agreement with the experimental observations.
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52.35.Qz Microinstabilities (ion-acoustic, two-stream, loss-cone, beam-plasma, drift, ion- or electron-cyclotron, etc.)

Two‐dimensional drift vortices and their stability

E. W. Laedke and K. H. Spatschek

Phys. Fluids 29, 133 (1986); http://dx.doi.org/10.1063/1.865990 (10 pages) | Cited 38 times

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Drift vortices in plasmas described by the Petviashvili equation in the case of strong temperature inhomogeneities or by the Hasegawa–Mima equation in the case of density gradients are investigated. Both equations allow for two‐dimensional vortex solutions. The models are reviewed and the forms of the vortices are discussed. In the temperature‐gradient case, the stationary solutions are only known numerically, whereas in the density gradient case analytical expressions exist. The latter are called modons; here the ground states are investigated. The result of a stability calculation is that both types of two‐dimensional solutions, for the Petviashvili equation as well as the Hasegawa–Mima equation, are stable. The methods used to prove this result are either direct (constructing Liapunov functionals) or indirect, and then based on variational principles.
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52.35.Kt Drift waves
52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)
52.25.Fi Transport properties

Dipole vortex solutions of magnetohydrodynamic equations describing microturbulence

M. Y. Yu and M. Lisak

Phys. Fluids 29, 143 (1986); http://dx.doi.org/10.1063/1.866019 (3 pages) | Cited 3 times

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The general set of low‐frequency nonlinear fluid equations describing microturbulence in an inhomogeneous warm plasma with a strong magnetic field is shown to possess two‐dimensional solitary vortex solutions.
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52.30.Cv Magnetohydrodynamics (including electron magnetohydrodynamics)
52.35.Bj Magnetohydrodynamic waves (e.g., Alfven waves)
52.35.Ra Plasma turbulence
52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)
47.65.-d Magnetohydrodynamics and electrohydrodynamics

Resistive ballooning modes driven by anomalous transport effects

A. K. Sundaram

Phys. Fluids 29, 146 (1986); http://dx.doi.org/10.1063/1.865969 (9 pages) | Cited 4 times

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Using Braginskii’s two fluid equations, the stability of resistive ballooning modes is examined in the presence of parallel thermal conduction, anomalous electron viscosity, and radial thermal conductivity. A generalized set of coupled second‐order differential equations in ϕ and ψ is derived in ballooning space and is solved to obtain analytical solutions in two interesting frequency regimes, SCs/ qR ≪ ‖ω‖ ≪ Cs/  qR and ‖ω‖ ≫ Cs/qR. It is shown that the anomalous thermal transport term excites the new m=1 resistive ballooning mode (‖ω‖ ≫ Cs/qR) with a large growth rate. The excitation of the m=2 type (or Δ′ driven) mode, on the other hand, is found to be strongly influenced by both anomalous electron viscosity and radial thermal conduction. Finally, the additional effect of parallel electron thermal conduction is shown to give new resistive ballooning modes with significantly large growth rates varying as fractional powers of anomalous electron viscosity and classical thermal conductivity.
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52.30.Cv Magnetohydrodynamics (including electron magnetohydrodynamics)
52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)
52.25.Fi Transport properties
52.55.Fa Tokamaks, spherical tokamaks

Calculation of the electrical conductivity of a partially ionized gas

S. W. Simpson

Phys. Fluids 29, 155 (1986); http://dx.doi.org/10.1063/1.865970 (6 pages)

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The theoretical calculation of electron transport properties in partially ionized gases using a Laguerre (or Sonine) polynomial expansion has met with convergence difficulties in certain cases, in particular, when there is a Ramsauer minimum in the electron‐neutral momentum transfer cross section. In this work, which deals specifically with electrical conductivity, it is shown that the difficulties can be avoided by choosing a (nonorthogonal) expansion that includes a limiting‐case analytical solution. With some simplifications, the approach leads to a practical formula for the conductivity involving generalized collision integrals. A formula based on a BGK‐type approximation to the electron–electron collision operator is also given. Calculated conductivities are found to agree well with the more sophisticated expansion method.
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52.25.Fi Transport properties
52.20.Fs Electron collisions

Isodynamic magnetohydrodynamic equilibria

Paul J. Channell

Phys. Fluids 29, 161 (1986); http://dx.doi.org/10.1063/1.865972 (6 pages) | Cited 6 times

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The class of isodynamic magnetohydrodynamic (MHD) equilibria in three dimensions is studied. A reduction of the problem to that of finding a force‐free equilibrium is carried out, and the resulting force‐free problem transformed, by means of a three‐dimensional hodograph technique, to a simple and elegant form. The flux surface equation in a potentially very useful form is split off in a further representation.
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52.30.Cv Magnetohydrodynamics (including electron magnetohydrodynamics)
02.30.Jr Partial differential equations

Alfvén wave heating of low‐beta, nonaxisymmetric, toroidal plasmas

J. A. Tataronis and H. R. Lewis

Phys. Fluids 29, 167 (1986); http://dx.doi.org/10.1063/1.865973 (5 pages) | Cited 3 times

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Solutions of the Alfvén wave continuum equation for a pressureless, magnetohydrodynamic plasma in nonaxisymmetric, toroidal geometry are presented. The analysis is based on the Hamiltonian form of the continuum equation along lines of force of the equilibrium magnetic field. Canonical transformation methods are invoked to determine approximate solutions of the continuum equation. The approximate solutions are used to identify the conditions that govern the Alfvén continuous spectrum in nonaxisymmetric toroidal geometry.
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52.50.Gj Plasma heating by particle beams
52.30.Cv Magnetohydrodynamics (including electron magnetohydrodynamics)
52.35.Bj Magnetohydrodynamic waves (e.g., Alfven waves)
52.55.Jd Magnetic mirrors, gas dynamic traps

Transport in driven plasmas

N. J. Fisch

Phys. Fluids 29, 172 (1986); http://dx.doi.org/10.1063/1.865974 (8 pages) | Cited 18 times

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A plasma in contact with an external source of power, especially a source that interacts specifically with highvelocity electrons, exhibits transport properties, such as conductivity, different from those of an isolated plasma near thermal equilibrium. This is true even when the bulk of the particles in the driven plasma is near thermal equilibrium. To describe the driven plasma, we derive an adjoint equation to the inhomogeneous, linearized, dynamic Boltzmann equation. The Green’s functions for a variety of plasma responses can then be generated. It is possible to modify the Chapman–Enskog [Mathematical Theory of Nonuniform Gases, 3rd ed., (Cambridge U.P., Cambridge, MA, 1970)] expansion in order to incorporate the response functions derived here.
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52.50.Gj Plasma heating by particle beams
52.25.Fi Transport properties
52.25.Dg Plasma kinetic equations
52.55.Fa Tokamaks, spherical tokamaks

Current in wave‐driven plasmas

Charles F. F. Karney and Nathaniel J. Fisch

Phys. Fluids 29, 180 (1986); http://dx.doi.org/10.1063/1.865975 (13 pages) | Cited 53 times

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A theory for the generation of current in a toroidal plasma by radio‐frequency waves is presented. The effect of an opposing electric field is included, allowing the case of time varying currents to be studied. The key quantities that characterize this regime are identified and numerically calculated. Circuit equations suitable for use in ray‐tracing and transport codes are given.
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52.50.Gj Plasma heating by particle beams
52.25.Fi Transport properties
52.55.Fa Tokamaks, spherical tokamaks
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