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Dec 1976

Volume 19, Issue 12, pp. 1843-2052

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Spatially decaying array of vortices

Harold L. Rogler and Eli Reshotko

Phys. Fluids 19, 1843 (1976); http://dx.doi.org/10.1063/1.861417 (8 pages) | Cited 4 times

Online Publication Date: 28 August 2008

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An analysis of two‐dimensional vorticity disturbances introduced at a grid and propagating downstream reveals that viscosity causes the x wavenumber to diminish and the phase speed to increase from their inviscid values, while also causing the amplitude to decay exponentially downstream. This decaying disturbance field introduces an adverse mean pressure gradient of strength O (1/RΛ), where RΛ is the Reynolds number based on freestream velocity and vortex diameter. Viscous effects introduce a discrepancy between the two‐point and two‐time velocity correlations when Taylor’s hypothesis r=U∞t is used to relate the space and time separations. This discrepancy arises because the vortices propagate at a phase speed c greater than the freestream. As RΛ→∞, c→U∞. If Taylor’s hypothesis is modified to r=ct, then the two‐point and two‐time correlations agree.
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47.10.-g General theory in fluid dynamics
47.20.Gv Viscous and viscoelastic instabilities

Rayleigh–Taylor instability in the presence of rotation

B. B. Chakraborty and Jyoti Chandra

Phys. Fluids 19, 1851 (1976); http://dx.doi.org/10.1063/1.861418 (2 pages) | Cited 5 times

Online Publication Date: 28 August 2008

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A heavy uniform fluid is superposed on a lighter one. The two uniform fluids are separated by a layer of transition of finite thickness in which the density increases exponentially in the vertical direction. The growth rate and the wavenumber of the mode of maximum instability are obtained when the whole system rotates uniformly about a vertical axis.
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47.15.G- Low-Reynolds-number (creeping) flows

Mechanism of tornado funnel formation

C. T. Hsu and B. Fattahi

Phys. Fluids 19, 1853 (1976); http://dx.doi.org/10.1063/1.861419 (5 pages) | Cited 8 times

Online Publication Date: 28 August 2008

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The mechanism of formation of a tornado funnel was observed in a specially designed facility for tornado simulation. In this facility, a tornado cyclone is first simulated at the exit of a vortex generator. Interaction of this vortex with a ground plane is found to be responsible for the funnel formation. Time‐averaged velocity measurements indicate that a smaller, but highly concentrated vortex core is developed upward from the ground plane due to the horizontal convergence motion induced by the above interaction. The flow visualization technique has revealed that the funnel cloud is the downward extension of its parent cloud inside this invisible concentrated vortex core. This is possibly the fluid mechanical phenomenon of axial flow reversal usually occurring in a converging or diverging swirling flow.
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92.60.Qx Storms
47.80.-v Instrumentation and measurement methods in fluid dynamics

Experiments on vortex stability

Param Indar Singh and Mahinder S. Uberoi

Phys. Fluids 19, 1858 (1976); http://dx.doi.org/10.1063/1.861420 (6 pages) | Cited 30 times

Online Publication Date: 28 August 2008

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The tip vortex of a laminar flow wing was studied at a sectional lift‐to‐drag ratio of 60. The vortex Reynolds number was Γ0/ν=7.8×104, where Γ0 is the total circulation and ν is the kinematic viscosity. At and near the wing the vortex core was turbulent with an axial jet. Downstream of the wing the jet rapidly dissipated and a wake developed in the core and intensity of turbulent velocities decreased. From 13 to 40 chord length periodic oscillations dominated the velocity fluctuations with little background turbulence. These instabilities had a symmetric and a helical mode with wavelength of the same order as the core diameter. In this range of distances along the vortex core the maximum axial, swirl, and fluctuating velocities vary slowly. At 40 chord lengths behind the wing there is a rapid change in these velocities. This change of state of the vortex core is accompanied by change of velocity fluctuations from periodic to turbulent. The core showed spatial excursions. Measurements up to 80 chord lengths downstream showed no self‐similar decay.
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47.15.Fe Stability of laminar flows

Statistical properties of the interface in the turbulent wake of a heated cylinder

John C. LaRue and Paul A. Libby

Phys. Fluids 19, 1864 (1976); http://dx.doi.org/10.1063/1.861421 (12 pages) | Cited 18 times

Online Publication Date: 28 August 2008

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The statistical properties of the interface between turbulent and nonturbulent fluid in the two‐dimensional wake of a heated cylinder are presented. The following are the principal results: distributions of the intermittency factor and crossing frequency; probability density functions of the durations of the passages of turbulent and nonturbulent fluid, indicating that the longer durations are exponentially distributed; the mean and rms slope angles of the interface at upstream and downstream edges, showing an asymmetry of the mean shape with a steeper downstream edge; and the relative number of overhangs at the two sets of edges, namely, 25% at the downstream edge and only 6% at the upstream edge.
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47.27.nb Boundary layer turbulence

Two‐point turbulence measurements downstream of a heated grid

Paavo Sepri

Phys. Fluids 19, 1876 (1976); http://dx.doi.org/10.1063/1.861422 (9 pages) | Cited 18 times

Online Publication Date: 28 August 2008

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Two‐point turbulence data from hot and cold wire anemometry are presented which describe the temperature and velocity fields downstream of a heated grid in a low speed wind tunnel. The velocity and temperature statistics are found to differ in that (a) the velocity energy spectrum has no inertial subrange while the temperature spectrum displays a significant region of −5/3 slope and (b) for lateral probe displacements the temperature coherence is larger than that of velocity. This result is surprising since both fields were generated by the same grid, and since the temperature field appears to have been passive. Phase measurements indicate that at least the lower turbulence wavenumbers move with equal convection velocities, thereby supporting Taylor’s hypothesis of quasi‐frozen pattern, but this convection velocity appears to be slightly greater than the mean fluid velocity. Cross correlations from the signals of two spatially separated probes are presented for both the velocity and temperature fields. In general, they agree with expectations derived from the theory of isotropic turbulence.
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47.27.Gs Isotropic turbulence; homogeneous turbulence

Shock formation distance in a pressure driven shock tube

E. M. Rothkopf and W. Low

Phys. Fluids 19, 1885 (1976); http://dx.doi.org/10.1063/1.861423 (4 pages) | Cited 2 times

Online Publication Date: 28 August 2008

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Measurements of the shock formation distance in a conventional pressure driven shock tube of 5.2 cm i.d. using two different diaphragm types and three different driver gases with a driven gas of air are presented. The shock formation distance is found to be approximately proportional to the effective opening time of the diaphragm and inversely proportional to the average speeds of sound of the driver and driven gases.
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47.40.Nm Shock wave interactions and shock effects

Equatorial propagation of axisymmetric magnetohydrodynamic shocks

Philip Rosenau and Shimshon Frankenthal

Phys. Fluids 19, 1889 (1976); http://dx.doi.org/10.1063/1.861424 (11 pages) | Cited 36 times

Online Publication Date: 28 August 2008

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Self‐similar solutions are presented for the equatorial propagation of axisymmetric, piston‐driven magnetohydrodynamic shocks into an inhomogeneous ideal gas permeated by a current‐free azimuthal magnetic field. Several regimes of magnetically dominated flow near the piston are possible, depending on the ambient density distribution of the unshocked gas. The strong hydrodynamic and magnetic gradients which permeate this flow indicate the need for a more refined treatment which will include the effects of diffusion due to dissipative phenomena. Global features of the flow are discussed and illustrated by numerical solutions. The use of reductive properties of the equations to simplify the problem in several special cases is indicated. Implications regarding the propagation of disturbances in stellar atmospheres are considered.
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47.65.-d Magnetohydrodynamics and electrohydrodynamics

Theory of a cylindrical probe in a collisionless magnetoplasma

J. G. Laframboise and J. Rubinstein

Phys. Fluids 19, 1900 (1976); http://dx.doi.org/10.1063/1.861425 (9 pages) | Cited 72 times

Online Publication Date: 28 August 2008

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A theory is presented for a cylindrical electrostatic probe in a collisionless plasma, when the probe axis is inclined at an angle ϑ to a uniform magnetic field. The theory is applicable to electron collection, and under more restrictive conditions, to ion collection. For a probe at space potential, the theory is exact in the limit when probe radius rp≪ Debye length λD. At attracting probe potentials, the theory yields an upper bound and an adiabatic limit for current collection. At repelling probe potentials, it provides a lower bound. The theory is valid if rp/λD and rp/ā, where ā is the mean gyroradius, are not simultaneously large enough to produce extrema in the probe sheath potential. The numerical current calculations are based on the approximation that particle orbits are helices near the probe, together with the use of kinetic theory to relate velocity distributions near the probe to those far from it. Probe characteristics are presented for ϑ from 0° to 90°, and for rp/ā from 0.1 to ∞. For ϑ=0°, the end‐effect current is calculated separately.
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52.70.Ds Electric and magnetic measurements
52.20.Dq Particle orbits

Free boundaries for plasmas in surface magnetic field configurations

Burton D. Fried, J. W. VanDam, and Y. C. Lee

Phys. Fluids 19, 1909 (1976); http://dx.doi.org/10.1063/1.861426 (15 pages) | Cited 4 times

Online Publication Date: 28 August 2008

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Exact analytic solutions for the plasma/magnetic‐field free‐boundary problem are derived for three planar, surface magnetic field configurations: the picket fence, the simple magnetic slab, and the staggered magnetic slab. In the ideal magnetohydrodynamic limit, conformal mapping techniques yield explicit closed‐form solutions for the free boundaries, the magnetic field lines, and the magnetohydrodynamic stability criteria. For configurations with plasma confined to one side of a continuous, sharp free boundary, it is found that with a given choice of the physical parameters (i.e., dimensions, current, and plasma pressure), there may be only one stable solution; only one unstable solution; two solutions, one stable and one unstable; or no solution at all. For the staggered slab, parameters can be chosen so that the conductors nearest the plasma are force free, an arrangement which could minimize the size of physical supports and the associated plasma loss in confinement applications.
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52.55.Jd Magnetic mirrors, gas dynamic traps
FREE

Kinetic processes in plasma heating by resonant mode conversion of Alfvén wave

Akira Hasegawa and Liu Chen

Phys. Fluids 19, 1924 (1976); http://dx.doi.org/10.1063/1.861427 (11 pages) | Cited 250 times

Online Publication Date: 28 August 2008

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An externally applied oscillating magnetic field (at a frequency near 1 MHz for typical tokamak parameters) resonantly mode converts to the kinetic Alfvén wave, the Alfvén wave with the perpendicular wavelength comparable to the ion gyroradius. The kinetic Alfvén wave, while it propagates into the higher density side of the plasma after the mode conversion, dissipates due to both linear and nonlinear processes and heats the plasma. If a magnetic field of 50 G effective amplitude is applied, approximately 10 MJ per cubic meter of energy can be deposited in 1 sec into the plasma. The heating rate here is faster than that in the transit time magnetic pumping by a factor of β−1.
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52.50.Gj Plasma heating by particle beams
52.25.Dg Plasma kinetic equations

Destabilization of hydromagnetic drift‐Alfvén waves in a finite pressure, collisional plasma

J. T. Tang and N. C. Luhmann

Phys. Fluids 19, 1935 (1976); http://dx.doi.org/10.1063/1.861410 (12 pages) | Cited 24 times

Online Publication Date: 28 August 2008

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The hydromagnetic drift mode of the coupled drift‐Alfvén wave is destabilized as a standing wave in a dense, current‐free plasma in the presence of a density gradient. When an axial electron current is drawn, a localized Alfvén mode propagating against the current is destabilized, in addition to the unstable drift mode now propagating along the current. The measured wave properties, dispersion, and dependence on plasma parameters are found to agree with the theory derived for a finite β, collisional plasma.
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52.35.-g Waves, oscillations, and instabilities in plasmas and intense beams
52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)

Two‐dimensional magnetohydrodynamic simulation of toroidal pinches II: Belt pinches

H. C. Lui and C. K. Chu

Phys. Fluids 19, 1947 (1976); http://dx.doi.org/10.1063/1.861411 (5 pages) | Cited 8 times

Online Publication Date: 28 August 2008

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The equations and method of solution used in a previous paper [Phys. Fluids 18, 1277 (1975)] for the simulation of toroidal screw pinches have been modified to simulate toroidal belt pinches and high‐beta tokamaks of rectangular cross sections. In addition to accounting for the geometric difference between a belt pinch and a screw pinch, this work permits the inclusion of vertical fields, while the previous work dealt only with conducting shells. Calculated results have been compared with the experimental results of the Belt Pinch I of Garching, and the agreement is again very good. The present code has been used extensively in the design and prediction of the Torus I experiment at Columbia.
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52.55.Ez Theta pinch
52.55.Fa Tokamaks, spherical tokamaks
52.55.Hc Stellarators, torsatrons, heliacs, bumpy tori, and other toroidal confinement devices
52.65.-y Plasma simulation

Frequency shift of a nonlinear plasma wave and its initial damping

José Canosa

Phys. Fluids 19, 1952 (1976); http://dx.doi.org/10.1063/1.861412 (6 pages) | Cited 9 times

Online Publication Date: 28 August 2008

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For large wave amplitudes, the frequency shift of a nonlinear plasma wave and its initial exponential damping are found to be approximately proportional to the square of the initial wave amplitude. Extensive Vlasov computations are in essential agreement with recent experimental results for wavenumber shifts and initial nonlinear dampings obtained at large amplitudes.
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52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)

Physical origins of the sideband instability

José Canosa and Alan Wray

Phys. Fluids 19, 1958 (1976); http://dx.doi.org/10.1063/1.861413 (9 pages) | Cited 5 times

Online Publication Date: 28 August 2008

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Over a large range of amplitudes of a main plasma wave, the growth rates of its lower sidebands are found to increase with amplitude significantly more strongly than previously reported. For special initial conditions of experimental interest, the Vlasov computations show that, as the main wave amplitude increases, the sideband growth rates first increase, then reach a peak, and finally start to decrease. These computational results, as well as those obtained for the sideband frequency separation, are explained quantiatively by simple theoretical calculations based on the model that sideband growth is due to a nonlinear resonance between the trapped electrons bouncing in the main wave and a test wave of appropriate frequency.
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52.35.-g Waves, oscillations, and instabilities in plasmas and intense beams

Stability of oblique modulation on an ion‐acoustic wave

Masashi Kako and Akira Hasegawa

Phys. Fluids 19, 1967 (1976); http://dx.doi.org/10.1063/1.861414 (3 pages) | Cited 29 times

Online Publication Date: 28 August 2008

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Modulation instability is shown to be a general property of a wave in a nonlinear and dispersive medium when the modulation is allowed in the direction oblique to that of the wave phase velocity. As an example a modulation on an ion‐acoustic wave is shown to be unstable even if this wave is modulationally stable in the case of parallel or perpendicular modulation.
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52.35.-g Waves, oscillations, and instabilities in plasmas and intense beams

Magnetic curvature and ion distribution function effects on lower hybrid drift instabilities

N. A. Krall and J. B. McBride

Phys. Fluids 19, 1970 (1976); http://dx.doi.org/10.1063/1.861415 (2 pages) | Cited 10 times

Online Publication Date: 28 August 2008

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Magnetic curvature is shown to have a relatively weak effect on lower hybrid drift instabilities, but ion temperature gradients can significantly alter stability in the broad sheath limit, ai≪Ln.
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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.Jd Magnetic mirrors, gas dynamic traps
52.55.Ez Theta pinch

Nonlinear effects on mode‐converted lower‐hybrid waves

H. H. Kuehl

Phys. Fluids 19, 1972 (1976); http://dx.doi.org/10.1063/1.861416 (3 pages) | Cited 24 times

Online Publication Date: 28 August 2008

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Nonlinear ponderomotive force effects on mode‐converted lower‐hybrid waves are considered. The nonlinear distortion of these waves is shown to be governed by the cubic nonlinear Schrödinger equation. The threshold condition for self‐focusing and filamentation is derived.
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52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)

Parametric instabilities with finite wavelength pump

Burton D. Fried, T. Ikemura, K. Nishikawa, and G. Schmidt

Phys. Fluids 19, 1975 (1976); http://dx.doi.org/10.1063/1.861428 (7 pages) | Cited 11 times

Online Publication Date: 28 August 2008

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The general problem of parametric instabilities driven by a finite wavelength pump is investigated. For the particular case of a Langmuir wave pump, it is shown that resonant decay instabilities (forward or backward scattering in the one‐dimensional case), with thresholds which vanish in the colisionless limit, can occur only for pump wavenumber k0 greater than the critical value [(m/M)1/2/γ]kD, where m and M are the electron and the ion mass, respectively, and γ is the specific heat ratio. For smaller wavenumbers, there is always a nonzero threshold, the instability being of modulation character at long wavelengths and almost pure growing for short wavelengths. Frequency locking for small k0 and wavenumber locking for large k0 are demonstrated. The results are generalized to the case where the coupled waves satisfy arbitrary dispersion relations and simple physical interpretations of the instabilities are given.
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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.Db Electromagnetic (nonlaser) radiation interactions with plasma

Nonlinear evolution of the sausage instability

David L. Book, Edward Ott, and Martin Lampe

Phys. Fluids 19, 1982 (1976); http://dx.doi.org/10.1063/1.861429 (5 pages) | Cited 18 times

Online Publication Date: 28 August 2008

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Sausage instabilities of an incompressible, uniform, perfectly conducting Z pinch are studied in the nonlinear regime. In the long wavelength limit (analogous to the ’’shallow water theory’’ of hydrodynamics), a simplified set of universal fluid equations is derived, with no radial dependence, and with all parameters scaled out. Analytic and numerical solutions of these one‐dimensional equations show that an initially sinusoidal perturbation grows into a ’’spindle’’ or cylindrical ’’spike and bubble’’ shape, with sharp radial maxima. In the short wavelength limit, the problem is shown to be mathematically equivalent to the planar semi‐infinite Rayleigh–Taylor instability, which also grows into a spike‐and‐bubble shape. Since the spindle shape is common to both limits, it is concluded that it probably obtains in all cases. The results are in agreement with dense plasma focus experiments.
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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.Ez Theta pinch
FREE

Numerical studies of nonlinear evolution of kink modes in tokamaks

Marshall N. Rosenbluth, D. A. Monticello, H. R. Strauss, and R. B. White

Phys. Fluids 19, 1987 (1976); http://dx.doi.org/10.1063/1.861430 (10 pages) | Cited 102 times

Online Publication Date: 28 August 2008

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A set of numerical techniques for investigating the full nonlinear unstable behavior of low‐β kink modes of given helical symmetry in tokamaks is presented. Uniform current density plasmas display complicated deformations including the formation of large vacuum bubbles provided that the safety factor q is sufficiently close to integral. Fairly large m=1 deformations, but not bubble formation, persist for a plasma with a parabolic current density profile (and hence shear). Deformations for m⩾2 are, however, greatly suppressed.
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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

Drift wave stabilization by a high frequency applied electric field

Irwin Weiss and Terry Morrone

Phys. Fluids 19, 1997 (1976); http://dx.doi.org/10.1063/1.861431 (8 pages) | Cited 2 times

Online Publication Date: 28 August 2008

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Dispersion relations are calculated for drift waves in an inhomogeneous plasma with an oscillatory electric field which is applied perpendicular to the constant magnetic field. Use is made of the Vlasov equation to calculate drift wave frequencies and growth rates. Electron currents, density, and temperature gradients and finite/infinitesimal ion gyroradius are included. It is determined that substantial reduction in the drift wave growth rate can be achieved using the ac electric field when its frequency is equal approximately to either the ion or electron cyclotron frequency.
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52.35.-g Waves, oscillations, and instabilities in plasmas and intense beams

Electron distribution function in a plasma generated by fission fragments

H. A. Hassan and Jerry E. Deese

Phys. Fluids 19, 2005 (1976); http://dx.doi.org/10.1063/1.861432 (7 pages) | Cited 9 times

Online Publication Date: 28 August 2008

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See Also: Erratum

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A Boltzmann equation formulation is presented for the determination of the electron distribution function in a plasma generated by fission fragments. The formulation takes into consideration ambipolar diffusion, elastic and inelastic collisions, recombination and ionization, and allows for the fact that the primary electrons are not monoenergetic. Calculations for He in a tube coated with fissionable material shows that, over a wide pressure and neutron flux range, the distribution function is non‐Maxwellian, but the electrons are essentially thermal. Moreover, about a third of the energy of the primary electrons is transferred into the inelastic levels of He. This fraction of energy transfer is almost independent of pressure and neutron flux.
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52.25.Kn Thermodynamics of plasmas
52.50.Dg Plasma sources

Spectroscopic measurements of the plasma temperature in the ZT−1 toroidal pinch

R. B. Howell and H. J. Karr

Phys. Fluids 19, 2012 (1976); http://dx.doi.org/10.1063/1.861433 (9 pages) | Cited 5 times

Online Publication Date: 28 August 2008

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The plasma temperature in the ZT‐1 toroidal pinch as been determined using spectral line Doppler broadening measurements facilitated by the use of analog‐to‐digital converters and an on‐line computer. The ZT‐1 system has been operated in a ’’derated’’ mode with peak toroidal currents of 30 to 110 kA, initial current rise times of 0.5 to 2.5×1011 A/sec, stabilizing toroidal magnetic fields of 0.06 to 0.3 T, and filling pressures of 2 to 30 Pa. Results of the plasma temperature measurements in this mode of operation are reviewed with the objective of determining the effects of scaling the machine parameters on the ion temperature with extension to results obtained in the earlier higher current ’’fast’’ mode operation. An analysis and experimental tests of several possible sources of error in the ion temperature measurements are given in an appendix.
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52.55.Ez Theta pinch
52.70.Kz Optical (ultraviolet, visible, infrared) measurements

Harmonic generation in a radially limited unstable beam‐plasma system

M. Weinfeld

Phys. Fluids 19, 2021 (1976); http://dx.doi.org/10.1063/1.861434 (8 pages) | Cited 1 time

Online Publication Date: 28 August 2008

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Harmonic generation by a three wave coupling mechanism has been studied in an unstable beam‐plasma system, whose dispersion is small at low frequencies due to its finite geometry. This mechanism has been observed in the case of resonant coupling (nondispersive mode), and in the case of quasi‐resonant coupling (weakly dispersive mode). The experimental results give fair agreement with hydrodynamic theory.
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52.40.Mj Particle beam interactions in plasmas
52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)
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