Volume 24, Issue 1, January 1981
Index of content:

Viscous buckling of thin fluid layers
View Description Hide DescriptionExperimental results are presented describing the buckling type of instability of thin layers of very viscous liquids subjected to linear shear flow. If the dimensionless shear stress in the layer exceeds a critical value, the layer will buckle in a manner similar to the buckling of a thin elastic plate. The stability criteria are obtained as a ratio of the destabilizing viscous stresses relative to the stabilizing surface tension and gravitational forces.

Generalization of the Kelvin–Kirchhoff equations for the motion of a body through a fluid
View Description Hide DescriptionThe classical Kelvin–Kirchhoff equations of motion of a rigid body in an inviscid, unbounded fluid are generalized for the case where other fixed boundaries are present. An interesting by‐product of the derivation is a relation, new to the authors, between the direction cosines of the body axes relative to fixed axes and kinematic parameters of the angular motion of the body.

The behavior of the density oscillator
View Description Hide DescriptionA density oscillator is studied which consists of two fluids of different density separated by a horizontal wall with the dense fluid at the upper side. A small hole connects the fluids, through which fluid is interchanged with distinct pulses. The pulse time is studied experimentally and a dimensional analysis is performed to find the relevant parameters governing the pulse time.

Extension of weakly nonlinear theory of solitary wave propagation
View Description Hide DescriptionA localized wave propagating horizontally at speed c through a fluid of arbitrary depth H which consists of a lower vertical layer (thickness L) with a weak‐depth‐dependent density below an upper layer of constant density is considered. The flow is characterized by Long’s equation for the stream function ψ. Exact analytic results are obtained for ψ through O[1/(h−1)] and the wave speed through order O[1/(h−1)^{2}], h=H/L. The terms in ψ of O[1/(h−1)^{2}] are obtained numerically. Using these results the range of validity of such a perturbative approach is examined.

Modulational instability and the Fermi‐Pasta‐Ulam recurrence
View Description Hide DescriptionThe long‐time behavior of the modulational instability of the nonlinear Schrödinger equation is investigated. Linear stability analysis shows that a finite amplitude uniform wave train is unstable to infinitesimal modulational perturbations with sufficiently long wavelengths while it is stable for perturbations with short wavelengths. Near the threshold for instability, the long‐time behavior of the unstable modulation is obtained by means of the multiple time scale technique. As a result, the Fermi–Pasta–Ulam recurrence is rediscovered, in agreement with recent experiments and with a numerical solution of the problem at hand.

Current generation with low‐frequency waves
View Description Hide DescriptionVarious types of traveling waves may be injected into a tokamak to continuously sustain the toroidal current. Interest in this problem arises from the possibility of operating tokamak reactors in the steady state. The low‐frequency waves most suitable for this task are identified in terms of the power cost for deployment in a reactor. Means of exciting these waves and tradeoffs with design criteria are discussed. A comparison is made with the alternative attractive regime of high‐frequency waves. Conclusions are based, in part, on the numerical solution of the two‐dimensional Fokker–Planck equation with an added quasi‐linear term due to the waves.

Finite‐wavelength oscillations of a diffuse‐profile, Vlasov‐fluid plasma column
View Description Hide DescriptionAn equation of motion is derived which describes finite‐wavelength (i.e., axial wavelength ∼ plasma radius) oscillations of a Vlasov‐ion, fluid‐electron plasma in a diffuse, near‐ϑ pinch geometry. The equations include simultaneous geometric and ion kinetic effects. For a screw‐pinch geometry, and with isothermal electrons, these equations are shown to be equivalent to the guiding‐center equations. Boundary conditions and a closed‐form solution for the sharp‐profile case are discussed.

Nonlinear magnetohydrodynamic stability
View Description Hide DescriptionWays have been found to significantly upgrade the resolution of an equilibrium and stability code that is based on the variational principle of ideal magnetohydrodynamics. Nonlinear saturation of ballooning modes for tokamaks has been demonstrated using a revised version of the code. Second stability regions are calculated from the nonlinear dependence of δW on β. An l=2,3 torsatron is described that appears to have no ideal magnetohydrodynamic instabilities for β as high as 5%.

Nonlinear evolution of drift cyclotron modes
View Description Hide DescriptionThe space‐time evolution of the drift‐cyclotron instability as influenced by the nonlinear shift in the ion‐cyclotron frequency is studied analytically, numerically, and by computer simulation. The analysis is specialized to the case of a single coherent wave with frequency near both a cyclotron harmonic and the ion diamagnetic frequency. Such an analysis is motivated by observations of large‐amplitude ion‐cyclotron waves in the 2XIIB mirror experiment, which were highly monochromatic and often exhibited these frequency characteristics. A nonlinear dispersion relation describing saturation of the instability is derived by means of a self‐consistent analytical solution of the Vlasov–Poisson equations to third order in the wave amplitude. Solitary wave and self‐similar solutions are obtained that describe nonlinear wave propagation. Numerical solution of a nonlinear evolution equation and particle simulation confirm the analysis.

Effects of toroidal coupling on the stability of tearing modes
View Description Hide DescriptionThe time evolution of tearing modes in toroidal geometry is studied in the low‐β and large aspect ratio limit. An initial value three‐dimensional computer code which numerically advances the reduced set of resistive magnetohydrodynamic equations is employed. Toroidicity has, in general, a destabilizing effect on tearing modes in this limit. A generalization of Δ′ formalism can be used to study the linear regime. The results obtained in this way are in very good agreement with the results from the initial value code. The nonlinear phase of the evolution is also followed numerically. In the case of strong interaction of different helicities, a larger region of stochastic magnetic field lines results than in the cylindrical geometry case.

Magnetic field diffusion and dissipation in reversed‐field plasmas
View Description Hide DescriptionA diffusion equation is derived which describes the evolution of a magnetic field in a plasma of arbitrary β and resistivity. The equation is valid for a one‐dimensional slab geometry, assumes the plasma remains in quasi‐equilibrium throughout its evolution (i.e., pressure balance), and does not include thermal transport. Scaling laws governing the rate of change of the magnetic energy, particle drift energy, and magnetic flux are calculated. It is found that the magnetic free energy can be substantially larger than the particle drift energy and can be an important energy reservoir in driving plasma instabilities (e.g., the lower‐hybrid‐drift instability). In addition, the effect of a spatially varying resistivity on the evolution of a reversed‐field plasma is studied. The resistivity model used is based upon the anomalous transport properties associated with the nonlocal mode structure of the lower‐hybrid‐drift instability. The relevance of this research to laboratory plasmas (e.g., theta pinches, reversed‐field theta pinches) and space plasmas (e.g., the earth’s magnetotail) is discussed.

Neoclassical transport in helically symmetric plasmas
View Description Hide DescriptionThe neoclassical rate of diffusion is calculated in such a way that the result is applicable to both helically symmetric and toroidally symmetric plasmas.

Convective cell formation and anomalous diffusion due to electromagnetic drift wave turbulence
View Description Hide DescriptionConvective cell formation and spectral cascade processes due to gravitational drift Alfvén waves are studied using a new type of model equation. Conservation relations are derived and explosive instability is found for systems near marginal finite β stability. This instability also remains when the effects of poor as well as favorable curvature regions are included, i.e., for ballooning modes. The anomalous diffusion due to convective cells and quasi‐linear effects are compared.

Kinetic theory of relativistic plasmas
View Description Hide DescriptionThe thermalization of particle kinetic motion by binary collisions is considered for a plasma with k T∼(10–100) m c ^{2}, where m is the electron mass. At this temperature, the principal mechanism for relaxation of electron motion is via radiationless electron‐electron collisions (Mo/ller scattering). Ions are nonrelativistic, but are energetic enough so that their Coulomb scattering can be treated in the Born approximation. Relaxation times are computed and Boltzmann‐equation Fokker–Planck operators are derived for the various binary‐collision processes. The expression for the rate of kinetic energy exchange between electron and ion gases is derived for the case where the gases are at different temperatures.

Relativistic theory of electron cyclotron resonance heating
View Description Hide DescriptionThe formal theory of the interaction of mildly relativistic electrons with a cyclotron resonant applied electromagnetic field described by geometric optics is developed. The electron distribution function is written as the sum of a quasi‐static part f _{0} and a high frequency part f _{1}. A quasi‐linear theory is employed to describe these. The resulting transport equation f _{0} is written to lowest significant order jointly in the reciprocal of the gyration frequency and the bounce frequency. It includes a relativistic collision term and a drag term associated with synchrotron emission. The linearized equations for f _{1} are solved, and an expression for the high‐frequency current is derived.

Temporal evolution of lower‐hybrid waves in the presence of ponderomotive density fluctuations
View Description Hide DescriptionThe propagation of lower‐hybrid waves in the presence of ponderomotive density fluctuations is considered. The problem is treated in two dimensions, and in order to be able to correctly impose the boundary conditions, the waves are allowed to evolve in time. The fields are described by i v _{τ}−F v _{ξ} dζ+v _{ζζ}+‖v‖^{2} v= 0, where v is proportional to the electric field, τ to time, and ζ and ξ measure distances across and along the lower‐hybrid ray. The behavior of the waves is investigated numerically. If the amplitude of the waves is large enough, the spectrum of the waves broadens and their parallel wavelength becomes shorter. The assumptions made in the formulation preclude the application of these results to the lower‐hybrid heating experiment on Alcator‐A. Nevertheless, there are indications that the physics embodied in this problem are responsible for some of the results of that experiment.

Electron heating in high intensity CO_{2} laser‐plasma interaction
View Description Hide DescriptionDetailed measurements of heated electron distributions generated in a short density scale length 10.6 μm laser plasma interaction in the intensity regime 10^{12}≲I(W cm^{−2})≲10^{14} are reported. Maximum electron energies up to 600 keV observed in this study appear to be inconsistent with characteristic maximum energies of electrons generated by resonant absorption in the warm plasma wave breaking limit. Furthermore, heated electron distributions from foils of thickness much less than the hot electron mean‐free‐path as well as from different Z targets suggest that reheating of very energetic electrons by multipassing through the resonant region does not occur; rather the highest‐energy electrons are preferentially accelerated in the backward direction with very large single pass energies. Plasma instabilities at the quarter critical as well as those at the critical are thought to be responsible for the generation of the high‐energy electron component. Finally, these studies show that the electron emission spectra are sensitive to the electrical properties of the target support structure.

Harmonic emission from adiabatically collapsing Langmuir solitons
View Description Hide DescriptionNumerical studies of radiation at 2ω_{ p } from a Langmuir envelope collapsing adiabatically in three dimensions show that the emissivity is higher than expected. A volume emissivity obtained from an approximate density of collapsing packets leads to favorable comparisons with measurements of type‐III solar radio bursts.

Cerenkov and anomalous Doppler effects in the relaxation of an electron beam
View Description Hide DescriptionThe interplay between the Cerenkov and anomalous Doppler interactions in the relaxation of a warm electron beam is investigated by numerical means. The most important feature in the interplay is found to be a nonelastic isotropization. A simple semianalytical model which allows one to estimate various quantities relevant to the relaxation process is also presented.

Equilibrium for cylindrical, magnetically insulated ion diodes assuming adiabatic turn‐on
View Description Hide DescriptionEquilibrium values of diode current are calculated for cylindrical, magnetically insulated vacuum ion diodes as functions of applied voltage, magnetic flux, and aspect ratio. The analysis is based upon the assumption of a slowly (adiabatically) applied voltage so that electrons are confined to a sheath near the cathode where they undergo E×B drifts in the electric field (E) and the magnetic field (B) in the anode–cathode gap. One‐dimensional cases in which the applied magnetic field is either axial or azimuthal are treated. A previous analysis, conducted under the assumption of cycloidal electron orbits, found that equilibrium solutions often do not exist. It is found that this difficulty is absent when E×B drift orbits are assumed.