Volume 25, Issue 5, May 1982
Index of content:

The meandering fall of paper ribbons
View Description Hide DescriptionThis paper discusses experimental observations of the meandering fall of light‐weight tissue paper ribbons. The photographs show that the ribbons assume a sinusoidal shape with a unique wavelength which scales with the thickness of the airstream entrained by the ribbon.

Hydromagnetic Rayleigh–Taylor instability of a rotating stratified fluid
View Description Hide DescriptionThe Raleigh–Taylor instability of an infinitely conducting, rotating, stratified fluid in the presence of a horizontal magnetic field is studied. The fluid density and the magnetic field strength are arbitrary functions of the vertical coordinate. An estimate of the upper bound for the growth rate of the instability of any unstable mode of disturbances of a given wavelength, which cannot occur unless the fluid density increases with height at some points, is obtained. This estimate is found to give a satisfactory result in a Rayleigh–Taylor instability model for which the dispersion relation is known exactly. A sufficient condition for stability of a disturbance is also obtained.

Interaction of small‐amplitude fluctuations with a strong magnetohydrodynamic shock
View Description Hide DescriptionThe interaction of small fluctuations with a strong magnetohydrodynamic shock is considered theoretically in the most general situation using an ideal model. An analytical treatment of the cutoff conditions for penetration of the upstream fluctuation through the bow shock is presented. The situation in which there exist unique solutions of the interaction is investigated quantitatively. Numerical examples of the interaction of waves in the solar wind with the terrestrial bow shock are given to illustrate the refracted waves to be found in the earth’s magnetosheath.

Resonant interaction of ion‐acoustic solitons in three‐dimensions
View Description Hide DescriptionThe two‐ and three‐dimensional collision of spherical ion acoustic solitons is experimentally observed. In each case, a new soliton is created which approximately satisfies the resonance conditions K̄_{α} = Σ^{ n } _{ i = 1}K̄_{ i }; Ω(K̄_{α}) = Σ^{ n } _{ i = 1} Ω(K̄_{ i }), n = number of colliding solitons. In three dimensions the soliton is localized along the axis of collision.

Particle transport in field‐reversed configurations
View Description Hide DescriptionParticle transport in field‐reversed configurations is investigated using a one‐dimensional, nondecaying, magnetic field structure. The radial profiles are constrained to satisfy an average β condition from two‐dimensional equilibrium and a boundary condition at the separatrix to model the balance between closed and open‐field‐line transport. When applied to the FRX‐B experimental data and to the projected performance of the FRX‐C device, this model suggests that the particle confinement times obtained with anomalous lower‐hybrid‐drift transport are in good agreement with the available numerical and experimental data. Larger values of confinement times can be achieved by increasing the ratio of the separatrix radius to the conducting wall radius. Even larger increases in lifetimes might be obtained by improving the open‐field‐line confinement.

Numerical studies of a field‐reversed theta‐pinch plasma
View Description Hide DescriptionA two‐dimensional magnetohydrodynamic computer code has been used to model the formation of a field‐reversed theta‐pinch plasma. The simulations have been performed in the parameter range of the recent FRX‐B experiments. It is found that an anomalous resistivity is required to explain the observed decay of reversed magnetic flux. The reconnection process is found to be sensitive to the density and amount of trapped reverse bias flux but insensitive to the assumed magnitude of anomalous resistivity. The reconnection process is more rapid and the axial plasma motion induced by reconnection is more pronounced for larger bias flux. If a straight coil is used instead of the usual shaped coil (end mirrors), full reconnection does not occur for low initial bias fields. These results are in general agreement with FRX‐B observations.

Dynamic magnetic x points
View Description Hide DescriptionTwo‐and‐one‐half dimensional magnetostatic and electromagnetic particle simulations of time‐varying magnetic x points and the associated plasma response are reported. The stability and topology depend on the geometry, irrespective of the plasma β. The electrostatic field and finite Larmor radius effects play an important role in current penetration and shaping of the plasma flow. The snapping of the field lines, and dragging of the plasma into, and confinement of the plasma at, an o point (magnetic island) is observed. Magnetic island coalescence with explosive growth of the coalescence mode occurs and is accompanied by a large increase in kinetic energy and temperature as well as the formation of hot tails on the distribution functions.

Finite β effects on the nonlinear evolution of the (m = 1; n = 1) mode in tokamaks
View Description Hide DescriptionThe stability and evolution of ISX‐B‐like plasmas are numerically studied using a reduced set of resistive magnetohydrodynamic equations. For a sequence of equilibria stable to ideal modes, the n = 1 mode changes from a tearing branch to a pressure‐driven branch as β_{ p } is increased. When this mode is unstable at low β, it is just the (m = 1; n = 1) tearing mode. Higher n modes also become linearly unstable with increasing β_{ p }; they are essentially pressure‐driven and have ballooning character. For low values of β, the instability is best described as a β_{ p } distortion of the (m = 1; n = 1) tearing mode. This mode drives many other helicities through toroidal and nonlinear couplings. As β_{ p } is increased, the growth of the m = 1 island slows down in time, going from exponential to linear before reconnection occurs. If β_{ p } is large enough, the island saturates without reconnection. A broad spectrum of other modes, driven by the (m = 1; n = 1) instability, is produced. These results agree with some observed features of magnetohydrodynamic activity in ISX‐B.

Effect of collisions on dc magnetic‐field generation in a plasma by resonance absorption of light
View Description Hide DescriptionThe importance of collisional ponderomotive effects on dc magnetic field generation is stressed. Computer simulations show that a weak rate of collision is sufficient to completely modify dc magnetic field generation in the resonant absorption of light, as compared with previous collisionless simulations. The agreement with theoretical predictions is shown.

Effect of turbulent diffusion on collisionless tearing instabilities
View Description Hide DescriptionThe direct interaction approximation is used to obtain the kinetic electron response in the presence of drift‐wave turbulence. Primary effects on the tearing mode equations are a diffusive broadening of the conductivity and an anomalous perpendicular viscosity. The first has a negligible effect on collisionless current channel modes. The viscous term, however, produces significant stabilization of the m = 1 mode for levels of turbulence much smaller than those expected.

Some nonlinear properties of drift‐cyclotron modes
View Description Hide DescriptionThe nonlinear evolution of the drift‐cyclotron instability has been investigated by means of analytical perturbation theory and computer simulations with a particle‐fluid hybrid code. The nonlinear modification of the electrostatic ion response in the presence of a single drift‐cyclotron mode is calculated. It is determined that a nonlinear shift in the effective ion cyclotron frequency can stabilize long‐wavelength modes (k a _{ i }∼N/κa _{ i }, ω≊ω_{*}≊Nω_{ c i }, N = ±1, ±2, ...) at fairly small amplitudes of the electrostatic potential ‖eφ/T _{ i }‖∼(m _{ e }/m _{ i }+ω^{2} _{ c i }/ ω^{2} _{ p i })^{1/2}κa _{ i }, where ω_{*} is the ion diamagnetic drift frequency, a _{ i } is the ion gyroradius, T _{ i } is the ion temperature, and κ^{−1} is the density‐gradient scale length. These saturated states exhibit nonlinear oscillations, and analysis indicates that these states are stable to perturbations arising from other long‐wavelength drift‐cyclotron modes. A shift of the ion cyclotron frequency is not the dominant nonlinear feature in the case of more unstable short‐wavelength modes [k a _{ i }∼(m _{ e }/m _{ i }+ω^{2} _{ c i } /ω^{2} _{ p i })^{−1/2}, ω≊Nω_{ c i }∼ω_{*}/2], and the nonlinear ion response is much more difficult to calculate. Simulations indicate that the short‐wavelength modes eventually trap ions, as has been seen in previous simulations of ion cyclotron instabilities and the closely related lower‐hybrid drift instability.

Eikonal expansion for an anisotropic plasma
View Description Hide DescriptionThe eikonal expansion developed by Weitzner and Batchelor is reformulated to study the cyclotron resonance in an inhomogeneous anisotropic plasma. It is shown that, to leading order, the functional dependences of the Poynting theorem, the cyclotron damping or growth [or the energy absorption (or emission)] rates on the unperturbed states are unaltered by anisotropy.

Warm relativistic electron fluid
View Description Hide DescriptionEquations of motion are derived for a warm relativistic collisionless electron fluid. In particular, the appropriate modification of the adiabatic equation of state is given. Use of the modified equation of state is illustrated by a calculation of the thermal correction to the zero‐temperature electron susceptibility tensor.

Free‐boundary high‐beta tokamaks. I. Free‐boundary equilibrium
View Description Hide DescriptionThe free‐boundary problem of a sharp‐boundary high‐β tokamak plasma inside a conducting shell is solved. This problem is reduced to solving Laplace’s equation on a domain with an unknown inner boundary. Centering this boundary with respect to the center of the shell is effected by means of a Moebius transformation which facilitates the use of the fast Fourier transformation. The method exploits Green’s theorem for the linear part of the problem which is the solution of Laplace’s equation with given boundaries. The nonlinear part consists of moving the plasma boundary until pressure balance is obtained. Fast convergence to accurate results is obtained through the use of a judiciously chosen damping factor determining the response of the plasma shape to changes in the poloidal field pressure. This allows for a complete scan of the two‐dimensional parameter space characterized by the plasma shift Δ and the plasma thickness a. Expressions are derived for the maximum permissible value of the poloidal beta.

Modulational instability of ion‐acoustic turbulence
View Description Hide DescriptionThe adiabatic interaction of low‐frequency ion‐acoustic‐like perturbations with high‐frequency ion‐acoustic turbulence is studied. The turbulence is found to become unstable provided certain conditions are met. These conditions are derived explicitly.

Theory of nonlinear ion‐electron instability
View Description Hide DescriptionA nonlinear theory describing the interaction of ion and electron ’’clumps’’ in a turbulent Vlasov plasma is presented. Numerical and analytical solutions predict that, in the presence of a drift v _{ d } of the average electron distribution, a spectrum driven by such fluctuations will be unstable to drifts which are appreciably below those required for the onset of a linear instability. The instability exists over a large range of the parameter space v _{ d } and T _{ e }/T _{ i } (the electron to ion temperature ratio); moreover, the growth rate is amplitude dependent being approximately proportional to the trapping time. These results are in good agreement with recent computer simulations.

Trapped particle induced ion solitary waves in a magnetized plasma
View Description Hide DescriptionSolitary ion acoustic and ion cyclotron waves due to the trapped electron nonlinearity are investigated for a magnetized plasma. The solitons obtained are narrower than those due to the convective nonlinearity.

Energy principle with global invariants
View Description Hide DescriptionA variational principle is proposed for constructing equilibria with minimum energy in a toroidal plasma. The total energy is minimized subject to global invariants which act as constraints during relaxation of the plasma. These global integrals of motion are preserved exactly for all ideal motions and approximately for a wide class of resistive motions. We assume, specifically, that relaxation of the plasma is dominated by a tearing mode of single helicity. Equilibria with realistic current density and pressure profiles may be constructed in this theory, which is also used here to study current penetration in tokamaks. The second variation of the free energy functional is computed. It is shown that if the second variation of any equilibrium constructed in this theory is positive, the equilibrium satisfies the necessary and sufficient conditions for ideal stability.

Scaling laws for single‐shell DT gas‐filled spherical targets for heavy‐ion fusion
View Description Hide DescriptionScaling relationships for high‐z shell DT gas‐filled spherical targets irradiated by a constant power heavy‐ion beam pulse are derived and compared with numerical calculations. Target performance as a function of incoming heavy‐ion particle energy is numerically determined and limitations on the analytical scaling relationships identified.

A new class of fluid instabilities: Vorticity‐induced waveforms on falling parabolic jets
View Description Hide DescriptionSteady streams issuing from horizontal circular ducts form stationary surface waves whose angular symmetry depends on the vorticity distribution within the duct. For Poiseuille flow, the observed sequences of wavelengths and frequencies are shown to be consistent with Crocco’s theorem: With fall the stream narrows, inducing secondary flow patterns dependent on the rate of narrowing. These findings suggest a method for obtaining scaling estimates of prevalent vorticity patterns in ducts.