Index of content:
Volume 24, Issue 12, December 1981

Modeling of vortex‐corner interaction using point vortices
View Description Hide DescriptionUsing point vortices, the interaction of a single vortex, as well as patterns of vortices, with a corner is examined; comparisons are made with corresponding experiments. Trajectories of a vortex swept past the corner can be well‐approximated, provided that a sufficiently weak strength of the vortex is specified. Calculations show that the dimensionless amplitude of the pressure fluctuation at the corner is very sensitive to small variations of the initial position of the vortex, and relatively insensitive to variations in strength; this finding has important consequences for recently observed amplitude modulation of pressure at, and velocity near, impingement. In fact, by prescribing transversely staggered patterns of vortices upstream of the corner, determined from experimental flow visualization, the form of the time‐averaged pressure and velocity spectra can be approximated. The most critical feature of these spectra, a well‐defined low‐frequency component(s), confirms the hypothesized mechanism associated with low‐frequency modulation observed in experiments. In addition, shortcomings of this method, primarily due to distributed vorticity inherent in laboratory vortices, are pointed out.

Shape of shock wave produced by a concentrated impact on a surface
View Description Hide DescriptionAn approximate similarity solution, derived by Raizer, of a concentrated impact (or intense explosion) at the boundary of a semi‐infinite volume of a perfect gas is used to determine the propagation velocity of the shock front as a function of its position. This velocity function is then used to obtain the shape of the propagating shock wave. It is shown that dish‐shaped shock fronts are formed when the movement of the gas at the surface is into the gas region and that cup‐shaped shock fronts are formed when the movement is out of the gas region. Comparison of these results with the shapes of explosions and meteorite craters are discussed.

Explosions on a gas‐vacuum interface
View Description Hide DescriptionA finite‐difference computer code is used to calculate the time development of an explosion on a gas‐vacuum interface. An analytic theory of the shape of the shock wave produced in the explosion is compared with the results of the computer simulation. The assumptions used in obtaining this analytic solution are verified, and the degree to which the variables describing the explosion are self‐similar is examined. Finally, certain consistency relations among the similarity exponents are tested.

Strong evaporation in half‐space: Integral transport solutions for one‐dimensional Bhatnagar–Gross–Krook model
View Description Hide DescriptionThe strong evaporation problem in half‐space for the one‐dimensional Bhatnagar–Gross–Krook model is considered, and it is shown that highly accurate results in the lowest approximation can be obtained by a judicious use of two different integral transport methods. Extensions of the work to higher order kinetic models is discussed, and some comments regarding obtaining accurate results for the Boltzmann equation by use of the generalized Maxwell method are also included.

Rarefaction shocks in spherical geometry
View Description Hide DescriptionThe uniform density region that develops behind a rarefaction shock in planar geometry (and attaches downstream to the outer expansion flow) is shown to be modified by geometric divergence into a region of self‐similar flow with nonvanishing gradients. A single set of equations is derived that describes both shock and similarity flow. The longest gradient scale length in the transition region is shown to vary linearly with the radius at the shock.

Kinetic theory of the plasma sheath transition in a weakly ionized plasma
View Description Hide DescriptionThe boundary layer of a weakly ionized rare gas discharge is treated using kinetic theory. A self‐consistent two‐scale analysis is performed and the exact presheath and sheath solutions are constructed for the following model: A plasma consisting of Boltzmann distributed electrons and singly charged atomic ions is in contact with a negative absorbing wall. The ion kinetics is dominated by charge exchange with cold neutrals (T _{0}≪T _{−}). The mean‐free‐path λ is constant and large compared with the electron Debye length λ_{D}. This model represents a collision‐dominated counterpart to the well known Tonks–Langmuir model of the collision‐free plasma.

Electron motion in a wave of slowly varying amplitude
View Description Hide DescriptionA theory of electron motion in a spatially sinusoidal wave of slowly varying amplitude is presented. On the basis of a reformulation of classical conservative motion, a set of approximate analytical, adiabatic solutions for the case of a temporally varying amplitude is derived. The set can describe electron transitions from trapped to untrapped conditions, or vice versa, and provides a formally unified description of both kinds of motion through Jacobi’s real transformation.

Damping of an electron plasma wave with detrapping of the electrons
View Description Hide DescriptionA self‐consistent calculation of the damping of an electron plasma wave is presented including, for the first time, the contribution of all electrons (untrapped, detrapped, and trapped). The effect of the detrapped electrons, which is strong at the higher damping rates, is to lower the first minimum of the amplitude after the initial stage of Landau damping and the general level of its oscillation afterward. At γ_{L}τ_{0}≊−0.75, a qualitative change is observed from amplitude oscillation to near‐plateau behavior. The existence of an important secularity in the Vlasov–Poisson system is pointed out.

Universal drift waves in a low‐aspect‐ratio spheromak geometry
View Description Hide DescriptionAn eigenmode equation for drift waves in low‐aspect‐ratio spheromak geometry is derived and solved numerically. In contrast to the large‐aspect‐ratio tokamak limit, the strong equilibrium variation involved here, allows partially localized modes at more than one value of the poloidal angle.

Dielectric tensor of a weakly relativistic, nonequilibrium, and magnetized plasma
View Description Hide DescriptionThe dielectric tensor of a weakly relativistic magnetized plasma is discussed for wave frequencies very close to the fundamental or higher harmonics of the electron or ion cyclotron frequency. Since in real situations the plasmas of interest are often in nonequilibrium states, the present paper generalizes Shkarofsky’s result to include three nonequilibrium features; namely, a loss‐cone distribution, anisotropic temperatures, and a field‐aligned drift. The discussion is motivated by current interest in the study of absorption and emission of radiation near the electron cyclotron harmonics.

Absorption and emission of extraordinary‐mode electromagnetic waves near cyclotron frequency in nonequilibrium plasmas
View Description Hide DescriptionTwo cases are investigated. In the first case the electron distribution is assumed to have a loss‐cone feature. It is found that the plasma can amplify electromagnetic waves near the electron cyclotron frequency, as expected. However, the growth rate is significantly larger than that used in an approximate calculation in which a slightly different physical model was considered. The present discussion is restricted to the fast extraordinary‐ mode radiation. One of the important conclusions is that, due to the combined relativistic and loss‐cone effects, the cutoff frequency can be lower than the electron cyclotron frequency, a result in contrast with that obtained with the usual ’’cold plasma’’ approximation. In the second case a drift motion along the ambient magnetic field is studied. It is found that the cyclotron damping can occur under certain conditions.

Electromagnetic radiation from an inhomogeneous plasma: Theory and experiment
View Description Hide DescriptionThe problem of the linear conversion of a Langmuir wave to a transverse electromagnetic wave in the presence of a density gradient has been solved numerically with appropriate boundary conditions. A reciprocity principle was found, allowing the deduction of solutions of this problem from those obtained from the transverse‐Langmuir conversion. This model has been applied to study the spectrum emitted from an inhomogeneous plasma, including the effect of the antenna radiation pattern. Experiments have been performed in a large unmagnetized dc discharge plasma (n _{ e }∼5×10^{10} cm^{−3}, T _{ e } = 1.3 eV, gradient scale length L = 100−1000 cm). The shape of the spectrum observed with a horn antenna agrees with the theoretical one, but the deduced level of Langmuir fluctuations is much higher than the thermal level. This enhancement is due to the presence of primary energetic (E = 60 eV) electrons.

Tearing‐mode stability properties of a diffuse anisotropic field‐reversed ion layer at marginal stability
View Description Hide DescriptionStability properties are investigated for purely growing (Reω = 0) tearing modes at marginal stability (Imω = 0) for a rotating, nonrelativistic cylindrically symmetric ion layer immersed in an axial magnetic field. The analysis is carried out within the framework of a Vlasov‐fluid model in which the electrons are described as a macroscopic cold fluid, and the layer ions are described by the Vlasov equation. Tearing‐mode stability properties are calculated numerically for azimuthally symmetric perturbations about an ion layer equilibrium with temperature anisotropy. The marginal stability eigenvalue equation is solved numerically for the perturbation amplitude and the normalized axial wavenumbers at marginal stability in terms of temperature anisotropy, layer radius, and magnetic field depression on the axis. It is found that the range of unstable wavenumbers decreases as T_{‖‖}/T _{⊥} is increased, and numerical estimates are made of the anisotropy required for complete stabilization.

Anomalous impurity ion transport due to magnetic fluctuations
View Description Hide DescriptionA positive ambipolar potential arising from large transport rates of electrons relative to ions in stochastic magnetic fields gives an outward force on the ions. It can reverse the neoclassical inward convection of the impurity ions in tokamaks. This process may by responsible for the observed decays of line emissions from high charge states of silicon in an impurity study experiment in Alcator.

Magnetic field generation in the underdense plasma
View Description Hide DescriptionMagnetic field generation in the underdense plasma due to the nonlinear interaction of an electromagnetic wave and a plasma is investigated. It is shown that the collisional absorption of the electromagnetic wave induces a nonlinear anistropy in the electron pressure tensor, which is responsible for a large source of the dc magnetic field. Six different regimes of interest appear, depending on four characteristic parameters related to the collision rate, the thermal effects, hydrodynamic convection, and the duration of the experiment. The theory is supported by a detailed comparison with recent microwave plasma experiments.

Ponderomotive effects and magnetic field generation due to short‐wavelength ion turbulence
View Description Hide DescriptionThe ponderomotive effects and magnetic field generation due to the nonlinear interaction of short‐wavelength ion turbulence with an electromagnetic wave are studied using the Dawson and Oberman model, which exactly describes the electron‐ion interaction in the limit of infinite ion‐to‐electron mass ratio. The differences from the results of the heuristic Landau model are stressed, in particular for nonthermal ion correlations and for an electromagnetic wave frequency slightly exceeding the electron plasma frequency.

Quasi‐linear saturation of coherent backward ionization modes
View Description Hide DescriptionQuasi‐linear saturation of the amplitude of coherent backward ionization waves is studied using the slowly varying parameters or Van der Pol’s method. The result is a nonlinear Landau equation which describes the growth and saturation of the wave amplitude. The analytical expression of the most important nonlinear saturation term predicts results in accordance with earlier experimental observations.

Necessary stability conditions for field‐reversed theta pinches
View Description Hide DescriptionToroidal systems of arbitrary cross section without toroidal magnetic field are analyzed via the double adiabatic fluid equations. Such systems are shown to be unstable if there exists one closed field line on which the average of κr B ^{2} is positive, where κ is the curvature. A similar criterion is derived for linear system and is applied to a noncircular z pinch.

Finite Larmor radius effect on thermosolutal instability of a plasma
View Description Hide DescriptionThe thermosolutal instability of a plasma in the presence of a transverse magnetic field is studied to include the effects of the finiteness of the ion Larmor radius. The finite Larmor radius and stable solute gradient are found to have stabilizing effects on stationary convection. The case of overstable modes is also considered wherein necessary conditions for the existence of overstability are derived.

Variational structure of the Vlasov equation
View Description Hide DescriptionThe variational structure of the Vlasov–Maxwell integral equations is derived for a plasma equilibrium having two ignorable coordinates. It is shown that the kernel of the Maxwell equations is a self‐adjoint integral operator. This operator may also be represented as a differential equation of arbitrary order. This representation is useful when the differential operator is truncated to finite order, yielding a system of intrinsically self‐adjoint differential equations.