Collisional Mode Research Articles

The stress tensor for simple shear flows of a granular material

The complete stress tensor has been measured using a computer simulation of an assemblage of rough, inelastic spheres in an imposed simple shear flow. Only five components of the stress tensor were found to be significantly different from zero. These represent the disperssive normal stresses τxx, τyyand τzzand the in-the-shear-plane shear stresses τxyand τyx; furthermore, the two off-diagonal stresses, τxyand τyx, were found to be equal so that the resultant stress tensor is symmetric. Two modes of microscopic momentum transport produce the final macroscopic stress tensor: the streaming or kinetic mode by which particles carry the momentum of their motion as they move through the bulk material, and the collisional mode by which momentum is transported by interparticle collisions. The contribution of each to the final result is examined separately. The friction coefficient, the ratio of shear to normal force, is shown to decrease at dense packings for both the collisional and streaming modes. Also observed were normal stress differences, both in and out of the shear plane, reflecting anisotropies in the granular temperature.

Velocity shear and fluctuations in the auroral regions of the ionosphere

The theory of collisional electrostatic modes that are driven unstable by a sheared field‐aligned ion flow velocity (i.e., Vi0 = V0(x)ez, B0 = B0ez) is presented. The instability threshold condition is |dV0/dx| > 2(1 + Ti/Te)νin, where νin is the ion‐neutral collision frequency, the characteristic longitudinal wavelengths λ∥ > λe, λe being the electron collision mean free path, and the transverse wavelengths λ⊥ > 21/2πρi, ρi being the ion gyroradius. The linearly unstable modes can account for fluctuations with transverse wavelengths ranging from a kilometer to several meters for only a moderate amount of shear (in the parallel velocity). Thus the instability provides a direct means of exciting almost the entire range of the observed (Basu et al., 1988) subkilometer scale irregularities in the auroral F region. The quasi‐linear analysis presented here shows that an effective viscosity describing momentum transport in the x direction can be derived in the presence of the shear‐driven fluctuations. The spatial profiles of the excited modes are evaluated from the solution of the appropriate dispersion equation. A model is proposed to describe the nonlinear evolution of the instability while ensuring that the characteristics of the turbulent fluctuation spectra resulting from it are consistent with those of the observed spectra.

Fluctuations associated with sheared velocity regions near auroral arcs

The theory of electrostatic, collisional plasma modes that are driven unstable by a sheared field‐aligned ion flow velocity (i.e., Vio=Vo(x)eˆz, Bo=Boêz) is presented. The threshold condition for the instability is V′o/νin>2 (1+Ti/Te) and the unstable modes are characterized by k∥«k⟂ such that λ∥(=2π/k∥)»λm and 2πρi<λ⟂(=2π/k⟂)≲Lv, where V′o=dVo/dx, νin is the ion‐neutral collision frequency, λm is the electron mean‐free‐path, ρi is the ion gyroradius and Lv is the velocity shear scale length. It is shown that the linear instability provides a direct means of exciting the observed [Basu et al., 1986, 1988] sub‐km scale irregularities in the auroral F‐region. The spectral characteristics of the density and the electric field fluctuations that can be inferred from the theory are found to be consistent with the observations.

Quasilinear energy balance of resistive modes

The global energy balance of collisional and semicollisional resistive modes is related term by term to the quadratic form derived from the linear dispersion equation. The partition of the released magnetic and electron pressure energy into dissipation, electron compression along field lines, and ion energy is explicitly discussed for internal kink modes, tearing modes, and resistive Alfvén modes.

Finite ion temperature effects on collisional and semicollisional tearing modes

The finite ion gyroradius effects on the stability of global tearing modes are studied using analytic and numerical techniques. It is found that in the collisional regime the modes are weakly stabilized compared to the zero ion temperature case. In the semicollisional regime, the stabilization is weaker than the polarization drift stabilization in the zero ion temperature case. There is overall a very minor finite ion temperature modification of the stability boundary in the plasma beta, collisionality space.

Numerical simulation of turbulent heat flows

Previous work on simulating turbulence in finite beta tokamak plasmas with isothermal models is extended to include temperature perturbations and temperature gradients in a collisional plasma. Turbulent heat as well as plasma flows are treated. The toroidal ion temperature gradient mode is the dominant source of turbulence in the model. Electron temperature gradients can stabilize the electron collisional drift mode and lead to reverse plasma flows (or thermal pinches). Magnetic electron heat conduction is found to be very small when correlations between temperature and magnetic field perturbations are considered. Quasilinear theory and mixing length rules are found to give a good representation of the results.

Excitation of electrostatic fluctuations by thermal modulation of whistlers

It is shown that a large‐amplitude field‐aligned whistler wave can excite nonducted whistler sidebands and nonresonant low‐frequency electrostatic collisional gradient drift modes. The effects of the ponderomotive force as well as the heating nonlinearities are included in the study of the modulational instabilities. For this case it is found that the heating nonlinearity dominates over the ponderomotive force effects, provided that the doppler‐shifted frequency of the collisional modes is smaller than the electron collision frequency. Our investigation can be relevant in understanding why enhanced density irregularities appear in the presence of electromagnetic heater waves in the lower part of the ionosphere.

The stress tensor in a two-dimensional granular shear flow

A computer simulation is used to make a detailed study of the stress tensor in a simple shear flow of two-dimensional disks. The stresses are shown to arise from two momentum-transfer mechanisms: the 'streaming’ or kinetic mode. By which momentum is carried by particles are they move through the bulk material: and the collisional mode, by which momentum is transferred from one point to another in the material by interparticle collisions. As might be expected, the results show that the streaming mode dominates at disperse packings and the collisional mode dominates at dense packings. The friction coefficient, the ratio of shear to normal forces, is shown to decrease at high particle packing for both the collisional and streaming modes of transport. Normal-stress differences are observed within the shear plane and are evident in both the streaming and collisonal parts.

Parametric excitation of collisional modes in the high‐latitude ionosphere

This paper points out that the nonlinear coupling of a large‐amplitude electromagnetic pump wave with convective plasma fluid instabilities is significantly enhanced by pump‐induced temperature changes. It should thus be easy to exceed the threshold values for parametric instabilities in several ionospheric heating experiments.

Role of Coulomb collisions in the equatorial F-region plasma instabilities

The effect of collisions on electrostatic instabilities driven by gravity and density gradients perpendicular to the ambient magnetic field is studied. Electron collisions tend to stabilize the short wavelength ( k y ϱ i ⪢ 1, where k y is the perpendicular wavenumber of the instability and ϱ i is the ion Larmor radius) kinetic interchange mode. In the presence of weak ion-ion collisions, this mode gets converted into an unmagnetized ion interchange mode which has maximum growth rate one order smaller than that of the collisionless mode. On the other hand, electron collisions can excite a long wavelength resistive interchange mode in a wide wavenumber regime ( 10 −3 ⩽ k y ϱ i ⋍ 0.3 ) with growth rates comparable to that of the collisional Rayleigh-Taylor mode. The results may be relevant to some of the spread F irregularities.

Stimulated scattering by collisional modes in the ionosphere

We point out that it ought to be possible to observe stimulated scattering by the two‐stream Farley‐Buneman and the gradient drift modes in the ionosphere with, for example, the radar facilities in northern Scandinavia.

Laboratory simulation of planetesimal collision

Low‐velocity impact experiment on rocks was performed to reveal a nature of collision process of planetesimals. Projectiles of mild steel (S15CK) and rocks (tuff and basalt) with velocities of 17 to 270 m/s were impacted against four kinds of rocks (tuff, basalt, granite, and dunite) with various shapes and sizes. Imparted energy divided by target mass Ei ranges 106 to 108 erg/g. The phenomena associated with the impact of mild steel against spherical targets were classified into five categories. The categories vary with increase in impact velocity as follows: (1) rebound of projectile with no contact damage on target, (2) rebound of projectile with fragmentary chips, and creations of a contact damage (crater?) and radiating fissures on target, (3) rebound of projectile with longitudinal splitting of target, (4) destruction of target with producing shatter conelike fragments, and (5) complete destruction of target. We cannot see any difference in the above categories between various kinds of rocks. The shape of the target also does not influence the categories significantly. The size distribution of the fragments produced by the complete destruction was well expressed by an inverse power law relation, n(l) dl ∝ l−α−1dl, where n(l) is the incremental number of the fragments within a linear increment dl, l is the size, and α is constant; α seems to increase with increase in Ei,. An empirical relationship between α and Ei was approximately given by α∝ log Ei. Empirical formulae for the maximum fragment mass normalized by the original target mass (Ml/Mt) and Ei were also presented for each rock type. It is found from the shape distribution of the fragments produced by the shatter cone type destruction that the length ratios between the shortest and the longest axes, c/a, and between the intermediate and the longest axes, b/a, were greater than 0.2 and 0.3, respectively. Average length ratio of a:b:c is roughly given by 2:√2:1. Collisional phenomena associated with low‐velocity impact experiment were shown to be different from those observed in the high‐velocity impact experiment. The difference in mechanical properties between projectile and target significantly influenced the collisional behavior. For collision between rocks with similar mechanical properties, relative size between projectile and target played an important role in determining the collisional mode: Only the smaller body was destroyed completely, and the larger body suffered no damage even if released Ei was enough to destroy both colliding bodies simultaneously. This observation suggests an important implication in the early stage of planetary formation, that runaway growth of the largest planetesimal takes place even when the collision condition prefers catastrophic destruction to other collision types.

Critical Conditions for Drift, Drift-Alfvén and Drift-Tearing Instabilities of Finite-β Plasma in Sheared Magnetic Field

Kinetic theory of low frequency instabilities of a current-carrying finite-β plasma in a sheared magnetic field is investigated. Drift mode, drift-Alfven mode and drift-tearing mode are identified both in collisionless and collisional regimes and each critical stability condition is obtained, We find a new electromagnetic localized mode driven by the plasma current in a finite-β plasma. Both the current and finite-β effects are required to excite this mode. It is also shown that, in the absence of the plasma current, any electromagnetic localized mode which is propagating in the direction of the electron diamagnetic drift is stable in the collisionless limit of a finite ion temperature plasma. For the collisional mode, stability holds if ω>ω * (electron diamagnetic frequency) or ω<0.

Nonlinear transport instability in a diffusive plasma

Fine structures of the unstable waves and the electron density profiles resulting from nonlinear spatial transport are studied numerically for a collisional flute mode instability. It is confirmed that the nonlinear transport produces regular pulse trains in both the electron density and wave amplitude profiles in the direction perpendicular to the electron drift (in the direction of the density gradient). Fourier spectral analysis of the nonlinear structure in that direction indicates two major spectral peaks. The main peak is the fundamental mode which leads to overall flattening of the density distribution in the entire unstable region. The second peak corresponds to the generated small-scale pulses resulting from a nonlinear instability associated with a strong wave-induced transport. In the steady state the pulses tend to merge and be smeared out to form a plateau in the density profile. These indicate the presence of two different, steady and pulsative, transports. Examination of the spatial structure finds that the pulse width is roughly the same size as the dominant wavelength of the primary wave which propagates in the direction of the drift. It is further found that the observed nonlinear mode can be explained by a nonlinear transport equation of the form nt+(n2)z+ky−2(n2)zzz=0, where ky is the wavelength of the dominant primary wave propagating in the direction of the drift and z denotes the coordinate perpendicular to the drift.

Thresholds of purely growing instabilities in the presence of an intense lower-hybrid pump wave

It is found that a wave near the lower-hybrid frequency may parametrically excite purely growing instabilities. Modulations by ion cyclotron and collisional modes are considered. The instability thresholds and the widths of the excited spectra are presented.

Filamentation instabilities associated with ordinary electromagnetic waves in collisional plasmas

It is shown that parametric coupling of an ordinary electromagnetic wave with purely damped convective cell modes as well as other collisional modes leads to purely growing perturbations in the direction perpendicular to the incoming radiation. The excitation of purely growing modes in a magnetized plasma is important because the resulting E × B drift can cause particle transport across the external magnetic field.

The collisional drift mode in a partly-ionized plasma

The structure of the drift instability is examined in several density regimes. Let λ e be the total electron mean free path, kz the wave-vector component along the magnetic field, and ν ⊥ / ν ∥ the ratio of perpendicular ion diffusion to parallel electron streaming rates. At low densities (kz ν e &gt; 1), the drift mode is isothermal, and should be treated kinetically. In the finite heat conduction regimethe drift instability threshold is reduced at low densities (v⊥/v‖&lt;0·1) and increased at high densities (v⊥/v‖&gt;0·1), as compared with the isothermal threshold. Finally, in the energy transfer limit (kzλe&lt;(m/M)½), the drift instability behaves adiabatically in a fully-ionized plasma, and isothermally in a partly-ionized plasma, for an ion-neutral to Coulomb collision frequency ratio vin/vii&gt;2(m/M)½ at Ti = Tc = Tn.

Nonlinear Saturation of the Trapped-Ion Mode

A nonlinear model of the collisional trapped-ion mode is presented, in which the energy in long-wavelength instabilities is transferred to short-wavelength modes which are then damped by ion-bounce resonances. Near marginal stability, the saturation of a single unstable Fourier mode is computed. Far from marginal stability, steady-state nonlinear solitary waves containing many Fourier modes are found. Particle transport is estimated in both cases.

Kinetic theory of a collision-dominated plasma

The kinetic theory of a classical electron gas is improved by taking into account higher-order collective effects describing collisional relaxation of the binary correlations. The resulting kinetic equation takes the form of a renormalized Balescu-Guernsey-Lenard equation, in which the free-particle propagators are replaced by collisional modes. This new kinetic equation conserves the total particle number, momentum and kinetic energy and satisfies an H-theorem. Finally we discuss briefly the most important modifications of the new collision term with respect to its BGL counterpart.

Cavity Detuning and Multimode Operation of an Optically Pumped Gas Laser

In an earlier paper, the problem of the optically pumped gas laser was treated within the formalism of a modified Boltzmann equation and the equation of radiative transfer. The laser cavity was tuned precisely to the center of the atomic lasing frequency and only one cavity mode was excited above threshold. These restrictions are now removed and the effects on the laser population inversion, gain, and power output are presented. In the limit that Doppler broadening dominates collisional broadening, it is found that the power output for the single-mode operation increases as the laser cavity is detuned, until an optimum occurs when the cavity detuning is equal to the collisional linewidth. In the limit that collisional broadening dominates Doppler effects, power output is independent of detuning, until the detuning approaches the collisional linewidth. For multimode operation, it is shown that power output increases approximately linearly with the number of modes in oscillation, provided that the Doppler width is much larger than the collisional width and mode interaction is not occurring. When collisional broadening dominates, no increase in total power output is achieved by allowing more than one mode to go into oscillation. Most of these results are well known, for low levels of lasing intensity, through the semiclassical analysis of Lamb. However, the formulation here provides a more quantitative evaluation of the relative roles of inhomogeneous and homogeneous broadening for all collisional linewidths and high lasing intensities.