Search

Helmholtz decomposition is applied to investigate the compressibility effect in hypersonic turbulent boundary layers. The dilatational component of flow plays an important role in the near-wall region, while the solenoidal component is dominant far from the wall. The cold wall condition significantly enhances the compressibility, and especially enhances compression motions near the wall.

Compressibility effects on pressure fluctuations in wall-bounded turbulent flows at various Mach numbers are investigated by deriving a pressure Poisson equation that allows splitting pressure fluctuations into rapid and slow terms, as in incompressible flows, and into additional mass-flux and viscous terms. The rapid and slow terms are further split into their solenoidal and dilatational components by using Helmholtz decomposition. The solenoidal components are found to be independent of Mach number and similar to incompressible flows, whereas the dilatational components increase quadratically with Mach number.

Finite dilatational effects in wall-bounded compressible turbulence are investigated. We find that finite correlation between the solenoidal and the dilatational parts of the velocity field account for a nonnegligible fraction of the turbulent shear stress near walls.

Pointwise zero velocity helicity density is shown not to prevent steady flows from acting as kinematic dynamos. We present numerical evidences that such flows can generate both small-scale magnetic fields as well as, by the magnetic $\alpha$-effect or negative eddy diffusivity mechanisms, large-scale ones. The flows are constructed as curls of analytically defined space-periodic steady solenoidal flows, whose vorticity helicity (i.e., kinetic helicity) density is everywhere zero.

Previous work (S. Davidovits and N. J. Fisch, "Sudden viscous dissipation of compressing turbulence," Phys. Rev. Lett., 116(105004), 2016) demonstrated that the compression of a turbulent field can lead to a sudden viscous dissipation of turbulent kinetic energy (TKE), and suggested this mechanism could potentially be used to design new fast-ignition schemes for inertial confinement fusion. We expand on previous work by accounting for finite Mach numbers, rather than relying on a zero-Mach-limit assumption as previously done. The finite-Mach-number formulation is necessary to capture a self-consistent feedback mechanism in which dissipated TKE increases the temperature of the system, which in turn modifies the viscosity and thus the TKE dissipation itself. Direct numerical simulations with a tenth-order accurate Pad\'e scheme were carried out to analyze this self-consistent feedback loop for compressing turbulence. Results show that, for finite Mach numbers, the sudden viscous dissipation of TKE still occurs, both for the solenoidal and dilatational turbulent fields. As the domain is compressed, oscillations in dilatational TKE are encountered due to the highly-oscillatory nature of the pressure dilatation. An analysis of the source terms for the internal energy shows that the mechanical work term dominates the viscous turbulent dissipation. As a result, the effect of the suddenly dissipated TKE on temperature is minimal for the Mach numbers tested. Moreover, an analytical expression is derived that confirms the dissipated TKE does not significantly alter the temperature evolution for low Mach numbers, regardless of compression speed. The self-consistent feedback mechanism is thus quite weak for subsonic turbulence, which could limit its applicability for inertial confinement fusion.

We introduce six families of three-dimensional space-periodic steady solenoidal flows, whose kinetic helicity density is zero at any point. Four families are analytically defined. Flows in four families have zero helicity spectrum. Sample flows from five families are used to demonstrate numerically that neither zero kinetic helicity density, nor zero helicity spectrum prohibit generation of large-scale magnetic field by the two most prominent dynamo mechanisms: the magnetic $\alpha$-effect and negative eddy diffusivity. Our computations also attest that such flows often generate small-scale field for sufficiently small magnetic molecular diffusivity. These findings indicate that kinetic helicity and helicity spectrum are not the quantities controlling the dynamo properties of a flow regardless of whether scale separation is present or not., Comment: 37 pages, 11 figures, 54 references

Background: The neutron β-decay asymmetry parameter A_0 defines the angular correlation between the spin of the neutron and the momentum of the emitted electron. Values for A_0 permit an extraction of the ratio of the weak axial-vector to vector coupling constants, λ≡gA/gV, which under assumption of the conserved vector current hypothesis (gV=1) determines gA. Precise values for gA are important as a benchmark for lattice QCD calculations and as a test of the standard model. Purpose: The UCNA experiment, carried out at the Ultracold Neutron (UCN) source at the Los Alamos Neutron Science Center, was the first measurement of any neutron β-decay angular correlation performed with UCN. This article reports the most precise result for A_0 obtained to date from the UCNA experiment, as a result of higher statistics and reduced key systematic uncertainties, including from the neutron polarization and the characterization of the electron detector response. Methods: UCN produced via the downscattering of moderated spallation neutrons in a solid deuterium crystal were polarized via transport through a 7 T polarizing magnet and a spin flipper, which permitted selection of either spin state. The UCN were then contained within a 3-m long cylindrical decay volume, situated along the central axis of a superconducting 1 T solenoidal spectrometer. With the neutron spins then oriented parallel or anti-parallel to the solenoidal field, an asymmetry in the numbers of emitted decay electrons detected in two electron detector packages located on both ends of the spectrometer permitted an extraction of A_0. Results: The UCNA experiment reports a new 0.67% precision result for A_0 of A_0=−0.12054(44)_(stat)(68)_(syst), which yields λ=gA/gV=−1.2783(22). Combination with the previous UCNA result and accounting for correlated systematic uncertainties produces A0=−0.12015(34)stat(63)syst and λ=gA/gV=−1.2772(20). Conclusions: This new result for A0 and gA/gV from the UCNA experiment has provided confirmation of the shift in values for gA/gV that has emerged in the published results from more recent experiments, which are in striking disagreement with the results from older experiments. Individual systematic corrections to the asymmetries in older experiments (published prior to 2002) were >10%, whereas those in the more recent ones (published after 2002) have been of the scale of

A tractor beam is a traveling wave that transports illuminated objects back to its source, opposite to the wave's direction of propagation, along its entire length. The requisite retrograde force arises when an object scatters the wave's momentum density downstream into the direction of propagation and then recoils upstream by conservation of momentum. Achieving this condition imposes constraints on the structure of the wave, which we elucidate in the Rayleigh limit, when the wavelength exceeds the size of the object. Continuously propagation-invariant modes such as Bessel beams do not satisfy these conditions at dipole order in the multipole expansion and so cannot serve as general-purpose long-ranged tractor beams. Modes with discrete propagation invariance, however, can act as first-order tractor beams. We demonstrate this by introducing a class of minimal solenoidal waves together with a set of design criteria that distinguish tractor beams that pull objects from repulsor beams that push them.

Charge separation induced by the chiral magnetic effect suggested that some ${\cal P}$- or ${\cal CP}$-odd metastable domains could be produced in a QCD vacuum in the early stage of relativistic heavy-ion collisions. Based on a multi-phase transport model, our results suggest that a domain-based scenario with final state interactions can describe the solenoidal tracker at RHIC detector (STAR) measurements of both same- and opposite-charge azimuthal angle correlations, $$, in Au+Au collisions at $\sqrt{s_{_{\rm NN}}}=200$ GeV. The occupancy factor of the total volume of domains over the fireball volume is small, which indicates that the size and number of metastable domains should be relatively small in the early stage of a quark-gluon plasma., Comment: 4 pages, 4 figures, 1 table, final published version

In a muon accelerator complex, a target is bombarded by a multi-MW proton beam to produce pions, which decay into the muons which are thereafter bunched, cooled, and accelerated. The front end of the complex captures those pions, then manipulates their phase space, and that of the muons into which they decay, to maximize the number of muons within the acceptance of the downstream systems. The secondary pion beam produced at the target is captured by a high field target solenoid that tapers down to a constant field throughout the rest of the front end. In this study we enhance the useful muon flux by introducing a new design of the longitudinal profile of the solenoid field at, and downstream of, the target. We find that the useful muon flux exiting the front end is larger when the field at the target is higher, the distance over which the field tapers down is shorter, and the field at the end of the taper is higher. We describe how the solenoid field profile impacts the transverse and longitudinal phase space of the beam and thereby leads to these dependencies.

It is shown that there exists a new class of cold-fluid corkscrewing elliptic beam equilibria for ultrahigh-brightness, space-charge-dominated beam propagation through a linear focusing channel consisting of uniform solenoidal, periodic solenoidal, and/or alternating-gradient quadrupole focusing magnets in an arbitrary arrangement including field tapering. The equilibrium beam density and flow velocity profiles and equilibrium self-electric and self-magnetic fields are determined by solving generalized beam envelope equations. In proper limits, such cold-fluid corkscrewing elliptic beam equilibria recover many familiar beam equilibria in beam physics, including the round rigid-rotor Vlasov beam equilibria in uniform and periodic solenoidal focusing fields and the Kapchinskij-Vladimirskij beam equilibrium in an alternating-gradient quadrupole focusing field. For beams with negligibly small emittance, the equilibrium solutions are validated with self-consistent simulations. Examples and applications of the present equilibrium beam theory are discussed. As an important application of the present equilibrium beam theory, a general technique is developed and demonstrated with an example to control large-amplitude density and flow velocity fluctuations (such as beam hollowing and halo formation) often observed in ultrahigh-brightness beams.

We report a measurement of the spin-dependent total cross section ratios {delta}{sigma}{sub T}/{sigma}{sub tot} and {delta}{sigma}{sub L}/{sigma}{sub tot} of the pp{yields}pp{pi}{sup 0} reaction between 325 and 400 MeV. The experiment was carried out with a polarized internal target in a storage ring. Nonvertical beam polarization was obtained by the use of solenoidal spin rotators. Near threshold, the knowledge of both spin-dependent total cross sections is sufficient to deduce the strength of certain participating partial waves, free of any model. (c) 1999 The American Physical Society.

Field theoretical renormalization group methods are applied to a simple model of a passive scalar quantity advected by the Gaussian non-solenoidal (``compressible'') velocity field with the covariance $\propto\delta(t-t')| x-x'|^{\epsilon}$. Convective range anomalous scaling for the structure functions and various pair correlators is established, and the corresponding anomalous exponents are calculated to the order $\epsilon^2$ of the $\epsilon$ expansion. These exponents are non-universal, as a result of the degeneracy of the RG fixed point. In contrast to the case of a purely solenoidal velocity field (Obukhov--Kraichnan model), the correlation functions in the case at hand exhibit nontrivial dependence on both the IR and UV characteristic scales, and the anomalous scaling appears already at the level of the pair correlator. The powers of the scalar field without derivatives, whose critical dimensions determine the anomalous exponents, exhibit multifractal behaviour. The exact solution for the pair correlator is obtained; it is in agreement with the result obtained within the $\epsilon$ expansion. The anomalous exponents for passively advected magnetic fields are also presented in the first order of the $\epsilon$ expansion., Comment: 31 pages, REVTEX file. More detailed discussion of the one-dimensional case and comparison to the previous paper [20] are given; references updated. Results and formulas unchanged

The properties and structure of second-order ~Cartesian! correlation tensors are derived for the general case of two solenoidal random vector fields. The theory is intended to describe homogeneous magnetohydrodynamic turbulence, with no assumed rotational or reflectional symmetries. Each correlation tensor can be written in terms of four scalar generating functions and the relationship of these functions to the potentials that generate the poloidal and toroidal components of the underlying vector fields is derived. The physical nature of the scalar functions is investigated and their true or pseudoscalar character is ascertained. In our general discussion we clarify several misleading statements dating back to Robertson’s original paper in the field @Proc. Camb. Philos. Soc. 36, 209 ~1940!#. It is also shown that using the one-dimensional correlation function, it is possible to obtain spectral information on the induced electric field in directions perpendicular to the measurement direction. @S1063-651X~97!09208-8#

A new rigid-rotor Vlasov equilibrium is obtained for an intense, axisymmetric charged-particle beam with uniform density in the radial direction propagating through a periodic solenoidal focusing field. The beam envelope equation is derived, and examples of periodically focused rigid-rotor Vlasov equilibria are presented. Statistical properties and possible applications of the present beam equilibrium are also discussed. {copyright} {ital 1997} {ital The American Physical Society}

The dynamics of continuous space-charge-dominated beams propagating through a periodic solenoidal focusing channel is studied using a test-particle model. It is shown that nonlinearities in the self-fields induce chaotic particle motion and beam halo formation for beams that are root-mean-square matched into the focusing channel but have nonuniform density profiles transverse to the direction of beam propagation. In particular, two parabolic density profiles are considered. For beams with hollow density profiles, it is found that excessive space charge at the edge of the beam induces two pairs of stable and unstable period-one orbits in the vicinity of the beam core envelope, and that the chaotic layer associated with the unstable period-one orbits allows particles to escape from the core to form a halo. On the other hand, for beams with hump density profiles (i.e., with high densities on the beam axis and low densities at the beam edge), it is found that excessive space charge on the beam axis induces an unstable fixed point on the axis and two stable period-one orbits off the axis inside the beam, and that the chaotic layer associated with the unstable fixed point is responsible for halo formation. In both cases, the halo ismore » found to be bounded by a Kolmogorov-Arnold-Moser surface. The ratio of halo to beam core envelope is determined numerically. {copyright} {ital 1997} {ital The American Physical Society}« less

A self-consistent two-dimensional model is used to investigate intense charged-particle beam propagation through a periodic solenoidal focusing channel, particularly in the regime in which there is a mismatch between the beam and the focusing channel. The present self-consistent studies confirm that mismatched beams exhibit nonlinear resonances and chaotic behavior in the envelope evolution, as predicted by an earlier envelope analysis [C. Chen and R. C. Davidson, Phys. Rev. Lett. 72, 2195 (1994)]. Transient effects due to emittance growth are studied, and halo formation is investigated. The halo size is estimated. The halo characteristics for a periodic focusing channel are found to be qualitatively the same as those for a uniform focusing channel. A threshold condition is obtained numerically for halo formation in mismatched beams in a uniform focusing channel, which indicates that relative envelope mismatch must be kept well below 20% to prevent space-charge-dominated beams from developing halos.

A 9[degree] solenoidal spin rotator or 5% partial Siberian Snake was used to successfully accelerate polarized protons for the first time to 10.8 GeV kinetic energy in the Brookhaven Alternating Gradient Synchrotron with partial depolarization due to the intrinsic resonance crossing. It was found that a 5% partial snake can effectively overcome 18 imperfection resonances in this energy range. We also observed an interference between the spin flip induced by an intrinsic resonance and linear coupling due to the solenoid field of the partial snake.

Omega-P, Inc., 258 Bradley Street, New Haven, Connecticut 06510, USA(Received 21 October 2010; published 31 January 2011)A nonlinear self-consistent theory has been constructed and used to investigate numerically thewakefield excitation in multilayered dielectric resonators by relativistic electron bunches. Analyticalexpressions for solenoidal and potential components of an excited electromagnetic field have beenderived. The excitation of a five-zone dielectric resonator by relativistic electron bunches was numericallyinvestigated and comparison was made between the longitudinal distribution of an axial electric field andthe results obtained previously for a corresponding problem in the waveguide formulation. The necessityof optimizing geometrical parameters of the resonator to reduce mode amplitudes nonresonant with abunch, and to obtain a symmetric distribution of the longitudinal electric field component in the drive andaccelerating channels, has been demonstrated.

In this Brief Report, a confinement physics analysis presented previously [D. D. Dolliver and C. A. Ordonez, Phys. Rev. E 59, 7121 (1999)] is extended to consider three-dimensional electric confinement instead of one-dimensional electric and two-dimensional magnetic confinement for thermal, high-charge-state ion plasmas in nested-well solenoidal traps. Self-consistent numerical results are presented.

We discuss the general problem of generating confining potentials for neutral matter by combining different kinds of static forces. The interactions taken into account are those having a linear or quadratic dependence on the modulus of an irrotational and solenoidal field. Particular attention is devoted to the combination of electric and magnetic forces for paramagnetic species. In this case, tight confinement can be achieved by enhancing and balancing the field gradients. Combined potentials turn out to be conveniently generated by configurations based on high permeability materials. Feasibility of combined electric-magnetic traps and guides for neutral alkali-metal atoms is discussed.

We explore a practical approach for designing ionization cooling channels with periodic solenoidal focusing. We examine the lattice characteristics in terms of the properties of the coils and the cell geometry. The peak magnetic field in the coils is an important engineering constraint in lattice design. We examine the dependence of the peak field, momentum passband locations, and the beta function on the coil parameters. We make a systematic examination of all allowed lattice configurations taking into account the symmetry properties of the current densities and the beta function. We introduce a unique classification for comparing cooling lattice configurations. While solutions with a single coil per cell illustrate most of the effects that are important for cooling channel design, the introduction of additional coils allows more flexibility in selecting the lattice properties. We look at example solutions for the problem of the initial transverse cooling stage of a neutrino factory or muon collider and compare our results with the properties of some published cooling lattice designs. Scaling laws are used to compare solutions from different symmetry classes.

The influence of helicity on the stability of scaling regimes, on the effective diffusivity, and on the anomalous scaling of structure functions of a passive scalar advected by a Gaussian solenoidal velocity field with finite correlation time is investigated by the field theoretic renormalization group and operator product expansion within two-loop approximation. The influence of helicity on the scaling regimes is discussed and shown in the plane of exponents $\epsilon-\eta$, where $\epsilon$ characterizes the energy spectrum of the velocity field in the inertial range $E\propto k^{1-2\epsilon}$, and $\eta$ is related to the correlation time at the wave number $k$ which is scaled as $k^{-2+\eta}$. The restrictions given by nonzero helicity on the regions with stable fixed points which correspond to the scaling regimes are analyzed in detail. The dependence of the effective diffusivity on the helicity parameter is discussed. The anomalous exponents of the structure functions of the passive scalar field which define their anomalous scaling are calculated and it is shown that although the separate composite operators which define them strongly depend on the helicity parameter the resulting two-loop contributions to the critical dimensions of the structure functions are independent of helicity. Details of calculations are shown., Comment: revtex4, 28 pages, 13 figures, final version published in Phys. Rev. E

We have measured the effects of high (0--4.5 T) magnetic fields on the operating conditions of 805 MHz accelerating cavities, and discovered that the maximum accelerating gradient drops as a function of the axial magnetic field. While the maximum gradient of any cavity is governed by a number of factors including conditioning, surface topology and materials, we argue that $\mathbf{J}\ifmmode\times\else\texttimes\fi{}\mathbf{B}$ forces within the emitters are the mechanism for enhanced breakdown in magnetic fields. The pattern of emitters changes over time and we show an example of a bright emitter which disappears during a breakdown event. We also present unique measurements of the distribution of enhancement factors, $\ensuremath{\beta}$, of secondary emitters produced in breakdown events during conditioning. We believe these secondary emitters can also be breakdown triggers, and the secondary emitter spectrum helps to determine the maximum operating field.

A general equation for the centroid motion of free, continuous, intense beams propagating off axis in solenoidal periodic focusing fields is derived. The centroid equation is found to be independent of the specific beam distribution and may exhibit unstable solutions. A new Vlasov equilibrium for off-axis beam propagation is also obtained. The properties of the equilibrium and the relevance of centroid motion to beam confinement are discussed.

We analyse the motion of a system of particles suspended in a fluid which has a random velocity field. There are coagulating and non-coagulating phases. We show that the phase transition is related to a Kramers problem, and use this to determine the phase diagram, as a function of the dimensionless inertia of the particles, epsilon, and a measure of the relative intensities of potential and solenoidal components of the velocity field, Gamma. We find that the phase line is described by a function which is non-analytic at epsilon=0, and which is related to escape over a barrier in the Kramers problem. We discuss the physical realisations of this phase transition., Comment: 4 pages, 3 figures

We present measurements of dark currents and x rays in a six cell 805 MHz cavity, taken as part of an rf development program for muon cooling, which requires high power, high stored energy, low frequency cavities operating in a strong magnetic field. We have done the first systematic study of the behavior of high power rf in a strong (2.5-4 T) magnetic field. Our measurements extend over a very large dynamic range in current and provide good fits to the Fowler-Nordheim field emission model assuming mechanical structures produce field enhancements at the surface. The locally enhanced field intensities we derive at the tips of these emitters are very large, (-10 GV/m), and should produce tensile stresses comparable to the tensile strength of the copper cavity walls and should be capable of causing breakdown events. We also compare our data with estimates of tensile stresses from a variety of accelerating structures. Preliminary studies of the internal surface of the cavity and window are presented, which show splashes of copper with many sharp cone shaped protrusions and wires which can explain the experimentally measured field enhancements. We discuss a 'cold copper' breakdown mechanism and briefly review alternatives. We also discuss amore » number of effects due to the 2.5 T solenoidal fields on the cavity such as altered field emission due to mechanical deformation of emitters, and dark current ring beams, which are produced from the irises by ExB drifts during the nonrelativistic part of the acceleration process.« less

Scaling exponents for structure functions of the velocity, density, and entropy are computed for driven supersonic flows for rms Mach numbers of order unity, with numerical simulations using the piecewise parabolic method algorithm on grids of up to 512(3) points. The driving is made up of either one or three orthogonal shear waves. In all cases studied, the compressible component of the velocity in the statistically steady regime is weaker than its solenoidal counterpart by roughly a factor of 6. Exponents for the longitudinal component of the velocity are comparable to what is found in the incompressible case and appear insensitive to the presence of numerous shocks. Scaling exponents of the transverse components of the velocity are comparable to those for the longitudinal component. Density and entropy structure functions display strong departures from linear scaling. Finally, the scaling of structure functions of the energy transfer is also given and compared with the Kolmogorov refined similarity hypothesis.

Two-pion correlation functions in Au+Au collisions at s{sub NN}=130 GeV have been measured by the STAR (solenoidal tracker at RHIC) detector. The source size extracted by fitting the correlations grows with event multiplicity and decreases with transverse momentum. Anomalously large sizes or emission durations, which have been suggested as signals of quark-gluon plasma formation and rehadronization, are not observed. The Hanbury Brown--Twiss parameters display a weak energy dependence over a broad range in s{sub NN}.

In this paper we study beam transport through a straight solenoidal channel using the single-particle and linear optics approach. We derive the single-particle invariants and show their use in an extended Courant-Snyder theory (1958) for a solenoidal coupled system. Matching between solenoidal channels and between solenoidal and quadrupolar channels is discussed. We give envelope solutions and illustrate them with some numerical examples. (15 refs).

A model of randomly advected solenoidal field is presented. The model is formally derived by a linearization of the Navier-Stokes equation with respect to the perturbation to a basic state and by assuming the characteristic time scale of the basic state to be very short. The model includes a nonlocal (in space) effect through a pressurelike term that keeps the advected field solenoidal, but still yields exact equations for multipoint moments. The advecting field is assumed to be statistically homogeneous and isotropic with zero mean and structure function with exponent xi. An analysis is made of the scaling of the steady second-order moments of the solenoidal field in two dimensions. The scaling exponent zeta(l) of the isotropic part (l=0) and the anisotropic part for the angular wave number l=2 is obtained analytically or numerically. The scaling of the isotropic part does not depend on whether the pressurelike term is present or not while the scaling of the anisotropic part is affected by the pressurelike term. There are two homogeneous similarity solutions with real positive exponents zeta(2) when xi>xi(c)(2) approximately 1.3. The same kind of analysis is also applied to a simplified two-point closure equation.

The spin correlation coefficent combinations A{sub xx}+A{sub yy} and A{sub xx}-A{sub yy}, the spin correlation coefficients A{sub xz} and A{sub zz}, and the analyzing power were measured for p(vector sign)p(vector sign){yields}d{pi}{sup +} between center-of-mass angles 25 degree sign {

The ..beta..-decay asymmetry of the free neutron is measured by use of a beam of polarized neutrons and a long solenoidal ..beta.. spectrometer with 4..pi.. solid angle for electron detection. The asymmetry parameter corrected for recoil and weak magnetism is A/sub 0/ = -0.1146 +- 0.0019, implying g/sub A//g/sub V/ = -1.262 +- 0.005 for the ratio of the axial-vector to the vector weak-coupling constants.

Stimulated emission in the high-gain regime from a cold, relativistic beam of electrons gyrating in a combined solenoidal and longitudinally polarized periodic wiggler magnetic field is considered as a source of short-wavelength radiation. The emitted wave frequency is Doppler upshifted in proportion to the wave number of the wiggler magnetic field. Amplification is due to a ponderomotive bunching force acting on the electrons in either the transverse or axial directions. Expressions for the linear growth rate are obtained; conditions for their validity and estimates for the saturated efficiency are given.

The already unified field theory of Rainich, Misner, and Wheeler is used to examine the solution of a geometry which is locally a static cylindrically symmetric geometry but whose global properties preclude this interpretation. It is shown that this geometry can be interpreted as a ''superposition'' of two static cylindrically symmetric universes. In particular, the fields found are those exterior to a line of current with mass and interior to a coaxial solenoidal current. Physically reasonable sources are shown to exist.

The streaming caused by an acoustic field, which has been previously considered by Eckart, is re-examined. The concept of mass flow is introduced, and a relaxational relation, between the excess pressure and the excess density, is employed. Further, various proofs of Eckart are modified. The results are essentially the same as those given by Eckart except that they apply to the mass flow instead of the particle velocity. Also, an explicit frequency dependence of the flow is obtained which explains Liebermann's result in ethyl formate. In general to have an average mass flow, viscosity is needed, and the time average of the product of the density and the particle velocity, to second order, is solenoidal. The proof regarding mass flow is not sufficiently general to preclude the possibility of flow under every acoustic field.

A gauge-invariant Lagrangian, leading to a hadronic electromagnetic current which to order $e$ is a linear combination of the vector mesons ${\ensuremath{\rho}}^{0}$, $\ensuremath{\varphi}$, and $\ensuremath{\omega}$, is constructed when the vector mesons are described by an antisymmetric second-rank tensor field.

The application of Nelson's stochastic quantization procedure to vector-meson field gives rise to a Gaussian vector field which coincides with the space components of the corresponding Euclidean vector field. However, some of the main features of Euclidean-Markov structures are absent in the stochastic vector-meson field.

Equations for the divergence of the vector and axial-vector currents follow from the assumptions of Lorentz invariance, locality, the chiral $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$ commutators between charges and charge densities, and the usual electromagnetic and weak Hamiltonians. The divergence conditions lead to derivations of the low-energy meson theorems which do not involve Schwinger terms.

It is well known that the Kepler problem admits two vector constants of the motion, the angular momentum and the Runge-Lenz vector. In this paper a generalization of the Runge-Lenz vector is found when a uniform constant electric field is present. The component of this vector in the direction of the external field is a constant of the motion.