Axial Harmonics Research Articles

Properties of the electrostatically driven helical plasma state

A novel plasma state has been found [Akçay et al., Phys. Plasmas 24, 052503 (2017)] in the presence of a uniform applied axial magnetic field in periodic cylindrical geometry. This state is driven by external electrostatic fields provided by helical electrodes with a (m=1,n=1) (helical) symmetry where m and n represent the poloidal and axial harmonics. The resulting plasma is a function of the cylinder radius r<rw and helical angle u=mθ−nζ, where θ is the poloidal angle, and ζ=z/R is the normalized axial coordinate in the context of a periodic cylinder. In this reference, the strongly driven form of the state was found to have a strong axial mean current density, with a mean-field line safety factor q0(r) just above the pitch of the electrodes m/n=1 in the interior, where the plasma is nearly force-free. However, at the edge the current density has a component perpendicular to the magnetic field B. This perpendicular current density drives nearly Alfvénic helical plasma flows, a notable feature of these states. This state is being studied for its possible application in DC electrical transformers. We present results on several issues of importance for these applications: the transient leading to the steady state; the twist and writhe of the field lines and their relation with the current density; the properties of the current density streamlines and length of the current density lines connected to the electrodes; the sensitivity to changes in the velocity boundary conditions; the effect of varying the radial resistivity profile; and the effects of a concentrated electrode potential.

Equations for a Heterogeneous Reactor with Effective Conditions on the Axial Boundaries of the Core

The equations of a heterogeneous reactor with effective conditions at the core boundary and end reflectors are derived. The numerical solution of these equations gives significant reduction of the axial harmonics and the computational time for calculation of reactors with a large spike in the thermal neutron flux in the end reflectors. The efficacy of the method is illustrated by an example of a calculation of a reactor with an intermediate neutron spectrum.

Features of the Distribution Process of the Electromagnetic Field Frequency Components in the High-Frequency Torch Discharge Plasma

The measurements of axial harmonics distribution of the radial component of the electric field have been made for a torch discharge burning in argon and argon-air mixture. The resonant character of the electric field fourth harmonic attenuation has been found. Using the Saha formula for the two-temperature plasma the electrons concentration and the electrons Langmuir frequency of the investigated plasma discharge have been evaluated.

Nonlinear Electrodynamic Effects of the Torch Discharge Argon Plasma

The measurements of the amplitude-frequency characteristics were done for the high-frequency torch discharge burning in argon. The resonant character of the electric field fourth harmonic attenuation was found. The axial harmonics distribution measurements of the electric field were done for the torch discharge burning in argon-air mixture. The electron concentration in the investigated plasma discharge was evaluated. The selective influence of the change in the electron concentration on the propagation process of the fourth harmonic of the electric field was found.

Calculating the spectrum of anisotropic waveguides using a spectral method

The computation of the spectrum of a waveguide with arbitrary anisotropy with spatial dependence is a challenging task due to the coupling between axial and azimuthal harmonics. This problem is tackled in cylindrical coordinates by extending a spectral method for the general case. By considering the matrix representation of the operator on the right-hand side of the governing equations, the latter are exactly reformulated as an infinite set of integro-differential equations. Essential part of this study is taking into account the coupling of different harmonics, which becomes evident from the kernels of these equations. Provided a waveguide is translationally invariant in the axial direction, the coupling of axial harmonics vanishes. A practical approximation and truncation procedure yields a generalized eigenvalue problem, which can be solved numerically to obtain the entire spectrum of the operator and to construct the dispersion curves for the eigenmodes. The spectral method is tested against the results from the measurements of dispersion curves for the monopole, dipole, and quadrupole normal modes of scaled boreholes in tilted transverse isotropy anisotropic rock sample. Besides, the comparison of dispersion curves calculated by the spectral method and those computed from the synthetic data is discussed.

Application of the spectral method for computation of the spectrum of anisotropic waveguides

Spectral method is formulated in cylindrical coordinates for the general case of waveguide with arbitrary anisotropy with the spatial dependence. According to the idea of this approach, matrix representation of operator in the right-hand side of governing equations is considered. As a result, the latter are cast into exact infinite set of integro-differential equations. Explicit expressions for their kernels expose coupling between axial and azimuthal harmonics. Coupling of axial harmonics vanishes in important case of waveguide with translational invariance in axial direction. It results in the set of differential equations, which is used to introduce practical approximation procedure. The latter yields generalized eigenvalue problem, which can be solved numerically for the spectrum of the operator. The spectrum is sorted according to eigenmodes’ properties. Thus dispersion curves of eigenmodes are constructed. Presented consideration can be adapted for waveguides of different physical nature (elastic, electromagnetic, etc.) and different geometry (rectangular, elliptical, etc.). Developed technique is verified by comparison with results of controlled laboratory measurements on anisotropic sample. Monopole, dipole and quadrupole normal modes for scaled borehole in anisotropic rock sample with TTI symmetry are considered. The comparison of spectral method results with the dispersion analysis of synthetic data is provided as well.

Axial flux harmonics affecting the parallel channel instabilities in boiling water reactors

Abstract The existence of boiling phenomena in boiling water-reactors makes these systems susceptible to instabilities. Neutronics coupled with the thermal-hydraulics complicate the problem further. Boiling water reactor instabilities are classified as global or local according to the spatial dependency of the power fluctuations. At present, numerical codes can simulate these instabilities rather well on event basis; but more general physical insight can be gained readily through analytical modeling. One of the earliest models applied to the problem is the parallel channel instability, so called because of the applied thermohydraulic boundary conditions. In this work a new type of instability model is proposed and studied by including the effect of the first axial harmonic of the flux. Thermal-hydraulics is represented with one-dimensional two-phase homogeneous equilibrium flow model equations. Analysis is performed with linearized equations in the frequency domain. Special emphasis is given to the motion of the boiling boundary. The results clearly indicate that the first axial harmonic of the flux affects the stability of the reactor adversely, and must be accounted for to make more realistic predictions.

Slotted cylindrical shell for use as a compact wiggler.

A compact wiggler offers the opportunity to reach high free-electron-laser frequencies with an electron beam of modest energy. We analyze here the properties of a conducting cylinder, slotted periodically on alternate sides, for use as a compact wiggler. In particular, we obtain the current distribution on the cylindrical surface, from which we obtain the magnetic fields inside the cylinder. We then expand this field into its multipole components, as well as into its axial harmonics. The resulting orbit dynamics are explored, leading to preferred choices for some of the design parameters. Finally, we examine the major consequences of possible fabrication errors, which suggest the tolerances and/or necessary compensation mechanisms required for a satisfactory wiggler design.

Computation of magnetic deflectors for electron beam lithography

The computation of magnetic deflectors by the finite element method is extended to the computation of tapered saddle coils. A general energy functional for any harmonic component of the field was derived, which enables the computation of any multipole made of saddle and toroidal coils in the presence of rotationally symmetric magnetic materials. In order to get the axial harmonics with high accuracy, a different expression for the cylindrical harmonics of the magnetic scalar potential was used, and analytical formulas for the integration of expressions in the energy functional containing higher powers of the radial coordinate were derived. A program defl for the computation of deflectors and other multipoles for mainframe computers and PC/ATs was written, and also a graphical display of the results is provided.

The variance of the temperature distribution in a reactor cell

Local variations in fuel packing density, fuel enrichment, bond-gap thickness, surface asperities, etc. give rise to potentially significant deviations in the temperature distribution in a reactor cell. Treating the second moments of the statistical variations of the fuel thermal conductivity, gap conductance, heat transfer coefficient from can to bulk coolant, etc. by means of specific variances the standard deviation of the temperature distribution is calculated. To account for the temperature dependence of the fuel thermal conductivity and to remover non-linearities in the equations describing the temperature deviations, a linearization approximation is adopted and the resulting equations are solved by means of an expansion over azimuthal harmonics utilizing radially dependent coefficients. In the main text, axial conduction effects are neglected in order to simplify algebraic expressions; however, in Appendix I it is shown how an expansion over axial harmonics enables the inclusion of axial conduction which is important in describing strong localized deviations. It is demonstrated that the standard deviation of a quantity that is a linear combination of the harmonics of the temperature has a variance that contains no cross-correlation between different harmonics.

Odd Harmonics of the Earth's Gravitational Field

VARIATIONS in the perigee radius have been observed for three satellites so far. The first of these is Vanguard I (1958 beta), which was used by O'Keefe et al. 1 in calculating the third and fifth axial harmonics of the Earth's field. The other two are Transit IB (1960 gamma) and Transit IIA (1960 eta). Nominal values of the pertinent orbit parameters of these three satellites are listed in Table 1. The data on Vanguard are from O'Keefe et al. 1, those on Transit IB are given by Guier2, and those on Transit IIA have not been published before.