A Variational Method for Analyzing Vortex Flows in Radar Scanned Tornadic Mesocyclones. Part I: Formulations and Theoretical Considerations

A variational method is formulated with theoretical considerations for analyzing vortex flows in Doppler radar scanned tornadic mesocyclones. The method has the following features. (i) The vortex center axis (estimated as a continuous function of time and height in the four-dimensional space) is used as the vertical coordinate, so the coordinate system used for the analysis is slantwise-curvilinear and non-orthogonal in general. (ii) The vortex flow (VF), defined by the three-dimensional vector wind minus the horizontal moving velocity of vortex center axis, is expressed in terms of the covariant basis vectors (tangent to the coordinate curves), so its axisymmetric part can be properly defined in that slantwise-curvilinear coordinate system. (iii) To satisfy the mass continuity automatically, the axisymmetric part is expressed by the scalar fields of azimuthally averaged tangential velocity and cylindrical stream-function and the remaining asymmetric part is expressed by the scalar fields of stream-function and vertically integrated velocity-potential. (iv) VF-dependent covariance functions are formulated for these scalar variables and then de-convoluted to construct the square root of background error covariance matrix analytically with the latter used to transform the control vector to precondition the cost-function. (v) The de-convoluted covariance functions and their transformed control variables satisfy two required boundary conditions (that is, zero vertical velocity at the lower rigid boundary and zero cross-axis velocity along the vortex center axis), so the analyzed VF satisfies not only the mass continuity but also the two boundary conditions automatically.

INFLUENCE OF THE MATRIX SHAPE ON THE TECHNOLOGICAL PARAMETERS OF THE DRAWING PROCESS

The results of computer modeling of the drawing of an axisymmetric part in matrices with a cylindrical and conical working part are presented. the influence of the shape of the working part of the matrix on the processes of defect formation and product quality is established.

Uncertainties in field-line tracing in the magnetosphere. Part I: the axisymmetric part of the internal geomagnetic field

The technique of tracing along magnetic field lines is widely used in magnetospheric physics to provide a "magnetic frame of reference'' that facilitates both the planning of experiments and the interpretation of observations. The precision of any such magnetic frame of reference depends critically on the accurate representation of the various sources of magnetic field in the magnetosphere. In order to consider this important problem systematically, a study is initiated to estimate first the uncertainties in magnetic-field-line tracing in the magnetosphere that arise solely from the published (standard) errors in the specification of the geomagnetic field of internal origin. Because of the complexity in computing these uncertainties for the complete geomagnetic field of internal origin, attention is focused in this preliminary paper on the uncertainties in magnetic-field-line tracing that result from the standard errors in just the axisymmetric part of the internal geomagnetic field. An exact analytic equation exists for the magnetic field lines of an arbitrary linear combination of axisymmetric multipoles. This equation is used to derive numerical estimates of the uncertainties in magnetic-field-line tracing that are due to the published standard errors in the axisymmetric spherical harmonic coefficients (i.e. gn0 ± δgn0). Numerical results determined from the analytic equation are compared with computational results based on stepwise numerical integration along magnetic field lines. Excellent agreement is obtained between the analytical and computational methods in the axisymmetric case, which provides great confidence in the accuracy of the computer program used for stepwise numerical integration along magnetic field lines. This computer program is then used in the following paper to estimate the uncertainties in magnetic-field-line tracing in the magnetosphere that arise from the published standard errors in the full set of spherical harmonic coefficients, which define the complete (non-axisymmetric) geomagnetic field of internal origin. Numerical estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, calculated here for the axisymmetric part of the internal geomagnetic field, should be regarded as "first approximations'' in the sense that such estimates are only as accurate as the published standard errors in the set of axisymmetric spherical harmonic coefficients. However, all procedures developed in this preliminary paper can be applied to the derivation of more realistic estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, following further progress in the determination of more accurate standard errors in the spherical harmonic coefficients.

Buoyant Convection During the Growth of Compound Semiconductors by the Liquid-Encapsulated Czochralski Process With an Axial Magnetic Field and With a Non-Axisymmetric Temperature

This paper treats the buoyant convection of a molten semiconductor in a cylindrical crucible with a vertical axis, with a uniform vertical magnetic field, and with a non-axisymmetric temperature. Most previous treatments of melt motions with vertical magnetic fields have assumed that the temperature and buoyant convection were axisymmetric. In reality, the temperature and resultant buoyant convection often deviate significantly from axisymmetry. For a given non-axisymmetric temperature, the electromagnetic suppression of the axisymmetric part of the buoyant convection is stronger than that of the non-axisymmetric part, so that the deviation from an axisymmetric melt motion increases as the magnetic field strength is increased. The non-axisymmetric part of the buoyant convection includes relatively strong azimuthal velocities adjacent to the electrically insulating vertical crucible wall, because this wall blocks the radial electric currents needed to suppress azimuthal velocities.

Fourier Analysis of Spiral Tracers. A Progress Report

Fourier analysis of the observed distribution of spiral tracers (e.g. HII regions) can yield a wealth of information about the spiral structure of a given galaxy. A description of the method has been given by Kalnajs (1975) and, in brief, involves decomposing the observed distribution into components with given angular periodicity m. Component m = 0 corresponds to the axisymmetric part, m = 1 to either a one-armed spiral or an asymmetry, m = 2 to a spiral or a bar, etc. Each component is then analysed into a superposition of logarithmic spirals. This does not imply that the observed spirals are assumed to be logarithmic, but rather that logarithmic spirals are convenient building blocks.

Source-sink flow in a gas centrifuge

We consider steady non-axisymmetric source-sink flow of a perfect gas in a rapidly rotating circular cylinder for the case in which the sources and the sinks are distributed on the top and bottom. We apply a linearized analysis to a small perturbation from the state of rigid-body rotation. We show that the radial pressure gradient plays an important role in the determination of the axisymmetric part of the flow field and that the effects of viscosity and thermal conductivity govern the overall configuration of the non-axisymmetric part. We also show that the transport of gas in the main inner flow is axial and that radial transport is confined to the horizontal boundary layers.