Showing papers on "Elliptic coordinate system published in 1984"

On the non-hydrostatic equations in pressure and sigma coordinates

Abstract: The non-hydrostatic equations governing the inviscid, adiabatic motion of a perfect gas are formulated using pressure as vertical coordinate; and M. J. Miller's 1974 approximate quasi-non-hydrostatic pressure coordinate equations are derived by applying a systematic scale analysis and power series expansion. The derivation makes clear that these equations are the pressure coordinate counterparts of the anelastic height coordinate equations obtained by Y. Ogura and N. A. Phillips in 1962. The two sets cannot be interconverted by coordinate transformation and so they are not physically equivalent; but the differences are small at the order of validity of both sets. Consideration of a quasi-hydrostatic approximation emphasizes the non-hydrostatic character of Miller's equations. Sigma coordinate quasi-non-hydrostatic equations are obtained by direct transformation of the pressure coordinate forms, and consistent energy equations are derived for both sets. Convenient diagnostic partial differential equations for the geopotential field are obtained for both pressure and sigma coordinate forms. As shown by Miller, the quasi-non-hydrostatic formulation does not permit vertically propagating acoustic waves. Horizontally propagating acoustic waves (Lamb waves) are in general allowed, but can be removed from the pressure coordinate system by applying suitable boundary conditions. Some aspects of the treatment of the Lamb wave problem are corrected in this study. The quasi-non-hydrostatic sigma coordinate system permits Lamb waves, but it may still be considered suitable for convective and (especially) mesoscale modelling with or without orography. The possible use of the quasi-non-hydrostatic system in large-scale theory and modelling is also discussed.

Superspace normal coordinates

Abstract: By using the geometric structure of the Wess-Zumino formulation of N=1 supergravity, a normal coordinate system is defined locally in curved superspace, and the normal coordinate expansions for the supervielbein and connection coefficients are derived.

Separation of Variables for Complex Riemannian Spaces of Constant Curvature. I. Orthogonal Separable Coordinates for S$_n$c and E$_n$c

Abstract: A complete classification of all orthogonal coordinate systems that admit a separation of variables for the null Hamilton-Jacobi equation in conformally flat complex Riemannian spaces is presented. This is a first step towards the complete solution of the problem for complex Riemannian spaces when, in general, the coordinates need not be orthogonal. A detailed prescription for constructing all such orthogonal coordinate systems is presented.

The comoving-frame equation of radiative transfer in a curvilinear coordinate system

Abstract: The comoving-frame equation of radiative transfer and moment equations are derived in orthogonal, curvilinear coordinates, inclusive of terms of orderv/c. The equation of radiative transfer, which contains the terms due to the effect of curvature of coordinate lines explicitly as well as those of Doppler shift and aberration, is the generalization of Castor's equation for spherical symmetry and of Buchler's equation for Cartesian coordinates. The moment equations agree with Buchler's.

Three-dimensional coordinate measuring method

Abstract: PURPOSE:To reduce considerably the time for positioning an object to be measured by determining the coordinate transformation matrix to the coordinate system intrinsic to the object to be measured from the coordinate system intrinsic to a measuring machine so that the automatic measurement complying with the coordinate system intrinsic to the object to be measured is made possible. CONSTITUTION:An optical displacement gage 4 of a reflection type is provided and a measurement is made by using a noncontacting type three-dimensional coordinate measuring machine added with the axes theta, beta, gamma for controlling the attitude of the gage 4 to the orthognal triaxial mechanism. An object to be measured is first installed and the three-dimensional coordinate value thereof is measured by the coordinate system Z0 (X, Y, Z) intrinsic to the measuring machine with the three points Q0, Q1, Q2 of the reference plane 6 to the object to be measured as reference points. The coordinate transformation matrix L to the coordinate system Z(x, y, z) intrinsic to the object to be measured is calculated from the coordinate system Z0(X, Y, Z) and the automatic measurement is started. The value (x) and target (X*) are compared by the coordinate system Z(x, y, z) and each axis is successively controlled until the value is brought within a criterion value. The coordinate value of (x, y, z) satisfying the criterion is filed as data and plotter display or the like is made by using freely such data.

Curvature Coordinates in Cosmology

Abstract: We develop cosmological theory from first principles starting with curvature coordinates ($R$,$T$) in terms of which the metric has the form $d{s}^{2}(R,T)=\frac{d{R}^{2}}{A(R,T)}+{R}^{2}d{\ensuremath{\Omega}}^{2}\ensuremath{-}B(R,T)d{T}^{2}$ The Einstein field equations, including cosmological constant, are given for arbitrary ${T}_{\ensuremath{ u}}^{\ensuremath{\mu}}$, and the timelike geodesic equations are solved for radial motion We then show how to replace $T$ with a new time coordinate $\ensuremath{\tau}$ that is equal to the time measured by radially moving geodesic clocks Cosmology is brought into the picture by setting ${T}_{\ensuremath{ u}}^{\ensuremath{\mu}}$ equal to the stress-energy tensor for a perfect fluid composed of geodesic particles, and letting $\ensuremath{\tau}$ be the time measured by clocks coincident with the fluid particles We solve the field equations in terms of ($R$,$\ensuremath{\tau}$) coordinates to get the metric coefficients in terms of the pressure and density of the fluid The metric on the subspace $\ensuremath{\tau}=\mathrm{const}$ is equal to $d{R}^{2}+{R}^{2}d{\ensuremath{\Omega}}^{2}$, and so is flat, with $R$ having the physical significance that it is a measure of proper distance in this subspace As specific examples, we consider the de Sitter and Einstein---de Sitter universes On an ($R$,$\ensuremath{\tau}$) spacetime diagram, all trajectories in an Einstein---de Sitter universe are emitted from $R=0$ at the "big bang" at $\ensuremath{\tau}=0$ Further, a light signal coming toward $R=0$ at some time $\ensuremath{\tau}g0$ will, in its past history, have started from $R=0$ at $\ensuremath{\tau}=0$, and have turned around on the line $2R=3\ensuremath{\tau}$ A consequence of this is a "tilting" of the null cones along the trajectory of a cosmological particle The turnaround line $2R=3\ensuremath{\tau}$ marks the transition where an $R=\mathrm{const}$ line changes from spacelike to timelike in character We show how to apply the techniques developed here to the inhomogeneous problem of a Schwarzschild mass imbedded in a given universe in the paper immediately following this one

Generation of surface-fitted coordinate grids

Abstract: An accurate numerical solution of the Navier-Stokes equations is strongly dependent on the description of the flow geometry. In the past, rectangular grids led to an ill description of curved boundaries and consequently hindered finite-difference solutions. Now, it is accepted that body-fitted coordinate systems give a much better description of complex flow geometries. The generation of body-fitted curvilinear coordinate grids in two dimensions by the solution of a nonlinear system of elliptic equations has been proposed Thompson. It has been extended to three dimensions and applied in practical configurations.

Error analysis and difference equations on curvilinear coordinate systems

Abstract: Publisher Summary This chapter discusses methods for deriving difference equations on curvilinear coordinate systems and the effect of coordinate systems on the solution. A computational grid must be constructed when solving partial differential equations by finite-difference or finite element methods. At present, there are many grid generation algorithms. The choice of algorithm will depend on the users' desire for control over properties, such as orthogonality of coordinate lines, location of grid points, and smoothness of grid point distribution. All of these properties may affect the accuracy of the numerical solution. Thereafter, the interpolation technique may affect the local truncation error as well as the convergence rate of iterative algorithms and the stability of explicit algorithms. A curvilinear coordinate system in the xy-plane is understood to be the image of a rectangular Cartesian coordinate system in a ξη-plane. The induced grid is, therefore, composed of quadrilateral cells and difference equations may be derived by transforming the partial differential equation to the ξη-plane. The degree of skewness in a nonorthogonal coordinate system must be limited to maintain the order of the numerical algorithm. An alternate method, the finite volume method, for approximating conservation laws has been widely used on curvilinear coordinate systems. A loss of accuracy in the numerical algorithm may occur if the grid is severely distorted. Therefore, when the region is too complicated, it may be advisable to partition the region and construct a separate curvilinear coordinate system for each subregion.