Equation In Spherical Coordinates Research Articles

Dimensional Analysis and the Buckingham Pi Theorem

Working from a simple example of a dimensional transformation, the essential elements are identified, and an abstract prototype transformation is defined. With the aid of careful definitions and a geometric interpretation of what happens in a dimensional transformation the Buckingham pi theorem is written down but not proved. The application of this theorem to the heat equation in spherical coordinates and one dimensional gas motion without boundary conditions serve to illustrate the usefulness of the theorem in physics.

Influence of migration and diffusion on the steady-state potentiostatic response at spherical electrodes

The effect of migration and diffusion on the steady-state potentiostatic response at spherical electrodes is studied numerically for radii of curvature 100 and 10 Å. The Poisson-Boltzmann equation in spherical coordinates is solved numerically for various concentrations of 1-1 inert electrolytes. The resulting potential-distance profiles are used to predict the degree of retardation of an electroactive pilot ion by the equilibrium diffuse double layer existing at spherical microelectrodes. Results illustrate that the electric field in the diffuse double layer can influence significantly the ionic flux to these small microelectrodes with normal charge densities on the electrode surface. For electrode radii of curvature of 0.1 μm or larger, this diffuse double-layer effect is essentially negligible under steady-state conditions.

Influence of migration and diffusion on the steady-state potentiostatic response at spherical electrodes

The effect of migration and diffusion on the steady-state potentiostatic response at spherical electrodes is studied numerically for radii of curvature 100 and 10 Å. The Poisson-Boltzmann equation in spherical coordinates is solved numerically for various concentrations of 1-1 inert electrolytes. The resulting potential-distance profiles are used to predict the degree of retardation of an electroactive pilot ion by the equilibrium diffuse double layer existing at spherical microelectrodes. Results illustrate that the electric field in the diffuse double layer can influence significantly the ionic flux to these small microelectrodes with normal charge densities on the electrode surface. For electrode radii of curvature of 0.1 μm or larger, this diffuse double-layer effect is essentially negligible under steady-state conditions.

Effect of spreading of material on the surface of a fluid—an exact solution

A material of small density is being supplied at constant rate at a point on the surface of a quiescent fluid. The material may be an immiscible, lighter fluid of higher viscosity, or may be composed of a stream of solid particles. After some time the material spreads out with constant thickness and a steady state is reached. We would like to know the effect on the fluid, due to the spreading of surface material. The problem is governed by the non-dimensional parameter B/ v where v is the viscosity and > B/ r is the velocity imparted by the spreading material to the fluid underneath. Depending on the range of B/ v, the Navier-Stokes equation in spherical coordinates admits one exact solution for positive B, and three different exact solutions for negative B. An exact boundary layer solution is found for B/ v ⪢ 1. A uniformly valid solution is then constructed. The Stokes solution for ¦ B/ v¦ ⪡ 1 is also determined. Both approximations are compared with the exact solution.

A Diagnostic Model for Equatorial Wave Disturbances: The Role of Vertical Shear of the Mean Zonal Wind

Abstract A model is developed to diagnose the response of the atmosphere to a known distribution of diabatic heating. The linearized primitive equations in spherical coordinates are reduced to a single partial differential equation relating the perturbation geopotential to the diabatic heating pattern. The model diagnoses the atmospheric perturbation due to a heating pattern of specified zonal wavenumber, frequency, and distribution in the meridional plane. The model is used to test the hypothesis that the observed westward propagating wave disturbances in the equatorial Pacific are Rossby waves driven by the latent heat release in the cloud clusters embedded within the waves. For a diabatic heating pattern designed to model the heating in cloud clusters the model duplicates many features of the observed waves. The computed perturbation meridional wind field has maxima in the upper and lower troposphere, separated by a relatively undisturbed region in the mid-troposphere. The structure of the disturbance ...

Numerical Solution to Laplace's Equation in Spherical Coordinates with Axial Symmetry

A numerical method for solving Laplace's equation in spherical coordinates for an axially symmetric geometry has been developed. Dirichlet boundary conditions on a closed surface yield a set of difference equations. The method of successive overrelaxation has been employed to find solutions to those equations. Application of the method is illustrated by finding the field configuration in a new type of electron gun used in a spherical electron monochromator.

Derivation of the Elliptic Condition for the Balance Equation in Spherical Coordinates

Derivation of the Elliptic Condition for the Balance Equation in Spherical Coordinates

Propagator matrix formulation of heat transfer in spherically stratified media

In the paper we have transformed the steady and unsteady conductive heat transfer differential equation in spherical coordinates into a system of first order differential equations and processed them by method of propagator matrices to extrapolate the known surface heat flux and temperature to any desired depth. The elements of propagator matrices have been summarised for various piecewise continuous conductivity and rate of heat generation functions to approximate inhomogeneities in the earth. In the analysis the rate of heat generation is either assumed to depend linearly upon temperature or correspond to first order irreversible chemical reactions.

NUMERICAL INTEGRATION OF A NINE-LEVEL GLOBAL PRIMITIVE EQUATIONS MODEL FORMULATED BY THE BOX METHOD

Abstract Based on the box method, finite-difference versions of a system of primitive equations in spherical coordinates are formulated for a spherical grid. Non-linear computational instability cannot occur in time integrations of these equations. Conservation of total mass is guaranteed by the finite-difference form of the continuity equation. The proposed scheme yields no fictitious sources of energy in the derivation of the difference formula for the budget of the total energy over the entire domain. The finite-difference equations for the budget of the relative and absolute angular momentum are not exact analogs of the continuous forms but nevertheless are very accurate. This system of primitive equations for a nine-level general circulation model of the atmosphere has been numerically integrated for 50 forecast days. The network of grid points covers the entire globe with nearly uniform spacing and has no artificial horizontal boundaries. The initial data were latitude-height-dependent zonal mean wi...

Converting Vector and Tensor Equations to Scalar Equations in Spherical Coordinates

Sunzriiary We show how to represent a vector field in a spherical shell in terms of three scalar fields, and a second-order tensor field in terms of nine scalar fields. We derive the scalar representations of the most common algebraic and differential operations on vector and second-order tensor fields, and apply the results to obtain scalar formulations of various tensor problems in the continuum mechanics of the Earth’s mantle. We give an exhaustive catalogue of all possible equilibrium stress fields in the mantle, we deduce the scalar equations of elastic-gravitational oscillation of a transversely isotropic, radially stratified, spherically symmetrical Earth, and we give the scalar convection equations for a viscous, self-gravitating, spherically symmetrical mantle with radially variable viscosity.

The “Radial Kinetic Energy” Term in the Schrödinger Equation for a Central Force

A straightforward derivation is given for the Hermitian operator pr ≡ 1r ħi δδrr corresponding to the radial component of linear momentum in the central force problem. The purpose is purely pedagogical; we believe that the radial term of the Schrödinger equation in spherical coordinates is more comprehensible when it appears in the form (pr2/2m)ψ = [(1r ħi δδrr)2/2m]ψ than when it appears, as it does in most textbooks, in the form −ħ22mr2 δδr[r2δδr]ψ. The latter expression is usually arrived at by a tedious transformation of variable done “in the appendix”; instead one may derive pr with a minimum of algebra and with the focus where it belongs, on the concepts of quantum mechanics.

Surveys for Spacecraft Tracking Sites

Special application of celestial surveying and surveying mathematics; mapping may be accomplished by establishing equation in spherical coordinates based on trajectory criteria; in evaluation of sites, it is desirable to determine acquisition and departure azimuths to spacecraft, and to measure horizon clearances so that effect of obstructions to tracking may be compared; geodetic and layout surveys.

PLANETARY WAVES IN THE ATMOSPHERE

Abstract A numerical method is presented for solving the non-linear barotropic vorticity equation in spherical coordinates. It is shown that, when the stream function is expressed as a sum of surface spherical harmonics, the barotropic vorticity equation gives rise to harmonic tendency equations which express the time rate of change of the harmonic coefficients as functions of all the harmonic coefficients. The harmonic tendency equations may be used in an iterative process to find the flow pattern at a future time from a knowledge of the harmonic coefficients at an initial time. The amount of computation is reduced greatly if the planetary waves are regarded as composed of perturbations superimposed on a steady zonal flow in which the angular velocity varies with colatitude. An example based on an actual synoptic situation is given. The effect of large scale lateral mixing is to curb the development of harmonics of large degree.

The Effects of Viscous Dissipation in the Spherical Acoustic Resonator

The problem of the spherical acoustic resonator containing a slightly viscous medium is considered. Derivation of the equations of sound waves in a viscous medium is carried out, and it is shown that separation of the particle velocity into a divergenceless vector and a curl-less vector gives rise to both a vector Helmholtz equation and a scalar Helmholtz equation. By using known solutions to these equations in spherical coordinates and imposing suitable boundary conditions, the frequencies of the free vibrations are calculated as well as the attenuation constants associated with each frequency. The effect of viscosity is, thus, seen to decrease slightly the natural frequencies and to cause a time rate of decay of the free vibrations. The attenuation constant for the purely radial modes of vibration is much smaller than that for modes having tangential velocity components.

The Generation of Harmonics by Small Circular Orifices

A spectral analysis was made of the sound field emanating from a small orifice set flush in the top of a table which acted as an infinite baffle. The orifice was driven at a single frequency at various intensities by an electromagnetic horn exciter, coupling through a small cavity below the baffle. At high intensities all harmonics up to the fortieth have been detected. Starting at a distance of one radius from the center of the orifice, the fundamental component of the sound pressure is observed to decrease at a rate of six decibels per distance doubled, indicating that the orifice acts as a spherical source whose radius is that of the orifice. The higher harmonic components decay at the same rate as the fundamental indicating that, for the levels considered, essentially no integrative interactions occur between the harmonics while progagating away from the orifice. Thus it appears that harmonic generation occurs locally at the orifice. The successive odd harmonics are invariably stronger than the immediately preceding even harmonics. Even at moderate levels, the third harmonic may lie less than five decibels below the fundamental component. A theory based upon the non-linear wave equation in spherical coordinates has been developed which accounts for the first two harmonics semiquantitatively. No account is taken of turbulence or rotational terms in the theory.