Toroidal Coordinates Research Articles

Stringlike solitons in toroidal coordinates

A classical field theory is studied in three space dimensions for the case in which the field variables range over a 2-sphere. Toroidal coordinates are found to be the most natural and lead to separation in the equations of motion. A number of finite energy solutions are indicated and correspond to structures extended in space.

677. 高ベー夕・トカマクにおける平衡トロイダル・プラズマのまわりのForce-Free磁場構造

For a sharp boundary equilibrium model of an axisymmetrical-toroidal dense plasma sunounded by a pressureless plasma with force-free fields, structure of force-free magnetic fields is studied by using analytical solutions of the Helmholtz equation for toroidal coordinates and Fourier decomposition. Geometrical form of the magnetic surfaces inside the separatrix surrounding the dense plasma is revealed for the poloidal beta βp and the aspect ratio of the dense plasma Rp/rp=coshσp which are characteristic of a high β tokamak.A condition of the equilibrium is determined and the maximum βp is given by βp=1/4(Rp/rp+rp/Rp)+1/2. The magnetic field equation ∇×H=αH is reduced to the Helmholtz equation(∇∇2+αα2)(Hφeiφ)=0 for the condition α=constant, which is solved without the approximation of the expansion about the inverse aspect ratio. The force-free magnetic fields can be evaluated in a high accuracy under the equilibrium condition of the toroidal dense plasma.For the force-free magnetic surfaces similar to the vacuum magnetic ones, by increasing-α or Rp/rp=coshσp, the separatrix surrounding the dense plasma is expanded and the layers of the magnetic surfaces inside the separatrix are increased. Thus, by decreasing Rp/rp for the fixed separatrix, the layers of the magnetic surfaces inside the separatrix are increased with the increase of -α.

Optical surface waves along a toroidal metallic guide

An approximate analysis shows that rays guided inside a torus oscillate sinusoidally around the equator. Expressing the Helmholtz equation in toroidal coordinates yields the equivalent index of refraction. The equation can be solved, yielding Airy functions in the direction perpendicular to the surface. Along the surface the equation is similar to that for media with a quadratic index profile, the Hermite-Gaussian solutions of which follow sinusoidal trajectories while oscillating in width. Experimental verification of these features is presented.

Numerical calculation of developing laminar flow in pipes of arbitrary curvature radius

AbstractElliptic forms of the Navier‐Stokes equations in rigorously formulated toroidal coordinates are solved numerically for laminar, incompressible, isothermal flow in curved pipes of small curvature radius and 90° deflection angle. As a special case, the flow through curved annuli is also considered in which the presence of the inner curved wall causes the appearance of two (rather than one) pair of cross‐stream vortices. Recirculation in the main flow direction is observed at the outer curvature wall of a pipe with Rc/d = 1 and Re = 1800 over a deflection angle corresponding to 9° ≤ θ ≤ 21°. Flow reversal is attributed to the large and unfavorable longitudinal pressure gradient prevailing at the outer curvature wall over the first half of the pipe. Calculations show that the cross‐steam flow in a strongly curved pipe can be as large as 80% of the bulk average velocity at θ = 45° but is reduced in magnitude between θ = 45° and 90°.The numerical procedure used for the present calculations is a special case of a more general method which will solve flows in arbitrary curvilinear orthogonal coordinates provided the appropriate scale factors and extra source terms are included. A detailed derivation of the difference equations in curvilinear orthogonal coordinates is, therefore, presented here.

The added-mass coefficients of a torus

The generalised added-mass coefficients of a torus in translatory and rotational motion in an inviscid incompressible fluid are obtained via an exact solution of Laplace's equation in toroidal coordinates. Of the six possible independent coefficients three are found to have nonzero, finite and separate values, due to symmetry. These are translation in, and perpendicular to the ring plane and rotation around a diameter. For translation normal to the ring plane, the added mass is somewhat larger than the mass of the torus of equal density. This coefficient tends to the torus mass for slender tori (large ratio of ring to core diameters). For translation in the ring plane the added mass tends to one half the torus mass, and for rotation the added inertia is approximately the torus moment of inertia for such slender tori. Simple relations for the added-mass coefficients as a function of the diameter ratio for general tori are also presented.

Ray Tracing in Axisymmetric Toroidal Coordinates

A method is presented for determining the trajectory of a particle moving along a straight line orbit in axisymmetric toroidal coordinate systems. These geometries occur in problems associated with neutral, neutron, and radiation transport studies in tokamak fusion devices. Numerical solutions of the equations describing the trajectory are performed in geometric configurations generated by the solution of the plasma equilibrium problem. An example problem of the deposition of a pencil beam of high-energy neutral particles in a tokamak plasma with a noncircular cross section due to the necessity of incorporating a divertor and the desirability of operating at a high plasma energy density is described.

Magnetohydrodynamic Stability of a Sharp Boundary Toroidal Plasma

In the toroidal coordinates, a stability analysis is presented for a toroidally symmetrical, sharp boundary plasma. The beta ratio β and aspect ratio, R p / r p , are assumed to be arbitrary. In the equilibrium, the maximum poloidal beta ratio β p is evaluated as Max β p =1/2 ( R p / r p +1). For the kink mode, Curves of β vs a safety factor q and curves of β p vs β in the marginal stability are described for the several aspect rations. Both curves show that the stable region for a fat plasma is wider than for a slender one. The curve of β vs q depends on the aspect ratio even at β=0, and the plasma is unstable for all values of q , if a value of β exceeds a critical value.

Transport Equations in Axisymmetric Toroidal Coordinates

A derivation is presented of the conservation law form of the single energy group transport equation in an axisymmetric toroidal coordinate system formed by rotating a nest of smooth, simply closed, plane curves of arbitrary parametric description about an axis which does not intersect the nest. This general equation may be used for generating equations specific to particular cross section geometries, or as the basis of a finite difference equation for the general case. The effect of both the toroidal and poloidal curvatures of the system are investigated, and criteria for the validity of cylindrical and planar approximations are established. The diffusion equation for this geometry is derived, and it is shown to be formally homologous to the ''r-theta'' cylindrical diffusion equation if the coordinate system is orthogonal and if the azimuthal coordinate, phi, is ignorable. (auth)

Torus topology of the static current loop field

The filamentary current loop of magnetostatics is naturally associated with toroidal topology rather than the usual cylindrical or spherical topology. The field near the wire is best explored in toroidal coordinates, but even the far magnetic dipole field may be found this way. The loop field becomes infinite near the wire just as does the endless straight wire field, but another, not well-known logarithmic infinity occurs that is proportional to the curvature. We demonstrate these facts and, in addition, mention a certain duality with the electric field of a ring of charge.

A minimum photon ?rest mass? ? Using Planck's constant and discontinuous electromagnetic waves

Reasons for taking1/2h/c 2 in cgs units as an equivalent in grams for the photon “rest mass” are given. Its numerical value of3.68×10 −48 g corresponds to the minimum mass equivalent energy for one half-cycle of an electromagnetic dipole field distribution, which is discontinuous. For the fluid models that are discussed, this field distribution corresponds somewhat to a hydrodynamic toroidal vortex which is stationary—if we use toroidal coordinates and assume that the ring origin has the radial velocity c, that the gauge is defined by the ring origin diameter, and that free space is represented by a two-fluid model (the fluids oppositely charged). There are mappings which can transform such toroidal entities (photons) into spherical ones. The toroidal entities are possible candidates for the role of “hidden variables.”

On the Exact Solution of the Stokes Equations for a Torus

The field disturbed by a torus placed perpendicular to a slowly streaming uniform flow of a viscous fluid is considered by use of toroidal coordinates. The drag exerted on the torus and the flux passing through it undergo special consideration accompanied with numerical evaluations.

On the slow viscous rotation of a body straddling the interface between two immiscible semi‐infinite fluids

Using toroidal coordinates, an exact solution is derived for the velocity field induced in two immiscible semi-infinite fluids possessing a plane interface, by the slow rotation of an axially symmetric body partly immersed in each fluid. The surface of the body is assumed to be formed from two intersecting spheres, or a sphere and a circular disc, with the circle of intersection of the composite surfaces lying in th interface.It is shown that when the rotating body possesses reflection symmetry about the plane of the interface of the fluids, the velocity field in either fluid is independent of the viscosities of the fluids. The torque exerted on the body is then proportional to the sum of the viscosities. Analytic closed-form expressions are derived for the torque when the body is either a sphere, a circular disc, or a tangent-sphere dumbbell, and for a hemisphere rotating in an infinite homogeneous fluid. Closed-form results are also given for an immersed sphere, tangent to a free surface. For other geometrical configurations, numerical values of the torque are provided for a variety of body shapes and two-fluid systems of various viscosity ratios.

Equilibrium Configuration of a Ring Plasma

In the earlier paper we treated a magnetohydrodynamic equilibrium of a long plasma cylinder by making use of the London equation and found that the plasma cylinder is almost confined in the radius of the several times the penetration depth r0 (= (me/μ0n0 e2) 1/2). We shall here discuss an equilibrium configuration of a ring form plasma, using the toroidal coordinates (r, ν, ψ) in which jr=Br=O (where j is the current density and B the magnetic field). Numerical calculations show that profiles of equi-macroscopic quantity, just as the equi-number density, the equi-magnetic field and so on, are shifted from the center of tube towards away from the ring axis.

The Dielectric Ring in a Uniform, Axial, Electrostatic Field

The exact solutions for the fields inside and outside a dielectric ring in a uniform, axial, electrostatic field are derived in the system of toroidal coordinates. Comparison is made with the perturbation solution generated by inverse aspect ratio expansion, and it is shown how the exact solution may be truncated to any desired accuracy. The cylindrical limits of the exact and truncated solutions are obtained.

Results of a computer simulation of an arc plasma in a curved discharge tube

In this paper are presented some results of numerical calculations of an arc plasma in a curved geometry. More about the mathematical methods (relaxation methods, transformation of the basic equations to toroidal coordinates) are to be found in earlier papers (Refs.4,6,7). In particular the effect of the curvature on thermal, magnetic, electric and dynamic quantities in a wall stabilized electric arc at atmospheric pressure will be discussed.

Tables of Toroidal Harmonics, I: Orders 0-5, All Significant Degrees

Abstract : The report contains two eleven-figure tables of the Legendre function of the second kind Q(superscript u)(sub v-1/2)(s) for integral values of u and v and s > 1. These functions, also known as toroidal harmonics, occur in the solution of potential problems involving a torus-shaped boundary, as well as in other practical areas. The parameter u, usually referred to as the 'order' is taken up to r and v (v - 1/2 is known as the 'degree') is allowed to vary from 0 to a value for which the absolute value of the function is less than 10 to the minus 12 power, for the largest value of u considered. The argument in first set of tables is taken as s and covers the range s = 1.1(.1)5. In the second set the argument is taken as cosh (n) which is a more natural parameter to use in the solution of the potential problem in toroidal coordinates. (Author)

Mixed boundary-value problems for an elastic half-space

In this paper the mixed problem for an isotropic, elastic half-space is considered. Boundary conditions are prescribed interior and exterior to a circular region of unit radius, and the state of stress is assumed to be axially symmetric. Several authors have treated this problem. Mossakovskii(1) considered a punch adhering to and indenting an elastic half-space. Has solution was obtained by introducing certain operators that transformed the half-space problem into a problem in plane potential theory. The method of linear relationship was used to solve this auxiliary problem and inverse operators returned the plane to the half-space. The general case of a circular, rigid punch adhering to a half-space was treated by Ufliand (2,3) and a solution was obtained through the use of toroidal coordinates and the Mehler-Fok integral transforms.

Circumferential axisymmetric free oscillations of thick hollowed tori

In this article, the effects of thickness and torus radii ratio on the axisymmetric circumferential natural vibrations of freely supported, hollowed tori are determined. The solutions for eigenfrequencies and eigenmodes are effected by finite difference approximations of the basic equations of elastic motion, written in terms of a set of toroidal coordinates. The results agree quite well with the special cases of toroidal membranes and infinite circular cylinders.

Polar Axisymmetric Free Oscillations of Thick Hollowed Tori

Because of its continually varying curvature, the solutions to problems involving the toroidal shell are complicated, and the literature is sparse compared with that of shells having zero or small Gaussian curvature, such as plates, cylindrical and conical shells, and shallow shells. Excellent bibliographies [1], [2] are available on problems of the toroidal shell. For the most part, however, toroidal shell investigations have been concerned with deformation, stress and stability analyses. There have been a few attacks on the dynamics of the toroidal shell (for example, [3]–[6]), but there remain many unanswered questions. The present article examines a problem in free vibrations of the freely supported toroidal shell by means of the exact elasticity equations, and the solution reduces to that of the infinite cylindrical shell as a special case. In earlier work [7], it was shown that the axisymmetric free oscillations of tori are made up of two independent classes of motion. The first of these, solved in [7], involves motion in the $\phi $-direction only (see Fig. 1) via a circumferential displacement $U_\phi $ which is a function of R and $\theta $. To be considered in the present paper is the second class of axisymmetric free vibration, consisting of motion in the polar R-$\theta $ plane, and described by the displacements $U_r $ and $U_\theta $, each a function of R and $\theta $. The surfaces of the torus are assumed to be stress-free. The three exact equations of elastic motion were written in [7] in terms of a set of toroidal coordinates. In this paper, the pair of these which govern polar motion are written in terms of finite differences. The four boundary conditions applicable to polar motion are also written in difference form. The solution to the difference equations is described, and some effects of difference spacing, thickness, torus radii ratio, and Poisson’s ratio on the natural vibrations of polar axisymmetric oscillations of free-free tori are found.

Eigenfunctions of the Hydrogen Atom in Momentum Space

The Schrödinger problem of the hydrogen atom is solved using toroidal coordinates in momentum space. The wavefunctions are expressible in terms of Jacobi polynomials and ordinary trigonometric functions.