Toroidal coordinates

Abstract: The indentation produced by an axially symmetrical punch bearing on the plane surface of an elastic half-space has been considered by Harding and Sneddon [1], who used Hankel transforms and a well-known pair of dual integral equations, and for the case of a spherical punch they took the indenting surface to be part of the approximating paraboloid of revolution. Chong [2], also using these dual integral equations has treated the case of a symmetrical punch of polynomial form and considers a two-termed expansion for a spherical punch. More recently, Payne [3] has given the exact solution for a spherical punch using either oblate spheroidal coordinates or toroidal coordinates.

Abstract: According to a general nonperturbative theory that describes atomic behavior in intense, high-frequency radiation fields, the atom becomes stable against decay by multiphoton ionization in the limit of high frequencies if the parameter ${\ensuremath{\alpha}}_{0}$=(I${/2)}^{1/2}$${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}2}$ (a.u.) (with I the intensity and \ensuremath{\omega} the frequency of the field) is kept constant, although otherwise unrestricted. We show that, under this condition, in the subsequent limit of strong fields (${\ensuremath{\alpha}}_{0}$ large), the Schr\"odinger equation describing the structure of the hydrogen atom in a laser field of circular polarization is separable in toroidal coordinates. Explicit asymptotic expressions are given for its energy eigenvalues and its eigensolutions. They correspond to a rapid decrease of the ionization potential and a drastic increase of the size of the atom with ${\ensuremath{\alpha}}_{0}$. For the binding energy of the ground state we find: \ensuremath{\Vert}${E}_{0}$\ensuremath{\Vert} =(1/2\ensuremath{\pi}${\ensuremath{\alpha}}_{0}$) (ln${\ensuremath{\alpha}}_{0}$+2.654284) (a.u.). A dramatic distortion of the shape of the atom is found, which in the strong field becomes a torus-shaped object. Furthermore, we introduce a classification of its states by strong-field quantum numbers. We show how the levels at low ${\ensuremath{\alpha}}_{0}$, characterized by the weak-field quantum numbers introduced earlier, and the levels at high ${\ensuremath{\alpha}}_{0}$, characterized by the strong-field quantum numbers, are correlated. We find that the energy spectrum in strong fields displays a multiplet structure. A comparison is made between our analytical results and those of a numerical calculation carried out earlier.

Abstract: The nonlinear gyrokinetic equations are frequently used as a basis for simulations of small-scale turbulence in magnetized toroidal plasmas. In this context, field-aligned coordinates are usually employed in order to minimize the number of necessary grid points. The present work proposes a system of Clebsch-type coordinates which does not depend on the existence of flux surfaces. The construction and use of these coordinates is explained, and the corresponding formulation of the nonlinear gyrokinetic equations is accomplished. This setup paves the way toward the investigation of nonaxisymmetric toroidal geometries, also in the region of magnetic islands as well as inside the ergodic layer where flux surfaces cease to exist. For testing purposes, in the axisymmetric, large aspect ratio case, the well-known s-α expressions are recovered for closed flux surfaces. Moreover, geometric data for a specific stellarator configuration are computed and discussed.

Abstract: A set of coordinates for simulations in toroidal magnetic geometry, called quasiballooning coordinates, is proposed and implemented. Quasiballooning coordinates are straight-field-line coordinates in which one of the coordinate directions is as close as possible to that of the magnetic field consistent with the near-parallel grid lines meshing exactly (without interpolation) as they cross the boundaries of the simulation region. This allows the true periodicity conditions in the toroidal-poloidal plane to be satisfied in a straightforward and seamless way, even for sheared magnetic fields. Quasiballooning coordinates are useful in the simulation of instabilities and turbulence of interest in fusion plasmas since the number of grid cells needed to represent structures that are elongated along the magnetic field with a given resoluton is greatly reduced compared with toroidal-poloidal or other nontwisting coordinates. For explicit codes, they allow shorter time steps, and it is anticipated that for particle codes, their use will naturally minimize the numerical noise. The key details necessary for the implementation of quasiballooning coordinates, both in finite-difference and pseudospectral fluid codes are presented, and a fluid code has been written. The advantages of quasiballooning coordinates are demonstrated by applying this code to turbulence driven by the ${\mathrm{\ensuremath{\upsilon}}}_{\mathrm{\ensuremath{\Vert}}}^{\ensuremath{'}}$ instability.