Toroidal Flux Surfaces Research Articles

Stochasticity reduction

A method for finding integrable Hamiltonian systems with one and a half degrees of freedom is introduced and applied to the problem of finding vacuum magnetic fields with rotational transform and a dense set of toroidal flux surfaces. In this method, the fixed points of the return map must be located, and the values of the corresponding residues calculated. The residues, being indicators of stochasticity, are systematically reduced by variation of the parameters of the system. Application of this method to the problem of finding integrable magnetic fields shows that it is practical, and it yields substantial improvement of the configuration. In particular, this method can result in an increase of the simple equilibrium-β estimate, ι2ε, by a factor of 2 or more.

Self-consistent solutions of the plasma transport equations in an axisymmetric toroidal system

A numerical method is presented for solving a recently derived ( S. P. Hirshman and S. C. Jardin, Phys. Fluids 22(1979) , 731) reduced set of equations describing two-dimensional transport in tokamak plasmas. The formulation exploits the different diffusion time scales by dividing each time advancement step into two parts. In the first part, the one-dimensional surface averaged partial differential equations are advanced implicitly. In the second part, the two-dimensional generalized differential equation for the toroidal flux surface velocity is inverted directly. Accurate efficient solutions are obtained with only explicit terms coupling these two steps. Solutions are presented illustrating the validity and the accuracy of this method.

High-n ballooning modes in highly elongated tokamaks

An analytic study of stability against high-n ballooning modes in highly elongated axi-symmetric plasmas is presented and compared with computational results. – From the equation for the marginal pressure gradient, it is found that local shear has an important effect on the stability of elongated plasmas, and that stability deteriorates through high elongation since the stabilizing effects of field line bending and local shear are reduced. The net contribution of the local shear to stability decreases with elongation for strong ballooning modes (eigenfunctions strongly localized near the outer edge of the toroidal flux surfaces) but increases for interchange modes (eigenfunctions more uniform along the flux surfaces). – The computational study of high-n ballooning modes in a highly elongated plasma reveals that lowering the aspect ratio and broadening the pressure profile enhance the marginal beta for βp less than unity but severely reduce the marginal beta for βp larger than unity.

Self-induced magnetic force of a toroidal current

A class of magnetostatic solutions is obtained for the magnetic field and current in a toroid. The solutions are characterized by nested toroidal flux surfaces whose cross-sections are concentric circles. The model distribution of the magnetic field and current allows a correct calculation of the typical self-induced magnetic force exerted by a current interacting with its own magnetic field in a toroidal configuration.

A fast method to calculate flux surfaces, applied to study helical island structures

Recently two technically simple methods to control plasma-wall interaction in tokamaks have been suggested; one being the helical divertor, which relies on the production of well-defined and well-positioned helical islands on a resonant q- surface , the other being the field line ergodization in the plasma boundary region only. For both methods resonant helical magnetic fields from external helical coils with currents typically 0.01 of the plasma current have to be applied. A fast numerical method is discribed which allows one to calculate toroidal magnetic flux surfaces more efficiently. This method is applied to map the above mentioned magnetic islands and ergodic regions.

On nonlinear axisymmetric equilibria in the magnetohydrodynamic approximation

The second order elliptic differential equation which describes the static equilibrium of an axisymmetric ideally conducting plasma toroid, in the magnetohydrodynamic approximation, may be transformed into an equivalent equation which admits at least one class of exact analytic solutions for which the original equation is essentially nonlinear. Necessary and sufficient conditions are given for the existence of systems in this category having closed toroidal flux surfaces and a result is proved which gives some insight into the mathematics by which the spatial variation of the hydrodynamic pressure can be represented by families of nested closed curves.