Integrable maps with non-trivial topology: application to divertor configurations

Abstract

We explore a method for constructing two-dimensional area-preserving, integrable maps associated with Hamiltonian systems, with a given set of fixed points and given invariant curves. The method is used to find an integrable Poincaré map for the field lines in a large aspect ratio tokamak with a poloidal single-null divertor. The divertor field is a superposition of a magnetohydrodynamic equilibrium with an arbitrarily chosen safety factor profile, with a wire carrying an electric current to create an X-point. This integrable map is perturbed by an impulsive perturbation that describes non-axisymmetric magnetic resonances at the plasma edge. The non-integrable perturbed map is applied to study the structure of the open field lines in the scrape-off layer, reproducing the main transport features obtained by integrating numerically the magnetic field line equations, such as the connection lengths and magnetic footprints on the divertor plate.

Fractal structures in nonlinear plasma physics

“…A divertor has the function of channelling the field lines from undesired regions to a specific target with the goal of removing undesirable energetic particles from the plasma edge [24][25][26][27][28]30]. The divertor surface is thus hit by these particles such that, if the field line traces are too narrowly concentrated, this could lead to overheating and even damage of portions of the collector plate.…”

“…The interest in creating (rather than avoiding) a chaotic region of field lines is the obtaining of a 'cold' boundary layer so as to uniformize heat and particle loadings on the tokamak wall, thus improving the confinement quality through a reduction of impurity releasing [22,23]. Another example of practical application of chaotic field lines in the fusion context is the divertor concept, which needs a chaotic region to drive out undesirable particles from the plasma edge to a collector plate [24][25][26][27][28][29][30].…”

Symplectic Maps for Diverted Plasmas

“…Here, we consider a safety factor profile q, as considered in [30], with a singularity at r = r * . This profile is polynomial for 0 ≤ r < r 95 and logarithmic for r 95 < r ≤ r * , where r 95 = 0.95r * :…”

“…In this section, we introduce a procedure to obtain an integrable map simulating a plasma equilibrium for a diverted tokamak [30,32,33].…”

Divertor map with freedom of geometry and safety factor profile

“…We used this method to obtain an integrable field line map compatible with the single null, and then obtained a nonintegrable map by adding a perturbation that describes an 'ergodic' limiter. The map obtained in [22] for a single-null divertor was based on a specific equilibrium plasma configuration with little freedom to choose different configurations.…”