Distribution transforms for guiding center orbit coordinates in axisymmetric tokamak equilibria

Abstract

Accurate distribution transforms between common 4D axisymmetric tokamak guiding center coordinates {energy,pitch,R,Z}, and orbit-space coordinates (made up of a particle's kinetic energy E, maximum radius in the tokamak Rm, pitch at the point of maximum radius pm, and time-normalised phase τ=t/torbit) are vital for the verification and implementation of orbit tomography. These transforms have proven difficult in the past due to discontinuities across topological boundaries in orbit-space. In this work we exhaustively investigate these transforms, comparing existing and novel transform methods in speed, accuracy and specific sources of error. A distribution transform that samples {energy,pitch,R,Z}-space along pre-computed orbit paths, and relies on Jacobians calculated with autodifferentiation, is benchmarked against a Monte Carlo sampling and binning algorithm. Our new transform demonstrates a hundredfold increase in speed, and better preserves natural discontinuities across the orbit trapped-passing boundary. A potentially damaging source of transform error caused by a Jacobian singularity that occurs for vanishingly small orbits is addressed, ensuring repeated transforms are well-behaved. The application of smoothing splines in {energy,pitch,R,Z}-space and orbit-space is also discussed. A Jacobian-based transform utilising thin-plate polyharmonic splines restricted to subdomains of similar orbit class is presented and benchmarked against its equivalent non-splined transform. This new smoothing transform correctly avoids interpolating across the trapped/passing boundary, doing so without the prohibitively slow computation and hyperparameter tuning required by previous orbit-space splines.