Neoclassical transport is a fundamental part of stellarator scenario development, since it is expected to account for an important part of the energy losses in a stellarator reactor. Therefore, its minimization is in the core or any process of stellarator optimization and design.

In a tokamak, the radial neoclassical transport is very small. The reason is that the average radial magnetic drift (the slow motion that separates particles from the original magnetic field lines) is zero: trapped particles move back and forth along the magnetic field lines and, while they do so, they drift outwards and inwards; in a tokamak, due to the exact axisymmetry of the magnetic configuration, this radial drift is zero on average. A generic stellarator is different: the value of the magnetic field on a flux surface lacks the simple symmetry existing in a tokamak. When a particle moves along the magnetic field line, it finds a complicated landscape of maxima and minima of the magnetic field strength, and the average magnetic drift is not zero. There are exceptions: one is the quasisymmetric stellarator, which as the tokamak has one direction of symmetry and, as a consequence of this, has equivalent transport properties. Quasisymmetric stellarators are a particular case of a class of devices whose average magnetic drift is zero: omnigenous stellarators. While perfect omnigenetiy is impossible to achieve, careful design of the magnetic coils may allow to obtain a magnetic configuration which is close enough to omnigeneity so that energy confinement is practically equivalent to that of a tokamak.

Neoclassical optimization is the process that, among other goals, tries to achieve magnetic configurations close enough to omnigeneity. The figure of merit in the optimization loop is usually the *effective ripple*, which is a proxy for the radial energy flux in a particular neoclassical transport regime at low collisionalities: the so-called 1/ν regime. This quantity goes to zero as the designed magnetic configuration approaches omnigeneity.

We have developed a code, the KiNetic Orbit-averaging Solver for Stellarators (KNOSOS) that is able to provide a fast and accurate estimate of neoclassical transport in several stellarator regimes apart from the 1/ν, and that can therefore be used for a more efficient stellarator optimization.

KNOSOS is available to be downloaded together with optimization suite STELLOPT

https://github.com/PrincetonUniversity/STELLOPT/tree/CIEMAT

and is employed in stellarator optimization at CIEMAT and also at the National Institute for Fusion Science (Toki, Japan). It has also been employed in predictive transport simulations with the suits Tango and Trinity.

- A. Bañón-Navarro, A. Di Siena, J.L. Velasco, F. Wilms, G. Merlo, T. Windisch, L.L. LoDestro, J.B. Parker, and F. Jenko. First-principles based plasma profile predictions for optimized stellarators. Nuclear Fusion, 63(5):054003, 2023. arxiv / PDF

Its user manual can be found at

https://github.com/joseluisvelasco/KNOSOS

The seminal papers are:

- J.L. Velasco, I. Calvo, F.I. Parra, and J.M. García-Regaña. KNOSOS: A fast orbit- averaging neoclassical code for stellarator geometry. Journal of Computational Physics, 418:109512, 2020. arXiv / PDF
- J L Velasco, I Calvo, F I Parra, V d’Herbemont, H M Smith, D Carralero, and T Estrada. Fast simulations for large aspect ratio stellarators with the neoclassical code KNOSOS. Nuclear Fusion, 61(11):11603, 2021. arxiv / PDF

and the code has been presented in several invited and contributed talks.

Preliminary calculations were also presented in:

- KNOSOS: a fast orbit-averaged neoclassical code for arbitrary stellarator geometry at the
*Kinetic Theory Working Group Meeting*, Madrid, Spain, 2018. - I Calvo, F I Parra, J L Velasco, and J M García-Regaña. Impact of main ion pressure anisotropy on stellarator impurity transport. Nuclear Fusion, 60(1):016035, 2019. arXiv / PDF

KNOSOS can also be applied to tokamaks with broken symmetry:

- X. Litaudon et al., EUROfusion-theory and advanced simulation coordination (E-TASC): programme and the role of high performance computing. Nuclear Fusion, 64(3):034005, 2022. PDF

KNOSOS is the result of a line of research aimed at the analytical description of optimized stellarators. Examples of this work can be found in:

- V d’Herbemont, F I Parra, I Calvo, and J L Velasco. Finite orbit width effects in large aspect ratio stellarators. Journal of Plasma Physics, 88(5):905880507, 2022. arxiv / PDF
- I Calvo, F I Parra, J L Velasco, and J A Alonso. The effect of tangential drifts on neoclassical transport in stellarators close to omnigeneity.
*Plasma Physics and Controlled Fusion*, 59(5):055014, 2017. arxiv / PDF - I Calvo, F I Parra, J A Alonso, and J L Velasco. Optimizing stellarators for large flows.
*Plasma Physics and Controlled Fusion*, 56(9):094003, 2014. arxiv / PDF - I Calvo, F I Parra, J L Velasco, and J A Alonso. Flow damping in stellarators close to quasisymmetry. Plasma Physics and Controlled Fusion, 57(1):014014, 2015. arxiv / PDF
- I Calvo, F I Parra, J L Velasco, and J A Alonso. Stellarators close to quasisymmetry.
*Plasma Physics and Controlled Fusion*, 55(12):125014, 2013. arxiv / PDF

KNOSOS has been employed in the analysis of experiments in W7-X and TJ-II, studying energy, particle and impurity transport. Some modules of the code have been employed for the characterization of energetic ion confinement.