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 freely available and open-source code. Its most up-to-date version can 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).

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 three invited and one contributed oral talks:

- KNOSOS, a fast neoclassical code for three-dimensional magnetic configurations. Oral talk at the
*28th IAEA Fusion Energy Conference*, Nice, France, 2021. - KNOSOS, a fast neoclassical code for three-dimensional magnetic configurations. Invited talk at the
*19th European Fusion Theory Conference,*Varenna, Italy, 2021. - KNOSOS, a fast neoclassical code for three-dimensional magnetic configurations: validation and optimization studies at the
*4th Asia Pacific Conference on Plasma Physics*, 2020 - Fast calculation of neoclassical transport of energy and impurities in arbitrary stellarator geometry with the code KNOSOS. Invited talk at the
*22nd International Stellarator-Heliotron Workshop,*Madison, EE.UU., 2019.

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