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

and is employed in stellarator optimization at the National Institute for Fusion Science (Toki, Japan).

Its user manual can be found at


The seminal papers are:

and the code has been presented in three invited and one contributed oral talks:

Preliminary calculations were also presented in:

KNOSOS can also be applied to tokamaks with broken symmetry:

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

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