The three-dimensional geometry of the stellarator magnetic configuration (and therefore its increased number of degrees of freedom with respect to the axisymmetric configuration of the tokamak) can be used to generate the equilibrium magnetic field by means of external coils, instead of requiring a large plasma current to generate part of it, as in the tokamak. This prevents the existence of current-driven plasma instabilities and gives a conceptually more straightforward path towards steady-state operation.

However, the magnetic configuration of a stellarator has to be designed very carefully for it to have confinement properties comparable to those of an axisymmetric tokamak. In a generic stellarator, trapped particle orbits have non-zero secular radial drifts and they leave the device in a short time. The stellarator is called omnigeneous if the magnetic configuration is chosen so that the secular radial drifts vanish. Omnigeneity guarantees that the neoclassical transport level of the stellarator is similar to that in a tokamak.

The proof of Cary and Shasharina [Cary-97a, Cary-97b] for the existence of omnigeneous magnetic fields implies, at the end of the day, that exact omnigeneity requires non-analiticity. Let us explain this in more detail. As shown in [Cary-97a, Cary-97b], analytic omnigeneous magnetic fields coincide with the set of quasisymmetric magnetic fields [Boozer-83, Nührenberg-88]. To the virtues of omnigeneity, quasisymmetry adds the vanishing of neoclassical damping in the quasisymmetric direction. Therefore, in quasisymmetric stellarators larger flow velocities can be attained. In principle, this makes the stellarator plasma prone to develop large flow shear, that is known to reduce turbulent transport [Connor-04]. However, the quasisymmetry condition is incompatible with the magnetohydrodynamic equilibrium equations in the whole plasma [Garren-91], and the stellarator can be made quasisymmetric only in a limited radial region. The mathematical obstructions to achieve quasisymmetry do not exist for omnigeneity. This is why we said above that a necessary condition for exact omnigeneity is non-analiticity; specifically, the discontinuity of some derivatives of second or higher order.

In the last few years, we have focused on quasisymmetric stellarators. Since exact quasisymmetry is impossible to achieve even mathematically, the interesting question is to formally determine when a stellarator can be considered to be sufficiently close to quasisymmetry in practice. In a series of papers, and for different collisionality regimes and geometric properties of the deviations, we have answered this question and we have computed the effect on flow damping and on neoclassical transport of the unavoidable deviations from quasisymmetry [Calvo-13, Calvo-14, Calvo-15].

Although, in principle, omnigenous configurations exist, designing and aligning coils that create a magnetic field with discontinuous derivatives at certain points in space is probably technically impossible. The existing evidence leads one to conclude that even small deviations of omnigeneity are able to produce non-negligible enhancement of neoclassical transport. In this project, we propose to generalize the techniques developed in [Calvo-13, Calvo-14, Calvo-15] for stellarators close to quasisymmetry to the more general case of stellarators close to omnigeneity. A preliminary result on the freedom to design exactly omnigeneous magnetic fields has been published [Parra-15].