Perturbing an axisymmetric magnetic equilibrium to obtain a quasi-axisymmetric stellarator

It is demonstrated that finite-pressure, approximately quasi-axisymmetric stellarator equilibria can be directly constructed (without numerical optimization) via perturbations of given axisymmetric equilibria. The size of such perturbations is measured in two ways, via the fractional external rotation and, alternatively, via the relative magnetic field strength, i.e. the average size of the perturbed magnetic field, divided by the unperturbed field strength. It is found that significant fractional external rotational transform can be generated by quasi-axisymmetric perturbations, with a similar value of the relative field strength, despite the fact that the former scales more weakly with the perturbation size. High mode number perturbations are identified as a candidate for generating such transform with local current distributions. Implications for the development of a general non-perturbative solver for optimal stellarator equilibria is discussed

Optimisation of out-vessel magnetic diagnostics for plasma boundary reconstruction in tokamaks

To improve the low frequency spectrum of magnetic field measurements of future tokamak reactors such as ITER, several steady state magnetic sensor technologies have been considered. For all the studied technologies it is always advantageous to place the sensors outside the vacuum vessel and as far away from the reactor core to minimize radiation damage and temperature effects, but not so far as to compromise the accuracy of the equilibrium reconstruction. We have studied to what extent increasing the distance between out-vessel sensors and plasma can be compensated for sensor accuracy and/or density before the limit imposed by the degeneracy of the problem is reached. The study is particularized for the Swiss TCV tokamak, due to the quality of its magnetic data and its ability to operate with a wide range of plasma shapes and divertor configurations. We have scanned the plasma boundary reconstruction error as function of out-vessel sensor density, accuracy and distance to the plasma. The study is performed for both the transient and steady state phases of the tokamak discharge. We find that, in general, there is a broad region in the parameter space where sensor accuracy, density and proximity to the plasma can be traded for one another to obtain a desired level of accuracy in the reconstructed boundary, up to some limit. Extrapolation of the results to a tokamak reactor suggests that a hybrid configuration with sensors inside and outside the vacuum vessel could be used to obtain a good boundary reconstruction during both the transient and the flat-top of the discharges, if out-vessel magnetic sensors of sufficient density and accuracy can be placed sufficiently far outside the vessel to minimize radiation damage.Comment: 36 pages, 17 figures, Accepted for publication in Nuclear Fusio

How well can VMEC predict the initial saturation of external kink modes in near circular tokamaks and l=2l=2 stellarators?

The equilibrium code, VMEC, is used to study external kinks in low $\beta$ tokamaks and $l=2$ stellarators. The applicability of the code when modelling nonlinear MHD effects is explored in an attempt to understand and predict how the initial saturation of the MHD mode depends on the external rotational transform. It is shown that helicity preserving, free boundary VMEC computations do not converge to a single perturbed solution with increasing spectral resolution. Additional constraints are therefore applied to narrow down the numerical resolution parameters appropriate for physical scans. The dependence of the modelled (4, 1) kink mode on the external rotational transform and field periodicity is then studied. While saturated states can be identified which decrease in amplitude with increasing external rotational transform, bifurcated states are found that contradict this trend. It was therefore not possible to use VMEC alone to identify the physical dependency of the nonlinear mode amplitude on the magnetic geometry. The accuracy of the VMEC solutions is nevertheless demonstrated by showing that the expected toroidal mode coupling is captured in the magnetic energy spectrum for stellarator cases. Comparing with the initial value code, JOREK, the predicted redistribution of poloidal magnetic energy from the vacuum to plasma region in VMEC is shown to be physical. This work is a first step towards using VMEC to study MHD modes in stellarator geometry.Comment: Submitted to Physics of Plasmas. The submission has been modified according to reviewer comment

Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time

The reconstruction of the equilibrium of a plasma in a Tokamak is a free boundary problem described by the Grad-Shafranov equation in axisymmetric configuration. The right-hand side of this equation is a nonlinear source, which represents the toroidal component of the plasma current density. This paper deals with the identification of this nonlinearity source from experimental measurements in real time. The proposed method is based on a fixed point algorithm, a finite element resolution, a reduced basis method and a least-square optimization formulation. This is implemented in a software called Equinox with which several numerical experiments are conducted to explore the identification problem. It is shown that the identification of the profile of the averaged current density and of the safety factor as a function of the poloidal flux is very robust

2D interpolation and extrapolation of discrete magnetic measurements with toroidal harmonics for equilibrium reconstruction in a Tokamak

International audienceWe present a method based on the use of toroidal harmonics and on a modelization of the poloidal field coils and divertor coils for the 2D interpolation and extrapolation of discrete magnetic measurements in a Tokamak. The method is generic and can be used to provide Cauchy boundary conditions needed as input by a fixed domain equilibrium reconstruction code like Equinox. It can also be used to extrapolate the magnetic measurements in order to compute the plasma boundary itself. The proposed method and algorithm are detailed in the paper and results from numerous numerical experiments are presented. The method is foreseen to be used in the real time plasma control loop on the WEST Tokamak