Ideal perturbed equilibria in tokamaks

Tokamaks are almost axisymmetric, but highly sensitive to a small non-axisymmetric magnetic field in the level of 10-4 compared to the axisymmetric magnetic field. The small non-axisymmetries can significantly degrade or improve the tokamak plasma performance. Only recently has the importance of understanding the plasma response to the small non-axisymmetries been appreciated. Since the non-axisymmetric field is almost static on the time scales of the equilibrium relaxation, the basic and fundamental understanding can be achieved by studying perturbed equilibria. The previous approach for these perturbed plasma states was to superpose the external vacuum field onto the axisymmetric equilibrium field. This is not self-consistent since it ignores the perturbed plasma currents arising as the plasma response.;Ideal Perturbed Equilibrium Code (IPEC), which has been developed to include the plasma response effects, is based on the DCON and the VACUUM stability codes. IPEC solves free-boundary ideal equilibria when axisymmetric equilibria are perturbed by small non-axisymmetric perturbations. The complications related to the external boundary conditions are efficiently handled using equivalent surface currents on a control surface instead of directly using the current sources in the external coils. As the internal boundary conditions, the ideal constraints are used at the resonant surfaces, which prevent the destruction of flux surfaces by magnetic islands. IPEC solutions include the perturbed field and the displacement throughout the entire region, but in particular, there are the two important pieces of information: (1) the singular currents that shield out the resonant field driving the islands, and (2) the variation in the field strength on the deformed magnetic surfaces, or equivalently along the perturbed magnetic field lines. IPEC can determine both the strength of the resonant field that is trying to open magnetic islands and the variation in the field strength, which is essential for the evaluation of non-ambipolar transport. These IPEC solutions have been tested in cylindrical force-free limit, and also benchmarked against a perturbed equilibrium code for stellarators and a non-linear MHD code.;Applications of IPEC are: (1) The opening of a magnetic island can stop the plasma rotation in a tokamak. This is called locking, which must be avoided for successful tokamak operations. The resonant field that tends to drive islands, when calculated by IPEC, gave a successful explanation for recent NSTX and DIII-D tokamak locking experiments, which were inconsistent when the plasma response was ignored. (2) IPEC results were also applied to study the parametric dependency of locking, based on data of three US major tokamaks, NSTX, DIII-D and C-MOD. This yielded different expectations and extrapolations to ITER, which will test the first burning plasma in a worldwide collaboration. (3) A non-axisymmetric variation in the field strength can strongly enhance rotational damping through non-ambipolar transport, or equivalently the Neoclassical Toroidal Viscosity (NTV). However, the previous approach using only the external field coupled with an asymptotic NTV calculations showed inconsistency between theory and experiment. In order to resolve the inconsistency, a new analytic treatment of NTV was developed to include important physics effects such as the particle precession and bounce-harmonic resonances. When the generalized NTV theory is coupled with the variation in the field strength calculated by IPEC, far better consistency is found between theory and experiment.;Non-axisymmetric magnetic perturbations can be used for plasma control. To do the control, both the resonant field and the NTV must be determined to avoid degrading the plasma performance. A new control scheme is presented based on the coupling between the resonant field driving islands and the external field. The proposed scheme can determine dominant external field to which various locations in the plasma are most sensitive. The similarity of the dominant external field at different locations in the plasma eases the mitigation of error field effects in tokamaks, but makes it difficult to selectively control different parts of the plasma. Nonetheless, IPEC can be used to determine what control is possible.;IPEC and the new NTV theory improved the understanding of plasma response to non-axisymmetries. However, IPEC does not at present calculate the effects in equilibria of the currents associated with the torque. The resonant field driving islands and NTV both produce toroidal torques. This is an important inconsistency when the toroidal torque becomes large. This effect could be included in IPEC calculations, and this enhancement is planned.