Models for calculation of the sideways force due to the kink modes in tokamaks

The solution of the sideways force problem is finally needed for the ITER project. The task became urgent when the extreme danger of such a force was perceived. The predictions were based on the so-called Noll's formula derived under some simplifications. One of them was the prescription of the plasma motion without testing its compatibility with the force balance condition. Later, an alternative approach has been proposed [D. V. Mironov and V. D. Pustovitov, Phys. Plasmas 24, 092508 (2017)], where the key element was the absence of an integral electromagnetic force on the plasma. Another important improvement was a proper treatment of the vacuum vessel wall. Now the extensions of the previously developed models leading to or supporting Noll's formula are proposed with the resistive wall reaction similarly incorporated. The main attributes of those approaches, the plasma displacements, are kept the same as in the original versions. Precisely, these are the plasma tilt or the ( 1 , 1 ) kink mode. Two forces are calculated with such displacements: on the plasma and on the vacuum vessel wall. The former is shown to be far from zero in the analyzed cases, violating thereby the force-free condition. This does not happen when this constraint goes first. It becomes a selection rule for allowable perturbations. These roughly resemble the tilt and (1,1) mode but differ from them, which changes the result dramatically. The maximal force that can be produced by such kink-like modes compatible with the force balance cannot reach even one tenth of Noll's force. The quantitative comparisons of the competing models are provided.