Tokamak elongation – how much is too much? Part 1. Theory

In this and the accompanying paper, the problem of the maximally achievable elongation${\it\kappa}$in a tokamak is investigated. The work represents an extension of many earlier studies, which were often focused on determining${\it\kappa}$limits due to (i) natural elongation in a simple applied pure vertical field or (ii) axisymmetric stability in the presence of a perfectly conducting wall. The extension investigated here includes the effect of the vertical stability feedback system which actually sets the maximum practical elongation limit in a real experiment. A basic resistive wall stability parameter,${\it\gamma}{\it\tau}_{w}$, is introduced to model the feedback system which although simple in appearance actually captures the essence of the feedback system. Elongation limits in the presence of feedback are then determined by calculating the maximum${\it\kappa}$against$n=0$resistive wall modes for fixed${\it\gamma}{\it\tau}_{w}$. The results are obtained by means of a general formulation culminating in a variational principle which is particularly amenable to numerical analysis. The principle is valid for arbitrary profiles but simplifies significantly for the Solov’ev profiles, effectively reducing the 2-D stability problem into a 1-D problem. The accompanying paper provides the numerical results and leads to a sharp answer of ‘how much elongation is too much’?