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

Abstract

In this and the accompanying paper the problem of the maximally achievable elongation κ in a tokamak is investigated. The work represents an extension of many earlier studies, which were often focused on determining κ limits due to (1) natural elongation in a simple applied pure vertical field or (2) 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 γτ 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 κ against 0 n = resistive wall modes for fixed γτ 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"?

MHD stability and disruptions in the SPARC tokamak

“…Theoretical works on vertical stability provide metrics for passive stability (Lazarus et al. 1990; Humphreys & Hutchinson 1993; Portone 2005) and for active stability (Freidberg, Cerfon & Lee 2015) that can be used to assess SPARC. Here we investigate the vertical field decay index normalized by the critical index across a database of stable and vertically unstable C-Mod discharges to validate the analytic theory and to relate a particular value of to an expected disruptivity.…”

An analytic scaling relation for the maximum tokamak elongation against n=0 MHD resistive wall modes

“…There have been many numerical investigations of the n = 0 MHD stability using different models (e.g. plasma surrounded by a perfectly conducting wall [11][12][13] or by a resistive wall [14][15][16][17]). However, these studies do typically not include the impact of the feedback system.…”

“…A key additional feature of the present calculations as compared to our previous study [18], making our scaling relations more widely applicable, is the generalization of the plasma equilibrium pressure and current profiles. In [17,18], the plasma profiles were restricted to the simple class of "Solov'ev profiles" [19], which have the advantage of leading to MHD equilibria with explicit analytic representations, but have the caveat that they correspond to pressure and current profiles that are relatively flat radially as compared to typical experimental profiles. Specifically, the Solov'ev profiles have an internal inductance of about l i 0.4, which is considerably smaller than the typical experimentally measured profiles characterized by l i > 0.7.…”

“…Specifically, we extend many earlier calculations of the n = 0 mode and numerically determine scaling relations for the maximum elongation as a function of dimensionless parameters describing (1) the plasma profile (β p and l i ), (2) the plasma shape ( and δ), (3) the wall radius (b/a) and (4) most importantly the feedback system capability parameter γτ w . These numerical calculations rely on a new formulation of n = 0 MHD theory we recently developed [Freidberg et. al.…”

Tokamak elongation – how much is too much? Part 2. Numerical results

“…In Paper II we convert the analytic formulation of the variational principle derived in Paper I (Freidberg, Cerfon & Lee 2015) into a form suitable for numerical analysis. A code has been written based on this analysis that allows us to quickly and accurately calculate the dependence of elongation and triangularity on inverse aspect ratio for various values of poloidal beta , wall radius and feedback parameter .…”