Necessary and Sufficient Condition for Hydromagnetic Stability of the Bennett Pinch

Abstract

Newcomb’s method for obtaining the necessary and sufficient condition for hydromagnetic stability is applied to the linear Bennett pinch of length 2πR. For the kink mode, the obtained necessary and sufficient condition for stability is β_z≤F(R⁄r₀)(r₀⁄R)², where β_z and r₀ are the toroidal β and the mean radius of the pinch, respectively. The function F(R⁄r₀) is positive definite and increases monotonically from F(0)=0 to F(∞)=1.0. It varies slowly for large R⁄r₀, i.e., it is larger than 0.9 if R⁄r₀>3.0. For the sausage mode, the necessary and sufficient condition for stability is β_z≤3.80. The Bennett pinch is hydromagnetically stable if both conditions are fulfilled. Tokamak T-3 fulfills these necessary and sufficient conditions.