Analysis of the MHD Instability Driving Mechanisms in
3D Heliotron and Quasi-Axisymmetric Systems

W.A.Cooper1), Y.Narushima2), K.Y.Watanabe2),
K.Yamazaki2), C.Suzuki2) and S.Okamura2)

1)Centre de Recherches en Physique des Plasmas,
Association Euratom-Suisse,
Ecole Polytechnique Fédérale de Lausanne,
CH 1015 Lausanne, Switzerland
2)National Institute for Fusion Science,
Toki, Gifu 509-5292, Japan

A module for the three dimensional (3D) ideal magnetohydrodynamic (MHD)code
TERPSICHORE has been formulated and implemented to investigate the driving and stabilising mechanism associated with global and local ideal MHD modes in 3D magnetic confinement systems. The energy principle that describes the MHD stability behaviour is expressed as δ W_p = (1/2)&integral; d3x (C2 - D|ξs|2), where C2 is the stabilising element due to the compression and bending of magnetic field lines, whereas the instability driving term is D|ξs|2 which can be separated into two components. One of these components is proportional to the pressure gradient responsible for local ballooning and Mercier modes. The second component is proportional to the parallel current density j.B/B2. This term is responsible for global internal and external kink modes. However, in finite β plasmas, the surface varying component of j.B/B2 is proportional to the pressure gradient. Therefore it becomes more complicated to discern its potential contribution to ballooning type modes. The flux surface averaged component of j.B/B2 is given, as calculated in a Boozer magnetic coordinate reference frame, by the expression [J(s)I'(s)-I(s)J'(s)]/[ψ'(s)J(s)-Φ'(s)I(s)], where J(s) and I(s) are the toroidal and poloidal current flux functions, respectively and Φ(s) and ψ(s) are the toroidal and poloidal magnetic flux functions, respectively. Primes (') indicate derivatives with respect to the radial variable s. This term constitutes the main mechanism that drives kink modes. In current free conditions, it vanishes. However, most stellarators have finite bootstrap currents at finite β that can provide free energy to drive kink modes. Applications to heliotron systems like LHD and quasiaxisymmetric devices like CHS-qa will be presented.