Tearing modes in toroidal geometry

Abstract

The separation of the cylindrical tearing mode stability problem into a resistive resonant layer calculation and an external marginal ideal magnetohydrodynamic (MHD) calculation (Δ′ calculation) is generalized to axisymmetric toroidal geometry. The general structure of this separation is analyzed and the marginal ideal MHD information (the toroidal generalization of Δ′) required to discuss stability is isolated. This can then, in principle, be combined with relevant resonant layer calculations to determine tearing mode growth rates in realistic situations. Two examples are given: the first is an analytic treatment of toroidally coupled (m=1, n=1) and (m=2, n=1) tearing modes in a large aspect ratio torus; the second, a numerical treatment of the toroidal coupling of three tearing modes through finite pressure effects in a large aspect ratio torus. In addition, the use of a coupling integral approach for determining the stability of coupled tearing modes is discussed. Finally, the possibility of using initial value resistive MHD codes in realistic toroidal geometry to determine the necessary information from the ideal MHD marginal solution is discussed.

The role of pressure flattening in calculating tearing mode stability

“…A computer code employing a basis function approach, similar to that used in Connor et al [3], has also been used and produces results consistent with this approach. The case with no flattening function was considered first.…”

“…where φ is the toroidal angle, p is the pressure, a prime denotes the derivative with respect to r, R 0 is the major radius at the magnetic axis and B 0 is the magnetic field there, following Connor et al [3] To facilitate the expansion we represent q(r) ≡ m n +δ∆q(t) where t = (r−r m )/δ and δ = δ/r m and expand inδ ≪ 1. (∆q(t) is related to the perturbation q (1) (t) appearing in (B.18) by ∆q(t) = r m tq (0) ′ (r m ) + q (1) (t).)…”

“…where the dependent variables are y = R 0 f ξ.∇r and Z = R 2 δB.∇φ/B 0 , with ξ the plasma displacement and δB the perturbed magnetic field, while Q, S, T, U, V, W, and X are equilibrium quantities (with T * the complex conjugate of T ) defined in Connor et al [3]:…”

“…In toroidal geometry poloidal harmonics are coupled together so that the external MHD solution produces a relationship between ∆ ′ m s. The external MHD equations therefore do not uniquely determine ∆ ′ m as they did in the cylinder. This relationship is given by Connor et al [3], although we use the notation of Fitzpatrick et al [4] where it is written as |∆ − E| = 0 (1) with ∆ the matrix of layer responses and E the matrix corresponding to the external MHD equations. As an example, we can look at the system with two rational surfaces where…”

Interaction of bootstrap-current-driven magnetic islands

“…where the local rotational transform for the primary island region is defined by n: n0w0 , f_(k) -2kK(k) nAmn (36) and Amn = Oq[(m/nq0 ) -1]. There is a resonance wherever k = kllv, where f_(k_tv) ---v/It.…”