VOLUME 87, NUMBER 3 PHYSICAL REVIEW LETTERS 16 JULY 2001 Loss of Second-Ballooning Stability in Three-Dimensional Equilibria C. C. Hegna 1 and S. R. Hudson 2 1 Departments of Physics and Engineering Physics, University of Wisconsin, Madison, Wisconsin 53706-1687 2 Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543 (Received 13 March 2001; published 27 June 2001) The effect of three-dimensional geometry on the stability boundaries of ideal ballooning modes is investigated. In particular, the relationship between the symmetry properties of the local shear and the magnetic curvature is addressed for quasisymmetric configurations. The presence of symmetry breaking terms in the local shear can produce localized ballooning instabilities in regions of small average magnetic shear which lower first-ballooning stability thresholds and can potentially eliminate the second stability regime. DOI: 10.1103/PhysRevLett.87.035001 PACS numbers: 52.35.Py, 52.30.–q, 52.55.Dy, 52.55.Hc Ballooning instabilities are short wavelength pressure driven ideal magnetohydrodynamic (MHD) modes [1] that limit plasma performance. For the stellarator class of magnetic confinement devices, ballooning stability crite- ria are often the most restrictive. Ballooning instabilities are driven by a pressure gradient in a region of configu- ration space with unfavorable magnetic field curvature. Since curvature varies on the magnetic surfaces of toroidal confinement devices, the eigenmode structure tends to lo- calize on the magnetic surface to facilitate access to the free energy source. Instability ensues when the destabi- lizing pressure/curvature drive is more virulent than sta- bilizing field line bending energy which is influenced by the local shear of the magnetic field line. The geometry of the magnetic field lines is important in describing the eigenmode structure and associated instability properties. Prior work [2] led to important insights into the nature of ballooning stability in tokamak configurations. A num- ber of authors looked at the stability properties of specific stellarator configurations [3,4], but it is difficult to draw general conclusions from these studies about the nature of MHD ballooning stability in three-dimensional configura- tions. In this paper, we attempt to understand some of the generic physics of ballooning stability relating to the role of three-dimensional (3D) magnetic geometry. A particularly important result of ballooning stability studies in axisymmetric configurations is the discovery of the second stability regime [2,5]. A key element in understanding the physics of the second stability regime is the role of the local shear [2]. In toroidal configu- rations, the local shear is influenced by pressure driven Pfirsch-Schlüter effects. To minimize the stabilizing effect of field line bending, the most unstable ballooning mode eigenfunctions reside in regions of small local shear. For the small pressure gradient, the low shear region lies on the large major radius side where the curvature is unfa- vorable. At higher pressure gradient, the region of small local shear moves away from the bad curvature region to- wards a region on the magnetic surface with favorable cur- vature. Consequently, at sufficiently high enough pressure gradient, the pressure modification of the stabilizing field line bending energy overcomes the destabilizing curvature drive, and the ballooning mode is stabilized. Typically, this second stability regime is limited to regions with small averaged magnetic shear; however, axisymmetric shaping and the aspect ratio affect quantitative estimates of the sta- bility boundary. Consequently, tokamaks with weak or re- versed magnetic shear have very good ballooning stability properties. As is shown in the following, the presence of symmetry breaking effects can have a dramatic impact on the “second stable” region. The difficult aspect of appreciating the role of three- dimensional shaping on ballooning instability is the generation of the equilibria themselves since there is no general prescription for the complete specification of 3D magnetostatic equilibria in a toroidal domain with concen- tric toroidal flux surfaces (surfaces upon which magnetic field lines lie). However, a method was developed [6] to generate a sequence of three-dimensional equilibria in a region localized to a magnetic surface; this generalizes previous work on local solutions to the Grad-Shafranov equation on asymmetric flux surfaces [2,7,8]. By appli- cation of this technique, one is able to calculate stability boundaries for modes localized to magnetic surfaces as functions of shaping and profile parameters. A particularly useful form for this analysis technique is the generation of stability boundaries as measured by ˆ s 2a curves, which are prominently used in tokamak research, where ˆ s and a are dimensionless measures of the flux surface averaged magnetic shear and pressure gradient, respectively [1,2]. Stability curves in an ˆ s 2a space can be generated for the prescribed three-dimensional equilibria as well [9]. A local 3D equilibrium is given by the specification of two flux surface profile quantities, the rotational transform i o and the magnetic coordinate mapping Xu, z  on the magnetic surface of interest c  c o , where u and z are any straight field line poloidal and toroidal angles. The choice of X is not completely free; it must satisfy con- straints. The interested reader is referred to Ref. [6] for a detailed discussion of the local model. It is often con- venient to specify the pressure gradient dpdc and the averaged magnetic shear didc on the surface as the two 035001-1 0031-9007 01 87(3) 035001(4)$15.00 © 2001 The American Physical Society 035001-1