On the nature of rotational shear stabilization in toroidal geometry and its numerical representation

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Abstract
In this paper a simple one-dimensional model problem is treated as a paradigm for understanding the nature of rotational shear stabilization in toroidal geometry and its numerical representation. The model is first formulated in a ballooning mode angle θ0 space, where the convective nature of the stabilization is clear. If θ0 is treated as a continuum variable, the slightest rotational shear eventually convects a ballooning mode through both the unstable and stable poloidal angles of a torus, resulting in a stable slab-like time-averaged growth rate and eigenmode growth rate. However, if θ0 is treated as a discrete variable, unstable eigenmodes remain at weak velocity shear. By transforming to real X space, we are able to physically interpret these unstable modes. Continuous θ0 formulations correspond to a radially infinite box and discrete θ0 formulations correspond to a finite box. If care is taken to make the rotational shear constant throughout the whole box, the unstable eigenmodes are found to be localized to the edge and the stable continuum-like modes are localized but distributed throughout the interior. In the case of constant rotational shear, it is found that a critical rotational shear exists for complete toroidal stabilization independent of representation (continuous or discrete θ0) or box size.