Generalization of the ballooning Delta ' to poloidally asymmetric Tokamak equilibria

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Abstract
The author shows that the conventional definition of the ballooning Delta ' (as the ratio of the coefficients of the small to large series solutions) may lead to divergences in the asymptotic solution of the ideal ballooning equation for Tokamaks. These problems occur for equilibria which are not up-down symmetric. In the limit of these divergences the terms in the small and large solutions are interchangeable and thus the definition of Delta ' becomes ambiguous. By mixing the solutions, the author shows how the divergences may be cancelled. The author illustrates that this 'mixing' is the correct procedure by evaluating the solution in an ideal inertial layer and matching onto the marginal ideal solution. The author shows that the conventional Delta ' is not meaningful as a measure of stability to ballooning modes when the series become divergent.