Generalization of the ballooning Δ ' to poloidally asymmetric tokamak equilibria

It is shown that the conventional definition of the ballooning Δ' (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 Δ' becomes ambiguous. By mixing the solutions, it is shown how the divergences may be cancelled. It is illustrated that this ''mixing'' is the correct procedure by evaluating the solution in an ideal inertial layer and matching onto the marginal ideal solution. It is shown that the conventional Δ' is not meaningful as a measure of stability to ballooning modes when the series become divergent. (author)