Drawing from decades-long-established results on tokamak equilibria (showing that flux surfaces closely surrounding the magnetic axis are ellipses), and using basic geometry and back-of-the-envelope calculations, recent theoretical results are revisited which have revealed that, for an externally imposed tilting, the elongation and hence the up–down asymmetry (UDA) of flux surfaces in the core of a tokamak plasma are inversely proportional to the toroidal current density flowing on axis. A figure of merit is also proposed to measure on-axis UDA, which stems from a simple geometrical analysis of elliptical flux surfaces and lies between zero and one. Moreover, it is shown that the current density on axis is bounded from below by the asymmetry-controlling coefficients themselves (which are essentially set by external shaping coils), meaning there is no way around it: enforcing stronger ellipticity from the outside can only result in increasing the minimum current density allowed in the core, whereas trying to make the latter go down is only possible with milder shaping coefficients. Such a severe structural constraint is simply the manifestation of a topological necessary condition to have closed flux surfaces nested around a tokamak magnetic axis and suggests that experimental attempts to push UDA from a strongly shaped plasma boundary into the core will remain mostly elusive.
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