Stream Helicity in Toroidal Domains

As the analogue of helicity, the concept of stream helicity proposed in 2007 provides a quantitative measurement for entanglement of streamlines for an incompressible flow field or magnetic field lines for a magnetic field. It is found that the helical-wave decomposition can serve as a convenient mathematical physics tool for the calculation of helicity and stream helicity. However, for a multiply connected domain such as a torus under the background of TOKAMAK, the analysis presents some peculiarities due to the nature of multiple-value of some involving potentials. Here we give a full derivation of stream helicity for such a case, together with some detailed analysis. Particularly, it is found that a maximum of absolute helicity under prescribed energy could be attained by a pure potential field.