  In [J. Cantarella, J. Parsley, J. Geom. Phys. 60:1127 (2010)] Cantarella and
Parsley introduced the notion of submanifold helicity. In the present paper we
investigate properties of surface helicity and in particular answer two open
questions posed in the aforementioned work: (i) We give a precise
mathematically rigorous physical interpretation of surface helicity in terms of
linking of distinct field lines. (ii) We prove that surface helicity is
non-trivial if and only if the underlying surface has non-trivial topology
(i.e. at least one hole).
  We then focus on toroidal surfaces which are of relevance in plasma physics
and express surface helicity in terms of average poloidal and toroidal windings
of the individual field lines of the underlying vector field which enables us
to provide a connection between surface helicity and rotational transform.
Further, we show how some of our results may be utilised in the context of coil
designs for plasma fusion confinement devices in order to obtain coil
configurations of particular "simple" shape.
  Lastly, we consider the problem of optimising surface helicity among toroidal
surfaces of fixed area and show that toroidal surfaces admitting a symmetry
constitute global minimisers.
