  We theoretically analyse a long constriction between the helical edge states
of a two-dimensional topological insulator. The constriction is laterally
tunnel-coupled to two superconductors and a magnetic field is applied
perpendicularly to the plane of the two-dimensional topological insulator. The
Josephson current is calculated analytically up to second order in the tunnel
coupling both in the absence and in the presence of a bias (DC and AC Josephson
currents). We show that in both cases the current acquires an anomalous
$4\pi$-periodicity with respect to the magnetic flux that is absent if the two
edges are not tunnel-coupled to each other. The result, that provides at the
same time a characterisation of the device and a possible experimental
signature of the coupling between the edges, is stable against temperature. The
processes responsible for the anomalous $4\pi$-periodicity are the ones where,
within the constriction, one of the two electrons forming a Cooper pair tunnels
between the two edges.
