  A junction between two boundaries of a topological superconductor (TSC),
mediated by localized edge modes of Majorana fermions, is investigated. The
tunneling of fermions across the junction depends on the magnetic flux. It
breaks the time-reversal symmetry at the boundary of the sample. The persistent
current is determined by the emergence of Majorana edge modes. The structure of
the edge modes depends on the magnitude of the tunneling amplitude across the
junction. It is shown that there are two different regimes, which correspond to
strong and weak tunneling of Majorana fermions, distinctive in the persistent
current behavior. In a strong tunneling regime, the fermion parity of edge
modes is not conserved and the persistent current is a $2\pi$-periodic function
of the magnetic flux. When the tunneling is weak the chiral Majorana states,
which are propagating along the edges have the same fermion parity. They form a
$4\pi$-phase periodic persistent current along the boundaries. The regions in
the space of parameters, which correspond to the emergence of $2\pi$- and of
$4\pi$-harmonics, are numerically determined. The peculiarities in the
persistent current behavior are studied.
