  We theoretically study a Josephson junction formed by two finite-size
$s$-wave SCs on a topological insulator under a magnetic field. At certain
conditions, the junction hosts the chiral Majorana modes enclosing the two
finite-size SCs. The interplay of the extended chiral Majorana modes and the
states inside the junction can results in nontrivial topological effects such
as the $2n \pi$ fractional AC Josephson effects predicted in
Ref.~\cite{ChoiSim} We show that the $2n \pi$ fractional AC Josephson effects
can occur in a realistic situation, such as the presence of the midgap states,
without requiring fine tuning of the parameters of the junction. We also find
that the Shapiro spikes of the junction show a rich structure in a wide range
of the AC voltage bias, facilitating experimental identification of the $2n
\pi$ fractional AC Josephson effects. Moreover, we discuss how to observe the
non-commutativity of the operations that braid the Majorana fermions of the
junction, by measuring the Josephson current. Finally, we study the state
evolution of the junction when the junction hosts a different number of
Majorana zero modes from the case of the $2n \pi$ fractional AC Josephson
effects.
