  We consider the zero-dispersion limit for the Benjamin-Ono equation on the
torus. We prove that when the initial data is a single well, the
zero-dispersion limit exists in the weak sense and is uniform on every compact
time interval. Moreover, the limit is equal to the signed sum of branches for
the multivalued solution of the inviscid Burgers equation obtained by the
method of characteristics. This result is similar to the one obtained by Miller
and Xu for the Benjamin-Ono equation on the real line for decaying and positive
initial data. We also establish some precise asymptotics of the spectral data
with the cosine initial data, justifying our approximation method, which is
analogous to the work of Miller and Wetzel concerning a family of rational
potentials for the Benjamin-Ono equation on the real line.
