  We consider $N$ non-intersecting Brownian bridges conditioned to stay below a
fixed threshold. We consider a scaling limit where the limit shape is
tangential to the threshold. In the large $N$ limit, we determine the limiting
distribution of the top Brownian bridge conditioned to stay below a function as
well as the limiting correlation kernel of the system. It is a one-parameter
family of processes which depends on the tuning of the threshold position on
the natural fluctuation scale. We also discuss the relation to the six-vertex
model and to the Aztec diamond on restricted domains.
