  In this paper, we investigate the long-time dynamics of the linearized 2-D
Euler equations around a hyperbolic tangent flow $(\tanh y,0)$. A key
difference compared to previous results is that the linearized operator has an
embedding eigenvalue, which has a significant impact on the dynamics of the
linearized system. For the first mode, the dynamics consists of there parts:
non-decay part related to the eigenspace associated with the embedding
eigenvalue, slow decay part due to the resolvent singularity, and fast decay
part related to the inviscid damping. For higher modes, the dynamics is similar
to the inviscid damping phenomena in the case without embedding eigenvalues.
