  In this paper, we consider the 2-dimensional non-viscous Oldroyd-B model. In
the case of the ratio equal 1~($\alpha=0$), it is a difficult case since the
velocity field $u(t,x)$ is no longer decay. Fortunately, by {observing the
exponential decay} of the stress tensor $\tau(t,x)$, we succeeded in proving
the global existence for this system with some large initial data. Moreover, we
give an unsteady result: when the ratio is close to 1~($a\rightarrow 0$), the
system is not steady for large time. This implies an interesting physical
phenomenon that the term $a\mathbb{D}u$ is a bridge between the transformation
of kinetic energy $u$ and elastic potential energy $\tau$, but this process is
transient for large time, which leads the instability.
