  In this paper, the decay and growth of localized wave packet (LWP) in
two-dimensional plane-Poiseuille flow are studied numerically and
theoretically. When the Reynolds number ($Re$) is less than a critical value
$Re_c$, the disturbance kinetic energy $E_k$ of LWP decreases monotonically
with time and experiences three decay periods, i.e. the initial and the final
steep descent periods, and the middle plateau period. Higher initial $E_k$ of a
decaying LWP corresponds to longer lifetime. According to the simulations, the
lifetime scales as $(Re_c-Re)^{-1/2}$, indicating a divergence of lifetime as
$Re$ approaches $Re_c$, a phenomenon known as "critical slowing-down". By
proposing a pattern preservation approximation, i.e. the integral kinematic
properties (e.g. the disturbance enstrophy) of an evolving LWP are independent
of $Re$ and single valued functions of $E_k$, the disturbance kinetic energy
equation can be transformed into the classical differential equation for
saddle-node bifurcation, by which the lifetimes of decaying LWPs can be
derived, supporting the $-1/2$ scaling law. Furthermore, by applying the
pattern preservation approximation and the integral kinematic properties
obtained as $Re<Re_c$, the Reynolds number and the corresponding $E_k$ of the
whole lower branch, the turning point, and the upper-branch LWPs with
$E_k<0.15$ are predicted successfully with the disturbance kinetic energy
equation, indicating that the pattern preservation is an intrinsic feature of
this localized transitional structure.
