  In this paper, we investigate the long-time behavior of the two-dimensional
incompressible Boussinesq system with kinematic viscosity in a periodic
channel, focusing on instability and asymptotic stability near hydrostatic
equilibria. Firstly, we prove that any hydrostatic equilibrium reveals
long-time instability when the initial data are perturbed in Sobolev spaces of
low regularity. Secondly, we establish asymptotic stability of the stratified
density, which is strictly decreasing in the vertical direction, under
sufficiently regular perturbations, proving that the solution converges to the
unique minimizer of the total energy.
  Our analysis is based on the energy method. Although the total energy
dissipates due to kinematic viscosity, such mechanism cannot capture the
stratification of the density. We overcome this difficulty by discovering
another Lyapunov functional which exhibits the density stratification in a
quantitative manner.
