  We use the Cole--Hopf transformation and the Laplace method for the heat
equation to justify the numerical results on enstrophy growth in the viscous
Burgers equation on the unit circle. We show that the maximum enstrophy
achieved in the time evolution is scaled as $\mathcal{E}^{3/2}$, where
$\mathcal{E}$ is the large initial enstrophy, whereas the time needed for
reaching the maximal enstrophy is scaled as $\mathcal{E}^{-1/2}$. These bounds
are sharp for sufficiently smooth initial conditions.
