  We consider the non-homogeneous generalised Burgers equation
  \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu
\frac{\partial^2 u}{\partial x^2} = \eta,\ t \geq 0,\ x \in S^1.
  Here f is strongly convex and satisfies a growth condition, \nu is small and
positive, while \eta is a random forcing term, smooth in space and white in
time.
  For any solution u of this equation we consider the quasi-stationary regime,
corresponding to t>=T_1, where T_1 depends only on f and on the distribution of
\eta. We obtain sharp upper and lower bounds for Sobolev norms of $u$ averaged
in time and in ensemble. These results yield sharp upper and lower bounds for
natural analogues of quantities characterising the hydrodynamical turbulence.
All our bounds do not depend on the initial condition or on t for t>=T_1, and
hold uniformly in \nu.
  Estimates similar to some of our results have been obtained by Aurell,
Frisch, Lutsko and Vergassola on a physical level of rigour; we use an argument
from their article.
