  We consider the question of growth of high order Sobolev norms of solutions
of the conservative surface quasi-geostrophic equation. We show that if $s>0$
is large then for every given $A$ there is exist small in $H^s$ initial data
such that the corresponding solution's $H^s$ norm exceeds $A$ at some time. The
idea of the construction is quasilinear. We use a small perturbation of a
stable shear flow. The shear flow can be shown to create small scales in the
perturbation part of the flow. The control is lost once the nonlinear effects
become too large.
