  We consider the kinematic fluctuation dynamo problem in a flow that is
random, white-in-time, with both solenoidal and potential components. This
model is a generalization of the well-studied Kazantsev model. If both the
solenoidal and potential parts have the same scaling exponent, then, as the
compressibility of the flow increases, the growth rate decreases but remains
positive. If the scaling exponents for the solenoidal and potential parts
differ, in particular if they correspond to typical Kolmogorov and Burgers
values, we again find that an increase in compressibility slows down the growth
rate but does not turn it off. The slow down is, however, weaker and the
critical magnetic Reynolds number is lower than when both the solenoidal and
potential components display the Kolmogorov scaling. Intriguingly, we find that
there exist cases, when the potential part is smoother than the solenoidal
part, for which an increase in compressibility increases the growth rate. We
also find that the critical value of the scaling exponent above which a dynamo
is seen is unity irrespective of the compressibility. Finally, we realize that
the dimension $d = 3$ is special, since for all other values of $d$ the
critical exponent is higher and depends on the compressibility.
