  We study the influence of the energy spectrum on the extent of the cascade
range of a passive tracer in turbulent flows. The interesting cases are when
there are two different spectra over the potential range of the tracer cascade
(in 2D when the tracer forcing is in the inverse energy cascade range, and in
3D when the Schmidt number Sc is large). The extent of the tracer cascade range
is then limited by the width of the range for the shallower of the two energy
spectra. Nevertheless, we show that in dimension $d=2,3$ the tracer cascade
range extends (up to a logarithm) to $\kappa_{d\text{D}}^{p}$, where
$\kappa_{d\text{D}}$ is the wavenumber beyond which diffusion should dominate
and $p$ is arbitrarily close to 1, provided Sc is larger than a certain power
(depending on $p$) of the Grashof number. We also derive estimates which
suggest that in 2D, for Sc${}\sim1$ a wide tracer cascade can coexist with a
significant inverse energy cascade at Grashof numbers large enough to produce a
turbulent flow.
