  While it is well-known that cosmic rays (CRs) can gain energy from turbulence
via second order Fermi acceleration, how this energy transfer affects the
turbulent cascade remains largely unexplored. Here, we show that damping and
steepening of the compressive turbulent power spectrum are expected once the
damping time $t_{\rm damp} \sim \rho v^{2}/\dot{E}_{\rm CR} \propto E_{\rm
CR}^{-1}$ becomes comparable to the turbulent cascade time. Magnetohydrodynamic
(MHD) simulations of stirred compressive turbulence in a gas-CR fluid with
diffusive CR transport show clear imprints of CR-induced damping, saturating at
$\dot{E}_{\rm CR} \sim \tilde{\epsilon}$, where $\tilde{\epsilon}$ is the
turbulent energy input rate. In that case, almost all the energy in large scale
motions is absorbed by CRs and does not cascade down to grid scale. Through a
Hodge-Helmholtz decomposition, we confirm that purely compressive forcing can
generate significant solenoidal motions, and we find preferential CR damping of
the compressive component in simulations with diffusion and streaming,
rendering small-scale turbulence largely solenoidal, with implications for
thermal instability and proposed resonant scattering of $E > 300$ GeV CRs by
fast modes. When CR transport is streaming dominated, CRs also damp large scale
motions, with kinetic energy reduced by up to to an order of magnitude in
realistic $E_{\rm CR} \sim E_{\rm g}$ scenarios, but turbulence (with a reduced
amplitude) still cascades down to small scales with the same power spectrum.
Such large scale damping implies that turbulent velocities obtained from the
observed velocity dispersion may significantly underestimate turbulent forcing
rates, i.e. $\tilde{\epsilon} \gg \rho v^{3}/L$.
