  We study the nonlinear growth of kinetic gyroresonance instability of cosmic
rays (CRs) induced by large scale compressible turbulence. This feedback of
cosmic rays on turbulence was shown to induce an important scattering mechanism
in addition to direct interaction with the compressible turbulence. The linear
growth is bound to saturate due to the wave-particle interactions. By balancing
increase of CR anisotropy via the large scale compression and its decrease via
the wave-particle scattering, we find the steady state solutions. The nonlinear
suppression due to the wave-particle scattering limit the energy range of CRs
that can excite the instabilities and be scattered by the induced slab waves.
The direct interaction with large scale compressible modes still appears to be
the dominant mechanism for isotropization of high energy cosmic rays (> 100
GeV).
