  In order for diffusive shock acceleration (DSA) to accelerate particles to
high energies, the energetic particles must be able to interact with magnetic
turbulence over a broad wavelength range. The weakly anisotropic distribution
of accelerated particles, i.e., cosmic rays (CRs), is believed capable of
producing this turbulence in a symbiotic relationship where the magnetic
turbulence required to accelerate the CRs is created by the accelerated CRs
themselves. In efficient DSA, this wave-particle interaction can be strongly
nonlinear where CRs modify the plasma flow and the specific mechanisms of
magnetic field amplification. Resonant interactions have long been known to
amplify magnetic fluctuations on the scale of the CR gyroradius, and Bell
(2004) showed that the CR current can efficiently amplify magnetic fluctuations
with scales smaller than the CR gyroradius. Here, we show with a multi-scale,
quasi-linear analysis that the presence of turbulence with scales shorter than
the CR gyroradius enhances the growth of modes with scales longer than the
gyroradius, at least for particular polarizations. We use a mean-field approach
to average the equation of motion and the induction equation over the ensemble
of magnetic field oscillations accounting for the anisotropy of relativistic
particles on the background plasma. We derive the response of the magnetized CR
current on magnetic field fluctuations and show that, in the presence of
short-scale, Bell-type turbulence, long wavelength modes are amplified. The
polarization, helicity, and angular dependence of the growth rates are
calculated for obliquely propagating modes for wavelengths both below and above
the CR mean free path. The long wavelength growth rates we estimate for typical
supernova remnant parameters are sufficiently fast to suggest a fundamental
increase in the maximum CR energy a given shock can produce.
