  Turbulence plays a very important role in determining the transport of energy
and particles in tokamaks. This work is devoted to studying the chaotic
diffusion in multi-scale turbulence in the context of the nonlinear
wave-particle interaction. Turbulent waves with different scales of
characteristic wavelengths can interact with the same group of charged
particles when their phase velocity is close to the velocities of the charged
particles. A multi-wavenumber standard mapping is developed to model the
chaotic diffusion in multi-scale turbulence. The diffusion coefficient is
obtained by calculating the correlation functions analytically. It is found
that the contribution of the largest scale turbulence dominates the deviation
from the quasi-linear diffusion coefficient. Increasing the overlap parameters
of the smaller scale turbulence by just the increasing the wavenumber cannot
make the diffusion coefficient to be the quasi-linear diffusion coefficient for
a finite wave amplitude. Especially, in two-scale turbulence, the diffusion
coefficient is mostly over the quasi-linear diffusion coefficient in the large
wavenumber (of the smaller scale turbulence) limit. As more scales of
components are added in the turbulence, the diffusion coefficient approaches
the quasi-linear diffusion coefficient. The results can also be applied to
other resonance-induced multi-scale turbulence in Hamiltonian systems with 1.5
or 2 degrees of freedom.
