  In this paper, the analytical form of the quasilinear diffusion coefficients
is modified from the Kennel-Engelmann diffusion coefficients to guarantee the
positive definiteness of its bounce average in a toroidal geometry. By
evaluating the parallel inhomogeneity of plasmas and magnetic fields in the
trajectory integral, we can ensure the positive definiteness and help
illuminate some non-resonant toroidal effects in the quasilinear diffusion.
When the correlation length of the plasma-wave interaction is comparable to the
magnetic field variation length, the variation becomes important and the
parabolic variation at the outer-midplane, the inner-midplane, and trapping
tips can be evaluated by Airy functions. The new form allows the coefficients
to include both resonant and non-resonant contributions, and the correlations
between the consecutive resonances and in many poloidal periods. The
positive-definite form is implemented in a wave code TORIC and we present an
example for ITER using this form.
