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We study transport in random undirectional wave-like velocity fields with nonlinear dispersion relations. For this simple model, we have several interesting findings: (1) In the absence of molecular diffusion the entire family of fractional Brownian motions (FBMs), persistent or anti-persistent, can arise in the scaling limit. (2) The infrared cutoff may alter the scaling limit depending on whether the cutoff exceeds certain critical value or not. (3) Small, but nonzero, molecular diffusion can drastically change the scaling limit. As a result, some regimes stay intact; some (persistent) FBM regimes become non-Gaussian and some other FBM regimes become Brownian motions with enhanced diffusion coefficients. Moreover, in the particular regime where the scaling limit is a Brownian motion in the absence of molecular diffusion, the vanishing molecular diffusion limit of the enhanced diffusion coefficient is strictly larger than the diffusion coefficient with zero molecular diffusion. This is the first such example that we are aware of to demonstrate rigorously a nonperturbative effect of vanishing molecular diffusion on turbulent diffusion coefficient.