  We study how the complex interplay between roughness, inertia, and diffusion
controls the tracer transport in rough channels. We first simulate the flow and
tracer transport over wide ranges of channel roughness, Reynolds number ($Re$),
and P\'eclet number ($Pe$) observable in nature. $Pe$ exerts a first-order
control on first-passage time distributions, and the effect of roughness on the
tracer transport becomes evident with the increase in $Re$. The interplay
between the roughness and $Re$ causes eddy flows, which intensify or suppress
anomalous transport depending on $Pe$. At infinite $Pe$, the late-time scaling
follows a universal power-law scaling, which is explained by conducting a
scaling analysis. With extensive numerical simulations and stochastic modeling,
we show that the roughness, inertia, and diffusion effects are encoded in
Lagrangian velocity statistics represented by velocity distribution and
correlation. We finally predict the anomalous transport using a stochastic
model that considers the Lagrangian velocity statistics.
