  David Goss developed a very general Fourier transform in additive harmonic
analysis in the function field setting. In order to introduce the Fourier
transform for continuous characteristic $p$ valued functions on $\mathbb{Z}_p$,
Goss introduced and studied an analogue of flows in finite characteristic. In
this paper, we use another approach to study flows in finite characteristic. We
recast the notion of a flow in the language of the classical umbral calculus,
which allows to generalize the formula for flows first proved by Goss to a more
general setting. We study duality between flows using the classical umbral
calculus, and show that the duality notion introduced by Goss seems a natural
one. We also formulate a question of Goss about the exact relationship between
two flows of a Drinfeld module in the language of the classical umbral
calculus, and give a partial answer to it.
