  \textit{Harmonic amoebas} are generalisations of amoebas of algebraic curves
immersed in complex tori. Introduced in \cite{Kri}, the consideration of such
objects suggests to enlarge the scope of tropical geometry. In the present
paper, we introduce the notion of harmonic morphisms from tropical curves to
affine spaces and show how these morphisms can be systematically described as
limits of families of harmonic amoeba maps on Riemann surfaces. It extends
previous results about approximation of tropical curves in affine spaces and
provides a different point of view on Mikhalkin's approximation Theorem for
regular phase-tropical morphisms, as stated e.g. in \cite{Mikh06}. The results
presented here follow from the study of imaginary normalised differentials on
families of punctured Riemann surfaces and suggest interesting connections with
compactifications of moduli spaces.
