  Morphogens are secreted signaling molecules that mediate tissue patterning
and growth of embryonic tissues. They are secreted in a localized region and
spread through the tissue to form a graded concentration profile. We present a
cell-based model of morphogen spreading that combines secretion in a local
source, extracellular diffusion and cellular trafficking. We introduce
hydrodynamic modes of morphogen transport and characterize the dynamics of
transport by dispersion relations of these dynamic eigenmodes. These dispersion
relations specify the characteristic relaxation time of a mode as a function of
its wavelength. In a simple model we distinguish two distinct dynamic modes
characterized by different timescales. We find that the slower mode defines the
effective diffusion and degradation as well as the shape of the concentration
profile in steady state. Using our approach we discuss mechanisms of morphogen
transport in the developing wing imaginal disc of the fruit fly
\textit{Drosophila}, distinguishing three transport regimes: transport by
extracellular diffusion, transport by transcytosis and a regime where both
transport mechanisms are combined.
