  Context. Indirect observations of the cosmic-ray electron (CRE) distribution
via synchrotron emission is crucial for deepening the understanding of the CRE
transport in the interstellar medium, and in investigating the role of galactic
outflows.
  Aims. In this paper, we quantify the contribution of diffusion- and
advection-dominated transport of CREs in the galaxy M51 considering relevant
energy loss processes.
  Methods. We used recent measurement from M 51 that allow for the derivation
of the diffusion coefficient, the star formation rate, and the magnetic field
strength. With this input, we solved the 3D transport equation numerically
including the spatial dependence as provided by the measurements, using the
open-source transport framework CRPropa (v3.1). We included 3D transport
(diffusion and advection), and the relevant loss processes.
  Results. We find that the data can be described well with the parameters from
recent measurements. For the best fit, it is required that the wind velocity,
following from the observed star formation rate, must be decreased by a factor
of 5. We find a model in which the inner galaxy is dominated by advective
escape and the outer galaxy is composed by both diffusion and advection.
  Conclusions. Three-dimensional modelling of cosmic-ray transport in the
face-on galaxy M51 allows for conclusions about the strength of the outflow of
such galaxies by quantifying the need for a wind in the description of the
cosmic-ray signatures. This opens up the possibility of investigating galactic
winds in face-on galaxies in general.
