  It is known that there are thermodynamic states for which the Gaussian-core
(GC) fluid displays anomalous properties such as expansion upon isobaric
cooling (density anomaly) and increased single-particle mobility upon
isothermal compression (self-diffusivity anomaly). We investigate how
temperature and density affect its short-range translational structural order,
as characterized by the two-body excess entropy. We find that there is a wide
range of conditions for which the short-range translational order of the GC
fluid decreases upon isothermal compression (structural order anomaly). The
origin of the structural anomaly is qualitatively similar to that of other
anomalous fluids and is connected to how compression affects static
correlations at different length scales. We find that the self-diffusivity of
the GC fluid obeys a scaling relationship with the two-body excess entropy that
is very similar to the one observed for a variety of simple liquids. One
consequence of this relationship is that the state points for which structural,
self-diffusivity, and density anomalies of the Gaussian-core fluid occur appear
as cascading regions on the temperature-density plane, a phenomenon observed
earlier for models of waterlike fluids. There are, however, key differences
between the anomalies of GC and waterlike fluids, and we discuss how those can
be qualitatively understood by considering the respective interparticle
potentials of these models. Finally, we note that the self-diffusivity of the
Gaussian-core fluid obeys different scaling laws depending on whether the
two-body or total excess entropy is considered. This finding, which deserves
more comprehensive future study, appears to underscore the significance of
higher-body correlations for the behavior of fluids with bounded interactions.
