  The motions of small-scale magnetic flux elements in the solar photosphere
can provide some measure of the Lagrangian properties of the convective flow.
Measurements of these motions have been critical in estimating the turbulent
diffusion coefficient in flux-transport dynamo models and in determining the
Alfven wave excitation spectrum for coronal heating models. We examine the
motions of internetwork flux elements in a 24 hour long Hinode/NFI magnetogram
sequence with 90 second cadence, and study both the scaling of their mean
squared displacement and the shape of their displacement probability
distribution as a function of time. We find that the mean squared displacement
scales super-diffusively with a slope of about 1.48. Super-diffusive scaling
has been observed in other studies for temporal increments as small as 5
seconds, increments over which ballistic scaling would be expected. Using
high-cadence MURaM simulations, we show that the observed super-diffusive
scaling at short temporal increments is a consequence of random changes in the
barycenter positions due to flux evolution. We also find that for long temporal
increments, beyond granular lifetimes, the observed displacement distribution
deviates from that expected for a diffusive process, evolving from Rayleigh to
Gaussian. This change in the distribution can be modeled analytically by
accounting for supergranular advection along with motions due to granulation.
These results complicate the interpretation of magnetic element motions as
strictly advective or diffusive on short and long timescales and suggest that
measurements of magnetic element motions must be used with caution in turbulent
diffusion or wave excitation models. We propose that passive trace motions in
measured photospheric flows may yield more robust transport statistics.
