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.
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