Quantifying the Difference Between the Flux-Tube Expansion Factor at the Source Surface and at the Alfvén Surface Using a Global MHD Model for the Solar Wind

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The potential field approximation has been providing a fast, and computationally inexpensive estimation for the solar corona's global magnetic field geometry for several decades. In contrast, more physics-based global magnetohydrodynamic (MHD) models have been used for a similar purpose, while being much more computationally expensive. Here, we investigate the difference in the field geometry between a global MHD model and the potential field source surface model (PFSSM) by tracing individual magnetic field lines in the MHD model from the Alfven surface (AS), through the source surface (SS), all the way to the field line footpoint, and then back to the source surface in the PFSSM. We also compare the flux-tube expansion at two points at the SS and the AS along the same radial line. We study the effect of solar cycle variations, the order of the potential field harmonic expansion, and different magnetogram sources. We find that the flux-tube expansion factor is consistently smaller at the AS than at the SS for solar minimum and the fast solar wind, but it is consistently larger for solar maximum and the slow solar wind. We use the Wang--Sheeley--Arge (WSA) model to calculate the associated wind speed for each field line, and propagate these solar-wind speeds to 1AU. We find a more than five hours deviation in the arrival time between the two models for 20% of the field lines in the solar minimum case, and for 40% of the field lines in the solar maximum case.