Parker Solar Probe (PSP) is providing an unprecedented view of the Suns corona as it progressively dips closer into the solar atmosphere with each solar encounter. Each set of observations provides a unique opportunity to test and constrain global models of the solar corona and inner heliosphere and, in turn, use the model results to provide a global context for interpreting such observations. In this study, we develop a set of global magnetohydrodynamic (MHD) model solutions of varying degrees of sophistication for PSPs first four encounters and compare the results with in situ measurements from PSP, Stereo-A, and Earth-based spacecraft, with the objective of assessing which models perform better or worse. All models were primarily driven by the observed photospheric magnetic field using data from Solar Dynamics Observatorys Helioseismic and Magnetic Imager (HMI) instrument. Overall, we find that there are substantial differences between the model results, both in terms of the large-scale structure of the inner heliosphere during these time periods, as well as in the inferred time-series at various spacecraft. The thermodynamic model, which represents the middle ground, in terms of model complexity, appears to reproduce the observations most closely for all four encounters. Our results also contradict an earlier study that had hinted that the open flux problem may disappear nearer the Sun. Instead, our results suggest that this missing solar flux is still missing even at 26.9 Rs, and thus it cannot be explained by interplanetary processes. Finally, the model results were also used to provide a global context for interpreting the localized in situ measurements.
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