The Solar Wind Speed Expansion Factor [${v}$–$f _{\text{s}}$] Relationship at the Inner Boundary (18 $\text{R}_{\odot }$) of the Heliosphere

Abstract

The accuracy of data-driven magnetohydrodynamics (MHD) models depends on accurate boundary conditions specified at the inner heliosphere. However, not all of the MHD parameters [$\boldsymbol{B}, \boldsymbol{v}, \rho , T$] are measurable close to the Sun at the present time, except the vector magnetic field [$\boldsymbol{B}$] at the photosphere. The solar wind speed [$\boldsymbol{v}$], which is probably most relevant to space-weather forecasting, is often modeled by the standard Wang–Sheeley (WS) formula, which is based on an inverse relationship between the solar wind speed [$\boldsymbol{v}$] at 1 AU and the expansion factor [$f_{\text{s}}$] estimated at 2.5 solar radii [$\text{R}_{ \odot }$], with the following generic form: ${v} = {v}_{1} + {v}_{2} f_{\text{s}}^{- \alpha }$ (where $\boldsymbol{v}$ is the solar wind speed at 18 $\text{R}_{\odot }$, $f_{\text{s}}$ is the magnetic-field expansion factor, and ${v}_{1}$, ${v}_{2}$, and $\alpha$ are three free parameters to be determined). While the WS formula uses “source projection” to determine the solar wind source, it does not treat the solar wind as plasma because it uses the solar wind speed observed at 1 AU to derive the empirical relationship. Thus, the resulting formula ignores the transport and acceleration of the solar wind as it propagates out into the heliosphere. The purpose of this study is to rectify this omission by using a numerical MHD simulation to find the optimal set of free parameters that relate the magnetic properties at the source surface to the plasma parameters at 1 AU. In addition to the expansion factor, conservation of mass [$\rho \boldsymbol{v}$], magnetic flux [$r ^{2} B$], and total pressure along the stream line are assumed to obtain the solar wind mass density, magnetic field, and temperature at 18 $\text{R}_{\odot }$. These parameters are used as the inner boundary conditions of our global three-dimensional MHD (G3DMHD) code to simulate solar wind plasma and field parameters out to ${\approx}\, 1~\mbox{AU}$. The simulation results are compared with the in-situ data from Wind to assess the accuracy. Such a procedure is repeated (880 times) to cover the three parameter regimes ($100 < {v}_{1} < 350~\mbox{km}\,\mbox{s}^{-1}$; $250 < {v}_{2} < 700~\mbox{km}\,\mbox{s}^{-1}$; and $0.2 < \alpha < 0.9$) to find the optimal set. The simulation is performed for the period of CR2082 [30 March 2019 to 27 April 2009]. It is found that $\boldsymbol{v} = 189 + 679 f_{\text{s}}^{-0.7}$ is the best formula to relate the solar wind speed at 18 $\text{R}_{\odot }$ to the expansion factor. Strictly speaking, this formula is most applicable for solar equatorial regions and near the times of solar minimum when there are few coronal mass ejection events.