Our model does not describe magnetic fields. Two different regions need to be distinguished where magnetic fields can play a role. First, a magnetic field can provide additional pressure in the disk wind and thus contribute to the external pressure \(P(z)\). Indeed, the region covered by our model is expected to be inside the Alvèn surface of the disk wind, so \(P(z)\) is probably magnetically dominated (further discussion and references in Section \ref{sect:boundary}), but for our model only the total value of \(P(z)\) matters, independent of the processes that contribute to the pressure.
Second, the stellar wind could be threaded by a stellar magnetic field. Fields on YSOs are often quite complex with a mixture of open and closed field lines \citep[e.g.][]{2011MNRAS.417..472D,2012MNRAS.425.2948D}. Qualitatively, closed field lines can either fill with coronal plasma or connect to the accretion disk and carry accretion funnels. Only those parts of the stellar surface connected to open field lines can launch a wind. Thus, the total mass loss rate would be reduced compared to a spherical wind. As a simple estimate we calculate the magnetic pressure \(P_{\textrm{mag}}=\frac{\boldsymbol{B}^2}{8 \pi}\) for a split monopole field with a field strength of 1 kG at \(r=R_\odot\) and compare it to the ram presure (eqn. \ref{eqn:Pofz}). Using the fiducial parameters from Table \ref{tab:fiducial} the ram presure dominates over the magnetic pressure already at 0.1 AU and since \(P_{\textrm{mag}} \propto \boldsymbol{B}^2 \propto r^{-4}\), while \(P_{\textrm{ram}} \propto r^{-2}\) (eqn. \ref{eqn:Pofz} and \ref{eqn:rho}) we can neglect the magnetic pressure of the stellar wind for our model which predicts typical radial distances on the AU scale.