MHD Simulations: Transition to Parker Spiral

Testing the Corotation Radius Hypothesis (Schulz 1973)

Context: These MHD simulation results explore the hypothesis proposed by Schulz (1973) that the smooth transition from solar-wind outflow along magnetic field lines in the co-rotating frame to formation of the Parker spiral occurs at a distance where the Ω×r co-rotation velocity overtakes the Alfvén speed.

The key prediction is that this transition should occur around 32 solar radii from the Sun, where the azimuthal velocity component reaches a maximum of approximately 64 km/s before declining in the Parker spiral regime.

Key Result: Plot 4 (VA/|vφ|) shows the corotation radius where this ratio equals 1.0, marking the transition from co-rotation dominated to Alfvén speed dominated dynamics.
1. Radial Magnetic Field Component
Shows Br/B as a function of radius for different azimuthal angles. In the co-rotating frame, magnetic field lines should be nearly radial (Br/B ≈ 1) until the corotation radius, where they begin to spiral and Br/B decreases.
Br/B vs Radius
2. Alfvén Speed Distribution
Spatial distribution of Alfvén speed (VA) in the equatorial plane. The Alfvén speed increases initially (rho falls more rapidly than B), then decreases modestly (or remains approximately constant based on B~r-1 and rho~r-2 arguments) out to 1 AU.
Alfven Speed Distribution
3. Azimuthal Velocity (vφ)
Shows the azimuthal velocity component in the equatorial plane. According to Schulz's hypothesis, this should reach a maximum around 32 R☉ (~64 km/s) before declining as the Parker spiral forms.
Azimuthal Velocity
4. Corotation Parameter (VA/|vφ|)
Critical plot: The white contour at VA/|vφ| = 1.0 marks the corotation radius where Alfvén speed equals the co-rotation velocity. This is where the transition to Parker spiral should begin according to Schulz (1973).
Corotation Parameter
5. Magnetic Field Lines and Corotation Surface
Magnetic field lines traced in the equatorial plane, colored by polarity (red=outward, blue=inward). The black contour shows the corotation radius. Field lines should transition from radial to spiral geometry at this boundary.
Field Lines and Corotation
6. Alfvén Mach Number (MA)
Ratio of radial velocity to Alfvén speed (MA = |vr|/VA). The white contour at MA = 1.0 shows the Alfvén critical surface where the solar wind becomes super-Alfvénic.
Alfven Mach Number
Initial Conclusions: Unfortunately, the simulation doesn't go out far enough to make any definitive conclusions about the transition. Plot 5 shows that most of the region in the equatorial plane is inside the r_c boundary. I'll try to get a solution where we merged the coronal and heliospheric solutions into a single, continuous dataset, or re-run this out out to 50 Rs, say.

References:
• Schulz, M. (1973). "Interplanetary Sector Structure and the Heliomagnetic Equator." Astrophysics and Space Science, 24, 371-383.
• Weber, E. J., & Davis, L. (1967). "The Angular Momentum of the Solar Wind." Astrophysical Journal, 148, 217-227.
• Kasper, J. C., et al. (2019). Parker Solar Probe observations confirming azimuthal velocity structure.