Even though Ω is only a function of Ψ, the physical connection between the two at the foot-point is not very obvious.9

9
They may be constrained by proper boundary conditions (e.g., Contopoulos et al. 2013).
 Likely most researchers, here we assume that they generally follow the ansatz
13





In the region approaching the polar axis, one expects that the magnetic flux vanishes and thus 





. Therefore, in the region near the polar axis, mathematically λ ≥ 0 is required to guarantee a finite value of Ω (notice that λ > 0 is likely unphysical). Magnetic field lines just around the polar axis would connect to the central CO, which is expected to rotate with a roughly constant angular velocity10

10
In the case of threading a BH, magnetic field lines at different polar angles may not rotate with exactly the same frequency; see discussion in Section 8.
 (see details below), implying λ = 0 for the case of magnetic field lines threading the CO. In the region where magnetic field lines are threading the AD, one usually expects λ  0. The choice of Φ, which determines the toroidal magnetic field, is important (e.g., Camenzind 1987; Sulkanen & Lovelace 1990). Physically, it is the rotation that develops a toroidal magnetic field and a poloidal electric field in the lab frame (or the inertial frame), which seems to imply that Φ would not be chosen arbitrarily and would be self-consistently determined by the MHD equations, given the Ψ and Ω specified (e.g., Beskin & Tchekhovskoy 2005; Contopoulos et al. 2013; Huang et al. 2019, 2020). To solve the rotation term, Equation (11), one can see that in the case of
14





the function Ψ can be variable-separated,
15





where the subscript “r” denotes “rotation.” The negative sign in Equation (14) exists to guarantee that there is always a swept-back magnetic field line with respect to the direction of rotation. This relation implies that the strength of the toroidal magnetic field would be proportional to the angular velocity and the enclosed magnetic flux, which can be reasonably understood because it is the rotation of the poloidal magnetic field (related to the magnetic flux) that produces the toroidal magnetic field. In terms of Equation (15), Equation (11) can be expressed as
16





where 





, 





, 





, and 





. The left-hand side of the above equation is only a function of r (the r component), and the right-hand side is only a function of θ (the θ component). Both equal the same constant. Let us set this constant as 





, a choice making the solution of the r component equation concise:
17





We introduce a new variable,
18





which makes the θ-component equation become
19





where 





 and 





. Let us first consider its asymptotic properties. In the case of θ ≪ 1 (keeping in mind 





 when 





), the leading-order terms give
20





It is clear that this equation has a solution of the form
21





Therefore, a magnetic field line forms a “general parabolic” configuration11

11
In the case of ν = 0, Ψ is only a function of θ, which presents a monopole solution. The case of ν = β leads to Ψ ∝ R, which gives a cylindrical solution.
 at θ ≪ 1, that is, Ψ ∝ rνθβ. Now, let us consider what value β might take. In order to do this, we have to consider the higher-order terms of Equation (19), which may be comparable to the nonrotation term. Therefore, we have to consider the original Equation (10). Let us substitute a general form of 





 (with coefficients a1, a2, a3... to be determined) into the original Equation (10). One gets the first two leading-order terms (note that a1 = 0):
22





It can be seen that, for any values of β, λ, and ν, one always has an a2 to make the second term vanish, whereas the first term yields β ≈ 2 (β = 0 corresponds to the nonrotation case). We note that this choice cannot guarantee that the higher-order terms vanish, so this solution is only an approximation. In the case of β  2ν (i.e., ν ≳ 1), the first term dominates over the second term in Equation (22), whereas in the case of β > 2ν (i.e., ν ≲ 1), the second term dominates over the first one. We therefore expect that the approximation β ≈ 2 would be more efficient in the former case (i.e., ν ≳ 1). In another aspect, from Equation (2), one has 





, which implies that the choice of β = 2 can avoid singularity or vanishing magnetic fields on the magnetic polar axis (either Bθ or Bϕ vanishes). This choice is also supported by some numerical simulations, which showed that this choice corresponds to a minimum torque (i.e., the least amount of toroidal magnetic fields) and is the one picked by a “real” system (see, e.g., Michel 1969; Contopoulos 1995; Narayan et al. 2007; Tchekhovskoy et al. 2008). Although the coefficient β ≈ 2 is derived asymptotically (θ ≪ 1), which must hold throughout the jet region because Φ, Ω, and Ψ are each conserved along magnetic field lines. In Appendix A, we come back to this question and present another proof of the relation Φ ≈ −2ΩΨ based on a more physical consideration, showing that this relation is only valid in the limit of a highly magnetized jet flow (e.g., Lyubarsky 2009).
Citation: Huang et al. 2019
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