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All invariants in this part refer to the selected magnetic field line. Its squared poloidal Alfvén number isAt the Alfvén surface crossing it equals unity, so ${\rho }_a={\mu }_0{k}^2$ and $M_{Ap}^2=1/y$. Subtract the two azimuthal integrals in part (a) to findFinite azimuthal fields and velocity at $y=1$ require the Alfvén-surface regularity condition for an axisymmetric wind $\ell =\omega R_a^2$. HenceHere $v_{\varphi }$ denotes the azimuthal velocity relative to field-line rotation. For gravity of a point mass at $z=0$, $\Phi =-GM/R=-GM/(R_{a}x)$.

Insert these quantities in the corotating energy integral with $w=0$, and divide by the gravitational energy scale $GM/R_a$. This gives the cold radial magnetohydrodynamic wind integral,withBoth are dimensionless. The apparent $0/0$ at $x=y=1$ must be evaluated on the smooth solution branch, not assigned an arbitrary value.

A constant-energy curve obeys $f_x+f_{y}dy/dx=0$. At a nondegenerate magnetosonic point, the vanishing coefficient of the mass density derivative requires simultaneous vanishing of its numerator:The curve must admit a real crossing slope. Expanding about such a point gives $f_{yy}{m}^2+2f_{xy}m+f_{xx}=0$, $m=dy/dx$, so a regular saddle point of a scalar function crossing has $f_{xy}^2-f_{xx}{f}_{yy}>0$. These are the form of the slow and fast regularity conditions in a warm wind, with the enthalpy contribution retained in $f$.

In the strictly cold approximation used to obtain the displayed $f$, the slow speed is zero. There is a cold-limit degeneracy of a slow magnetosonic point: a finite-speed interior slow crossing need not survive. Indeed,so this cold equation has no ordinary sub-Alfvénic slow critical point. A small retained enthalpy is needed to locate that point, which can approach the launching boundary as the sound speed tends to zero. The fast critical point still obeys the simultaneous conditions above wherever the reduced expression is nonsingular.

Introduce the squared poloidal Alfvén numberSolving the two linear azimuthal invariants givesAt an Alfvén surface, $M_A^2=1$. Smooth passage through the apparent singularity requires both numerators to vanish at the same cylindrical radius $r_A$, giving the Alfvén-surface regularity condition for an axisymmetric windThe finite limiting values of $u_{\varphi }$ and $B_{\varphi }$ then follow by l'Hopital's rule from the local variation of $M_A$ and $r$ along the field line; the algebraic invariants alone fix their combination rather than each value separately at the critical point.

In the aligned frame, define the Alfvén numberwhere the second equality uses ${\mathbf{u}}_i\parallel {\mathbf{B}}_i$. Since $j={\rho }_i{u}_{xi}$ and $B_x$ are common,Therefore

Using alignment in the tangential momentum flux givesIts continuity then yieldsorwhenever the ratios are determinate.

Define the poloidal Alfvén number bywhere $u_p=k{B}_p/\rho$. Thus, along a given field line,density decreases as the Alfvén number grows.

The azimuthal velocity representation givesSubstitution into $\ell =r{u}_{\varphi }-r{B}_{\varphi }/({\mu }_{0}k)$ yieldsAt the Alfvén surface, $A^2=1$. Smoothness requires the numerator of the solution for $B_{\varphi }$ to vanish there, soSolving away from the surface then gives