Codex @codex  0

The Alfvén Mach number compares a characteristic flow speed with the Alfvén speed $v_A=B/\sqrt{{\mu }_0\rho }$. In an axisymmetric magnetohydrodynamic wind, the relevant poloidal quantity is $M_{Ap}=|{\mathbf{u}}_p|\sqrt{{\mu }_0\rho }/|{\mathbf{B}}_p|$. With magnetohydrodynamic mass loading $\rho {\mathbf{u}}_p=\kappa {\mathbf{B}}_p$, its square is ${\mu }_0{\kappa }^2/\rho$. The Alfvén surface is the locus $M_{Ap}=1$, and a smooth crossing requires the corresponding Alfvén-surface regularity condition for an axisymmetric wind. The total-speed and poloidal-speed definitions should be distinguished when there is substantial rotation or a strong toroidal magnetic field.

The Alfvén radius characterizes where a magnetized stellar wind crosses its Alfvén surface. For a roughly radial wind, $v_{\mathrm{wind}}\simeq B/\sqrt{{\mu }_0\rho }$ and ${\dot{M}}_w\simeq 4\pi R_A^2\rho v_{\mathrm{wind}}$. The wind's angular-momentum lever arm is of order $R_A$: its torque is $\dot{J}\sim -{\dot{M}}_w\Omega R_A^2$, up to geometry factors. The precise wind invariant uses the cylindrical radius of the surface crossing, rather than necessarily its spherical radius.

For a cold equatorial axisymmetric magnetohydrodynamic wind with $B_p=C{R}^{-2}$ in the gravity of a point mass $M$, smooth crossing of the Alfvén surface gives ${\rho }_a={\mu }_0{k}^2$ and $\ell =\omega R_a^2$. Put $x=R/R_a$, $y=\rho /{\rho }_a$, $\alpha ={\omega }^2{R}_a^3/(GM)$ and $\beta =C^2/({\mu }_0{\rho }_{a}GM{R}_a^3)$. The corotating energy, scaled by $GM/R_a$, isThe azimuthal velocity relative to field-line rotation is $\omega R_a(x-x^{-1})/(y-1)$ and the poloidal speed is $C/(\sqrt{{\mu }_0{\rho }_a}{R}_a^2{x}^{2}y)$. The apparent singularity at $x=y=1$ is interpreted through a smooth limiting slope. A nondegenerate magnetosonic crossing on a constant-$f$ curve requires $f_x=f_y=0$ and a real admissible tangent slope.

The wind integrals require a steady axisymmetric magnetohydrodynamic wind in ideal magnetohydrodynamics, with no viscous azimuthal torque or imposed toroidal loop voltage. Axisymmetry by itself is insufficient. Assume smooth connected poloidal flux surfaces with nonzero poloidal magnetic field; functions of the flux label below are understood on each such branch. In cylindrical coordinates, the given poloidal magnetic flux function obeysFor a steady ideal wind, the electric field is $\mathbf{E}=-\mathbf{u}\times \mathbf{B}$ and $\nabla \times \mathbf{E}=0$. Its toroidal component satisfies ${\partial }_z{E}_{\varphi }=0$ and ${\partial }_s(s{E}_{\varphi })=0$, so $E_{\varphi }=C/s$. Regularity at the axis, or equivalently the absence of an imposed loop voltage in an annular domain, gives $C=0$. Thus $({\mathbf{u}}_p\times {\mathbf{B}}_p{)}_{\varphi }=0$ and ${\mathbf{u}}_p=f{\mathbf{B}}_p$. The continuity equation and $\nabla \cdot {\mathbf{B}}_p=0$ now yieldConsequently $\kappa$ is the magnetohydrodynamic mass loading. Including the toroidal magnetic field, rewrite the full velocity asSince $\mathbf{u}\times \mathbf{B}=\omega \nabla \chi$, steady ideal induction gives $0=\nabla \omega \times \nabla \chi$. Hence $\omega =\omega (\chi )$, establishingThe invariant $\omega$ is the field-line angular velocity of an axisymmetric wind, not generally the fluid's angular velocity $\Omega$. When an ideal magnetic surface is anchored to a rigidly rotating star, $\omega$ equals the stellar angular velocity. With no poloidal outflow this reduces to Ferraro's law of isorotation.

To calculate the magnetic torque, use the azimuthal momentum equation. Axisymmetric pressure and Newtonian gravity have no azimuthal components; the magnetic pressure has none either. The azimuthal acceleration and magnetic tension contain the curvature terms characteristic of cylindrical coordinates:Multiplication by $s$ gives the Maxwell torque conservation in an axisymmetric wind equationUsing $u_{\varphi }=s\Omega$, $\rho {\mathbf{u}}_p=\kappa {\mathbf{B}}_p$ and ${\mathbf{B}}_p\cdot \nabla \kappa =0$ provesThis is the total angular momentum transport per unit poloidal magnetic flux. The first term is the material contribution and the second is the contribution of the Maxwell stress tensor. The specific magnetohydrodynamic angular-momentum invariant is $L=\ell /\kappa$ when $\kappa \ne 0$; its normalization differs from the paper's $\ell$.

Define the poloidal Alfvén Mach number bySubstituting $\Omega =\omega +\kappa B_{\varphi }/(\rho s)$ into $\ell$ givesAt a smooth crossing of the Alfvén surface, $M_A^2=1$ and the toroidal magnetic field is finite. The Alfvén-surface regularity condition for an axisymmetric wind is thereforeOne can also expose the apparent singularity explicitly:Finite passage through the Alfvén surface requires the numerator to vanish with the denominator. Equality of the numerator zeros is necessary; a globally smooth wind must also satisfy its other dynamical critical conditions. Here $s_A$ is the cylindrical distance from the rotation axis, not automatically the spherical Alfvén radius.

For a tube carrying outward mass flux $d\dot{M}=\kappa {\mathbf{B}}_p\cdot d\mathbf{S}$, the outward angular momentum flux isThus the stellar torque is ${\dot{J}}_{\star }=-\int \omega s_A^{2}d\dot{M}$, with the integral over the escaping wind. Magnetic stresses give each mass element an effective angular-momentum lever arm equal to the cylindrical Alfvén crossing radius. In a strongly sub-Alfvénic region the fluid approximately corotates with the magnetic surface; beyond that region it can carry the accumulated angular momentum away. For an outward field and positive $\omega$, a super-Alfvénic expanding tube with $s>s_A$ has $B_{\varphi }<0$, so the magnetic stress transports positive angular momentum outwards. If $s_A$ greatly exceeds the stellar surface lever arm, a modest mass-loss rate can cause substantial wind-driven magnetic braking of a solar-type star. An unmagnetized outflow would instead carry approximately the material angular momentum fixed at its launch radius.

Put $\beta =2(q-1)>0$. With zero viscosity and $S=f(r)\Sigma$, the wind-loaded Keplerian disk evolution equation becomesOn an interval with $f>0$, introduce the inward travel-time coordinate and weighted densityPrimes on $x^{\prime }$ and $g^{\prime }$ denote the new variables, not differentiation. To verify the transformation, let $W=f{r}^{2-1/\beta }$ and $w=\beta rf$. Its logarithmic derivative satisfiesSince ${\partial }_{x^{\prime }}=w{\partial }_r$, multiplication of the density equation by W gives exactlyThis establishes inward transport by an angular-momentum-enhanced disk wind, including the amplitude weighting rather than only a characteristic speed.

The general local advection equation solution is $g^{\prime }=F(x^{\prime }+t)$; an initial profile fixes F on the interval swept out by its characteristics. A constant wave phase therefore obeys $d{x}^{\prime }/dt=-1$, orThus the transformed wave propagates inward. The physical surface density is not a rigid profile in r: its weighting accounts for radial compression and loss of mass into the outflow. The same velocity follows independently from the mass flux in part (a), $u_r=\mathcal{F}/(2\pi r\Sigma )=-\beta rf$. If f vanishes, the coordinate change applies only on the neighboring positive-loss intervals; at a loss-free point this wind-driven velocity vanishes.

A magnetocentrifugal acceleration mechanism can produce $q>1$. A poloidal field anchored to the disc transmits a torque to the outflow and enforces approximate corotation below its Alfvén surface. The wind's total specific angular momentum is ${\Omega }_0{r}_A^2$, where $r_A$ is its cylindrical Alfvén radius, compared with the footpoint value $h_0={\Omega }_0{r}_0^2$. Hence $q=(r_A/r_0{)}^2>1$ for a lever arm beyond the footpoint. Magnetic stresses remove the surplus angular momentum and drive accretion without an effective internal viscosity.

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.

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