Axisymmetric magnetohydrodynamic wind (axisymmetric MHD wind)

A steady axisymmetric magnetohydrodynamic wind follows nested magnetic-flux surfaces and has conserved mass loading, field-line angular velocity, angular momentum, entropy, and Bernoulli quantities along each surface.

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.

Assume zero toroidal loop voltage in smooth steady ideal magnetohydrodynamics, so the poloidal flow aligns with the poloidal magnetic field. Write $\rho {\mathbf{u}}_p=\eta {\mathbf{B}}_p$. The continuity equation makes $\eta$ constant along a magnetic line. The induction equation gives the conserved field-line angular velocity ${\Omega }_F=\Omega -\eta B_{\varphi }/(\rho R)$. The azimuthal momentum equation is $\rho {\mathbf{u}}_p\cdot \nabla (R^2\Omega )={\mathbf{B}}_p\cdot \nabla (R{B}_{\varphi })/{\mu }_0$; hence $L=R^2\Omega -R{B}_{\varphi }/({\mu }_0\eta )$ is also constant along the line.

With $M_{Ap}^2={\mu }_0{\eta }^2/\rho =|{\mathbf{u}}_p{|}^2/v_{Ap}^2$, eliminating $B_{\varphi }$ givesFor $M_{Ap}^2\ll 1$ and nonsingular invariants, $\Omega ={\Omega }_F+O(M_{Ap}^2)$ and matter approximately corotates along a field line. With exactly zero poloidal flow, induction directly gives Ferraro's law of isorotation. The approximation requires the correction $M_{Ap}^2({\Omega }_F-L/R^2)$ to be small on the segment in question.

For a pressure-free flowing poloidal flux surface, subtracting its angular velocity times the azimuthal momentum equation from the mechanical energy equation shows that ${ϵ}_J$ is constant. It equals poloidal kinetic energy plus $(u_{\varphi }-R\omega {)}^2/2+\Phi -R^2{\omega }^2/2$. This is the total magnetohydrodynamic Bernoulli invariant minus $\omega$ times the total magnetohydrodynamic angular-momentum invariant, not the total energy alone.

In steady axisymmetric ideal magnetohydrodynamics, induction makes $\omega$ constant on regular connected poloidal flux surfaces. It describes field-line rotation. The material angular velocity is $u_{\varphi }/R=\omega +k{B}_{\varphi }/(\rho R)$ and differs when the toroidal magnetic field contributes. If the poloidal magnetic field vanishes, the flux-surface integration requires separate treatment.

For ${\mathbf{B}}_p=\nabla \psi \times \nabla \varphi$, surfaces $\psi =\text{constant}$ are poloidal magnetic surfaces and $2\pi \psi$ is magnetic flux up to the choice of axial reference value.

For an axisymmetric poloidal magnetic field ${\mathbf{B}}_p=\nabla \Psi \times \nabla \varphi$, the ideal magnetohydrodynamic induction equation gives ${\partial }_t\Psi +{\mathbf{u}}_p\cdot \nabla \Psi =0$, after fixing the time-dependent additive constant in $\Psi$. A regular axial reference makes $2\pi \Psi$ the enclosed magnetic flux. This scalar statement of magnetic flux freezing holds even when rotation also produces a toroidal magnetic field.

A regular level surface of the axisymmetric poloidal magnetic flux function contains both the poloidal and toroidal magnetic field directions. Conservation along field lines can be expressed locally as a function of the flux label on connected regular surfaces. Degenerate points with no poloidal magnetic field and disconnected branches with the same label require separate treatment.

If $\Psi (R,0)={\Psi }_1+{\Psi }_0(R/R_0{)}^{\delta }$ and the surface inclination to the vertical is a radius-independent $\alpha$, then $B_z=\delta {\Psi }_0{R}^{\delta -2}/R_0^{\delta }$ and $B_R=B_z\tan \alpha$. The solenoidal vector field condition givesThe constant offset in the poloidal magnetic flux function has no effect on the magnetic field.

When $\rho {\mathbf{u}}_p=k{\mathbf{B}}_p$, the flux-surface function $k$ is the mass flux per unit poloidal magnetic flux.

For steady axisymmetric ideal flow with only poloidal components, induction implies $\mathbf{u}\times \mathbf{B}=C{\mathbf{e}}_{\varphi }/R$. Axis regularity, or zero toroidal electromotive force, sets $C=0$. Continuity then gives $\rho \mathbf{u}=k(\psi )\mathbf{B}$, locally on connected magnetic surfaces of the poloidal magnetic flux function. Without the zero-circulation condition, annular domains can support steady cross-field flow.

Ideal induction in a steady axisymmetric flow makes each magnetic surface rotate with a constant field-line angular velocity $\omega (\psi )$.

The total specific angular momentum transported by matter and magnetic stress isand is constant on each magnetic surface.

In a steady axisymmetric magnetohydrodynamic wind, axisymmetric pressure and Newtonian gravity exert no azimuthal torque. The azimuthal momentum equation is $\rho {\mathbf{u}}_p\cdot \nabla (s{u}_{\varphi })={\mu }_0^{-1}{\mathbf{B}}_p\cdot \nabla (s{B}_{\varphi })$. Mass conservation and the solenoidal vector field condition imply $\nabla \cdot {\mathbf{F}}_J=0$. With magnetohydrodynamic mass loading $\rho {\mathbf{u}}_p=\kappa {\mathbf{B}}_p$, the transported angular momentum per unit poloidal magnetic flux is $\ell =\kappa s{u}_{\varphi }-s{B}_{\varphi }/{\mu }_0$. It is constant on connected regular poloidal flux surfaces. The usual specific magnetohydrodynamic angular-momentum invariant is $L=\ell /\kappa$, provided $\kappa \ne 0$; confusing these two normalizations loses a factor of mass loading in the wind torque.

Along a steady axisymmetric ideal-MHD wind, the total specific energyis constant on each magnetic surface. The last term is the electromagnetic energy transported per unit mass flux.

In aligned steady ideal flow with $p=c_s^2\rho$ and constant isothermal sound speed, the Lorentz force density has no field-parallel component. Projecting momentum along the magnetic field makes $u^2/2+\Phi +c_s^2\ln (\rho /{\rho }_0)$ constant on each connected magnetic surface. The arbitrary reference density changes only the additive constant.

An Alfvén surface is where the poloidal flow speed equals the poloidal Alfvén speed. Smooth passage through it imposes a regularity condition relating field-line angular velocity and angular momentum.

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.

If $r_A$ is the cylindrical radius at which $M_A^2={\mu }_0{k}^2/\rho =1$, finite azimuthal velocity and magnetic field requireThis cancels the simultaneous numerator and denominator zeros in the wind's azimuthal integrals.

If $r^2|B_p|\to F$ and $u_p\to u_{\infty }$ along an open magnetic surface, thenWhen enthalpy and gravity vanish asymptotically, the Bernoulli invariant becomes

Magnetocentrifugal acceleration occurs when matter tied to a rotating inclined magnetic field moves outward under the effective centrifugal force. For a Keplerian disc, cold launching is possible when the field tilts by more than $30^{\circ }$ from the vertical.

For a cold corotating field line anchored in a circular point-mass orbit, the effective potential has meridional Hessian ${\omega }^2\mathrm{diag}(-3,1)$. Along a straight line inclined by $\alpha$ to the vertical its quadratic curvature is ${\omega }^2(1-4{\sin }^2\alpha )$. Negative curvature gives the strict local launching criterion. Equality is marginal and global escape requires additional information.

At exactly thirty degrees the quadratic curvature vanishes, but the specified straight-line point-mass effective potential is downhill for small positive displacement at cubic order. Thus the strict negative-curvature criterion does not exclude every nonlinear one-sided marginal launch. An exactly stationary bead remains at equilibrium until displaced.

In a flattened power-law gravitational potential $\Phi ={\Phi }_0{R}_0^{\beta }(R^2+{\lambda }^2{z}^2{)}^{-\beta /2}$, a magnetic surface corotating with its disk footpoint has an effective potential with meridional Hessian matrix ${\Omega }_f^2\mathrm{diag}[-(\beta +2),{\lambda }^2]$ there. A cold outward displacement along a field inclined by $\alpha$ to the vertical is downhill when ${\tan }^2\alpha >{\lambda }^2/(\beta +2)$. Equality is marginal and can depend on higher derivatives and field-line curvature; this is a local launching criterion, not a guarantee of a global escaping wind.