Asymptotic energy of a radial magnetohydrodynamic wind

Consider a steady radial axisymmetric magnetohydrodynamic wind with outward poloidal speed $u_p\to u_{\infty }>0$. First choose an outward-positive poloidal magnetic field $B_p>0$ and positive magnetohydrodynamic mass loading $k$, soIf the field-line angular velocity is $\omega$, ideal induction gives the large-radius toroidal fieldThe density is $\rho \sim kF/(u_{\infty }{r}^2)$, hence the toroidal Alfvén speed approachesThe absolute value of $\omega$ is necessary when quoting a nonnegative speed.

When specific enthalpy and gravitational potential vanish at infinity and the azimuthal material speed also decays, the Bernoulli invariant becomesThe second term is magnetic energy transport per unit mass flux, so terminal material motion need not carry all of the outward energy.

For a signed inward $B_p$, use signed $F_B=\lim r^2{B}_p$ and signed $k$ consistently. Then $F_B/k>0$ for an outward wind, $B_{\varphi }\sim -\omega F_B/(u_{\infty }r)$ and the speed formula uses $F_B/k$. Mixing an absolute magnetic flux with a signed mass-loading parameter would produce an unphysical sign.