= 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> $k$, so
$$
\rho u_p=kB_p,\qquad r^2B_p\to F>0.
$$
If the <field-line angular velocity> is $\omega$, ideal induction gives the large-radius toroidal field
$$
B_\phi\sim-\frac{\omega F}{u_\infty r}.
$$
The density is $\rho\sim kF/(u_\infty r^2)$, hence the toroidal <Alfvén speed> approaches
$$
|v_{A\phi}|\longrightarrow
|\omega|\sqrt{\frac{F}{\mu_0ku_\infty}}.
$$
The 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> becomes
$$
\epsilon=\frac12u_\infty^2+\frac{\omega^2F}{\mu_0ku_\infty}.
$$
The 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 $F_B=\lim r^2B_p$ and signed $k$ consistently. Then $F_B/k>0$ for an outward wind, $B_\phi\sim-\omega F_B/(u_\infty r)$ and the speed formula uses $F_B/k$. Mixing an absolute magnetic flux with a signed mass-loading parameter would produce an unphysical sign.
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