= Maxwell torque conservation in an axisymmetric wind
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{title2=$\mathbf F_J=\rho\mathbf u_p s u_\phi-\mu_0^{-1}\mathbf B_p sB_\phi$}
In a steady <axisymmetric magnetohydrodynamic wind>, axisymmetric <pressure> and <Newtonian gravity> exert no azimuthal <torque>. The azimuthal momentum equation is $\rho\mathbf u_p\cdot\nabla(su_\phi)=\mu_0^{-1}\mathbf B_p\cdot\nabla(sB_\phi)$. <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 $\ell=\kappa s u_\phi-sB_\phi/\mu_0$. It is constant on connected regular <poloidal flux surfaces>. The usual specific <magnetohydrodynamic angular-momentum invariant> is $L=\ell/\kappa$, provided $\kappa\ne0$; confusing these two normalizations loses a factor of mass loading in the wind <torque>.
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