Regularity at a magnetosonic point

If a steady wind equation has the form $(w^2-c_{\mathrm{f}}^2)(w^2-c_{\mathrm{s}}^2)w^{\prime }/w=\mathcal{N}$, a smooth nondegenerate crossing requires $\mathcal{N}=0$ at the corresponding magnetosonic critical speed. Subsequent differentiation or the full system determines the admissible slope. The numerator condition alone does not prove that a global smooth wind exists.

In a strictly cold flow governed by ideal magnetohydrodynamics the sound speed vanishes, so the slow magnetosonic wave speed is zero. A nonzero-speed outflow therefore has no ordinary finite interior slow crossing. For the cold radial magnetohydrodynamic wind integral, $f_y=-\alpha (x-x^{-1}{)}^2/(y-1{)}^3-\beta /(x^4{y}^3)<0$ on the sub-Alfvénic branch $y>1$, assuming $\beta >0$. A small retained specific enthalpy is needed to locate a finite slow point; the zero-temperature limit can move it to the launch boundary. The usual simultaneous numerator and denominator regularity conditions still apply to nondegenerate critical points.