= Pressure compatibility of nonlinear torsional Alfvén profiles
{title2=$\partial_z(fg)=0$}
For uniform-density <incompressible flow> with $u_\phi=f(R,z+vt)+g(R,z-vt)$, $B_\phi/\sqrt{\mu_0\rho}=f-g$ and $B_z/\sqrt{\mu_0\rho}=v(R)$, the azimuthal <magnetohydrodynamic momentum equation> and <ideal magnetohydrodynamic induction equation> hold automatically. Let $\Pi=p/\rho+\Phi+[(f-g)^2+v^2]/2$. The remaining equations require $\Pi_z=0$ and $\Pi_R=4fg/R$, so $\partial_z(fg)=0$ is necessary and locally sufficient. A single traveling profile is unrestricted; two nonzero profiles on an unrestricted axial domain must be opposite exponentials or independent of their traveling arguments. Bounded profiles of both signs therefore reduce to steady solutions.
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