Magnetohydrodynamic self-consistent exact helical solutions

We consider the idealized case of a one-component plasma with aligned fluid velocity and current density. Constant density and pressure as well as zero external magnetic field are also assumed. We show that suitably determined axially symmetric helical current densities within a straight infinite cylinder are exact self-consistent solutions of magnetohydrodynamics. Self-consistent here means that the magnetic field is the field produced by the current density itself. The equation of motion gives a nonlinear differential equation that relates the axial vz(ρ) and the azimuthal vψ(ρ) velocities as functions of radial distance ρ. Prescribing one of these gives a specific solution for the other. The solutions can be understood as a set of helix-shaped charged particle trajectories that spiral self-consistently through the magnetic field that they themselves give rise to. Four different specific exact solutions are given: (i) a single particle outside a rectilinear line current, (ii) current on a thin cylinder, (iii) current density with constant angular velocity and (iv) current density with constant axial velocity, both within a cylinder of finite radius