Stability of the transonic plasma flow in the magnetic nozzle

Transonic acceleration of plasmas driven by thermal pressure occurs in converging-diverging magnetic field configurations such as magnetic nozzles in plasma propulsion devices and magnetic mirror systems for fusion applications. A similar mechanism is responsible for the acceleration of the solar wind where the nozzle effect is created by a combination of the diverging magnetic field and the gravity effect. Wave propagation along the stationary transonic non-uniform flow is of interest for some applications. Alternatively, this problem can be viewed as a problem of the linear stability of the transonic stationary flow. The analysis of the linear stability of plasma flow in the magnetic nozzle using the spectral method suffers from spectral pollution. For purely subsonic and supersonic velocity profiles the spurious solutions can be filtered out using the convergence test. However, the eigenvalue problem for the transonic velocity profile has an additional difficulty due to the presence of the singularity in the eigenvalue problem which occurs at the sonic point. Using the analytical expansion near the singular point, regular solutions to the polynomial eigenvalue problem can be constructed and the regular eigenfunctions can be built in the whole domain by using the shooting method. We show that regular eigenfunctions are spectrally stable. However, the spatial profiles of the eigenfunction can be interpreted as convective amplification of the perturbation entering the transonic acceleration region.