The stability analysis of magnetohydrodynamic equilibria: Comparing the thermodynamic approach with the energy principle

Ralf Peter Brinkmann

Phys. Fluids B 1, 987 (1989)


This paper is a contribution to the stability analysis of current?carrying plasmas, i.e., plasma systems that are forced by external mechanisms to carry a nonrelaxing electrical current. Under restriction to translationally invariant configurations, the thermodynamic stability criterion for a multicomponent plasma [Phys. Rev. Lett. 56, 143 (1986); Phys. Fluids 30, 3713 (1987)] is rederived within the framework of nonideal magnetohydrodynamics (MHD). The chosen dynamics neglects scalar resistivity, but allows for other types of dissipation effects both in Ohm’s law and in the equation of motion. In the second section of the paper the thermodynamic stability criterion is compared with the ideal MHD based energy principle of Bernstein et al. [Proc. R. Soc. London Ser. A 244, 17 (1958)]. With the help of Schwarz’s inequality it is shown that the former criterion is always more ‘‘pessimistic’’ than the latter, i.e., that thermodynamic stability implies stability according to the MHD principle, but not vice versa. This result confirms the physical plausible idea that dissipational effects tend to weaken the stability properties of current-carrying plasma equilibria by breaking the constraints of ideal MHD and allowing for possibly destabilizing effects such as magnetic field line reconfiguration.