On the stability of an inhomogeneous magnetized plasma

Abstract

It is known that a magnetized plasma with a sharp boundary parallel to the external magnetic field may be unstable if there are forces which cause a drift of charged particles along the boundary of the plasma (see [1]). The reasons for these forces arising (inhomogeneity in the magnetic field in space, curvature of the lines of force, graviational field, rotation of the plasma) do not play an essential part; the important thing is that the drift caused by them depends on the sign of the charge of the particles. Thus, if the force is directed from the plasma into the vacuum, the plasma will be unstable with respect to disturbances with a wave of sufficient length (this is called “canalized instability”). If the force is directed from the vacuum into the plasma, there is no instability. Attempts are made in [2]–[6] to associate this kind of instability with the inhomogeneity of the plasma. This method of approach in particular allows us to consider a plasma with a blurred boundary. However, although in [2] the problem is solved logically for the simplest model of a plasma (assumed to be cold, incompressible and ideally conducting), [3]–[6] are far from rigorous from the mathematical point of view. Assuming that the wave length is much greater than the scale of inhomogeneity of the plasma the authors omit the derivatives of unknown functions in the differential equations and reduce the problem to algebraic equations. Thus the question of the limits of applicability of the results of these works requires additional mathematical investigation. In this paper it is our aim to study the stability of an inhomogeneous cold plasma, starting from the equation of two fluid hydrodynamics and Poisson's equation for the potential of an electric field. We introduce a hypothetical gravitational field as the cause of the particle drift. We shall consider the one-dimentional problem. The results we obtain are illustrated by the use of examples.