Relativistically covariant warm charged fluid beam modeling

Abstract

A relativistically covariant moment description of the relativistic Vlasov equation is presented and is applied to a warm charged particle beam. Two truncation schemes of the moment equations are proposed yielding models that are distinct both mathematically and physically. The first simply neglects the heat flow tensor S μνλ and recovers what is essentially the model of Newcomb. The distinctions between Newcomb’s formulation and the formulation presented in this paper are described. The second method incorporates fully one of the relativistic constraints on the moments and assumes S μνλ nonzero, but no fourth moments are introduced. The first leads to a hyperbolic system of partial differential equations, whereas the second is partially elliptic and partially hyperbolic. Thus, although the second system appears better on physical grounds, it leads to a mathematically ill-posed system and cannot be used. The second model is rejected. The first system is used to analyze anew the Bennett pinch and space-charge neutral steady flows. It is found that the usual assumptions regarding particular elements of the pressure tensor Θij are incomplete and require modification. For the nonrotating and rigidly rotating Bennett pinch steady flows, it is found that the beam rest frame pressure tensor is diagonal if and only if all of the diagonal elements are equal. Only when the axial and radial pressures are equal can one invoke the standard assumption of isotropy in a plane perpendicular to the axial guide magnetic field. For rigidly rotating space-charge neutral steady flows, the same conclusion holds but now from the perspective of the laboratory frame. For both types of steady flows in the case of rotational shear, Θij is generally nondiagonal and anisotropic in the respective reference frames. An arbitrary warm beam steady flow must satisfy the constraints imposed by the generalized ‘‘equations of state’’ given in this paper in order to be physically legitimate.