Quasi-static evolution of a dissipative plasma column in vacuum

Abstract

We consider the following problem: how far, in what sense and with what practical applications might the analysis of the quasi-static evolution of a (strictly) dissipative plasma column in vacuum be carried out independently of computational means (however necessary at a later stage) in the framework of a conveniently simplified, yet significant, mathematical model. We show that the the (two-region) evolution problem of concern can be reduced to the (numerical) solution of just three nonlinear non-autonomous evolution equations in three unknowns, sayẊ=Ẋ(X,t), withX≡ (X1,X2,X3), where the corresponding mapping (X,t)↣Ẋcan be made effective at the cost of inverting a non-canonical Fredholm operator of the second kind that is completely defined for the givenX=X(t) andt. The assumption of sufficiently slow evolution allows us to work with the static-equilibrium momentum equation. Under axial (or cylindrical) symmetry, we know that the ‘radial’ equilibrium condition leads to a semi-linear elliptic equation in the meridional (or (x, y)) plane. The Fredholm operator arises from the inversion of the Laplacian appearing in that equation.