Nearby-fluids equilibria. I. Formalism and transition to single-fluid magnetohydrodynamics

Abstract

The nearby-fluids model is an improvement of the basic two-fluid flowing equilibrium system developed elsewhere [L.C. Steinhauer, Phys. Plasmas 6, 2734 (1999)]. The two-fluid system is a singular perturbation problem in which the usually small ion-inertia length (ion skin depth) gives rise to a small parameter. This has frustrated attempts to find practical equilibria. The nearby-fluids ordering assumes that the ion and electron flow surfaces are close to (“nearby”) each other but do not coincide exactly. This eliminates the singularity and “softens” the stiff differential equations, thus facilitating numerical solution. Previous treatments of two-fluid equilibria did not find a smooth transition to the single-fluid magnetodynamics (MHD) model in the small ion-inertia-length limit. This difficulty is only partly eliminated by recognizing that the electric field is ordered “large” in MHD. A small but nonzero ion-inertia length produces several important effects, including the appearance of a transition layer between the “closed” and “open” field regions of a field-reversed configuration plasma.

Ertel's vorticity theorem and new flux surfaces in multi-fluid plasmas

“…Of course, Ertel surfaces can exist there as well, and this was already exploited in the equilibrium study of Ref. 22, but the fuller set of boundary conditions was not yet worked out.…”

Hamiltonian formulation and analysis of a collisionless fluid reconnection model

“…Steinhauer has proposed a method for dealing with this problem that he named the "nearby fluid" approximation. [46,47] In the following section we outline a similar approach to solving Eqs. (52)-(53).…”

“…The second difficulty is that the right-hand sides are singular in the limits d e ≪ L and d i ≪ L, where L is a macroscopic scale length. [40,41,46] That is, for macroscopic equilibria the derivatives in Eqs. (52)- (53) are multiplied by a small parameter, so that these equations form a stiff system.…”

Jump conditions in transonic equilibria

“…Two curves are shown, for the values on the two sides of the discontinuity ("In" corresponds to the left of the discontinuity in Figs. [14][15][16][17]. Both errors decrease with the increasing resolution.…”

“…The second one is being developed: the next step in transonic equilibrium calculations will be the use of a two-fluid model, in which we expect the transonic discontinuity to acquire a width of the order of the ion skin depth. 17 Future results will give a further improvement of the model and understanding of transonic equilibria. @Bðq 0 ; wÞ @q ¼ 0:…”