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Fully Magnetized Plasma Flow in a Magnetic Nozzle Mario Merino and Eduardo Ahedo

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Fully Magnetized Plasma Flow in a Magnetic Nozzle Mario Merino and Eduardo Ahedo Fully magnetized plasma flow in a magnetic nozzle Mario Merino and Eduardo Ahedo Citation: Physics of Plasmas 23, 023506 (2016); doi: 10.1063/1.4941975 View online: http://dx.doi.org/10.1063/1.4941975 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/23/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A magnetic nozzle calculation of the force on a plasma Phys. Plasmas 19, 033507 (2012); 10.1063/1.3691650 On plasma detachment in propulsive magnetic nozzles Phys. Plasmas 18, 053504 (2011); 10.1063/1.3589268 Status of Magnetic Nozzle and Plasma Detachment Experiment AIP Conf. Proc. 813, 465 (2006); 10.1063/1.2169224 Magnetohydrodynamic scenario of plasma detachment in a magnetic nozzle Phys. Plasmas 12, 043504 (2005); 10.1063/1.1875632 Magnetic‐Laval‐Nozzle Effect on a Magneto‐Plasma‐Dynamic Arcjet AIP Conf. Proc. 669, 306 (2003); 10.1063/1.1593926 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 163.117.179.242 On: Thu, 18 Feb 2016 17:44:42 PHYSICS OF PLASMAS 23, 023506 (2016) Fully magnetized plasma flow in a magnetic nozzle Mario Merinoa) and Eduardo Ahedob) Equipo de Propulsion Espacial y Plasmas (EP2), Universidad Carlos III de Madrid, Leganes, Spain (Received 8 December 2015; accepted 2 February 2016; published online 18 February 2016) A model of the expansion of a plasma in a magnetic nozzle in the full magnetization limit is presented. The fully magnetized and the unmagnetized-ions limits are compared, recovering the whole range of variability in plasma properties, thrust, and plume efficiency, and revealing the differences in the physics of the two cases. The fully magnetized model is the natural limit of the general, 2D, two-fluid model of Ahedo and Merino [Phys. Plasmas 17, 073501 (2010)], and it is proposed as an analytical, conservative estimator of the propulsive figures of merit of partially magnetized plasma expansions in the near region of the magnetic nozzle. VC 2016 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4941975] I. INTRODUCTION finite ion magnetization covers all cases of practical interest, as extremely high magnetic fields are required to keep ions From the inception of the magnetic nozzle1 (MN), there magnetized into the far downstream expansion region; and has been a growing interest in their capability to accelerate a second, the FMIL is not a regular limit of the DIMAGNO plasma “contactlessly” in advanced electric propulsion. formulation. In this limit, the character of ion equations Examples of next-generation plasma thrusters based on MNs degenerates as the Mach characteristic lines no longer that are actively being developed include the Helicon Plasma carry any useful information, thus requiring a different inte- Thruster2,3 (HPT), the Electron-Cyclotron-Resonance gration approach. Nonetheless, being able to compare the thruster4,5 (ECRT), the Applied-field MPD thruster6 unmagnetized-ions limit (UMIL) and the FMIL would pro- (AF-MPDT), and the Variable Specific Impulse Rocket vide valuable insight on the physics of the MN, as they are (VASIMR).7 In the usual design, a MN consists in an axi- the extremes of the operation range of the device. symmetric, divergent magnetic field that guides the super- Following this motivation, the paper presents and dis- sonic expansion of a hot plasma jet. The quasineutral plasma cusses the FMIL of the 2D fluid formulation of Ref. 9. gains axial kinetic energy at the expense of its internal Contrary to the finite ion-magnetization case, the FMIL energy, thanks to the self-consistent ambipolar electric field. results in a set of completely algebraic equations and thus is The reaction to the magnetic forces that shape the plasma simple and fast to compute. The UMIL and the FMIL are expansion is felt on the magnetic circuit of the thruster as then compared to quantify the influence of the level of ion “magnetic thrust.” MNs are also a topic of research in the field of advanced plasma material processing,8 where their magnetization on the 2D plasma expansion and the propul- non-propulsive use allows to control the flux of plasma to sive figures of merit of the MN. In particular, it is shown that the substrate. the thrust gain is rather independent of the ion magnetization In a previous work,9 the DIMAGNO model of the degree in the near expansion region, and that the FMIL plasma expansion in a divergent MN was presented. This 2D yields a conservative value of the thrust gain and the plume two-fluid model treats electrons as a fully magnetized spe- efficiency for all possible ion magnetizations. In view of this cies, whereas heavier ions, on the other hand, are allowed to result, the 2D FMIL solution is proposed for the quick esti- have arbitrary magnetization. The method of characteristics mation of the propulsive performance of the MN in the is used to integrate the hyperbolic equations of supersonic general case. ions. DIMAGNO was instrumental to study the acceleration In this regard, there are several models in the literature 9,10 that present some degree of similarity with the FMIL model mechanisms and forces on the plasma, the development 1 of electric currents and electric fields, and the downstream presented here. First, Andersen et al. applied the well- detachment of the plasma from the MN.11–13 This model has known paraxial (i.e., quasi-1D) model of the expansion of a revealed the richness of 2D phenomena in the expansion, gas in a channel of slowly varying area to a MN, providing which are unaccessible to paraxial 1D models. The algebraic expressions for the axial profiles of the mean ion DIMAGNO model was also used to investigate the condi- velocity and plasma density. This model coincides with the tions for the formation of double-layer structures14 and the radially averaged equations of the DIMAGNO model and effects of electron cooling and ion thermal energy.15 can be used to estimate thrust with good accuracy within the While the finite ion magnetization regimes have already near expansion of the MN. However, the radial variability of been analyzed, the fully magnetized ions limit (FMIL) plasma properties, the development of electric currents in the remains to be explored. There are two reasons for this: first, plasma, plume efficiency, and plasma detachment cannot be computed with a quasi-1D model. Second, Fruchtman 16 a)Electronic mail: [email protected] and [email protected] et al. also studied the paraxial approximation. Beyond b)Electronic mail: [email protected] reproducing the solution of Andersen et al., they postulated a 1070-664X/2016/23(2)/023506/7/$30.00 23, 023506-1 VC 2016 AIP Publishing LLC Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 163.117.179.242 On: Thu, 18 Feb 2016 17:44:42 023506-2 M. Merino and E. Ahedo Phys. Plasmas 23, 023506 (2016) self-similar expansion of the plasma density radial profile in u?e ¼ 0. Then, the parallel and perpendicular components of an attempt to recover some of the radial features of the Eq. (3) provide two conservation laws along electron expansion and derive a formula for the magnetic thrust. streamtubes Third, Little and Choueiri17 took the 2D DIMAGNO model n of Ref. 9 as their starting point and applied several simplify- Te ln À e/ ¼ HeðÞw ; (4) ing assumptions in order to derive an approximate, “less n0 numerical” solution which can be used for quick computa- @ lnðÞn=n0 @/ tions of the plasma response and the propulsive figures of euheB ¼Te þ e @1? @1? merit. @He dHe The rest of the paper is structured as follows. Section II À ¼rB ; (5) @1 dw derives the FMIL model from the general fluid equations and ? briefly discusses the numerical solution. Section III explores where w is the streamfunction of the applied magnetic field and compares the plasma properties and propulsive perform- (i.e., w rB1 ), and H w =e is the thermalized poten- ance at the UMIL and FMIL. Section IV discusses the r ¼ ? À eð Þ tial of each streamtube. Finally, using the solenoidal charac- applicability and limitations of the FMIL to describe plasma ter on the B field (rB ¼ 0), the electron continuity expansions with finite ion magnetizations and comments fur- equation in Eq. (1) can be written as a third conservation law ther on the differences with the works in the literature enum- along w erated above. Finally, conclusions of this work are gathered in Section V. nuke=B ¼ GeðwÞ; (6) II. 2D MODEL OF THE FULLY MAGNETIZED PLASMA EXPANSION with GeðwÞ being the electron-to-magnetic flux ratio at each streamtube. The derivation of the FMIL model commences with the From the four scalar ion equations, however, only the complete 2D equations of the full two-fluid model of Ref. 9. first integrals associated to the conservation of canonical Where not specified, the conventional notation defined angular momentum and mechanical energy along ion stream- therein will be used here too. The model describes the tubes (which can differ from the electron/magnetic stream- steady-state expansion of hot electrons, “e,” and single- tubes) can be found,9 and in general, a numerical scheme charged cold ions, “i,” in an externally applied magnetic must be employed to integrate the ion motion in the meridio- field, B. The plasma is assumed to be quasineutral nal plane. The method of characteristics was chosen in Ref. (ni ’ ne n), collisionless and low-beta (i.e., negligible 9, which exploits the fact that physical information is carried induced magnetic field effects), and to have a clean lateral only along the characteristic lines in the hyperbolic problem boundary with vacuum.
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